Probing correlated phases of bosons in optical lattices via trap squeezing

We theoretically analyze the response properties of ultracold bosons in optical lattices to the static variation of the trapping potential. We show that, upon an increase of such potential (trap squeezing), the density variations in a central region, with linear size of>~ 10 wavelengths, reflect that of the bulk system upon changing the chemical potential: hence measuring the density variations gives direct access to the bulk compressibility. When combined with standard time-of-flight measurements, this approach has the potential of unambiguously detecting the appearence of the most fundamental phases realized by bosons in optical lattices, with or without further external potentials: superfluid, Mott insulator, band insulator and Bose glass.

We theoretically analyze the response properties of ultracold bosons in optical lattices to the static variation of the trapping potential. We show that, upon an increase of such potential (trap squeezing), the density variations in a central region, with linear size of 10 wavelengths, reflect that of the bulk system upon changing the chemical potential: hence measuring the density variations gives direct access to the bulk compressibility. When combined with standard time-of-flight measurements, this approach has the potential of unambiguously detecting the appearence of the most fundamental phases realized by bosons in optical lattices, with or without further external potentials: superfluid, Mott insulator, band insulator and Bose glass. Ultracold gases in optical lattices offer the unique opportunity of literally implementing fundamental lattice models of strongly correlated quantum many-body systems, either bosonic or fermionic, traditionally considered as "toy" models for the description of complex condensed matter systems [1,2]. In the particular case of ultracold bosons realizing the Bose-Hubbard (BH) model, recent experimental developments have led to the spectacular demonstration of the Mott insulating (MI) phase with controllable filling [3,4,5], and even more recent developments in laser trapping offers the possibility of realizing further fundamental insulating phases, such as a band insulator (BI) in a commensurate superlattice [6] or a Bose glass (BG) in an incommensurate superlattice or in a laser-speckle potential [7,8].
Two main technical aspects limit to date the possibility of the experiments to retrieve full information on the true bulk behavior of the model Hamiltonian implemented in the system. One aspect is the presence of a parabolic trapping potential which imposes a spatial variation of the filling and hence a spatial modulation of the local behavior exhibited by the system. A second aspect is represented by the typical measurements performed on the system. The detection of strongly correlated phases is typically based on the measurement of correlation functions (phase correlations via time-of-flight measurements [3] and density-correlations via noise-correlation analysis [9]) and on site-occupation statistics [5,10]. Latticemodulation spectroscopy offers the possibility of measuring the dynamic structure factor at zero transferred momentum [8,11] but it has the drawback of probing the global response of the inhomogeneous system, and of being subject to a low-energy cutoff imposed by the duration of the experiment. This aspect prevents e.g. the unambiguous observation of the BG, which does not have a special signature in correlation functions, but it is unambiguously marked by the absence of a gap in the excitation spectrum.
The purpose of this paper is to propose a technique trap-squeezing spectroscopy -which circumvents these two limitations at once, taking advantage of the parabolic trapping to extract the bulk behavior of the model implemented in the system and in particular its lowest particle-hole excitation energy. Two fundamental observations are at the basis of this proposal. On the one hand, the slowly varying nature of the parabolic potential guarantees the validity of the local-density approximation (LDA) [12,13], particularly close to the potential minimum. Hence the density around the trap center mimics the behavior of the bulk system over an extended region of space of linear size of several ( 10) lattice spacings, and consequently the average density in this region can be accessed via laser microscopy [14]. On the other hand, this central average density can be controlled via the trapping potential in very much the same way as the chemical potential controls the density of a bulk system in the grand-canonical ensemble. In particular the trapping potential is by far the lowest-energy potential to which the system is coupled (with trapping frequencies as low as ∼10 Hz), and measuring the response of the central density to small variations of such potential allows to directly probe the low-energy response of the bulk system.
We theoretically investigate the trap-squeezing spectroscopy in the one-dimensional (1d) BH model in a parabolic potential plus an external superlattice potential, (1) Here g i = g i (α, φ) = cos 2 (2πα i + φ) − 1/2 is a one-color superlattice potential. J and U are experimentally controlled via the height of the primary optical lattice, while V 2 is controlled via the height of a secondary optical lattice [6,8]. In the following we make the fundamental assumption that V t can be varied independently of the other parameters, which is possible by applying an extra dipolar trap to the system created by an additional running laser wave.
