Kilohertz-driven Bose-Einstein condensates in optical lattices
We analyze time-of-flight absorption images obtained with dilute Bose-Einstein condensates released from shaken optical lattices, both theoretically and experimentally. We argue that weakly interacting, ultracold quantum gases in kilohertz-driven optical potentials constitute equilibrium systems characterized by a steady-state distribution of Floquet-state occupation numbers. Our experimental results consistently indicate that a driven ultracold Bose gas tends to occupy a single Floquet state, just as it occupies a single energy eigenstate when there is no forcing. When the driving amplitude is sufficiently high, the Floquet state possessing the lowest mean energy does not necessarily coincide with the Floquet state connected to the ground state of the undriven system. We observe strongly driven Bose gases to condense into the former state under such conditions, thus providing nontrivial examples of dressed matter waves.
keywords:Bose-Einstein condensates, optical lattices, time-periodic forcing, Floquet theory, dressed matter waves, periodic thermodynamics
Pacs:03.75.Lm, 67.85.De, 67.85.Hj, 05.30.Jp
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There is a growing interest in ultracold atoms confined in time-periodically driven optical lattices. It was pointed out already in 1997 that a metal-insulator-like transition undergone by ultracold atoms in quasiperiodic optical lattices can be induced by changing the amplitude of a time-periodic driving force (Drese and Holthaus, 1997). But the field became truly active only 10 years later, after the unambiguous experimental observation of dynamical tunneling suppression with Bose-Einstein condensates, and even of driving-induced reversal of the sign of the tunneling matrix element, by the Pisa group (Lignier et al., 2007), and after related single-particle experiments performed with driven double-well potentials in Heidelberg (Kierig et al., 2008). Several groups then have monitored the dynamics under the combined action of both a homogeneous time-independent and a time-periodic driving force, either with non-condensed cold atoms (Ivanov et al., 2008; Alberti et al., 2009), or with Bose-Einstein condensates (Sias et al., 2008; Haller et al., 2010); such dynamics may be viewed as an analog of photon-assisted tunneling (Eckardt et al., 2005a). The Pisa group subsequently also reported the coherent control of the superfluid-to-Mott insulator transition in shaken three-dimensional optical lattices (Zenesini et al., 2009).
The recent detection of spontaneous breaking of time-reversal symmetry with fast-oscillating triangular optical lattices by Struck et al. (2011) constitutes a further milestone in this line of research. Moreover, photon-assisted tunneling in a strongly correlated Bose gas has very recently been investigated, possibly allowing for applications to topological physics and quantum computing (Ma et al., 2011). Also active control of correlated tunneling in ac-driven optical superlattices has been observed, providing a novel approach to the realization of spin models (Chen et al., 2011).
On the theoretical side, several studies indicate promising future perspectives: Eckardt and Holthaus (2008b) have suggested to perform avoided-level-crossing spectroscopy with ultracold Bose gases in strongly driven optical lattices, while Eckardt et al. (2010) have worked out a proposal for simulating frustrated quantum antiferromagnetism with oscillating triangular lattices. Later Tokuno and Giamarchi (2011) have pointed out that a small periodic phase modulation of an optical lattice can give direct access to the system’s conductivity. Furthermore, Tsuji et al. (2011) have argued that ac forcing may even change the interparticle interaction from repulsive to attractive, thereby enabeling one to simulate an effectively attractive Hubbard model with a temperature below the superconducting transition temperature.
On a first superficial glance, ultracold atoms in shaken optical lattices, with shaking frequencies on the order of a few to tens of kilohertz, might appear as typical examples of nonequilibrium systems. But this naive view is not correct. Quantum systems governed by a time-periodic Hamiltonian possess a particular basis, consisting of Floquet states, with respect to which one finds occupation numbers which remain constant in time, with the periodic time-dependence already being incorporated into the basis states themselves. Thus, a weakly interacting quantum gas subjected to time-periodic forcing, and possibly exposed to some sort of noise, will be statistically characterized by a time-independent distribution of Floquet-state occupation numbers. What is that equilibrium distribution?
In the present tutorial article we develop this line of thought, and discuss experimental data, obtained with Bose-Einstein condensates in kilohertz-shaken optical lattices, which we interpret as evidence for Floquet-based dynamics. We proceed as follows: In Sec. 2 we review basic elements of the Floquet picture, and speculate about the existence of analogs of the Bose-Einstein and the Fermi-Dirac distribution function for isolated, time-periodically driven quantum gases. We then turn in Sec. 3 to the realization of periodically shaken optical lattices, extending the analysis of Madison et al. (1997), and pay particular attention to the distinction between the co-moving frame of reference, in which the trapped quantum gas experiences a spatially homogeneous inertial force, and the laboratory frame, in which the measurements are performed. In Sec. 4 we briefly introduce the driven Bose-Hubbard model (Eckardt et al., 2005b). In Sec. 5 we calculate momentum distributions expected in time-of-flight absorption imaging for matter waves occupying a single Floquet state. We consider two particular situations: An interacting gas periodically driven at high frequencies, and a noninteracting gas subjected to a force with arbitrary time-dependence. Here we also point out certain characteristic differences between time-of-flight absorption images obtained from shaken lattices utilizing the inertial force, and the corresponding images acquired when the lattice remains at rest, while the force is exerted, e.g., through the harmonic modulation of a levitation gradient (Haller et al., 2010). In Sec. 6 we tie the various strands together and report our measurements, which suggest the existence of Floquet condensates occupying the “lowest” available Floquet state. While we consider only one comparatively simple experimental setting, it appears likely that our findings are of a more general nature. In the final Sec. 7 we briefly spell out our main conclusions.
