Nonequilibrium dynamics of bosonic atoms in optical lattices: Decoherence of many-body states due to spontaneous emission

We analyze in detail the heating of bosonic atoms in an optical lattice due to incoherent scattering of light from the lasers forming the lattice. Because atoms scattered into higher bands do not thermalize on the time scale of typical experiments, this process cannot be described by the total energy increase in the system alone (which is determined by single-particle effects). The heating instead involves an important interplay between the atomic physics of the heating process and the many-body physics of the state. We characterize the effects on many-body states for various system parameters, where we observe important differences in the heating for strongly and weakly interacting regimes, as well as a strong dependence on the sign of the laser detuning from the excited atomic state. We compute heating rates and changes to characteristic correlation functions based on both perturbation-theory calculations and a time-dependent calculation of the dissipative many-body dynamics. The latter is made possible for one-dimensional systems by combining time-dependent density-matrix-renormalization-group methods with quantum trajectory techniques.

Figure 1

Schematic picture of the leading-order heating processes via scattering of laser photons in a red-detuned (left) and a blue-detuned (right) lattice in the Lamb-Dicke limit (see text). In a red-detuned lattice the fastest process returns the scattered atom to the lowest band, while this process is strongly suppressed in the blue-detuned lattice. In both cases there are higher-order processes in the Lamb-Dicke parameter $\eta$ of the lattice, in which the atoms are scattered into higher Bloch bands. In the limit of a deep lattice, these processes occur at identical rates in the red-detuned and blue-detuned cases.

Figure 2

Plot of the function $F(\mathit{\xi })$ for $\mathit{\xi }$ parallel to $\mathit{d}$ [see Eq. (21) for the definition of $F$]. Note that the localized form of this function reflects the localization of atoms in spontaneous emission events.

Figure 3

Comparison between the total process rate ${\Gamma }_{\mathrm{scatt}}/{\gamma }_0$ (solid lines) and the rate for processes back to the lowest band (dashed lines) for $\Delta <0$ (upper lines) and a $\Delta >0$ (lower lines) in the ground state of a 1D lattice (using DMRG ground states for Bosons in a 1D lattice: $V_x=V_y=30E_R,\hspace{0.5em}{V}_z=10E_R$). The rates include scattering from all three lattice-generating beams.

Figure 4

Relative rate of change of the (integrated) single-particle density matrix $S_{1\mathrm{D}}(z_1,z_2)=\int \int \mathit{dxdy}\langle {\mathrm{\psi ̂}}^{†}(x,y,z_1)\mathrm{\psi ̂}(x,y,z_2)\rangle$ in an effectively 1D lattice ($V_x=V_y=30E_R$; $V_z=10E_R$) generated by a red $(a)$ and a blue $(b)$ detuned laser. In the tightly bound transversal directions the atoms are assumed to be in the lowest band. Scattering from all three lattice-generating laser beams is taken into account (with weights corresponding to the lattice depths). (The lattice constant is denoted by $a$.)

Figure 5

Momentum distribution (in arbitrary units) of the ground state of noninteracting particles in a 1D lattice of depth $V=10E_R$ (black) and after a time ${\gamma }_{0}t=0.5$ if the lattice is generated by a red laser (red, dashed). The inset shows the central peak initially (black, solid) and after ${\gamma }_{0}t=0.5$ in a red (red, dashed) and in a blue lattice (blue, dashed-doted) (36 particles on 36 sites).

Figure 6

Development of the mean energy in a system with 36 particles on 36 sites, at $U/J=3$ just below the superfluid to Mott-insulator transition and $\gamma =0.01J$, computed from Eq. (28) via quantum-trajectories methods (solid blue line), compared with the equivalent result from first-order perturbation theory (dashed red line). Statistical error bars are shown based on the stochastic average over 1000 trajectories.

Figure 7

Momentum distribution (in arbitrary units) calculated for a 1D lattice (36 particles on 36 sites) with $U/J=3$ (i.e., in the superfluid regime) and a scattering rate $\gamma =0.01J$ from Eq. (28) via quantum-trajectories methods in a lattice of depth $V=10E_R$. Averages are taken over 1000 trajectories. The scale of the momenta axis is $1/a$, where $a$ is the lattice constant.

