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Interference pattern and visibility of a Mott insulator

We analyze theoretically the experiment reported in [F. Gerbier et al, cond-mat/0503452], where the interference pattern produced by an expanding atomic cloud in the Mott insulator regime was observed. This interference pattern, indicative of short-range coherence in the system, could be traced back to the presence of a small amount of particle/hole pairs in the insulating phase for finite lattice depths. In this paper, we analyze the influence of these pairs on the interference pattern using a random phase approximation, and derive the corresponding visibility. We also account for the inhomogeneity inherent to atom traps in a local density approximation. The calculations reproduce the experimental observations, except for very large lattice depths. The deviation from the measurement in this range is attributed to the increasing importance of non-adiabatic effects.

We analyze theoretically the experiment reported in [F. Gerbier et al, cond-mat/0503452], where the interference pattern produced by an expanding atomic cloud in the Mott insulator regime was observed. This interference pattern, indicative of short-range coherence in the system, could be traced back to the presence of a small amount of particle/hole pairs in the insulating phase for finite lattice depths. In this paper, we analyze the influence of these pairs on the interference pattern using a random phase approximation, and derive the corresponding visibility. We also account for the inhomogeneity inherent to atom traps in a local density approximation. The calculations reproduce the experimental observations, except for very large lattice depths. The deviation from the measurement in this range is attributed to the increasing importance of non-adiabatic effects. The superfluid to Mott insulator (MI) transition undergone by an ultracold Bose gas in an optical lattice has attracted much attention in the recent years as a prototype for strongly correlated quantum phases [1,2,3,4]. A key observable in these systems is the interference pattern observed after releasing the gas from the lattice and letting it expand for a certain time of flight. Monitoring the evolution of this interference pattern not only reveals the superfluid-to-MI transition [2,4], but also allows for example the detection of number-squeezed states in the lattice [5,6], or the observation of collapse and revivals of coherence due to atomic interactions [7]. Because of its experimental importance, a quantitative understanding of this interference signal is crucial to characterize quantum phases of bosons in optical lattices.
Although no interference pattern is expected for a uniform array of Fock states (what we call a "perfect" Mott Insulator) [29], a finite visibility is nevertheless observed in experiments above the insulator transition [2,4,8], in agreement with numerical calculations [9,10]. We have studied this phenomenon experimentally, and shown that despite its insulating nature that forbids long-range coherence, a MI still exhibits short-range coherence at the scale of a few lattice sites [8]. This can be attributed to the structure of the ground state for finite lattice depths, which consists of a small admixture of particle/hole pairs on top of a perfect MI. A qualitative model based on a lowest-order calculation of the ground state wavefunction was also presented in our previous work, which reproduced the main trend and order of magnitude of the observed visibility.
In the present paper, we would like to present a more precise calculation that includes higher order corrections (see also [11]). We describe a MI state at zero temperature using the Random Phase Approximation (RPA), already introduced in Refs. [12,13,14,15,16]. Instead of the path integral approach used by these authors, we obtain here the RPA Green's function using a different method inspired by Hubbard's original treatment of the fermionic model [17,18]. Taking the experimental geometry and the inhomogeneous particle distribution into account, we find good agreement with our experimental data, which provides further support for the physical picture presented above.
The paper is organized as follows. In section I, we recall the description of ultracold atoms in an optical lattice by the Bose-Hubbard model, and discuss the inhomogeneous shell structure that develops in an external confining potential. Section II presents the calculation of the interference pattern observed after free expansion of the atom cloud and its link with the quasi-momentum distribution. The main results are presented in sections III and IV, where we respectively present the calculation of the interference pattern in the uniform case using the RPA, and extend it to the inhomogeneous case to compare to the experimental data of [8]. Details of the calculation are described in the appendix.

