Experimental Validation of Fully Quantum Fluctuation Theorems Using Dynamic Bayesian Networks

Fluctuation theorems are fundamental extensions of the second law of thermodynamics for small systems. Their general validity arbitrarily far from equilibrium makes them invaluable in nonequilibrium physics. So far, experimental studies of quantum fluctuation relations do not account for quantum correlations and quantum coherence, two essential quantum properties. We here apply a novel dynamic Bayesian network approach to experimentally test detailed and integral fully quantum fluctuation theorems for heat exchange between two quantum-correlated thermal spins-$1/2$ in a nuclear magnetic resonance setup. We concretely verify individual integral fluctuation relations for quantum correlations and quantum coherence, as well as for the sum of all quantum contributions. We further investigate the thermodynamic cost of creating correlations and coherence.

Figure 1

Schematic representation of the experimental system. (a) Two qubits made of the nuclear spins-$1/2$ of ${}^1\mathrm{H}$ and ${}^{13}\mathrm{C}$ of ${}^{13}\mathrm{C}$-labeled chloroform diluted in Acetone-d6 are placed in a NMR magnetometer that produces a high intensity magnetic field ($B_0$) in the longitudinal direction using a superconducting magnet. The liquid sample inside a 5 mm glass tube and maintained at room temperature (about 298 K) is placed at the center of the magnet within the radio frequency coil of the probe head. The magnet is composed of a superconducting coil (that produces a magnetic field strength of 11.75 T) placed in a thermally shielded vessel and immersed in liquid He (consequently, the temperature along the coil is lower than 10 K). Vacuum chambers and a liquid N chamber are part of the thermal isolation, and are used to maintain the temperature of the superconducting coil stable. (b) Experimental pulse sequence used to implement thermal interaction between the two qubits. The brown (green) square represents $x$ ($y$) rotations by the indicated angle. The black vertical lines indicate free evolution under the scalar coupling, ${\mathcal{H}}_{\mathrm{J}}^{\mathrm{HC}}=(\pi \hslash /2)J{\sigma }_z^{\mathrm{H}}{\sigma }_z^{\mathrm{C}}$, between the ${}^1\mathrm{H}$ and ${}^{13}\mathrm{C}$ nuclear spins with durations $2\tau /\pi J$ and $\tau /\pi J$. We perform a total of 22 samplings of the interaction time $\tau$ in the interval 0 to 2.32 ms.

Figure 2

Detailed quantum fluctuation theorem with and without initial quantum correlations. (a) Forward and (b) reverse heat distributions, $P_f(Q)$ and $P_r(-Q)$, as a function of time in the presence of initial quantum correlations between the two qubits. (c) Forward and (d) reverse heat distributions in the absence of initial correlations between the qubits. Symbols represent data and solid lines are simulations [34]. Error bars are evaluated by a Monte Carlo sampling of the standard deviation. (e) Heat exchange fluctuation theorem, Eq. (4), for $t=1.88\mathrm{ms}$, with (purple dots) and without (green triangles) initial correlations. In the absence of initial correlations, the Jarzynski-Wójcik fluctuation relation, $P_f^{\mathrm{JW}}(Q)/P_r^{\mathrm{JW}}(-Q)=\exp (Q\mathrm{\Delta }\beta )$, is obeyed, while the generalized theorem, $P_f(Q)/P_r(-Q)=\exp (Q\mathrm{\Delta }\beta )/\mathrm{\Psi }(Q)$, that fully accounts for quantum correlations and quantum coherence with $\mathrm{\Psi }(Q)\ne 1$ holds in their presence (solid and dashed lines are guides to the eye). (f) The function $\mathrm{\Psi }(Q)$ may be directly evaluated as the ratio $\mathrm{\Psi }(Q)=[P_f^{\mathrm{JW}}(Q)P_r(-Q)]/[P_r^{\mathrm{JW}}(-Q)P_f(Q)]$ (shown here for $t=1.88\mathrm{ms}$).

Figure 3

Integral fluctuation theorems with initial correlations. The individual contributions from classical and quantum correlations, $J_l$ and $I_l$, as well as from quantum coherence $C_l$, and the relative entropies ${\mathrm{\Sigma }}_j$, for the two qubits ($l=0$, 1 and $j=A$, $B$), and $\gamma$, separately verify the quantum integral fluctuation theorem (3). At the same time, the sum of all the quantum contributions $\sigma =-Q_A\mathrm{\Delta }\beta -I_0+I_1+{\mathrm{\Sigma }}_A+{\mathrm{\Sigma }}_B-\gamma$ obeys the integral fluctuation relation (2).

Figure 4

Energetic cost of creating quantum correlations and coherence. In the absence of initial correlations, $I_0=0$, the rates of quantum correlation and coherence generation are upper bounded by the rate of (dimensionless) energy change, ${\dot{I}}_1\le \mathrm{\Delta }\beta {\dot{Q}}_A$ and ${\dot{C}}_1\le \mathrm{\Delta }\beta {\dot{Q}}_A$. The derivative $\dot{X}(t_k)$ at time $t_k$ is evaluated as the central difference $X(t_{k+1})-X(t_{k-1})$.