Quantum simulation of expanding space-time with tunnel-coupled condensates

We consider two weakly interacting quasi-1D condensates of cold bosonic atoms. It turns out that a time-dependent variation of the tunnel-coupling between those condensates is equivalent with the spatial expansion of a one-dimensional toy-Universe with regard to the dynamics of the relative phase field. The dynamics of this field is governed by the quantum sine-Gordon equation. Thus, this analogy could be used to 'quantum simulate' the dynamics of a scalar, interacting quantum field on an expanding background. We discuss, how to observe the freezing out of quantum fluctuations during an accelerating expansion in a possible experiment. We also discuss an experimental protocol to study the formation of sine-Gordon breathers in the relative phase field out of quantum fluctuations.

We consider two weakly interacting quasi-1D condensates of cold bosonic atoms. It turns out that a time-dependent variation of the tunnel-coupling between those condensates is equivalent with the spatial expansion of a one-dimensional toy-Universe with regard to the dynamics of the relative phase field. The dynamics of this field is governed by the quantum sine-Gordon equation. Thus, this analogy could be used to 'quantum simulate' the dynamics of a scalar, interacting quantum field on an expanding background. We discuss, how to observe the freezing out of quantum fluctuations during an accelerating expansion in a possible experiment. We also discuss an experimental protocol to study the formation of sine-Gordon breathers in the relative phase field out of quantum fluctuations. The recent progress in coherently controlling systems of cold atoms (e.g., [1][2][3][4] , see also [5] and references therein), stimulated a lot of research concerned with employing these experimental systems to 'quantum simulate' prototypical quantum many-body models (see e.g., [6,7]). Particularly fascinating are ideas concerned with simulating quantum many-body physics on curved spacetime ('analog gravity') (see, e.g., [8][9][10][11][12][13][14] and [15], as well as references therein) connecting concepts and techniques from cosmology and condensed matter (see [16]).
Here, it is argued that a pair of tunnel-coupled, quasi-1D, bosonic condensates (as realized in [17] and related experiments) can be employed for simulating an interacting, scalar quantum field on top of an expanding 1+1 dimensional space-time (Fig. 1a,c). This scalar field is represented by the relative phase field between the condensates. As argued in [18,19], at low energies, its dynam-ics is described by the quantum sine-Gordon model. It turns out that in this setup, one can simulate the expansion of the 1D toy-Universe simply by varying the tunnelamplitude according to a suitable protocol (Fig. 1b). In the experiments (e.g., [17]) the tunnel-amplitude itself can be largely tuned and the field dynamics can be directly visualized by means of matter-wave interferometry. The 'quantumness' of the relative phase-field dynamics depends on the interaction strength within each condensate (which is easily tunable, e.g., by adapting the 1D condensate density). As far as we are dealing with non-linear dynamics, we consider the weakly interacting, 'semiclassical' limit (e.g., [19][20][21]) and employ the truncated Wigner approximation (TWA) (e.g., [20]). In contrast, the proposed quantum simulator would allow to explore also the deep quantum regime -in fact, this is its main purpose.
In the following, we start with deriving an effective Hamiltonian description of an interacting quantum field on an expanding 1+1 dimensional space time. In a second step, we introduce the experimental 'quantum simulator' -the tunnel-coupled condensates -and eventually discuss two of the possible effects, which could be explored.
The first effect is the well known 'freezing' of quantum fluctuations and the related 'cosmological particle production' during an accelerating expansion. This purely linear, though most fundamental effect is made responsible for the structure formation in the very early universe [22]. In the present experiments, one can only perform single measurements of the relative phase field per run. Fortunately, this mode freezing also manifests itself in the spatial fluctuation spectrum of the field and thus, in principle, is detectable with single measurements per run. Although, strictly speaking, there is no need for a quantum simulation of the exactly solvable linear dynamics, observing the freeze-out of quantum fluctuations in the experiment would nevertheless be exciting (see for instance [9,10]) and constitute an important check on the setup.
