Signatures of coherent initial ensembles on all work moments

TL;DR

The paper shows that coherence in the initial state distribution of a driven dissipative qubit leaves a measurable imprint on the full distribution of work, not just the mean. By adopting a non-intrusive operational work definition and tracking quantum trajectories via Kraus operators, it derives a moment generating function and a hierarchy of equations that reveal how initial coherence shifts higher-order work moments and reduces fluctuations under monotonic driving. It also establishes a quantum generalization of the Jarzynski equality with a coherence-induced bound parameter $\xi$, which tightens the mean-dissipation bound and connects to absolute irreversibility. In the slow-driving limit, the authors derive a modified fluctuation-dissipation relation showing coherence can enhance thermodynamic precision without extra dissipative cost. These results position initial coherence as a resource for precision thermodynamics and open avenues for minimizing dissipation in information-processing tasks like finite-time erasure.

Abstract

Standard treatments of quantum work using projective energy measurements erase initial coherence and alter the dynamics, thereby failing to capture the thermodynamic effects of coherent superpositions of energy eigenstates in an ensemble of initial states. In this article, we use an operational work definition that is non-intrusive, applying it to the case of a driven dissipative qubit, where the qubit's initial preparation comprises coherent superposition states, while the driving is coherence-less. We derive an evolution equation for the moment generating function for this work, faithfully capturing the thermodynamic signature of coherent superpositions in the initial ensemble. We demonstrate that different initial ensembles that correspond to the same density matrix upon ensemble average, while having the same average work, display different work fluctuations. For monotonic driving, we show that fluctuations are maximum for coherence-less initial ensembles. As an application, we consider quantum bit-erasure in finite time and demonstrate significantly different work statistics for erasing a classical bit of information versus a Haar random initial ensemble. Our results indicate that coherence in the initial ensemble can be utilized as a resource for thermodynamic precision without incurring additional dissipative work costs. We also obtain a generalized fluctuation theorem that establishes a new quantum lower bound on the mean dissipated work. This bound, counterintuitively, is also applicable to a "classical" initial ensemble with the same initial density matrix and is connected to quantum absolute irreversibility.

Figures (4)

• Figure 1: The schematic depicts a quantum trajectory of a qubit driven with $H_t = E_t \ket{e}\!\!\bra{e}$. The initial state $\ket{\psi_0}_i$ is sampled with a probability $p_i$. It undergoes a non-radiative decay on the surface of the Bloch sphere (red dashed line) until it has its first jump to, say, $\ket{g}$ at time $t_k$ (curved blue arrow). The subsequent quantum jumps between $\ket{g}$ and $\ket{e}$ proceed with Poisson rates dictated by the Kraus operators. The initial state label $i$ identifies the entire trajectory and helps in calculating the correct work statistics, as explained in the text.
• Figure 2: Average work ($\mu =\braket{W}$), standard deviation ($\sigma = \braket{(W-\mu)^2}^{1/2}$), and skewness ($\kappa_3 = \braket{(W-\mu)^3}$) of work cost plotted against total runtime/duration $\tau$. For (a)-(c), an optimal protocol that minimizes mean work cost $\mu$ in a given time interval $[0,\tau]$ is considered, such that the average population in the excited state at $t=0$ is $1/2$, and at $t=\tau$ is $0.01$---this corresponds to optimal erasure with a fixed error of $1\%$. A minimum $\tau$ is required to achieve this erasure fidelity, and different values of $\tau$ correspond to how "fast" erasure is carried out. Large $\tau$ corresponds to a slow erasure process. For (d)-(f), a protocol with constant slope $E_t=t/2$ is considered in the interval $[0,\tau]$. The three lines in each figure correspond to three different initial ensembles $\mathscr{D}_{\rm EG}$, $\mathscr{D}_{\rm PM}$, and $\mathscr{D}_{\rm Haar}$. The symbols correspond to the results of Monte Carlo simulations. We choose $\gamma_t = \gamma_0 = 0.1$ and $\beta =1$.
• Figure 3: The mean dissipated work $W_{\rm diss}$ (independent of $\mathscr{D}$) is plotted as a function of protocol runtime/duration $\tau$ with an initial ensemble with $p_e=p_g=1/2$ (dashed line) for four different protocols: (a) $E_t = t$, (b) $E_t = t^{1/2}$, (c) $E_t = t^{1/3}$, and (d) $E_t=\mathrm{tanh}(2t)$. The quantum lower bound of $W_{\rm diss}$, $-\ln(1-\xi)/\beta$, is presented for two different coherent decompositions with $\rho_0 = \mathbb{I}/2$. For any given protocol, the mean dissipated work for all the decompositions (including $\mathscr{D}_{\rm EG}$) is the same. $\mathscr{D}_{\rm PM}$ produces the thick line in the figure. The thin line is produced by the equiprobable distribution of $\ket{g}/2 \pm \sqrt{3}\ket{e}/2$. We choose $\gamma_t = \gamma_0 = 0.1$ and $\beta=1$.
• Figure 4: In (a), we plot the classical deviation from FDR that goes to zero as runtime $\tau$ increases (cl: $\dot\sigma^2_W - 2\beta^{-1}W_{\rm diss}$ with dotted line). The deviation from FDR due to coherence in the decomposition $\mathscr{D}_{\rm PM}$ (qu: $\dot\sigma^2_W - 2\beta^{-1}W_{\rm diss}$ with dashed line) clearly deviates from the classical behavior. The theoretically predicted deviation from classicality in the slow-driving regime for $\mathscr{D}_{\rm PM}$ is given by the solid line. The drive protocol is $E_t = t/\tau$ with $t \in [0,\tau]$. We chose the so-called Ohmic bath $\gamma_t = 0.1 E_t$ as it retains effects of coherence for large $\tau$. In the inset of (a), we demonstrate with $\gamma_t = \gamma_0 = 0.1$ that the deviation from the classical result and the theoretical prediction all tend to $0$ at long times. For (b) and (c), we consider protocols $E_t=0.1t$ and $E_t = 0.1t^{1/3}$ respectively with runtimes $\tau$. We plot $-\partial_\tau \ln(1-\xi)$ and compare it with the modification to the FDR $\beta^2 \mathfrak{\bar{a}} \lambda \dot E^2$ as a function of runtime $\tau$. $\beta=1$ for all the plots.