Non equilibrium quantum dynamics in ultra-cold quantum gases

Non equilibrium quantum dynamics in ultra-cold quantum gases

Ehud Altman Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, 7610001, Israel
Abstract

Advances in controlling and measuring systems of ultra-cold atoms provided strong motivation to theoretical investigations of quantum dynamics in closed many-body systems. Fundamental questions on quantum dynamics and statistical mechanics are now within experimental reach: How is thermalization achieved in a closed quantum system? How does quantum dynamics cross over to effective classical physics? Can such a thermal or classical fate be evaded? In these lectures, given at the Les Houches Summer School of Physics ”Strongly Interacting Quantum Systems Out of Equilibrium”, I introduce the students to the novel properties that make ultra-cold atomic systems a unique platform for study of non equilibrium quantum dynamics. I review a selection of recent experimental and theoretical work in which universal features and emergent phenomena in quantum dynamics are highlighted.

Contents

I Introduction to ultra-cold quantum gases

A lot of the recent work on ultra cold atomic systems is focused on using them as simulators to investigate fundamental problems in condensed matter systems. However it is important to note that many-body systems of ultracold atoms are not one-to-one analogues of known solid state systems. These are independent physical systems that have their own special features, and present new types of challenges, and opportunities. One of the main differences between ensembles of ultracold atoms and solid state systems is in the experimental tools available for characterizing many-body states. While many traditional techniques of condensed matter physics are not easily available (e.g. transport measurements), there are several tools that are unique to ultracold atoms. This includes time-of-flight expansion technique, interference experiments, molecular association spectroscopy, and many more. In some cases, these tools have been used to obtain unique information about quantum systems. One of the most striking examples is single site resolution in optical lattices Bakr et al. (2009); Sherson et al. (2010), which allowed exploration of the superfluid to Mott transition at the unprecedented level. Another example was measurements of the full distribution functions of the contrast of interference fringes in low dimensional condensates, which provided information about high order correlation functions. Polkovnikov et al. (2006); Gritsev et al. (2006); Hadzibabic et al. (2006); Gring et al. (2012)

An area in which ultracold atoms can make unique contributions is understanding non-equilibrium dynamics of quantum many-body systems. Several factors make ultracold atoms ideally suited for the study of dynamical phenomena.

  • Convenient timescales. Characteristic frequencies of many-body systems of ultracold atoms correspond to kilohertz. System parameters can be modified and properties resulting from this dynamics can be measured at these timescales (or faster). It is useful to contrast this to dynamics of solid state systems, which usually takes place at Giga and Terahertz frequencies, which are extremely difficult for experimental analysis.

  • Isolation from the environment. Ultra-cold atomic gases are unique in being essentially closed, almost completely decoupled from an external environment. More traditional condensed matter systems by contrast are always strongly coupled to some kind of thermal bath, usually made of the phonons in the crystal. This distinction does not matter much if the systems are at thermal equilibrium because of the equivalence of ensembles. However, it leads to crucial differences in the non equilibrium dynamics of the two systems. While the dynamics in standard materials is generically overdamped and classical in nature, atomic systems can follow coherent quantum dynamics over long time scales. A corollary of this is that it is also much harder for systems of ultra-cold atoms to attain thermal equilibrium. Hence understanding dynamics is important for interpreting experiments even when ultracold atoms are used to explore equilibrium phases.

  • Rich toolbox. One of the difficulties in studying nonequilibrium dynamics of many-body systems is the difficulty of characterizing complicated transient states. Experimental techniques that have been developed for ultracold atoms recently are well suited to this challenging task. Useful tools include local resolution Bakr et al. (2009); Sherson et al. (2010), measurements of quantum noiseAltman et al. (2004); Fölling et al. (2005) and interference fringe statistics Polkovnikov et al. (2006); Gritsev et al. (2006); Hadzibabic et al. (2006); Gring et al. (2012).

Before proceeding let me list several fundamental questions that can be addressed with ultra cold atomic systems and that we will touch upon.

Emergent phenomena in quantum dynamics– The main theme in the study of complex systems in the last century has been to identify emergent phenomena and understand the universal behavior they exhibit. Examples of successful theories of universal phenomena in equilibrium phases include the Ginsburg- Landau-Wilson theory of broken symmetry phases and critical phenomena, the theory of Fermi liquids, and more recently, the study of quantum hall states, spin liquids and other topological phases. In addition there is a growing understanding of emergent universal phenomena in classical systems out of equilibrium. This includes for example the phenomena of turbulence, coarsening dynamics near phase transitions, the formation of large scale structures such as sand dunes and Raleigh-Bernard cells, dynamics of active matter such as flocking of animals and more. Yet there is very little understanding of non equilibrium many-body phenomena that are inherently quantum in nature. It is an interesting open question whether the dynamics of complex systems can exhibit emergent universal phenomena in which quantum interference and many-body entanglement play an important role.

Thermalization and prethermalization in closed systems– Intimately related to establishment of emergent phenomena is the question of thermalization and the approach to equilibrium in closed quantum systems. Cold atomic system are ideal for testing the common belief that generic many-body systems ultimately approach thermal equilibrium. There are however exceptions to this rule. Integrable models, for example, fail to come to thermal equilibrium because the dynamics is constrained by an infinite set of integrals of motion. In general integrable models require extreme fine tuning. Nevertheless systems of ultra-cold atoms in one dimensional confining potentials often realize very-nearly integrable models. The reason for this is that the interactions naturally realize almost pure two body contact interactions. This fact has been used to demonstrate breaking of integrability in the Lieb-Lininger model of one-dimensional bosons with contact interactionsKinoshita et al. (2006). This experiment has motivated a lot of theoretical and numerical investigations to characterize the non thermal steady state that is reached in such cases. The latter is usually characterized using a generalized Gibbs ensemble (GGE) that seeks the equilibrium (maximum entropy) state subject to the infinite number of constraints set by the values of the integrals of motion in the initial stateRigol et al. (2008).

The systems which realize nearly integrable models usually do include small perturbations that break integrability. It is believed (though not proved) that these perturbations will eventually lead to true thermalization. Hence the (GGE) is considered to be a long lived prethermalized state. An intersting question that we will touch upon concerns the time scale for relaxational dynamics to take over and lead to true thermal equilibrium.

Absence of thermalization and many-body localization – A body of recent work starting with a theoretical suggestion Basko et al. (2006); Gornyi et al. (2005) (and much earlier conjecture by Anderson Anderson (1958)) points to quantum many-body systems with quenched disorder as providing a generic alternative to thermalization. That is, in contrast to integrable models, lack of thermalization is supposed to be robust to a large class of local perturbations in such closed systems. This phenomenon is known as many-body localization (MBL) Important questions that are beginning to be addressed with ultra-cold atomic systems Kondov et al. (2013); Schreiber et al. (2015) concern the dynamical behavior in the MBL state and the nature of the transition from MBL to conventional ergodic dynamics. If there is a critical point controlling the transition from a thermalizing to a non thermalizing state, does it involve singularities in correlation functions similar to those that dominate quantum critical phenomena at equilibrium?