2 The quest for Floquet condensates
The notion of Bose-Einstein condensation of an almost ideal gas (Einstein, 1924, 1925; London, 1938) appears to be inextricably linked to the energy eigenstates of some single-particle Hamiltonian: When the temperature of a gas of weakly interacting Bose particles becomes lower than a certain critical temperature, the gas “condenses” into the single-particle ground state, meaning that this ground state becomes macroscopically occupied. It might seem that this concept cannot be applied to explicitly time-dependent systems, for which, in general, there are no stationary states. Yet, there is one notable exception: If a single-particle Hamiltonian is periodic in time, so that for some period , then Floquet’s theorem (Floquet, 1883; Kuchment, 1993) suggests the existence of a set of distinguished solutions , known as Floquet states, to the time-dependent Schrödinger equation ; these Floquet states are in many ways analogous to the usual energy eigenstates of time-independent Hamiltonian operators (Shirley, 1965; Zel’dovich, 1967; Ritus, 1967; Sambe, 1973; Breuer and Holthaus, 1991; Chu and Telnov, 2004). They have the particular form
where the functions inherit the period of the underlying Hamiltonian; the index denotes a set of quantum numbers specifying the state. The quantities , which determine the phase factors accompanying the time-evolution of the Floquet states (1) in the same manner as energy eigenvalues of time-independent systems determine the phase factors expressing the evolution of their energy eigenstates, are dubbed quasienergies (Zel’dovich, 1967; Ritus, 1967).
There is a compelling reason for considering these Floquet states. Namely, at each instant the set of Floquet functions is complete in the physical Hilbert space on which acts. Hence, any solution to the time-dependent Schrödinger equation admits an expansion of the form
with time-independent coefficients . Therefore, under conditions of perfectly coherent evolution the occupation probabilities of the Floquet states remain constant, despite the periodic time dependence of the Hamiltonian. These coefficients then keep a memory of how the system has been prepared, that is, how the time-periodic forcing has been turned on in the past, whereas the Floquet states and their quasienergies are independent of this history.
If, however, the evolution is not coherent, but the system is in contact with some sort of thermal environment or heat bath, what would be the equilibrium distribution of the Floquet-state occupation probabilities which establishes itself in the long run, irrespective of the particular initial condition? An important step towards a general answer to this question has been taken by Breuer et al. (2000), who have studied anharmonic oscillators subjected to strong time-periodic forcing while being weakly coupled to thermal degrees of freedom provided by a surrounding, and who have shown that the corresponding quasistationary density matrix actually is diagonal in the Floquet representation. This line of investigation has recently been taken up by Ketzmerick and Wustmann (2010), with a detailed view on the classical-quantum correspondence, after general aspects of such “periodic thermodynamics” had been discussed by Kohn (2001).
Turning now from the single-particle dynamics to that of an ideal gas of identical quantum particles subjected to some periodically time-dependent external influence, such as a dilute gas of Bose particles stored in a confining potential which is modulated periodically in time, an even more intriguing question poses itself: What are the statistics of the Floquet-state occupation numbers? In other words, what is the replacement for the familiar Bose-Einstein (or Fermi-Dirac, when dealing with Fermions) distribution function in a periodically time-dependent setting?
In order to appreciate what lies behind this question, let us briefly retrace the steps which led Einstein (1925) to what is now known as Bose-Einstein statistics, by now having become basic knowledge of statistical physics (Pathria, 1996): Consider an isolated ideal Bose gas consisting of particles possessing a total energy , being kept in some large confinement. In view of the corresponding denseness of the single-particle spectrum we divide the energy axis into small intervals, the th such interval containing single-particle states with practically identical energy . We then denote the number of particles occupying the states in the th interval as , and ask for the most probable set of these cell occupation numbers. All admissible sets obviously have to comply with the two constraints
the latter holding approximately, reflecting the above coarse-graining procedure, but to good accuracy. The number of microstates associated with a given set is
because the binomial coefficients quantify the number of possibilities to distribute indistinguishable particles over states “with repetition”, as is characteristic for Bosons. Assuming that both and , and employing Stirling’s formula, one then obtains the entropy
The searched-for most probable occupation numbers maximize this expression, subject to the two constraints (3) and (4). Introducing a Lagrangian multiplier to account for the conservation of the particle number, and a further multiplier to account for the conservation of energy, we are thus led to the variational problem
After brief calculation, this gives
and thus results in the Bose-Einstein distribution
quantifying the most probable occupation number of an individual single-particle state with energy under microcanonical conditions. Finally, the multiplier is identified with , with denoting Boltzmann’s constant and being the temperature of the gas, while is related to its chemical potential (Einstein, 1925; Pathria, 1996).
