[{"category":"Post","content":" Enrico Fermi. A portrait photographed during my visit to the Scuola Normale Superiore. Enrico Fermi is the best Italian physicist of the 20th century and is widely known for his work in nuclear physics. He began bombarding nuclei with neutrons in Rome and later settled at the University of Chicago after the Second World War, where he worked until his death in 1954. Besides being a Quantum pioneer, he was one of the leading physicists in the Manhattan Project. And he is immortalized in the minds of young physicists through the name of Fermions- subatomic particles with half integer spin. I first learned about Fermi as a 13-year-old from the book 50 Best Ideas in Physics. One of these was the Fermi paradox, the argument that evidence of extraterrestrial intelligence should be apparent because the universe is so vast and old. Obviously, Fermi knew about relativity, so the paradox can’t simply be resolved by the possibility of aliens outside our light cone, since even sub-relativistic travel could provide more than enough time to colonize our own galaxy. And, as the aforementioned paradox illustrates, Fermi is known for his predictive skills on the fly, which rely on order-of-magnitude estimation. Another legendary example is how he estimated the strength of the first atomic bomb, nicknamed The Gadget. He did so by dropping several small pieces of paper and measuring how far the blast wind displaced them. His result of 10 kilotons of TNT was of the same order of magnitude as the later estimate of about 25 kilotons. This kind of estimation is a powerful tool for a physicist and reflects the strong physical intuition of people who do it well. Whether in Quantum Mechanics or in General Relativity, order-of-magnitude estimation has been the messenger of great discoveries, with the dimensional estimate behind the proposed power bound associated with Dyson being a good example at today’s frontier. My fascination with Fermi grew when I read The Pope of Physics as a high schooler and learned about his intellectual upbringing in Pisa at the Scuola Normale Superiore, to be specific. Spaniards conquering Normale The entrance. Pisa can be considered the birthplace of modern physics, thanks to Galileo. He was one of the pioneering European scientists to combine rigorous mathematical underpinnings with observational methods, paving the way for all theoreticians today. Not only did he conduct experiments with falling objects, reportedly from the famous Leaning Tower of Pisa, and formulate the principle of relativity, but he also made discoveries, as in the case of Jupiter’s moons. The Leaning Tower of Pisa and the cathedral. And over the centuries, Pisa remained an intellectual hub of Italian physics, where prominent scholars studied up until the 19th century. Then, following Napoleon’s conquest of Tuscany, the Scuola Normale Superiore was established in 1810. Much like its twin, the École Normale Supérieure in Paris, it is a highly selective, intimate institution where the country’s best students are chosen through a rigorous entrance exam to learn in an intensely challenging environment. A 17-year-old Enrico Fermi took this very exam in 1918. For the essay section on the characteristics of sound, Fermi derived the partial differential equation for a vibrating rod rather than writing a standard descriptive answer. He then continued with Fourier analysis, showing a mathematical maturity far beyond his age and leaving his examiner, Giulio Pittarelli, astounded. Over the years, I have heard the legend of this derivation, so below I work through it, starting from the photo I personally took of the essay’s first page. The physics entrance examination, 14 November 1918. We limit our focus to the small transverse vibrations of a uniform elastic rod clamped at one end and free at the other. The governing partial differential equation is $$ \\frac{\\partial^2 y}{\\partial t^2} + a^2 \\frac{\\partial^4 y}{\\partial x^4} = 0 $$ where $y(x,t)$ is the transverse displacement, and the stiffness parameter is defined as $$ a^2 = \\frac{EI}{m} $$ with $E$ representing the modulus of elasticity, $I$ the area moment of inertia of the cross-section, and $m$ the mass per unit length. Applying separation of variables, we expand the sine component of the displacement into a sum of spatial modes $u(x)$ and temporal harmonics oscillating at angular frequencies $k$: $$ y(x,t) = \\sum u(x) \\sin(kt) $$ Taking the second partial derivative with respect to time and the fourth with respect to space yields $$ \\begin{aligned} \\frac{\\partial^2 y}{\\partial t^2} \u0026= -\\sum k^2 u \\sin(kt) \\\\ \\frac{\\partial^4 y}{\\partial x^4} \u0026= \\sum \\frac{d^4 u}{dx^4} \\sin(kt) \\end{aligned} $$ Substituting these back into the wave equation shows that the spatial profile $u(x)$ of each mode must satisfy the fourth-order ordinary differential equation $$ \\frac{d^4 u}{dx^4} - \\beta^4 u = 0 $$ where the spatial parameter is defined by $\\beta^4 = \\frac{k^2}{a^2}$. The roots of the characteristic equation $r^4 - \\beta^4 = 0$ are $\\pm \\beta$ and $\\pm i\\beta$, producing the general solution for the spatial mode: $$ \\begin{aligned} u(x) ={}\u0026 C_1 \\cosh(\\beta x) + C_2 \\sinh(\\beta x) \\\\ \u0026+ C_3 \\cos(\\beta x) + C_4 \\sin(\\beta x) \\end{aligned} $$ We apply the boundary conditions for the rod fixed at one end. At the clamped base ($x = 0$), both the displacement and slope must be zero: $$ \\begin{aligned} u(0) = 0, \u0026\\quad C_1 + C_3 = 0, \\\\ \u0026\\quad C_3 = -C_1 \\\\[4pt] u'(0) = 0, \u0026\\quad \\beta (C_2 + C_4) = 0, \\\\ \u0026\\quad C_4 = -C_2 \\end{aligned} $$ This leaves two unknown coefficients in the spatial profile: $$ \\begin{aligned} u(x) ={}\u0026 C_1 (\\cosh(\\beta x) - \\cos(\\beta x)) \\\\ \u0026+ C_2 (\\sinh(\\beta x) - \\sin(\\beta x)) \\end{aligned} $$ At the free tip of the rod ($x = L$), the physical constraints demand that both the bending moment and the transverse shearing force be zero, meaning the second and third spatial derivatives must vanish: $$ \\begin{aligned} u''(L) ={}\u0026 \\beta^2 [ C_1 (\\cosh \\beta L + \\cos \\beta L) \\\\ \u0026+ C_2 (\\sinh \\beta L + \\sin \\beta L) ] = 0 \\\\[6pt] u'''(L) ={}\u0026 \\beta^3 [ C_1 (\\sinh \\beta L - \\sin \\beta L) \\\\ \u0026+ C_2 (\\cosh \\beta L + \\cos \\beta L) ] = 0 \\end{aligned} $$ $$ \\begin{pmatrix} \\cosh(\\beta L) + \\cos(\\beta L) \u0026 \\sinh(\\beta L) + \\sin(\\beta L) \\\\ \\sinh(\\beta L) - \\sin(\\beta L) \u0026 \\cosh(\\beta L) + \\cos(\\beta L) \\end{pmatrix} \\begin{pmatrix} C_1 \\\\ C_2 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix}. $$ For the rod to vibrate, $C_1$ and $C_2$ cannot both be zero. Therefore, the determinant of their coefficients in this linear system must equal zero: $$ \\begin{aligned} \u0026 (\\cosh(\\beta L) + \\cos(\\beta L))^2 \\\\ \u0026- (\\sinh(\\beta L) + \\sin(\\beta L)) \\\\ \u0026\\quad\\times(\\sinh(\\beta L) - \\sin(\\beta L)) = 0 \\end{aligned} $$ Expanding this expression: $$ \\begin{aligned} \u0026\\cosh^2(\\beta L) + 2\\cosh(\\beta L)\\cos(\\beta L) \\\\ \u0026+ \\cos^2(\\beta L) - \\sinh^2(\\beta L) \\\\ \u0026+ \\sin^2(\\beta L) = 0 \\end{aligned} $$ When we apply the fundamental identities $\\cosh^2(z) - \\sinh^2(z) = 1$ and $\\cos^2(z) + \\sin^2(z) = 1$, the equation collapses perfectly into a transcendental characteristic equation: $$ \\begin{aligned} 2 + 2\\cosh(\\beta L)\\cos(\\beta L) \u0026= 0 \\\\ \\cosh(\\beta L)\\cos(\\beta L) + 1 \u0026= 0 \\end{aligned} $$ Setting $\\mu = \\beta L$, the roots of $\\cos(\\mu) = -1/\\cosh(\\mu)$ dictate the permitted eigenvalues of the rod. The first three non-zero roots are numerically evaluated as: $$ \\begin{aligned} \\mu_1 \u0026\\approx 1.875, \\\\ \\mu_2 \u0026\\approx 4.694, \\\\ \\mu_3 \u0026\\approx 7.855 \\end{aligned} $$ Recalling that $\\beta^4 = k^2 / a^2$, the angular frequencies of the vibrating rod are given by: $$ k_n = \\frac{\\mu_n^2}{L^2} \\sqrt{\\frac{EI}{m}} $$ Because the frequency scales with the square of the roots, the overtones do not fall at integer multiples of the fundamental frequency, generating the characteristic inharmonic sound of a metallic bar. Geometry examination, 13 November 1918. Algebra examination, 12 November 1918. I am not aware of all the exam setup details, including what references Fermi had available. However, if he wrote down the fourth-order equation from memory and solved it as a 17-year-old in an exam setting, this is more than impressive, and the legend behind this essay truly deserves the hype. A page of handwritten calculations among the examination papers. Fermi took the exam near his home in Rome, just after the First World War. Returning to the Scuola Normale, before Napoleon’s conquest, the building was essentially a palace for noble knights training to defend against external forces, primarily the Ottomans. Today, the former school library, used as a conference room, houses historical documents from the Salviati family archive. Books and the Scuola’s banner inside the Palazzo della Carovana. The glass ceiling inside the Scuola. As you can see here, the beautiful glass roof is surrounded by coats of arms of the families whose descendants trained in this very palace. And when you follow the stairs right up from there, you arrive at Fermi’s room, which overlooks the square where the Scuola sits. Fermi’s room. Although physics, the crown of positive sciences, is purely objective and independent of where it is practiced, a rich intellectual spirit often inspires one to think more deeply about nature and the laws that govern it. I can see this is certainly the case for the SNS and Pisa in general, making it entirely unsurprising that this environment fostered such greats as Galileo and Fermi. Galileo and a telescope shaped like the Leaning Tower, on a wall in Pisa. The photos I include here were all taken by me during a conference visit to Pisa in September 2026. ","date":"2026-09-28","dateLabel":"September 28, 2026","readingTime":8,"section":"posts","summary":"A visit to the Scuola Normale Superiore, Fermi’s handwritten entrance examinations, and the mathematics of a vibrating rod.","tags":["Enrico Fermi","Pisa","history","physics"],"timestamp":1790553600,"title":"Fermi's Pisa","url":"/posts/fermis-pisa/"},{"category":"Presentation","content":"Talk given at QUEST 2026 — Quantum Gravity, Early Universe, Space-Time and Theoretical Frontiers, Pisa, Wednesday September 23, 2026. 📊 Singularities as Solitons — Pisa Download  Singularities as Solitons — Pisa Singularities as Solitons Pisa Singularities as Solitons Eren Erberk Erkul Wed, September 23, Pisa The emergence of solitons Zabusky \u0026 Kruskal · Physical Review Letters (1965), Fig. 1 KdV Equation Vacuum fields “Every measurement can yield only an average value of the amplitude in a very small region of space and during a very short interval of time.” Werner Heisenberg, 1930 The Physical Principles of the Quantum Theory Vacuum is Dispersive Can dispersion balance collapse? The scalaron Scalaron dispersion Free slow-envelope limit The scalaron soliton Dispersion and binding Exact gravitational evolution Contraction and re-expansion Gradient support at strong compactness Strong compactness The Lamarina A regular lapse well The wall Radiation The core power bound A Field-Theoretic Origin? Bezrukov \u0026 Shaposhnikov (2008) Are Singularities Solitons? Thank you","date":"2026-09-23","dateLabel":"September 23, 2026","readingTime":1,"section":"posts","summary":"Talk given at QUEST 2026 in Pisa on singularities as solitons, September 23, 2026.","tags":[],"timestamp":1790121600,"title":"Singularities as Solitons — Pisa","url":"/posts/singularities-as-solitons-pisa/"},{"category":"Presentation","content":"3MT Competition — METU Engineering Day 🥇 1st Place Winner — METU Engineering Day 3MT (Three Minute Thesis) Competition, 20 May 2026. 📊 Holographic Spectral Alignment — 3MT Competition Download Poster \u0026 Long Presentation 📄 Holographic Spectral Alignment — Poster Download 📊 Holographic Spectral Alignment — Long Presentation (PHYS400) Download  Holographic Spectral Alignment Holographic 3MT 1 A Holographic Shortcut for 3D Orientation Eren Erberk Erkul · Murat Temiz Physics \u0026 Electrical and Electronics Engineering Middle East Technical University ≈ 100× faster 2–5° accuracy Results HSA poster Eren Erberk Erkul \u0026 Murat Temiz Middle East Technical University , Ankara, Turkey Department of Physics \u0026 EEE · 2026 Recover R ∈ SO(3) that aligns an observed object to a reference. A Fourier transform wrapped around the sphere. Robotics \u0026 pose · molecular reconstruction · computer vision all on band-limited data on the sphere. VOLUMETRIC: ✓ global \u0026 reliable ✗ but O(L⁴) — costly ℓ — angular frequency; LOCAL (ICP): ✓ cheap per step ✗ but needs init · local minima m — azimuthal order. m → east–west nodal lines ℓ−|m| → north–south circles. High |m| modes hug the equator the part of the boundary hologram reads best! WE WANT: deterministic · global · fast. Nodal patterns Yℓm (ℓ = 0…4) THE BULK-BOUNDARY MAP T give it a 3-D 2 …and it just SLIDES! W I ST ! Gauge quadrupole, β small boundary hologram H[y] Holography 5 6 Fourier shift on the boundary Rotation → shift THE BULK Rotation Simple Shift! Read α FFT phase correlation Polar search 1-D in β, γ per trial Compose R undo the gauge Analogy is a two way street. The boundary map is lossy (~4/L). Pushing this purely geometric bound into black hole thermodynamics predicts a logarithmic area correction to the Bekenstein–Hawking entropy. agree! S E L O H K even BLAC Schur's lemma + Cauchy–Schwarz set c from the bulk symmetry group Same structure without gravity! All four theorems verified numerically. Measured vs. predicted within ~1%. HSA overtakes the O(L⁴) baseline at L=16; 8× faster by L=48 as the gap grows like L/log L. On real proteins (PDB) antipodal flips drop 62–78% to 0%, staying 2–3× faster. References: Kostelec–Rockmore ’08 · Driscoll–Healy ’94 · Kuglin–Hines ’75 · Besl–McKay ’92 · Kovacs–Wriggers ’02 · ’t Hooft ’93 · Almheiri–Dong–Harlow ’15 · Cover–Thomas ’06 · Cohen et al. ’18. Stanford bunny: HSA+ICP is 100% reliable at 40 ms, the fast and reliable corner, ~16× under FPFH+RANSAC. HSA METU Phys400 presentation METU · PHYSICS 400 FINAL PRESENTATION Holographic Spectral Alignment Recovering 3-D orientation from boundary information Eren Erberk Erkul Advisor Murat Temiz Middle East Technical University, Ankara Thursday, 11 June 2026 The problem: 3-D orientation Recover the rotation R ∈ SO(3) that aligns an observed object to a reference Robotics \u0026 pose Molecular reconstruction Computer vision band-limited data on the sphere Global vs. fast — the gap Volumetric ✓ global \u0026 reliable ✗ O(L⁴) — costly Local methods ✓ cheap per step ✗ needs init · local minima We want: deterministic · global · fast A holographic viewpoint Bulk: SO(3) Boundary: a 1-D hologram Rotation ⟷ shift Spherical harmonics: Fourier on a sphere Expand band-limited data A wrapped Fourier mode ℓ — angular frequency m — azimuthal order Reading the nodal patterns m → east–west lines ℓ−|m| → north–south circles high |m| hugs the equator How rotations act The Holographic Expression heat-kernel taper — damps high ℓ Algorithm 1 · Gauge quadrupole — β small 2 · Holography boundary hologram H[y] 3 · Rotation → shift Fourier shift on the boundary 4 · Read α FFT phase correlation 5 · Polar search 1-D in β, γ per trial 6 · Compose R undo the gauge golden-section over β Numerical results Faster — 8× at L = 48 Fast \u0026 reliable vs. FPFH A Reciprocal Prediction a log-area correction to the Bekenstein–Hawking law TAKE-HOME 3-D bulk ⟷ 1-D hologram Deterministic · global · no initialization Thank you Any Questions? Eren Erberk Erkul · Advisor: Murat Temiz Middle East Technical University · Physics 400 References 1. Kostelec \u0026 Rockmore — FFTs on the rotation group · J. Fourier Anal. Appl. (2008) 2. Driscoll \u0026 Healy — FFTs \u0026 convolutions on the 2-sphere · Adv. Appl. Math. (1994) 3. Kovacs \u0026 Wriggers — Fast rotational matching · Acta Cryst. D (2002) 4. Kuglin \u0026 Hines — The phase-correlation image-alignment method (1975) 5. Horn — Extended Gaussian images · Proc. IEEE (1984) 6. ’t Hooft (1993) · Susskind (1995) — the holographic principle 7. Cover \u0026 Thomas — Elements of Information Theory, 2nd ed. (2006) HSA poster final final Eren Erberk Erkul \u0026 Murat Temiz Middle East Technical University , Ankara, Turkey Department of Physics \u0026 EEE · PHYS400 Final Project · 2026 Recover R ∈ SO(3) that aligns an observed object to a reference. Robotics \u0026 pose · molecular reconstruction · computer vision all on band-limited data on the sphere. VOLUMETRIC: ✓ global \u0026 reliable ✗ but O(L⁴) — costly LOCAL (ICP): ✓ cheap per step ✗ but needs init · local minima WE WANT: deterministic · global · fast. A Fourier transform wrapped around the sphere . Nodal patterns Yℓm (ℓ = 0…4) ℓ — angular frequency; m — azimuthal order. m → east–west nodal lines ℓ−|m| → north–south circles. High |m| modes hug the equator the part of the boundary hologram reads best! 2 5 6 Analogy is a two way street. The boundary map is lossy (~4/L). Pushing this purely geometric bound into black hole thermodynamics predicts a logarithmic area correction to the Bekenstein –Hawking entropy. Schur's lemma + Cauchy–Schwarz set c from the bulk symmetry group Same structure without gravity! All four theorems verified numerically. Measured vs. predicted within ~1%. HSA overtakes the O(L⁴) baseline at L=16; 8× faster by L=48 as the gap grows like L/log L. On real proteins (PDB) antipodal flips drop 62–78% to 0%, staying 2–3× faster. Stanford bunny: HSA+ICP is 100% reliable at 40 ms, the fast and reliable corner, ~16× under FPFH+RANSAC. References: Kostelec – Rockmore ’08 · Driscoll–Healy ’94 · Kuglin –Hines ’75 · Besl –McKay ’92 · Kovacs– Wriggers ’02 · ’t Hooft ’93 · Almheiri –Dong–Harlow ’15 · Cover–Thomas ’06 · Cohen et al. ’18. Gauge Holography Rotation → shift Read α Polar search Compose R THE BULK Rotation Simple Shift! quadrupole, β small boundary hologram H[y] Fourier shift on the boundary FFT phase correlation 1-D in β, γ per trial undo the gauge …and it just SLIDES! give it a 3-D TWIST! even BLACK HOLES agree! THE BULK-BOUNDARY MAP","date":"2026-05-20","dateLabel":"May 20, 2026","readingTime":1,"section":"posts","summary":"🥇 1st Place — METU Engineering Day 3MT, plus the full PHYS400 poster and long presentation.","tags":[],"timestamp":1779235200,"title":"Holographic Spectral Alignment","url":"/posts/holographic-spectral-alignment/"},{"category":"Presentation","content":" 📄 What is a Horizon? — Experimental Projects (WIZ) Download  What is a Horizon? WIS Experimental Projects What is a Horizon Experimental Projects WIZ Experimental Projects Course Weizmann Institute of Science What is a Horizon? A Fibre Optics Perspective Eren Erberk Erkul 14 March 2026 Alles Gute zum Geburtstag, Herr Einstein! Abstract. This work contains the simulations and theoretical analysis I carried out during the semester break for the Experimental Projects course in the MSc curriculum at the Weizmann Institute of Science, for the fiber-optics tabletop Hawking-radiation experiment directed by Prof. Ulf Leonhardt. Acknowledgment. I am grateful to Mattan Gelvan for providing the initial simulation code, his clear explanations, and his enthusiasm. I also wish to thank my mentor, Ulf Leonhardt, who originated the ideas presented here and holds the copyright to this document. Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Contents 1 Theoretical 2 1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Theoretical Speculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Model, notation, and numerical grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 FFT grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Governing equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Simulation 7 config.