In these lecture notes I focus on universal phenomena in dynamics that can and have been investigated using ultra cold atomic systems. Of course I cannot cover everything, so I picked some illustrative examples that I find interesting from a fundamental perspective and have also been directly investigated experimentally using ultra-cold atomic systems. In section II I discuss the dynamics of ultra-cold atom interferometers, primarily in one dimension, as case studies of prethermalization and thermalization dynamics. In section III I turn to the dynamics of bosons in optical lattices. I review the theoretical understanding of the quantum phase transition from a superfluid to a Mott insulator and then discuss emergent dynamical phenomena that have been studied in the vicinity of this transition. In particular I will discuss the nature of the Higgs amplitude mode that emerges near the critical point as well as the modes for decay of super currents due to the enhanced quantum fluctuations near the Mott transition. In section IV I discuss far from equilibrium dynamics involving a quench of system parameters across the superfluid to Mott insulator phase transition. Section V reviews some of the recent work on quenches in one dimensional dimensional optical lattices. Finally, in section VI I turn to review the recent progress in theoretical understanding of many-body localization as well as recent experiments that investigated this phenomenon. I should note that in these notes I include the very significant recent progress on many-body localization that took place after the lectures were delivered. I have also tried to make each section self contained so that they can be read independently of each other. However if the reader is not familiar with the physics of Bosons on optical lattices and the Mott transition it would help to read section III before section IV.

Ii Dynamics of ultra-cold atom interferometers: quantum phase diffusion

Atomic condensates have been used early on as interferometers, utilizing the matter waves for making precision measurements of accelerations. Such experiments constitute a natural and simple example of a dynamical quench experiment, which beside its practical applications raise fundamental questions in non-equilibrium quantum physics. Hence this example will serve here to illustrate several key concepts, which occur frequently in non-equilibrium studies of cold atoms.

An idealized interferometer experiment consists of a macroscopic condensate separated into two wells and prepared with a well-defined relative phase between the wells. When the wells are disconnected the relative phase is expected to evolve in time under the influence of the potential difference between the two wells. Following this evolution the phase is measured directly by releasing the atoms from the trap and observing the interference pattern established after a time of flight.

One method to realize such a preparation protocol was to start with a one condensate in an elongated potential well and raise a radio-frequency induced potential barrier to split the condensate into two one dimensional condensatesSchumm et al. (2005). A schematic setup of this nature is illustrated in Fig. 1. Alternatively, instead of using two separate wells as the two arms of the interferometer one can use two internal states of the atoms, as done in Ref. Widera et al. (2008). The system is prepared with a condensate in a single internal state, say . A two photon transition is used to a induce a rotation to the state , this is the analog of the coherent splitting in the double well realization. Thereafter the system is let to evolve freely under the influence of the Hamiltonian with no coupling between the two internal state. Finally the coherence between the two states remaining after a time is determined through a Ramsey type measurement. That is, measuring how much of the state population is restored following an rotation of the internal state.

Figure 1: Typical setup for an interferometer measurement. A condensate is split into two decoupled wells such that the relative condensate phase between the wells is initially well defined . The two condensates are let to evolve under the influence of the local potentials acting on them for time . At time they are released from the trap and their expansion leads to an interference pattern that is observed on a light absorption image.

In reality the relative phase is not determined solely by the external potential difference between the two wells (or two internal spin states). The phase field of an interacting condensate should be viewed as a quantum operator with it’s own dynamics. The intrinsic quantum evolution leads to uncertainty in the relative phase, which grows in time and eventually limits the accuracy of the interferometric measurement. Thus even if the condensates are subject to exactly the same potential, the relative phase measured by the interference pattern between them will have a growing random component. The rate and functional form with which this uncertainty grows is a hallmark of the many-body dynamics. Fig. 2 is an example of an experiment performed by the Vienna group Hofferberth et al. (2007) that shows the distribution of relative phase determined in repeated measurements after different times from the initial preparation.

Figure 2: (a) Example of fringes from interfering one-dimensional condensates taken at varying times after the condensate had been coherently split Hofferberth et al. (2007). (b) The distribution of relative phase extracted from the interference fringes grows with time. (Reproduced from Ref. Hofferberth et al. (2007))

It is interesting to point out the essential difference between the problem we consider here and dephasing of single-particle interference effects as seen, for example, in mesoscopic electron systems. In the latter case the dephasing of a single electron wave-function is a result of the interaction of the single electron with a thermal bath, which consists of the other electrons in the Fermi liquid or of phonons. Because the bath is thermal and the energy of the injected electron is low, the dephasing problem is ultimately recast in terms of linear response theory Stern et al. (1990).

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 splitting scheme, which then evolves under the influence of a completely different Hamiltonian, that of the split system. It is the first example we encounter of a quantum quench. Dephasing, from this point of view, is the process that takes the system to a new steady (or quasi-steady) state. In this respect, the ultracold atom interferometer is a useful tool for the study of non equilibrium quantum dynamics. One of our goals is to classify this dynamics into different universality classes.

ii.1 Phase diffusion in the single-mode approximation

Let us start the theoretical discussion from the simplest case of perfect single mode condensates initially coupled by a large Josephson coupling that locks their phases. The Hamiltonian which describes the dynamics of the relative phase is given in the number-phase representation by

(1)

Here is the charging energy of the condensate , where is the chemical potential and the average particle number in the condensate. Alternatively we can express the charging energy using the contact interaction and the condensate volume as . The relative particle number and relative phase are conjugate operators obeying the commutation relation .

At time the Josephson coupling is shut off and the time evolution begins subject to the interaction hamiltonian alone. The system is probed by releasing the atoms from the trap after a time evolution over a time subject to the Hamiltonian of decoupled wells . The aim is to infer the coherence between the two condensates from the emerging oscillating pattern in the density profile.

Here it is important to note that such information can only be inferred from averaging the density profile (interference fringes) over many repetitions of the experiment. A single time-of-flight image, originating from a pair of macroscopic Bose condensates, exhibits density modulations of undiminishing intensity but with possibly a completely random phaseAndrews et al. (1997); Castin and Dalibard (1997); Polkovnikov et al. (2006). If the condensates are completely in phase with each other, then is fixed and the average interference pattern taken over many experimental runs would be the same that seen in a single shot. On the other hand, if the relative phase is completely undetermined, the average interference pattern would be vanishing. Therefore the correct measure of coherence is the average interference amplitude remaining after averaging over many shots, which is directly related to the quantum expectation value .

The time evolution of the relative phase subject to the Hamiltonian (1) with follows the obvious analogy to spreading of a wave-packet of a particle with mass , the relative phase fluctuation follows a ballistic evolution

(2)

This implies a Gaussian decay of the Fringe amplitude with time

(3)

with the dephasing time scale . Several important insights may already be gleaned from this simple model.

  1. Quantum phase diffusion in a true condensate is a finite size effect. Indeed in the proper thermodynamic limit the broken symmetry state with well defined phase is infinitely long lived.

  2. Quantum phase diffusion is driven by interaction. Setting eliminates it.

  3. The rate of phase diffusion depends on the initial state. A narrower initial particle number distribution entails longer dephasing time.

The last point implies that the dephasing time depends crucially on the preparation protocol of the interferometer. In one extreme the initial state is prepared with maximal coherence. That is, a state with all particles in a symmetric superposition between the two wells. In the limit of large particle number N, we can take this state to have a gaussian relative number distribution between the wells with a variance . Accordingly the phase diffusion time in this case is . The

As we’ll see below, such a state can be prepared by a rapid ramp-down of on a scale faster than the characteristic scale set by the chemical potential . We can increase the coherence time by preparing an initial state with a narrower number distribution. Due to the number phase uncertainty relation, we must pay by having a larger initial phase distribution and hence achieve lower accuracy in phase determination at short times. In quantum optics terminology this is called a squeezed state.