I. BOSE-HUBBARD HAMILTONIAN
In this section, we briefly recall the theoretical description of an ultracold atomic gas trapped in an optical lattice. The optical lattice potential, which results from the superposition of three orthogonal and independent pairs of counterpropagating laser beams, can be written as Here V 0 is the lattice depth, k L = 2π/λ L is the laser wavevector, λ L is the laser wavelength and m is the atomic mass. As usual, we measure V 0 in units of the single-photon recoil energy E R = h 2 /2mλ 2 L . The lattice potential has a simple cubic periodicity in three dimensions, with a lattice spacing d = λ L /2 ≈ 425 nm in our case. As shown in [1], the behavior of the atomic system in such a potential can be described by the Bose-Hubbard model, defined by the hamiltonian Here the operatorâ † i creates an atom at site i,n i =â † iâ i is the on-site number operator, and the notation i, j restricts the sum to nearest neighbors only. The relative strength between the tunneling matrix element t and the on-site interaction energy U is controlled by the depth V 0 of the periodic potential which confines the atoms [30]. The phase diagram of this hamiltonian is well known: The system is in a MI state within characteristic lobes in a t/U versus chemical potential µ phase diagram, and in a superfluid state outside of these lobes [19].
In the experiments, an additional potential V ext (r) is superimposed to the lattice potential, leading to a spatially varying chemical potential across the cloud. This favors the formation of a "wedding cake" structure of alternating MI and superfluid shells, which reflects the phase diagram of the Bose-Hubbard model [1,9,19,20]. The external potential is due to a combination of a magnetic potential in which the condensate is initially formed and of an optical potential due to the Gaussian shape of the lattice beams. To a good approximation, it can be considered as a harmonic potential with trapping frequency where ω m is the frequency of the magnetic trap, assumed isotropic, and where w is the waist (1/e 2 radius of the intensity profile) of the lattice beams, assumed identical for all axes. For large lattice depths, the confinement is mainly due to the optical part.
In the current experiments, this external potential varies slowly across the lattice. In this limit, the shell structure can be calculated in a local density approximation, which assumes a known relation n h [µ] between the density n and the chemical potential µ for the homogeneous system. Then, the coarse-grained density [31] for the inhomogeneous system is calculated as n(r) = n h [µ − V ext (r)]. The chemical potential is fixed by the relation N = d (3) r n(r). For a fixed lattice depth and atom number, we calculate numerically the relation n h [µ] using mean-field theory at zero temperature, i.e. in the mean-field ground state [12,21]. We then repeat the steps outlined above, varying the chemical potential until the target atom number is obtained within 0.1 %. For all calculations, the values ω m = 2π × 15 Hz and w = 130 µm are used, which match our experimental parameters. We show in Fig. 1 an example of such a calculation for a lattice depth of V 0 = 18 E R and N = 2.2 × 10 5 atoms.
The presence of the external potential significantly affects the atom distribution in the lattice, which is determined by the competition between interaction and potential energy. On the one hand, expanding the cloud minimizes the density and the interaction energy, and on the other, contracting it minimizes the potential energy, as in conventional harmonic traps [22,23]. The latter is favored at low atom numbers, where only a n 0 = 1 shell forms. When more atoms are added, the radius of the unity filled MI region increases until a critical atom number, which we estimate to be 7 × 10 4 atoms for our parameters, is reached. Above this critical atom number, a higher density core appears near the trap center. A MI with 2 atoms per site is then obtained near the trap center if the lattice depth is above the critical value V 0 ≈ 14.7 E R . This value has been calculated using the boundary derived in [12], where z = 6 is the number of nearest neighbors in 3 dimensions. In the specific example shown in Fig. 1, we have chosen V 0 = 18 E R and N = 2.2 × 10 5 atoms, so that both n 0 = 1 and n 0 = 2 MI are present. Similarly, we calculate for our experimental parameters that an n 0 = 3 shell is also present for atom numbers larger than 2.7 × 10 5 , and lattice depths larger than 16 E R . Density

II. INTERFERENCE PATTERN
We now turn to the description of the interference pattern observed after release of the atom cloud from the optical lattice and a period of free expansion. From an absorption image of such a pattern, the phase coherence of the atomic sample can be directly probed. The density distribution of the expanding cloud after a time of flight t f can be calculated as [3,9,24] which mirrors the momentum distribution of the original cloud. Momentum space and real space in the image plane are related by the scaling factor t f /m -independent of the lattice parameters. The envelope |w| 2 is the Fourier transform of the Wannier function in the lowest Bloch band. When each potential well is approximated as an harmonic potential, the Wannier function is the corresponding gaussian ground state wavefunction. The envelope function in Eq. (5) then reads where w t f = t f /mw 0 , and where w 0 is the size of the on-site Wannier function. Finally, we have defined the quantity When k is restricted to the first Brillouin zone, S(k) is nothing else than the quasi-momentum distribution. Information about the many-body system is contained in this quantity, which is periodic with the periodicity of the reciprocal lattice 2π/d. Thus, to predict the interference pattern and compare to the experiments, our goal is to calculate S(k) for a given lattice depth and density.