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Wavelength and frequency conversion . . . . . . . . . . . . . . . . . . . . . . . . . 7 Pump and probe relative amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Grid sizes and step counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Pulse width conversion for sech profiles . . . . . . . . . . . . . . . . . . . . . . . . 8 Dispersion model and propagation constant . . . . . . . . . . . . . . . . . . . . . . 8 Pump velocity and co-moving frequency . . . . . . . . . . . . . . . . . . . . . . . . 9 Kerr prefactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Configuration presets: fast demo vs paper-like . . . . . . . . . . . . . . . . . . . . . 9 solver.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Time and frequency grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Positive-frequency projector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Analytic-signal Kerr functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Spectral right-hand side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Split-step propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Legacy wrapper: propagate_Probe_Pulse . . . . . . . . . . . . . . . . . . . . . 12 main.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Building the initial pump–probe field . . . . . . . . . . . . . . . . . . . . . . . . . . 12 From co-moving frequency to laboratory branches . . . . . . . . . . . . . . . . . . . 13 Selecting the UV branch used in the comparison plots . . . . . . . . . . . . . . . . . 13 Co-moving theory targets and their laboratory images . . . . . . . . . . . . . . . . . 14 Stimulated subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Peak extraction near each theory branch . . . . . . . . . . . . . . . . . . . . . . . . 15 i Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Template construction and correlation metric . . . . . . . . . . . . . . . . . . . . . 15 Single-run - One ring to rule them all!! . . . . . . . . . . . . . . . . . . . . . . . . . 17 Probe-wavelength sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Interpretation of main.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 plots.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Main visualization: Hawking_plots . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Plotting summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Supplementary modules not used in the final results 25 fme.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Spectral building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Forward propagation right-hand side . . . . . . . . . . . . . . . . . . . . . . . . . . 26 fwm.py . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Spectral pump intensity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Co-moving frequency in the reduced model . . . . . . . . . . . . . . . . . . . . . . 27 Runge–Kutta integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Four-wave mixing link to the main simulation . . . . . . . . . . . . . . . . . . . . . 28 Analytic bound-state style functions . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Results 30 ii Experimental Projects Course — Weizmann Institute of Science 1 Eren Erberk Erkul Theoretical Background The concept of a horizon in Einstein’s relativity is a paradoxical concept. It appears in an interrelated class of divergencies, namely in the boundary of black holes as a solution of the zero of the lapse function of a given black-hole metric, in the Hawking–Hartle cosmological horizon, and from the Unruh effect. Although the spontaneous particle creation shared between these horizons is suggestive, there still appears to be no complete generalisation of the whole class of horizons under the same mathematical structure. Moreover, we still lack the theory of Quantum Gravity, and these spontaneous particle creations are derived using nonrigorous semiclassical techniques. Hence, in the literature, it is widely accepted that these calculations are clues for a greater theory that still awaits us ever since Hawking’s initial work [5]. So, the concept of gravitational horizon is not yet fully established and generalised. So in this project, the aim was to explore what we can learn about the concept of gravitational horizon from its optical counterpart, and what it may teach us about the general properties of such spontaneous particle creation. To do this, one needs a strong dictionary between the two domains in order to translate back and forth the mathematical structure. It is already established that general relativity and classical electrodynamics have a strong mathematical commonality in their structure. Hence, in this report, we will take as given that the relevant optical solutions have a direct counterpart in the horizon kinematics of general relativity, even though this is not obvious at first [2, 6]. Mathematically, we follow the Hamiltonian formulation of the optical analogue developed by Prof. Ulf Leonhardt. Briefly, the idea is the following. One describes the pump and probe fields by amplitudes 𝐴1 and 𝐴2 , whose analytic-signal components are 𝑎 1 and 𝑎 2 . In the co-moving frame the evolution is generated by a Hamiltonian 𝐻 = 𝐻1 + 𝐻2 + 𝐻int , (1.1) where 𝐻1 and 𝐻2 are the free-propagation Hamiltonians of the pump and probe, respectively, and 𝐻int is the Kerr interaction Hamiltonian of the form 𝐻int = 16𝜅 𝐴12 𝐴22 . (1.2) When expanded in terms of 𝑎 𝑚 and 𝑎 ∗𝑚 , this interaction contains several elementary processes, but the most important ones here are \u0010 \u0011 2 𝐻𝑆 = 4𝜅 𝑎 ∗1 𝑎 1 𝑎 ∗2 𝑎 2 , 𝐻 𝑅 = 2𝜅 𝑎 ∗1 𝑎 1 𝑎 ∗2 + 𝑎 (1.3) 2 2 . The first term 𝐻𝑆 gives the refractive-index shift produced by the pump, namely the cross-phase modulation that creates the effective horizon. The second term 𝐻 𝑅 is the pair-creation term: it is the part of the interaction that generates the Hawking partner channel and, correspondingly, the backreaction on the pump. Thus both ingredients are needed: 𝐻𝑆 establishes the moving medium, while 𝐻 𝑅 is the actual radiation-generating process. The pump pulse induces, through the Kerr effect, a refractive-index perturbation 𝑛(𝜔, 𝜏) = 𝑛0 (𝜔) + 𝛿𝑛(𝜏), (1.4) where 𝜏 = 𝑡 − 𝑧/𝑢 is the retarded time in the frame of the pump moving with velocity 𝑢. The local 1 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul propagation constant is then 𝑛(𝜔, 𝜏) 𝜔 . 𝑐 (1.5) 𝜔′ = 𝜔 − 𝑢 𝛽(𝜔, 𝜏) (1.6) 𝛽(𝜔, 𝜏) = In the co-moving frame the quantity plays the role of a conserved Hamiltonian. Equivalently one may write H (𝜏, 𝜔) = 𝜔 − 𝑢 𝛽(𝜔, 𝜏). (1.7) Equation (1.7) acts as the effective co-moving ray Hamiltonian generating the phase-space dynamics in (𝜏, 𝜔). This is the central quantity of the optical horizon picture, because the branch structure in the moving frame is generated by it. The corresponding Hamilton equations are 𝑑𝜏 𝜕H 𝜕𝛽 =− = −1 + 𝑢 , 𝑑𝑧 𝜕𝜔 𝜕𝜔 𝑑𝜔 𝜕H 𝜕𝛽 = = −𝑢 . 𝑑𝑧 𝜕𝜏 𝜕𝜏 (1.8) The first equation determines the drift of a mode relative to the moving pump profile, while the second equation describes how the mode’s frequency changes in a nonuniform Kerr background. Since 𝜕𝛽 1 = , 𝜕𝜔 𝑣 𝑔 (1.9) the first Hamilton equation in Eq. (1.8) becomes 𝑑𝜏 𝑢 = −1 + . 𝑑𝑧 𝑣𝑔 (1.10) Therefore the turning-point or horizon condition is simply 𝑑𝜏 =0 𝑑𝑧 ⇐⇒ 𝑣 𝑔 = 𝑢. (1.11) So the optical horizon is the point where the group velocity of the probe relative to the medium equals the velocity of the pump pulse. At that point the probe can no longer escape across the moving Kerr profile. More precisely, the relevant group velocity is the one in the Kerr-shifted medium established by the pump, so the horizon is created by 𝐻𝑆 while the actual Hawking pair generation is governed by 𝐻 𝑅 . As shown in Figure 1, when the probe pulse approaches the trailing edge of the pump it gets stuck there akin to a black-hole horizon. The same statement may be written directly as a constant-𝜔′ scattering problem: 𝜔 − 𝑢 𝛽(𝜔, 𝜏) = 𝜔0′ . (1.12) For a fixed value of 𝜔0′ , Eq. (1.12) can admit multiple laboratory-frequency roots 𝜔. In that sense, the horizon problem is naturally a mode-conversion problem between different branches that share the same co-moving invariant. This is exactly the mathematical structure that later appears in the numerical code when we solve for the NHR (Negative Hawking Radiation) and DRR (Double Resonant Radiation) branches from a fixed 𝜔′ . 2 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Pump (a) E 2 ∝ ∆n vg′ \u003e vg Probe vg vg′ z Optical Event Horizon Pump (b) Probe vg z Figure 1: Kinematic representation of the fibre-optical event horizon. (a) A faster probe pulse catches up to a slower pump pulse. (b) The probe pulse coincides with the trailing edge of the pump. The Kerr-induced refractive-index change creates an effective horizon where the probe can no longer advance. Near the horizon, one expands the Hamiltonian around a turning point (𝜏ℎ , 𝜔 ℎ ) satisfying 𝜕H = 0. 𝜕𝜔 ( 𝜏ℎ , 𝜔ℎ ) H (𝜏ℎ , 𝜔 ℎ ) = 𝜔0′ , (1.13) Then, to the lowest nontrivial order, one gets 1 H − 𝜔0′ ≈ H 𝜔 𝜔 (𝜔 − 𝜔 ℎ ) 2 + H𝜏 (𝜏 − 𝜏ℎ ), 2 (1.14) which is the local normal form of a turning-point Hamiltonian. Equation (1.14) is the origin of the familiar Airy-type scattering behaviour and is the mathematical reason the horizon acts as a converter between different asymptotic branches. In other words, the optical Hawking problem is not mysterious at the formal level, it is a dispersive Hamiltonian scattering problem with a moving inhomogeneous background. Hence, in the experiment, one launches a weak probe on one asymptotic branch and lets the moving Kerr profile scatter it into the other allowed roots with the same co-moving invariant 𝜔′ . The ultraviolet channels that are later labeled NHR and DRR are therefore not inserted by hand, but arise intrinsically from the branch structure of the Hamiltonian relation. At the same time, from the Hamiltonian point of view, the existence of these partner channels is not only a kinematic statement about the branches, but also a dynamical one where the horizon-forming term 𝐻𝑆 and the radiation-generating term 𝐻 𝑅 act together, the former establishing the effective moving background and the latter producing the Hawking pair and its backreaction. Hence in this project our aim is to analyze whether the UV spectrum directly resembles that of Hawking radiation. 3 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Theoretical Speculations Here, our aim is not to rederive the optical horizon itself, but to ask whether the mathematical structure identified above may be more general. In particular, we ask whether Hawking-like spectral leakage may arise in a broader class of systems governed by suitable partial differential equations with moving inhomogeneous backgrounds and nontrivial branch structure. This is precisely why the fibre-optical system is such a useful laboratory model as it isolates the mathematical core of the horizon process without requiring the full machinery of quantum gravity. Figure 2: Schematic of the Penrose process. An incident particle with initial energy 𝐸 0 \u003e 0 enters the ergosphere and splits. One fragment acquires negative energy 𝐸 2 \u003c 0 and falls past the event horizon. The remaining fragment escapes to infinity with amplified energy 𝐸 1 \u003e 𝐸 0 , thereby extracting rotational energy from the black hole. The Penrose process is a mechanism for extracting energy from a rotating Kerr black hole. Roughly speaking, radiation or matter falling towards a black hole is blueshifted with respect to an observer at infinity, which already suggests that a rotating black hole can in principle act as an energy source. Penrose therefore proposed a gedanken experiment in which a particle enters the ergosphere and splits into two parts: one falls through the horizon with negative energy relative to infinity, while the other escapes with greater energy than the original incident particle, so that energy is extracted from the black hole’s rotation while the total energy remains conserved. Inspired by this idea, Yakov Zel’dovich generalised the mechanism to classical rotating dissipative systems beyond the Kerr geometry [3, 4]. Shortly afterwards, his student Starobinsky extended the analysis further and showed that analogous amplification phenomena could arise for wave scattering in rotating backgrounds more generally. Zel’dovich also argued that quantum fluctuations in such systems should lead to spontaneous emission. This line of thought played an important conceptual role in the development of Hawking’s later work on black-hole radiation [5, 9]. Hence, one should attempt to approach the question in the spirit of Zel’dovich. Hawking radiation may well reflect a more general kinematical structure already present in certain classical systems that exhibit superradiant or horizon-like behaviour, with quantum theory determining the spontaneous character of the emission. It is then natural to ask whether other dynamical systems, equipped with the appropriate boundary conditions and branch structure, may also produce Hawking-like radiation. Hence, in the optical system, the moving Kerr profile provides a classical Hamiltonian scattering problem whose branch conversion imitates the kinematical structure of horizon physics. The proposed idea, then, is that the gravitational setting and the fibre-optical setting share the same essential mathematical structure: the gravitational well is mirrored by the optical background, while 4 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Figure 3: Schematic of Zel’dovich superradiance. A wave incident on a rotating absorbing body can undergo amplification under the condition 𝜔 \u003c 𝑚Ω, illustrating the classical rotational energy-extraction mechanism that later became central to the broader discussion of Hawking-like emission. the moving Kerr perturbation plays the role of the travelling inhomogeneity that drives redshift, mode conversion, and branch scattering. In this sense, there is a near one-to-one correspondence between the mathematical problem underlying Penrose-type horizon physics and that of the fibre-optics experiment proposed here. What the numerics test later is precisely whether the stimulated ultraviolet output tracks those Hamiltonian branches in the manner predicted by the theory. However, due to the constraints of this course’s time frame, we will delay the theoretical conjecture outlined here to a later time. Model, notation, and numerical grids The simulation is formulated as a one-dimensional propagation problem. The field evolves along the fibre coordinate 𝑧, while its temporal structure is resolved on a finite retarded-time window 𝑡. Numerically, we work with the analytic signal 𝑎(𝑡, 𝑧) rather than the real field itself, so that only positive laboratory frequencies are propagated explicitly. This is the natural representation for the code, since dispersion acts diagonally in frequency space while the Kerr response is evaluated in time. Units Internally we use 𝑡 in femtoseconds, 𝑧 in micrometers, 𝜔 in radians per femtosecond, and 𝑐 in micrometers per femtosecond. With this convention, wavelengths are naturally expressed in micrometers and the conversion 2𝜋𝑐 (1.15) 𝜔= 𝜆 is used directly throughout the implementation. 5 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul FFT grids With 𝑁𝑡 samples over a time window 𝑇, the discrete time and frequency grids are Δ𝑡 = 𝑇 , 𝑁𝑡 \u0010 𝑁𝑡 \u0011 𝑡𝑛 = 𝑛 − Δ𝑡, 2 𝜔 𝑘 = 2𝜋 fftfreq(𝑁𝑡 , Δ𝑡). (1.16) The code uses the FFT pair associated with the discretization in Eq. (1.16). The linear dispersive step is therefore applied directly on the 𝜔 𝑘 grid, whereas the nonlinear response is evaluated in the time domain and then transformed back to frequency space. The positive-frequency condition of the analytic signal is enforced by projection on the same 𝜔 𝑘 grid. Governing equation The propagated spectral field 𝐴(𝜔, 𝑧) = F [𝑎(𝑡, 𝑧)] (1.17) obeys an analytic-signal UPPE-type equation of the form \u0010 \u0011 𝜕𝑧 𝐴(𝜔, 𝑧) = −i 𝛽eff (𝜔) 𝐴(𝜔, 𝑧) − i 𝜃 (𝜔) F {N [𝑎]} , + (1.18) where (·)+ ≡ P+ denotes projection onto positive laboratory frequencies. Here 𝛽eff (𝜔) is either the laboratory propagation constant 𝛽(𝜔) or, when the co-moving option is used, the shifted quantity 𝛽(𝜔) − 𝜔/𝑢. The factor 𝜃 (𝜔) is the frequency-dependent Kerr prefactor in the normalization adopted by the code. The nonlinear term is implemented as N [𝑎] = 𝑎 3 + 3|𝑎| 2 𝑎 + 3|𝑎| 2 𝑎 ∗ . (1.19) This is the form actually used by the numerical solver and is fully consistent with the theory described in previous section. In practice, the code alternates between frequency space and time space implying the dispersive part is updated spectrally, while the nonlinear source is constructed in time and returned to frequency by FFT. In the stimulated runs, the evolution law in Eq. (1.18) is solved once for the pump alone and once for the pump plus probe, and the ultraviolet signal of interest is then extracted from their difference. 6 Experimental Projects Course — Weizmann Institute of Science 2 Eren Erberk Erkul Simulation config.py Wavelength and frequency conversion 1 2 def omega0(self, lambda_um: float) -\u003e float: return 2.0 * np.pi * self.c / float(lambda_um) 3 4 5 6 7 8 9 def lambda_from_omega(self, omega): omega = np.asarray(omega, dtype=float) lam = np.full_like(omega, np.inf, dtype=float) m = omega != 0.0 lam[m] = 2.0 * np.pi * self.c / omega[m] return lam Code 1: config.py: omega0 and lambda conversion This establishes the dictionary between the laboratory language of wavelengths and the numerical language of angular frequencies. 𝜔(𝜆) = 2𝜋𝑐 . 𝜆 (2.1) The 𝜔 = 0 bin is mapped to 𝜆 = ∞, which prevents division by zero while ensuring no finite-𝜔 physics is altered. Pump and probe relative amplitudes 1 2 3 4 5 6 7 def pump_probe_amplitudes(self) -\u003e tuple[float, float]: P1 = float(self.P_pump_mW) * float(self.pump_coupling_eff) P2 = float(self.P_probe_mW) if P1 \u003c= 0: return 1.0, 0.0 ratio = np.sqrt(max(P2, 0.0) / P1) return 1.0, float(ratio) Code 2: config.py: pump_probe_amplitudes This sets the relative scaling between the two input envelopes. The pump amplitude is fixed to 1 in the internal units of the repository, and the probe amplitude is set by a square-root power ratio. There is no SI calibration step here; this is a controlled way to ensure that the probe is weaker than the pump with the same relative hierarchy as the experiment. Grid sizes and step counts 1 2 def dt(self) -\u003e float: return float(self.T_window) / float(self.Nt) 3 4 5 6 7 def zsteps(self) -\u003e int: if self.dz \u003c= 0: return 1 return int(np.round(self.z_total / self.dz)) Code 3: config.py: dt and zsteps 7 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Propagation is explicit in 𝑧, so the step size Δ𝑧 sets accuracy and cost. Time is resolved by FFT, so Δ𝑡 = 𝑇/𝑁𝑡 sets spectral resolution and bandwidth. In practice, 𝑇 must contain the full nonlinear interaction without wrap-around, and 𝑁𝑡 must support the UV features without aliasing. Pulse width conversion for sech profiles 1 2 def fwhm_to_T0_sech(self, fwhm_fs: float) -\u003e float: return float(fwhm_fs) / 1.763 Code 4: config.py: FWHM to T0 conversion For 𝐼 (𝑡) = sech2 (𝑡/𝑇0 ), the intensity full-width at half-maximum is FWHM ≈ 1.763 𝑇0 , hence 𝑇0 = FWHM/1.763. This is what controls the bandwidth of the carrier-modulated envelope in a reproducible way. Dispersion model and propagation constant 1 2 3 4 5 6 7 8 def n(self, omega): omega = np.asarray(omega, dtype=float) if self.n_func is not None: return self.n_func(omega) denom = (self.w_res**2 - omega**2) denom = np.where(np.abs(denom) \u003c 1e-6*self.w_res**2, np.sign(denom)*1e-6*self.w_res**2, denom) return self.n0 + self.chi/denom 9 10 11 12 13 14 def beta(self, omega): omega = np.asarray(omega, dtype=float) if self.truncate_beta: return 0.5 * self.beta2 * omega**2 return self.n(omega) * omega / self.c Code 5: config.py: refractive index and beta The main architecture uses 𝑛(𝜔) 𝜔 . 