When trying to adapt this reasoning to time-periodically driven quantum gases, and to determine the corresponding Floquet-state occupation numbers, we can immediately carry over the first constraint (3): The sum of all occupation numbers has to equal the total number of particles. This may seem trivial, but it is not, because already here we are making essential use of Floquet theory: It is only because the coefficients in the expansion (2) are time-independent that we can assign occupation numbers to the Floquet states; only then can one ask for the associated distribution function.
It is, however, not obvious what becomes of the second constraint (4), because energy eigenvalues do not exist for a time-periodic system, and the quasienergies of the Floquet states cannot be considered as their proper substitutes in this context. To understand the quasienergy concept more deeply, we insert the Floquet states (1) into the time-dependent Schrödinger equation, and deduce
The conspicuous notation employed here, i.e., writing the Floquet functions with a double right angle instead of as in Eq. (1), has a deep significance. Namely, Eq. (10) has to be regarded as an eigenvalue equation for the quasienergies; within the Floquet framework, this eigenvalue equation takes over the role played for time-independent systems by the stationary Schrödinger equation. Unlike the latter, this quasienergy eigenvalue equation (10) lives in an extended Hilbert space of -periodic functions (Sambe, 1973). In contrast to the usual physical setting, in which “time” emerges as an evolution variable, in that extended space time plays the role of a coordinate, and therefore needs to be integrated over in the associated scalar product: Denoting the momentary scalar product of two -periodic functions and in the physical space as , their scalar product in the extended space reads (Sambe, 1973)
Thus, we write when considering a Floquet function in the physical Hilbert space, as in Eqs. (1) and (2), whereas refers to that same function when viewed as an element of the extended space, as in Eq. (10).
This apparently formal observation has an important, physically meaningful consequence. Suppose that is a solution to the eigenvalue equation (10) with quasienergy , define , and let be an arbitrary (positive, zero, or negative) integer. Then also is a -periodic solution, with quasienergy . For these two Floquet functions obviously are orthogonal with respect to the scalar product (11), and represent different eigenfunctions in the extended space. On the other hand, when returning to the physical space and forming the actual Floquet states (1), one has
so that both solutions represent the same physical state. Hence, a Floquet state does not correspond to a single solution to Eq. (10), but rather to a whole class of such solutions, labeled by the index , while individual representatives of such a class are distinguished by the integer . This subtlety already reflects itself in the expansion (2): Although all functions , with ranging over all state labels and extending over all integers, are required for the completeness relation in the extended space, only one representative from each class is required in Eq. (2), where no sum over the “photon index” appears. By the same token, the quasienergy of a physical Floquet state is determined only up to an integer multiple of the “photon” energy . In accordance with the analogous terminology used in solid-state physics, the quasienergy spectrum is said to consist of an infinite set of identical “Brillouin zones” of width , covering the entire energy axis, each state placing one of its quasienergy representatives in each zone. This means, in particular, that there is no natural “quasienergy ordering”: Without additional specification it is meaningless to ask whether one given Floquet state lies “below” another. And there is still a further complication of a more mathematical nature: Because one finds one quasienergy representative of each state in each Brillouin zone, the quasienergy spectrum is “dense”, and it may be technically difficult to decide whether one has a dense pure point spectrum, so that the expansion (2) can be taken literally and the system is stable, possessing a quasiperiodic wave function, or whether there is an absolutely continuous spectrum, allowing for diffusive energy growth (Bunimovich et al., 1991; Howland, 1992a). We bypass this problem by restricting ourselves to stable systems with a pure point spectrum, such as the forced anharmonic oscillators with superquadratic potentials investigated by Howland (1992b).
These considerations clearly reveal that, although there are Floquet-state occupation numbers, the required analog of the energy constraint (4) for time-periodically forced microcanonical close-to-ideal Bose gases cannot involve quasienergies directly. However, each Floquet state possesses a mean energy (Fainshtein et al., 1978)
which obviously is independent of the choice of the representative . If we now repeat the coarse-graining procedure employed in the formulation of the energy constraint (4), that is, if we divide the energy axis into small cells, the th cell now containing Floquet states having almost the same mean energy , and if denotes the corresponding sets of cell occupation numbers, then it seems reasonable to demand
for an isolated system, thus obtaining a constraint which effectively may replace the previous Eq. (4). Here we assume that each driven particle sees the other driven particles as its heat bath, and undergoes subsequent short relaxation events only on a time scale which is large compared to the period . In general, the justification of this constraint may require the consideration of particular physical setups, and some more specifications may be needed. But if we tentatively accept this constraint (14) as it stands, it is immediately clear how to proceed: We assign the entropy (6) to a time-periodically forced isolated Bose gas with Floquet-state occupation numbers , account for the particle number conservation (3) again by introducing a Lagrangian multiplier , and incorporate the conservation of the mean energy (14) with the help of a further Lagrangian multiplier . Then the most probable set of Floquet-state occupation numbers is determined by the variational equation
Because this problem exactly parallels the problem (7) already considered by Einstein (1925), we can directly take over the solution: Under the above conditions, the most probable set of Floquet-state occupation numbers is
is the expected occupation number of an individual Floquet state with mean energy . Of course, this reasoning can easily be adpated to Fermions, leading to the familiar “plus” sign in the denominator (Pathria, 1996).
