2 + 1 dimensional de Sitter universe emerging from the gauge structure of a nonlinear quantum system

Berry phases and gauge structures are fundamental quantum phenomena. In linear quantum mechanics the gauge field in parameter space presents monopole singularities where the energy levels become degenerate. In nonlinear quantum mechanics, which is an effective theory of interacting quantum systems, there can be phase transitions and hence critical surfaces in the parameter space. We find that these critical surfaces result in a new type of gauge field singularity, namely, a conic singularity that resembles the big bang of a 2 + 1 dimensional de Sitter universe, with the fundamental frequency of Bogoliubov excitations acting as the cosmic scale, and mode softening at the critical surface, where the fundamental frequency vanishes, causing a causal singularity. Such conic singularity may be observed in various systems such as Bose-Einstein condensates and molecular magnets. This finding offers a new approach to quantum simulation of fundamental physics.

SUPPLEMENTARY INFORMATION for 2+1 dimensional de Sitter universe emerging as the gauge structure of a nonlinear quantum system

Note 1. Two-mode model for asymmetric double well BECs
The many-body Hamiltonian that describes interacting bosons confined by an external potential V ext (r) is where Ψ (r) and Ψ † (r) are the boson field operators that annihilate and create a particle at r respectively and V (r − ′ r ) is the two-body interatomic potential [S1]. In the Heisenberg representation for the field operators, the time evolution of the field operator is determined by the Heisenberg equation In a dilute ultracold atomic gas, only the elastic binary collisions between individual atoms are relevant. The binary collisions are characterized by a single s-wave scattering length a , which is irrelevant to the expressions of the two-body potential. Hence, we can replace the two-body potential V (r − ′ r ) with an effective interaction gδ (r − ′ r ) , which results in i! ∂Ψ(r,t) ∂t = − ! 2 ∇ 2 2m + V ext (r) + gΨ † (r,t)Ψ(r,t) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥Ψ (r,t) , where the coupling constant g is related to the scattering length through g = 4π! 2 a / m . When BEC occurs, we can replace the field operator Ψ (r,t) with its mean-field value Φ(r,t) ≡ 〈Ψ(r)〉 and obtain the time-dependent Gross-Pitaevskii equation for the condensate wave Here | ∫ Φ(r,t) | 2 dr = N is the number of condensed atoms. In a double-well potential, the Bose-Einstein condensate wave function Φ(r,t) can be written as a superposition of two time-independent spatial wave functions φ 1 (r) and φ 2 (r) that are localized in each well Φ(r,t) = N [ψ 1 (t)φ 1 (r) +ψ 2 (t)φ 2 (r)], where ψ 1 (t) and ψ 2 (t) are the time-dependent modal amplitudes. The condensate wave functions in the two wells φ 1 (r) and φ 2 (r) are assumed to be real valued functions satisfying the orthonormal condition Hence |ψ 1 | 2 and |ψ 2 | 2 represent the occupation probabilities for the two modes, and the normalization condition for the condensate wave function | ∫ Φ(r,t) | 2 dr = N leads to the conservation of the occupation probabilities, |ψ 1 | 2 + |ψ 2 | 2 = 1. Substitution of Eqs.
(A3) and (A4) into Eq. (A2) immediately yields i ! ψ 1 = ε 1 ψ 1 + Kψ 2 + U 1 ψ 1 2 ψ 1 + U 12 (ψ 1 2 ψ 2 * + 2 ψ 1 2 ψ 2 ) where the parameters ε i , K , U i , U ij and I are given by the following overlap Here ε i are the single-mode energies, K is the tunneling rate of atoms between the two wells, and U i are the on-site interaction energies. These parameters are the same as those defined for the standard two-mode model [S3]. The remaining parameters U 12 , U 21 and I include all the mixed terms in the spatial wave functions, and thus they are present only when the spatial wave functions have small but non-zero density on the other side. They were first introduced by Ananikian and Bergeman to include a renormalized tunneling rate to provide better agreement with numerical simulations and experimental results [S4]. To show this, we rewrite Eqs. (A5a)-(A5b) according to the canonical formalism as i ! ψ k = ∂H / ∂ψ k * , where the classical Hamiltonian is The physical meaning of the parameters U 12 , U 21 and I can be understood from the Hamiltonian, where the U 12 ψ 1 2 term contributes an interaction-assisted tunneling to the Hamiltonian and similarly for the U 21 ψ 2 2 term. The last two terms in the Hamiltonian have different originsthe first term 2I ψ 1 2 ψ 2 2 represents the inter-well interaction and the second term I 2 (ψ 1 *2 ψ 2 2 +ψ 1 2 ψ 2 *2 ) is the pair tunneling energy.
Quantitative analysis of the population variations of the two condensates and the macroscopic tunneling effects can be performed using the canonical formalism. Let us make the substitution ψ k = p k e iθ k , where p 1 and p 2 are the fractional populations of the Bose atoms at the two wells and θ 1 and θ 2 are the phases on the two sides of the barrier. If we define θ = θ 2 − θ 1 , Eqs. (A5a)-(A5b) can be written as where ′ ε 1 = ε 1 + U 1 p 1 + Ip 2 , ′ ε 2 = ε 2 + U 2 p 2 + Ip 1 are the single-mode energies modified by the nonlinear interactions, ′ K = K + U 12 p 1 + U 21 p 2 is the tunneling energy modified by the overlap of the spatial wave functions, ′ K 1 = ′ K + 2U 12 p 1 and ′ K 2 = ′ K + 2U 21 p 2 . The first pair of equations implies ! p 1 = − ! p 2 , which comes from the conservation of total population, p 1 + p 2 = 1. The atomic current cross the barrier is N! p 1 , or − N! p 2 . When p 1 and p 2 are approximately the same, the atomic current would be given by J = J 0 sinθ + I 0 sin 2θ , where J 0 = N ′ K and I 0 = NI / 2 , and the phase evolution is which can be derived from the Hamiltonian [S6, S7] Here the coefficients Δ , ε , α , β and γ are given by Δ = 2K + U 12 + U 21 , Note that the nonlinear interactions produce a temporal change in the tunneling energy, and the tunneling energy Δ + β p is proportional to the population imbalance.
In a symmetric double well, we expect ε 1 = ε 2 , U 1 = U 2 and U 12 = U 21 , which implies ε = β = 0 . Then the time evolution for p and θ is governed by . By contrast, for the case when θ ≠ 0 or π , the fixed points of Eqs. (A10a)-(A10b) are given by p = 0 and cosθ = −Δ / α . The oscillations of an initial population imbalance and phase difference are described by In the absence of the bare tunneling parameter Δ , the frequency becomes α (α − γ ) .
We now discuss the time evolution of the condensates in an asymmetric double well.
For sinθ = 0 , the fixed points of Eqs. (A9a)-(A9b) are solved by where the plus and minus signs correspond to θ = 0 and π respectively. For sinθ ≠ 0 , the fixed points of the dynamics are determined by which has the solution p = βΔ − αε αγ − β 2 and 1− p 2 cosθ = βε − γΔ αγ − β 2 . Specifically, for the case where ε = Δ = 0 , the fixed points are located at ( p,θ ) = (0,±π / 2) , which implies a vanishing population imbalance and a π / 2 phase difference. Near the fixed points ( p,θ ) = (0, ±π / 2) , the time evolution of the population imbalance and the relative phase is determined by ! p = ±β p − αθ and ! θ = γ p ∓ βθ . Evidently, the atomic current is proportional to a linear combination of the population imbalance and the relative phase, and the relative phase is proportional to a linear combination of the population imbalance and the relative phase itself. For given values of α , β and γ , the time evolution of p and θ are sinusoidal, which can be easily observed from

