Particle statistics and lossy dynamics of ultracold atoms in optical lattices

Experimental control over ultracold quantum gases has made it possible to investigate low-dimensional systems of both bosonic and fermionic atoms. In closed one-dimensional systems there are many similarities in the dynamics of local quantities for spinless fermions and strongly interacting “hard-core” bosons, which on a lattice can be formalized via a Jordan-Wigner transformation. In this study, we analyze the similarities and differences for spinless fermions and hard-core bosons on a lattice in the presence of particle loss. The removal of a single fermion causes differences in local quantities compared with the bosonic case because of the different particle exchange symmetry in the two cases. We identify deterministic and probabilistic signatures of these dynamics in terms of local particle density, which could be measured in ongoing experiments with quantum gas microscopes.

Figure 1

(a) Diagram of a loss event in an optical lattice on site $i$ with probability ${\gamma }_i$. The density hole created will propagate through normal tunneling processes and will delocalize over time. (b) Evolution of the particle density for bosons ${\widehat{n}}_i^b$, fermions ${\widehat{n}}_i^f$, and the normalized difference of these, $\mathrm{\Delta }{n}_i$, as a function of time. In this case, loss occurs on site $i=10$ on a lattice with $M=20$ from an initial product state with a single particle on each site, so a single sign is applied to the fermionic wave function and both profiles remain identical: $\mathrm{\Delta }{n}_i=0$.

Figure 2

(a) Evolution of the difference in density distribution $\mathrm{\Delta }{n}_i$ for a system with $M=20, n_0=1, D=100, dt=0.001, J=1, {\tau }_0=1, {\delta }_M=0$. (b) Same as (a) with ${\tau }_0=2$ and $n_0=0.5$. (c) Same as (a) with ${\tau }_0=2$ and ${\delta }_M=4$. (d) Same as (a) with ${\tau }_0=2, n_0=0.5$ and ${\delta }_M=4$. These calculations are performed beginning from a product state with the corresponding densities indicated above: a single particle on each lattice site ($n_0=1$) or every odd site ($n_0=0.5$).

Figure 3

(a) Evolution of the weighted difference in the entanglement entropy $\mathrm{\Delta }S$ at every lattice bipartition $\tilde{M}\in [1,M-1]$ for a system with $M=20, n_0=1, D=100, dt=0.001, J=1, {\tau }_0=1, {\delta }_M=0$. (b) Same as (a) with ${\tau }_0=2$ and $n_0=0.5$. (c) Same as (a) with ${\tau }_0=2$ and ${\delta }_M=4$. (d) Same as (a) with ${\tau }_0=2, n_0=0.5$ and ${\delta }_M=4$. These calculations are performed beginning from a product state with the corresponding densities indicated above: a single particle on each lattice site ($n_0=1$) or every odd site ($n_0=0.5$).

Figure 4

(a) Comparison of bosonic and fermionic evolution of the middle site density $\langle {\widehat{n}}_{M/2}\rangle$ for a system with $M=16, n_0=0.5, J=1, {\gamma }_l=0.01, {\gamma }_d=0$; numerical parameters are $dt=0.001$ and $D=200$. The inset shows the total particle number $\langle {\widehat{n}}_T\rangle$ to provide some guidance over the evolution of the total occupation in the lattice as losses occur. (b) Same as (a) for the normalized total density fluctuations ${\sum }_i{\sigma }_{n_i}/M$. These calculations are performed beginning from a charge density wave at half-filling, with a particle on each even-numbered site.

Figure 5

(a) Comparison of bosonic and fermionic evolution of the middle site density $\langle {\widehat{n}}_{M/2}\rangle$ for a system with $M=16, n_0=0.5, J=1, dt=0.001, {\gamma }_l=0.01, {\gamma }_d=0$. (b) Same as (a) with ${\gamma }_d=0.01$. (c) Comparison of bosonic and fermionic evolution of the imbalance $\mathcal{I}$ with same parameters as (a). (d) Same as (c) with ${\gamma }_d=0.01$. These calculations are performed beginning from a charge density wave at half-filling, with a particle on each odd-numbered site. Note that these functions are rapidly oscillating, and that each point represents a snapshot of the values on a regularly spaced grid in time.