These deliberations may have some profound physical implications. To begin with, a time-periodically forced, stable Bose gas manifestly does not constitute a nonequilibrium system, as one might naively assume, but rather an equilibrium one, with equilibrium parameters and . Most noteworthy, these parameters depend on the form of the periodic time-dependence, because so do the Floquet states, and hence their mean energies (13), which enter into the modified constraint (14). Therefore, when writing
in analogy to the familiar relation , the parameter introduced here is a temperature-like quantity which can be varied by changing, e.g., the strength of an external time-periodic force; in general, will differ from the usual temperature the gas would have if there were no such forcing. Likewise, when setting
is the corresponding, forcing-dependent chemical potential.
It follows that there are Floquet condensates: If the -temperature is sufficiently low, a time-periodically forced close-to-ideal Bose gas condenses into the single-particle Floquet state possessing the lowest mean energy. This deduction immediately suggests a further possibility: Suppose that the Hamiltonian under consideration has the natural form , where represents a time-periodic force acting on the unperturbed system with adjustable strength. In the regime of perturbatively weak forcing the mean-energy ordering of the Floquet states of is likely to be the same as the energy ordering of the eigenstates of from which they have developed, but in the nonperturbative regime this is no longer guaranteed; here the Floquet state possessing the lowest mean energy not necessarily is connected to the -ground state. Under such conditions the Floquet state into which an ultracold, time-periodically forced Bose gas condenses is determined by the strength of the forcing, and may change when the latter is varied.
We close this section by emphasizing that the above reasoning hinges on two key issues: The assignment of occupation numbers to the Floquet states, which is borne out by theory, and the adoption of the mean-energy constraint (14) as a building principle for a microcanonical Floquet ensemble, which may require further thoughts. Important pieces of evidence supporting our theoretical deductions are provided by the experimental observations reported in Sec. 6 below. In order to facilitate the interpretation of our measurements, we first lay out some important ingredients in Secs. 3 to 5.
3 The experimental setup: Shaken optical lattices
A one-dimensional (1D) optical lattice potential, which we write in the form
is generated by two linearly polarized counterpropagating beams of laser radiation with wave number , its depth being proportional to the laser intensity (Morsch and Oberthaler, 2006). Here the lattice coordinate refers to the laboratory frame of reference. Considering an atom of mass moving in this potential, the lattice depth is measured in units of its single-photon recoil energy ,
typical optical lattices are up to tens of recoil energies deep. For example, when working with Rb in an optical lattice made up from laser light with wavelength nm one obtains eV. Thus, the typical depth of an optical lattice is 11 orders of magnitude lower than that of the lattices encountered in traditional solid-state physics. This implies that the characteristic frequencies of ultracold atoms in optical lattices fall into the lower kilohertz regime.
The pairs of counterpropagating beams needed to create optical lattices in one, two, or three dimensions can either be realized by splitting a laser beam in two, or by retro-reflecting a beam off a mirror. In the former case, a small frequency difference introduced between the two splitted laser beams with the help of acousto-optic modulators makes the lattice move with velocity (Niu et al., 1996; Ben Dahan et al., 1996; Madison et al., 1998; Lignier et al., 2007; Sias et al., 2008). Therefore, in the laboratory frame of reference an atom then experiences the potential
When using retro-reflected beams, lattice motion can be achieved by mounting the mirror on a piezo-electric actuator, so that it shifts according to some prescribed protocol (Zenesini et al., 2009; Alberti et al., 2009). As the shift range of such devices is in the micrometer regime, this method is suitable for inducing an oscillatory motion as needed for shaking lattices, but not for applying a constant acceleration.
In the following steps, the Schrödinger equation for the translational motion of an atom in the shifted lattice, governed by the Hamiltonian
is subjected to a unitary transformation to the frame of reference co-moving with the lattice. First, the corresponding shift in position is implemented by means of the unitary operator
thus leading to the new Hamiltonian
Next, the shift of momentum showing up here is compensated through
Finally, the merely time-dependent energy shift appearing in is removed by the operator
The full transformation to the co-moving frame is then realized by the combined operation , giving
denoting the single-particle Hamiltonian pertaining to the undriven lattice, and
introducing a homogeneous inertial force acting in the co-moving frame, determined by the lattice motion according to
In particular, when a purely sinusoidal frequency shift with an initial phase is suddenly applied at ,
for , so that the lattice is shaken in the laboratory frame with the amplitude
According to Eq. (23), one has . Therefore, this protocol (38) effectuates a sudden jump of the lattice’s velocity at time , unless or . By virtue of Eq. (37), the force felt by the atoms in the co-moving frame of reference then is composed of a monochromatic oscillating drive acting for , and of a delta-like kick acting at the moment of turn-on: Formally, one obtains
where denotes the Heaviside function, and the driving amplitude is given by
Needless to say, in reality the delta-kick experienced at has a finite sharpness, determined by the short-time details of the actual velocity jump. Nonetheless, this kick has an important experimental consequence, as will be discussed later.