Note 2. Derivation of the Hannay angle
Here we consider an integrable classical system described by a Hamiltonian are the canonical conjugate coordinates, and λ = {λ µ } represent the slowly varying parameters. The classical adiabatic theorem [S11] ensures that if the system is initially located at a torus in the N-dimensional phase space with action I = {I i } , it will remain on the torus with the same values of I. The evolution of the conjugate angle coordinates θ = {θ j } is determined by the generating function S (α ) (q, I;λ(t)) according to where the superscript α labels the branches where p is a single-valued function of q .
After the canonical transformation from {p,q} to {I,θ} , the new Hamiltonian From the definition of F(θ, I;λ) , we have Substituting Eq. (B4) into Eq. (B2), the Hamiltonian ′ H (θ, I,t) becomes As p and q are now both single-valued functions of I and θ , we can remove the superscript α of p(θ, I;λ(t)) . Now, using the Hamilton equations for the angle variables dθ / dt = ∂ ′ H / ∂I and integrating from 0 to T , we obtain [S12] where ω ≡ {ω j } with ω j (I;λ) ≡ ∂H (I;λ) / ∂I j are the instantaneous frequencies for fixed parameters, and the third term as a whole are the shifts in the angles θ = {θ j } in response to the adiabatic changes of parameters. To simplify the integration, we average out the fast oscillations by integrating over the torus where the integration is performed over a closed curve C in the parameter space. The first term inside the parenthesis ∂F / ∂λ is just a gradient and vanishes identically after integration. Therefore, the shifts in the angle variables can be transformed into an integral over a surface A whose boundary is C in the parameter space where W is the angle 2-form determined by A first example of the angle 2-form is the generalized harmonic oscillator whose Hamiltonian is described by Substituting Eq. (B10a)-(B10b) into Eq. (B8), we immediately obtain After a short calculation, we get the final expression of the angle 2-form for a generalized harmonic oscillator

