Emergence of spatially extended pair coherence through incoherent local environmental coupling

We demonstrate that quantum coherence can be generated by the interplay of coupling to an incoherent environment and kinetic processes. This joint effect even occurs in a repulsively interacting fermionic system initially prepared in an incoherent Mott insulating state. In this case, coupling a dissipative noise field to the local spin density produces coherent pairs of fermions. The generated pair coherence, while metastable, is long lived and spatially extended. This conceptually surprising approach provides a path towards a better control of quantum many-body correlations.

Figure 1

Time evolution of the momentum distribution of local pairs: $C_{\mathbf{k}}=\frac{1}{V}{\sum }_{\mathbf{r},\mathbf{d}}{e}^{-ia\mathbf{k}·\mathbf{d}}\langle {\widehat{c}}_{\mathbf{r}\downarrow }^{†}{\widehat{c}}_{\mathbf{r}\uparrow }^{†}{\widehat{c}}_{\mathbf{r}+\mathbf{d}\uparrow }{\widehat{c}}_{\mathbf{r}+\mathbf{d}\downarrow }\rangle$. A chain of $L=36$ lattice sites is prepared at $t=0$ in a perfect Mott insulating state where pair correlations are absent. A fast buildup in occupation of the momenta except close to $k=\frac{\pi }{a}$ takes place at short times (plotted on a linear time scale). Then, over time, all momenta except $k=\frac{\pi }{a}$ become homogeneously occupied, signaling the generation of the coherence of pairs over longer distances (plotted on a logarithmic scale). The evolution is obtained using the effective diffusion equation (3) with $\frac{U}{\hslash \Gamma }=1.5$ and ${\gamma }_0=\frac{8J^2}{{\hslash }^2\Gamma }$.

Figure 2

Left: Example of the effective creation and diffusion of pair correlations. The system evolution is described here using a locally factorized representation of the density matrix: Each shaded circle corresponds to a local element of this density matrix. Right: Within the adiabatic elimination method, the evolution is based on the effective coupling of two states (lower and upper states) of the decoherence-free subspace via a virtual excitation (center). The virtual state is reached by the hopping process and can decay with $\Gamma$ and dephase due to interaction $U$. Here this process is shown for a state with no pair correlations (lower state) connected to a state containing pair correlations (upper state) through the creation process (box) presented on the left panel.

Figure 3

$C_d\left(t\right)$ as a function of time $t$ as described by the diffusion equation (3). The chain is prepared in a perfect Mott insulator with $L=36$ sites and evolves with $\frac{U}{\hslash \Gamma }=1.5$. Arrows mark the $t\to \infty$ limit. The double occupancy, $C_{d=0}$, and nearest-neighbor pair correlation, $C_{d=1}$, raise quickly and feed the delayed increase of correlations at large distances. Inset: evolution of the diffusion constant as a function of time. $D\left(t\right)$ becomes time independent as the double occupancy, $C_{d=0}$, saturates.

Figure 4

${\tilde{C}}_d\left(t\right)$, the staggered pair correlations, are shown as a function of distance, $d$, for three different times in a chain of $L=36$. We compare the solution of the diffusion equation (3) with time-dependent density matrix renormalization group (DMRG) simulations. For both cases the initial conditions are taken from ground-state DMRG calculations at $U=12J$. We use $\frac{U}{\hslash \Gamma }=1.5$ and in the time-dependent DMRG $U=12J$. Inset: Symbols represent the variance of the pair correlation distribution vs time. Lines are linear fits for $10<{\gamma }_{0}t<20$.

Figure 5

The variance of the distribution of the staggered pair correlations ${\tilde{C}}_d\left(t\right)$ with distance for different dissipative strength $\frac{\hslash \Gamma }{J}$ at constant $U=12J$ in a chain of length 36. The linear rise of the variance at larger times is in accordance with a diffusive formation of the pair correlations. Symbols are taken from DMRG simulations, and straight lines are linear fits $\overline{d^2}=2D_{\text{eff}}t+\text{const.}$ for values with $\overline{d^2}>2.5$. Inset: $D_{\text{eff}}$ as a function of dissipative coupling at $U=12J$ (symbols). The straight line corresponds to $D(t\to \infty )$ derived within perturbation theory and agrees well with the numerical results for large coupling strengths.