We study the above model via Stochastic Series Expansion quantum Monte Carlo [19] at low temperatures (capturing the T = 0 behavior) and in the grandcanonical ensemble, namely we simulate the Hamiltonian H µ = H − µ i n i where the chemical potential µ is fine tuned to get the desired average number of particles N , and in this way it becomes a function of the other Hamiltonian parameters µ = µ(V t , N, J, U, V 2 ). According to LDA, the average density at the center of the trap n C =: 1/|C| i∈C n i (where the region C will be defined later) reproduces closely that of a homogeneous system (V t = 0) at a chemical potential µ. Hence controlling µ via one of the other parameters V t , N , J, U , and V 2 , allows to control n C . In particular, if µ is controlled by changing V t , namely by trap squeezing, while holding all the other parameters fixed, one has access to the compressibility for the bulk Hamiltonian H 0 (J, U, V 2 ), estimated via κ = ∂n C /∂µ. The control on the chemical potential µ via trap squeezing requires the detailed knowledge of the function µ = µ(V t , N, J, U, V 2 ). Such a function can be accurately sampled via quantum Monte Carlo, given that its values are the result of the fine-tuning procedure of the chemical potential required to achieve a desired average N . Fig. 1 shows µ = µ(V t , N, ...) for different cases of the BH model without external potentials, V 2 = 0, and for an applied incommensurate superlattice potential with strength V 2 = U and incommensurability parameter α = 0.7714... identical to that of the experiment of Ref. 8. The behavior of µ for a commensurate superlattice with α = 3/4 is found to be nearly identical to that of the incommensurate case. In absence of a superlattice, and for weakly interacting bosons, Thomas-Fermi (TF) theory [15] would predict the following scaling for the chemi- Fig. 1 shows that, at fixed U and for all the cases considered, µ is a homogeneous function of the combination x = N 2/3 V 1/3 t ; in particular, even for large U/J ratios it suprisingly verifies the TF prediction of a linear dependence on x; significant deviations are observed only in the low-density and high-(U/J) case, where the hardcore boson regime sets in [16]. Moreover, for sufficiently low U/J the data for different U 's collapse on the same universal curve where f µ is essentially a straight line. A linear fit for the lowest-(U/J) data gives f µ (x) = −1.225(13) + 0.817 (2)x for all the cases considered, namely in presence or in absence of a superlattice. We notice that the crude TF prediction would give f µ (x) = [(d + 2)Γ(d/2 + 1)/(2π d/2 )] 2/(2+d) x, which is well off the numerical data. For higher U/J we observe that the (U/J) 2/3 scaling is still essentially obeyed by the derivative ∂µ/∂x but not by the intercept µ(x = 0); hence the slight disagreement between the scaling curve and the data for U/J = 20 in Fig. 1(a) and U/J = 30 in Fig. 1(b). Therefore we obtain a universal prediction for the dependence of the effective chemical potential in the center of the trap on the experimentally controllable parameters J, U , V t and N for a large range of their values, and for the extreme case of d = 1 where the applicability of mean-field theory is in doubt. Similar results are then expected to hold a fortiori for the cases d = 2, 3. Hence we can firmly conclude that the chemical potential in the center of the trap represents a well controlled experimental parameter.
Armed with this prediction, we can then move on to simulate the outcome of a trap-squeezing experiment, where the central density n C is monitored as a function of the trapping potential V t . We start from the case of the 1d BH model without any superlattice, for which we consider a boson number N = 100 in a variable-frequency trap and with fixed repulsion U/J = 20. Fig. 2 shows the evolution of the central density n C averaged over a region C containing 10 − 20 sites as a function of the chemical potential µ(V t ), and compared with the data for the bulk system. It is evident that, for a sufficiently low µ (namely for sufficiently low V t ), the bulk density curve is very well reproduced (in this case for n c ≤ 2). The deviation of n C from the bulk value reveals that the truly homogeneous region in the trap center has become smaller than the C region, a fact that can be simply cured by increasing the number of particles and decreasing the trapping potential so as to leave µ ∼ V 1/3 t N 2/3 fixed. The succession of incompressible plateau regions at integer filling and compressible regions in the n c (µ) curve marks the alternation between incoherent MI and coherent superfluid (SF) behavior, as also revealed by the (global) coherent fraction n k=0 = (1/N ) ij b † i b j . Remarkably, when the effective chemical potential in the trap center overcomes the Mott gap, a few particles can be transfered from the wings to the center into a locally SF state, and this gives rise to a violent increase in the coherent fraction with a very sharp kink. The width of the integer-filling plateaus corresponds to that of the MI lobes in the phase diagram of the 1d BH model: hence this kind of measurement allows to reconstruct that phase diagram with high accuracy, and to extract the particle (hole) gap at any point as the minimal chemical potential variation required to increase (decrease) the density. In particular, trap squeezing probes the density-driven transition from MI to SF, which is in a different universality class [1] with respect to the transition driven by the J/U ratio and probed so far in experiments [3,11,20]. Moreover we emphasize the high tolerance of the method to the variation of the size of C, which corresponds to the size of the focus of the imaging laser.
