Dissipative quantum dynamics of fermions in optical lattices: A slave-spin approach

We investigate the influence of a Markovian environment on the dynamics of interacting spinful fermionic atoms in a lattice. To explore the physical phenomena occurring at short times, we develop a method based on a slave-spin representation of fermions that is amenable to the investigation of the dynamics of dissipative systems. We apply this approach to two different dissipative couplings that can occur in current experiments: a coupling via the local density and a coupling via the local double occupancy. We complement our study based on this method, with results obtained using the adiabatic elimination technique and with an exact study of a two-site model. We uncover that the decoherence is slowed down by increasing either the interaction strength or the dissipative coupling (the Zeno effect). We also find, for the coupling to the local double occupancy, that the final steady state can sustain single-particle coherence.

Figure 1

Schematic of the adiabatic elimination treatment. The slow diagonal states ${\rho }_{\vec{m}}^{\vec{m}}$ and ${\rho }_{{\vec{m}}_{l,\uparrow }}^{{\vec{m}}_{l,\uparrow }}$ are connected via excitations (arrow $J$) to the fast decaying state ${\rho }_{{\vec{m}}_{l,\uparrow }}^{\vec{m}}$ on which both the interaction, $U$, and dissipation, $\gamma$, terms can act.

Figure 2

Time evolution of the normalized kinetic energy for a dissipative coupling $\hslash \gamma =0.1J_{\text{eff}}$ to the local density and for various interaction strengths in (a) linear scale and (b) log scale. These results are obtained within the slave-spin approach solving Eqs. (36, 37, 38, 39). The evolution begins from the metallic ground state of the mean-field Hamiltonian describing the charge sector [Eq. (33)] with $Z(t=0)=1$, 0.44, and 0.098 for $U/J_{\text{eff}}=0$, 1.5, and 1.9, respectively.

Figure 3

Time evolution of the normalized kinetic energy for different dissipative couplings to the local density for an interaction strength $U=J_{\text{eff}}$ in (a) linear scale and (b) log scale. The results are obtained within the slave-spin approach solving Eqs. (36, 37, 38, 39). Here again the evolution begins from the metallic ground state of the mean-field Hamiltonian describing the charge sector [Eq. (33)] with $Z(t=0)=0.75$ for $U=J_{\text{eff}}$. The two dotted lines denote pure exponential decay $\propto e^{-\gamma t}$ for $\hslash \gamma /J_{\text{eff}}=0.01$ and $0.1$.

Figure 4

Real (a) and imaginary (b) parts for ${\lambda }_{n,2}$ (solid black), ${\lambda }_{n,3}$ (dotted orange), ${\lambda }_{n,4}$ (dashed blue), and ${\lambda }_{n,\mathrm{Zeno}}$ (dotted red) vs $U/J$ for a system of two fermions (one spin up and one spin down) on two sites with dissipative coupling $\hslash \gamma /J=1$ to the local density. The solid thin lines correspond to analytical predictions for the real parts, $-8\gamma J^2/[{\left(\hslash \gamma \right)}^2+U^2]$ (red) and $-2\gamma$ (blue), and for the imaginary parts, $U/\hslash$ (orange) and $-U/\hslash$ (black). The real parts of ${\lambda }_{n,2}$ and ${\lambda }_{n,3}$ are both equal to $-J/\hslash$, whereas the imaginary parts of both ${\lambda }_{n,4}$ and ${\lambda }_{n,\mathrm{Zeno}}$ are null.

Figure 5

Real (a) and imaginary (b) parts for ${\lambda }_{n,2}$ (solid black), ${\lambda }_{n,3}$ (dotted orange), ${\lambda }_{n,4}$ (dashed blue), and ${\lambda }_{n,\mathrm{Zeno}}$ (dotted red) vs $\hslash \gamma /J$ for a system of two fermions (one spin up and one spin down) on two sites with interaction strength $U/J=0.1$ and dissipative coupling to the local density. The solid thin lines correspond to analytical predictions for the real parts, $-8\gamma J^2/[{\left(\hslash \gamma \right)}^2+U^2]$ (red) and $-2\gamma$ (blue). The imaginary parts of both ${\lambda }_{n,4}$ and ${\lambda }_{n,\mathrm{Zeno}}$ are null.

Figure 6

Time evolution of the kinetic energy for a two-site two-fermion system with a dissipative coupling $\hslash \gamma /J=3$ to the local density. This figure highlights the presence of two regimes: a short-time exponential decay with a rate proportional to $\gamma$ and a long-time decay dominated by the rate ${\lambda }_{n,\text{Zeno}}$. The black dash-dotted lines are the corresponding Zeno predictions.

Figure 7

Time evolution of the normalized kinetic energy for an interaction strength $U=J_{\text{eff}}$ and different dissipative couplings to the local double occupancy. The results are obtained within the slave-spin approach solving Eqs. (44, 45, 46, 47, 48, 49, 50). The evolution begins from the metallic ground state of the mean-field Hamiltonian describing the charge sector [Eq. (33)] with $Z(t=0)=0.75$ for $U=J_{\text{eff}}$.

Figure 8

Real (a) and imaginary (b) parts for ${\lambda }_{d,1}$ (dashed blue), ${\lambda }_{d,2}$ (black), ${\lambda }_{d,3}$ (dotted orange), and ${\lambda }_{n,\mathrm{Zeno}}$ (dotted red) vs $\hslash \gamma /J$ for two fermions (one spin up and one spin down) on two sites with $U/J=3$ and dissipative coupling to the local double occupancy. The solid thin lines correspond to analytical predictions for the real parts, $-24\gamma J^2/[{\left(\hslash \gamma \right)}^2+U^2]$ (red), $-\gamma /2$ (orange), $-\gamma$ (blue), and for the imaginary parts, $-U/\hslash$ (black) and $U/\hslash$ (orange). The imaginary parts of both ${\lambda }_{d,1}$ and ${\lambda }_{d,\mathrm{Zeno}}$ are null.

Figure 9

Time evolution of the normalized kinetic energy for a dissipative coupling $\hslash \gamma /J_{\text{eff}}=10$ to the local double occupancy and different interaction strengths. The results are obtained within the slave-spin approach solving Eqs. (44, 45, 46, 47, 48, 49, 50). Here again the evolution begins from the metallic ground state of the mean-field Hamiltonian describing the charge sector [Eq. (33)] with $Z(t=0)=0$, 0.8, and 0.098 for $U/J_{\text{eff}}=0$, 0.8, and 1.9, respectively.

Figure 10

Real (a) and imaginary (b) parts for ${\lambda }_{d,1}$ (dashed blue), ${\lambda }_{d,2}$ (black), ${\lambda }_{d,3}$ (dotted orange), and ${\lambda }_{n,\mathrm{Zeno}}$ (dotted red) vs $U/J$ for two fermions (one spin up and one spin down) on two sites with dissipative coupling $\hslash \gamma /J=10$ to the local double occupancy. The solid thin lines correspond to analytical predictions for the real parts $-24\gamma J^2/[{\left(\hslash \gamma \right)}^2+U^2]$ (red), $-\gamma /2$ (orange), $\gamma$ (blue), and for the imaginary parts $-U/\hslash$ (black) and $U/\hslash$ (orange). The imaginary parts of both ${\lambda }_{d,1}$ and ${\lambda }_{d,\mathrm{Zeno}}$ are null.