Quantum simulation of particle pair creation near the event horizon

Abstract Though it is still a big challenge to unify general relativity and quantum mechanics in modern physics, the theory of quantum field related with the gravitational effect has been well developed and some striking phenomena are predicted, such as Hawking radiation. However, the direct measurement of these quantum effects under general relativity is far beyond present experiment techniques. Fortunately, the emulation of general relativity phenomena in the laboratory has become accessible in recent years. However, up to now, these simulations are limited either in classical regime or in flat space whereas quantum simulation related with general relativity is rarely involved. Here we propose and experimentally demonstrate a quantum evolution of fermions in close proximity to an artificial black hole on a photonic chip. We successfully observe the acceleration behavior, quantum creation, and evolution of a fermion pair near the event horizon: a single-photon wave packet with positive energy escapes from the black hole while negative energy is captured. Our extensible platform not only provides a route to access quantum effects related with general relativity, but also has the potentiality to investigate quantum gravity in future.

I. Constructing gravitational filed using photonic waveguide lattice with nonuniform coupling coefficients Inspired by transformation optics, we use femtosecond laser written photonic waveguide lattice to mimic gravitational field and investigate the dynamic evolution of boson in close proximity of black hole. Firstly, we start by considering the line element of a two-dimensional Schwarzschild spacetime: where M is the mass of black hole, and nature units have been adopted (G = c = 1). In our work, we are interested in the short distance behavior of Hawking radiation near the horizon (ρ ≈ 2M ), the Schwarzschild metric can be written as: where α = 1/2r s , r s = 2M , r(ρ) = 8M (ρ − 2M ). What is more interesting is that the Schwarzschild metric near the horizon of black hole has the same form as the line element of the Rindler [1].
While the quantum evolution of boson for null geodesic satisfies ds = 0, which means |dr/r| = αdt, then we can obtain the evolution function of boson near to black hole as where r 0 is initial position. The quantum evolution depends on the curvature of black hole α, which means that the quantum evolution is faster near the event horizon of a black hole with small radius.
For the spacetimes based on Eq. (S1), the evolution of boson similar to that in an optical media with an effective refraction index based on transformation optics [2][3][4] [ Fig. S1(c)]: where g 00 = α 2 r 2 , g 11 = −1. In our work, we use evanescently coupled photonic waveguide lattice with designed coupling coefficient to achieve the required inhomogeneous effective refractive index. The dynamics of single-photon wave packet in photonic waveguide lattice can be described by a set of coupled discrete Schrodinger equations, which are derived from Schrodinger-type paraxial wave equation by employing the tight-binding approximation: where ϕ m is the complex field amplitude of site m, z is the propagation distance along the waveguides mapping the time variable, β 0 is on site energy of each waveguide, and parameter κ m represents the coupling strength between the adjacent sites. Taking coupling coefficients as κ m = κ m+1 = κ and substituting the complex field amplitude with the plane wave solution ϕ m = A · exp(iβ r md + iβ z z), we can obtain the dispersion connecting transverse and longitudinal dynamics as [ Fig. S1(b)] : where A is the amplitude of plane wave, β r and β z are transverse and longitudinal wavevector respectively, d is waveguide spacing (r = md). After photon evolves in the such a waveguide over distance ∆z, each transverse component gains a phase Φ = β z (β r )∆z, and the corresponding transverse shift of a wave centered around β r is ∆r = ∂Φ/∂β r = (∂β z /∂β r ) ∆z.
Due to that the propagation distance z in the coupled waveguide equation plays the role as the time t in Schrodinger equation, hence when exciting the coupled waveguide with the transverse wavevector as around β r = π/2d, we can define the velocity of wavepacket in such system as And the refractive index of such photonic lattice is Comparing Eq. (S3) and (S7), to constructe gravitational field of 1+1 dimensional Schwarzschild black hole in evanescently coupled waveguides, the coupling coefficients should satisfy where we take discrete the continuous function with lattice r = md. Therefore, by constructing