𝑐 In the present implementation, the simplest complete effective profile is taken to be 𝛽(𝜔) = 𝑛(𝜔) = 𝑛0 + 𝜒 𝜔2res − 𝜔2 , (2.2) (2.3) where 𝑛0 is the background refractive index, 𝜒 sets the strength of the dispersive correction, and 𝜔res is the effective resonance frequency of the model. This is a simplified Lorentz-type resonance profile, chosen as the minimal dispersive model capable of producing a nontrivial ultraviolet branch structure and a resonant behaviour in the spectrum. It should therefore be understood as an effective profile rather than as a fully realistic fit to the experimental fibre, and in future iterations it should be replaced by a more accurate experimentally calibrated dispersion model. The quadratic truncation exists only for the legacy demo (Mattan’s first version) and is not the dispersion used in the Hawking/backreaction runs. The clipping around the model pole is a numerical regularization of the effective index, introduced only to avoid divergence when 𝜔 approaches 𝜔res . 8 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Pump velocity and co-moving frequency 1 2 def u_pump(self) -\u003e float: return self.c / float(self.n_g) 3 4 5 6 7 def omega_comoving(self, omega): omega = np.asarray(omega, dtype=float) u = self.u_pump() return (1.0 - self.n(omega) * u / self.c) * omega Code 6: config.py: pump speed and co-moving frequency This implements the co-moving branch label introduced in the theoretical section: \u0010 𝑛(𝜔)𝑢 \u0011 𝜔′ (𝜔) = 𝜔 − 𝑢 𝛽(𝜔) = 1 − 𝜔, 𝑐 𝑢= 𝑐 . 𝑛𝑔 (2.4) All theory targets in 𝜔′ are mapped back to laboratory 𝜔 by solving Eq. (2.4) for 𝜔. Kerr prefactor 1 2 3 4 5 6 def kerr_theta(self, omega): omega = np.asarray(omega, dtype=float) n = self.n(omega) theta = omega * (self.chi3 * self.chi3_scale) / (16.0 * self.c * n) theta = np.where(omega == 0.0, 0.0, theta) return theta Code 7: config.py: Kerr theta The solver writes the nonlinear source as a spectrum multiplied by 𝜃 (𝜔). In the analytic-signal conventions used here, 𝜔 𝜒 (3) 𝜃 (𝜔) = . (2.5) 16 𝑐 𝑛(𝜔) The scale factor chi3_scale is the knob that sets nonlinear strength in this repository’s internal units. Configuration presets: fast demo vs paper-like 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 def paper_config(fast: bool = True) -\u003e Config: cfg = Config() cfg.truncate_beta = False cfg.n_func = cfg.generate_n_func() cfg.lambda_pump = 0.800 cfg.lambda_probe = 1.400 cfg.fwhm_pump = 8.0 cfg.fwhm_probe = 30.0 cfg.z_total = 7000.0 cfg.n_g = float(cfg.n(cfg.omega0(cfg.lambda_pump)) + 5e-6) cfg.use_comoving_frame = True cfg.enforce_analytic = True if fast: cfg.Nt = 2**13 cfg.T_window = 1600.0 cfg.dz = 10.0 else: cfg.Nt = 2**15 cfg.T_window = 3176.0 9 Experimental Projects Course — Weizmann Institute of Science 20 21 Eren Erberk Erkul cfg.dz = 0.5 return cfg 22 23 24 def default_config() -\u003e Config: return paper_config(fast=True) Code 8: config.py: paper_config and default_config The presets control whether one runs a fast demonstration or a heavier run closer to the paper’s numerical section. The key toggles are the grid size 𝑁𝑡 , the time window 𝑇, and the propagation step 𝑑𝑧. In both cases, the same dispersion and the same co-moving mapping are used, so the comparison logic remains consistent. However, due to the limitations of my computer or other reasons that I am not aware of, my computer couldn’t produce a consistent result for the fourth plot, hence the results are based on the fast configuration. In the conclusion, I contemplate this point in greater detail. solver.py Time and frequency grids 1 2 3 def make_time_grid(cfg): dt = cfg.dt() return (np.arange(cfg.Nt) - cfg.Nt//2) * dt 4 5 6 7 def make_omega_grid(t): dt = float(t[1] - t[0]) return 2.0*np.pi*np.fft.fftfreq(t.size, d=dt) Code 9: solver.py: time grid and omega grid The time grid is centred so delays are symmetric and FFT phases behave cleanly. The frequency grid is the exact companion of that discretization. The solver uses the FFT bins implied by 𝑡 for both dispersion and projection. Positive-frequency projector 1 2 3 4 def project_positive(A_w, omega): out = np.zeros_like(A_w) out[omega \u003e= 0.0] = A_w[omega \u003e= 0.0] return out Code 10: solver.py: positive-frequency projection This implements (P+ 𝐴) (𝜔) = Θ(𝜔) 𝐴(𝜔). (2.6) In the analytic-signal formulation this is the enforcement that the propagated field contains only positive laboratory frequencies, while the conjugate contribution is carried explicitly by the nonlinear term. 10 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Analytic-signal Kerr functional 1 2 3 def kerr_nonlinearity_time(a_t): abs2 = np.abs(a_t)**2 return a_t**3 + 3.0*abs2*a_t + 3.0*abs2*np.conj(a_t) Code 11: solver.py: Kerr nonlinearity in time This is the full cubic structure used by the Hawking/backreaction code: N [𝑎] = 𝑎 3 + 3|𝑎| 2 𝑎 + 3|𝑎| 2 𝑎 ∗ . (2.7) These terms are the computational channels for four-wave mixing between pump and probe bands, including conjugate mixing required by the analytic-signal convention. Spectral right-hand side 1 2 3 4 5 6 7 def rhs_uppe(A_w, omega, beta_eff, theta, cfg): a_t = np.fft.ifft(A_w) NL_t = kerr_nonlinearity_time(a_t) NL_w = np.fft.fft(NL_t) if cfg.enforce_analytic: NL_w = project_positive(NL_w, omega) return -1j*(beta_eff*A_w + theta*NL_w) Code 12: solver.py: spectral right-hand side This is the direct discretization of \u0010 \u0011 𝜕𝑧 𝐴 = −i 𝛽eff (𝜔) 𝐴 − i 𝜃 (𝜔) F {N [𝑎]} . + (2.8) The order “IFFT → build N → FFT → project” is the numerical implementation of the projected nonlinear source in Eq. (2.8). Split-step propagation 1 2 3 4 beta = cfg.beta(omega) if cfg.use_comoving_frame: beta = beta - omega/cfg.u_pump() theta = cfg.kerr_theta(omega) 5 6 7 8 9 10 11 12 13 14 15 16 17 lin_half = np.exp(-1j*beta*dz*0.5) for _ in range(zsteps): A_w = A_w * lin_half a_t = np.fft.ifft(A_w) NL_t = kerr_nonlinearity_time(a_t) NL_w = np.fft.fft(NL_t) if cfg.enforce_analytic: NL_w = project_positive(NL_w, omega) A_w = A_w + (-1j*dz)*(theta*NL_w) if cfg.enforce_analytic: A_w = project_positive(A_w, omega) A_w = A_w * lin_half Code 13: solver.py: split-step propagation loop 11 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul The linear part is exactly solvable in frequency space as multiplication by 𝑒 −i𝛽 ( 𝜔)Δ𝑧 . The nonlinear part is local in time, so it is evaluated in 𝑡 after IFFT. The co-moving option replaces 𝛽(𝜔) by 𝛽(𝜔) − 𝜔/𝑢, which is the spectral form of removing pump-frame phase advance. Legacy wrapper: propagate_Probe_Pulse 1 2 3 4 5 6 7 8 9 10 11 12 def propagate_Probe_Pulse(t, omega, in0_t, out0_t, in0_w, out0_w, cfg, d: float = 1.0, iteration_index: int = 0, u_t = None, method: str = \"Linear\", return_time: bool = True): if method == \"Linear\": beta = cfg.beta(omega) phi = np.exp(1j * beta * d) in1_w = in0_w * phi out1_w = out0_w * phi if return_time: return in1_w, out1_w, np.fft.ifft(in1_w), np.fft.ifft(out1_w) return None Code 14: solver.py: propagate_Probe_Pulse legacy wrapper This code corresponds to the simplest linear propagation picture for the older reduced demo of Mattan. For a monochromatic component of angular frequency 𝜔, the field evolves as 𝐴(𝜔, 𝑧) = 𝐴(𝜔, 0) 𝑒 𝑖𝛽 ( 𝜔) 𝑧 , (2.9) 𝜕 𝐴(𝜔, 𝑧) = 𝑖 𝛽(𝜔) 𝐴(𝜔, 𝑧). 𝜕𝑧 (2.10) which solves So each spectral component propagates independently and acquires only a linear dispersive phase. However, in the actual Hawking/backreaction runs this linear monochromatic evolution is no longer sufficient. There the code propagates the analytic signal with both the dispersive term and the nonlinear Kerr source, which is what allows mode conversion, four-wave mixing, and the appearance of the NHR and DRR channels. main.py Building the initial pump–probe field 1 2 def sech(x): return 1.0 / np.cosh(x) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 def build_initial_fields(cfg, t): pump_amp, probe_amp = cfg.pump_probe_amplitudes() T0_1 = cfg.fwhm_to_T0_sech(cfg.fwhm_pump) T0_2 = cfg.fwhm_to_T0_sech(cfg.fwhm_probe) w1 = cfg.omega0(cfg.lambda_pump) w2 = cfg.omega0(cfg.lambda_probe) env1 = pump_amp * sech((t - cfg.pump_delay) / T0_1) env2 = probe_amp * sech((t - cfg.probe_delay) / T0_2) a1 = env1 * np.exp(1j * w1 * t) a2 = env2 * np.exp(1j * w2 * t) return { \"a0_pump_t\": a1, \"a0_probe_t\": a2, 12 Experimental Projects Course — Weizmann Institute of Science \"a0_both_t\": a1 + a2, \"w_pump\": w1, \"w_probe\": w2, 17 18 19 20 Eren Erberk Erkul } Code 15: main.py: sech and initial-field construction The input analytic signal is written as the sum of two carrier-modulated sech pulses, 𝑎(𝑡, 0) = 𝑎 1 (𝑡) + 𝑎 2 (𝑡), \u0010𝑡 − 𝑡𝑗 \u0011 ei𝜔 𝑗 𝑡 . 𝑎 𝑗 (𝑡) = 𝐴 𝑗 sech 𝑇0, 𝑗 (2.11) Here 𝐴 𝑗 is the internal amplitude, 𝑡 𝑗 is the delay, and 𝑇0, 𝑗 is the sech width obtained from the chosen FWHM. The function returns the pump-only, probe-only, and combined fields, together with the carrier frequencies 𝜔1 and 𝜔2 , so that the rest of the code uses exactly the same initial spectral labels as the propagated field. From co-moving frequency to laboratory branches 1 2 3 4 5 6 7 8 9 10 def solve_lab_omega_from_comoving(cfg, omega_prime_target, omega_max, ngrid=20000): w = np.linspace(1e-6, float(omega_max), int(ngrid)) f = cfg.omega_comoving(w) - float(omega_prime_target) jumps = np.where(np.diff(np.sign(f)) != 0)[0] roots = [] for j in jumps: a, b = w[j], w[j + 1] root = brentq(lambda ww: float(cfg.omega_comoving(ww) - omega_prime_target), a, b) roots.append(float(root)) return sorted(roots) Code 16: main.py: solving 𝜔′ (𝜔) = 𝜔★′ The theory is organized in terms of the co-moving conserved label introduced previously. For a fixed target value 𝜔★′ , the code solves 𝜔′ (𝜔) = 𝜔★′ (2.12) for the corresponding laboratory frequency 𝜔. Because dispersion can make 𝜔′ (𝜔) non-monotone, one co-moving target can correspond to several laboratory roots. Numerically, the code samples 𝑓 (𝜔) = 𝜔′ (𝜔) − 𝜔★′ , detects sign changes, and then applies Brent’s method (is essentially a root finding algorithm that combines multiple methods) inside each bracket. The output is therefore not a single branch but a list of admissible laboratory-frequency branches. Selecting the UV branch used in the comparison plots 1 2 3 4 5 6 7 8 9 def choose_root_in_window(cfg, roots, lam_min=0.18, lam_max=0.50): candidates = [] for w in roots: lam = float(cfg.lambda_from_omega(w)) if np.isfinite(lam) and (lam_min \u003c= lam \u003c= lam_max): candidates.append((w, lam)) if not candidates: return np.nan return max(w for w, _ in candidates) Code 17: main.py: selecting the plotted branch 13 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul This extra selection step is important. Since the theory comparison is plotted in a UV wavelength window, the code does not use an arbitrary laboratory root. Instead, after finding all solutions of Eq. (2.12), it keeps the branch that lies inside the plotting window 𝜆min ≤ 𝜆(𝜔) ≤ 𝜆max , 𝜆(𝜔) = 2𝜋𝑐 . 𝜔 (2.13) In the present implementation this is the 0.18–0.50 𝜇m window. This prevents the sweep panel and the single-run UV panel from accidentally following different mathematical branches of the same co-moving relation. Co-moving theory targets and their laboratory images 1 2 3 def hawking_theory(cfg, lambda_probe_um): w1 = cfg.omega0(cfg.lambda_pump) w2 = cfg.omega0(lambda_probe_um) 4 5 6 w1_p = float(cfg.omega_comoving(w1)) w2_p = float(cfg.omega_comoving(w2)) 7 8 9 wN_p = -w2_p wB_p = w1_p - 2.0 * w2_p 10 11 12 13 omega_max = np.pi / cfg.dt() * 0.98 roots_N = solve_lab_omega_from_comoving(cfg, wN_p, omega_max) roots_B = solve_lab_omega_from_comoving(cfg, wB_p, omega_max) 14 15 16 wN = choose_root_in_window(cfg, roots_N, lam_min=0.18, lam_max=0.50) wB = choose_root_in_window(cfg, roots_B, lam_min=0.18, lam_max=0.50) 17 18 19 lamN = float(cfg.lambda_from_omega(wN)) if np.isfinite(wN) else np.nan lamB = float(cfg.lambda_from_omega(wB)) if np.isfinite(wB) else np.nan 20 21 22 23 24 25 26 27 28 29 30 31 32 33 return { \"w_pump\": w1, \"w_probe\": w2, \"wprime_pump\": w1_p, \"wprime_probe\": w2_p, \"wprime_NHR\": wN_p, \"wprime_DRR\": wB_p, \"omega_NHR\": wN, \"omega_DRR\": wB, \"lambda_NHR_um\": lamN, \"lambda_DRR_um\": lamB, \"c\": cfg.c, } Code 18: main.py: theory targets and UV branch selection The theoretical NHR and DRR targets are first built in the co-moving frame: ′ 𝜔NHR = −𝜔2′ , ′ 𝜔DRR = 𝜔1′ − 2𝜔2′ . (2.14) These are then mapped back to laboratory frequencies by solving the Doppler relation for 𝜔. The ′ ′ function returns both levels of description: the co-moving targets 𝜔NHR , 𝜔DRR , and the selected laboratory frequencies and wavelengths 𝜔NHR , 𝜔DRR , 𝜆NHR = 14 2𝜋𝑐 , 𝜔NHR 𝜆DRR = 2𝜋𝑐 . 𝜔DRR (2.15) Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul This makes the theory comparison concrete where the invariants are formed in the moving frame, but the final comparison is done in the laboratory spectrum where the simulation is actually measured. Stimulated subtraction 1 2 3 A_pump_w = propagate_uppe(init[\"a0_pump_t\"], cfg)[\"a1_w\"] A_both_w = propagate_uppe(init[\"a0_both_t\"], cfg)[\"a1_w\"] S_stim = np.maximum(np.abs(A_both_w)**2 - np.abs(A_pump_w)**2, 0.0) Code 19: main.py: stimulated UV signal The plotted UV signal is not the raw output spectrum but the probe-induced excess, \u0010 \u0011 𝑆stim (𝜔) = max | 𝐴both (𝜔)| 2 − | 𝐴pump (𝜔)| 2 , 0 . (2.16) This subtraction removes the background generated by the pump alone and isolates the spectral weight that appears only when the probe is present. The point of the max is not to reduce the actual components of the true signal, but simply to suppress small negative residuals produced by finite-grid subtraction and numerical noise. Peak extraction near each theory branch 1 2 3 4 5 6 7 8 def extract_peak_near(omega, spectrum, omega0, width=0.25): if not np.isfinite(omega0): return np.nan mask = (omega \u003e 0.0) \u0026 (omega \u003e omega0 - width) \u0026 (omega \u003c omega0 + width) if not np.any(mask): return np.nan j = np.argmax(spectrum[mask]) return float(omega[mask][j]) Code 20: main.py: branch-local peak extraction Once theory predicts a branch center 𝜔★, the simulation peak is defined as the local maximum of the stimulated spectrum in a neighborhood of that theory value: 𝜔peak = arg max 𝑆stim (𝜔). (2.17) | 𝜔− 𝜔★ |\u003c𝑤 This is important in the sweep. The goal is not to find the global maximum of the entire UV spectrum, but to ask whether the simulated signal near the predicted NHR or DRR location actually follows the corresponding theoretical branch as the probe wavelength is varied. Template construction and correlation metric 1 2 3 4 5 6 def gaussian_template(omega, centers, sigma=0.25): tmp = np.zeros_like(omega, dtype=float) for c in centers: if np.isfinite(c): tmp += np.exp(-0.5 * ((omega - c) / sigma) ** 2) return tmp 7 8 def spectral_correlation(sim, ref): 15 Experimental Projects Course — Weizmann Institute of Science 9 10 11 12 13 14 15 16 17 18 19 Eren Erberk Erkul sim = np.asarray(sim, dtype=float) ref = np.asarray(ref, dtype=float) m = np.isfinite(sim) \u0026 np.isfinite(ref) if np.sum(m) \u003c 10: return np.nan a = sim[m] - np.mean(sim[m]) b = ref[m] - np.mean(ref[m]) den = np.sqrt(np.sum(a * a) * np.sum(b * b)) if den == 0.0: return np.nan return float(np.sum(a * b) / den) Code 21: main.py: theory template and correlations The theoretical UV spectrum used for visual comparison is represented by a two-peak Gaussian template centered at the predicted NHR and DRR frequencies. If those centers are 𝜔NHR and 𝜔DRR , then the template has the form \u0015 \u0014 \u0015 (𝜔 − 𝜔DRR ) 2 (𝜔 − 𝜔NHR ) 2 + exp − . 𝑇 (𝜔) = exp − 2𝜎 2 2𝜎 2 \u0014 (2.18) Although the stimulated spectrum is not available in closed analytic form, the code represents it numerically as sampled values 𝑆𝑖 = 𝑆stim (𝜔𝑖 ) on the FFT frequency grid. \u0001 \u0001 ¯ 𝑇𝑖 − 𝑇¯ 𝜌spec = √︂ h , \u0001 2 i hÍ \u00012i Í ¯ ¯ 𝑖 𝑆𝑖 − 𝑆 𝑖 𝑇𝑖 − 𝑇 Í 𝑖 𝑆𝑖 − 𝑆 (2.19) The scalar spectral correlation is defined by Eq. (2.19). It is the normalized inner product between the stimulated simulation spectrum and the theory template after subtracting the means. It is therefore a shape-comparison metric rather than an absolute-power comparison of a direct Hawking formula. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 def time_xcorr(x, y, dt): x = np.asarray(x, dtype=float) y = np.asarray(y, dtype=float) x = x - np.mean(x) y = y - np.mean(y) n = x.size m = 2 * n X = np.fft.fft(x, n=m) Y = np.fft.fft(y, n=m) corr = np.fft.ifft(X * np.conj(Y)).real corr = np.concatenate((corr[-(n - 1):], corr[:n])) lags = np.arange(-(n - 1), n) delay = lags * dt if np.max(np.abs(corr)) \u003e 0: corr_n = corr / np.max(np.abs(corr)) else: corr_n = corr return {\"delay_fs\": delay, \"corr\": corr, \"corr_norm\": corr_n} 19 20 21 22 23 24 25 26 def spectral_xcorr(x, y, domega): out = time_xcorr(x, y, domega) return { \"shift_radfs\": out[\"delay_fs\"], \"corr_spec\": out[\"corr\"], \"corr_spec_norm\": out[\"corr_norm\"], } 16 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Code 22: main.py: time and spectral cross-correlation The time-domain cross-correlation is computed after filtering the stimulated complex spectrum around the NHR and DRR bands, transforming each band back to time, and forming the corresponding intensities 𝐼NHR (𝑡 𝑛 ) and 𝐼DRR (𝑡 𝑛 ). In the code, the mean values are first removed, 𝐼˜NHR (𝑡 𝑛 ) = 𝐼NHR (𝑡 𝑛 ) − 𝐼 NHR , 𝐼˜DRR (𝑡 𝑛 ) = 𝐼DRR (𝑡 𝑛 ) − 𝐼 DRR , and the discrete linear cross-correlation is then evaluated as ∑︁ 𝐶𝑘 = 𝐼˜NHR (𝑡 𝑛 ) 𝐼˜DRR (𝑡 𝑛+𝑘 ), (2.20) 𝑛 with delay axis Δ𝑡 𝑘 = 𝑘 Δ𝑡. The implementation uses the standard FFT method with zero-padding to obtain the linear, rather than circular, correlation. Afterward the correlation is normalized by its maximum absolute value, 𝐶 𝑘(norm) = 𝐶𝑘 , max 𝑗 |𝐶 𝑗 | so the result indicates whether the two selected UV channels are temporally localized in the same nonlinear event, independent of their absolute scale. The spectral version is mathematically the same construction applied to arrays indexed by 𝜔 rather than 𝑡, so its horizontal axis is a spectral shift Δ𝜔 𝑘 = 𝑘 Δ𝜔. Single-run - One ring to rule them all!! 1 2 3 4 def run_single(cfg): t = make_time_grid(cfg) omega = make_omega_grid(t) init = build_initial_fields(cfg, t) 5 6 7 res_pump = propagate_uppe(init[\"a0_pump_t\"], cfg) A_pump_w = res_pump[\"a1_w\"] 8 9 10 res_both = propagate_uppe(init[\"a0_both_t\"], cfg) A_both_w = res_both[\"a1_w\"] 11 12 13 14 theory = hawking_theory(cfg, cfg.lambda_probe) template = gaussian_template(omega, [theory[\"omega_NHR\"], theory[\"omega_DRR\"]], sigma=0.25) theory[\"template_uv\"] = template 15 16 17 18 19 S_stim = np.maximum(np.abs(A_both_w) ** 2 - np.abs(A_pump_w) ** 2, 0.0) A_stim_w = A_both_w - A_pump_w if cfg.enforce_analytic: A_stim_w = project_positive(A_stim_w, omega) Code 23: main.py: propagation and stimulated spectrum in a single run The single-run code first builds the common time and frequency grids introduced above, then propagates two initial conditions through the same UPPE solver, namely the pump alone and the pump plus probe. The theory prediction is generated for the same probe wavelength as the propagated input, using the NHR 17 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul and DRR branch construction discussed previously. At that point the code has three key spectral objects: 𝐴pump (𝜔), 𝐴both (𝜔), 𝑆stim (𝜔). (2.21) Here 𝑆stim (𝜔) is precisely the stimulated difference spectrum already defined in Eq. (2.16). In addition, it constructs an approximate complex stimulated field 𝐴stim (𝜔) ≈ 𝐴both (𝜔) − 𝐴pump (𝜔), (2.22) which is then used for bandpass filtering and time-domain correlation analysis. This subtraction is only an operational metric; the physically plotted spectrum remains 𝑆stim from Eq. (2.16). A small drawback is that 𝐴stim (𝜔) is not itself a directly measured observable, but rather a convenient constructed field used for the analysis which is also done in the analysis of Prof. Leonhardt. 