Irreversibility and the Arrow of Time in a Quenched Quantum System

Irreversibility is one of the most intriguing concepts in physics. While microscopic physical laws are perfectly reversible, macroscopic average behavior has a preferred direction of time. According to the second law of thermodynamics, this arrow of time is associated with a positive mean entropy production. Using a nuclear magnetic resonance setup, we measure the nonequilibrium entropy produced in an isolated spin-$1/2$ system following fast quenches of an external magnetic field. We experimentally demonstrate that it is equal to the entropic distance, expressed by the Kullback-Leibler divergence, between a microscopic process and its time reversal. Our result addresses the concept of irreversibility from a microscopic quantum standpoint.

Figure 1

A quantum system (with Hamiltonian ${\mathcal{H}}_0^F$) is initially prepared in a thermal state ${\rho }_0^{\mathrm{eq}}$ at inverse temperature $\beta$. It is driven by a fast quench into the nonequilibrium state ${\rho }_{\tau }^F$ along a forward protocol described by the Hamiltonian ${\mathcal{H}}_t^F$. In the backward process, the system starts in the equilibrium state ${\rho }_{\tau }^{\mathrm{eq}}$ corresponding to the final Hamiltonian ${\mathcal{H}}_{\tau }^F$ and is driven by the time-reversed Hamiltonian ${\mathcal{H}}_t^B={\mathcal{H}}_{\tau -t}^F$ to ${\rho }_{\tau }^B$. The entropy production $⟨\mathrm{\Sigma }⟩$ at time $t$ is given by the Kullback-Leibler divergence between forward and backward states ${\rho }_t^F$ and ${\rho }_{\tau -t}^B$ [cf. Eq. (1)].

Figure 2

We show a section of the magnetometer employed in our NMR experiment. A superconducting magnet, which produces a high intensity magnetic field ($B_0$) in the longitudinal direction, is immersed in liquid He, surrounded by liquid N in another vacuum separated chamber, in a thermally shielded vessel. The liquid sample (inside a 5 mm glass tube) is placed at the center of the magnet within the rf coil of the probe head. A digital electronic time-modulated pulse induces a transverse rf field ($B_1$) that drives the ${}^{13}\mathrm{C}$ nuclear spins out of equilibrium. In the forward (backward) protocol, the rf coil can perform (retrieve) work on (produced by) the nuclear spin sample. Depending on the speed of the modulation of the driving field, irreversible entropy is produced. The sketch is not in scale and has been stripped of unnecessary technical details of the apparatus.

Figure 3

(a),(b) Evolution of the Bloch vector of the forward [backward] spin-$1/2$ state ${\rho }_t^F$ [${\rho }_{\tau -t}^B$] during a quench of the transverse magnetic field, obtained via quantum state tomography. A sampling of 21 intermediate steps has been used. The initial magnetization (gray arrow) is parallel to the external rf field, aligned along the positive $x$ [$y$] axis for the forward [backward] process. The final state is represented as a red [blue] arrow. (c) Polar projection (indicating only the magnetization direction) of the Bloch sphere with the trajectories of the spin. Green lines represent the path followed in a quasistatic ($\tau \to \infty$) process.

Figure 4

(a) Black dots represent the measured negative and positive values of the entropy production $\mathrm{\Sigma }$ of the spin-$1/2$ system after a quench of the transverse magnetic field of duration $\tau =100\mu \mathrm{s}$. The mean entropy production (red dashed line) is positive, in agreement with the second law. (b) Average entropy production $⟨\mathrm{\Sigma }⟩$ (dashed lines) evaluated through the probability distribution $P(\mathrm{\Sigma })$, and Kullback-Leibler divergence $S({\rho }_t^F\parallel {\rho }_{\tau -t}^B)$ between forward and backward states ${\rho }_t^F$ and ${\rho }_{\tau -t}^B$ (dots), reconstructed through the tomographic measurements, as a function of time for three different quench durations $\tau =100\mu \mathrm{s}$ (blue), $500\mu \mathrm{s}$ (green), and $700\mu \mathrm{s}$ (red). Good agreement (within experimental uncertainties represented, respectively, by error bars and shadowed regions) between the two quantities is observed, quantifying the arrow of time (1). (c) Mean entropy production $⟨\mathrm{\Sigma }⟩$ (blue dots) and its linear response approximation $⟨{\mathrm{\Sigma }}_{\mathrm{LR}}⟩$ (green dots) as a function of the quench time. The difference between the two values, especially for fast quenches (small $\tau$), suggests the systematic deviation of the experiment from the linear response regime. The dashed lines represent the results of numerical simulations.