The second feature is the generation of localized, macroscopic structures during the expansion out of quan-tum fluctuations. This pattern formation involves the full non-linearity of the underlying sine-Gordon field theory and was also observed in the static case [19]. In contrast to [19], for an exponentially fast expansion this happens only for small enough expansion rates. At large expansion times, these patterns seem to turn into standing sine-Gordon breathers simply drifting apart from each other. We argue how to detect signatures of this 'Hubble' drift experimentally. In cosmology, e.g., dealing with the preheating following inflation, excitations like this (e.g., in the scalar inflaton field) are sometimes denoted as 'oscillons' (e.g., [23][24][25][26][27][28]). A full experimental quantum simulation would allow investigating the formation and persistence of these excitations on an expanding background, even in regimes where quantum effects become very important for the dynamics.
Quantum field on curved space-time.
-The space-time action of a classical, scalar field theory in 1+1 dimensions is given by [22] (c = 1, = 1) where g µν denotes the metric (with g µν g νγ = δ γ µ ), ∂ µ χ = ∂χ/∂x µ and V (χ) is an arbitrary potential. We are interested in an homogeneous, spatially expanding spacetime described by the Friedmann-Robertson-Walker metric (FRW) (see for instance [22], ds 2 = g µν dx µ dx ν ) Here, x denotes co-moving coordinates which are related to physical coordinates x ph via the scale parameter a(τ ) as x ph = ax. The time τ denotes the cosmological time, i.e., the proper time of a co-moving observer. In reality, the dynamics of a is determined by Einstein's equations.
Tunnel coupled condensates as quantum simulator. -Here, we propose a quantum simulation of the field χ(x, η) for the special case of a sine-Gordon potential V = −m 2 0 β −2 cos(βχ). This potential has several interesting properties: The corresponding field theory is interacting and integrable. Second, the sine-Gordon potential appears in the so-called 'natural inflation' scenario [31]. Third, the sine-Gordon potential supports the formation of 'quasibreathers' [19] (in the cosmology literature denoted as 'oscillons', e.g., [25,26,28,32] and references therein).
The quantum simulator (Fig. 1a) consists of two tunnel-coupled quasi-1D condensates of cold, bosonic atoms (e.g., [17]) with a time-dependent tunnel amplitude t ⊥ . The laboratory time is identified with the conformal time η. At low energies, the dynamics of the relative phase field βφ(x, t)/ √ 2 = (φ 1 −φ 2 )/ √ 2 can be described by the quantum sine-Gordon model [18] (the sound velocity v s = 1) The relative phase field and the relative density varia- form a canonical pair. The tunnel amplitude enters the mass term m 2 (η) = 2β 2 ρ 0 t ⊥ (η), where ρ 0 is the mean density per condensate. The Luttinger liquid description should be reliable as long as the typical lengthscale of Eq. (5), set by √ v s /m, is much larger than the healing length of the condensates ξ h [18,19]. This can always achieved by choosing a sufficiently small tunnel amplitude t ⊥ (η). The parameter β is related to the Luttinger parameter K as β = 2π/K. For weak interactions β 1. This is the limit we are considering here. It can be shown that β plays the role of Plank's constant [20] and β 1 corresponds to the semiclassical limit of the quantum sine-Gordon model (see also, e.g., [33]). The analysis here (as far as we are dealing with non-linear dynamics) is based on the semiclassical TWA. However, in the experiment one can go deep into the quantum regime corresponding to larger β (a rather broad range of values up to K ∼ 50 is realizable, e.g. [34]). The following identifications connect the quantum simulator and the quantum field theory on an expanding background. Identifying the fieldsχ ↔φ and m 2 (η) = m 2 0 a 2 (η) [and thus a(η) ↔ t ⊥ (η)/t ⊥ (0)], the dynamics of the relative phase field simulatesχ in conformal time and co-moving coordinates. In the remainder, we will always argue in terms of the fieldφ, i.e., we analyze Eq. (5).
