Simulating cosmological supercooling with a cold atom system

We perform an analysis of the supercooled state in an analogue to an early universe phase transition based on a one dimensional, two-component Bose gas. We demonstrate that the thermal fluctuations in the relative phase between the components are characteristic of a relativistic thermal system. Furthermore, we demonstrate the equivalence of two different approaches to the decay of the metastable state: specifically a non-perturbative thermal instanton calculation and a stochastic Gross--Pitaevskii simulation.


I. INTRODUCTION
In its early stages, our universe was filled with hot, relativistic plasma that cooled through all of the major energy thresholds of fundamental particle physics, undergoing several changes of phase between different physical regimes. At the most extreme, the universe may have undergone first order transitions, characterised by metastable, supercooled states and the nucleation of bubbles. Bubbles of a new matter phase would produce huge density variations, and unsurprisingly first order phase transitions have been proposed as sources of gravitational waves [1,2] and as sources of primordial black holes [3,4]. Despite the importance of this phenomenon, we have no experimental test of the basic theory. In this paper, we propose that a thermal supercooled state, analogous to a relativistic system, can be realised in a Bose gas experiment. Phase transitions in fundamental particle physics can be associated with a Klein-Gordon field in an effective potential. At high temperatures, the field fluctuates about the minimum value of the potential representing a high temperature phase. As the temperature drops, the minimum of the potential changes to represent the low temperature phase, but the field can become trapped in a metastable state. Extreme supercooling can even lead to a zero-temperature metastable 'false vacuum' state.
The idea of using analogue systems for cosmological processes comes under the general area of modelling the "universe in the laboratory" [5,6]. So far, analogue systems have mostly been employed to test ideas in perturbative quantum field theory [7,8], but the non-perturbative phenomenon of false vacuum decay has recently been discussed, with theoretical descriptions of vacuum decay of atomic [9] and relativistic systems [10] at zero temperatures. Among possible analogue systems, (quasi-)one-dimensional ultracold Bose gases have emerged as an outstandingly versatile experimental platform for probing many-body quantum dynamics [11][12][13].
Fialko et al. [14,15] proposed an actual experiment to simulate the relativistic vacuum decay in a cold atom system. Their system consists of a Bose gas with two different spin states of the same atom species in an optical trap. The two states are coupled by a microwave field. By modulating the amplitude of the microwave field, a new quartic interaction between the two states is induced in the time-averaged theory which creates a non-trivial ground state structure as illustrated in Fig. 1.
However, this proposal has a drawback. A parametric instability on small wavelengths causes a classical decay of the metastable state [10,16]. In order to produce a metastable state, some form of dissipative mechanism would have to act on small scales to damp out the parametric resonance. A possible mechanism to achieve this could be thermal damping in the system. Furthermore, in (quasi-)one-dimensional experiments, the low temperatures needed to achieve a phasecoherent condensate are likely to be challenging. To open a path for future experiments, an understanding of how this proposal behaves in the thermal "cross-over" regime of a quasione-dimensional Bose gas is essential.
In this paper we demonstrate that in this regime the bubble nucleation dynamics are correctly reproduced by numerical modelling using a stochastic projected Gross-Pitaevskii equation (SPGPE) [17][18][19][20][21]. Theoretical studies have suggested that classical field methods such as this are valid when k B T is large compared to the maximum energy per mode ω k [20]. We shall show that the stochastic approach agrees with semiclassical predictions based on non-equilibrium thermal field theory of a relativistic Klein Gordon system. This agreement applies not only to the correlation functions, but also to the non-perturbative decay rate of a metastable state.

II. SYSTEM
Our system is a one-dimensional, two-component Bose gas of atoms with mass m. The two components are different spin states of the same species, coupled by a time-modulated microwave field. The Hamiltonian is given by where the field operator ψ has two components ψ i , i = 1, 2. Fialko et al. [14,15]  The field potential V plotted as a function of the relative phase of the two atomic wave functions, ϕ. The metastable phase is at the minimum ϕ = π and the stable phase is at the global minimum ϕ = 0. The difference in energy density between these phases is ∆V. over timescales longer than the modulation timescale can lead to an interaction potential of the form where σ {x,y} are Pauli matrices. The potential includes the chemical potential µ, equal intra-component s-wave interactions of strength g between the field operators (we assume inter-component s-wave interactions are negligible), and a microwave-induced interaction with strength µ 2 . The final term comes from the averaging procedure, and introduces a new parameter λ, dependent on the amplitude of the modulation. The trapping potential used to confine the condensate has been omitted in order to isolate the physics of vacuum decay. In principle, a quasi-one-dimensional ring trap experiment could realize the uniform system we study. The terms proportional to 2 are responsible for the difference in energy between the global and local minima of the energy, and we require to be small. The global minimum represents the true vacuum state and the local minimum represents the false vacuum. The true vacuum is a state with ψ 1 = ψ 2 and the false vacuum is a state with ψ 1 = −ψ 2 . The condensate densities of the two components at the extrema are equal to one another, and given by ψ † , in terms of the mean density ρ 0 = µ/g.
Throughout this paper, we will make use of the healing length ξ = /(mgρ 0 ) 1/2 and the sound speed c = /(mξ). Together, these define a characteristic frequency ω 0 = c/ξ. The dimensionless form of the potential constructed from these parameters becomesV = V/( ω 0 ρ 0 ). If we now introduce the relative phase ϕ between the spin components, such that ψ 1 ≈ ρ 0 e iϕ/2 and ψ 2 ≈ ρ 0 e −iϕ/2 , then the potential becomeŝ as shown in Fig. 1, with ∆V = 4 ω 0 ρ 0 2 . In the experimental proposal, the system is initially prepared in the metastable phase at a temperature T . In one dimension, the physics of Bose gases critically depends on the dimensionless interaction strength parameter ζ = (ρ 0 ξ) −2 and the temperature [22][23][24][25]. We consider the weakly interacting case, ζ 1. A phase-fluctuating quasi-condensate, in which density fluctuations are suppressed, appears at temperatures below the cross-over temperature 1 The gas remains degenerate up to a temperature of order T D = ζ −1/2 T CO > T CO .