As a dimensonless measure of the shaking or driving strength we introduce the quantity
where is the lattice constant. Thus, with driving frequencies on the order of the recoil frequency , and modulation amplitudes (40) on the order of the lattice constant, one can easily reach the nonperturbative regime . It would be quite hard to realize corresponding conditions in laser-irradiated traditional solids without introducing, e.g., additional polarization effects, or even damaging the sample. Therefore, strongly shaken optical lattices may also be viewed as “strong-field simulators” which allow one to study even superstrong-field-induced multiphoton-like processes in periodic potentials, such as interband transitions, in their purest form (Arlinghaus and Holthaus, 2010).
4 The driven Bose-Hubbard model
In principle, the time-periodic motion (39) always induces transitions between the unperturbed energy bands of the optical lattice. However, if the energy scale associated with the modulation frequency remains small compared to the energy gap between the lowest two bands, and if the driving amplitude remains sufficiently low, the dynamics of driven ultracold atoms remain restricted to the lowest band at least to good approximation. Since the band structure of a cosine lattice (20) is determined by the characteristic values of the Mathieu equation (Slater, 1952), the known expansions of these values (Abramowitz and Stegun, 1965) can be employed for obtaining estimates of the gap width, resulting in
for shallow lattices, and
for fairly deep ones. Fig. 1 demonstrates that these two approximations indeed provide a reasonable estimate of the exact band gap for all , if one switches from the shallow-lattice result (44) to the deep-lattice formula (45) at . As a figure of merit, the exact gap width is for a lattice with depth .
In our experiments (Lignier et al., 2007; Sias et al., 2008; Eckardt et al., 2009; Zenesini et al., 2009) we work with Bose-Einstein condensates of Rb consisting of about atoms in a shaken 1D optical lattice ( nm or nm) with depths ranging from 3 to 10, employing scaled shaking frequencies between roughly 0.1 and 2 (corresponding to between 0.3 and 6 kHz). Under such conditions the single-band approximation is viable, provided the forcing is not too strong, since crossing the gap would require higher-order multiphoton-like transitions (Arlinghaus and Holthaus, 2010). In order to model these driven, interacting many-body systems we resort to the co-moving frame of reference, and then follow the standard route (Jaksch et al., 1998; Jaksch and Zoller, 2005; Bloch et al., 2008). Employing a basis of site-localized Wannier functions pertaining to the lowest Bloch band, and assuming a reasonable lattice depth , it suffices to retain only the hopping matrix element connecting neighboring sites. The accuracy of this approximation has been assessed quantitatively by Boers et al. (2007) and by Eckardt et al. (2009); for instance, when the neglected matrix element connecting next-to-nearest neighbors is actually smaller than by a factor of about . Moreover, the van der Waals length of alkali atoms typically amounts to just a few nanometers (Bloch et al., 2008) and thus is significantly smaller than the lattice constant , so that only the on-site interaction among the atoms has to be accounted for; this is done in terms of a parameter which quantifies the interaction energy of one pair of atoms occupying the same lattice site (Schneider et al., 2009). In short, when adopting these three approximations (single-band, nearest-neighbor, and on-site), an ultracold gas of bosonic atoms in a time-periodically shaken optical lattice is described in the co-moving frame by the explicitly time-dependent many-body Hamiltonian111We use the “hat”-symbol to indicate operators acting on the bosonic Fock space (Eckardt et al., 2005b; Creffield and Monteiro, 2006)
is the standard Bose-Hubbard model for a 1D lattice (Fisher et al., 1989; Jaksch et al., 1998; Jaksch and Zoller, 2005), with denoting the bosonic annihilation operator for atoms occupying the Wannier state at the th site; is the number operator for that site. Recalling that the position operator mediating the inertial force built into the single-particle Hamiltonian (36) translates, on the many-body level, into , with for the distance between two sites, the time-periodic forcing now is introduced through
where the strength is related to the forcing amplitude (42) by . Here we disregard an initial phase , which would refer to a sudden turn-on of the force, as in Eq. (38). This driven 1D Bose-Hubbard model (46) may be extended to higher spatial dimensions (Zenesini et al., 2009); we remark that approximate expressions relating the parameters and to the lattice depth have been provided by Zwerger (2003) and by Bloch et al. (2008).
5 Interference patterns produced by Floquet states
It is essential to observe that the existence of Floquet states (1) hinges solely on the periodicity of the given Hamiltonian in time. Thus, besides the single-particle Floquet states which form the basis of the statistical considerations in Sec. 2, there also are Floquet states for periodically time-dependent, interacting many-body systems, such as periodically driven Bose gases, which incorporate both the periodic time-dependence and all interaction effects.
