Small quench dynamics as a probe for trapped ultracold atoms

Finite systems of bosons and/or fermions described by the Hubbard model can be realized using ultracold atoms confined in optical lattices. The ground states of these systems often exhibit a coexistence of compressible superfluid and incompressible Mott insulating domains. We analyze such systems by studying the out-of-equilibrium dynamics following a weak sudden quench of the trapping potential. In particular, we show how the temporal variance of the site occupations reveals the location of spatial boundaries between compressible and incompressible regions. The feasibility of this approach is demonstrated for several models using numerical simulations. We first consider integrable systems, hard-core bosons (spinless fermions) confined by a harmonic potential, where space-separated Mott and superfluid phases coexist. Then, we analyze a nonintegrable system, a $J\text{−}V\text{−}{V}^{\prime }$ model, with coexisting charge density wave and superfluid phases. We find that the temporal variance of the site occupations is a more effective measure than other standard indicators of phase boundaries such as a local compressibility. Finally, to make contact with experiments, we propose a consistent estimator for such temporal variance. Our numerical experiments show that the phase boundary is correctly spotted using as little as 30 measurements. Based on these results, we argue that analyzing temporal fluctuations is a valuable experimental tool for exploring phase boundaries in trapped atom systems.

Figure 1

Verification of Eqs. (4) and (5), which predict ${\chi }_{AB}\propto L$ and ${\mathrm{\Delta }}_B{\mathcal{A}}^2\propto L^2$ when $q=1$. A best fit to algebraic scaling gives ${\mathrm{\Delta }}_B{\mathcal{A}}^2\propto L^{2.0}$ and ${\chi }_{AB}\propto L^{1.1}$. We consider a tight binding model at half filling with both the observable and perturbation set to $\widehat{A}=\widehat{B}={\sum }_i{(-1)}^i{\widehat{n}}_i$. ${\mathrm{\Delta }}_B{\mathcal{A}}^2$ and ${\chi }_{AB}$ are made dimensionless by dividing by their values at $L_0=102$, i.e., we plot ${\tilde{\chi }}_{\mathrm{AB}}={\chi }_{\mathrm{AB}}\left(L\right)/{\chi }_{\mathrm{AB}}\left(L_0\right)$ and ${\mathrm{\Delta }}_B{\tilde{\mathcal{A}}}^2={\mathrm{\Delta }}_B{\tilde{\mathcal{A}}}^2\left(L\right)/{\mathrm{\Delta }}_B{\tilde{\mathcal{A}}}^2\left(L_0\right)$.

Figure 2

(a) “Wedding cake” site occupation profile of hard-core bosons in a one-dimensional harmonic trap described by Eq. (6). The system consists of $L=500$ sites and $N=250$. The Hamiltonian parameters are $\lambda =10,\epsilon =0.2$, and $\delta \lambda =L^{-2}$ ($J=1$ throughout). The phase boundaries between the Mott plateau located at the trap center and the adjacent superfluid regions can be detected by the conventional local compressibility ${\kappa }_i$ (red) and by the temporal variance of the site occupations $\mathrm{\Delta }{\mathcal{N}}_i^2$ (green) introduced in this work. (b) A closer look at the superfluid region for the system shown in (a) reveals temporal variance peaks at the interface between the superfluid and the Mott insulator. (c) Finite-size scaling of the maximum temporal variance of the site occupations and of the compressibility vs $L$ for the Hamiltonian in Eq. (6). We find $\mathrm{\Delta }{\mathcal{N}}_{\max }^2\propto L^{0.83}$ and ${\kappa }_{\max }\propto L^{0.05}$. Both quantities in this plot are made dimensionless by dividing by their values at $L_0=50$, i.e., ${\tilde{\kappa }}_{\max }={\kappa }_{\max }\left(L\right)/{\kappa }_{\max }\left(L_0\right)$ and $\mathrm{\Delta }{\tilde{\mathcal{N}}}_{\max }^2=\mathrm{\Delta }{\mathcal{N}}_{\max }^2\left(L\right)/\mathrm{\Delta }{\mathcal{N}}_{\max }^2\left(L_0\right)$. (d) Dependence of the normalized temporal variance $(\mathrm{\Delta }{\mathcal{N}}_i^2/\delta {\lambda }^2)$ on the quench amplitude $\delta \lambda$ for the Hamiltonian in Eq. (6) with all other parameters as in (a).

