Intrinsic dephasing in one-dimensional ultracold atom interferometers

Abstract

Quantum-phase fluctuations prevent true long-range phase order from forming in interacting 1D condensates, even at zero temperature. Nevertheless, by dynamically splitting the condensate into two parallel decoupled tubes the system can be prepared with a macroscopic relative phase, facilitating interferometric measurement. Here, we describe a dephasing mechanism whereby the quantum-phase fluctuations, which are so effective in equilibrium, act to destroy the macroscopic relative phase that was imposed as a nonequilibrium initial condition. We show that the phase coherence between the condensates decays exponentially with a dephasing time that depends on intrinsic parameters: the interaction strength, sound velocity, and density. Interestingly, significant temperature dependence appears only above a cross-over scale T∗. In contrast to the usual phase diffusion, which is essentially an effect of confinement and leads to Gaussian decay, the exponential dephasing caused by fluctuations is a bulk effect that survives the thermodynamic limit.

The existence of a macroscopic phase facilitates observation of interference phenomena in Bose-Einstein condensates. For example, an interference pattern involving a macroscopic number of particles arises when a pair of condensates is let to expand freely until the two clouds spatially overlap. Since the pioneering experiment by Andrews et al. (1), which demonstrated this effect, it has been a long-standing goal to construct matter wave interferometers based on ultracold atomic gases. Such devices have promising applications in precision measurement (2) and quantum information processing (3), as well as fundamental study of correlated quantum matter. The key requirement for interferometric measurement is deterministic control over the phase difference between the two condensates. That is, the relative phase must be well defined and evolve with time under the sole influence of the external forces that are the subject of measurement. This requirement was not met, for example, in the setup of ref. 1, where the two condensates were essentially independent and the relative phase between them was initially undetermined.

One way to initialize a system with a well defined phase is to construct the analogue of a beam splitter whereby a single condensate is dynamically split into two coherent parts, which serve as the two “arms” of the interferometer (4–7). In recent experiments such a split was achieved by raising a potential barrier along the axis of a quasi 1D condensate (5). The split is applied slowly compared with the transverse frequency of the trap, but fast compared with the longitudinal time scales. Thus each atom is transferred in this process to the symmetric superposition between the two traps without significantly changing its longitudinal position. An illustration of the procedure is given in Fig. 1. In repeated experiments the condensates are released from the trap to probe the interference at various times after the split. Immediately after the split, the two condensates are almost perfectly in phase. However, repeated measurements at longer times show that the phase distribution becomes gradually broader until it becomes uniformly distributed on the interval [0, 2π]. The accuracy of interferometric measurements is limited by such dephasing.

Fig. 1.

Interferometer setup. At time t = 0, a single 1D condensate is split into two coherent parts by raising a high potential barrier along the condensate axis. After a wait time t, the condensates are released from the trap and let to expand. After a sufficient expansion time, the cloud is imaged by a probe beam sent along the condensate axis, and the phase of the resulting interference pattern is recorded. This process is repeated many times to obtain a phase distribution at a number of different wait times t.

In this article, we develop a theory of dephasing in interferometers made of 1D Bose gases. We assume that such systems can be well isolated from the noisy environment, so that this source of decoherence (8) is set at bay. Another dephasing mechanism that has been discussed extensively in the context of condensates in double wells is “quantum phase diffusion” (7, 9–11). The uncertainty in the particle numbers in each well brought about by the split entails a concomitant uncertainty in the chemical potential difference, which leads to broadening of the global relative phase on a time scale of ${\tau }_D\sim ħ\sqrt{N}/\mu$. Here, N is the total particle number and μ is the chemical potential. Note that this is essentially a finite size effect; τ_D diverges in the proper thermodynamic limit in which the trap frequency is taken to zero while the density is kept constant. Here, we show that dephasing in quasi 1D systems is dominated by a mechanism that involves quantum fluctuations of the local relative-phase field rather than the global-phase difference between the condensates. In contrast to the usual phase diffusion this is a bulk mechanism that survives the thermodynamic limit. Indeed, it is well known that in 1D liquids quantum-phase fluctuations are extremely effective and prevent the formation of long-range order. Here, we show how these fluctuations destroy long-range order in the relative phase, which was imposed on the system by the initial conditions. We account for the phase fluctuations by using a quantum hydrodynamic description of the Bose liquids (12). The phase coherence between the condensates is then shown to decay exponentially in time. The dephasing time we obtain is a simple function of intrinsic parameters: the dimensionless interaction strength, density, and sound velocity. These predictions are verified and extended to finite temperatures, using direct simulation of the dynamics in the microscopic Hamiltonian. The results are also compared with the dephasing time measured in ref. 5 and are found to be in quantitative agreement.