5.1 Signatures of interacting shaken Bose gases
Here we are concerned, in particular, with the Floquet states of the driven Bose-Hubbard model (46), and their experimental signatures. In order to solve the quasienergy eigenvalue problem (10) for this model, the quasienergy operator has to be diagonalized in the associated extended Hilbert space (Sambe, 1973), as reviewed in Sec. 2. To this end, let denote an admissible set of site-occupation numbers . The physical many-body Hilbert space then is spanned by the set of all Fock states
where is the “empty-lattice” state. A possible basis of the extended space would be provided by the products with integer . However, it is more useful here to employ the basis of Floquet-Fock states (or “dressed Fock states”) given by
which already diagonalize for vanishing interwell hopping strength (Eckardt et al., 2005b; Eckardt and Holthaus, 2007). Invoking the scalar product (11), in this basis the matrix of the quasienergy operator has the structure
where is a time-independent Bose-Hubbard Hamiltonian of the familiar form (47), but with the hopping matrix element multiplied by a zero-order Bessel function with the argument , resulting in the effective hopping strength
The operator , which is off-diagonal with respect to the photon index , contains nearest-neighbor couplings scaled by higher-order Bessel functions (Eckardt et al., 2005b; Eckardt and Holthaus, 2007). Thus, the diagonalization of the quasienergy matrix (51) constitutes a tremendous problem of a kind not usually considered in traditional many-body physics: Infinitely many -blocks, each one corresponding to the full Hamiltonian matrix of a time-independent Bose-Hubbard model (47) with modified hopping matrix elements (52), are shifted against each other in energy by integer multiples of , and are coupled by -blocks. The latter blocks, in their turn, embody hopping elements which are multiplied by Bessel-function factors with indices reflecting the distance of the respective block from the main diagonal. In effect, these -couplings cause multiphoton-like resonances among the states described by the shifted diagonal blocks (Eckardt and Holthaus, 2008b). While the physics of this problem has not yet been explored in full generality, its high-frequency regime is comparatively transparent. Namely, if is large compared to the two energy scales and (while remaining still small compared to the band gap of the underlying optical lattice, so that the single-band treatment remains applicable), it is a good approximation to neglect the couplings induced by altogether, so that the driven system (46) reduces to an undriven system described by , meaning that the effect of the time-periodic force essentially is to “renormalize” the hopping matrix element according to Eq. (52). With the -couplings out of action, all -blocks are equivalent, and returning from the extended to the physical Hilbert space is tantamount to considering only one such block from the outset. Since the ratio governs the superfluid-to-Mott insulator transition intrinsic to the Bose-Hubbard model (Fisher et al., 1989; Zwerger, 2003; Bloch et al., 2008), the renormalization of in response to high-frequency forcing implies that it is possible to induce that transition through adjusting the parameters of the driving force, while keeping the lattice depth and hence constant (Eckardt et al., 2005b; Eckardt and Holthaus, 2008a). This type of coherent control over the superfluid-to-Mott insulator transition has been demonstrated in a pioneering experiment by Zenesini et al. (2009). It involves adiabatic following of the -ground state when the driving amplitude is slowly changed; this has meanwhile been studied in detail with the help of numerical simulations by Poletti and Kollath (2011).
As in the time-independent case, experimental information about the many-body state is obtained by time-of-flight absorption imaging: Switching off the time-periodically shaken lattice potential at some moment , then letting the matter wave expand (neglecting interaction effects during that expansion), and finally recording its density in space after a sufficiently long expansion time yields the momentum distribution (Zwerger, 2003; Bloch et al., 2008)
where is the annihilation operator for a free-particle state with momentum in the direction of the lattice, and is the Fourier transform of the Wannier function pertaining to the lowest Bloch band; again, denotes the lattice constant. Thus, apart from the factor the observed momentum distribution is given by the Fourier transform of the system’s one-particle density matrix . Let us now assume that for a given driving amplitude , and hence for a given value of , an eigenstate of with energy takes the form
with certain coefficients . Under conditions such that the high-frequency approximation detailed above is valid, the corresponding many-body state in the driven optical lattice is then obtained by replacing the Fock states (49) in this expansion (54) by the Floquet-Fock states (50) with , say, resulting in
Now the only nonvanishing matrix elements are those with for ; together with and . This implies
which allows us to take the phase factor appearing on the right-hand side of Eq. (56) out of the sum, giving
Inserting this result into the representation (53), we immediately obtain the momentum distribution provided by a single Floquet state of the driven Bose-Hubbard model (46) in the high-frequency regime:
where, by assumption, is the associated energy eigenstate of .
However, an important step is still missing in order to connect theory with experiment: The driven Bose-Hubbard model, with the inertial force being incorporated through the driving term (48), refers to the co-moving frame of reference as considered in Sec. 3, whereas measurements usually are performed in the laboratory frame. Thus, in order to obtain the momentum distribution as recorded by an observer in the laboratory frame, we still have to invert the transformation (30). This is done in the general case by writing
for the driven Bose-Hubbard model with forcing (48), as corresponding to a frequency variation between the counterpropagating laser beams, or to lattice motion , keeping in mind the relation (42) for the driving amplitude, together with the definition . Therefore, we finally have
This is quite a significant observation: The momentum distribution of a matter wave occupying a many-body Floquet state in a time-periodically shaken optical lattice (39) in the high-frequency regime equals that of the associated energy eigenstate of , which is obtained from the undriven Bose-Hubbard Hamiltonian (47) through replacing the hopping matrix element by as defined in Eq. (52), insofar as exactly the same Fourier transform of the one-particle density matrix appears in both cases. Hence, even though the position of the lattice is periodically shifted, the peak pattern observed in the laboratory frame does not move. The effect of the time-periodic shift is seen only in the envelope of that pattern, given by the Fourier transform of the Wannier function, the argument of which is modulated periodically in time in accordance with Eq. (61). Thus, apart from this modulation the experimental signature of the superfluid-to-Mott insulator transition occurring in shaken optical lattices upon changing the driving amplitude (Zenesini et al., 2009) is the same as that of the transition occuring in a stationary lattice in response to a variation of its depth (Zwerger, 2003; Bloch et al., 2008).