III. STOCHASTIC GROSS PITAEVSKII EQUATION
Stochastic Gross-Pitaevskii equations (SGPEs) are widely used for modeling atomic gases at and below the condensation temperature [17-20, 26, 27]. Here, we use the simple growth stochastic projected Gross-Pitaevskii equation (SPGPE) [19,20], which has been successfully used to model experimental phase transitions [28,29]. Extension of the SPGPE to spinor and multi-component condensates is described in Ref. [21].
For convenience, from this point in the paper we use ξ as the length unit and ω −1 0 as the time unit. We also rescale the wave function by replacing ψ → ρ 1/2 0 ψ, and measure the temperature in units of T CO . In these units, the form of SPGPE we use is Here the complex fields ψ j describe the well-occupied, lowmomentum modes of the system (the c-field region), and the projector P eliminates modes above the momentum cut-off k cut = 2ρ 0 ξT . We also considered other values of k cut , to ensure our results were not overly sensitive to the choice of momentum cut-off. The noise source η is a Gaussian random field with correlation function and the potential Typically, we set the dimensionless dissipation rate γ = 10 −2 . Values of γ from O(10 −4 ) to O(10 −2 ) have been used in previous work that made direct comparisons to experiment [28][29][30][31], making this a reasonable choice. We comment on the effect of γ later in the text. Our SPGPE simulations use a one dimensional grid of size L = 240ξ with periodic boundaries and spacing ∆x = 0.4ξ. We set ρ 0 ξ = 100. Our simulations were executed using the software package XMDS2 [32].
Averaged quantities were calculated over 1000 stochastic realizations. Some care should be exercised when applying the SPGPE in reduced dimensions, since a three-dimensional thermal cloud is assumed [33]. For a gas confined in a transverse harmonic trap of frequency ω ⊥ , the simple growth SPGPE above is valid in 1D with dimensionally-reduced interaction strength g = 2 a s ω ⊥ provided ω ⊥ k B T and, in principle, µ ω ⊥ . In practice µ ω ⊥ is sufficient: 1D S(P)GPE equilibrium states were investigated in Refs. [34,35] and shown to be an excellent match to quasi-1D atom-chip experiments in this regime [36,37] 2 . The temperature, T φ = 2 ρ 0 /(mk B L), at which phase coherence is attained across the entire system is typically much lower than this. Crucially however, the relative phase ϕ has an effective potential barrier that assists phase coherence in the relative phase at higher temperatures than T φ , as we shall see in the following results.

A. Equilibrium correlations
The correlation functions for fluctuations about the stable and metastable phases provide an essential check on the validity of the numerical modelling, and also elucidates the relation between fluctuations in the SGPE and the Klein-Gordon field ϕ. As shown in Appendix A, small fluctuations in the relative phase ϕ of the two components induced by the SGPE have a thermal Klein Gordon correlation function where the Klein-Gordon mass m ϕ = 2 (λ 2 ± 1) 1/2 , for the stable and metastable phases respectively. The correlation function about the stable phase computed from SPGPE simulations is shown in Fig. 2. As expected, at low temperatures we have complete agreement with the Klein-Gordon result. At higher temperatures, non-linear effects are increasingly important, until the state becomes completely phase incoherent, in analogy to symmetry restoration in fundamental particle physics. At intermediate temperatures, we can restore the agreement against the theoretical result by introducing an 'effective' coupling' λ eff , as shown in the inset in Fig. 2.