1 2 fN = np.exp(-0.5 * ((omega - theory[\"omega_NHR\"]) / 0.40) ** 2) fB = np.exp(-0.5 * ((omega - theory[\"omega_DRR\"]) / 0.40) ** 2) 3 4 5 aN_t = np.fft.ifft(A_stim_w * fN) aB_t = np.fft.ifft(A_stim_w * fB) 6 7 8 IN = np.abs(aN_t) ** 2 IB = np.abs(aB_t) ** 2 9 10 corr = time_xcorr(IN, IB, cfg.dt()) Code 24: main.py: bandpass filters and time-domain correlation The two Gaussian filters 𝑓 𝑁 (𝜔) and 𝑓 𝐵 (𝜔) isolate narrow neighborhoods around the predicted NHR and DRR bands, in the same spirit as the two-peak template of Eq. (2.18). After inverse Fourier transform, the code obtains two band-limited time-domain signals whose intensities are then cross-correlated according to the discrete construction described in Eq. (2.20). In other words, the code asks whether the two theoretically selected UV channels are generated with similar temporal localization. A second small drawback is that the Gaussian windows although common in stochastic analysis are choices rather than unique physical objects, so their widths affect the output. 1 2 3 4 5 6 7 8 9 10 11 result = { \"t\": t, \"omega\": omega, \"a0_pump_t\": init[\"a0_pump_t\"], \"a0_probe_t\": init[\"a0_probe_t\"], \"a0_both_t\": init[\"a0_both_t\"], \"A_pump_w\": A_pump_w, \"A_both_w\": A_both_w, \"S_stim\": S_stim, \"A_stim_w\": A_stim_w, } 12 13 14 pos = omega \u003e 0.0 corr[\"spectral_corr\"] = spectral_correlation(S_stim[pos], template[pos]) 15 16 17 18 19 20 21 22 if np.isfinite(theory.get(\"omega_NHR\", np.nan)) and np.isfinite(theory.get(\"omega_DRR\", np.nan)): wmin = min(theory[\"omega_NHR\"], theory[\"omega_DRR\"]) - 2.0 wmax = max(theory[\"omega_NHR\"], theory[\"omega_DRR\"]) + 2.0 spec_mask = (omega \u003e 0.0) \u0026 (omega \u003e wmin) \u0026 (omega \u003c wmax) else: lam = cfg.lambda_from_omega(omega) spec_mask = (omega \u003e 0.0) \u0026 (lam \u003e 0.18) \u0026 (lam \u003c 0.50) 23 24 if np.any(spec_mask): 18 Experimental Projects Course — Weizmann Institute of Science 25 26 27 28 29 30 Eren Erberk Erkul domega = float(omega[1] - omega[0]) spec = spectral_xcorr(S_stim[spec_mask], template[spec_mask], domega) corr.update(spec) jmax = int(np.argmax(spec[\"corr_spec_norm\"])) corr[\"spec_shift_at_max\"] = float(spec[\"shift_radfs\"][jmax]) corr[\"spec_maxcorr\"] = float(spec[\"corr_spec_norm\"][jmax]) 31 32 return result, theory, corr Code 25: main.py: spectral metric and return values The function finally computes two spectral metrics. The first is the scalar correlation coefficient between the stimulated spectrum and the theory template, namely the quantity defined in Eq. (2.19). The second is the full spectral cross-correlation curve, obtained from the same discrete construction applied on the spectral grid rather than the time grid. From this the code extracts the shift at maximum overlap. If the theory template were centered perfectly on the simulated UV peaks, the maximum would occur close to zero shift. Thus the single-run execution provides not only the propagated fields, but also a compact measure of how well the observed UV structure aligns with the branch prediction. The remaining drawback is that all of these are finite-grid measures, so the precise values of the correlation and shift still depend on the chosen numerical resolution and spectral window. Probe-wavelength sweep 1 2 def run_sweep(cfg, lambda_list_um): original_lambda_probe = cfg.lambda_probe 3 4 5 t = make_time_grid(cfg) omega = make_omega_grid(t) 6 7 8 9 init0 = build_initial_fields(cfg, t) res_pump = propagate_uppe(init0[\"a0_pump_t\"], cfg) A_pump_w = res_pump[\"a1_w\"] 10 11 12 13 14 15 16 17 18 out = { \"lambda_probe_um\": [], \"theory_NHR_um\": [], \"theory_DRR_um\": [], \"sim_NHR_um\": [], \"sim_DRR_um\": [], \"spectral_corr\": [], } Code 26: main.py: sweep setup and pump reuse The sweep reuses a single pump-only propagation and then varies only the probe wavelength. This is computationally efficient and also conceptually clean, because the comparison is always made against the same pump background. The only changing external parameter is 𝜆2 , the probe wavelength. 1 2 for lam2 in lambda_list_um: cfg.lambda_probe = float(lam2) 3 4 5 6 init = build_initial_fields(cfg, t) res_both = propagate_uppe(init[\"a0_both_t\"], cfg) A_both_w = res_both[\"a1_w\"] 7 8 9 theory = hawking_theory(cfg, lam2) S_stim = np.maximum(np.abs(A_both_w) ** 2 - np.abs(A_pump_w) ** 2, 0.0) 10 11 sep = abs(theory[\"omega_NHR\"] - theory[\"omega_DRR\"]) 19 Experimental Projects Course — Weizmann Institute of Science 12 Eren Erberk Erkul w_width = max(0.03, 0.45 * sep) 13 14 15 wN_sim = extract_peak_near(omega, S_stim, theory[\"omega_NHR\"], width=w_width) wB_sim = extract_peak_near(omega, S_stim, theory[\"omega_DRR\"], width=w_width) 16 17 18 lamN_sim = float(cfg.lambda_from_omega(wN_sim)) if np.isfinite(wN_sim) else np.nan lamB_sim = float(cfg.lambda_from_omega(wB_sim)) if np.isfinite(wB_sim) else np.nan 19 20 21 22 template = gaussian_template(omega, [theory[\"omega_NHR\"], theory[\"omega_DRR\"]], sigma=0.25) pos = omega \u003e 0.0 c_spec = spectral_correlation(S_stim[pos], template[pos]) 23 24 25 26 27 28 29 out[\"lambda_probe_um\"].append(float(lam2)) out[\"theory_NHR_um\"].append(float(theory[\"lambda_NHR_um\"])) out[\"theory_DRR_um\"].append(float(theory[\"lambda_DRR_um\"])) out[\"sim_NHR_um\"].append(lamN_sim) out[\"sim_DRR_um\"].append(lamB_sim) out[\"spectral_corr\"].append(c_spec) 30 31 32 cfg.lambda_probe = original_lambda_probe return out Code 27: main.py: branch tracking across the sweep For each probe wavelength, the code constructs the corresponding theory branches, propagates the pump+probe field, and then extracts the simulated peaks near those theory predictions. The extraction window is not fixed globally; instead it is chosen from the theory branch separation, \u0001 𝑤 width = max 0.03, 0.45 |𝜔NHR − 𝜔DRR | , (2.23) so that the two search windows remain local to their respective theory branches and do not collapse onto one another when the branch separation changes. The function then stores 𝜆2 , th 𝜆th NHR , 𝜆 DRR , sim 𝜆sim NHR , 𝜆 DRR , (2.24) together with the spectral correlation score used in the final comparison panel. Restoring the original probe wavelength at the end keeps the configuration object well-defined after the sweep. Interpretation of main.py The logic of main.py is therefore the following. First, it constructs the two-colour analytic input field. Second, it computes the co-moving theory targets and maps them into the laboratory UV window where the simulation is read out. Third, it compares pump-only and pump+probe propagation in order to isolate the stimulated spectral contribution. Fourth, it evaluates whether the simulated UV bands lie near the predicted NHR and DRR branches for one probe wavelength and then across a full probe-wavelength sweep. In this way the module acts as the bridge between the propagation solver and the final physics figures hence it is where the abstract mathematical branch picture becomes a concrete comparison between predicted and simulated spectral bands. 20 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul plots.py Main visualization: Hawking_plots 1 2 3 4 def Hawking_plots(result_single, theory_single, corr_single, sweep=None, filename=\"hawking_output.png\"): t = result_single[\"t\"] omega = result_single[\"omega\"] c = theory_single[\"c\"] 5 6 7 lam = _omega_to_lambda_um(omega, c) pos = omega \u003e 0.0 8 9 10 11 S_pump = np.abs(result_single[\"A_pump_w\"]) ** 2 S_both = np.abs(result_single[\"A_both_w\"]) ** 2 S_stim = np.maximum(S_both - S_pump, 0.0) 12 13 14 15 16 17 idx = np.argsort(lam[pos]) lam_p = lam[pos][idx] Sp = S_pump[pos][idx] Sb = S_both[pos][idx] Ss = S_stim[pos][idx] 18 19 20 lam_NHR = theory_single.get(\"lambda_NHR_um\", None) lam_DRR = theory_single.get(\"lambda_DRR_um\", None) 21 22 fig, axes = plt.subplots(4, 1, figsize=(10, 12), constrained_layout=True) Code 28: plots.py: extracting the plotted spectra The plotting routine begins by converting the propagated positive-frequency grid from angular frequency to wavelength, 2𝜋𝑐 𝜆(𝜔) = , 𝜔 \u003e 0, (2.25) 𝜔 and then restricting attention to the physical positive-frequency sector. The three spectral quantities passed to the figure are 𝑆pump (𝜔) = | 𝐴pump (𝜔)| 2 , 𝑆both (𝜔) = | 𝐴both (𝜔)| 2 , \u0001 𝑆stim (𝜔) = max 𝑆both − 𝑆pump , 0 . (2.26) After that, the arrays are sorted by wavelength so that the horizontal axis is visually intuitive. This sorting step is purely for presentation: the simulation itself is performed on the native FFT 𝜔-grid. 1 2 3 4 5 6 7 axes[0].plot(t, np.real(result_single[\"a0_pump_t\"]), label=\"Re[pump] @ z=0\") axes[0].plot(t, np.real(result_single[\"a0_probe_t\"]), label=\"Re[probe] @ z=0\") axes[0].plot(t, np.real(result_single[\"a0_both_t\"]), label=\"Re[pump+probe] @ z=0\", alpha=0.7) axes[0].set_xlabel(\"retarded time 𝜏 (fs)\") axes[0].set_ylabel(\"Re[a(𝜏)]\") axes[0].grid(True, alpha=0.3) axes[0].legend(loc=\"upper right\") Code 29: plots.py: panel 1 — input fields in time Panel 1 shows the real parts of the initial analytic-signal fields at 𝑧 = 0. This panel is not an output metric but an input reference. It makes clear which two carrier-modulated pulses are launched into the solver and how their sum is arranged in the retarded-time window. In other words, the first panel displays the initial condition 𝑎(𝑡, 0) = 𝑎 pump (𝑡) + 𝑎 probe (𝑡) (2.27) whose subsequent propagation generates the UV response shown later. 21 Experimental Projects Course — Weizmann Institute of Science 1 2 3 4 5 Eren Erberk Erkul uv_mask = (lam_p \u003e= 0.18) \u0026 (lam_p \u003c= 0.50) lam_uv = lam_p[uv_mask] Sp_uv = Sp[uv_mask] Sb_uv = Sb[uv_mask] Ss_uv = Ss[uv_mask] 6 7 8 9 10 11 12 13 14 15 16 17 18 axes[1].plot(lam_uv, _normalize(Sp_uv), label=\"pump-only |A(𝜔)|2 (UV norm)\") axes[1].plot(lam_uv, _normalize(Sb_uv), label=\"pump+probe |A(𝜔)|2 (UV norm)\") axes[1].plot(lam_uv, _normalize(Ss_uv), label=\"stimulated (diff) (UV norm)\") axes[1].set_xlim(0.18, 0.50) axes[1].set_xlabel(\"wavelength 𝜆 (um)\") axes[1].set_ylabel(\"normalized spectral intensity (UV)\") axes[1].grid(True, alpha=0.3) if lam_NHR is not None: axes[1].axvline(lam_NHR, linestyle=\"--\", linewidth=1.2, label=\"theory NHR\") if lam_DRR is not None: axes[1].axvline(lam_DRR, linestyle=\":\", linewidth=1.2, label=\"theory DRR\") axes[1].legend(loc=\"upper right\") Code 30: plots.py: panel 2 — UV spectrum and theory markers Panel 2 is the central spectral comparison. The code restricts the wavelength axis to the UV window 0.18 𝜇m ≤ 𝜆 ≤ 0.50 𝜇m, (2.28) because this is the region where the branch-selected NHR and DRR partners are expected in the present simulation. Each spectrum is normalized separately by the plotting helper, so the panel compares spectral shape and peak location rather than absolute power. The vertical lines mark the theory-predicted laboratory wavelengths obtained from the co-moving branch calculation in main.py. This panel therefore answers the most direct question in the simulation: whether the stimulated UV signal appears in the same spectral region as the predicted partner branches. 1 2 3 4 5 6 7 8 9 10 if (\"shift_radfs\" in corr_single) and (\"corr_spec_norm\" in corr_single): axins = axes[1].inset_axes([0.62, 0.10, 0.35, 0.35]) axins.plot(corr_single[\"shift_radfs\"], corr_single[\"corr_spec_norm\"]) if \"spec_shift_at_max\" in corr_single: axins.axvline(corr_single[\"spec_shift_at_max\"], linestyle=\"--\", linewidth=1.0) axins.set_title(\"spectral xcorr\", fontsize=8) axins.set_xlabel(\"Δ𝜔 (rad/fs)\", fontsize=7) axins.set_ylabel(\"xcorr\", fontsize=7) axins.tick_params(axis=\"both\", labelsize=7) axins.grid(True, alpha=0.3) Code 31: plots.py: inset in panel 2 — spectral cross-correlation The inset refines the previous comparison. Instead of showing only the theory markers, it plots the full spectral cross-correlation between the stimulated UV spectrum and the two-peak theory template. If that cross-correlation were maximized at Δ𝜔 = 0, (2.29) then the template would already be centered optimally on the observed UV structure. A nonzero maximizing shift means that the simulation and the theory template agree best after a small spectral displacement. 1 2 3 4 5 axes[2].plot(corr_single[\"delay_fs\"], corr_single[\"corr_norm\"], label=\"xcorr(I_NHR, I_DRR)\") axes[2].set_xlabel(\"delay (fs)\") axes[2].set_ylabel(\"normalized cross-correlation\") axes[2].grid(True, alpha=0.3) axes[2].legend(loc=\"upper right\") 22 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Code 32: plots.py: panel 3 — time-domain cross-correlation Panel 3 visualizes the normalized time-domain cross-correlation defined previously in Eq. (2.20). Since the code normalizes by the maximum absolute value, the panel is a structural metric rather than an absolute-amplitude one. A peak near Δ𝑡 = 0 indicates that the two selected UV channels are localized around the same nonlinear interaction event in time. 1 2 3 4 5 6 if sweep is not None: lam_probe = np.asarray(sweep[\"lambda_probe_um\"], dtype=float) th_N = np.asarray(sweep[\"theory_NHR_um\"], dtype=float) th_B = np.asarray(sweep[\"theory_DRR_um\"], dtype=float) sim_N = np.asarray(sweep[\"sim_NHR_um\"], dtype=float) sim_B = np.asarray(sweep[\"sim_DRR_um\"], dtype=float) 7 8 9 10 11 12 13 14 15 axes[3].plot(lam_probe, th_N, marker=\"o\", linestyle=\"--\", label=\"theory NHR\") axes[3].plot(lam_probe, th_B, marker=\"o\", linestyle=\":\", label=\"theory DRR\") axes[3].plot(lam_probe, sim_N, marker=\"x\", linestyle=\"-\", label=\"sim NHR\") axes[3].plot(lam_probe, sim_B, marker=\"x\", linestyle=\"-\", label=\"sim DRR\") axes[3].set_xlabel(\"probe wavelength 𝜆2 (um)\") axes[3].set_ylabel(\"UV peak wavelength (um)\") axes[3].grid(True, alpha=0.3) axes[3].legend(loc=\"upper right\") Code 33: plots.py: panel 4 — theory and simulation across the sweep When a sweep is provided, panel 4 becomes the branch-tracking panel. Its horizontal axis is the input probe wavelength 𝜆 2 , and its vertical axis is the UV output wavelength extracted either from theory or from the stimulated spectrum. What is plotted is therefore the comparison th sim sim 𝜆2 ↦−→ 𝜆th NHR , 𝜆 DRR , 𝜆 NHR , 𝜆 DRR . (2.30) This is the most stringent visual test in the figure, because it checks not only whether the UV response exists at one probe wavelength, but also whether the observed peak locations move with 𝜆 2 in the same way as the theoretical branches. 1 2 3 4 5 6 7 8 9 10 def _corr(a, b): m = np.isfinite(a) \u0026 np.isfinite(b) if np.sum(m) \u003c 2: return np.nan A = a[m] - np.mean(a[m]) B = b[m] - np.mean(b[m]) den = np.sqrt(np.sum(A * A) * np.sum(B * B)) if den == 0: return np.nan return float(np.sum(A * B) / den) 11 12 13 14 rN = _corr(sim_N, th_N) rB = _corr(sim_B, th_B) axes[3].set_title(f\"peak-location similarity: corr(NHR)={rN:.3f}, corr(DRR)={rB:.3f}\") Code 34: plots.py: panel 4 title — similarity scores The panel title gives an additional scalar summary. For each branch separately, the code computes a normalized correlation coefficient between the simulated and theoretical wavelength arrays. If the simulated and theoretical curves had exactly the same shape up to an overall affine offset in level, the coefficient would be close to 1. Thus these numbers serve as indicators of how well the simulation tracks the theory across the entire sweep. 23 Experimental Projects Course — Weizmann Institute of Science 1 2 3 4 Eren Erberk Erkul else: template = theory_single.get(\"template_uv\", None) if template is None: template = np.zeros_like(Ss) 5 6 7 8 9 10 11 12 13 Tt = _normalize(template[pos][idx]) axes[3].plot(lam_p, _normalize(Ss), label=\"stimulated UV spectrum (norm)\") axes[3].plot(lam_p, Tt, label=\"theory template (norm)\") axes[3].set_xlim(0.18, 0.35) axes[3].set_xlabel(\"wavelength 𝜆 (um)\") axes[3].set_ylabel(\"normalized amplitude\") axes[3].grid(True, alpha=0.3) axes[3].legend(loc=\"upper right\") 14 15 16 17 fig.savefig(filename, dpi=200) print(f\"Saved {filename}\") plt.show() Code 35: plots.py: alternative panel 4 when no sweep is given If no sweep is supplied, the fourth panel is repurposed into a single-run overlay between the stimulated UV spectrum and the normalized theory template. In that case the panel does not compare branch trajectories across probe wavelength, but instead compares the shape of the measured UV response against the two-peak theoretical ansatz at fixed input parameters. Plotting summary The logic of Hawking_plots is therefore layered. Panel 1 records the launched initial condition. Panel 2 isolates the UV output and places it against the theory-predicted NHR and DRR wavelengths, with an inset quantifying spectral alignment under shifts. Panel 3 tests whether the two selected UV channels are temporally correlated. Panel 4 either tracks theory versus simulation across a probe-wavelength sweep or, if no sweep is provided, overlays the stimulated UV spectrum with the corresponding theory template. 24 Experimental Projects Course — Weizmann Institute of Science 3 Eren Erberk Erkul Supplementary modules not used in the final results fme.py The following two modules are based on Mattan’s initial approach to the problem. During the initial study I went over this code as well, so I am including the explanations here for completeness even though these modules are not used in the final stimulated Hawking/backreaction plots. Spectral building blocks 1 2 def k_of_omega(omega, cfg): return cfg.n(omega) * np.asarray(omega) / cfg.c 3 4 5 6 7 8 9 def Kz_FME(omega, k_perp, cfg): k = k_of_omega(omega, cfg) k = np.asarray(k) k_perp = np.asarray(k_perp) den = np.where(k == 0.0, np.inf, k) return k - (k_perp ** 2) / (2.0 * den) 10 11 12 13 14 15 def Q_FME(omega, k_perp, cfg): omega = np.asarray(omega) n = np.asarray(cfg.n(omega)) den = np.where(n == 0.0, np.inf, n) return omega / (cfg.c * den) Code 36: fme.py: spectral blocks This module implements a forward Maxwell-type approximation. The basic spectral quantity is 𝑘 (𝜔) = 𝛽(𝜔) = 𝑛(𝜔) 𝜔 , 𝑐 (3.1) which is the same dispersive ingredient used elsewhere in the repository. The function k_of_omega simply builds this effective propagation constant from the refractive-index model. The next quantity, 𝐾 𝑧FME (𝜔, 𝑘 ⊥ ) = 𝑘 (𝜔) − 2 𝑘⊥ , 2 𝑘 (𝜔) (3.2) is the forward longitudinal wave number appearing in the paraxial or forward Maxwell approximation. It comes from expanding √︃ 𝑘𝑧 = 2 𝑘 (𝜔) 2 − 𝑘 ⊥ (3.3) 2 𝑘⊥ . 2𝑘 (3.4) for small transverse wave number 𝑘 ⊥ , namely 𝑘𝑧 ≈ 𝑘 − So this object is the longitudinal propagation operator for a forward-propagating mode with frequency 𝜔 and transverse structure 𝑘 ⊥ . The quantity 𝑄 FME (𝜔, 𝑘 ⊥ ) = 𝜔 𝑐 𝑛(𝜔) (3.5) is the spectral coupling factor multiplying the polarization source. In this reduced Maxwell form, the 25 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul material response is therefore split into two pieces: a propagation term 𝐾 𝑧FME and a source prefactor 𝑄 FME . Although this module is not part of the final Hawking code, it is useful because it shows that the same dispersive medium can be written in a forward Maxwell language rather than only in the analytic-signal UPPE language. Forward propagation right-hand side 1 2 3 4 5 6 7 8 def rhs_dE_dz(Ehat, omega, k_perp, P_hat, cfg): Kz = Kz_FME(omega, k_perp, cfg) term_field = 1j * Kz * Ehat if P_hat is None: return term_field Q = Q_FME(omega, k_perp, cfg) source = 1j * Q * P_hat / (2.0 * cfg.eps0) return term_field + source Code 37: fme.py: forward propagation RHS The actual forward equation implemented here is 𝑄 FME (𝜔, 𝑘 ⊥ ) ˆ 𝜕 𝐸ˆ = 𝑖 𝐾 𝑧FME (𝜔, 𝑘 ⊥ ) 𝐸ˆ + 𝑖 𝑃, 𝜕𝑧 2𝜀 0 (3.6) where 𝐸ˆ (𝜔) is the spectral electric field and ˆ 𝑃(𝜔) is the spectral polarization source. Thus the structure is again that of a diagonal linear propagation term plus an optional driven term. At the level of numerical ˆ architecture this resembles the UPPE solver, but the object being propagated here is a Maxwell field 𝐸, and the source is an externally supplied polarization ˆ 𝑃, not the analytic-signal Kerr nonlinearity used in the main simulation. 