It may be useful to point out that the experimental signatures differ from the above description if the optical lattice is not shaken, but kept at rest, while the force is induced by means of the time-periodic modulation of a levitation gradient which “stirs” the condensate, as done in the experiments by Haller et al. (2010). In this latter situation the model (48) actually describes the driving force in the very reference frame in which the momentum distribution is recorded, so that one obtains an oscillating interference pattern given directly by Eq. (59). In contrast, in the experiments reported by Lignier et al. (2007), Sias et al. (2008), Eckardt et al. (2009), Zenesini et al. (2009), and in Sec. 6 below, the lattice is shaken in the laboratory frame according to Eq. (39). Hence, here the “micromotion” is taken out of the interference pattern by means of Eq. (61), which connects the momentum in the co-moving frame to the momentum observed in the laboratory frame. The fact that the resulting interference pattern does not move in the laboratory frame (Eckardt et al., 2005b; Eckardt and Holthaus, 2007, 2008a) may sometimes facilitate its interpretation.
5.2 Signatures of ideal shaken Bose gases
With respect to the Floquet condensates envisioned in Sec. 2, the case of an ideal Bose-Einstein condensate in a shaken optical lattice now is of particular interest. For vanishing interaction the time-dependent Schrödinger equation for the model (46) can easily be solved exactly for any type of forcing described by a driving term
without requiring a specific time-dependence of the force. Therefore, we abandon time-periodic forces and the Floquet picture for the moment being, and consider an initial-value problem instead: We assume that vanishes for , and is switched on sharply at , but is arbitrary otherwise. Generalizing the previous Eq. (41), we thus impose a force
where the second term accounts for the sudden velocity jump of the lattice from to an arbitrary value at . Moreover, we consider a lattice with sites and disregard finite-size effects, so that the operator
creates a particle in the Bloch state with quasimomentum . We then assume that the initial state at is an ideal -particle condensate occupying such a Bloch state. Even if might be the only experimentally realistic option here, we do not impose this restriction at this point. For , after the force has been turned on, the resulting -particle wave function can then be written in the form
Here the creation operator , given by
refers to a so-called Houston state, also known as accelerated Bloch state (Houston, 1940; Eckardt et al., 2009). This nomenclature stems from the fact that the time-dependent wave number appearing here has to obey the “semiclassical” acceleration law
Therefore, using the particular connection (64) between the inertial force and the lattice motion, and requiring that be equal to the wave number of the initial state for , we have
for , having properly accounted for the delta-kick at the moment of turn-on. Finally, the expression
Therefore, the interference pattern provided by an ideal Houston condensate in time-of-flight absorption imaging again is stationary in the laboratory frame, and the peak positions are not affected by the force: Even after the force (64) has been switched on, the interference peaks are still permanently centered around the wave numbers , as they had been for the unforced initial condensate. It needs to be stressed that this peculiar feature is crucially dependent on the delta-kick which accompanies the sudden turn-on of the force in the co-moving frame.
In particular, let us now consider a monochromatic oscillating force switched on instantaneously at with starting phase according to Eq. (41), which leads to
for . In this case there is a close relation between the -particle Houston states (66), which have been constructed for as solutions of an initial-value problem, and -particle Floquet states, which presuppose a perfectly time-periodic force acting at all times . In order to obtain these Floquet states, one only has to extend , as given by Eq. (73), to all , and use this expression in Eq. (67), now to be considered for all . Then the emerging “extended” Houston states (66) have the basic form (1): By construction, is -periodic, with , but the energy integral in the exponent of Eq. (67) is not. However, writing
with the help of the one-cycle-averaged energies
the first of these exponentials is -periodic and therefore part of the Floquet functions, whereas the occurrence of the averages (74) in the second exponential, accompanied by the time , allows one to identify them as quasienergies. The Houston-Floquet states thus found are particularly simple examples of spatiotemporal Bloch waves, incorporating both the spatial periodicity of the lattice and the temporal periodicity of the driving force on equal footing. They are labeled by the same quantum numbers as the customary, time-independent Bloch waves to which they reduce in the absence of the drive, while their time-evolution, apart from the time-periodic motion incorporated into the moving wave numbers , is specified by the quasienergies (Arlinghaus and Holthaus, 2011). Observing that one again encounters here the effective hopping matrix element introduced in Eq. (52), the quasienergy-quasimomentum dispersion relation takes the form
which differs from the original dispersion relation (70) of the undriven lattice only through the replacement of by . That same replacement had been met before in the context of the interacting many-body system described by the driven Bose-Hubbard model (46), when constructing the approximate effective Hamiltonian pertaining to the high-frequency regime. In contrast, in the noninteracting case considered here no approximations have been made; Eq. (75) holds exactly for all driving frequencies.