The simplest way to achieve squeezing is to split the condensates more slowly. In the slow extreme of adiabatic splitting the system ends up in the ground state with completely undetermined relative phase (and vanishing number fluctuations). This state is infinitely long lived, but of course useless for phase determination. More generally we can split the condensates on a time scale . The frequency scale we should compare to in order to asses whether this drive is slow or fast is the instantaneous gap in the system given roughly by the Josephson frequency . As long as the system to a good approximation remains in the instantaneous ground state and the dynamics is essentially adiabatic. However from some point during the split when , the dynamics should rather be viewed as a sudden split. The effective initial state is the ground state of the system at the break-point .

Using a Harmonic approximation of the junction, , the number fluctuation in the initial state is easily found to be . We then obtain the phase-diffusion time of the squeezed state . Because of the squeezing, the minimal phase uncertainty is now , larger than the uncertainty in the case of the (fast) maximally coherent split.

The single mode phase diffusion considered in this section was first discussed in the context of Bose condensates by Leggett and SolsLeggett and Sols (1991). It is however a special case of the more general problem of spontaneous symmetry breaking as discussed originally by Anderson in Ref. Anderson (1952). A single mode Hamiltonian emerges as a description of the dynamics of the uniform broken symmetry order parameter. In this case it is the dynamics of the order parameter, or condensate phase. The dynamics is governed by an effective mass which grows heavier with the system volume. Hence the phase becomes static due to the infinite mass in the thermodynamic limit allowing spontaneous symmetry breaking. In the next section we will see under what conditions the condensate zero mode can indeed be safely separated from the linearly dispersing Goldstone modes. In particular we’ll show how the multi mode dynamics can dominate the dephasing in interferometers composed of low dimensional condensates.

ii.2 Many modes: hydrodynamic theory of phase diffusion in low dimensional systems

The single mode approximation described above is expected to be a reasonable description of bulk three dimensional condensates. In this case internal phase fluctuations, quantum or thermal, are innocuous since they do not destroy the broken symmetry. The mode responsible for restoring the symmetry in a finite system is the uniform mode and therefore only its dynamics has to be considered. However, many interference experiments have been performed with highly elongated, essentially one dimensional condensatesSchumm et al. (2005); Jo et al. (2007); Hofferberth et al. (2007); Widera et al. (2008). In this case zero point fluctuations due to long-wavelength phonons destroy the broken symmetry even in the thermodynamic limit. It is expected that these phonons will have a crucial impact on the loss of phase coherence.

The model Hamiltonian to consider is a direct generalization of the single mode Hamiltonian considered in the previous section

(4)

Here are the Hamiltonians of the individual one (or higher) dimensional condensates and the Josephson coupling now operates along the entire length (or area) of the condensates. As in the single mode case the system is prepared with strong Josephson coupling, which is thereafter rapidly shut off. In the way this scheme is commonly implemented Schumm et al. (2005); Jo et al. (2007) the system actually starts as a one dimensional condensate in a single tube, which is then split into a double tube over a timescale slow compared to transverse energy levels in the tube. At the same time can be fast compared to the chemical potential of the condensate, leading to preparation of a coherent state of the relative phaseSchumm et al. (2005). Alternatively can be larger than the inverse chemical potential, leading to preparation of a squeezed phase stateJo et al. (2007).

We want to track the ensuing dynamics of the relative phase following this preparation stage. Again, this is done by releasing the atoms from the trap at fixed times after the split and inspecting the resulting ”matter wave” interference fringes in the density profile. As in the single mode case interference fringes can be seen even when the two condensates are independent. The correct measure of the coherence between the two condensates is the average of the fringe amplitude over many repeats of the experiment, directly related to the expectation value . What can be conveniently measured in an experiment is in fact the integral of this quantity over a certain averaging length:

(5)

This quantity of course vanishes for independent condensates. It is important to distinguish from the quantities and that would give a non-vanishing value (dependent on ) even for independent condensates Polkovnikov et al. (2006). Fig. 2 shows an example of interference fringes and the distribution of phases of the quantity measured in repeated experiments.

Here, in computing the time evolution of the aversge fringe amplitude we will closely follow the analysis of Ref. Bistritzer and Altman (2007). There, a detailed theoretical description of the dynamics is obtained by approximating the exact Hamiltonians by the low energy Harmonic-fluid description of the phase fluctuations (phonons) in the superfluidPopov (1972)

(6)

Here is the phase field of the condensate or and is the canonical conjugate smooth density field. and are the macroscopic phase stiffness and compressibility of the condensate respectively. These should in general be considered as phenomenological parameters extracted from experiment. However, for weakly interacting Bose gases the compressibility is only weakly renormalized from the bare value given by the inverse contact interaction while the stiffness is directly related to the average density as . The short range cutoff of this hydrodynamic theory is the healing length of the condensate . It will sometimes be convenient, to express quantities in terms of the Luttinger parameter that determines the power-law decay of correlations in 1d.

We note that the anharmonic terms neglected here are irrelevant for the asymptotic low energy response in the ground state, yet they may influence the decoherence dynamics at long times. This will be considered in the next section.

It is convenient to transform to the relative and ”center of mass” phase variables and their respective conjugate momenta and . Within the harmonic-fluid description the relative and ”center-of mass” coordinates are decoupled. This is not changed by the dynamic Josephson term, which does not involve the ”center of mass” modes at all. Since the observable we are interested in (5) involves only the relative phase, we can forget about the center of mass modes at this level of approximation.

After the Josephson coupling is turned off the relative phase evolves under the influence of a purely Harmonic Hamiltonian. In Fourier space the different spatial modes of the relative degrees of freedom are conveniently decoupled and we have

(7)

Assuming that the initial state is a Gaussian wave-function, the time evolution of the mean fringe amplitude, our proxy for phase coherence, is easily computed from the Harmonic theory

(8)
(9)

To make further progress we need the form of the initial phase distribution, which depends on the preparation scheme.

ii.2.1 Fast split scheme

As in the single mode case we begin with a discussion of a fast split. In this case all particles remain in the symmetric superposition between the two wells. Such an initial state is particularly simple to write in the number phase representation if there are many particles per length unit that serves as a cutoff to the hydrodynamic theory. This is equivalent to the weak coupling requirement . In this case the Poisson statistics of the particle number is well approximated with Gaussian statistics with . Accordingly the phase wave-function is Gaussian in this limit and the variance should be plugged into (11).

We convert the sum over all in (8) to an integral, while separating out the uniform component, which leads to a unique contribution. Following a simple change of variables we then have

(10)

Where is the surface area of a unit sphere in dimensions and . The first term in the integrand saturates to a constant at long times. The second term depends crucially on the spatial dimension . The contribution of the zero mode together with the integral over the internal phonons then gives:

(11)

where .

