Main

To deduce the bulk properties of homogeneous systems from the observed properties of non-uniform systems, local density approximation (LDA) naturally comes to mind. In this approximation, the properties of a non-uniform system at a given point are deduced from their bulk values assuming an effective local chemical potential. To the extent that LDA is valid, determining bulk thermodynamic quantities as functions of chemical potentials amounts to determining their spatial dependencies in confining traps. In present experiments with ultracold atomic gases, column-integrated density (or density for two-dimensional (2D) experiments) is the only local property that can be accessed. No other thermodynamic quantities have been measured because there are no clear ways to access them. Here, we show that by studying changes in density caused by external perturbations, one can access the quantities mentioned above from density data. The deduction of superfluid density is particularly important, as it is a fundamental quantity that has eluded measurement since the discovery of Bose–Einstein condensation.

Our first step is to use the density near the surface of the quantum gas as a thermometer. Within LDA, the density is n(x)=n(μ(x),T), where n(μ,T) is the density of a homogeneous system with temperature T and chemical potential μ,μ(x)=μ−V (x); is a harmonic trapping potential with frequencies ωi and M is the mass of the atom. Near the surface, the density is sufficiently low that one can carry out a fugacity expansion to obtain

where is the thermal wavelength and kB is the Boltzman constant. For a p-component quantum gas in a single trap, we have α=p. If the quantum gas is in the lowest band of a cubic lattice with hopping integral t and lattice spacing d, then α=p(λ/d)3[I0(2t/kBT)]3, where I0(x) is the Bessel function of the first kind (see the Methods section). The corresponding column density (with r=(x,y)) is

Equation (2) has been widely used to determine μ and T of quantum gases in single traps but not yet for gases in optical lattices, as the density at the surface in such cases is very low. The lack of accurate thermometry in optical lattices has been the bottleneck for extracting information from present experiments. For example, it has prevented mapping out the phase diagram of the Bose–Hubbard model at finite temperature despite many years of studies. It has also given rise to concern about heating effects in current optical lattice experiments11,12,13,14. To make use of the asymptotic forms in equations (1) and (2), we need imaging resolutions comparable to a few lattice spacings (typically a few micrometres). Recently, the density of a 3D quantum gas has been imaged using a focused electron beam with extremely high resolution (0.15 μm; ref. 15). Furthermore, advances in optical imaging techniques have also enabled resolutions from a few lattice spacings10 to even one lattice spacing (M. Greiner, reported in APS March Meeting, 2009). These developments show that the capability to determine μ and T accurately using density measurements at the surface is now in place.

Before proceeding, we would like to point out that LDA has been verified in a large number of boson and fermion experiments9,10 and numerical calculations16,17. For the rest of our discussions, we shall assume that LDA is valid. One might also worry about poor signal-to-noise ratios for the density near the surface. However, by averaging over a surface layer of thickness of one or two lattice spacings, one can obtain a considerable enhancement of the signal-to-noise ratio even at the surface10,17.

With μ and T determined from the surface density, one readily obtains the equation of state n(ν,T) by identifying it with n(x), where x is given by V (x)=μ−ν. In present experiments on 3D systems, only column density is measured. To deduce n(x) from , one can use the inverse Abel transform in the case of cylindrically symmetric samples9, or a method developed by E. Mueller (private communication). The latter method first constructs the pressure P from , and then n(x) from the pressure. It works as follows: from the Gibbs–Duham equation,

we have if T is constant. By integrating the column density along dy dz, and noting that dydz=−(2π/M ωyωz)dμ for given x, we have

Again applying the Gibbs–Duham relation, we then get the 3D density (see Fig. 1)

Figure 1: An illustration of the method to construct pressure from column density.
figure 1

The 3D quantum gas is represented by a ellipse. The rod represents the column density collected in the experiment. By integrating the column density along the y direction, as shown in the rectangle, one obtains , and hence P(x,0,0) from equation (3). The density n(x,0,0) can be obtained by differentiating P(x,0,0) as in equation (4).

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As singularities of thermodynamic potentials show up in the equation of state, boundaries between different phases can be identified in the density profile. Recall that first-order and continuous phase transitions correspond to discontinuities in the first- and higher-order derivatives of P. Equation (3) implies that n and s are discontinuous across a first-order phase boundary, whereas the slope of dn/dμ and ds/dT are discontinuous for higher-order phase boundaries. The discontinuity in n has been used in a recent experiment to determine the first-order phase boundary in spin-polarized fermions near unitarity9. As dn(x,0,0)/dx∝dn(μ(x,0,0),T)/dμ, a higher-order phase boundary will show up as a discontinuity of the slope of the density. The presence of such a discontinuity has also been seen in Monte Carlo studies17.