26 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul fwm.py Spectral pump intensity model 1 2 3 4 5 6 7 8 9 def Pump_intesity_omegadomain(omega, cfg): omega = np.asarray(omega) I_w = np.zeros_like(omega) for i, w in enumerate(omega): if w == 0: I_w[i] = 0 else: I_w[i] = cfg.I_pump * cfg.tau0 * (np.pi * cfg.tau0 * omega[i]) / (np.sinh(np.pi * cfg.tau0 * omega[i] * 0.5)) return I_w Code 38: fwm.py: pump intensity in frequency space This function provides a closed-form spectral envelope for the pump in a reduced mixing model. The structure 𝜋𝜏0 𝜔 \u0001 𝐼 𝜔 ∝ 𝜏0 (3.7) sinh 𝜋 𝜏20 𝜔 is the familiar Fourier-domain shape associated with a localized hyperbolic-secant-type profile. Thus, rather than propagating the full pump dynamically as in the main UPPE simulation, this reduced model inserts a fixed spectral pump weight directly in frequency space. Co-moving frequency in the reduced model 1 2 3 4 def omega_prime(omega, cfg): if cfg.truncate_beta: return cfg.sign * cfg.L_source * omega ** 2 return omega * (1 + cfg.n(omega) / cfg.n_g) Code 39: fwm.py: omega prime mapping This is the reduced model’s version of a co-moving invariant. In the truncated case the code replaces the dispersion by an effective quadratic law, 𝜔′ ∼ sign 𝐿 source 𝜔2 , (3.8) while in the non-truncated version it uses \u0013 𝑛(𝜔) 𝜔 =𝜔 1+ . 𝑛𝑔 ′ \u0012 (3.9) The precise form differs from the UPPE treatment, but conceptually the role is the same: 𝜔′ labels the branches that can mix in the moving-frame scattering picture. 27 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Runge–Kutta integrator 1 2 3 4 5 6 7 8 9 10 11 12 13 def RK4(omega_prime, A0_w, u_t, zeta, h, cfg): omega_prime = np.asarray(omega_prime) def RHS(A_w, zeta): A_t = np.fft.ifft(A_w) NL_t = u_t * A_t NL_w = np.fft.fft(NL_t) return 1j * omega_prime * A_w + 1j * cfg.NL_source * NL_w k1 = RHS(A0_w, zeta) k2 = RHS(A0_w + 0.5*h*k1, zeta + 0.5*h) k3 = RHS(A0_w + 0.5*h*k2, zeta + 0.5*h) k4 = RHS(A0_w + h*k3, zeta + h) A1_w = A0_w + (h/6.0)*(k1 + 2*k2 + 2*k3 + k4) return A1_w Code 40: fwm.py: RK4 step This reduced model is evolved as an ordinary differential equation in the variable 𝜁. The right-hand side has the form \u0002 \u0003 𝑑𝐴 𝜔 = 𝑖 𝜔′ (𝜔) 𝐴 𝜔 + 𝑖 𝑁L F 𝑢 𝑡 F −1 ( 𝐴 𝜔 ) , (3.10) 𝑑𝜁 where 𝑢 𝑡 is the time-domain potential or pump profile and 𝑁L is the reduced nonlinear-source strength. The update is then performed with the classical fourth-order Runge–Kutta scheme, 𝐴1 = 𝐴0 + \u0001 ℎ 𝑘 1 + 2𝑘 2 + 2𝑘 3 + 𝑘 4 . 6 (3.11) This module is therefore structurally an ODE solver in 𝜁, unlike the split-step spectral propagator used in the main UPPE code. Four-wave mixing link to the main simulation In the main UPPE simulation, the Kerr cubic term generates new frequency components through mixing of pump and probe bands. In frequency space, this is four-wave mixing implemented numerically as FFT-domain convolution of products in time. The partner UV bands identified as NHR and DRR correspond to specific mixing channels organized by the co-moving invariant 𝜔′ . The simulation makes this visible by using a seeded probe and plotting the stimulated difference spectrum; the sweep then checks whether the extracted UV peaks track the predicted 𝜔′ -mapped branches as 𝜆 2 changes. Analytic bound-state style functions 1 2 def energy(cfg, n: int): return -(cfg.L * cfg.I_ratio/(2*cfg.L_d)) * (cfg.s - n)**2 3 4 5 6 def psi(cfg, n, x): return (1 / np.cosh(np.sqrt(cfg.I_ratio)*x)**(cfg.s - n) * eval_jacobi(n, cfg.s - n, cfg.s - n, np.tanh(np.sqrt(cfg.I_ratio)*x))) Code 41: fwm.py: energy and eigenfunction These functions define an analytic discrete spectrum and its associated eigenfunctions in the reduced model: 𝐿 𝐼ratio 𝐸𝑛 = − (𝑠 − 𝑛) 2 , (3.12) 2𝐿 𝑑 28 Experimental Projects Course — Weizmann Institute of Science and 𝜓 𝑛 (𝑥) = √ 1 cosh 𝐼ratio 𝑥 (𝑠−𝑛,𝑠−𝑛) \u0001 𝑠−𝑛 𝑃𝑛 Eren Erberk Erkul \u0010 √︁ \u0001\u0011 tanh 𝐼ratio 𝑥 . (3.13) These do not enter the main stimulated Hawking/backreaction execution loop; they are included as optional theoretical structure that can be compared qualitatively to mode families if desired. 29 Experimental Projects Course — Weizmann Institute of Science 4 Eren Erberk Erkul Results In this report we outlined the idea of simulating the experimental setup carried out by Prof. Leonhardt’s group. The aim was not just to produce ultraviolet light in a generic nonlinear process, but to see whether the stimulated UV response follows the branch structure predicted by the co-moving Hamiltonian picture described in the theoretical section. The plots we have obtained are the following. Figure 4: Reduced-model output from Mattan Gelvan’s original code. Figure 4 corresponds directly to Mattan Gelvan’s original code. It already shows the general intuition of the problem, namely that a moving optical background can act as a scattering medium for the probe and generate new spectral structure. However, this code is based on a reduced description testing the initial conditions and does not yet contain the final analytic-signal UPPE treatment used in the main part of this report. So this plot should be understood as the original motivation and not as the final quantitative result. Figure 5: Stimulated ultraviolet spectrum with theory markers for the NHR and DRR branches. Figure 5 is the most important result in the report. The stimulated ultraviolet spectrum defined in Eq. (2.16) develops spectral weight near the theory-predicted NHR and DRR branches. This is the main evidence that the simulated branch-conversion picture is working in the intended way. The figure therefore supports the interpretation that the ultraviolet output is not a generic by-product of nonlinear broadening, but a structured response tied to the predicted branch locations. Figure 6 isolates the temporal metric introduced in Eq. (2.20). The near-zero-delay peak indicates that the NHR and DRR channels are generated in the same temporal interaction region rather than being unrelated spectral features. This gives additional support to the interpretation that the two ultraviolet outputs belong to the same branch-conversion process. Figure 7 is in some sense the strongest test, because it asks whether the extracted ultraviolet peaks move systematically with the probe wavelength in the same way as the theoretical branches. In the figure the theory and simulation both remain in the same ultraviolet region, and the simulated NHR and DRR peaks 30 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Figure 6: Normalized time-domain cross-correlation between the extracted NHR and DRR ultraviolet channels. Figure 7: Theory–simulation comparison of NHR and DRR ultraviolet peak wavelengths versus probe wavelength. track the same overall tendency as the predicted branch curves. So this confirms that the observed UV structure is not a fixed accidental feature of the spectrum, but is tied to the co-moving branch matching of the theory. Taken together, Figures 5–7 support the following claim. In the present fibre-optical model, a weak seeded probe stimulates ultraviolet output bands which appear near the theory-predicted NHR and DRR branches, remain temporally correlated, and move with the probe wavelength in a way consistent with the branch structure of the co-moving Hamiltonian picture [7]. However, one shortcoming of this simulation is that we were not able to carry out the full paper-like numerical run under the finest grid and propagation conditions. In particular, the final results shown here were obtained on the numerically stable fast grid rather than on the heavier paper-like grid. Concretely, instead of using 𝑁𝑡 = 215 , 𝑇 = 3176 fs, 𝑑𝑧 = 0.5 𝜇m, (4.1) we used 𝑁𝑡 = 213 , 𝑇 = 1600 fs, 𝑑𝑧 = 10 𝜇m. (4.2) Therefore the actual discretization used in the reported simulation was Δ𝑡 = 𝑇 ≈ 0.195 fs, 𝑁𝑡 Δ𝜔 = 2𝜋 ≈ 3.93 × 10−3 rad/fs, 𝑇 𝜔Ny = 𝜋 ≈ 16.1 rad/fs, Δ𝑡 (4.3) whereas the paper-like grid would have given Δ𝑡 ≈ 0.0969 fs, Δ𝜔 ≈ 1.98 × 10−3 rad/fs, 31 𝜔Ny ≈ 32.4 rad/fs. (4.4) Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul Similarly, with total propagation length fixed at 𝑧 total = 7000 𝜇m, (4.5) 𝑧 total = 700 𝑑𝑧 (4.6) the fast run uses only 𝑁𝑧 = propagation steps, while the paper-like run would require 𝑁 𝑧 = 14000. (4.7) Hence the present simulation has coarser temporal resolution, coarser spectral resolution, a lower ultraviolet cutoff, and a less refined split-step propagation in 𝑧. Mathematically, this means that the present results should be interpreted as a stable qualitative demonstration of branch conversion and stimulated UV generation, but not yet as a numerically converged reproduction of the full paper-level simulation. In addition, we also did not use a fully calibrated experimental dispersion relation, exact SI-calibrated nonlinear material parameters, the full spatial structure of the real fibre, or delayed nonlinear effects beyond the instantaneous Kerr-type model used here. Instead, we used a one-dimensional analytic-signal propagation model with an effective refractive-index function 𝑛(𝜔), scaled nonlinear strength, and sech input pulses. This should still capture the main mechanism of branch conversion and stimulated ultraviolet generation, but for a complete quantitative comparison with experiment one would need to repeat the analysis on the finer paper-like grid together with the exact measured fibre parameters and full experimental operating conditions. Theoretically, we also outlined a more general conjecture. Starting from the spirit of Zel’dovich’s superradiant ideas, the suggestion is that Hawking-like spectral leakage may not be restricted only to gravitational systems, but may arise more generally in classical systems whose governing partial differential equations admit a moving inhomogeneous background, a conserved co-moving invariant, and a turning-point branch-conversion structure. In that sense the present optical system should be viewed not as an imitation in a loose metaphorical sense, but as a concrete mathematical laboratory for studying the horizon mechanism itself. 32 Experimental Projects Course — Weizmann Institute of Science Eren Erberk Erkul References [1] L. M. Procopio, R. Aguero-Santacruz, D. Bermudez, and U. Leonhardt, Observation of the backreaction of stimulated Hawking radiation in an optical analogue (to be published). [2] T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-Optical Analog of the Event Horizon,” Science 319, 1367–1370 (2008). [3] Ya. B. Zel’dovich, “Generation of Waves by a Rotating Body,” JETP Letters 14, 180–181 (1971). [4] Ya. B. Zel’dovich, “Amplification of Cylindrical Electromagnetic Waves Reflected from a Rotating Body,” Soviet Physics JETP 35, 1085–1087 (1972). [5] S. W. Hawking, “Particle Creation by Black Holes,” Communications in Mathematical Physics 43, 199–220 (1975). [6] D. Bermudez and U. Leonhardt, “Hawking spectrum for a fiber-optical analog of the event horizon,” Physical Review A 93, 053820 (2016). [7] J. Drori, Y. Rosenberg, D. Bermudez, Y. Silberberg, and U. Leonhardt, “Observation of Stimulated Hawking Radiation in an Optical Analogue,” Physical Review Letters 122, 010404 (2019). [8] S. Robertson and U. Leonhardt, “Frequency shifting at fiber-optical event horizons: The effect of Raman deceleration,” Physical Review A 81, 063835 (2010). [9] C. Adami, “Paradox No More: How Stimulated Emission of Radiation Preserves Information Absorbed by Black Holes,” arXiv preprint arXiv:2502.05642 [gr-qc] (2025). 33","date":"2026-03-16","dateLabel":"March 16, 2026","readingTime":1,"section":"posts","summary":"PDF presentation on horizon concepts and WIS experimental projects.","tags":[],"timestamp":1773619200,"title":"What is a Horizon? WIS Experimental Projects","url":"/posts/what-is-a-horizon-wis-experimental-projects/"},{"category":"Presentation","content":"Presentation slides from the Nonlinear Optics course. 📄 Nonlinear Optics Presentation — Interband Berry Phase Download  Nonlinear Optics Presentation – Interband Berry Phase Nonlinear Optics Presentation Nonlinear Optics • 20 January 2026 Interband Berry Phase in Laser-Driven Crystals HHG as an internal interferometer in solids Eren Erberk Erkul nature.com/articles/s41586-023-06828-5 The Origin: Curvature Carl Friedrich Gauss Geometric Phase Intuition Parallel Transport on a Sphere Rotated! γ= I A(R) · dR Phase from path geometry The Quantum Connection Sir Michael Berry (1984) Berry phase γ= I A(R) · dR phase from the path geometry The Core Concept • HHG → Internal Interferometer • Target: Interband Geometric Phase • Phase → Interference Spectrum Mechanism Time = Loop Size in Reciprocal Space 2. Acceleration (e iϕ ) Conduction Band 1. Tunneling Valence Band 3. Recombination XUV Photon The Measured Phase Z γint = (Ac − Av ) · dk − ∆ϕd intraband connection + dipole-phase jump Ellipticity bends the loop Γ-M Interband Berry phase Interference Oscillations 8/13 Curvature → Drift ṙ = ∇k εn (k) − k̇ × Ωn (k) anomalous velocity controls recombination overlap Control field reveals curvature Drift Symmetry Broken Interband Berry phase 10/13 Circular dichroism CD(ω) = I+ (ω) − I− (ω) I+ (ω) + I− (ω) helicity dependence as a geometric fingerprint Chiral response in HHG l na % CD Sig 70 Interband Berry phase 12/13 Take-home • HHG as sub-cycle interferometer • Resolves interband geometric phase • Curvature → Drift \u0026 Dichroism Questions","date":"2026-01-20","dateLabel":"January 20, 2026","readingTime":1,"section":"posts","summary":"Presentation slides from the Nonlinear Optics course on the interband Berry phase.","tags":[],"timestamp":1768867200,"title":"Nonlinear Optics Presentation – Interband Berry Phase","url":"/posts/nonlinear-optics-presentation-interband-berry-phase/"},{"category":"Post","content":"This page contains a draft manuscript and an assessment note provided as supporting material for my application to the Caltech PhD program in Physics. Private materials — please do not distribute or repost. The documents below are end-to-end encrypted. Enter the passphrase to decrypt and view them in your browser — the files are never served or stored in readable form. 🔒 Encrypted materials — enter passphrase to decrypt Decrypt 📄 Manuscript (Draft) — Holographic Spectral Alignment: Fast Deterministic 3D Orientation via Boundary Holograms Locked 📄 Research Assessment Statement — Asst. Prof. Murat Temiz (METU) Locked ","date":"2025-12-15","dateLabel":"December 15, 2025","readingTime":1,"section":"posts","summary":"This page contains a draft manuscript and an assessment note provided as supporting material for my application to the Caltech PhD program in Physics. Private materials — please do not distribute or repost. The documents below are end-to-end encrypted. Enter the passphrase to decrypt and view them in your browser — the files are never served or stored in readable form. 🔒 Encrypted materials — enter passphrase to decrypt Decrypt ","tags":[],"timestamp":1765756800,"title":"Holographic Spectral Alignment — Draft","url":"/posts/holographic-spectral-alignment-draft/"},{"category":"Presentation","content":"Talk given at METU, December 3, 2025. 📄 METU Talk — Einstein's Equations in Electromagnetic Media Download  METU Talk — Einstein's Equations in Electromagnetic Media METU Talk Einstein Equations Einstein’s Equations in Electromagnetic Media Encoding General Relativity Inside Optical Materials Eren Erberk Erkul Wednesday, 3 December 2025 METU Department of Electrical and Electronics Engineering Entropy S =− X pi log pi i Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Telegrapher’s Equation LC ∂t2 V + (RC + LG ) ∂t V + RG V − ∂x2 V = 0 τ ∂t2 Φ + ∂t Φ − D ∇2 Φ = 0 Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Holography Inspired Object Detection Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Requiem for Gravity Requiem for Gravity From Newton to Einstein 1687: The First Theory of Gravity Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism 1687: The First Theory of Gravity F = − GMm r2 Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism The Orbits of the Planets F =− GMm r2 ⇒ conic-section trajectories Orbits correspond to conic sections. Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Speed of Light c = 299 792 458 m s−1 Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism The Addition of Velocities vnew = v = speed of object v +u 1 + uv c2 · u = your speed Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Einstein’s Question How fast does gravity move? Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Einstein’s Field Equations Rµν − 12 R gµν = 8πG Tµν c4 Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism The Mathematics of Curved Space ds 2 = − f (r ) c 2 dt 2 + f (r )−1 dr 2 + r 2 dΩ2 f (r ) = 1 − 2GM , rc 2 dΩ2 = dθ2 + sin2 θ dϕ2 . Schwarzschild geometry. Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Invisibility Devices Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism ∇ × E = −∂t B, ∇·B = 0, ∇ × H = ∂t D + J, ∇·D = ρ Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism ∇ × E = −∂t B, ∇·B = 0, ∇ × H = ∂t D + J, g × H, c B = µ0 µ H − D = ε0 ε E + ∇·D = ρ g ×E c Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism ∇ × E = −∂t B, ∇·B = 0, ∇ × H = ∂t D + J, g × H, c B = µ0 µ H − D = ε0 ε E + √ ij ij ε =µ =− −g ij g , g00 gi = ∇·D = ρ g ×E c g0i g00 Units: SI; c explicit; ε0 , µ0 explicit; signature (−+++); g is the magneto-electric vector. Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Maxwell is blind to overall scale gµν 7→ Ω2gµν Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Virtual vs. Physical Space A coordinate deformation (left) is realised by a graded εij , µij profile (right). Light rays follow geodesics of the virtual metric while propagating through an ordinary laboratory sample. Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Spacetime transformations in moving media gi = gilab = g0i g00 n2 − 1 ui 1 − u 2n2/c 2 Motion mixes E and B via g0i . Flow/rotation/sound ⇒ frame-dragging analogue. Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism The Big Picture Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism The Idea Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Optical Metric in ADM Form Transformation optics ADM (3 + 1) split ds 2 = g00 dt 2 + 2g0i dt dx i + gij dx i dx j ds 2 = −α2 dt 2 + γij (dx i + β i dt)(dx j + β j dt) Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism ADM Constraint Equations (3) R + K 2 − Kij K ij = 16πG ρ \u0001 Dj K ij − γ ij K = 8πG j i Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Visualising the ADM Constraints Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism ADM Evolution Equations ∂t γij = −2αKij + Di βj + Dj βi \u0001 ∂t Kij = −Di Dj α + α (3)Rij + KKij − 2Kik K k j + . . . Conformal gauge g00 = −1, shift β i = 0: γij = (det ε)−1/5 εij . Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Self-Gravity of Light EM source terms \u0001 1 ρ = 2 D ·E + B ·H 2c 1 j = 2E ×H c E.g., radiation dominated era. Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism What about sources? 3+1 Electromagnetism Thorne \u0026 Macdonald (1982) Membrane formalism Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Weak-Field Approximation Small perturbations gij = δij + hij , |hij | ≪ 1 β i = 0 ⇒ Kij = − 21 ∂t hij Linearised constraints give wave equations. Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Optical Analogue of Gravitational Waves Linearised constraints ∂j ∂k hjk − ∇2h = 0 ∂j ḣ ij − ∂ i ḣ = 0 TT gauge (two radiative modes) 1 2 TT ∂t hij − ∇2hijTT = 0 2 c Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Plane-Wave Solution Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Another Einstein↔Maxwell Lens: DGREM (Most et al. 2022) Maxwell d Aµ −→ Fµν = dA d ⋆F = ⋆J homogeneous / inhomogeneous Phys. Rev. D 105 (2022) 124038 Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Another Einstein↔Maxwell Lens: DGREM (Most et al. 2022) ⇐⇒ Maxwell d DGREM d Aµ −→ Fµν = dA θâ −→ F â = dθâ d ⋆F = ⋆J d u â = t â + κT â homogeneous / inhomogeneous Nester–Witten / Sparling Phys. Rev. D 105 (2022) 124038 Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Limitations Eren Erberk Erkul — METU EEE — Einstein ↔ Electromagnetism Take-Home Message Gravity ⇐⇒ Optics Take-Home Message Gravity ⇐⇒ Optics Transformation optics + ADM \u0001 split gµν ⇐⇒ εij , µij , g Take-Home Message Gravity ⇐⇒ Optics Transformation optics + ADM \u0001 split gµν ⇐⇒ εij , µij , g Weak-field limit ⇒ tabletop gravitational waves. Design materials ≡ design geometry. Acknowledgement This work was carried out under the mentorship and companionship of Prof. Ulf Leonhardt","date":"2025-12-03","dateLabel":"December 3, 2025","readingTime":1,"section":"posts","summary":"Talk given at METU on Einstein’s field equations in the context of electromagnetic media.","tags":[],"timestamp":176472e4,"title":"METU Talk — Einstein's Equations in Electromagnetic Media","url":"/posts/metu-talk-einsteins-equations-in-electromagnetic-media/"},{"category":"Presentation","content":"Talk given at the ISTA Hosten Group Meeting, Wednesday September 10, 2025. 📄 ISTA Group Presentation — Einstein's Equations in Electromagnetic Media Download 📄 ISTA Board Presentation — ADM Formalism Download  ISTA Hosten Group Meeting Talk — Einstein's Equations in Electromagnetic Media ISTA Group Presentation Einstein’s Equations in Electromagnetic Media Encoding General Relativity Inside Optical Materials Eren Erberk Erkul Wednesday, 10 September 2025 ISTA Hosten Group Requiem for Gravity Requiem for Gravity From Newton to Einstein 1687: The First Theory of Gravity Eren Erberk Erkul — Einstein ↔ Electromagnetism 1687: The First Theory of Gravity F = − GMm r2 Eren Erberk Erkul — Einstein ↔ Electromagnetism The Orbits of the Planets F =− GMm r2 ⇒ conic-section trajectories Orbits correspond to conic sections. Eren Erberk Erkul — Einstein ↔ Electromagnetism Speed of Light c = 299 792 458 m s−1 Eren Erberk Erkul — Einstein ↔ Electromagnetism The Addition of Velocities vnew = v = speed of object v +u 1 + uv c2 · u = your speed Eren Erberk Erkul — Einstein ↔ Electromagnetism Einstein’s Question How fast does gravity move? Eren Erberk Erkul — Einstein ↔ Electromagnetism Einstein’s Field Equations Rµν − 12 R gµν = 8πG Tµν c4 Eren Erberk Erkul — Einstein ↔ Electromagnetism The Mathematics of Curved Space ds 2 = − f (r ) c 2 dt 2 + f (r )−1 dr 2 + r 2 dΩ2 f (r ) = 1 − 2GM , rc 2 dΩ2 = dθ2 + sin2 θ dϕ2 . Schwarzschild geometry. Eren Erberk Erkul — Einstein ↔ Electromagnetism Invisibility Devices Eren Erberk Erkul — Einstein ↔ Electromagnetism ∇ × E = −∂t B, ∇·B = 0, ∇ × H = ∂t D + J, g × H, c B = µ0 µ H − D = ε0 ε E + √ ij ij ε =µ =− −g ij g , g00 gi = ∇·D = ρ g ×E c g0i g00 Units: SI; c explicit; ε0 , µ0 explicit; signature (−+++); g is the magneto-electric vector. Eren Erberk Erkul — Einstein ↔ Electromagnetism Maxwell is blind to overall scale gµν 7→ Ω2gµν Eren Erberk Erkul — Einstein ↔ Electromagnetism Virtual vs. Physical Space A coordinate deformation (left) is realised by a graded εij , µij profile (right). Light rays follow geodesics of the virtual metric while propagating through an ordinary laboratory sample. Eren Erberk Erkul — Einstein ↔ Electromagnetism Spacetime transformations in moving media gi = gilab = g0i g00 n2 − 1 ui 1 − u 2n2/c 2 Motion mixes E and B via g0i . Flow/rotation/sound ⇒ frame-dragging analogue. Eren Erberk Erkul — Einstein ↔ Electromagnetism The Big Picture Eren Erberk Erkul — Einstein ↔ Electromagnetism The Idea Eren Erberk Erkul — Einstein ↔ Electromagnetism Optical Metric in ADM Form Transformation optics ADM (3 + 1) split ds 2 = g00 dt 2 + 2g0i dt dx i + gij dx i dx j ds 2 = −α2 dt 2 + γij (dx i + β i dt)(dx j + β j dt) Eren Erberk Erkul — Einstein ↔ Electromagnetism ADM Constraint Equations (3) R + K 2 − Kij K ij = 16πG ρ \u0001 Dj K ij − γ ij K = 8πG j i Eren Erberk Erkul — Einstein ↔ Electromagnetism Visualising the ADM Constraints Eren Erberk Erkul — Einstein ↔ Electromagnetism ADM Evolution Equations ∂t γij = −2αKij + Di βj + Dj βi \u0001 ∂t Kij = −Di Dj α + α (3)Rij + KKij − 2Kik K k j + . . . Conformal gauge g00 = −1, shift β i = 0: γij = (det ε)−1/5 εij . Eren Erberk Erkul — Einstein ↔ Electromagnetism Self-Gravity of Light EM source terms \u0001 1 ρ = 2 D ·E + B ·H 2c 1 j = 2E ×H c E.g., radiation dominated era. Eren Erberk Erkul — Einstein ↔ Electromagnetism What about sources? 3+1 Electromagnetism Thorne \u0026 Macdonald (1982) Membrane formalism Eren Erberk Erkul — Einstein ↔ Electromagnetism Weak-Field Approximation Small perturbations gij = δij + hij , |hij | ≪ 1 β i = 0 ⇒ Kij = − 21 ∂t hij Linearised constraints give wave equations. Eren Erberk Erkul — Einstein ↔ Electromagnetism Optical Analogue of Gravitational Waves Linearised constraints ∂j ∂k hjk − ∇2h = 0 ∂j ḣ ij − ∂ i ḣ = 0 TT gauge (two radiative modes) 1 2 TT ∂t hij − ∇2hijTT = 0 2 c Eren Erberk Erkul — Einstein ↔ Electromagnetism Optical Analogue of Gravitational Waves Hamiltonian constraint ∂j ∂k hjk − ∇2 h = 0, Momentum : ∂j ḣij − ∂ i ḣ = 0. TT-gauge isolates two radiative degrees: 1 2 TT ∂ h − ∇2 hijTT = 0. c 2 t ij Identify hijTT = δεTT ij to obtain the optical wave equation for dielectric perturbations. Eren Erberk Erkul — Einstein ↔ Electromagnetism Plane-Wave Solution Eren Erberk Erkul — Einstein ↔ Electromagnetism Another Einstein↔Maxwell Lens: DGREM (Most et al. 2022) ⇐⇒ Maxwell d DGREM d Aµ −→ Fµν = dA θâ −→ F â = dθâ d ⋆F = ⋆J d u â = t â + κT â homogeneous / inhomogeneous Nester–Witten / Sparling after Phys. Rev. D 105 (2022) 124038 Eren Erberk Erkul — Einstein ↔ Electromagnetism Take-Home Message Gravity ←→ Optics Transformation optics + ADM \u0001 split gµν ←→ εij , µij , g Weak-field limit ⇒ tabletop gravitational waves. Design materials ≡ design geometry. Limitations Eren Erberk Erkul — Einstein ↔ Electromagnetism Acknowledgement This work was carried out under the mentorship and companionship of Prof. Ulf Leonhardt ISTA Board ADM Formalism Eren Erberk Erkul 9/9/25 1. Spacetime Foliation and Diffeomorphism Invariance The idea of describing space-time as a spatial state in a given solution space and evolving it with time, reminiscent of the Hamiltonian formulation of quantum mechanics, exists; rightly so, it is called the Hamiltonian formulation of General Relativity, or ADM, after its founders, Arnowitt, Deser, and Misner. However, unlike evolving in Hilbert space, which has a trivial global definition for time, this is not as trivial in general relativity. In Newtonian gravity, this is called Newton–Cartan (NC) spacetime, honoring Henri Cartan for his contributions to our understanding of Newtonian spacetime (𝒩 ); it embodies this split absolutely. It is structured as a fiber bundle with a base space 𝐸1 (absolute time) and fibers 𝐸3 (Euclidean space). The foliation structure is fixed, which, like in QM, makes the spacetime absolute. However, in GR the spacetime manifold (ℳ, 𝑔𝜇𝜈 ) lacks a preferred time structure, meaning the Cauchy Problem, the fancy name for the initial value problem for PDEs, lacks a global definition; hence, not every spacetime can be assumed to be represented this way. But, to a certain degree, every spacetime can, and this formalism of general relativity forms the basis of numerical relativity. Thus, the 3+1 decomposition ’artificially’ foliates ℳ by choosing a scalar time function 𝑡: ℳ \u001b R × Σ𝑡 , defining a stack of 3D spatial hypersurfaces Σ𝑡 equipped with an induced Riemannian metric 𝛾𝑖𝑗 (𝑡, 𝑥). Worldline 𝑡3 Σ𝑡3 (𝐸 3 in NC) 𝑡2 Σ𝑡2 (𝐸 3 in NC) 𝑡1 Σ𝑡1 (𝐸 3 in NC) Time (𝑡) Figure 1: Foliation of spacetime ℳ into spatial hypersurfaces Σ𝑡 . 1 2. The Role of Diffeomorphisms. A key feature of GR is diffeomorphism invariance: the physical structure of spacetime (the geometry 𝐺) is independent of the coordinates used. This diffeomorphism invariance created a significant conceptual dilemma during the development of general relativity, especially as first raised by Einstein himself and then continued by Hilbert. Their concern was that, because of this invariance, if a spacetime is curved in the presence of a source, it would be topologically and algebraically equivalent to one with no source. So, they believed this created an impossibility to formulate a proper theory. So, mathematically, every solution of general relativity is pointwise equivalent to every other. However, in GR, what is important is not the relation between these absolute entities, which are grid points of spacetime, but rather how these points are connected to each other. These geometric connections within GR, known as the Christoffel coefficients, are the cause of the description of ‘force’, not the absolute relation between the points, which is a dramatic difference from the classical picture and a structure lacking in quantum field theory as well. This is also why some physicists don’t consider GR as a description of a fundamental source, but instead a pseudoforce. To be more exact, the choice of foliation is a gauge choice. Moving or relabeling the spatial slices is merely a coordinate transformation (a diffeomorphism) that preserves the underlying spacetime structure. So, in simple terms, by aligning the surfaces parallel to each other, I can always return to the original Minkowski spacetime. And this operation I did, moving these surfaces right or left, this “map” is called a coordinate transformation. Arbitrary Coordinates Aligned Coordinates Diffeomorphism Figure 2: A coordinate transformation (diffeomorphism) can align spatial hypersurfaces. If spacetime is flat, this alignment yields Minkowski coordinates. 2 3. The 3+1 Decomposition: Kinematics and Metric The geometry of a chosen foliation is described by the Lapse function 𝛼 and the Shift vector 𝛽 𝑖 . 𝛽 𝜇 𝑑𝑡 Σ𝑡+𝑑𝑡 𝛼𝑛 𝜇 𝑑𝑡 𝑑𝜏 = 𝛼 𝑑𝑡 𝑡 𝜇 𝑑𝑡 Σ𝑡 Figure 3: Geometry of the 3+1 decomposition. The coordinate time vector 𝑡 𝜇 splits into normal evolution (lapse 𝛼𝑛 𝜇 ) and tangential evolution (shift 𝛽𝜇 ). The coordinate time vector 𝑡 𝜇 = (𝜕𝑡 )𝜇 is decomposed as 𝑡 𝜇 = 𝛼𝑛 𝜇 + 𝛽 𝜇 , with 𝑛 𝜇 the future-pointing unit normal to Σ𝑡 . The lapse relates coordinate time 𝑑𝑡 to the proper time 𝑑𝜏 measured by an observer moving orthogonally to the slices: 𝛼 = −𝑔 𝜇𝜈 ∇𝜇 𝑡 ∇𝜈 𝑡 \u0001 −1/2 . The shift 𝛽 𝑖 describes how spatial coordinates are dragged when moving from slice to slice. The full spacetime metric 𝑔𝜇𝜈 is reconstructed from the spatial metric 𝛾𝑖𝑗 , the lapse, and the shift: 𝑑𝑠 2 = −(𝛼2 − 𝛽 𝑖 𝛽 𝑖 )𝑑𝑡 2 + 2𝛽 𝑖 𝑑𝑥 𝑖 𝑑𝑡 + 𝛾𝑖𝑗 𝑑𝑥 𝑖 𝑑𝑥 𝑗 , 3 𝛽 𝑖 = 𝛾𝑖𝑗 𝛽 𝑗 . 4. Rigid - Galilean and Newtonian Limits In GR, the collection of fibers through base space can be viewed as the collection of inertial frames, where a suitable choice of coordinates (an orientation of foliation) can always locally flatten each fibre into an inertial one. The key difference between the trivial bundles of GR and those of Newton–Cartan theory is an extra degree of freedom that allows the spatial foliation to twist and curl dynamically. Hence, Newtonian foliations are absolute entities; lacking the radiative Transverse–Traceless (TT) part of the curvature. Relatedly, both Newtonian and Einsteinian bundles differ from the Galilean bundle (which is rigid) in that the extra freedom now resides in the fibers themselves (allowing for curvature). We can recover classical spacetime structures by taking specific limits of the 3+1 variables. Galilean Limit (Flat) Newtonian Limit 𝛼 = 1, 𝛽 𝑖 = 0, 𝛾𝑖𝑗 = 𝛿 𝑖𝑗 . Spacetime is flat; inertial paths are straight. 𝛽 𝑖 = 0, 𝛾𝑖𝑗 = 𝛿 𝑖𝑗 , 𝛼 ≈ 1 + Φ. Spatial slices are flat, but paths bend due to gravity. Figure 4: Comparison of the Galilean and Newtonian limits. In GR, the spatial slices Σ𝑡 can possess intrinsic curvature (non-flat 𝛾𝑖𝑗 ), and the foliation can twist (non-zero shift 𝛽 𝑖 ). This extra shift is why gravitational waves can be present in Einsteinian gravity but not in Newtonian gravity. 4","date":"2025-09-10","dateLabel":"September 10, 2025","readingTime":1,"section":"posts","summary":"Talk at the ISTA Hosten Group Meeting, Wednesday September 10, 2025.","tags":[],"timestamp":1757462400,"title":"ISTA Hosten Group Meeting Talk — Einstein's Equations in Electromagnetic Media","url":"/posts/ista-hosten-group-meeting-talk/"},{"category":"Post","content":"This post presents a framework for categorizing the different types of analogies that arise between physical systems in mathematical physics. The central question: when does an interrelation between theories become strong enough to warrant the label duality rather than mere analogy? Zeroth Degree Analogy Intuitive-level connections without formal mathematical correspondence. Example: Planck looking through a window, watching water droplets — this inspired the concept of quantization. The connection is suggestive, not formal. First Degree Analogy One degree of freedom correspondence in state variables; the base spaces are isomorphic. Example: Plebanski’s 1960 formulation, which stores metric information in electromagnetic properties. This idea was foundational to transformation optics. Second Degree Analogy System B mimics the evolution of System A; equivalent PDE classes. This represents the highest degree achievable for analogue systems. Third Degree Analogy Beyond first and second degree conditions. Includes a master equation generating A’s full range within B — constituting a “dual” rather than a mere analogue. This requires homeomorphic and isomorphic domains and ranges. The distinction between analogy and duality in mathematical physics hinges on whether interrelations between theories become sufficiently strong to warrant the stronger terminology. ","date":"2025-07-30","dateLabel":"July 30, 2025","readingTime":1,"section":"posts","summary":"A framework for categorizing analogies between physical systems — from intuitive connections to full mathematical dualities.","tags":["physics","mathematics","analogy","duality"],"timestamp":1753833600,"title":"Classification of Different Analogies in Mathematical Physics","url":"/posts/classification-of-analogies-in-mathematical-physics/"},{"category":"Presentation","content":"Talk given at the TAPIR / Caltech SXS Group Meeting, Monday July 7, 2025. 📄 TAPIR Presentation — Einstein's Equations in Electromagnetic Media Download 📄 Caltech Poster — EEE Download 📄 Board Notes Download  TAPIR Caltech SXS Group Meeting Talk TAPIR Presentation WHO again? Eren Erberk Erkul — Einstein ↔ Electromagnetism Einstein’s Equations in Electromagnetic Media Encoding General Relativity Inside Optical Materials Eren Erberk Erkul Monday, 7 July 2025 Caltech TAPIR Invisibility Devices Eren Erberk Erkul — Einstein ↔ Electromagnetism ∇ × E = −∂t B, ∇·B = 0, ∇ × H = ∂t D + J, ∇·D = ρ The covariant free-space Maxwell equations are equivalent to electromagnetism in a material medium (Tamm, 1924; Plebanski, 1960). D = εE + w × H, c B= µ w H− ×E εc 2 c √ ij ij ε =µ = −g ij g , g00 wi = g0i g00 Eren Erberk Erkul — Einstein ↔ Electromagnetism Virtual vs. Physical Space A coordinate deformation (left) is realised by a graded εij , µij profile (right). Light rays follow geodesics of the virtual metric while propagating through an ordinary laboratory sample. Eren Erberk Erkul — Einstein ↔ Electromagnetism Spacetime transformations in moving media wi = g0i g00 =⇒ wilab = n2 − 1 ui 1 − u 2 n2 /c 2 Motion couples E and B through g0i . Flow, rotation, or sound waves mimic frame dragging. Eren Erberk Erkul — Einstein ↔ Electromagnetism The Big Picture Eren Erberk Erkul — Einstein ↔ Electromagnetism The Idea Eren Erberk Erkul — Einstein ↔ Electromagnetism Optical Metric in ADM Form Transformation optics ADM (3 + 1) split ds 2 = g00 dt 2 + 2g0i dt dx i + gij dx i dx j ds 2 = −α2 dt 2 + γij (dx i + β i dt)(dx j + β j dt) Identify the optical metric parameters via g00 = −α2 + γij β i β j , g0i = γij β j , gij = γij . Eren Erberk Erkul — Einstein ↔ Electromagnetism ADM Constraint Equations (3) R + K 2 − Kij K ij = 16πG ρ, \u0001 Dj K ij − γ ij K = 8πG j i ρ and j i are supplied by the electromagnetic energy density and Poynting vector once the mapping is complete. Eren Erberk Erkul — Einstein ↔ Electromagnetism Visualising the ADM Constraints Eren Erberk Erkul — Einstein ↔ Electromagnetism ADM Evolution Equations ∂t γij = −2αKij + Di βj + Dj βi , \u0001 ∂t Kij = −Di Dj α + α (3) Rij + KKij − 2Kik K k j + . . . Set g00 = −1 (α = 1) and identify the optical metric γij (x) = det ε εij \u0001−1 . Eren Erberk Erkul — Einstein ↔ Electromagnetism Self-Gravity of Light Electromagnetic field as source: ρ= \u0001 1 D ·E + B ·H , 2 2c j = 1 E × H. c2 This scenario naturally arises, e.g., in the early radiation-dominated universe. Eren Erberk Erkul — Einstein ↔ Electromagnetism Weak-Field Approximation Assume gij = δij + hij , |hij | ≪ 1. Gauge choice β i = 0 gives Kij = − 12 ∂t hij . Linearising the constraints then leads to familiar wave equations. Eren Erberk Erkul — Einstein ↔ Electromagnetism Optical Analogue of Gravitational Waves Hamiltonian constraint ∂j ∂k hjk − ∇2 h = 0, Momentum : ∂j ḣij − ∂ i ḣ = 0. TT-gauge isolates two radiative degrees: 1 2 TT ∂ h − ∇2 hijTT = 0. c 2 t ij Identify hijTT = δεTT ij to obtain the optical wave equation for dielectric perturbations. Eren Erberk Erkul — Einstein ↔ Electromagnetism Plane-Wave Solution Eren Erberk Erkul — Einstein ↔ Electromagnetism Another Einstein↔Maxwell Lens: DGREM (Most et al. 2022) ⇐⇒ Maxwell d DGREM d Aµ −→ Fµν = dA θâ −→ F â = dθâ d ⋆F = ⋆J d u â = t â + κT â homogeneous / inhomogeneous Nester–Witten / Sparling after Phys. Rev. D 105 (2022) 124038 Eren Erberk Erkul — Einstein ↔ Electromagnetism Why DGREM helps simulators First-order \u0026 hyperbolic ∂t Dkâ − ∇ × H â = − Jkâ Flux-conservative core ∂t U + ∇·F (U) = 0 Machine-precision constraints ∇·D â = ρâ , ∇·B â = 0 Eren Erberk Erkul — Einstein ↔ Electromagnetism Two roads, one goal Eren Erberk Erkul — Einstein ↔ Electromagnetism Take-Home Message Gravity ←→ Optics Transformation optics + ADM \u0001 split gµν ←→ εij , µij , w Weak-field limit ⇒ tabletop gravitational waves. Design materials ≡ design geometry. Acknowledgement This work was carried out under the mentorship and companionship of Prof. Ulf Leonhardt Acknowledgement I gratefully thank Prof. Saul Teukolsky \u0026 Prof. Elias Most for giving me the opportunity to be a visiting researcher at TAPIR. Caltech Poster EEE EINSTEIN’S EQUATIONS IN ELECTROMAGNETIC MEDIA Eren Erberk Erkul \u0026 Ulf Leonhardt Middle East Technical University Weizman Institute of Science Blueprint - Caltech 06/24/25 Bending Gravity into Light We turn Einstein’s theory inside out, transforming gravitational dynamics into the language of electromagnetic materials. By extending Plebanski’s 1960 map, we encode Einstein’s entire ADM system within a dielectric medium. The result? Real gravitational physics modeled as engineered optical media: light itself becomes the sculptor of spacetime geometry! Splitting Spacetime into Slices How does gravity evolve? ADM formalism answers clearly: it splits spacetime into 3-D slices evolving in time. Each slice has geometry described by Einstein’s equations become constraints that each slice must satisfy: Hamiltonian constraint Momentum constraint Geometry changes slice by slice, just like frames in a film. Optics Mirrors Gravity A stationary optical metric in transformation optics, mirrors ADM’s gravitational slicing: This optical–ADM match hints that engineered media can encode full gravitational dynamics, the spark for our study. The Key Equation Plebanski’s groundbreaking insight: This equation bridges Einstein’s gravity and Maxwell’s electromagnetism. It’s the hidden symmetry we use to turn