It is interesting to point out the essential difference between the problem we consider and dephasing of single-particle interference effects as considered, for example, in mesoscopic electron systems. As in our problem, phase coherence in mesoscopic systems can be defined by an interferometric measurement (13). In a Fabry–Perot setup, oscillations of the conductivity as a function of applied Aharonov–Bohm flux vanish exponentially with system size, thereby defining a characteristic dephasing length. Because what is measured is ultimately the DC conductivity, the problem may be recast in terms of linear response theory (14). By contrast, dynamic splitting of the condensate in the ultracold atom interferometer takes the system far from equilibrium, and the question of phase coherence is then essentially one of quantum dynamics. The system is prepared in an initial state determined by the ground state of a single condensate, which then evolves under the influence of a completely different Hamiltonian, that of the split system. Dephasing, from this point of view, is the process that takes the system to a new steady (or quasi-steady) state.

Results

Hydrodynamic Theory.

We start our analysis by considering the hydrodynamic theory that describes low-energy properties of 1D Bose liquids (12). The hydrodynamic Hamiltonian for a pair of decoupled condensates is that of two decoupled Luttinger liquids (we set ħ = 1 throughout):

where c is the sound velocity and K is the Luttinger parameter that determines the decay of correlations at long distance. ∏_α is the density fluctuation operator conjugate to the phase ϕ_α. The smooth component of the Bose field operator is given by ${\psi }_{\alpha }\approx \sqrt{\rho }\exp (i{\varphi }_{\alpha })$, where ρ is the average density (12).

The operator corresponding to the interference signal between the two condensates is given by (15):

where L is the imaging length. For a pair of decoupled condensates at equilibrium, 〈A〉 = 0, while 〈A²〉 > 0 (15). Therefore, the interference pattern displayed in repeated experiments has a finite amplitude but its phase is completely random. On the contrary, in this work we consider a pair of condensates that are prepared out of equilibrium with a well defined relative phase. In this case 〈A〉 is expected to be nonvanishing at the time of the split and decay in time as the fluctuation in relative phase grows.

Calculation of the time evolution of 〈A〉 is greatly simplified by the fact that the hydrodynamic theory is quadratic and by the decoupling in the Hamiltonian of “center of mass” and relative phase fields ϕ₊ = (ϕ₁ + ϕ₂)/2 and ϕ₋ = ϕ₁ − ϕ₂. In other words, the Hamiltonian can be rewritten as a sum of two commuting harmonic terms H = H₊(ϕ₊, ∏₊) + H₋(ϕ₋, ∏₋), where ∏_± are the conjugate momenta of ϕ_±. Moreover, the splitting process described above ensures that the initial state may be factorized as ψ[ϕ₊, ϕ₋] = ψ₊[ϕ₊] × ψ₋[ϕ₋]. Because during the split all atoms are simply transferred to the symmetric superposition without changing their axial state, the wave function ψ₊[ϕ₊] is identical to the wave function of the single condensate before the split. If this condensate is at finite temperature ψ₊ is replaced by the appropriate density matrix.

On the other hand, ψ₋ is determined by the splitting process and will generally be strongly localized around ϕ(x) = 0. In fact, we can find the approximate form of this wave function, which will be an important input for the time-dependent calculation.