Scale parameter and initial state. At η = 0, we start in the ground state of massive phonons with a small mass m 0 . In particular, m 0 is chosen much smaller than the UV cutoff 1 ∼ 1/ξ h (all TWA simulations are performed on a lattice with lattice constant set to one, while keeping m 0 a(η) 1 throughout the whole simulation). For finite system size and β 1, the center-of-mass mode (COM mode) Φ(η) ≡ 1 L´L 0 dxφ(x, η) needs some special attention. For L/ξ h 1, we can treat the COM mode classically such thatφ(x, η) Φ(η) + δφ(x, η). Before, the expansion starts, the COM mode is tuned to some value Φ(0) Φ 0 with βΦ 0 ∈ [0, π] (Φ (0)/m 0 0). As argued in [19], such an initial state can be achieved by slowly splitting a single condensate followed by applying a potential gradient between the condensates to tune Φ 0 .
Freezing out of quantum fluctuations. -One of the fascinating results of modern cosmology is that the structure formation in the very early Universe seems to have been seeded by quantum fluctuations [22]. This result is truly amazing, as on cosmological scales zero-point fluctuations are tiny. However, it seems that an exponential expansion of the very early Universe (inflationary stage) led to a 'freezing' of quantum fluctuations and stretched them to cosmological scales. One can reformulate this basic mechanism in a condensed matter language (e.g., [16]). In this terminology, the inflationary expansion corresponds to a rapid, non-adiabatic 'quench' (see, e.g., [35]) producing a large number of excitations ('cosmological particle creation', cf. [10]).
Formation of breathers out of quantum fluctuations. -While the freezing out of quantum fluctuations is a purely linear effect, we now discuss a feature, which heavily relies on interactions. From now on, we consider a slow expansion H/m 0 1 and finite 0 < Φ 0 ∼ π/β (see Fig. 3a). At short times η > 0, the global relative phase Φ(η) performs Josephson oscillations (see, e.g., [19]). However, it is well known that these are parametrically unstable against small, spatial fluctuations (e.g. [19,41,42]). Linearizing around Φ, one findŝ φ k + [k 2 + m 2 0 a 2 (η) cos βΦ]φ k 0. In the static case [19], this parametric drive leads toφ k (η) ∼ e Γ(k,Φ0)η with Γ |k| 2 sin 2 (βΦ 0 /2) − k 2 /m 2 [42](displayed for βΦ 0 1). In turn the fluctuations δφ 2 grow and at some point the linearization breaks down. In the semiclassical limit β 1, it was demonstrated [19] that the non-linearity of the sine-Gordon equations leads to the formation of localized patterns in the fieldφ(x, η). These patterns were identified with 'quasibreather'-solutions of the classical sine-Gordon equation [19], which in contrast to usual breathers have a finite lifetime. The 'quasibreather'-solutions could also explain the formation of these excitations out of phononic quantum fluctuations.
Here, for a slow, exponential expansion (in proper time τ ), one observes the formation of similar excitations only for m 0 /H exceeding a certain, almost sharp threshold (Fig. 3c, cf. also [28]), which depends on β and Φ 0 . To see this, first note that as long as the linearization of the sine-Gordon equation applies, all field fluctuations are suppressed asφ k ∼ 1/ √ a (including the COM mode). Furthermore, in the course of time, resonant modes can get shifted out of resonance as a consequence of the expansion. For a slow expansion, replacing k → k/a and Φ 0 → Φ 0 / √ a (cf. [28,42]), one finds that φ k (η)φ −k (η) ∝ a −1 exp 2´η 0 dη aΓ( k a , Φ0  Fig. 3a). While the COM mode is damped by the expansion, these structures turn into standing sine-Gordon breathers with a constant amplitude (cf. Fig. 3b), which, in physical coordinates simple drift away from each other. This 'Hubble expansion' is mirrored by the linear decay of the correlation function as ∝ (1 − ηH) at late times. Once the expansion stops at η f , the correlation function becomes almost constant. Here: β = 0.1, H/m0 = 0.02, m0 = 0.05, L = 800 and βΦ0 = 0.7π.
exceed a certain value for some η (see Fig. 3c and [43]). Numerically, we find that this value is of the order O(10 −1 ).
