Techniques to measure quantum criticality in cold atoms

We describe how rescaling experimental data obtained from cold atom density profiles can reveal signatures of quantum criticality. We identify a number of important questions which can be answered by analyzing experimental data in this manner. We show that such experiments can distinguish different universality classes and that the signatures are robust against temperature, noise, and finite system size.

Figure 1

Quantum critical crossovers in the Bose-Hubbard model. (a) Normal (Mott insulator) to superfluid phase boundaries for (left to right) temperatures $T/U=0.00,0.06,0.12,\dots ,0.96$, calculated via finite temperature Gutzwiller mean field theory in $d=2$. The parameters $t$, $\mu$, and $U$ are the tunneling rate, chemical potential, and on-site interaction energy of the Bose-Hubbard model, respectively. Paths (1,2) are governed by the dilute Bose gas universality class, and path (3) (passing through the tip) is governed by the $O(2)$ universality class. (b) Slice through the quantum phase transition at fixed $t/U$. Slices with fixed $\mu /U$ are similar. The abbreviations “SF,” “MI,” and “QC” denote the superfluid, Mott insulator-like normal fluid, and quantum critical regime, respectively, with the Mott insulator being strictly defined only at $T=0$. The shaded region indicates where the physics is governed by the quantum critical point. The dashed lines represent smooth crossovers between the qualitatively distinct “QC” and “low temperature” regions. Deep in the former, $T$ is the only relevant energy scale.

Figure 2

Extracting universal behavior and dynamical critical exponents from density profiles of the trapped one-dimensional Bose-Hubbard model. (a) Exact density profiles of the one-dimensional harmonically trapped hard-core Bose-Hubbard model for $N=70$ particles at temperatures $\mathrm{T̄}\equiv T/t=0.1,0.24,0.38$, with larger temperatures corresponding to lower central density. Here, $r$ is the radial displacement in the trap and $\ell$ is the lattice spacing. These density profiles are nonuniversal; for example, they depend on temperature. (b) Our construction for obtaining universal scaling curves applied to this system, plotting $\mathrm{\kappa ̄}{\mathrm{T̄}}^{d/z}$ versus $\mathrm{n̄}/{\mathrm{T̄}}^{1/z}$ (defining $\mathrm{\kappa ̄}\equiv \kappa -{\kappa }_0$, $\mathrm{n̄}\equiv n-n_0$, and $\mathrm{T̄}=T/t$) for this $d=1$, $z=2$ transition and temperatures $\mathrm{T̄}=0.1,0.17,0.24,0.31,0.38$ (from closest to the shaded grey band to farthest). The compressibility is approximated by $\kappa =\partial n/\partial \mu \approx (\partial n/\partial r)/(m\omega {}^{2}r),$ where $\omega$ is the trap frequency, and $\partial n/\partial r$ is obtained by numerically differentiating the density. Lower temperatures display a larger region of collapse. We observe good collapse up to $T~0.25t$, and see that the analysis accurately reproduces the homogeneous infinite system’s scaling curve (shaded gray line) within $\lesssim 10\%$ for $T\lesssim 0.05t$ for the transition near $n=1$ ($\mathrm{n̄}<0$) and for $T\lesssim 0.15t$ for the transition near $n=0$ ($\mathrm{n̄}>0$). This collapse occurs even for drastically different density profiles obtained by adjusting the trap depth in place of temperature (not shown). With moderately larger particle numbers ($N~200$, not shown), the simulated data at low temperatures even more accurately reproduces the infinite homogeneous system’s universal scaling function (the extracted curve lies within the shaded gray region).

Figure 3

Extracting universal behavior and dynamical critical exponents from density profiles of the trapped two-dimensional Bose-Hubbard model. Each panel shows an application of our analysis procedure to simulated density profiles (as described in text) for temperature $T/t=1/4,1/2,1,2,$ and $4$ using the data in a density range $|n-1|<0.15$. The symbols $\mathrm{n̄}$, $\mathrm{\kappa ̄}$, and $\mathrm{T̄}$ are defined in the legend of Fig. 2. Lower temperature curves are identified by noting that they span a wider $\mathrm{n̄}$ range. On the left (a and c), we plot $\mathrm{\kappa ̄}$ versus $\mathrm{n̄}{\mathrm{T̄}}^{-1}$, while on the right (b and d) we plot $\mathrm{\kappa ̄}{\mathrm{T̄}}^{-1}$ versus $\mathrm{n̄}{\mathrm{T̄}}^{-2}$: these will show collapse if the dynamical critical exponent is, respectively, $z=2,1$. Top (a and b) shows data with $t/U=0.0400$, which should be described by the dilute Bose gas universality class ($z=2$). Bottom (c and d) shows data with $t/U=0.0585$, which should be better described by the $O(2)$ rotor model universality class ($z=1$): the tip of the $n=1$ Mott lobe in the homogeneous system is at $t/U=0.0593$. As expected, we see collapse in (a) and (d). The scatter in data points corresponds to stochastic noise in our Monte Carlo simulations amplified by the differentiation. This noise is of comparable size to what would be seen in an experiment.

Figure 4

Universality classes of the Bose-Hubbard model. (a) Annotated zero temperature Bose-Hubbard model phase diagram (cf. Fig. 1). Transitions via paths (1-2) are described by the dilute Bose gas (DBG) universality class while path (3) is described by the $O(2)$ universality class. Shaded regions schematically depict where each universality class holds: DBG physics governs the (blue) region along the lobe edge and $O(2)$ rotor physics governs the (orange) “bowtie”-shaped region near the tip. The entire (green) region in the area around the tip—including both the bowtie and part of the lobe edge—is described by the finite density $O(2)$ model. (b) Finite temperature phase diagram along path (4). At zero temperature, this path crosses two quantum phase transitions going from a superfluid to a Mott insulator fs and back. Each phase transition is in the DBG universality class and displays a quantum critical fan, inside of which the temperature sets the only length scale. $O(2)$ physics is recovered in the particle-hole symmetric regions where the quantum critical fans overlap. The “finite density $O(2)$” model universally governs this entire phase diagram including all of the crossovers (dashed lines) and phase transitions (solid lines) shown in this panel.