III. QUASI-MOMENTUM DISTRIBUTION IN THE HOMOGENEOUS MOTT INSULATOR
For simplicity, we consider first the case of uniform filling in the lattice, i.e. an integer number n 0 of atoms per site, and we assume the system to be at zero temperature and in the insulating phase. In the limit of zero tunneling, the ground state wavefunction is a perfect MI, i.e. a product of number states at each site, and its Green function G 0 can be calculated exactly (see appendix). The lowest-lying excited states of the system are "particle" and "hole" states, where a supplementary particle is added (respectively removed) at one lattice site. Creating these excitations costs a finite interaction energy, respectively E (+) = Un 0 and E (−) = U(n 0 − 1) [19].
To calculate the quasi-momentum distribution for a finite tunneling t, many-body techniques can be applied to obtain the single-particle Green function, G(k, ω). Using a path integral approach, several authors [12,13,14,15,16] have been able to calculate the Green function of the Mott insulator within the RPA, The poles E (±) k of the Green function are the quasi-particle energies [12] In Eq. (9), t k = −2t ν=x,y,z cos(k ν d) is the dispersion relation for a free particle in the tight-binding limit, and D k [n 0 ] = t 2 k + 4t k U(n 0 + 1 2 ) + U 2 . The particle weight is In the Appendix, we present an alternative derivation of (8) based on the equation of motion method, which follows closely Hubbard's method [17,18]. Here we will simply comment on the physical picture behind this approach. The RPA considers that the particle/hole nature of the low-lying excitations is not significantly changed by introducing a finite tunneling (in technical terms, the self energy remains approximately the same as in the t → 0 limit). The first effect of tunneling is to introduce a finite amount of particle/hole components in the the many-body ground state wavefunction. In the form given in [8], corresponding to a first order calculation, a particle/hole pair necessarily occupies two neighboring lattice sites due to the particular form of the tunneling hamiltonian. Through higher-order tunneling processes captured by the Green function (8), the particle and the hole forming the pair can tunnel independently. As a result, the pair acquires a mobility through the lattice, and may even "stretch" over a few lattice sites. This mobility acquired by particle/hole pairs is reflected in the modified dispersion relation (9), which explicitly includes the band structure. Note finally that higher order excitations, corresponding to occupation numbers n 0 ± 2, n 0 ± 3, ... are neglected. At zero temperature, such excitations become important only very close to the superfluid transition where the MI is destroyed.
The quasi-momentum distribution can be directly deduced using the general relation S(k) Using (8), one has [15] S(k) = n 0 To first order in t/U, this reduces to also obtained in [8] by calculating the many-body wave function perturbatively. We find that the two predictions rapidly converge. For example, they differ by less than 10 % for U/zt > 6.6, 11.6 and 16.9 for n 0 = 1, 2 and 3, respectively. These values have to be compared to the respective critical values for MI formation, U/zt = 5.83, 9.89, 13.93. This indicates that the coherence beyond nearest neighbors is rather rapidly lost as one goes further into the MI phase. However, the visibility itself remains finite in a substantial range of U/zt, implying a persistent short-range coherence.