Figure 6

Comparison of bosonic (dashed line) and fermionic (solid line) evolution of the time-block averaged imbalance $\overline{\mathcal{I}}={\sum }_i^{i+N_{\delta t}}\left|\mathcal{I}\left(t_i\right)\right|/N_{\delta t}$ for a system with $M=16, n_0=0.5, J=1, dt=0.001, N_{\delta t}=300, {\gamma }_l=0.01, {\gamma }_d=0$, and variable off-site interaction strength $V$. The imbalance average drops over time in the presence of interaction but remain distinguishable for both species. Inset: Total density fluctuation for the same parameters, included for the purpose of comparison. Here all lines overlap while we observe relevant differences in the imbalance. These calculations are performed beginning from a charge density wave at half-filling, with a particle on each odd-numbered site.

Figure 7

(a) Evolution of the difference in density distribution $\mathrm{\Delta }{n}_i$ for a system with $M=30, n_0=0.5, D=128, dt=0.001, J=1, {\tau }_0=1, {\delta }_M=0$. (b) Same as (a) for a system with $M=20$ with no loss at $t=0$, a first loss is induced at $t={\tau }_0=1$ and a second loss occurs at $t={\tau }_1=2{\tau }_0$. In contrast with the case of the first loss chosen at $t=0$, differences between bosons and fermions are observed from the first loss event.

Figure 8

(a) Comparison of bosonic and fermionic evolution of the middle site density $\langle {\widehat{n}}_{M/2}\rangle$ for a system with $M=16, n_0=0.5, J=1, {\gamma }_l=0.01, {\gamma }_d=0$; numerical parameters are $dt=0.001$ and $D=200$. The inset shows the total particle number $\langle {\widehat{n}}_T\rangle$ to provide some guidance over the evolution of the total occupation in the lattice as losses occur. (b) Same as (a) for the normalized total density fluctuations ${\sum }_i{\sigma }_{n_i}/M$. These calculations are performed beginning from a charge density wave at half-filling, with a particle on each even-numbered site. Note that these functions are rapidly oscillating, and that each point represents a snapshot of the values on a randomly spaced grid in time. The data include statistical error bars, which are contained within the point markers in most of the cases.

Figure 9

(a) Closed system evolution of the middle site density $\langle {\widehat{n}}_{M/2}\rangle$ for a system with $M=16, n_0=0.5, J=1, {\gamma }_l=0, {\gamma }_d=0$; numerical parameters are $dt=0.001$ and $D=150$. In the absence of loss both fermionic and bosonic results are identical, fluctuating around $n_0$. Inset: Averaged density difference between fermions and bosons; as predicted this quantity is zero. (b) Same as (a) for the imbalance $\mathcal{I}$. Again, fermions and bosons show the same profile, fluctuating around zero as expected for a nondisordered system. Inset: Short-time imbalance (blue) compared with the analytic result (red) that can be derived for free fermions. Note that the disagreement occurs at time $\sim M/2J$ that corresponds to the required time by an excitation to travel through the whole lattice, which is a finite-size effect.

Figure 10

Evolution of the middle site density fluctuation from the expected long-time average $|\langle {\widehat{n}}_{M/2}\rangle -\langle {\widehat{n}}_T/M\rangle |/\langle {\widehat{n}}_T\rangle$ for a system with $M=16, n_0=0.5, J=1, {\gamma }_l=0.01, {\gamma }_d=0$, and a closed system (${\gamma }_l=0$). We provide time-block averages (lines) of the full data set (shades) for better visualization. We observe how the fermionic deviation remains approximately constant for the studied time, while the bosons and the closed system approach the long-time expected average with the former exhibits smaller deviations.