Experimental study of the thermodynamic uncertainty relation

A cost-precision trade-off relationship, the so-called thermodynamic uncertainty relation (TUR), has been recently discovered in stochastic thermodynamics. It bounds certain thermodynamic observables in terms of the associated entropy production. In this Rapid Communication, we experimentally study the TUR in a two-qubit system using an NMR setup. Each qubit is prepared in an equilibrium state, but at different temperatures. The qubits are then coupled, allowing energy exchange (in the form of heat). Using the quantum state tomography technique we obtain the moments of heat exchange within a certain time interval and analyze the relative uncertainty of the energy exchange process. We find that generalized versions of the TUR, which are based on the fluctuation relation, are obeyed. However, the specialized TUR, a tighter bound that is valid under specific dynamics, is violated in certain regimes of operation, in excellent agreement with analytic results. Altogether, this experiment-theory study provides a deep understanding of heat exchange in quantum systems, revealing favorable noise-dissipation regimes of operation.

Figure 1

(a) Molecular structure of the two-qubit NMR spin system, sodium fluorophosphate. The NMR active spin-1/2, ${}^{19}\mathrm{F}$ and ${}^{31}\mathrm{P}$ nuclei in the molecule, labeled as qubit 1 and qubit 2, respectively, are coupled by the Hamiltonian (11) with the coupling strength $J_{12}=868\mathrm{Hz}$. (b) Pulse sequence to realize heat exchange coupling Hamiltonian ${\mathcal{H}}_{XY}$ in Eq. (6). The pulses are applied on qubits 1 and 2 in a time ordered manner from left to right. The black and white narrow solid bars represent $\pi$ and $\pi /2$ pulses, respectively, with the phases mentioned above them. $1/2J_{12}$ represents the free evolution delay. The white box represents the $\theta$ (in rad) angle pulse about the $y$ axis.

Figure 2

(a) First three cumulants of heat exchange, along with a measure for the S-TUR, as a function of the inverse temperature of qubit 1 ${\beta }_1$; ${\beta }_2=0$. Measurements (square, cross, diamond, and circle symbols) are constructed with the help of Eq. (5), and are compared to the theory (solid, dark-dashed, light dashed-dotted, and dotted lines), Eq. (8). (b) Comparison between different bounds, showing that the S-TUR (experiment: cross; theory: dashed line) provides the tightest lower bound to $\frac{{\langle Q^2\rangle }_{\tau }^c}{{\langle Q\rangle }_{\tau }^2}$ (experiment: square; theory: solid line). Experimental results are obtained from state tomography, yielding ${\langle Q\rangle }_{\tau }$, which is used to calculate the entropy production. Theoretical results are based on Eq. (8). Parameters are $J\tau =1/8$ and ${\nu }_0=\pi /20$ (${\omega }_0=2\pi {\nu }_0$). Error bars are obtained by repeating the experiments eight times.

Figure 3

Same as Fig. 2 but at $J\tau =1/4$ leading to ${\mathcal{T}}_{\tau }\left(J\right)>2/3$, therefore the violation of the S-TUR.

Figure 4

Cumulants of heat exchange and the S-TUR as a function of $J\tau$ for $J=1$ Hz, ${\beta }_1{\omega }_0=2.02$, and ${\beta }_2=0$. Other parameters are the same as in Fig. 2.