Intrinsic dephasing in onedimensional ultracold atom interferometers
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Edited by Paul C. Martin, Harvard University, Cambridge, MA, and approved April 27, 2007 (received for review October 9, 2006)
Abstract
Quantumphase fluctuations prevent true longrange 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 quantumphase 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 crossover 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 BoseEinstein 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 longstanding 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.
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
It is interesting to point out the essential difference between the problem we consider and dephasing of singleparticle 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 quasisteady) state.
Results
Hydrodynamic Theory.
We start our analysis by considering the hydrodynamic theory that describes lowenergy 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
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 〈A ^{2}〉 > 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_{+} = (ϕ_{1} + ϕ_{2})/2 and ϕ_{−} = ϕ_{1} − ϕ_{2}. 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 timedependent calculation.
Consider a region of size ξ
_{h}
, which is the length scale on which the hydrodynamic variables are defined. Let n_{1}
and n_{2}
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_{−}
= (n_{1}
− n_{2}
)/2, with an uncertainty of order
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=0}
^{2} ∝ t^{2}
, which results in a Gaussian decay of the coherence: (t) = A_{0}
exp(−t^{2}
/τ_{L}
^{2}), where
It is interesting to note that one can also express the time scale associated with the finite size as
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 longwavelength, lowenergy correlations. However, it is natural to question the validity of the hydrodynamic description for computing timedependent 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 lowenergy 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 temperaturedependent 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
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 crossover temperature T*, the dephasing time gains significant temperature dependence. The crossover 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 shortdistance 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 temperatureindependent 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.
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):
Taking the scattering length of rubidium87 atoms a_{s} = 105a _{0}, 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)/2πI_{0} (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 longrange 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 longrange 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 numbersqueezed 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 quantumphase 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 crossover 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 longrange order even at finite temperatures. In this case the longrange 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. Twodimensional systems can, in principle, sustain longrange 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 quasisteady) state is not yet thermal equilibrium. In particular, we find that the offdiagonal 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 _{i} ψ*_{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 ρ_{0}. 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 timedependent 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
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.
Acknowledgments
We thank N. Davidson, E. Demler, S. Hofferberth, A. Polkovnikov, J. Schmiedmayer, T. Schumm, and J. H. Thywissen for useful discussions. This work was partially supported by the U.S.–Israel Binational Science Foundation and an Alon fellowship (E.A.).
Footnotes
 ^{‡}To whom correspondence should be addressed. Email: rafi.bistritzer{at}weizmann.ac.il

Author contributions: R.B. and E.A. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

↵ § This length scale may be obtained by solving the GP boundary problem in which the wave function increases from zero at x = 0 to its bulk value at x → ∞ (19).

↵ ¶ The interaction is activated by using U(t) = U[1 + tanh(λt)]/2, where λ controls the degree of adiabaticity.
 Abbreviations:
 TWA,
 truncated Wigner approximation;
 GP,
 Gross–Pitaevskii.
 © 2007 by The National Academy of Sciences of the USA
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