curved space into clear glass. When Light Creates Gravity By treating the field’s own energy ρ and momentum j as the matter terms, we complete Einstein’s ADM equations inside the medium. - Effective energy density - Effective momentum flux The medium now hosts genuine gravitational self-interactionsthe self-gravity of pure electromagnetic fields! DESIGNER'S REMARKS Result - Engineered dielectrics replicate gravitational dynamics. Limitations - Model fixes α and omits strong fields, loss, and dispersion. Outlook - Next-gen metamaterials could let us test geometrodynamics in the lab. Optical Analog of Gravity Waves In the weak-field limit our mapped system produces metric ripples that obey Identifying these with dielectric fluctuations, we obtain optical copies of gravitational waves. Thus, gravitational-wave physcics becomes a tabletop optical experiment! TAPIR Board I / £ J","date":"2025-07-08","dateLabel":"July 8, 2025","readingTime":1,"section":"posts","summary":"Talk at the TAPIR / Caltech SXS Group Meeting on Einstein’s equations in electromagnetic media.","tags":[],"timestamp":1751932800,"title":"TAPIR Caltech SXS Group Meeting Talk","url":"/posts/tapir-caltech-sxs-group-meeting-talk/"},{"category":"Post","content":"The cosmos lives within everyday moments. I was sitting in the METU EE Canteen, watching an aquarium. Air bubbles rose through the water — their motion, viewed from different angles through different glass panels, created perspective shifts. And I realized: this is relativity in a fish tank. Buoyancy and Gravity The bubbles rising through water behave in a way that mirrors projectile motion under gravity. Buoyancy in water acts as an analogue to gravitational force — the medium changes, but the physics rhymes. We don’t yet fully understand what we mean by the mass of the object — the concept hides layers of complexity. Viewing the aquarium from different panels is analogous to viewing spacetime from different reference frames. Einstein showed us that perspective isn’t just geometry; it’s physics. The question remains open: what is gravity, really? Perhaps observing bubbles in water brings us one step closer to the answer. ","date":"2025-05-10","dateLabel":"May 10, 2025","readingTime":1,"section":"posts","summary":"The cosmos lives within everyday moments — observing an aquarium at METU and drawing parallels to gravitational physics.","tags":["astronomy","philosophy","physics","science","universe"],"timestamp":1746835200,"title":"Gravitational Aquarium","url":"/posts/gravitational-aquarium/"},{"category":"Post","content":"Introduction In 1905, Einstein formulated how atomic motion causes both friction and fluctuations in Brownian motion, establishing the deep connection between dissipation and fluctuations. The fluctuation-dissipation theorem tells us that any system which dissipates energy must also exhibit fluctuations — and the two are quantitatively related. Quantum Perspective Quantum fluctuations differ fundamentally from classical ones. Vacuum fluctuations arise through the Heisenberg uncertainty relations and field operator decomposition. Even in the vacuum state — what we might naively call “empty space” — quantum fields exhibit irreducible fluctuations. Experimental Evidence An ETH Zurich experiment by Jaquil Feist used nonlinear crystals at near-absolute-zero temperatures to demonstrate vacuum noise effects on light polarization. Quantum Optics Applications Absorption The output amplitude for absorption: $$B = \\sqrt{1 - \\eta}, A + \\sqrt{\\eta}, A_0$$ Amplification $$B = G, A + \\sqrt{G - 1}, A_0$$ where $G$ is the gain factor. Lindblad Equation The master equation for density matrix evolution, incorporating irreversible processes through jump operators: Conclusion Photon absorption statistics follow binomial distributions, connecting the quantum formalism back to classical probability. References Einstein, A. (1905). Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik, 17(8), 549–560. Lindblad, G. (1976). On the generators of quantum dynamical semigroups. Communications in Mathematical Physics, 48(2), 119–130. Leonhardt, U. (2010). Essential Quantum Optics: From Quantum Measurements to Black Holes. Cambridge University Press. ","date":"2025-03-02","dateLabel":"March 2, 2025","readingTime":2,"section":"posts","summary":"From Einstein’s 1905 Brownian motion to quantum vacuum fluctuations and the Lindblad equation.","tags":["quantum optics","physics","thermodynamics"],"timestamp":1740873600,"title":"Fluctuation-Dissipation Theorem","url":"/posts/fluctuation-dissipation-theorem/"},{"category":"Post","content":" Drawn by Eren Erberk Erkul with the assistance of DALL·E A single rebellious light wave departed from its fellow waveguides, striking the wall of a Victorian-era house in Cambridge. The dim study was as quiet as ever, but today, something was different. Charles Darwin, timeless and contemplative, held a letter in his hands, the seal freshly broken. It was from a young man — a naturalist named Alfred Russel Wallace — whose words trembled with ambition. Darwin’s eyes skimmed the letter, and the room seemed darker with each line he read. Wallace had arrived at conclusions so eerily familiar to Darwin that he had been too cautious to share. The timing of this discovery was both thrilling and terrifying. How could this be? Darwin thought. He is hitting the wall I’ve stood before, staring into the same darkness. The rebellious light from the small gap in his curtains caught his eye, reflecting his restless thoughts. It danced erratically across the wall, unbound, breaking from its expected path. A single wavelet, he thought, trying to escape through the smallest of cracks. How could it exist on its own? Could it act like something so solid? The idea seemed absurd. But Darwin shrugged it off — thankfully, he wasn’t a physicist. He didn’t need to waste his time on such matters. But Darwin quickly dismissed it with relief — physics was not his burden. Yet, the analogy struck a deeper chord. In his carefully constructed life, he had chosen safety — chosen to let his revolutionary ideas lie dormant, like that trapped beam of light. He thought that I had become a respected figure, an expert among my peers, a Fellow of the Royal Society. But am I not just like this single wave, hitting a wall, unable to shine the whole room? Darwin’s days of adventure felt distant now. His youth was spent journeying to the Galapagos and observing nature’s wonders, which seemed like the fading light of the past. The dusty old notebook he once filled with radical thoughts lay untouched for years. That notebook, he reflected, was my waveguide. But it hit a wall long ago. I, too, hit a wall — choosing to live a quiet, respected life. I’ve blended in so well with the world around me that I’ve almost forgotten what it means to rebel. But the letter in his hands was a reminder — a stark one. Wallace had dared to push forward with the same ideas, threatening to shine first. The wave was no longer alone. The urgency was now unbearable. This is it, Darwin realized. This is my moment. Wallace is bold, but I cannot let him shine brighter than I. His heart raced, not out of jealousy but out of a sense of inevitability. I am not just a single wave, he thought, gripping the edges of the letter. I am an orchestra. If he opened the curtain now, he wouldn’t just be a solitary beam breaking through — he would be a force strong enough to tear down the entire wall if necessary. He placed Wallace’s letter aside and picked up his old, dusty notebook. Its pages, though long untouched, still held the same burning ideas. As he opened the curtains fully, the light flooded in, and in the brilliant glow, he could almost hear the cry of that once lonely wave, now joined by many others. Darwin’s eyes fell upon the publishing house in the heart of the town; with that vision, he knew what had to be done. He wouldn’t be a follower any longer — he would lead the way, just as he had always known he could. My beginning was to understand all beginnings, My end will teach them where it all starts, To show we’re the same in the pile of life, Though we seem so far apart. ","date":"2024-09-30","dateLabel":"September 30, 2024","readingTime":4,"section":"posts","summary":"A creative reimagining of Charles Darwin receiving Wallace’s letter — told through light, curtains, and the courage to publish.","tags":["charles-darwin","darwin","evolution","history","science"],"timestamp":1727654400,"title":"Through the Curtain — Darwin","url":"/posts/through-the-curtain-darwin/"},{"category":"Post","content":" This article explains how dispersion emerges from a granular spacetime model. Following Prof. Ulf Leonhardt’s framework, spacetime operates as a lattice structure where neighboring nodes interact as harmonic oscillators — mirroring condensed matter physics, similar to the Ising model where only neighboring points interact. Part One — Composite Harmonic Oscillator Starting with a simple harmonic oscillator equation, the derivation extends to coupled oscillators where displacement depends on nearest neighbors. The analysis combines equations for $Y_{i-1}$ and $Y_{i+1}$ influences on $Y_i$ to create a composite equation. Part Two — Taylor Expansion Taylor expansions of $Y_{i-1}$ and $Y_{i+1}$ around $Y_i$ with respect to time, where $\\delta$ represents the spacing between nodes. Only even-order terms survive when summing the series. The second derivative corresponds to standard wave equations; fourth derivatives account for dispersive medium corrections. Part Three — Dispersive Term Derivation A plane wave solution is assumed: $$y = A e^{i(kx - \\omega t)}$$ Subsequent derivatives are calculated and substituted into the composite equation. Through algebraic simplification, the dispersion relation emerges: $$\\omega^2 = \\frac{(4Z/m)\\sin^2(k\\delta/2)}{1 + (Z\\delta^2/m)\\sin^2(k\\delta/2)}$$ This shows frequency-dependent wavenumber behavior characteristic of dispersive media. Concluding Remarks Dispersion naturally arises from discrete lattice structures without requiring external analogies. Bose–Einstein condensates exhibit positive dispersion (versus negative in standard oscillator systems), suggesting they behave like oscillators with purely imaginary distances. Dispersion can be understood as something far more fundamental — a direct consequence of the granular structure of spacetime itself. ","date":"2024-09-10","dateLabel":"September 10, 2024","readingTime":2,"section":"posts","summary":"How dispersion emerges naturally from a granular spacetime model — deriving dispersion relations from coupled oscillators on a lattice.","tags":["physics","spacetime","dispersion","condensed matter"],"timestamp":1725926400,"title":"Dispersion as a Consequence of Discrete Space-Time","url":"/posts/dispersion-as-a-consequence-of-discrete-space-time/"},{"category":"Post","content":" Stone Waterfalls in front of METU Yüksel Proje Lecture Hall A thought experiment exploring the fundamental connection between time and gravitation — what if they share a common origin? ","date":"2024-04-01","dateLabel":"April 1, 2024","readingTime":1,"section":"posts","summary":"A thought experiment on the origins of time and gravitation, inspired by the stone waterfalls at METU.","tags":["physics","time","gravitation","thought-experiment"],"timestamp":1711929600,"title":"On the Origins of Time and Gravitation — A Thought Experiment","url":"/posts/on-the-origins-of-time-and-gravitation/"},{"category":"Post","content":"Introduction Quantum Circuit Notation (QCN) is proposed as a novel framework for classical circuit analysis, inspired by the mathematical formalism of quantum mechanics. By representing circuit elements as vectors in Hilbert space, we can bring the power of quantum mechanical techniques to bear on classical problems. Background Dirac’s bra-ket notation provides an elegant language for describing states in Hilbert space. The key objects are: Ket $|\\psi\\rangle$ — a state vector Bra $\\langle\\phi|$ — a dual vector Inner product $\\langle\\phi|\\psi\\rangle$ — a scalar This notation, originally developed for quantum mechanics, turns out to be remarkably useful for circuit analysis. QCN Framework Circuit elements are represented as vectors in Hilbert space using Dirac notation: Resistances as ket vectors $|R\\rangle$ Currents as bra vectors $\\langle I|$ Voltage as the inner product $\\langle V|R|I\\rangle$ Hilbert Space Elements The five key components of the framework: State vectors — representing circuit states Operators — representing circuit transformations Inner product — computing physical quantities Superposition — combining circuit states Evolution operators — describing time-dependent circuits Discussion QCN facilitates the application of quantum mechanical mathematical techniques to classical problems, potentially simplifying complex circuit analyses. Applications may extend to signal processing and control theory. Conclusion This framework opens up classical circuit theory to the full toolkit of Hilbert space methods. Further research is needed to refine the theoretical framework and explore its practical applications. ","date":"2023-11-26","dateLabel":"November 26, 2023","readingTime":2,"section":"posts","summary":"Proposing QCN as a novel framework for classical circuit analysis inspired by quantum mechanics and Dirac’s bra-ket notation.","tags":["quantum mechanics","circuit theory","hilbert space","physics"],"timestamp":1700956800,"title":"Quantum Circuit Notation (QCN) — A Hilbert Space Approach to Classical Circuit Theory","url":"/posts/quantum-circuit-notation-qcn/"},{"category":"Post","content":" \"Time Lord constrained by the black hole\" / Edited by Eren Erberk Erkul with DALL·E assistance Time’s relentless passage shapes our existences, forging bonds and memories. But can time be changed? Can its fixed points be overwritten? Minkowski Spacetime In special relativity, spacetime is a four-dimensional manifold where time is not separate from space but woven into it. Events are points in this manifold — and once an event exists in the past light cone, it is as immutable as geometry. Black holes take this further: at the event horizon, the roles of space and time swap. Falling past the horizon, you can no more avoid the singularity than you can, outside, avoid tomorrow. The Waters of Mars In the Doctor Who episode The Waters of Mars, even the Doctor — a Time Lord — cannot alter fixed points in time. The episode is a meditation on temporal inevitability: some events are so load-bearing in the structure of history that changing them would collapse everything. The physics agrees. In Minkowski spacetime, the causal structure is absolute. You cannot send a signal faster than light; you cannot reach back and rewrite what has already entered your past light cone. Time is not a river we float along. It is the geometry of the universe itself — and geometry does not bend to wishes. ","date":"2023-10-25","dateLabel":"October 25, 2023","readingTime":2,"section":"posts","summary":"On the immutability of time — from Minkowski spacetime to black holes, with a detour through Doctor Who.","tags":["time","physics","doctor-who","philosophy"],"timestamp":1698192e3,"title":"Time Lord Victorious","url":"/posts/time-lord-victorious/"},{"category":"Post","content":" Eren Erberk Erkul with DALL·E assistance Group theory is the mathematical language of symmetry — and symmetry is the deepest organizing principle in physics. From Noether’s theorem linking symmetries to conservation laws, to the gauge groups of the Standard Model, to the representation theory underlying quantum mechanics — groups are everywhere in modern physics. This post explores the fundamental connection between abstract algebra and the physical world. ","date":"2023-05-12","dateLabel":"May 12, 2023","readingTime":1,"section":"posts","summary":"On the role of symmetry and group theory as the mathematical language of modern physics.","tags":["group theory","physics","mathematics","symmetry"],"timestamp":1683849600,"title":"Group Theory in Physics","url":"/posts/group-theory-in-physics/"},{"category":"Post","content":" Eren Erberk Erkul with DALL·E assistance A scientific treatise on the understanding of character and consciousness. What makes you you? Is it the arrangement of atoms in your brain? The electrochemical signals racing along neurons? The memories accumulated over a lifetime? Consciousness remains one of the deepest puzzles at the intersection of physics, neuroscience, and philosophy. This essay explores what science can — and cannot — tell us about the origins of character. ","date":"2023-05-12","dateLabel":"May 12, 2023","readingTime":1,"section":"posts","summary":"A scientific treatise on the understanding of character and consciousness.","tags":["consciousness","science","philosophy"],"timestamp":1683849600,"title":"Why You Are Who You Are","url":"/posts/why-you-are-who-you-are/"},{"category":"Post","content":"Is it possible to manipulate dimensions? The TARDIS — being “bigger on the inside” — contradicts known physical laws. But the question itself opens doors. The Question of Dimensions We perceive the world in three spatial dimensions plus time. But our biological perception is limited: humans see 2D images projected as 3D by the brain. What does it mean for a dimension to exist? String Theory and Extra Dimensions String theory proposes additional spatial dimensions — compactified, curled up at scales far too small to observe directly. If extra dimensions exist, could they be expanded in a localized region? Creating extra space inside a smaller area seems to require manipulating dimensions — an unexisting concept in physics. Einstein’s relativity tells us that spacetime can curve, stretch, and warp — but creating genuinely new spatial volume from nothing requires something beyond current theory. Conclusion Current physics cannot explain how to create interior space within smaller external areas, making TARDIS-like technology currently impossible. But the history of physics is a history of impossible things becoming possible. Further Reading Quantum Gravity — Springer Wormholes — Quanta Magazine TARDIS — Doctor Who Doctor Who — BBC YouTube — Related Video ","date":"2023-04-03","dateLabel":"April 3, 2023","readingTime":1,"section":"posts","summary":"Is it possible to manipulate dimensions? Examining the TARDIS through the lens of mathematical physics.","tags":["physics","doctor-who","dimensions","string-theory"],"timestamp":168048e4,"title":"TARDIS — Time and Relative Dimensions in Space","url":"/posts/tardis-time-and-relative-dimensions-in-space/"},{"category":"Post","content":"William Rowan Hamilton was a 19th-century Irish mathematician from Dublin — an alcoholic poet with emotional struggles, particularly regarding an unrequited love named Catherine. He became a professor of astronomy at Trinity College Dublin while still an undergraduate. But his greatest legacy would come from a flash of inspiration on a bridge. The Bridge The breakthrough occurred on Broome Bridge in Dublin. Hamilton had been struggling for years with the mathematics of triples — trying to find a way to multiply three-dimensional numbers the way complex numbers work in two dimensions. On that bridge, he realized a fourth component was needed. The key insight: you need four dimensions, not three. Quaternions $$\\mathbf{i}^2 = \\mathbf{j}^2 = \\mathbf{k}^2 = \\mathbf{i}\\mathbf{j}\\mathbf{k} = -1$$ Quaternions — the mathematics behind quantum mechanics and 3D computational graphics — are the descendants of the inspirations acquired from human emotions. Hamilton carved the equation into the stone of Broome Bridge in his excitement. That act of mathematical vandalism gave us the tools that now power everything from video games to spacecraft navigation. Conclusion Revolutionary scientific breakthroughs emerge from emotional