Consider a region of size ξ_h, which is the length scale on which the hydrodynamic variables are defined. Let n₁ and n₂ be the operators corresponding to the particle number in each of the condensates within this region. The splitting process delocalizes the particles between the two condensates, resulting in a roughly Gaussian distribution of the relative particle number n₋ = (n₁ − n₂)/2, with an uncertainty of order $\sqrt{(n_1+n_2)/4}=\sqrt{\rho \xi h/2}$. The phase ϕ defined on this grain in the hydrodynamic theory is canonically conjugate to n₋. Therefore, the wave function in the phase representation is approximately Gaussian with 〈ϕ(x)ϕ(x′)〉 ≈ δ(x −x′)/(2ρ). That is:

From here on we drop the subscript − where no ambiguity may arise.

We are now ready to compute 〈A〉, which depends only on ϕ₋. The wave function ψ[ϕ₋] evolves in time under the influence of the harmonic Hamiltonian:

The wave function remains Gaussian at all times:

where η_q(ϕ_q, ϕ*_q, t) is a pure phase and

The expectation value 〈A〉 = in the Gaussian wave function (5) is given by:

In the thermodynamic limit (L → ∞) the summation in Eq. 7 can be converted to an integral to obtain:

The last equality is valid at times t > ξ_h/c, when the integrals become independent of the high momentum cutoff. In addition, we require that the particle number in a healing length ξ_h is large, i.e., ρξ_h ≫ 1, this is always satisfied at weak coupling K ≫ 1. Thus we find that the phase coherence decays exponentially:

with the characteristic time

Note that the dephasing time τ diverges, as it must, in the noninteracting limit (K → ∞ and c → 0). Clearly, the initial state induced by the split is an eigenstate of noninteracting particles; therefore, all observables, including the coherence, must be time independent in this limit.

For a finite system, the discreteness of the sum in Eq. 7 must be taken into account at times t ≫ L/c. In such a case, the dominant contribution to g(t) arises from the q = 0 term. This term yields a ballistic broadening of the phase ϕ_q=0² ∝ t², which results in a Gaussian decay of the coherence: (t) = A₀ exp(−t²/τ_L²), where ${\tau }_L=\sqrt{\tau L/2c}$.

It is interesting to note that one can also express the time scale associated with the finite size as ${\tau }_L\alpha \sqrt{N}/\mu$. Written in this way, it is clear that the Gaussian term is just the quantum-phase diffusion discussed in the context of double-well systems (7, 9). It stems from the uncertainty relation between the uniform component of the relative phase and the difference in total particle numbers between the wells. The uniform component of the interaction, which drives the ballistic broadening of ϕ_q=0 scales like 1/L, making the Gaussian term irrelevant in large systems. By contrast, the exponential dephasing caused by the internal modes is length independent. The fact that these modes produce exponential dephasing is special to 1D interferometers. A similar analysis of two parallel planar condensates yields logarithmic divergence in Eq. 8, which implies power law decay of the coherence as ∝ (ξ_h/ct)^α. We note that in this case K is not dimensionless. The power α is nonuniversal and depends on the short distance cutoff. In three dimensions, on the other hand, the only contribution to dephasing is the usual phase diffusion caused by finite size and there is no bulk contribution.

It is striking that the hydrodynamic theory developed in this section predicts a dephasing time that is independent of temperature. Temperature enters the initial condition only through the density matrix for the symmetric degrees of freedom, whereas the phase coherence between the condensates depends on the evolution of the relative phase. Because the symmetric and antisymmetric fields are completely decoupled within the harmonic theory, the temperature associated with the symmetric degrees of freedom does not affect the dephasing. However, it is clear that the full microscopic Hamiltonian contains anharmonic terms that do couple those degrees of freedom. At equilibrium the nonlinear terms are irrelevant (in the renormalization group sense) and do not affect the asymptotic long-wavelength, low-energy correlations. However, it is natural to question the validity of the hydrodynamic description for computing time-dependent properties. In particular, here the system is prepared out of equilibrium in a state with extensive energy relative to the ground state. Then it is a priori unclear that the low-energy correlations given by the hydrodynamic theory are sufficient to describe the dynamics. In the next section we shall test the predictions of the hydrodynamic theory and obtain temperature-dependent corrections to the dephasing by computing the dynamics within a microscopic model of the twin condensates.

