Minimal-backaction work statistics of coherent engines

Abstract

Determining the work statistics of quantum engines is challenging due to measurement backaction. We here show that a dynamic Bayesian network-based measurement scheme, which preserves quantum coherence within an engine cycle, is minimally invasive, in the sense that the averaged measured state over one cycle exactly coincides with the unmeasured state. It therefore provides a general framework to investigate energy exchange statistics in quantum machines. This stands in contrast to the standard two-point measurement protocol, whose backaction can be so strong that it generally fails to reproduce the average work output of a coherent motor. It may even alter its mode of operation, causing it to cease functioning as an engine under observation. We further demonstrate that recently proposed universal fluctuation bounds do not necessarily apply to coherent machines.

Figures (9)

• Figure 1: Entropic distance, $D_\textrm{KL} (P_{\textrm{DBN}} || P_{\textrm{TPM}})$, between the work distributions of the dynamic Bayesian network (DBN) approach and the two-point measurement (TPM) scheme versus the relative entropies of coherence, $\mathcal{C}_{1,3}$, after the two isochores in a three-level quantum Otto cycle. The largest difference is observed for maximal coherence. It vanishes for zero transverse driving strength $g=0$ (inset). Parameters are $\omega_{\mathrm{h}} = 10$, $T_{\mathrm{h}} = 14$, $\omega_{\mathrm{c}} = 0.5$, $T_{\mathrm{c}} = 0.1$, $g = 9$, $\lambda_{\mathrm{c}} = \lambda_{\mathrm{h}} = \lambda$.
• Figure 2: Trace distance, $d(\langle \rho \rangle, \tilde{\rho}_1)$, between the averaged measured state $\langle \rho\rangle$ and the unmeasured steady state $\tilde{\rho}_1$ for the two measurement protocols (TPM and DBN). It vanishes for dynamic Bayesian networks, showing that the latter minimizes measurement backaction. Same parameters as in Fig. 1.
• Figure 3: Operation regimes of the quantum Otto machine for the two measurement schemes. a) Unmeasured and DBN-measured machines have identical mean work and heat, and hence display identical behavior. b) By contrast, the TPM-measured machine exhibits radically modified regimes due to measurement backaction. Same parameters as in Fig. 1.
• Figure 4: The fluctuation bound, $\eta^{(2)}/ \eta_{C}^{2}\leq 1$, Eq. (5), can be violated both for the TPM and DBN protocols owing to quantum coherence within the engine cycle for small thermalization rates $\lambda$, in the quasistatic regime. Parameters are $\omega_{\mathrm{h}} = 1$, $T_{\mathrm{h}} = 1.2$, $\omega_{\mathrm{c}} = 0.85$, $T_{\mathrm{c}} = 1$, $g = 0.06$, and $\lambda_{\mathrm{c}} = \lambda_{\mathrm{h}} = \lambda$.
• Figure S1: Work probability distributions $P (w)$ of the engine cycle computed using TPM (red) and DBN (blue), with their respective means, $\langle w \rangle_{\mathrm{TPM}}$ and $\langle w \rangle_{\mathrm{DBN}}$, compared to the unmeasured value of work $W$. a) In the generic case, where coherences in the energy basis are dynamically induced, $g = 9$, and the working substance does not fully thermalize, $t_{\mathrm{h}} = t_{\mathrm{c}} = 0.92$, both probability distributions are different and only the one computed with DBN reproduces the correct mean value for the work. b) In the absence of dynamically induced coherences, $g = 0$ and $t_{\mathrm{h}} = t_{\mathrm{c}} = 0.92$, or c) if the working substance thermalizes, $g = 9$ and $t_{\mathrm{h} , \mathrm{c}} \to \infty$, TPM and DBN are equivalent. Parameters are $\omega_{\mathrm{h}} = 10$, $T_{\mathrm{h}} = 14$, $\omega_{\mathrm{c}} = 0.5$, $T_{\mathrm{c}} = 0.1$, $g = 9$, $t_{\mathrm{h}} = t_{\mathrm{c}} = 0.92$.