Heat and Work Along Individual Trajectories of a Quantum Bit

We use a near quantum limited detector to experimentally track individual quantum state trajectories of a driven qubit formed by the hybridization of a waveguide cavity and a transmon circuit. For each measured quantum coherent trajectory, we separately identify energy changes of the qubit as heat and work, and verify the first law of thermodynamics for an open quantum system. We further establish the consistency of these results by comparison with the master equation approach and the two-projective-measurement scheme, both for open and closed dynamics, with the help of a quantum feedback loop that compensates for the exchanged heat and effectively isolates the qubit.

Figure 1

Evaluating heat and work along single quantum trajectories. (a) Schematic of the qubit system, drive, and homodyne detection. (b) Work (blue), heat (red), and energy (green) along a single trajectory. The discrete time step resolution is $\delta t=20\mathrm{ns}$, the smallest compatible with the detection bandwidth. (c) The total energy along a single quantum trajectory (green) compared to the total energy as determined from an ensemble of projective measurements at each time point (circles). The error bars indicate the standard error of the mean. (d) Projective measurements binned and averaged according to the sum of the work and heat contributions ${\tilde{P}}_{0,0}^W+{\tilde{P}}_{0,0}^Q$. The error bars indicate the standard error of the mean based on the number of occurrences ($N$) for each value of ${\tilde{P}}_{0,0}^W+{\tilde{P}}_{0,0}^Q$ (inset).

Figure 2

Comparison of stochastic and average heat and work quantities. (a) Individual heat and work trajectories, $\tilde{Q}$, and $\tilde{W}$, are displayed as transparent red and blue traces. The mean of these individual trajectories, $⟨\tilde{Q}⟩$, and $⟨\tilde{W}⟩$, are displayed as dashed lines which are in good agreement with the mean values from the master equation, $\overline{Q}$ and $\overline{W}$, Eq. (3), solid lines. (b) Distributions of $\tilde{Q}$ and $\tilde{W}$ at evolution time $\tau =6\mu \mathrm{s}$.

Figure 3

Quantum feedback loop. (a) Instantaneous heat and feedback work along a single trajectory. The feedback work has been time shifted by 20 ns to account for the time delay in the feedback circuit. The anticorrelation ($r=-0.68$) of heat and feedback work is evident in the scatter plot (b). (c) Average of the instantaneous contribution of heat and feedback to the transition probability for ${10}^4$ experimental iterations. (d), Parametric plot of ${\tilde{P}}_{0,0}^W$ versus ${\tilde{P}}_{0,0}^Q$ (red) and ${\tilde{P}}_{0,0}^{F+Q}$ (blue) showing how the feedback cancels the heat, narrowing, and shifting the distribution toward zero for $\tau =6\mu \mathrm{s}$.

Figure 4

Work along trajectories with and without feedback. (a) the instantaneous work $\delta \tilde{W}$ along a single trajectory in the presence of feedback (blue) and the open loop configuration (no feedback) (red) is compared to the calculated instantaneous work expected for pure unitary evolution (green). (b),(c) The correlation between the instantaneous work $\delta \tilde{W}$ and the work for a unitary evolution along a quantum trajectory. The linear regression fit (black lines) show correlation (slope 0.44, estimated error 0.03) when the feedback loop is employed, and no correlation in the open loop configuration (slope 0.007, estimated error 0.04).