The fact that the quasienergies (75) for the driven, noninteracting Bose-Hubbard model could be calculated by taking the time-averages (74) further suggests that these quasienergies do coincide with the corresponding mean energies (13), as considered in Sec. 2. Indeed, taking a single-particle Houston-Floquet state as constructed above, one easily confirms the identity
so that here the mean energies of the Floquet states actually equal the quasienergies (75), disregarding their “”-multiplicity. In the case of a general -periodic single-particle Hamiltonian with Floquet states (1) this is not the case, since then
as discussed in detail by Fainshtein et al. (1978).
Interestingly, the quasienergy band (75) “collapses” when equals a zero of the Bessel function (Holthaus, 1992), implying that an arbitrary single-particle wave packet driven under such conditions cannot spread, but reproduces itself -periodically. This phenomenon, termed “dynamic localization” (Dunlap and Kenkre, 1986), has recently been observed with dilute Bose-Einstein condensates in driven optical lattices (Lignier et al., 2007; Eckardt et al., 2009; Arlinghaus et al., 2011). With a view towards future applications, the dependence of the quasienergy band width on the driving strength also has been identified as a means of controlling transport in systems with attractive pairing interactions (Kudo et al., 2009).
Once again, it is instructive to compare the above results, obtained for a shaken optical lattice, to the corresponding physics when the lattice is at rest, while the condensate is stirred by a harmonically modulated levitation gradient (Haller et al., 2010). Then it is actually possible to impose an instantaneously turned-on oscillating force
without the additional delta-kick present in Eq. (41), which necessarily appears when an inertial force is abruptly turned on. The moving wave numbers , which previously had been determined in Eq. (69) as solutions to the equation of motion (68), now have to be replaced by the solutions to the corresponding equation , giving
for , instead of Eq. (73) above. Therefore, one formally has
with a wave-number shift
which means that an additional momentum is imparted on the condensate particles when the stirring force is switched on rapidly, as described by Eq. (78). Thus, besides the micromotion the resulting time-of-flight absorption images would also show a shift of the peak positions. Both of these features are absent in experiments with shaken lattices: It really matters whether the condensate is “shaken” or “stirred”!
We note that Kudo and Monteiro (2011a, b), focusing on stirring forces of the form (78), have suggested to introduce effective dispersion relations which also incorporate the initial phase . While one is free to adopt this viewpoint, it seems to obscure the conceptual simplicity of the Floquet approach: The quasienergy band (75) characterizes the time-periodically driven system as such, regardless of the way the drive has been switched on, that is, independent of the phase which parametrizes the sudden turn-on (78), or of any other parameters which specify other, equally possible turn-on protocols. Because this quasienergy band (75) actually consists of eigenvalues of the quasienergy operator it even provides a dispersion relation in the usual sense of solid-state physics which allows one to compute group velocities by taking its derivative, properly evaluated at that quantum number around which the wave packet is centered (Arlinghaus and Holthaus, 2011).
With regard to shaken condensates, we may summarize our considerations as follows: If a perfectly ideal Bose-Einstein condensate in an optical lattice initially occupies a Bloch state with wave number , and then is abruptly being shaken and thus subjected to the force (41) under single-band conditions, it permanently populates a single Floquet state labeled by the same wave number , regardless of the phase .
6 Experimental findings
In order to substantiate the relevance of the above ideal-gas considerations for laboratory experiments, we took time-of-flight absorption images of dilute Rb-condensates released from shaken 1D optical lattices (39), employing various driving frequencies and amplitudes. In Fig. 2 we display typical results obtained for a lattice with depth , driven with frequency kHz and scaled amplitude . Under such conditions the single-band approximation is well justified; according to Eq. (52), one has . The drive was turned on quickly with a starting phase , as modeled by Eq. (41), and the images were taken 10 driving cycles later. Hence, provided that the solution (66) to the initial-value problem for the ideal gas discussed in Sec. 5 describes the real laboratory system correctly, Eqs. (71) and (72) tell us that for any value of there should be stationary interference maxima at , or
using the recoil momentum as reference scale. This expectation is fully confirmed in Fig. 2. In all three cases considered there, with , , and , the central peak is located at , as corresponding to a condensate occupying the Houston-Floquet state associated with the minimum of the quasienergy band (75). In addition there are side peaks at , corresponding to the width of the quasimomentum Brillouin zone. The left panel of Fig. 2 depicts the time-resolved evolution of the positions of the respective interference maxima during a single cycle of the driving force. As anticipated, these positions remain practically constant in time, apart from apparent slight wigglings. The right panel of Fig. 2 shows the modulation of the height of the side peaks during one driving cycle, which stems from the oscillating time-dependence of the argument of the factor determining the envelope of the interference pattern in accordance with Eq. (62). For all values of the starting phase we observe practically the same signals, which is in line with the surmise that they are signatures of the same state.
Therefore, we interpret the data displayed in Fig. 2 as experimental signatures of Floquet condensates: The existence of a stationary, stable and lasting interference pattern indicates that the driven Bose gas actually tends to occupy a single Houston-Floquet state. The fact that this pattern is centered around