Numerical Results.

For the purpose of numerical calculations we shall consider a lattice Hamiltonian, the Bose–Hubbard model on twin chains:

Here b_αi^† creates a boson on site i of chain α. The model (11) can also describe continuum systems, such as the one in ref. 5, if the average site occupation n̄ = 〈n_i〉 is much less than unity.

We are interested in the time evolution of the expectation value (t) = Σ_i〈b_1i^†(t)b_2i(t)〉. At zero temperature the average is taken over the wave function of the condensate at the time of the split. However, we can also address the situation in which the condensate is initialized at finite temperature. In this case the average is taken with respect to the density matrix of the system at the time of the split. The numerical calculation is performed by using the semiclassical truncated Wigner approximation (TWA; cf. refs. 16, 17, and 23). The essence of this approach is to integrate the Gross–Pitaevskii (GP) equations starting from an ensemble of initial conditions, which are given quantum weights according to the initial density matrix. Details of the method are given in Methods:TWA.

To compare the numerical results to the prediction of the hydrodynamic theory we use the fact that at weak coupling there are simple relations between the parameters of the microscopic model (11) and those of the hydrodynamic theory (1) (18). In particular, the value of the Luttinger parameter is given by $K\approx \pi \sqrt{2J\overline{n}U}$, and the sound velocity is $c=\sqrt{2J\overline{n}U}$. The short distance cutoff of the hydrodynamic theory may also be extracted from Eq. 11, by finding the length over which amplitude fluctuations decay. It is given by $l_h=\sqrt{2}{\xi }_h\approx a\sqrt{4J/U\overline{n}}$.^§ Increasing n̄ decreases this length scale until it reaches its lower bound, which is one lattice constant a. When l_h ≫ a the model (11) in effect describes a continuum system, whereas for a large site occupancy l_h → a it describes a lattice of coupled condensates. We perform simulations in both regimes. However, for technical reasons, described in Methods:TWA, the temperature range we were able to cover in the continuum regime was somewhat limited. The temperature of the condensate before the split is parameterized by measuring the thermal correlation length ξ_T ∝ 1/T associated with the decay of the correlation function 〈b_i^†b_i+r〉.

The numerical results are summarized in Figs. 2 and 3. The dephasing time, τ, is extracted by fitting (t) to Eq. 9. The results at low temperatures give a dephasing time τ that is almost temperature independent and in good agreement with the hydrodynamic result (10). On the other hand, above a cross-over temperature T*, the dephasing time gains significant temperature dependence. The cross-over scale is set by the condition ξ_T* ∼ l_h (equivalently T* ∼ μ). This criterion is not surprising, given that the hydrodynamic theory is expected to break down when correlations decay over a length scale shorter than its short-distance cutoff. The agreement between the numerical results and the hydrodynamic theory at low temperatures is highly nontrivial in view of the fact that the two approaches are based on entirely different approximations. Most importantly, the temperature-independent result (10) relied on the decoupling of symmetric and antisymmetric degrees of freedom within the harmonic theory (1). Such decoupling does not exist in the microscopic model (11) nor in the TWA dynamics. In the next section we shall compare the results of the hydrodynamic theory to experimental data.

Fig. 2.

Dephasing. Decay of the coherence is calculated with the TWA for U = 1, J = 1, and n̄ = 7.5 and compared with the result of the hydrodynamic theory (dots). The time axis is scaled to the hydrodynamic dephasing time τ₀ (10). Lines correspond to different initial temperatures parameterized by the correlation length ξ_T that existed before the split.

Fig. 3.