Let us pause to discuss this result. The contribution of the zero mode is the same in all dimensions. It gives rise to a Gaussian decay on a time scale that diverges in the thermodynamic limit. This is precisely our result obtained in the previous section from the single mode approximation. Now in addition we have included the internal modes at the Gaussian level. In three dimensions we see from (11) that the phase fluctuations do not contribute to the time dependence at long times. This is tantamount to the statement that in three dimensions broken symmetry is dynamically stable in the thermodynamic limit. On the other hand, in the coherence decays even in the thermodynamic limit due to the internal phonons. This is the dynamical counterpart of the Mermin Wagner theorem, showing how the internal (Gaussian) phase fluctuations dynamically destroy long range order. The decay can be traced back to an infrared divergence of the integral.

ii.2.2 Prethermalization in the fast split scheme

The harmonic model we are considering now is the simplest example of an integrable model. The different modes are decoupled from each other and their occupations are conserved quantities. Hence the model does not in general relax to thermal equilibrium. It is interesting to ask what is the nature of the steady state the system reaches after it has dephased. Approach to such a non-thermal steady state is commonly referred to as prethermalization. The name stems from the expectation that there is a much longer time scale over which the dephased state would finally relax to a true thermal state. Thus the prethermalized state is a long lived quasi-steady state established in the system before the onset of true thermalization. In our case the final relaxation to the thermal state is governed by the enharmonic terms. This analysis is postponed to the next section.

Within the Gaussian approximation, the full information on the steady state is held in the two point correlation function (in the relative phase sector):

(12)

This expression is directly analogous to the formula for the fringe amplitude (8). Similarly the important contribution to the integral comes from the second term in (11) leading to, for ,

(13)

This integral exhibits completely different behavior depending on whether the point at the position and time is inside or outside the ”light-cone” emanating from the other point which we have set to at at the time of the split . For the last cosine term is rapidly oscillating and averages to zero. In this case we have and therefore . This agrees with the expectation that for there was no time for information to propagate between the two points and therefore the correlation function can be decoupled into the independent expectation values at the two points.

On the other hand at long times we make a different change of variables using as an integration variable to obtain:

(14)

Now the last cosine term is rapidly oscillating () and we have a time independent result . This implies an exponential decay of the phase correlations in the steady state with a correlation length .

Such steady state behavior mimics the exponential decay of phase correlations in equilibrium at a finite temperature, which is characterized by the correlation length . Thus we can assign an effective temperature to the prethermalized state by comparing the correlation length in it to the correlation length in thermal equilibrium.

It is important to note that the effective temperature is completely independent and, in fact, has nothing to do with the real temperature of the condensate prior to the split. The original, thermal degrees of freedom of that condensate are projected following the rapid split onto the symmetric density and phase modes of the split condensate. The state of the relative degrees of freedom, accessible to interference experiments, is a pure state fully determined by the splitting process. Within the harmonic fluid theory the symmetric and anti-symmetric sectors are decoupled, and so within this approximation the temperature of the original condensate does not affect the dynamics.

The prethemalized state reached following a fast split of a one dimensional condensate has been seen in beautiful experiments by Vienna group using atom chipsGring et al. (2012); Smith et al. (2013). These results are based on an earlier theoretical proposal by Kitagawa et. al. Kitagawa et al. (2011) showing that the prethermalized state can be fully characterized by the distribution of the amplitude of fringes seen in an interference experiment. These experiments confirm the applicability of the harmonic theory in the accessible time-scales.

ii.2.3 Finite split rate

A dynamic split done on a timescale can be treated in complete analogy to our treatment of a single mode. For simplicity we separate the two stages of the dynamics. First we treat the splitting process in which the Josephson coupling is gradually turned off. The outcome of this process provides the initial state for calculating the dynamics at later times under the influence of the hamiltonian of the decoupled tubes. Here we give a brief summary of a detailed calculation along these lines found in Bistritzer (2007).

To address the splitting dynamics we expand the Josephson coupling to quadratic order to obtain a time dependent mass term for the relative phase mode We then assess for each mode separately whether the splitting on a time-scale is slow or fast compared with the characteristic frequency of this mode. For modes at wave vectors the split is effectively adiabatic. Therefore these modes end up in the quantum ground state of the final, fully split Hamiltonian. Modes at lower wave-vectors develop adiabatically only to the point that their frequency is and they remain frozen for the rest of the splitting process.

ii.3 Beyond the Harmonic fluid description of one-dimensional phase diffusion

The harmonic fluid hamiltonian (6) is a fixed point of the renormalization group in a one dimensional system, which describes a stable quantum phase of matter known as a Luttinger liquid Haldane (1981). Hence, as long as the macroscopic (fully renormalized) values of and are used, the harmonic Hamiltonian provides an asymptotically exact description of the ground state and low energy excitations of the system. The neglected anharmonic terms affect the long distance correlation and low frequency response only through the renormalization of the quadratic coefficients and from their bare values.

But in spite of being irrelevant for the linear response of the system in the ground state, the non linear terms can influence the long time dynamics of the interferometer, which starts in a state of finite energy density after the split. As discussed above, the splitting scheme lead to excitation of modes at all wave vectors, which then evolve independently within the quadratic theory (6). Only the non-linear terms can break this integrability through the coupling they induce between modes at different wavectors. Moreover, the non linear terms lead to coupling between the relative and total density fluctuations of the two split condensates. Recall that the relative fluctuations are produced in a pure state by the splitting scheme while the total density fluctuations had already existed in the original single condensate and presumed to be thermal. Thus the non-linear terms give rise, effectively, to coupling with a thermal bath.

ii.3.1 Self consistent phonon damping

The long time limit of the evolution of the relative phase due to the equilibration with the thermal bath was considered by Burkov et. al in Ref. Burkov et al. (2007). The central assumption in this approach is that the relative mode has nearly reached thermal equilibrium with the center of mass modes at a final temperature . We note that the time scale it takes the system to reach this near equilibrium regime is set by different processesArzamasovs et al. (2013), on which I will comment later on.

The damped relative phase mode near thermal equilibrium can be described by phenomenological Langevin dynamics

(15)

Because the system is near thermal equilibrium the noise satisfies the fluctuation-dissipation theorem with . Eq. (15) is simply a model of damped phonons with being the width of the phonon peaks in the structure factor. We expect that the final temperature will be somewhat higher than the initial temperature of the unsplit condensate because of the energy added to the system in the split process. For a sudden split the added energy density is , i.e. chemical potential per healing length.

The kinematic conditions of linearly dispersing phonons in one dimension lead to divergence of the damping rate calculated in to one loop order. A finite result is obtained, however, if a damping rate is inserted into the phonon Green’s function at the outset and then calculated self consistently. Such a self consistent calculation, first done by Andreev Andreev (1980) (see also Burkov et al. (2007)), gives a non-analytic dependence of on :

(16)

This expression is valid for wave-vectors in the linearly dispersing regime, i.e. and for weak interactions so that the Luttinger parameter .

The solution for can be formally written in terms of the Green’s function of the damped Harmonic oscillator

(17)

Where

(18)

Now by plugging in the -correlated noise near equilibrium we can compute the relative phase fluctuations

(19)

Using the result (with ) from Eq. (16) we find

(20)

where we have used the relation valid for weak interaction . This result implies decay of phase coherence as

(21)

with . is the Luttinger parameter in the weak coupling limit. The non analytic time dependence of the phase coherence stems directly from the non analyticity of the momentum dependent damping rate.

It is interesting to note that the decoherence driven by thermal fluctuations (21) , being a stretched exponential, is slower than the dephasing driven by quantum dynamics, which, we have seen, depends exponentially on time. The physical reason behind this somewhat counter intuitive result is that the thermally excited phonons provide a damping mechanism that slows down the unitary phase evolution.

ii.3.2 Kardar-Parisi-Zhang scaling

The result (21) relies on the scaling of the phonon damping rate derived using a self consistent diagrammatic approachAndreev (1980). Because the approach is not rigorously controlled it would be good to understand the scaling in a different way.

