Measurement-dependent corrections to work distributions arising from quantum coherences

For a quantum system undergoing a unitary, process work is commonly defined based on the two projective measurement protocols, which measures the energies of the system before and after the process. However, it is well known that projective measurements disregard quantum coherences of the system with respect to the energy basis, thus removing potential quantum signatures in the work distribution. Here we consider weak measurements of the system's energy difference and establish corrections to work averages arising from initial system coherences. We discuss two weak measurement protocols that couple the system to a detector, prepared and measured either in the momentum or the position eigenstates. Work averages are derived for when the system starts in the proper thermal state versus when the initial system state is a pure state with thermal diagonal elements and coherences characterized by a set of phases. We show that by controlling only the phase differences between the energy eigenstate contributions in the system's initial pure state, the average work done during the same unitary process can be controlled. By changing the phases alone, one can toggle from regimes where the system absorbs energy, i.e., a work cost, to the ones where it emits energy, i.e., work can be drawn. This suggests that the coherences are additional resources that can be used to manipulate or store energy in a quantum system.

Figure 1

System $S$ and detector $D$ are initially uncorrelated in a state ${\rho }_S\left(0\right)\otimes {|\varphi \rangle }_D\langle \varphi |$. The global unitary (1) couples them at time 0, and then evolves the system alone with the time-dependent Hamiltonian ${\widehat{H}}_S\left(t\right)$, while not affecting the detector. $S$ and $D$ are coupled again at time $\tau$, resulting in the final joint state ${\rho }_{SD}\left(\tau \right)$. The measurement $M_D$ of the detector then reads out the work done on the system during its evolution between times 0 and $\tau$. Here we discuss two variations of the initial detector state and its final measurement, Protocols 1 and 2, and find that work one associates with the system depends on this choice.

Figure 2

Work measured with the phase measurement (Protocol 1) for the qubit example discussed in Sec. 4. The unitary transformation $V_S\left(\tau \right)$ is characterized by $\vec{n}=\{0.83,0,0.55\}$ and $\delta =1$. The curves are for the initial state $|\mathrm{\Psi }\rangle$ with phase $\phi =4$ (yellow dashed curve), $\phi =1$ (red dot-dashed curve) and $\phi =0$ (green dotted curve), respectively. For comparison, the blue solid curve shows the expectation values when the initial system state was a thermal state ${\rho }_S^{eq}$. (a) Average exponentiated work, given by Eq. (8), over $\beta \mathrm{\Delta }$. Deviations from the JE (blue solid curve) are significant. (b) The average work, given by Eq. (11), for system evolutions starting from initial superposition states $|\mathrm{\Psi }\rangle$, deviates from the work for the initial thermal state for small $\beta \mathrm{\Delta }$. All curves converge at $\beta \to \infty$, i.e., in the low-temperature limit.

Figure 3

Expectation values of work done on the qubit discussed in Sec. 4 when the qubit is weakly measured using the position of the detector (Protocol 2). The variance of the detector's initial state determined by ${\tilde{\varphi }}_{\sigma }\left(x\right)$ is $\sigma =\lambda \mathrm{\Delta }$. The unitary transformation $V_S\left(\tau \right)$ is characterized by $\vec{n}=\{0.83,0,0.55\}$ and $\delta =1$. Curves shown correspond to the initial state being $|\mathrm{\Psi }\rangle$ with phases $\phi =0$ (yellow dashed curve), $\phi =1$ (green dotted curve), and $\phi =4$ (red dot-dashed curve), and for thermal initial state ${\rho }_S^{eq}$ (solid blue curve). (a) The average exponentiated work for initial state $|\mathrm{\Psi }\rangle$ and thermal state ${\rho }_S^{eq}$ converges to the JE value of 1 at small $\beta \mathrm{\Delta }$, indicated by the horizontal black line. However, differences due to initial coherences of the qubit state become pronounced for increasing $\beta \mathrm{\Delta }$. (b) The average work when the initial state is $|\mathrm{\Psi }\rangle$ shows clear dependence on the phase $\phi$ and differs from the thermal state curve over the whole range of $\beta \mathrm{\Delta }$. (c) The difference between the average work done on the system evaluated for initial pure state $\langle W\rangle$ and initial thermal state ${\langle W\rangle }^{eq}$, as a function of $\sigma /\lambda \mathrm{\Delta }$. Here, $\phi =0$ is chosen for the pure initial state, and $\beta =1/\mathrm{\Delta }$ for both initial state choices.