Thermalization in a Coherently Driven Ensemble of Two-Level Systems

We investigate the coherent quantum time evolution of a driven mesoscopic chain of two-level systems that interact via the van der Waals interaction in their excited state. The Hamiltonian is the sum of a classical lattice gas Hamiltonian and an off-diagonal driving term without classical counterpart. Starting from a product state we observe—beyond a certain interaction strength—thermalization of the system with respect to observables of the classical lattice gas. This transition can be studied experimentally with Rydberg atoms, ions, or polar molecules. We suggest how to experimentally determine the temperature of the thermal state which should allow for thermometry of the internal degrees of freedom of cold Rydberg gases whose external dynamics is frozen.

Figure 1

(a) Temporal evolution of the mean excitation number $⟨N⟩(t)$ and the correlation functions $g_{12}(t)$ and $g_{13}(t)$ ($L=20$, $V=5\Omega$, $\mu =0$). All quantities reach their steady-state—characterized by small amplitude oscillations—in the interval ${10}^1<\Omega t<{10}^2$. (b) Expectation value of the number of excited particles $⟨N⟩$ (blue) and its fluctuations $\Delta N$ (red), studied over the time interval $47<\Omega t<50$, as a function of the interaction strength. The thin lines correspond to 20 instantaneous snapshots taken during that period. Thick solid lines represent the average over that time interval. For very small $V$ no relaxation into a steady state occurs as large oscillations of the $⟨N⟩$ and $\Delta N$ are present. For large values of $V$ the instantaneous values of $⟨N⟩$ and $\Delta N$ differ only very little from the averaged ones. Moreover, the fluctuations $\Delta N$ are small. We refer to this as the thermal regime.

Figure 2

(a) Entropy function for $L=20$. Shown is $S(n,\epsilon )-\log \dim$, where dim is the total number of states of the system. The interaction energy is coarse grained over the interval ${\epsilon }_k=[kV-V/2,kV+V/2]$. (b) Evolution of $⟨N⟩$ into equilibrium for $V=5\Omega$. (c) The logarithm of $p_{\epsilon }/g(\epsilon )$ is given by the dotted line. The red solid line represents a linear fit.

Figure 3

(a) Inverse temperature as a function of the interaction strength after a propagation time of $\Omega t=250$ ($L=20$). The error bars delimit the 95% confidence interval. (b) Mean density $⟨N⟩/L$ of the classical lattice gas in thermal equilibrium plotted as a function of the interaction strength $V$ (red or dashed curve). The inverse temperature was taken from panel (a). The black solid curve shows the result if next-nearest-neighbor interactions are accounted for perturbatively. For comparison the mean density which is obtained from the quantum calculation is shown (blue circles).

Figure 4

Mean number of excited spins, $⟨N⟩$ (blue), and its variance, $⟨N^2⟩-⟨N{⟩}^2$ (red) for $L=20$. At $\Omega t=50$ the chemical potential $\mu$ is ramped up linearly from 0 to $8\Omega$. The thick curves result from smoothing the data. The inset shows the inverse temperature calculated from the fluctuation-dissipation theorem (3). The dashed curve is calculated using the classical lattice gas model at a chemical potential $\mu (t)$ and the corresponding temperature of the inset.