Figure 3

(a) Unit cell averaged site occupancy in the presence of a staggered potential Eq. (10). This is a system with $L=500,N=150$, and parameters $\lambda =10,\epsilon =0.2,\delta \lambda =1/L^2,V_0=1.5$. (b) A closer look at the superfluid region for the system shown in (a) reveals temporal variance peaks at the interface between the superfluid and the Mott insulator. (c) Finite-size scaling of the maximum temporal variance of the site occupations and of the compressibility vs $L$ for the Hamiltonian in Eq. (10). We find $\mathrm{\Delta }{\mathcal{N}}_{\max }^2\propto L^{0.80}$ and ${\kappa }_{\max }\propto L^{0.14}$. Both quantities in this plot are made dimensionless by dividing by their values at $L_0=50$, i.e., ${\tilde{\kappa }}_{\max }={\kappa }_{\max }\left(L\right)/{\kappa }_{\max }\left(L_0\right)$ and $\mathrm{\Delta }{\tilde{\mathcal{N}}}_{\max }^2=\mathrm{\Delta }{\mathcal{N}}_{\max }^2\left(L\right)/\mathrm{\Delta }{\mathcal{N}}_{\max }^2\left(L_0\right)$.

Figure 4

(a) Spatial profile of the temporal variance of the site occupations $\mathrm{\Delta }{\mathcal{N}}_i^2$ and of the local compressibility ${\kappa }_i$ for the model in Eq. (13). We initialize the system with 19 sites and 5 particles in the ground state with parameters $J=1,V=8.0,V^{\prime }=0.5,$ and $\lambda =0.1225$. The quench is performed by changing the trap potential from $\lambda$ to $\lambda +\delta \lambda$ with $\delta \lambda =0.0061$. (b) Dependence of the variance on the quench amplitude $\delta \lambda$.

Figure 5

(a) Distributions of the site occupations ${\mathcal{N}}_i\left(t\right)$ at sites near the interface between the CDW-I and the superfluid phase. (b) and (d) Distribution function of the site occupation at a site deep in the CDW-I regime (site $i=1$) and at a site at the edge of the CDW-I domain (site $i=10$), respectively. (c) and (e) Time depend-ence of ${\mathcal{N}}_i\left(t\right)$ corresponding to (b) and (d), respectively. These results are obtained from simulations with the same Hamiltonian and system parameters as in Fig. 4. Each ${\mathcal{N}}_i\left(t\right)$ is sampled at $N=4\times {10}^4$ random times uniformly distributed in $[0,T]$ with $T=40\hslash /J$.

Figure 6

Estimating the temporal variance. (a) An example of the temporal variance estimator $v$ at each site for $N_S=3$ and $N_T=10$. All parameters are the same as for Fig. 4. We compute $v$ independently for each of the $L=19$ sites of the system. (b) Scaling of the variance of the estimator $v$ at the site $i=10$ with $N_T$, for different values of $N_S$. The fit shows $\mathsf{var}\left(s_i\right)\sim N_T^{-0.97}$ for the $N_S$ values considered, in accordance with our prediction that $v$ is a consistent estimator. (c) Results of a numerical experiment to compute $p_M$, the probability that $\left\{v_i\right\}$ and the exact variance have maxima at the same sites $(i=9,10)$. ${10}^4$ samples of $\left\{v_i\right\}$ were used to compute these probabilities.