We now turn to entropy density s(x), which is useful for identifying phases. For example, for a spin-1/2 fermion Hubbard model, if s(x) is far below kBln2 per site in a Mott phase, this is strong evidence for spin ordering. To obtain s=(dP/dT)μ, we need to generate two slightly different configurations of P(x) with different T and calculate their difference at the same μ. To do this, we change the trap frequency ωx adiabatically to a slightly different value ωx′(ωx′=ωx+δ ωx,δ ωx≪ωx). Both μ and T will then change to a slightly different value, for example, to μ′ and T′ (ref. 11). One can then measure the column density of the final state and construct its pressure function P(x,0,0). The entropy density of the initial state along the x axis is

where x and x′ are related to each other as follows (see Fig. 2):

Figure 2: An illustration of the scheme for determining entropy density s(x).
figure 2

The pressure curve P(x,0,0) and effective chemical potential μ(x) of the initial state with temperature T are shown as black lines. The corresponding quantities of the final state are shown as grey lines. The final equilibrium state is generated from the initial state by changing the trap frequency from ω to ω′ adiabatically. To find s(x), we find the position x′ related to x, with an identical effective local chemical potential, by equating μ(x,0,0)=μ′(x′,0,0). The pressures at x and x′ are denoted as P and P′ in the figure. s(x) is given by equation (5).

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We next consider superfluid density ns. It is a quantity particularly important for 2D superfluids18,19,20, as the famous Kosterlitz–Thouless transition is reflected in a universal jump in superfluid density. Without a precise determination of ns, interpretation of experimental results, be they based on quantum Monte Carlo simulations or on features of an interference pattern, will be indirect. Here, we propose a scheme to measure the inhomogeneous superfluid density in the trap. For a superfluid, we have21,22

where ns is the superfluid number density and w=vs−vn;vs and vn are the superfluid and normal fluid velocity, respectively. The term μo corresponds to the chemical potential in the vn=0 frame. A direct consequence of equation (6) is that

For a potential rotating along with frequency . If Ω is below the frequency for vortex generation, vs=0 and w2=Ω2r2. As w varies in space, we cannot apply the method developed for s(r). Instead, one can use the following procedure: let n(i)(x) be the density of a stationary system (with temperature T and chemical potential μ(i)) in a cylindrical trap with transverse frequency and longitudinal frequency ωz. Within LDA, we have, n(i)(x)=n(μ(i)(x);T;w=0), where

We then rotate this system with frequency Ω along , and adjust to , so that the temperature remains T. The chemical potential then becomes μ(f), and the density of this final state is n(f)(x)=n(μ(f)(x),T,w), where

For small w2, we have

We then write n(μ(f)(x),T,0)=n(μ(i)(x*),T,0)=n(i)(x*), where x*=(x*,y*,z*) is the point that satisfies μ(f)(x)=n(i)(x*). Specifically, we can choose , and μ(i)(x*,0,0)≡μ(f)(x,0,0). Using equation (7), we have

where ns(x,y,z)=ns(μ(f)(x),T,0). Integrating equation (8) over z and y, and noting that when x is constant, we have

Equation (9) gives ns in terms of the column densities of the initial and final state. The above formula continues to hold for non-axisymmetric traps (with ). (See also Supplementary Material for the expression for the 2D case, and an alternative scheme for obtaining ns(x).)

Our method can also be applied to obtain other important thermodynamic properties such as the staggered magnetization and the contact density of a strongly interacting fermion gas. For the latter, see Supplementary Materials. In quantum simulations of the fermion Hubbard model using two-component fermions in optical lattices23,24, the measurement of the staggered magnetization will be crucial for identifying the antiferromagnet. Consider an antiferromagnet in a cubic lattice with a staggered magnetic field, , where x=(nx,ny,nz)d are the lattice sites, ni are integers, d is the lattice spacing and is the magnitude of the staggered field. The Hamiltonian for a homogeneous system is , where HH is the Hubbard Hamiltonian, is the staggered-magnetization operator and m(x) is the spin operator at x. Antiferromagnetism corresponds to as . It is straightforward to show that

The staggered field h(x) has been produced recently25. To reduce spontaneous emission and hence heating, one can use a low-intensity laser and hence a weak field . Note that even a weak field can produce large changes in density in the spatial region close to the antiferromagnetic phase boundary, where bulk spin susceptibility diverges. So, measuring the responses to can locate the phase boundary.

As , we need to generate two configurations of P with different while fixing μ and T. We begin with an initial state with , and determine its μ, T and pressure P(x,0,0) as discussed above. We then turn on a weak adiabatically. At the same time, we adjust ω to a new value ω′, so that the temperature of the final state remains fixed at T, and the chemical potential is changed to . We then construct the pressure P′(x,0,0) of the final state. By noting that for any point (x′,0,0) in the final state, one finds a corresponding point (x,0,0) in the initial state such that their effective chemical potentials are identical, that is, μ(x,0,0)≡μ−(1/2) M ω2x2=μ′−(1/2)M ω′2x′2≡μ′(x′,0,0). We then have

The success in deducing the properties of bulk homogeneous systems hinges on two key factors. The first is the ability to determine the density, temperature and chemical potential of the trapped system with high accuracy. The second is to come up with algorithms to deduce the bulk properties of interest from the density data of non-uniform systems. A combination of precision measurements and specifically designed algorithms, aiming to uncover the properties of bulk systems both qualitative and quantitative will be crucial in realizing the full power of quantum simulation.

Methods

Equation (1) is derived as follows: near the surface, the gas is in the low-fugacity limit. The number density can then be obtained by fugacity expansion. To the lowest order in fugacity, the number of particles per site of a (homogeneous p-component) quantum gas in a cubic lattice is

where the k-sum is over the first Brillouin zone, is the lattice constant and t is the tunnelling integral. In the continuum limit, the above expression becomes

where I0(x) is the Bessel function of the first kind. The number of particles per unit volume is then

where is the thermal wavelength.