An alternative viewpoint was provided in Ref. Kulkarni and Lamacraft (2013). In this paper Kulkarni and Lamacraft suggested a possible connection between the one dimensional condensate dynamics at finite temperature and the Kardar-Parisi-Zhang (KPZ) equation commonly used to describe randomly growing interfaces. If indeed the dynamics belonged to the same universality class it would immediately imply as a consequence the anomalous dynamical scaling obtained in the self consistent calculation.

Following Ref. Kulkarni and Lamacraft (2013) we illustrate the relation to KPZ dynamics starting from the hydrodynamic equations of the condensate

(22)

These are the Euler and continuity equations with the velocity field the density and the contact interaction. Linearizing these equations with respect to the velocity field and the deviation from the average density, leads to a wave-equation describing phonons of the system moving to the right or left at the speed of sound . Note that beyond the linear approximation, the local speed of sound of the fluid depends on the local density . Now define the ”chiral velocities” as the fluid velocities measured with respect to the local sound velocity of left and right moving waves: . In terms of the chiral velocities the equations of motion can be written as:

(23)

Kulkarni and Lamacraft pointed to the similarity between these equations and two copies of the, so called, noisy Burgers equation. This would be precisely the situation if we could replace the right hand sides of the equations by fields providing Gaussian white noise to the left hand sides. As it stands however, we can think of the left moving modes , randomly occupied with the Bose distribution at temperature , as a random force field acting on the right movers and similarly of the right movers as providing the noise to the dynamics of the left moving modes. With this approximation the coupled equations reduce to two effectively independent noisy burgers equations.

The connection between the noisy burgers equation and the KPZ equation is well known. It is easily demonstrated by representing the chiral velocities as spatial derivatives of chiral phases , just as the actual velocity is related to the condensate phase . Plugging these relations into the Burgers equations and integrating over gives . There is also a dissipative term generally present in the KPZ equation. It is absent here because we started from non dissipative Hamiltonian dynamics. Dissipation however would be generated upon coarse graining because of the coupling between high and low wave vector modes induced by the non linearity. Hence in the long wavelength limit we expect the dynamics to be governed by the KPZ equation:

(24)

The height field (here the chiral phases ) is known to obey the dynamic scaling . Hence also the phase field obeys the same scaling, which leads to the stretched exponential decay of the coherence (21).

ii.4 Discussion and experimental situation

Above we have arrived at two seemingly conflicting results. The harmonic fluid theory of quantum phase diffusion predicts exponential decay (8) of phase coherence in a long one dimensional system, while the hydrodynamic theory of thermal relaxation predicts a stretched exponential decay (21). Which will be seen in experiments? Direct comparison between the thermal and quantum time scales and suggests that as long as the temperature of the relevant bath is smaller than the chemical potential then the quantum phase diffusion should dominate. By the time the thermal relaxation sets in the phase system is then essentially dephased.

Moreover, the calculation of the thermal phase relaxation time summarized in the previous section assumes that the process can be described in terms of an effective hydrodynamic theory of a nearly thermalized system. However the system in question is initially very far from equilibrium and, being a one-dimensional system of bosons with short-range interactions, it is very nearly described by an integrable model (Lieb-Liniger model). Clearly the equilibration rate should be proportional to some measure of integrability breaking, which is absent in the expression for . Breaking of integrability in an elongated Bose gas with a tight transverse confinement stems from effective three particle interactions mediated by virtual occupation of transverse excited statesMazets et al. (2008). Therefore the integrability breaking should be controlled by the ratio of the interaction energy (chemical potential) to the transverse confinement frequency.

Recently, Arzamasovs et. al. Arzamasovs et al. (2013) computed the thermalization rate of a nearly integrable, weakly interacting one dimensional Bose gas, by expanding around the integrable Lieb-Liniger model. Their result can be expressed as

(25)

From this it is clear that in the relevant experimental regime of and temperature range the system does not equilibrate. The dynamics should be well described by quantum phase diffusion leading to a prethermalized state.

The first experiments, which measured the dephasing dynamics seemed to contradict this conclusion, reporting initially a streched exponential decay of the phase coherenceHofferberth et al. (2007) with the exponent expected from the thermal dephasing process. However, improved experiments of the same group done with a very similar apparatus found excellent agreement with prethermalization driven by the harmonic fluid dynamicsGring et al. (2012) (See Ref. [Smith et al., 2013] for a detailed reanalysis explaining the problems in the original experiments, which led to the missnterpretation).

Iii Dynamics of ultra-cold bosons in optical lattices

The natural regime of ultra-cold Bose gases is that of weak interactions. In this regime, the gas is well described by the dynamics of a non-fluctuating classical field through Gross-Pitaevskii equation. Even in low dimensions, where fluctuation effects are inevitably important at long wavelengths they are hard to observe in small traps. One approach to enhance fluctuation effects and potentially reach novel quantum phases and dynamics is to load the atoms into optical lattice potentials generated by standing waves of laser light.

The lattice has two important effects on the quantum gas. One effect is to increase the effective mass of the atoms and thereby quench the kinetic energy with respect to the interactions. The second important effect is to break Galilean invariance. This liberates the quantum gas from strong constraints, thus opening the way to realizing alternative quantum phases and new modes of dynamics. In this regard the observation of a quantum phase transition of bosons in an optical lattice from a superfluid to a Mott insulator Greiner et al. (2002a) has been an important milestone in the study of ultra cold atomic systems. This experiment has lead to intense theoretical and experimental work to understand the dynamics of strongly correlated lattice bosons, and later also of fermions.

The superfluid to insulator transition of lattice bosons was first discussed by Giamarchi and SchulzGiamarchi and Schulz (1988) in the context of one dimensional systems and soon after by Fisher et. al. in Ref. Fisher (1992) for higher dimensional systems. Investigation of this physics with ultra-cold atoms in optical lattices was first proposed by Jaksch et. al. Jaksch et al. (1998). In most of this section I review the basic theory of the superfluid to Mott insulator transition and describe recent theoretical and experimental work, which advanced our understanding of the universal dynamical response near the critical point. Discussion of far from equilibrium dynamics in this regime is postponed to sections III.3 and IV.

iii.1 Mott transition in two and three dimensional lattices

Figure 3: Zero temperature phase diagram of the Bose Hubbard model. The grey areas are the incompressible Mott phases with integer filling . The horizontal dashed lines (red) correspond to phase transitions occurring at constant commensurate filling; they are described by the relativistic critical theory (32). In contrast, the transitions tuned along the vertical dashed lines (blue) are described by the Gross-Pitaefskii action (29), which also describes the weak interaction regime. The latter can be thought as being near a transition tuned by chemical potential from a vacuum state to the weakly interacting BEC.