Close to the creation threshold, once created, these breather-like excitations persist at the position, where they were 'born' out of quantum fluctuations. This is in contrast to the 'quasibreathers' observed in the static case [19] (cf. also [32]). In the long-time limit, the homogenous part of the field Φ is damped away ∝ 1/ √ a. We find good numerical evidence that the localized excitations, however, are robust against the expansion and can be well described as standing (classical) sine-Gordon breathers φ B (see Fig. 3b). Their typical distance is set by the maximally amplified wavelength before the nonlinearity sets in, ending the parametric amplification. It seems that at late times (a 1), the only effect of the 'adiabatic' expansion (H/m 0 1) on breathers is a trivial shrinking of the breather period and width (both ∝ 1/m 0 a(η)) in co-moving coordinates, while their amplitude stays approximately constant. This can be understood realizing that the amplitude of a classical sine-Gordon breather βφ B is solely determined by the breather parameter ϕ ∈ [0, π/2] (maxβφ B = (2π − 4ϕ), see e.g., [44]). In the quantum sine-Gordon model, this parameter gets quantized [45], i.e., it is promoted to a quantum number. However, it is well known that for a slow change of system parameters (by slow, here, we understand Ω −2 B ∂Ω B /∂η 1, where Ω B = m 0 a(η) sin ϕ is the instantaneous breather frequency), quantum numbers (and thus the breather amplitude) are approximatively preserved (cf. [46]). While for large a 1 the number and amplitude of breathers remain constant (per experimental run), they simply move apart from each other in physical coordinates. This 'Hubble expansion', e.g., can be observed in the (experimentally accessible) equal time correlation C φφ (x, η) = φ (x, η)φ(0, η) − φ (0, η) 2 (Fig. 4). Under the assumption that at late times, the fieldφ can be described as a set of independent (standing) sine-Gordon breathers with fixed amplitudes, one obtains that The linear suppression of C φφ is a direct consequence of the decreasing breather density in physical coordinates. From a condensed matter point of view the observation that a suitable protocol for the tunnel-amplitude prepares a state consisting of independent, standing sine-Gordon breathers is interesting by itself. While the analysis here is based on semiclassical considerations (numerically on the TWA) reliable for β 1, the proposed quantum simulator could for instance test the stability of classical sine-Gordon breathers against quantum fluctuations for larger β ∼ 1 (cf. [47]). Furthermore, a quantum simulation could give insight in the excitation of 'oscillonic' patterns (as discussed in the cosmology community, e.g., [26][27][28]32]) in a scalar quantum field (such as the inflaton field or even the Higgs field) during a spatial expansion. Conclusions.
-We demonstrate that tuning the tunnel-amplitude between a pair of tunnel-coupled 1D condensates, the relative phase field can simulate an interacting quantum field on an expanding 1+1 space time. The proposed 'quantum simulator' should be realizable with present cold atom setups. As examples of the quantum many-body dynamics, which could be investigated, we discussed the freezing out of phonon modes and the creation of sine-Gordon breathers out of quantum fluctuations during an exponential FRW-expansion. While the discussion here is restricted to the semiclassical limit of the underlying quantum sine-Gordon model, the 'quantum simulator' is meant to explore the deep quantum regime.
Acknowledgements. -We thank R. Schützhold for fruitful discussions related to this work. Financial support by the Emmy-Noether program is gratefully acknowledged.