Temperature dependence of the dephasing time. Dephasing time τ as a function of the thermal correlation length, which is a measure of the inverse temperature before the split. Data are taken in different regimes of microscopic parameters. All data are seen to collapse on approximately the same curve when the dephasing time is scaled to the theoretical time scale (10) and the thermal correlation length scaled by l_h.

Comparison with Experiments.

The dephasing time (10) is written in terms of parameters of the hydrodynamic theory. To compare with experiments we have to translate these parameters to numbers that are relevant to a specific experimental situation. For this purpose we will consider the setup of the experiments described in ref. 5.

This system consists of bosons with contact interactions parameterized by a dimensionless interaction strength γ. At weak coupling, γ ≪ 1, the following relations hold (18): $K=\pi /\sqrt{\gamma }$ and $c=h(\rho /M)\sqrt{\gamma }$, where M is the mass of an atom. Substituting these into 10 we obtain:

Now consider a 1D tube geometry with transverse trap frequency ω_⊥. If the oscillator length l_⊥ is much larger than the s wave scattering length a_s, as is the case in ref. 5, we can use the approximate expression γ ≈ 2Mω_⊥a_s/ħρ (18, 20). This allows us to write the dephasing time by using parameters that are easily determined in the experiment:

Note that ρ is the density per condensate in the split system, that is half the density of the initial condensate.

Taking the scattering length of rubidium-87 atoms a_s = 105a₀, the transverse trap frequency ω_⊥ = 2π × 2.1 kHz and density of ≈50 atoms per μm (J. Schmiedmayer, personal communication), we obtain a dephasing time τ ≈ 4.3 ms. To estimate the dephasing time in the experiment (5) we use the data for the phase broadening assuming a von Mises distribution of the phase [f(θ) = exp(κcosθ)/2πI₀(κ)]. This gives τ ≈ 2 ms, slightly shorter than our theoretical estimate. We note that data points marked by squares in Fig. 3 correspond to parameters relevant to the experiment. Unfortunately, the temperature in that experiment is not known to a good accuracy. In the future it would be interesting to look for the universal temperature dependence implied by the numerical results.

Discussion

It is well known that quantum fluctuations prevent long-range phase order from forming in 1D Bose liquids. This phenomenon is most conveniently described within the framework of the Luttinger liquid or hydrodynamic theory. Here, we used this framework in a nonequilibrium situation to show how quantum fluctuations destroy long-range order that was imposed on the system as an initial condition. The outcome is a simple formula describing exponential decay of phase coherence in interferometers made of 1D condensates. The dephasing time found in this way provides a fundamental limit on the accuracy of such interferometers. We did not discuss the situation in which the interferometer is prepared in a number-squeezed initial state. Such an initial condition would slow the usual phase diffusion process (6, 11) and is expected to similarly affect the bulk mechanism discussed here.

Interestingly the dephasing time measured in ref. 5 is only slightly shorter than the prediction of the hydrodynamic theory. We therefore conclude that quantum-phase fluctuations were probably a dominant dephasing mechanism in that experiment.

The validity of the Luttinger liquid framework out of equilibrium is not ensured a priori. We therefore test its predictions against numerical simulations of the microscopic model by using the TWA. At low temperatures, the simulation results were essentially temperature independent and in good agreement with the hydrodynamic theory. On the other hand, at temperatures above a cross-over scale set by the chemical potential, the dephasing time displayed considerable temperature dependence. Indeed the Luttinger liquid theory is expected to break down above this temperature scale.

Phase fluctuations are expected to be weaker at higher dimensions. For example, 3D condensates can support long-range order even at finite temperatures. In this case the long-range order in the relative phase imposed as an initial condition of the interferometer is resistant to the phase fluctuations. The situation with planar condensates is more delicate. Two-dimensional systems can, in principle, sustain long-range order at equilibrium at zero temperature. However, the initial condition of the interferometer drives the system out of equilibrium. The hydrodynamic theory yields power law dephasing in this case.