Superfluidity in two and three dimensions is perturbatively stable to introducing a weak lattice potential regardless of the interaction strength and independent of the commensurability of the potential with the particle density. To allow the establishment of an insulating phase it is therefore important to work with a strong lattice potential with wide gaps between the lowest and the second Bloch bands. In this case, if both the temperature and the interaction strength are smaller than the band gap, we can use the well known Bose Hubbard model as a microscopic description of bosons in the lowest Bloch band

(26)

The chemical potential is used to control the average lattice filling . For simplicity I will not consider here the effects due to the global trap potential. The values of model parameters expressed in terms of the microscopic couplings are given by:

(27)
(28)

where is the wave vector of the lattice, the -wave scattering length corresponding to the two body interaction between atoms and is the recoil energy. For details on the effect of the trap and derivation of the model parameters I refer the reader to Refs. Jaksch et al. (1998); Bloch et al. (2008). Where the parameter values are

The fact that there must be a phase transition in the ground state of the model as a function of the relative strength of interaction at integer lattice filling can be established by considering the two extreme limits. First consider the limit of weak interactions. For vanishing interaction strength (and all the bosons necessarily condense at the bottom of the tight binding band. Weak interactions compared to the band-width, i.e. cannot excite particles far from the band bottom. Therefore in this regime the system can be described by an effective continuum action (Gross-Pitaevskii)

(29)

where is the effective mass in the lowest Bloch band, and is the lattice constant. Hence the physics in this regime is identical to that of a weakly interacting superfluid in the continuum. The effect of the lattice is only to renormalize parameters.

Consider now the opposite regime of strong interactions or small hopping . In particular let us start with the extreme limit of decoupled sites (i.e. vanishing hopping ). At integer lattice filling the system of decoupled has a unique ground state where each site is occupied by exactly bosons. The elementary excitations above this ground state are gapped (with a gap ), and consist of particles (sites with bosons) and holes (sites with bosons). The gap makes this state robust against introducing a small hopping matrix element . The elementary excitations gain a dispersion with band width . However these excitations remain gapped and retain their charge quantum number as long as .

The gap to charge excitations implies that the zero temperature Mott phase is incompressible. A chemical potential couples with an opposite sign to particle and to hole excitations, decreasing the gap of one species while increasing it for the other. But as long as both excitations remain gapped, the lattice filling cannot change with chemical potential. Thus the compressibility .

From the above consideration one can infer that the Mott phases are established as lobes in the space of chemical potential and hopping , characterized by a constant integer filling (see Fig. 3). Upon changing the chemical potential the upper (or lower) boundary of the phase is reached when either the particle (or hole) excitations condense. The density begins to vary continuously upon further increase (or decrease) of the chemical potential beyond the boundary with the compressible phase. The critical theory, which describes this transition, tuned by the chemical potential, is just the field theory (29) where the Bose field here describes the low energy particle or hole excitations at the upper or lower phase boundary respectively.

The transition can also be tuned by varying the tunneling strength (or the interaction) at a fixed integer density. In this case the particle and hole excitations must condense simultaneously, which enforces particle hole symmetry at the critical point. This in turn implies emergent relativistic (i.e. Lorentz invariant) theory in the vicinity of the quantum phase transition, which in the gapped phase indeed must describe particle and anti-particle excitations of equal mass.

More directly, emergence of Lorentz invariance at the tip of the Mott lobe can be established as follows (see e. g. Sachdev (2001)). We write down a more general theory than (29)

(30)

In general if both and are non vanishing we can neglect at the critical point as it is it is irrelevant by simple power-counting compared to the term. However we now show that must vanish at the tip of the Mott lobe leaving us with a relativistic theory. By requiring invariance under the uniform Gauge transformation (i.e. redefinition of the energy):

(31)

we find that . Thus at the tips of the Mott lobes where reaches a minimum as a function of , the coefficient must vanish. Although the special transition at the tip may seem highly fine tuned it is actually realized naturally in a uniform system with a fixed (integer) average filling.

iii.2 The Higgs resonance near the superfluid to Mott insulator transition

Figure 4: Mexican hat potential that describes the dynamics of the order parameter in the superfluid phase. The longitudinal and transverse fluctuations and are two independent modes in the effective relativistic description of the critical point (32), the gapless Goldstone mode and the Gapped Higgs excitation. In the Gross-Pitaevskii action (29) by contrast, the same two fluctuations form a canonically conjugate pair and therefore make up a single mode, the Goldstone mode of the superfluid.

We now discuss how the existence of a quantum phase transition from a superfluid to a Mott insulator impacts the dynamics of bosons in an optical lattice. As discussed above, near the critical point, where the diverging correlation length far exceeds the lattice spacing, the dynamics is described by an effective relativistic field theory

(32)

where is the sound velocity. One of the most interesting consequences of this emergent critical theory is the appearance of a new gapped excitation analogous to the Higgs resonance in particle physics.

The collective mode structure in the superfluid phase near the Mott transition can be understood by considering the classical oscillations of the order parameter field about its equilibrium broken symmetry state. Note that both the critical action (32) valid near the superfluid to Mott transition at integer filling and the Gallilean invariant action (29) of the weakly interacting condensate describe the motion of the order parameter field on a mexican hat potential as illustrated in Fig. 4. Naively it looks like in both cases there are two modes, a soft Goldstone mode corresponding to fluctuations of the order parameter along the degenerate minimum of the potential and a massive fluctuation of the order parameter amplitude. This static picture is misleading. It hides a crucial difference in the dynamics which stems from the different kinetic terms.

In the Gallilean invariant (Gross-Pitaevskii) dynamics the amplitude of the order parameter is proportional to the particle density, i.e. . Plugging this into the kinetic term of (29) we get . Hence the amplitude and phase mode are in this case a canonical conjugate pair which together make up only one excitation mode, the gapless Goldstone mode (or phonon) of the superfluid. On the other hand, in the relativistic theory the phase and amplitude fluctuations are not canonical conjugates. This is easy to see by expanding in small fluctuations around a classical symmetry breaking solution. We take , where is a real classical saddle point solution and the real fields and represent fluctuations in the massive and soft directions of the potential. A quadratic expansion of the action (32) in these fluctuation fields gives

(33)

Hence in the relativistic theory (33) and represent two independent harmonic modes: a massless Goldstone mode and a massive amplitude mode with a mass that vanishes at the critical point. The decoupling between phase and amplitude (at least in the quadratic part of the theory) is possible because near the Mott transition the order parameter amplitude is unrelated to the particle density. Indeed the amplitude vanishes even as we tune the system to the Mott transition at constant particle density.

The order-parameter amplitude mode is closely analogous to the famous Higgs particle in the standard model of particle physics. In the standard model the order parameter is charged under a local Gauge symmetry, therefore the Goldstone mode is replaced, through the Anderson-Higgs mechanism by a massive Gauge boson. What is left of the order parameter dynamics is then only the amplitude fluctuations, which make up the Higgs Boson. In our case because the condensate is Gauge-neutral there is no ”Higgs-mechanism”, Goldstone modes remain and coexist with the Higgs amplitude mode.

The gapless Goldstone modes provide a decay channel, for example through the coupling obtained upon expanding the original action to cubic order, which can lead to broadening of the Higgs resonance. This observation naturally raises the question if the amplitude mode is observable as a sharp resonance near the critical point. The answer to this question, which turns out to be interesting and subtle was formulated only recently in Refs.Podolsky et al. (2011); Podolsky and Sachdev (2012); Gazit et al. (2013). The theory was confirmed when the Higgs was first observed in a system of ultra-cold atoms near the superfluid to Mott insulator transitionEndres et al. (2012). Below I review the theoretical understanding and the measurement scheme used to detect and characterize the Higgs mode.