Having shown that trap squeezing allows to reconstruct the phase diagram of the bulk Bose-Hubbard model, we generalize this approach to probe other phases of correlated bosons in an optical lattice. To this end we consider N = 100 trapped bosons in an additional commensurate superlattice potential [17] with α = 3/4, fixed phase φ = 0, and strength V 2 = U = 20J, such that it overcomes the MI gap and hence it removes the MI , φ = 0). All symbols and notation as in Fig. 2. Notice that the deviation of the data for C = 10 sites from the bulk ones is due to the fact that the C region does not contain an integer number of periods of the superlattice potential.
phase; the insulating phase which is left for large U/J is a BI with fractional, commensurate fillings (2n + 1)/4 (n = 0, 1, ...). Fig. 3 shows the alternation of phases in the center of the trap under trap squeezing as revealed by the central density, and compared to the bulk result; similarly to the MI-SF transition, the BI-SF alternation is clearly evidenced. The density plateaus correspond to the formation of incompressible BI region in the trap center, an event associated with a significant lowering of the global coherence in the system, as shown by the n k=0 curve; the coherence is suddenly increased when the BI gap is overcome by the chemical potential and particles are transfered into a locally SF state in the center. The situation changes drastically when tuning slightly the superlattice parameter from the commensurate value α = 3/4 to the incommensurate value α = 0.7714.. realized in recent experiments [8]. In this case, for a strong superlattice V 2 = U and for small J/U the ground state of the system changes from SF to incompressible incom-mensurate band insulator (IBI) and to compressible BG upon changing the chemical potential [13]. Fig. 4 shows the variation under trap squeezing for the central density averaged over random fluctuations of the spatial phase, n C φ . This average is intrinsic in current experimental setups, where the phase φ can change from shot to shot, and it is essential for the central region of the trap to sample the full statistics of the quasi-periodic potential and hence to mimic the bulk behavior of the system [13]. Indeed we observe that n C φ reproduces very well the bulk behavior for low enough density. In striking contrast to the previous two cases of no superlattice and of a commensurate superlattice, the n C φ curve exhibits extended compressible regions for which the coherent fraction does not vary upon changing the chemical potential. This corresponds to transfer of particles at no energy cost from the wings to the center of the trap into localized states which do not contribute to the coherent fraction of the system: this fact provides smoking-gun evidence for the appearence of a BG state in the center of the trap [18]. Moreover the joint information coming from the central density and the global coherent fraction enables to experimentally probe the incompressible IBI behavior and the compressible SF behavior.
In summary, we have proposed an experimental method (trap squeezing spectroscopy) to directly extract bulk properties of strongly correlated bosons from measurements on a trapped system -a fundamental requirement in the future perspective of quantum simulations of complex quantum systems realized with cold atoms. The method relies on a simple, universal relationship between the trapping potential and the effective chemical potential for the particles in the trap center, which we numerically elucidate in the case of the Bose-Hubbard model realized in optical lattices. Measuring the response of the central density in the trap to the variation of the trapping potential provides direct access to the compressibility of the infinite system, a piece of information which is not directly accessible to current experimental setups and which is crucial to extract the energy gap over the ground state of the Hamiltonian implemented in the system. The method offers the possibility to extract the phase diagram of the Bose-Hubbard model with high resolution. Most remarkably, the joint measurement of the compressibility and of the coherent fraction (obtained via time-of-flight techniques [3]) provides clear evidence for the realization of a Bose-glass state in the center of the trap. We have demonstrated this property in the case of an incommensurate superlattice as recently realized in experiments [8], although the same technique can be applied to different realizations of random or pseudo-random potentials [7].
From the experimental point of view, this method requires the application of an extra dipolar trap whose strength can be controlled independently of that of the optical lattice, and the measurement of the optical depth of the cloud over a region of order ∼ 10 wavelengths (∼ 10µm) of the optical lattice in all three spatial directions. The measurements of the central density and of the coherent fraction cannot typically be performed in the same shot, so that special care is needed in mantaining the number of particles N fixed from shot to shot to achieve the same experimental conditions. This can be typically achieved by post-selecting only those measurements with the same total N in the trap. Useful discussions with N. Bar-Gill, L. Fallani, C. Fort, and M. Rigol are gratefully acknowledged.