II. The Dirac equation near the event horizon of black hole
Besides the dynamic behavior of photon as boson, we can also investigate the evolution of fermion pair generated by Hawking Radiation in close proximity to black hole. We start with the Dirac equation in the 1 + 1 dimensional spacetime for a massless particle is where µ is 0 for time and 1 for spatial 1 dimensional direction, ϕ = (a, b) T (T stands for transpose) is a two-component spinor, γ µ are the Dirac gamma matrices, and the covariant derivative ∆ µ = ∂ µ + Ω µ . Considering the metric Schwarzschild black hole according to Eq. (S1), we obtain the element of metric as g 00 = (αr) 2 , Then we can get the calculated Christoffel symbol as the Einstein convention is adopted here where the repeated indices are summed over. We have for nonzero Christoffel symbol as The calculated spin connection ω a bv can be directly obtained as where e a µ (we use the convention that latin indices a, b are used to label local inertial coordinates and greek indices µ, v for general coordinates) is vielbein and satisfy the equation Then we obtain e0 0 = 1/αr, e1 1 = 1. Based on Eq. (S13), the nonzero spin connection is Using the spin connection, we can obtain the calculated spinor Ω v = 1 4 ω abv σ ab , where σ ab = γ a , γ b /2. Then nonzero spinor is Therefore, the Eq. (S9) can be written as After choosing γ 0 = σ z , γ 1 = iσ y , the above equation can be simplified as where h is Hamiltonian density.
III. The quantum evolution of single-photon wave packet in close proximity to an artificial black hole In order to investigate the quantum evolution of single-photon wave packet near the event horizon of black hole, we assume that the Eq. (S17) has the solution ϕ = 1 where k r is the wavevector of the plane wave, E is the frequency of the plane wave, and then the Eq. (S17) can be written as (S19) Therefore, the solution has two energy state E = ±αk r r. For positive energy state E + = αk r r, the solution is ϕ + (r, t) = 1 √ 2r 1 1 e ikrr−iE + t ; and for negative energy state E − = −αk r r, In order to consider the dynamic behavior of single-photon wave packet with positivenegative energy solution close to the black hole, the time-depended position operator in the Heisenberg pictures can be written as We consider a single-photon Gaussian wave packet of positive energy state as ϕ + (r, 0) = N dk r · 1 √ 2r e −w 2 (kr−k 0 ) 2 /2 e ikrr 1 1 where N is normal coefficient, and the packet centered at k 0 is characterized by a width of w.
A direct calculation gives the evolution of the wave packet as r + (t) = ϕ + (r, 0) |r H (t)| ϕ + (r, 0) = ϕ + (r, 0) |σ x sinh(αt)r + cosh(αt)r| ϕ + (r, 0) The single-photon Gaussian wave packet of negative energy state is and the dynamic evolution can be described as (S24) By comparing the time evolution of wave packet with different energy, we obtain that the positive energy state has positive velocity and exponentially escapes away from black hole, while the negative state propagates toward and eventually stops around the black hole. The evolution process is analogy of Hawking Radiation. Owing to vacuum fluctuations, particleantiparticle pair is generated close to the event horizon of a black hole. One of the particles with negative energy falls into the black hole while the other escapes. In order to preserve total energy, the particle that fell into the black hole must have had a negative energy (with respect to an observer far away from the black hole). This causes the black hole to lose mass to an outside observer; it would appear that the black hole has just emitted a particle.

IV. Discrete Dirac equation near the event horizon of black hole in the waveguide lattice
In order to discrete the continuous Hamiltonian in the photonic lattice, we start single-particle Hamiltonian h with [5]: Figure S2: The schematic of optical lattice. a m and b m respectively corresponds to waveguide in the upper layer and lower layer, β 0 is on site energy of each waveguide, κ 0 is the nearest neighbor coupling coefficient in same layer, κ 1m and κ 2m are coupling coefficient between the upper and lower layer.
Considering two-component spinor ϕ = (a, b) T , the above equation is written as: Furthermore, we discrete the continuous function with lattice as shown Fig. S2: The discrete Hamiltonian is Then we obtain coupling equation in the optical lattice: After takingã m = i 2m a m ,b m = i 2m+1 b m , the above equation is:

V. The inhomogeneous coupling between bi-layer waveguide
Furthermore, if we construct bi-layer lattice fabricated by femtosecond laser written technology as shown in Fig. S2, considering the nearest neighbor, the coupled wave equation can be described as: where a m and b m respectively corresponds to waveguide site in the upper layer and lower layer, β 0 is on site energy of each waveguide, κ 0 is the nearest neighbor coupling coefficient in same layer, κ 1m and κ 2m are coupling coefficient between the upper and lower layer. When assuming the approximated solution in local lattice, then we obtain a m±1 = exp(±iβ r d)a m , b m±1 = exp(±iβ r d)b m , the above equation can be written as: If assuming , the above equation can be written as: In order to clearly clarify the positive and negative state in the waveguide, the dispersion relation at the site m in the momentum space according to Eq. (S30) can be given as: In Fig. S4, we show the two modes maintain separated with the change of parameters κ 1m and κ 2m .
Due to translational symmetry in the propagation, the β z is a conserved quantity. In the initial Figure S4: The spectrum of the bi-layer waveguide lattice. The two modes maintain separated with the change of parameters κ 1m . The inset (i) and (ii) give the dispersion relation of the fabricated bi-layer lattice, where κ 0 is adopted as 0.15 mm −1 , β 0 is uniform and is adopted as 0 in calculation, κ 1 is adopted as 0.5 mm −1 for inset (i) and 1.0mm −1 for inset (ii) respectively, and κ 2 = κ 1 + 0.005mm −1 . For the whole lattice, κ 1 varies from 0.5 to 1.0 mm −1 in step of 0.005mm −1 .
time of the exciting waveguide, β z is taken as β z = β 0 −2κ 0 . Then the group velocity of positive and negative wave packet is: v + = ∂β + z /∂β r = αm, v − = ∂β − z /∂β r = −αm. (S35) The Eq. (S34) is corresponding to Eq. (S21) and Eq. (S23) well. All these equations unveil that the positive energy state escapes away from black hole with increasing positive velocity, while the negative energy state propagates toward to black hole with decreasing negative velocity.