IV. COMPARISON WITH THE EXPERIMENTS
To compare with the experiments reported in [8], several features have to be taken into account. First, only the column density n is accessible experimentally, i.e. the density integrated along the probe line-of-sight (which we take here parallel to the z axis). Second, the visibility is experimentally deduced from two points according to To eliminate the Wannier envelope, n max and n min are measured from two points at the same distance from the cloud center (see Fig. 2), so that the envelope automatically cancels out. For example, n max is found at point (2π/d, 0) and n min at (2π/ √ 2d, 2π/ √ 2d). This reduces the visibility compared to the usual definition. In the theoretical calculation it is straightforward to account for these two effects.
The third effect, the shell structure of the MI, is handled here in an approximate way. In the numerical calculations, the shell distribution always includes small regions with noninteger filling, which the theory above cannot handle. However, these domains are small, and have a strongly depleted superfluid component, so that we do not expect them to have a large effect on the visibility. Therefore, we approximate the density distribution by a "ziggurat"-like profile, where only 2π/d MI shells are present. The actual extension of each shell is calculated as if t were zero, taking the external potential into account [25]. In Fig. 1, we compare the profile in this approximation (dotted line) with the numerically calculated one (solid line). For large lattice depths, both agree reasonably. Note that the density profile still depends weakly on the lattice depth through the external confinement [see Eq. (3)].
The momentum distribution deduced from Eqs. (5,10) is averaged over the distribution of atoms to compare with the experimental data (see [8] for details on the experiment). The results are plotted versus lattice depth in Fig. 3, for two different atom numbers in the lattice. For the lowest atom number N = 2.2 × 10 5 , we calculate that only n 0 = 1 and n 0 = 2 shells are present. For the largest N = 5.6 × 10 5 , a core with n 0 = 3 atoms per site is also present. Note that in the latter case, the actual density distribution might deviate more from the calculated one, due to three-body losses in the n 0 = 3 region. We find that the calculation agrees with the measured visibility within 20 % for V 0 ≤ 22 E R . The theory curves terminate when the MI shell with highest filling disappears, as it is replaced by a large superfluid core not described by our theory. Note that the calculation does not include any free parameter.
However, we consistently find that the calculated value lies below the measured visibility for large lattice depths V 0 ≤ 22 E R . Moreover, the deviation increases with increasing lattice depths, which shows that the superfluid shells play little role in determining the visibility for such large lattice depths, as assumed in our calculation. In Fig. 4, the fractional deviation of the calculated visibility from the measured one is plotted versus lattice depth for four data sets. Remarkably, although the atom numbers are rather different from one data set to another we find a common trend in the data. On the other hand, this observation also suggests that a breakdown of adiabaticity occurs for the particular ramp used in the experiments to increase the laser intensity to its final value, a point already identified in [8]. We conclude that, perhaps surprisingly, the visibility in the MI may be a sensitive probe of the many-body dynamics of the superfluid-to-insulator transition.

V. CONCLUSION
In conclusion, we have derived in this paper a theoretical expression for the interference pattern of a Mott insulator after release from the optical lattice and a time of flight. Our calculations take deviations from perfect filling due to a finite tunneling into account, and use a simplified but realistic model of the shell structure of the MI. Good agreement with our experimental data reported in [8] is found, at least for moderate lattice depths. For very large lattice depths, an increasing de- viation points to non-adiabatic effects in the conversion from a condensate to an insulating state, which could in principle be studied by the method presented here. Nevertheless, in view that no free parameter is included in the theory, we conclude that the momentum distribution (10) describes the system well. This supports the physical picture of the system as a (dilute) gas of partice/hole pairs, mobile through the lattice, on top of a regularly filled Mott insulator. Furthermore, the validity of the RPA to describe their behavior is qualitatively verified.
Our calculation neglects entirely the superfluid component, which is correct only for large lattice depths where the system is almost completely insulating. Recently, several authors [15,26,27] have proposed to modify the standard mean-field description [19,21] to better account for long-and short-range coherence. It would be interesting to compare the predictions of those approaches with our data for lower lattice depths, where the system is expected to be a strongly depleted superfluid, and therefore amenable neither to a Bogoliubov-like description nor to a strongly interacting one as provided in this paper. Also, an investigation of finite temperature effects [13] would be useful. In particular, an interesting question is to know whether the visibility measurements presented here could be used for thermometry in the lattice.
We would like to thank Dries van Oosten, Paolo Pedri and Luis Santos for useful discussions. Our work is supported by the Deutsche Forschungsgemeinschaft (SPP1116), the European Union under a Marie-Curie Excellence Grant and AFOSR. FG acknowledges support from a Marie-Curie Fellowship of the European Union. strongly interacting limit. We comment on this approximation in the test. Making this approximation yields which has a typical RPA form. Using the explicit result for G 0 , we obtain after some algebra Eq. (8) in the text, which explicitly displays particle and hole components.