As a first attempt to address the questions of decay of the Higgs mode and asses if it is visible as a peak in some linear response measurement we may be tempted to compute the self energy of the longitudinal mode . To lowest order in the cubic coupling the Matsubara self energy is given by

(34)

In real frequency this implies in two dimensions. One may worry that the low frequency divergence in the above self energy would wash away or completely mask any peaked response associated with the Higgs mode. But this conclusion is incorrect. As I explain below, the above self energy does not reflect the intrinsic decay of the amplitude mode.

To clarify whether a particular mode would show up as a resonance in a linear response measurement it is important to first specify what is the perturbation the system is responding to. A common probe in some systems with broken symmetry is one that couples directly to the order parameter. We can write a perturbation of this type as , where is a vector (e.g. magnetic) order parameter. In magnetic systems, Neutron scattering acts as a vector probe of this type. In the case of a superfluid we can think of the complex order parameter as a two component vector . The component of the external vector field acting parallel to the ordering direction (longitudinal component) couples to the amplitude fluctuation . Thus the self energy (34), with infra-red divergent spectrum, is related to the linear response of the system to a longitudinal vector probe, i.e. . But the vector perturbation is not a natural one to apply in ultra-cold atomic systems because such a perturbation has to violate charge conservation.

A simple and direct scheme to measure dynamical response of ultra-cold atoms involves periodic modulation of the optical lattice potentialSchori et al. (2004). For atoms in the lowest Bloch band, described by the Hubbard model (26), the lattice modulation translates to a modulation of the hopping amplitude . Since the lattice modulation does not break the symmetry this probe can only couple to scalar terms in the critical action (32). In particular because the lattice strength is the microscopic parameter used to tune the transition, its modulation translates to a modulation of the tuning parameter , i.e. the perturbation Hamiltonian is .

In order to describe the response to the scalar probe it is convenient to use the polar representation of the order parameter . At quadratic order the amplitude and phase fluctuations and play the same role as the longitudinal and transverse fluctuations and respectively. The scalar probe couples linearly to in the same way as the longitudinal probe couples to . However the cubic coupling of the amplitude mode to the phase fluctuations is of the form , i.e. the amplitude fluctuations couples to gradients of the phase and not directly to the phase. Indeed this must be the case by the symmetry. The self energy of the scalar amplitude fluctuation is then given by a similar loop diagram as that of the longitudinal fluctuation (34), but the additional gradients now cancel the denominators to give an infrared convergent result .

It is instructive to reconsider the longitudinal susceptibility in the polar representation as well. As before let us assume, without loss of generality, that the broken symmetry order parameter is real. The longitudinal fluctuation of the order parameter can be written directly in the polar representation

(35)

From this expression it is clear that the longitudinal fluctuation is not a pure amplitude fluctuation ; it is contaminated by pairs of Goldstone modes through the term. A longitudinal probe therefore directly excites this gapless continuum leading to the infrared divergent response we have obtained above. Compared to the longitudinal probe, the more physical scalar probe also gives a cleaner signature of the Higgs amplitude mode Podolsky et al. (2011).

Having shown that the response to a scalar probe is infra-red convergent, I now briefly review what is known about the full line-shape and how the Higgs resonance manifests in it. Of particular interest is the question whether a peak associated to the Higgs resonance appears in the scaling limit. We broaden our discussion a bit and consider the response to the scalar probe on both sides of the critical point. In the Mott side there is only a gapped excitation at , which therefore cannot decay and corresponds to a real pole at . We shall now use rather than to parameterize the deviation from the critical point on both sides of the transition.

Figure 5: (a) Energy of the Higgs excitation in the superfluid phase () and of the gapped particle-hole excitation in the Mott insulator () measured in Ref. Endres et al. (2012) shows softening of the modes on approaching the critical point. (b) Examples of the measured spectra. The Higgs mode is seen as a sharp leading edge rather than a peak because of in homogenous broadening in the harmonic trap. Figure reproduced fro Ref. Endres et al. (2012).

The pertinent question concerning the response in the (ordered) superfluid side of the transition is whether a peak appears also on this side in the scaling limit. If it does then the frequency at the peak must also vanish as upon approaching the critical point. Equivalently, the scalar susceptibility of the superfluid is expected to follow a scaling form with a universal scaling function with a peak at (see Podolsky and Sachdev (2012)). Theoretical analyses using Recent numerical resultsPollet and Prokof’ev (2012); Gazit et al. (2013); Rançon and Dupuis (2014) as well a theoretical analysisPodolsky and Sachdev (2012); Katan and Podolsky (2015) indeed find such a a scaling form. It is important to note, however, that neither method is fully controlled. In particular, the quantum Monte-Carlo simulations performed for this system can give a Matsubara response function, while the analytic continuation to real frequencies requires uncontrolled approximations or otherwise is an exponentially hard problem. Therefore an experimental test of the theoretical and numerical predictions is needed.

An experiment aimed at testing some of the above predictions was reported in Ref. Endres et al. (2012) using bosons in a two dimensional optical lattice. Here, the heating rate due to weak lattice modulations was measured to high accuracy. Typical response spectra and the characteristic mode frequency obtained from them are reproduced here in Fig. 5. Note that these spectra exhibit a sharp edge followed by a continuum rather than a peak. This behavior is associated with the inhomogeneity of the trap. The leading edge stems from the response in the middle of the trap, which is closest to the critical point. The scaling of the leading edge upon approaching the transition from either side of it agrees quantitatively with the numerical predictions. In particular the ratio of excitation frequency in the superfluid to that in the Mott side is found to be , consistent with state of the art numerics Gazit et al. (2013); Rançon and Dupuis (2014) and markedly different from the mean field prediction . The conclusion from the theoretical analyses and the experiment is that the Higgs mode in two spatial dimensions is visible and the spectral peak associated with it survives in the scaling limit.

iii.3 Dynamical instability and decay of super flow in an optical lattice

Bosons tend to condense and become superlfuids at low temperature. In the last section we have seen that bosons in an optical lattice can lose their superfluid properties at zero temperature through phase quantum transition into the Mott insulating state. I have discussed how this transition manifests in the dynamical linear response of the system at equilibrium. In this section I will discuss the breakdown of superfluidity that occurs due to flow of a super current in the lattice. This is a highly non-equilibrium route for breakdown of superfluidity. In particular I will focis on the interplay between the Mott transition and the critical current in the superfluid phase.

If a superfluid in a Galilean invariant system is carrying a current then this current can be removed by moving to the Galilean frame of the superfluid. Hence without disorder or a lattice potential to break translational symmetry, a state with any magnitude of current is necessarily stable . Here I will focus on the effect of the lattice. In particular I will ask what is the critical current above which superfluidity breaks down in the different regimes and under what conditions this is indeed the critical current is sharply defined.

iii.3.1 Mean field critical current

Let me start with a mean field description of super-currents. Consider a condensate described by the order parameter , where is a lattice site. A state with uniform current, say in the direction, is described by a uniform phase twist, . The super-current density is then

(36)

where is the effective mass of a particle on the lattice, and is the phase change across a link in the direction.

By looking into this expression we can anticipate many of the results that I will discuss in more detail below. In a weakly interacting condensate at the superfluid density is just the total atom density , essentially independent of the phase twist . Then the maximal current that the system can carry is and it occurs when the phase twists by over a lattice constant (). This result can be understood by considering the single particle dispersion , where is the lattice momentum. A condensate with phase twist is obtained by condensing the bosons into the state with lattice momentum . When exceeds the local effective mass at that momentum turns negative. The situation near the Mott transition is different. The system there is highly sensitive to increase of the effective mass , which leads to vanishing of . It would , in the same way, be sensitive to increasing the local mass . Hence in this regime is a decreasing function of and we reach the critical (maximal) current at a much smaller value of the phase twist .