Figure 8

Comparison of the decay of off-diagonal correlations in the single-particle density matrix $S(i,j)=\langle b_i^{†}{b}_j\rangle$ as a function of time. These are computed from the master equation for the lowest band [Eq. (28)] with $\gamma /J=0.01$ for 24 particles on a 24-site lattice by combining quantum-trajectories methods with t-DMRG. (a) Comparison of the decay of off-diagonal elements $S(i,i+x)$ when $U/J=2$ (superfluid regime, upper lines) and $U/J=10$ (Mott-insulator regime, lower lines). In each case we show correlations at separation distances $x=1$ (solid lines) and $x=2$ (dashed lines), each of which are averaged over $i$. We see that the correlations for the Mott-insulator state at $U/J=10$ are almost constant, whereas for $U/J=2$, the decay becomes more rapid than at initial times. (b) Comparison of $S(i,i+2)$ averaged over $i$ and normalized to the value at time $t=0$, $|S(i,i+2)|[t=0]$ for $U/J=2,4,6,8,10$ (seen here from bottom to top). The dashed line shows the corresponding result from perturbation theory. Each computation was averaged over 1000 trajectories, and error bars are shown in (b). For (a), the statistical errors fit inside the line thickness.

Figure 9

Particle number statistics for the lowest band and one of the three first excited bands, computed for a noninteracting system as a function of time using a Gutzwiller ansatz for the system density operator. Initially, the statistics in the lowest band are Poissonian, corresponding to a local coherent state. The statistics in the lowest band change toward an exponential decay with $n_0$ as a function of time, while the population in the exited band increased [calculated with ${\gamma }^{(\mathit{0},0,0,0)}=0.01\mathit{zJ}$ in an isotropic 3D lattice ($z=6$) of depth $V=10E_R$].

Figure 10

Particle number statistics for the lowest band and one of the three first excited bands, computed for an interacting system with $U/(\mathit{zJ})=1$ as a function of time using a Gutzwiller ansatz for the system density operator. As in Fig. 9, we choose ${\gamma }^{(\mathit{0},0,0,0)}=0.01\mathit{zJ}$ in an isotropic 3D lattice ($z=6$) of depth $V=10E_R$.

Figure 11

Time evolution of the mean number of particles per lattice site in the lowest and first excited bands, computed using a Gutzwiller ansatz for the system density operator and plotted as a function of time for different interaction strengths. As the rate of population of higher bands is independent of the interaction strength, we see from the similarity of results for interacting and noninteracting systems that processes returning the atoms to the lowest band are very slow, leading to a lack of thermalization of energy transferred to the system by transferring particles to higher bands. As for Fig. 9, we choose ${\gamma }^{(\mathit{0},0,0,0)}=0.01\mathit{zJ}$ in an isotropic 3D ($z=6$) lattice of depth $V=10E_R$.

Figure 12

Evolution of the condensate density $|\psi (t)|{}^2$, computed using a Gutzwiller ansatz for the system density operator under Eq. (38) for different interaction parameters. The initial states are the corresponding Gutzwiller ground states. The different initial states have different nonvanishing condensate densities, which in all cases decay on the time scale set by the localization rate in the lowest band [calculated with ${\gamma }^{(\mathit{0},0,0,0)}=0.01\mathit{zJ}$ in an isotropic 3D lattice ($z=6$) of depth $V=10E_R$].

Figure 13

Evolution of the mean total energy per lattice site, computed using a Gutzwiller ansatz for the system density operator. The initial values depend on the interaction strength of the system. The increase in energy is the same in all cases, as we see in Sec. 3a2. Since we have included only one excited band, the total energy saturates, leading to the deviation from the linear increase of the exact solution. The thin dashed lines indicate the increase in energy as calculated in Sec. 3a2 [calculated with ${\gamma }^{(\mathit{0},0,0,0)}=0.01\mathit{zJ}$ in an isotropic 3D lattice ($z=6$) of depth $V=10E_R$].

Figure 14

Increase of entropy per lattice site $S[\sigma ]=-\mathbf{Tr}\{\sigma \ln \sigma \}$ for different interactions, computed using a Gutzwiller ansatz for the system density operator. Starting in the ground state, the entropy is initially zero. Due to the different initial states, the increase in entropy is higher for lower interaction parameters, reflecting the more significant change in more weakly interacting states due to spontaneous emission events [calculated with ${\gamma }^{(\mathit{0},0,0,0)}=0.01\mathit{zJ}$ in an isotropic 3D lattice of depth $V=10E_R$].