Entropy production and information fluctuations along quantum trajectories

Employing the stochastic wave function method, we study quantum features of stochastic entropy production in nonequilibrium processes of open systems. It is demonstrated that continuous measurements on the environment introduce an additional, nonthermal contribution to the entropy flux, which is shown to be a direct consequence of quantum fluctuations. These features lead to a quantum definition of single trajectory entropy contributions, which accounts for the difference between classical and quantum trajectories and results in a quantum correction to the standard form of the integral fluctuation theorem.

Figure 1

Diagram showing the connection between the probability density, the density matrix, and their dynamics generated, respectively, by the propagator $T[\chi ,t_2;\psi ,t_1]$ and by the dynamical map $\Lambda (t_2,t_1)$.

Figure 2

Pictorial representation of a forward quantum trajectory (dark blue states) and its backward counterpart (light green states) consisting of $N=3$ jumps (vertical red arrows). The backward process is characterized by jumps through the channels $\{{\gamma }_{i_k}^b,B_{i_k}=A_{i_k}^{†}\}$ and by deterministic evolution periods according to ${\mathcal{U}}_{\mathrm{eff}}(t_k,t_{k+1})=U_{\mathrm{eff}}^{†}(t_{k+1},t_k)$.

Figure 3

$\langle e^{-{\sigma }_f}\rangle$ (black dots), evaluated over ${10}^4$ quantum trajectories for the direct photodetectionlike scheme for a driven two-level atom interacting with a reservoir of modes, for ten different values of driving amplitude and relaxation rates $\delta t{\omega }_0=8k\times {10}^{-4},\delta t{g}_1=4.8k\times {10}^{-4},\mathrm{and}\delta t{g}_2=8k\times {10}^{-5}$ for $k=1,\cdots ,10$ and having fixed the other parameters as $\frac{\delta t}{{\tau }_1}=1.3\times {10}^{-3},\frac{\delta t}{{\tau }_2}={10}^{-3},\frac{\delta t}{\tau }=2.7\times {10}^{-3}$, and $\frac{\delta t}{T}=8\times {10}^{-4}$. The red full line is a quadratic function of $k$ roughly interpolating numerical data and their increasing trend.

Figure 4

$\langle e^{-{\sigma }_f}\rangle$ (black dots), evaluated over ${10}^4$ quantum trajectories for the homodynelike scheme for a driven two-level atom interacting with a reservoir of modes, for ten different values of $\beta =k{e}^{i3\pi /5},k=1,\cdots ,10$ and with fixed system parameters as $\delta t{\omega }_0=5.6\times {10}^{-4},\delta t{g}_1=4\times {10}^{-4},\delta t{g}_2=2.4\times {10}^{-4},\frac{\delta t}{{\tau }_1}=1.3\times {10}^{-3},\frac{\delta t}{{\tau }_2}={10}^{-3}$, and $\frac{\delta t}{\tau }=2.7\times {10}^{-3}$. The red full line is a fourth degree polynomial function of $\left|\beta \right|$ roughly interpolating numerical data and their increasing trend.

Figure 5

$\langle e^{-{\sigma }_f}\rangle$ (black dots), evaluated over $3\times {10}^4$ quantum trajectories for the direct photodetectionlike scheme for a driven two-level atom interacting with a thermal reservoir of modes, for eight different values of temperature such that the average thermal photon number is $\langle N\rangle =0.2+0.3k,k=0,\cdots ,7$ and having fixed the other parameters as $\frac{\delta t}{\tau }=2.7\times {10}^{-3},{\omega }_0\delta t=8\times {10}^{-4}$, and $\frac{\delta t}{T}=8\times {10}^{-4}$. The red full line is a quadratic function of $\langle N\rangle$ roughly interpolating numerical data and their increasing trend.