inspiration rather than pure logic. Hamilton’s story is proof. Note (added 08.12.2024): Historical sources dispute some aspects of Hamilton’s personal life. Some references suggest he was a devoted family man rather than unhappily married. The emotional framing of this essay is literary, not biographical. Further Reading Hamilton — St Andrews Mathematics Hamilton — Royal Irish Academy Quaternions — Wolfram MathWorld ","date":"2023-03-16","dateLabel":"March 16, 2023","readingTime":2,"section":"posts","summary":"The story of William Rowan Hamilton — an alcoholic poet who invented quaternions on a bridge in Dublin.","tags":["mathematics","history","quaternions","physics"],"timestamp":1678924800,"title":"William Rowan Hamilton — Forgotten Genius","url":"/posts/william-rowan-hamilton-forgotten-genius/"},{"category":"Post","content":" Eren Erberk Erkul with DALL·E assistance Einstein’s theory of relativity represents a revolutionary departure from classical mechanics. It emerged from Gedanken (thought) experiments rather than pure mathematics — making it accessible even at a high school level. Birth of Relativity: Newton, Mach, Einstein Galileo Galilei — the first scientific founder of the relativity principle Isaac Newton — formalized it through his law of inertia, but treated space and time as absolute quantities Ernst Mach — challenged Newtonian absoluteness, arguing motion requires a frame of reference and cannot exist in empty space \"Newton's Bucket\" — Eren Erberk Erkul with DALL·E assistance Einstein was heavily influenced by Mach’s philosophy. His teenage obsession with the nature of light and spacetime crystallized into the 1905 breakthrough — achieved while he was working as a patent clerk at age 26. Postulates of Special Relativity The laws of physics are the same in all inertial frames of reference. The speed of light in vacuum $c$ is the same for all observers, regardless of the motion of the source or observer. From these two postulates, the entire structure of special relativity follows. The Most Famous Equation: $E = mc^2$ Mass and energy are equivalent and related by: $$E = mc^2$$ A small amount of mass corresponds to an enormous amount of energy, since $c \\approx 3 \\times 10^8 , \\text{m/s}$. Conclusion Relativity’s significance lies in transforming time from an independent background into a key player in the mechanics of the universe. Understanding time is crucial to grasping unified field theory. Further Reading World Science U — Special Relativity Einstein Papers — Princeton Feynman Lectures — Caltech Newton’s Bucket — St Andrews Ernst Mach — Stanford Encyclopedia Perimeter Institute — Quantum Gravity ","date":"2023-03-04","dateLabel":"March 4, 2023","readingTime":2,"section":"posts","summary":"An introduction to Einstein’s special relativity — from Galileo and Newton through Mach to the 1905 revolution.","tags":["einstein","relativity","physics"],"timestamp":1677888e3,"title":"Special Relativity","url":"/posts/special-relativity/"},{"category":"Post","content":" Eren Erberk Erkul with DALL·E assistance The Case for Physics Physics explains matter at quantum scales and cosmic scales through relativity. Chemistry depends on the Standard Model. Biology relies on chemistry. Major advances in biology stem from physics discoveries — imaging technology, X-rays, NMR. Richard Dawkins once called biology a “junior science” compared to physics. Whether or not you agree, the hierarchy of reduction is hard to dispute. The Case for Neuroscience Neuroscience could theoretically simulate the thought processes of great physicists to discover new equations. But this doesn’t make neuroscience more fundamental — physical laws existed before human consciousness. The argument involves deep philosophical considerations, but within a materialist framework, the brain is made of atoms — and atoms are the domain of physics. The Case for Mathematics Mathematics represents coherent logical thought and elegant systems. But mathematics only becomes fundamental when it explains reality. The abstract theorems of mathematics today are the physics of tomorrow — but the question is whether this will continue. Further Reading Feynman Lectures on Physics — Chapter 3 Where Math Meets Physics — Penn Today ","date":"2023-03-04","dateLabel":"March 4, 2023","readingTime":1,"section":"posts","summary":"Which science is the most fundamental — physics, neuroscience, or mathematics?","tags":["physics","neuroscience","mathematics","philosophy"],"timestamp":1677888e3,"title":"What is the Most Fundamental Science?","url":"/posts/what-is-the-most-fundamental-science/"},{"category":"Page","content":"I’m Eren Erberk Erkul, a physicist and electrical \u0026 electronics engineer interested in general relativity, quantum field theory, and applied mathematics as a whole. This website, Effective Field, presents my work in physics, technology, and thought experiments extending to various disciplines. The title is inspired by effective field theories in physics, which describe systems at a specific scale without requiring complete knowledge of the underlying structure. Find Me LinkedIn YouTube ","date":"0001-01-01","dateLabel":"January 1, 0001","readingTime":1,"section":"","summary":"About Eren Erberk Erkul","tags":[],"timestamp":-62135596800,"title":"About","url":"/about/"},{"category":"Page","content":"A question, an idea, or just a hello? Leave this field empty Your email So I can write back to you. Subject Message Send message → You can also find me on LinkedIn and YouTube. ","date":"0001-01-01","dateLabel":"January 1, 0001","readingTime":1,"section":"","summary":"Contact Eren Erberk Erkul","tags":[],"timestamp":-62135596800,"title":"Contact","url":"/contact/"},{"category":"Page","content":" Presentations Talks, lectures, and competition slides — open each entry to view the embedded slides or recording. Singularities as Solitons — Pisa Talk given at QUEST 2026 in Pisa on singularities as solitons QUEST 2026 · Pisa · September 2026 Holographic Spectral Alignment 🥇 1st Place 3MT (METU Engineering Day), plus the full PHYS400 poster \u0026 long presentation METU · May 2026 · Winner METU Talk — Einstein's Equations in EM Media Talk given at METU on Einstein's field equations in the context of electromagnetic media METU · General Relativity · December 2025 ISTA Hosten Group Meeting Talk Talk at the ISTA Hosten Group Meeting on Einstein's equations in electromagnetic media ISTA · Quantum Optics · September 2025 TAPIR / Caltech SXS Group Meeting Talk Talk on Einstein's equations in electromagnetic media at TAPIR and the SXS group meeting Caltech · Numerical Relativity · July 2025 Nonlinear Optics — Interband Berry Phase Course presentation on the interband Berry phase and its role in nonlinear optical responses Nonlinear Optics · Topology · January 2026 What is a Horizon? — WIS Experimental Projects PDF presentation on the concept of a horizon in general relativity and WIS experimental projects WIS · Horizons · March 2026 More presentations coming soon.","date":"0001-01-01","dateLabel":"January 1, 0001","readingTime":0,"section":"","summary":"Talks, lectures, and competition slides by Eren Erberk Erkul","tags":[],"timestamp":-62135596800,"title":"Presentations","url":"/presentations/"},{"category":"Page","content":"","date":"0001-01-01","dateLabel":"January 1, 0001","readingTime":0,"section":"","summary":"Academic publications by Eren Erberk Erkul","tags":[],"timestamp":-62135596800,"title":"Publications","url":"/publications/"},{"category":"Page","content":" Simulations Interactive physics visualisations — open in full screen for the best experience. de Sitter Space Interactive 5D embedding — explore the geometry of de Sitter spacetime with real-time controls General Relativity · Cosmology More simulations coming soon.","date":"0001-01-01","dateLabel":"January 1, 0001","readingTime":0,"section":"","summary":"Interactive physics simulations by Eren Erberk Erkul","tags":[],"timestamp":-62135596800,"title":"Simulations","url":"/simulations/"},{"category":"simulations","content":"de Sitter Space ☰ Coordinate Patches Global Poincaré (Expanding) Static Contracting Boundary / Holography Yellow curves — null geodesics White ring — throat ($\\tau = 0$) Dots — comoving observers Tessellated disks — $\\mathscr{I}^\\pm$ Coordinates × Reset Wireframe Observers Auto-Spin Boundaries Click regions to explore · Drag to rotate · Arrow keys to pan · Scroll to zoom Designed by EEE Embedding $$u = \\\\sinh\\\\tau \\\\qquad w = \\\\cosh\\\\tau\\\\,\\\\cos\\\\alpha \\\\qquad r = \\\\cosh\\\\tau\\\\,\\\\sin\\\\alpha$$ Constraint (5D Minkowski) $$\\\\eta_{AB} = \\\\mathrm{diag}(+1,\\\\,-1,\\\\,-1,\\\\,-1,\\\\,-1)$$ $$-u^2 + w^2 + |\\\\vec{x}|^2 = \\\\ell^2 = 1$$ Induced Metric $$ds^2 = d\\\\tau^2 - \\\\cosh^2\\\\!\\\\tau\\\\;d\\\\Omega_3^2$$ $$d\\\\Omega_3^2 = d\\\\alpha^2 + \\\\sin^2\\\\!\\\\alpha\\\\;d\\\\phi^2$$ Eliminate $w$ $$w\\\\,dw = u\\\\,du - r\\\\,dr$$ $$dw^2 = \\\\frac{(u\\\\,du - r\\\\,dr)^2}{u^2 + 1 - r^2}$$ Scale factor $a(\\\\tau) = \\\\cosh\\\\tau$ is minimal at the throat $\\\\tau = 0$. Range: $\\\\tau \\\\in (-\\\\infty,+\\\\infty)$, $\\\\;\\\\alpha \\\\in [0,\\\\pi]$ Flat Coordinates $$u = \\\\sinh\\\\tau + \\\\tfrac{1}{2}\\\\,e^\\\\tau\\\\,\\\\rho^2$$ $$w = -\\\\cosh\\\\tau + \\\\tfrac{1}{2}\\\\,e^\\\\tau\\\\,\\\\rho^2$$ $$r = e^\\\\tau\\\\,\\\\rho$$ Metric (Inflationary) $$ds^2 = d\\\\tau^2 - e^{2\\\\tau}\\\\!\\\\left(d\\\\rho^2 + \\\\rho^2\\\\,d\\\\Omega^2\\\\right)$$ Patch $$u - w = e^\\\\tau \u003e 0$$ Spatial sections are flat $\\\\mathbb{R}^3$ . Hubble rate $H = \\\\dot{a}/a = 1$. The metric of inflationary cosmology . Comoving observers sit at $\\\\rho = \\\\rho_0 = \\\\mathrm{const}$. Static Coordinates ($\\\\rho \\\\leq 1$) $$u = \\\\sqrt{1-\\\\rho^2}\\\\;\\\\sinh t \\\\qquad r = \\\\rho$$ $$w = -\\\\sqrt{1-\\\\rho^2}\\\\;\\\\cosh t$$ Time-Independent Metric $$ds^2 = (1-\\\\rho^2)\\\\,dt^2 - \\\\frac{d\\\\rho^2}{1-\\\\rho^2} - \\\\rho^2\\\\,d\\\\Omega^2$$ Killing vector $\\\\partial/\\\\partial t$ — metric is time-independent . Cosmological Horizon $$g_{tt} = (1 - \\\\rho^2) \\\\;\\\\xrightarrow{\\\\rho\\\\to 1}\\\\; 0$$ At $\\\\rho = 1$: signals cannot reach the static observer at $\\\\rho = 0$. Geodesic Equation $$\\\\frac{d^2 x^\\\\mu}{d\\\\lambda^2} + \\\\Gamma^\\\\mu_{\\\\;\\\\alpha\\\\beta}\\\\,\\\\frac{dx^\\\\alpha}{d\\\\lambda}\\\\,\\\\frac{dx^\\\\beta}{d\\\\lambda} = 0$$ Free-falling observers ($\\\\rho = \\\\mathrm{const}$ in Poincaré) drift through this static frame. Past Poincaré Patch $$u + w \u003c 0 \\\\qquadtime-reversal of expanding patch)$$ Metric $$ds^2 = d\\\\tau^2 - e^{2\\\\tau}\\\\!\\\\left(d\\\\rho^2 + \\\\rho^2\\\\,d\\\\Omega^2\\\\right)$$ Coverage $$\\Expanding \\\\cup \\Contracting = \\dS \\\\setminus \\\\{\\null boundaries\\\\}$$ The universe contracts . The boundary between patches is a pair of null surfaces (yellow curves). dS/CFT Correspondence $$\\\\mathcal{Z}_{\\grav}[\\dS_{d+1}] \\\\;\\\\sim\\\\; \\\\mathcal{Z}_{\\CFT}[\\\\mathscr{I}^+]$$ Strominger (2001): quantum gravity in dS is dual to a Euclidean CFT on the future boundary $\\\\mathscr{I}^+ \\\\cong S^3$. Group Structure Isomorphism $$\\\\mathrm{Isom}(\\\\mathrm{dS}_4) = SO(1,4) \\\\;\\\\cong\\\\; \\\\mathrm{Conf}(S^3)$$ $$\\\\mathrm{Isom}(\\\\mathrm{AdS}_5) = SO(2,4) \\\\;\\\\cong\\\\; \\\\mathrm{Conf}(\\\\mathbb{R}^{3,1})$$ Generator Decomposition $\\\\dim = 10$ $$\\\\underbrace{M_{AB}}_{\\ambient SO(1,4)} \\\\;\\\\xrightarrow{\\boundary} \\\\;\\\\bigl\\\\{\\\\,\\\\underbrace{P_a}_{3},\\\\;\\\\underbrace{M_{ab}}_{3},\\\\;\\\\underbrace{D}_{1},\\\\;\\\\underbrace{K_a}_{3}\\\\,\\\\bigr\\\\}$$ Conformal Algebra $\\\\mathfrak{so}(1,4)$ $$[D,\\\\,P_a] = P_a \\\\qquad [D,\\\\,K_a] = -K_a$$ $$[K_a,\\\\,P_b] = 2\\\\bigl(\\\\eta_{ab}\\\\,D - M_{ab}\\\\bigr)$$ $$[M_{ab},\\\\,P_c] = \\\\eta_{bc}\\\\,P_a - \\\\eta_{ac}\\\\,P_b$$ $P_a$: translations $M_{ab}$: rotations $D$: dilation $K_a$: special conformal. The same algebra acts as isometries in the bulk and as conformal symmetries on $\\\\mathscr{I}^+$ . Boundary Operator Map $$\\\\langle\\\\mathcal{O}\\\\rangle_{\\CFT} = \\\\lim_{\\\\tau\\\\to\\\\infty} e^{\\\\Delta\\\\tau}\\\\,\\\\phi(\\\\tau,\\\\vec{x})$$ AdS/CFT Analogy $$\\AdS: \\\\;\\\\langle\\\\mathcal{O}\\\\rangle = \\\\lim_{r\\\\to\\\\infty} r^\\\\Delta\\\\,\\\\phi(r,x)$$ $$\\dS: \\\\;\\\\langle\\\\mathcal{O}\\\\rangle = \\\\lim_{\\\\tau\\\\to\\\\infty} e^{\\\\Delta\\\\tau}\\\\,\\\\phi(\\\\tau,x)$$ Escher \u0026 the Poincaré Disk The tessellated disks at $\\\\mathscr{I}^\\\\pm$ visualize the conformal boundary where the holographic dual lives. Similar to Escher's Circle Limit paintings, which are based on the Poincaré disk model of hyperbolic space, where figures shrink toward the boundary but remain the same size in the intrinsic geometry. This is the UV/IR connection ; small distances on the boundary encode large-scale bulk geometry. The boundary tessellation encodes all information in the bulk spacetime. Holographic Entropy $$S_{\\dS} = \\\\frac{\\Area(\\horizon)}{4\\\\,G_N} = \\\\frac{\\\\pi\\\\,\\\\ell^2}{G_N}$$","date":"","dateLabel":"","readingTime":0,"section":"simulations","summary":"Interactive 5D embedding: explore the geometry of de Sitter spacetime with real-time controls.","tags":["Simulation","General Relativity","Cosmology"],"timestamp":0,"title":"de Sitter Space — Interactive 5D Embedding","url":"/desitter.html"},{"category":"publications","content":"The Unruh Effect in Relativistic Fluids Erkul, Eren Erberk We identify the relativistic-fluid counterpart of the Unruh effect, in which a comoving probe measures a Thermodynamic Unruh temperature. Frame changes in first-order hydrodynamics are recast as a local, time-dependent hyperbolic rotation in a Rindler-style state space where the instantaneous map between frames is the Thermodynamic Boost and its proper-time variation defines the Thermodynamic Acceleration, which results in an Unruh-like thermal spectrum. To leading order, the Thermodynamic Unruh temperature is frame-independent and universal across out-of-equilibrium relativistic fluid descriptions, from Israel-Stewart to modern theories. 2511.12366","date":"2025-11-15","dateLabel":"November 15, 2025","readingTime":0,"section":"publications","source":"https://inspirehep.net/literature/3084086","summary":"We identify the relativistic-fluid counterpart of the Unruh effect, in which a comoving probe measures a Thermodynamic Unruh temperature. Frame changes in first-order hydrodynamics are recast as a local, time-dependent hyperbolic rotation in a Rindler-style state space where the instantaneous map between frames is the Thermodynamic Boost and its proper-time variation defines the Thermodynamic Acceleration, which results in an Unruh-like thermal spectrum. To leading order, the Thermodynamic Unruh temperature is frame-independent and universal across out-of-equilibrium relativistic fluid descriptions, from Israel-Stewart to modern theories.","tags":["Publication","Research"],"timestamp":1763164800,"title":"The Unruh Effect in Relativistic Fluids","url":"/publications/#publication-3084086"},{"category":"publications","content":"Singularities as Solitons? Quantum Vacuum Architecture of Black Holes Erkul, Eren Erberk We propose that black holes are soliton-esque objects, where gravitational collapse is balanced by quantum vacuum dispersion, modeled via R+αR^{2} gravity. Classical singularities are replaced by oscillating, finite-radius cores, thereby evading static no-go theorems. The event horizon is replaced by the Lamarina, a surface of maximum redshift whose surface geometry yields Hawking-like radiation with corrections. The Raychaudhuri equations impose a Dyson-type ceiling on the maximum radiated power (P_{\\infty} \\lesssim c^{5}/G), while effective field theory matching dictates a universal minimum Lamarina radius set by the dispersion scale. 2510.06428","date":"2025-10-07","dateLabel":"October 7, 2025","readingTime":0,"section":"publications","source":"https://inspirehep.net/literature/3064804","summary":"We propose that black holes are soliton-esque objects, where gravitational collapse is balanced by quantum vacuum dispersion, modeled via R+αR^{2} gravity. Classical singularities are replaced by oscillating, finite-radius cores, thereby evading static no-go theorems. The event horizon is replaced by the Lamarina, a surface of maximum redshift whose surface geometry yields Hawking-like radiation with corrections. The Raychaudhuri equations impose a Dyson-type ceiling on the maximum radiated power (P_{\\infty} \\lesssim c^{5}/G), while effective field theory matching dictates a universal minimum Lamarina radius set by the dispersion scale.","tags":["Publication","Research"],"timestamp":1759795200,"title":"Singularities as Solitons? Quantum Vacuum Architecture of Black Holes","url":"/publications/#publication-3064804"},{"category":"publications","content":"Dispersion in Analogue Gravity Erkul, Eren Erberk Leonhardt, Ulf Analogue models of gravity, from Newton to Unruh, have evolved from simple fluid--mechanical models to sophisticated modern experiments. They have shown the robustness of Hawking radiation and highlighted the role of dispersion for quantum fields in curved space. We speculate whether some underlying dispersion may also play a role in explaining the cosmological constant and in resolving the cosmological tensions. 2510.02542","date":"2025-10-02","dateLabel":"October 2, 2025","readingTime":0,"section":"publications","source":"https://inspirehep.net/literature/3062875","summary":"Analogue models of gravity, from Newton to Unruh, have evolved from simple fluid--mechanical models to sophisticated modern experiments. They have shown the robustness of Hawking radiation and highlighted the role of dispersion for quantum fields in curved space. We speculate whether some underlying dispersion may also play a role in explaining the cosmological constant and in resolving the cosmological tensions.","tags":["Publication","Research"],"timestamp":1759363200,"title":"Dispersion in Analogue Gravity","url":"/publications/#publication-3062875"},{"category":"publications","content":"Einstein's equations in electromagnetic media Erkul, Eren Erberk Leonhardt, Ulf In this paper, we extend Plebanski's mapping to encode the Einstein equations in Arnowitt-Deser-Misner (ADM) form within a bianisotropic electromagnetic medium. We realise this by translating the ADM constraints and evolution equations into dynamical conditions on the medium's constitutive parameters. These transformed equations are then linearised in vacuum to derive gravitational-wave analogues as perturbations of the optical medium. In this paper, we extend Plebanski's mapping to encode the Einstein equations in ADM form within a bianisotropic electromagnetic medium. We realise this by translating the ADM constraints and evolution equations into dynamical conditions on the medium's constitutive parameters. These transformed equations are then linearised in vacuum to derive gravitational-wave analogues as perturbations of the optical medium. EPL 154 16003 2026 2508.11300 10.1209/0295-5075/ae58cf 10.1209/0295-5075/ae58cf","date":"2025-08-15","dateLabel":"August 15, 2025","readingTime":0,"section":"publications","source":"https://inspirehep.net/literature/2960945","summary":"In this paper, we extend Plebanski's mapping to encode the Einstein equations in Arnowitt-Deser-Misner (ADM) form within a bianisotropic electromagnetic medium. We realise this by translating the ADM constraints and evolution equations into dynamical conditions on the medium's constitutive parameters. These transformed equations are then linearised in vacuum to derive gravitational-wave analogues as perturbations of the optical medium.","tags":["Publication","Research"],"timestamp":1755216e3,"title":"Einstein's equations in electromagnetic media","url":"/publications/#publication-2960945"}]