Note 3. Derivation of the curvature 2-form for the de Sitter space
Here we consider the vielbein formalism of the connection and curvature in classical general relativity [S13, S14]. In the vielbein formalism, the metric tensor is expressed as where η αβ is the conventional Minkowski metric; µ , ν are world indices; and α , β are Lorentz indices. The fundamental quantities are the vielbeins e µ α , which form an n × n invertible matrix and satisfy the orthogonality conditions e α µ e ν α = δ ν µ , An immediate application of this formalism is to express the connection and curvature in terms of the vielbeins, or equivalently the vielbein 1-form e α = e µ α dx µ . The connection and the curvature are then defined by where ω β α = ω βµ α dx µ is the connection 1-form which is anti-symmetric in the Lorentz indices, ω αβ = −ω βα , and R β α = 1 2 R βµν α dx µ ∧ dx ν is the curvature 2-form. Suppressing the Lorentz indices, we get Cartan's first and second structure equations It should be noted that, for a local Lorentz transformation e = Λ! e , the connection and the curvature transform as where the metric of the Minkowski space is given by For the coordinate choice Z = cosht , T = sinht coshψ , X = sinht sinhψ cosφ and Y = sinht sinhψ sinφ , the metric describes a homogeneous and isotropic open universe where a(t) = sinht is the cosmological scale factor. The homogeneous and isotropic two-dimensional surfaces at constant t provide a natural slicing of spacetime. In this slicing, the metric can be written as ds 2 = −dt 2 + γ ij (x,t)dx i dx j , where γ ij (x,t) are the spatial components of the metric tensor, which are explicitly written as γ 11 = a 2 and γ 22 = a 2 sinh 2 ψ . In these coordinates, the components of the extrinsic curvature are simply given by K ij ≡ 1 2 ∂ t γ ij and hence we have K 11 = a ! a and K 22 = a ! asinh 2 ψ . The mean curvature K is the trace of the extrinsic curvature K ij where γ is the determinant of the spatial metric tensor γ ij . As the mean curvature K is proportional to the Hubble parameter H ≡ ! a / a , it can be regarded as a measure of spatial expansion of the expanding universe. The vielbein 1-forms can be obtained directly from the metric A straightforward computation yields where the connection 1-forms satisfy ω a 0 = ω 0 a and ω b a = −ω a b . Here a and b denote spatial indices. Employing Cartan's second structure equation, we obtain Similarly, the curvature 2-forms satisfy R a 0 = R 0 a and R b a = − R a b . As the curvature 2-forms obey the simple relation R β α = e α ∧ e β , the Riemann curvature tensors in the vielbein formalism are given by a and R bab a = R aba b . Thus the Ricci tensors in the vielbein formalism are R 00 = R 010 1 + R 020 2 = −2, (C16) The above shows that the Ricci tensors obey the simple relation R αβ = 2η αβ . As a result, the scalar curvature R becomes We would now like to express the curvature 2-forms R 1 0 , R 2 0 and R 2 1 in terms of T , X and Y . If we denote the coordinates of the unit hyperboloid satisfying These immediately lead to which is the area element of the unit hyperboloid. If t is now allowed to vary over After summation, we have the relation Recognizing T 2 − X 2 − Y 2 as the cosmological scale factor a(t) , we obtain the formula As a remark, for a closed curve C = ∂S in the parameter space T , X , Y ( ) , the integration of the 2-form a −2 R 2 1 over S always gives the area enclosed by the projection of C on the unit hyperboloid. Now, a straightforward computation gives As a result, we obtain the simple formula where H ≡ ! a / a is the Hubble parameter, γ is the determinant of the spatial metric tensor γ ij and ! γ is the determinant of the spatial metric tensor evaluated on the unit hyperboloid. Hence the 2-form R 2 0 is proportional to the difference between the renormalized area element in the horizontal XY-plane and its projection on the unit hyperboloid, with γ being the normalized factor. Similarly, we have the formula where L ≡ X 2 + Y 2 = asinhψ . Evidently, the 2-form R 1 0 is proportional to the area element in the vertical TL-plane.
As a final remark, the other two curvature 2-forms R 1 0 and R 2 0 are not directly related to the angle 2-form W , but they can still be expressed in terms of T , X and a / a is the Hubble parameter and g ≡ det g µν is the determinant of the metric tensor. Hence, the 2-form R 1 0 is proportional to the area element in the TL-plane and the 2-form R 2 0 is proportional to the difference between the renormalized area element in the XY-plane and its projection on the unit hyperboloid.

Note 4. Relationship between Bogoliubov excitation spectrum and the fundamental frequencies
Bogoliubov excitations are collective excitations that involve the motion of the whole condensate. For such a motion, the condensate wave function Φ(r,t) can be written as a superposition of the time-independent condensate ground sate and the small amplitude harmonic perturbations on the condensate ground sate where Φ(r) is the condensate ground state, u(r) and v(r) are the small amplitude oscillations, namely Bogoliubov excitations, on the condensate ground state, and ω is a frequency from the corresponding Bogoliubov excitation spectrum. In the following, we will show by explicit calculation that the Bogoliubov excitation spectrum is identical to the fundamental frequency of periodic orbits around the fixed points.
Expanding ψ k = p k e iθ k around the fixed points, we obtain ψ k = ψ k (1+ δ p k + δθ k ) , where δ p k and δθ k are the harmonic oscillations around the fixed points. For fixed parameters, as a result of Eqs. (B10a)-(B10b), the harmonic oscillations surrounding the fixed points ( p,θ ) = (0,±π / 2) can be expressed as where I is the action variable, which is the area enclosed by a given periodic orbit, and ω ≡ αγ − β 2 is the fundamental frequency around the fixed points. Substitution of Eqs.