B. Bubble nucleation
In a first order regime, we expect to see exponential decay of the metastable state, triggered by bubble nucleation events. 2 Corrections to improve accuracy beyond the 1D SGPE in the regime µ ∼ ω ⊥ without resorting to full 3D are possible and discussed in Ref. [34]. In this section we present numerical results which confirm this prediction, and we show agreement with a semi-classical, non-perturbative instanton approach.
In order to model bubble nucleation using the the SPGPE, we must initialize the system in the metastable state. In most runs we do this by placing the fields ψ j in the fluctuationfree metastable state at time t = 0, allowing the noise term in the SPGPE [Eq. (5)] to rapidly build up thermal fluctuations. We also verified that equivalent results are produced by first allowing the fields to thermalize with a high potential barrier (λ = 1.8) and then instantaneously reducing λ, since this latter procedure is closer to a likely experimental protocol. A signature of bubble formation in an individual trajectory is given by the spatial average cos ϕ exceeding −1 + ∆, where ∆ = 0.2 is chosen to be larger than the typical fluctuations of cos ϕ due to thermal noise in the system. An example is shown in Fig. 3. Running many stochastic trajectories and computing the probability, P, of remaining in the metastable state results in an exponential decay curve, also shown in Fig. 3. A fit to the exponential form P = ae −Γt over the time intervals seen to be exhibiting exponential decay yields the decay rate Γ. Figure 4 shows the decay rate Γ for several values of T and λ. Uncertainties on Γ, reflecting the statistical uncertainty arising from the trajectory averaging, are computed by a bootstrap resampling approach [38].
The semi-classical model of bubble decay is based on an instanton calculation, where the equations are solved in imaginary time τ to give an instanton solution ψ b . For thermal scenarios, the imaginary time coordinate is taken to be periodic, Below, the logarithm of the probability, P, of remaining in the metastable state. In this case the system was first allowed to equilibrate at λ = 1.8, a barrier large enough that bubble nucleation was negligible, after which the barrier was reduced to λ = 1.4. Inset, the spatial average of cos ϕ for ten different runs. The nucleation time is taken to be when cos ϕ > −1 + ∆, where ∆ = 0.2 in this example.
with period β = /(k B T ). The instanton solution approaches the metastable state at large distances, and for a purely thermal transition the solution is independent of τ.
The full expression for the nucleation rate of vacuum bubbles in a volume V is [39,40], where B denotes the difference in action between the instanton and the metastable state divided by . The pre-factor A 0 depends on the change in the spectra of the perturbative modes induced by the instanton. This should only depend mildly on temperature, so we will treat this term as an undetermined constant.
The exponent is explicitly In the Klein-Gordon approximation, the decay exponent simplifies to There is good agreement when the decay constant Γ < γ, and this can be extended to higher Γ by using an effective coupling λ eff (lower plot).
where the factor α(λ) is defined by Note that the dependence in (12) disappears if we rescale x → x/ and use Eq. (3).
The values of α(λ) for a Klein-Gordon model have been obtained recently in Ref. [41]. A comparison between the instanton and stochastic approaches is shown in Fig. 4. They agree very well in the region were Γ < γ, which we interpret as the nucleation rate having to be less that the relaxation rate of the thermal ensemble. Remarkably, the two approaches also agree over a wider range if we replace the coupling λ by an 'effective' value λ eff .
Finally, we note that we repeated a sample of our SPGPE simulations with lower dissipation rate γ = 5 × 10 −3 . We find that the rate Γ is dependent on γ. However, the results are still well-fitted by the instanton approach, but with a different prefactor A 0 , as would be expected from the theory of dissipative tunnelling in quantum mechanics [42].

V. CONCLUSION
The quasi-condensed thermal Bose gas described above would serve as a laboratory analogue to an early universe, supercooled phase transition. We show that the SPGPE can be used to model the system, and that where overlap with instanton calculations is possible there is agreement between the predictions of the two approaches.
As an example experimental configuration, we consider one of the experimental setups proposed by Fialko et al. [15], which is based on tuning the interactions between two Zeeman states of 7 Li 3 . Based on the average scattering length, suitable experimental parameters would be 5 × 10 4 atoms in a quasi-1D optical trap [43] of length 90 µm and transverse frequency 2π × 66 kHz. The interaction strength ζ = 10 −4 (as in Fig. 4), and the cross-over temperature T CO = 215 µK. In this context the results in Fig. 4 correspond to temperatures from around 3.2 µK to 6.4 µK, where bubble nucleation should be observable. Interestingly, the phase correlation length at this temperature is less than the length of the gas, but the relative phase correlation length is larger due to the effective Klein-Gordon mass.
In future work, we will be simulating a modulated (i.e., time varying) potential in order to investigate the effects of thermal damping on parametric instabilities. We will also extend our results to two dimensions and include realistic trapping potentials where there is a possibility that the boundaries of the trap affect bubble nucleation.
Data supporting this publication is openly available under a Creative Commons CC-BY-4.0 License on the data.ncl.ac.uk site [44].