To see how the instability occurs in the weakly interacting limit we have to consider the relevant equation of motion in this regime, which is the lattice Gross Pitaevskii equation:

(37)

This mean field description provides an accurate description of the dynamics for . The idea now is to linearize the equation in the fluctuations and around the uniform current solution . For solving these linear equations simply gives the linearly dispersing phonons (Bogoliubov modes ) of the superfluid. However, beyond the critical value there are eigenmodes that take imaginary values, indicating a dynamical instabilityWu and Niu (2001).

It is interesting to convert the critical current for dynamical instability to a critical flow velocity. In doing so we get . This is much larger than Landau’s criterion, which gives the much smaller sound velocity, as the maximal stable flow velocity. In terms of the collective modes, Landau’s criterion corresponds to the point where the excitation with negative momenta turn negative because of the doppler shift. Negative frequencies, unlike imaginary ones, do not in by themselves imply an instability. The idea of Landau’s criterion is that if we add static impurities to the system, then they will induce scattering that creates pairs of negative and positive frequency modes that will lead to decay of the super current. But in a pure lattice system, the current will decay only when the dynamical instability is reached. The dynamical instability in a weakly interacting lattice condensate has been seen experimentally in Ref. Fallani et al. (2004).

Let us now turn to the strongly interacting regime, near the Mott transition, where the correlation length . Here the dynamics is described by the effective continuum relativistic action (32). Treating this theory within the mean field approximation, we obtain the saddle-point equation of motion

(38)

where I have rescaled time and the field: , and . In these units we can write the correlation length as . This equation admits the stationary solutions:

(39)

where denote the spatial coordinates transverse to . From this it is obvious that the solutions disappear and therefore the current carrying states cannot be stable for phase twist . To find the precise critical twist for a dynamical instability I expand Eq. (38) following Ref. Altman et al. (2005) in small fluctuations around the stationary solutions. In the long wavelength limit this gives two modes with frequencies:

(40)
(41)

The first mode is the generalization of the amplitude (Higgs) mode to the finite current () situation. It is gapped and has positive frequency unless . The second mode is the gapless phase (phonon) mode. This mode becomes unstable for . We see that the critical momentum vanishes as upon approaching the critical current.

The behavior of the critical current near the Mott transition in a three dimensional optical lattice has been investigated in experimentMun et al. (2007). A current was induced by moving the optical lattice, whereas the main observable was the existence of a sharp peak in the momentum distribution function as measured in a standard time of flight expansion scheme. The critical momentum was the was determined as the flow momentum at which the condensate peak disappeared from the time-of flight absorption image. Fig. 6 reproduced from Ref. Mun et al. (2007) shows the measured critical current versus the lattice depth, which is in excellent agreement with the mean field result discussed above.

Figure 6: Measured critical momentum (phase twist) as a function of the lattice strength near the superfluid to Mott transition. The red line is the mean field approximation for the critical momentum discussed in this section (Adapted from Ref. Mun et al. (2007)).

iii.3.2 Current decay below the critical current

So far we discussed the breakdown of superfluidity at the critical current within a mean field analysis. This seems to be a good enough approximation to describe experiments in three dimensional lattices Mun et al. (2007). However it is of fundamental interest to ask whether there is a possibility for current to decay, due to fluctuations, even when the flow is slower than the critical flow.

The fact that the flow below the critical current is a stable solution of the classical equation of motion implies that in order to decay, the field configuration must tunnel through a classically forbidden region of phase space. In field theory such tunneling of a macroscopic field configuration out of a meta-stable state was first termed by Coleman as the ”Fate of the false vacuum”Coleman (1977a) (see also the erratum Coleman (1977b) as well as the followup paper discussing quantum corrections to the semiclassical theory Callan Jr and Coleman (1977)). In our case, the false vacuum is the state with a twist; it must go over an action barrier corresponding to creation of phase slip or soliton in space-time to unwind the twist. A similar approach was taken earlier to describe the current decay due to thermal activation of phase slips in superconducting wires below the critical currentLanger and Ambegaokar (1967); McCumber and Halperin (1970)

For now, calculation of the tunneling probability of a multi-dimensional field configuration looks like a formidable problem. However, following Ref. Polkovnikov et al. (2005) we will simplify it by restricting ourselves to the asymptotic behavior of the decay rate near the classical critical current, i.e. as from below, where the action barrier is low. I will show that in this regime the pertinent information about the solution can be extracted from a simple scaling ansatz.

To understand how the scaling approach works consider first the toy problem of escape from a metastable state in single particle quantum mechanics (or field theory in dimensions). As usual to facilitate a semiclassical approximation of the tunneling path we rotate the action to imaginary time

(42)

Here is the potential, which for a small barrier can generically be written as a cubic function . The limit , at which the barrier vanishes and the particle at becomes classically unstable, is analogous to reaching the classical critical current in the field theory above. Within the semi-classical theory the decay rate is given by where is the saddle-point action and is an ”attempt-rate” obtained from integrating over the Gaussian fluctuations around the Saddle-point (see e.g. Alexander and Simons (2010) for a complete pedagogical treatment of this problem).

We now pull the dependence out of the action (42) by applying the following rescaling and . This leads to the action

(43)

Consequently the decay rate in the semi-classical approximation is given by , where is just a number independent of . In this way we have managed to obtain the parametric dependence on the vanishing barrier hight without solving the saddle-point equations. This will be needed only to obtain the number . In the case of the toy model this number can be computed exactly because there is an exact solution to the saddle-point equations, while in more interesting situations it can be approximated using a variational ansatz.

Let me now turn to the real situation at hand. The first task is to write down an action, analogous to (42) that can describe the tunneling of the field configuration out of the current carrying state close to the critical current. I start with effective action of the Mott transition in imaginary time, using units such that . Furthermore I rescale all lengths and times with the correlation length, i.e. . In these units

(44)

Next, following Polkovnikov et al. (2005) we express using the fluctuations around the stable current carrying state with phase twist (note that I am using which describes the twist, or momentum, in units of appropriate for the above action)

(45)

Here the coordinate is the direction of the current, while denotes all other coordinates, including imaginary time, transverse to the current. In order to allow possibility for tunneling across the small barrier we must expand the action to cubic order in the fluctuations and . The two fluctuations are decoupled at the quadratic level after the transformation

(46)

Now the gapped amplitude fluctuation can be disregarded and we are left with a cubic action that describes the action barrier in terms of the phase mode

(47)

where we have denoted . After apply the rescaling

(48)

we obtain to leading order in

(49)

From this we conclude that the current decay rate is

(50)

where is a number, to be discussed below, that can be obtained with a variational calculation. The physical picture behind this scaling solution is the following. The rescaling was done after identifying natural scales in the action (47) that become singular at the classical critical twist. The natural length in the direction of the current is and in the transverse and time direction it is . Rescaling with these lengths amounts to the postulate that the critical instanton has these spatial extent in the respective directions of space time. In addition the energy barrier scales as . Putting all this together we see that the instant on action must behave as .

The precise value of the action depends on the detailed shape (functional form) of the instanton, which is encapsulated in the constant . These constants have been obtained using a variational calculation in Ref. Polkovnikov et al. (2005). The results are