Measuring Entanglement Growth in Quench Dynamics of Bosons in an Optical Lattice

We discuss a scheme to measure the many-body entanglement growth during quench dynamics with bosonic atoms in optical lattices. By making use of a 1D or 2D setup in which two copies of the same state are prepared, we show how arbitrary order Rényi entropies can be extracted by using tunnel coupling between the copies and measurement of the parity of on-site occupation numbers, as has been performed in recent experiments. We illustrate these ideas for a superfluid-Mott insulator quench in the Bose-Hubbard model, and also for hard-core bosons, and show that the scheme is robust against imperfections in the measurements.

Figure 1

(a) Measurement of $n=2$ Rényi entropy for bosons in an optical lattice. First, two instances of the many-body state are produced (shown here for a single site in a 1D chain or 2D plane). Then tunneling is switched off within each copy, and the barrier between the copies is lowered to realize a beam-splitter operation between the copies. Finally, the parity of the atom number is measured at each site. This measurement is repeated to obtain expectation values for the swap operator $V_2$, from which the Rényi entropy can be computed (see the text). (b) Example measurement outcome for a single shot on a quantum chain. Here the measurement result for the whole system swap operator $V_2^{\{1,\dots ,7\}}$ is 1, since the total number of particles in copy 2 is even. For the swap of the first three sites $V_2^{\{1,2,3\}}$ is $-1$, since this number is odd.

Figure 2

Entanglement buildup and measurement, for both (i) soft-core bosons in 1D after a quench from a Mott insulator at $U/J=10$ to a superfluid at $U/J=1$ and (ii) hard-core bosons with tunneling $J$, beginning from an initial state where every second lattice site is occupied. (a) Quenches in small systems with 8 particles on 8 lattice sites, showing von Neumann and Rényi entropies for a bipartite splitting in the center of the system; (b) the same as (a), but for a system of 30 particles on 30 sites, and also including $2S_2-S_3$ (dotted line). (c) The number of single shot measurements ($\#$) required to determine $S_n$ for the Bose-Hubbard quench in (b) with a relative accuracy $\sigma$. For $2S_2-S_3$, measurements are distributed between ($S_2$) and ($S_3$) to minimize the total number. (d) Limitations of TDMRG simulations for the Bose-Hubbard quench based on different bond dimensions $D$ [10], shown here with $D=2^l$ for $4\le l\le 9$, with corresponding bound $S_{\mathrm{VN}}=l$. Note that we use logarithms to base 2 throughout.

Figure 3

Errors $\Delta =[\mathrm{tr}({\rho }_{a\dots M}^2)-⟨\prod _{j=a}^M{e}^{i\pi {\widehat{n}}_j}{⟩}_t]/\mathrm{tr}({\rho }_{a\dots M}^2)$ that are introduced by an imperfect beam-splitter operation, determined via TDMRG simulation of the Bose-Hubbard quench in Fig. 2a and the resulting measurement process. (a) The effect of timing errors, with $T=\pi /(4J_{12})+ϵ$; (b) error introduced by a finite interaction strength $u_{ϵ}=U_{ϵ}/J_{12}$ during the measurement.

Figure 4

Deviations resulting from slightly different evolutions in the two copies, $\Delta =[\mathrm{tr}({\rho }_{a\dots M}^2)-⟨\prod _{j=a}^M{e}^{i\pi {\widehat{n}}_j}{⟩}_t]/\mathrm{tr}({\rho }_{a\dots M}^2)$. (a) Results for slightly different tunneling amplitudes in each copy after the Mott insulator-superfluid quench from Fig. 2a. (b) Results for a case where each copy has an additional harmonic trapping potential with ${\omega }_1=2\times 10J(N/2{)}^{-2}$ for the first copy, and we otherwise perform the same quench as in (a).