Finally, we point out that the dephasing process considered here is a mechanism that brings the system, from a nonequilibrium state imposed by the initial conditions to a new steady state. According to the hydrodynamic theory, this steady (or quasi-steady) state is not yet thermal equilibrium. In particular, we find that the off-diagonal correlations along the condensates are distinctly nonthermal at steady state. It is an interesting question whether there are processes, much slower than dephasing, that eventually take the system toward thermal equilibrium. This issue can be addressed by experiments. After the phase has randomized, correlations along the condensates can be measured by using analysis methods developed in refs. 15 and 21. Thus, we propose that interferometric experiments can serve as detailed probes to address fundamental questions in nonequilibrium quantum dynamics, supplementing measurements of global properties previously used to touch on these issues (22).

Methods: TWA

Let us first briefly review the strictly classical approximation to the dynamics. The usual procedure is to write the Heisenberg equations of motion for b_i and b_i^† using Eq. 11 and then replace these operators by complex classical fields ψ_i and ψ_i^∗. This leads to the lattice GP equation:

Given the initial condition ψ_α,i(0) one can integrate the GP equations to find the value of the fields at any time t and obtain _cl(t) = Σ_iψ*_1i(t)ψ_2i(t). If the split is fully coherent ψ_1i(0) = ψ_2i(0). Because the evolution in the two chains is described by identical equations, this equality persists to all subsequent times. Thus in the absence of external noise sources, the classical dynamics cannot account for dephasing. By contrast, quantum fluctuations provide an intrinsic dephasing mechanism.

Quantum corrections modify the dynamics in two ways. First, they introduce “quantum noise” to the initial conditions. The fields ψ_αi and ψ*_αi originate from noncommuting quantum operators b_αi and b_αi^†, which cannot be determined simultaneously. Therefore, the unique classical initial condition should be replaced by an ensemble of initial conditions characterized by a quantum distribution. Second, the classical trajectories determined by Eq. 14 are supplemented by additional quantum paths.

It can be shown that the leading quantum correction to the dynamics enters through the initial conditions (16). The essence of TWA is to integrate the GP equations, starting from an ensemble of initial conditions, which are given quantum weights derived from the initial density matrix ρ₀. Within this approximation:

Here

is the Wigner representation of the initial density matrix of the system and |ψ〉 denotes a coherent state with eigenvalue ψ of the boson annihilation operator. We note that ρ_w should not be thought of as a probability distribution, because in general it can assume negative values.

The main difficulty in applying the recipe (15) to compute (t) is the need to find the Wigner distribution of the split condensate at the time of the split. To overcome this problem we use the following procedure. Rather than tackling a split interacting system we start the calculation from a single noninteracting condensate, where we have an exact expression for ρ_w (16). The evolution of the Wigner distribution from a noninteracting system to an interacting one is done using the TWA by slowly increasing the interaction constant, U, from zero to the desired value^¶. The heating induced by the time-dependent Hamiltonian is controlled by the rate at which U is increased. The next step is the split of the condensate. It involves the doubling of the degrees of freedom at each site. Because of the way the split is carried out, the field ψ_i of the single condensate is simply copied to the symmetric field ${\psi }_{+,i}\equiv ({\psi }_{1i}+{\psi }_{2i})/\sqrt{2}$. The new degree of freedom ψ_−,i is chosen at this point from a Wigner distribution:

which corresponds to the vacuum of the boson $b_{-,i}=(b_{1i}-b_{2i})/\sqrt{2}$. The calculation is continued by using the dynamics of the twin condensates.

The condensate (before the split) is prepared at various temperatures by using two different methods: (i) initializing the noninteracting condensate in a finite temperature density matrix, and (ii) varying the rate by which the interactions are switched on to induce a controlled amount of heating. We verified that the final result for (t) depends only on the correlation length ξ_T and not on the method used to achieve finite temperature.

We note that simulations in the continuum regime (ξ_h much larger than a lattice constant) require an extremely slow activation of the interaction to avoid heating, which limited the temperature range we could study in this regime.

Abbreviations

TWA

truncated Wigner approximation

GP

Gross–Pitaevskii.