Reshaping the Quantum Arrow of Time

TL;DR

The paper tackles the quantum arrow of time by formalizing forward and backward trajectories under continuous quantum measurements and introducing a Hamiltonian, $H_{\text{meas}}$, that can reproduce monitored dynamics. By adding a measurement-based feedback term, $H_{\text{fback}}^{\mathcal{X}}$, the authors demonstrate controllable reshaping of the arrow, including stretching, blurring, or reversing it, quantified via $\ln \mathcal{R}$. They further show two anomalous thermodynamic applications: simulating backward-in-time open dynamics through emulation of Lindblad evolution and a continuous measurement engine that can extract energy pumped by the monitoring process, with regimes where energy flow is reversed or engineed. Collectively, the work reveals that the perceived direction of time in quantum systems is not absolute but can be manipulated by measurement and feedback, with clear experimental pathways and connections to Maxwell’s demon-like behavior.

Abstract

While the microscopic laws of physics are often symmetric under time reversal, most natural processes that we observe are not. The emergent asymmetry between typical and time-reversed processes is referred to as the arrow of time. In quantum physics, an arrow of time emerges when a sequence of measurements is performed on a system. We introduce quantum control tools that can yield dynamics more consistent with time flowing backward than forward. The control tools are based on the explicit construction of a Hamiltonian that can replicate the stochastic trajectories of a monitored quantum system. Such Hamiltonian can reverse the effect of monitoring and, via a feedback process, generate trajectories consistent with a reversed arrow of time. It can also be used to simulate the backward-in-time dynamics of an open quantum system. Finally, we design a feedback-driven continuous measurement engine powered by the energy pumped into the system by the monitoring process. We show the engine can operate under experimentally realizable conditions with feedback delay and finite-efficiency measurements.

Figures (4)

• Figure 1: Replicating stochastic quantum trajectories. Simulations of the evolution of a monitored qubit's $z_t = \operatorname{\textnormal{Tr}}\left( {\rho_t \sigma_z} \right)$ component as a function of time. For the simulations, $A = \sigma_z$, $H = \omega \sigma_y/2$, $\omega \tau = 2\pi$, and $\tau/dt = 10^3$. One of the two plots corresponds to the simulation of the stochastic dynamics of a monitored qubit. The other curve was generated by the Hamiltonian $H_\textnormal{meas}$ in Eq. \ref{['eq:Hmeas']}. The stochastically generated original trajectory is reproduced by $H_\textnormal{meas}$. [Plot (a) is the original stochastic trajectory and (b) is the one reproduced by $H_\textnormal{meas}$.]
• Figure 2: Reshaping the quantum arrow of time. Normalized histograms of the log relative likelihood $\ln \mathcal{R}_\mathcal{X}$ for $10^6$ realizations of stochastic trajectories for different values of the arrow-modifying factor $\mathcal{X}$. We consider a qubit where $A = \sigma_z$ is continuously monitored for a time $T = \tau$, with $\omega \tau = 8\pi$ and $\tau/dt = 10^3$. The areas of the blue regions characterize the fraction of trajectories $\textnormal{Prob}[\rightarrow]$ for which a forward arrow of time is more consistent than a backward one. For $\mathcal{X} = \{-4,-3,0,1\}$, the simulations show the respective values $\overline{ \ln \mathcal{R}_\mathcal{X}} \approx \{-0.516, -0.022, 1.524, 2.046\}$, illustrated with vertical dashed lines in the histograms. Equation \ref{['eq:arrow-steeredSimple']} yields similar values of $\overline{ \ln \mathcal{R}_\mathcal{X}} \approx \{-0.5,0,1.5,2\}$. While $\mathcal{X} = 1$ stretches time's arrow [plot (b)], $\mathcal{X} = -3$ prevents the arrow from manifesting [plot(c)], and $\mathcal{X} = -4$ reverses its direction [plot (d)].
• Figure 3: [Plot (a)] The black continuous curve denotes the averaged evolution over many realizations of a monitored qubit's $z_t = \operatorname{\textnormal{Tr}}\left( {\rho_t \sigma_z} \right)$ coordinate as a function of time, with $\omega \tau = 2\pi$ and $\tau/dt = 10^3$. The blue dots represent the evolution obtained by averaging over trajectories reproduced by $H_\textnormal{meas}^\xi$. That is, the blue dots are a Hamiltonian emulation of the dynamics of the open system. The red dots are the average of the time-reverse trajectories, driven realizations of $-H -H_\textnormal{meas}^\xi$ from the final to the initial states of the forward processes. This illustrates the Hamiltonian emulation of the backward-in-time dynamics of an open quantum system. [Plot (b)] The von Neumann $S(\overline{\rho_{\xi,t}})$ entropy of the averaged state $\overline{\rho_{\xi,t}}$ increases in the emulation of the open dynamics, but decreases in the emulation of the backward-in-time dynamics.
• Figure 4: We consider the two-level continuous measurement engine introduced in Sec. \ref{['sec:engine']}. Without feedback, the trajectory-averaged energy of the qubit increases according to $\overline{\langle H \rangle_t} = e^{-t/(2\tau)} \langle H \rangle_0$ [see paragraph before Eq. \ref{['eq:EnergyMeas']} and note that $\langle H \rangle_0 < 0$ for the system to act as an engine]. Feedback with $H_\textnormal{fback}^\mathcal{X}$ and $\mathcal{X} = -1$ slows down the rate at which energy increases. The energy that would have been pumped into the system is thus extracted by the agent performing feedback. Figures (a) and (b) show the system's internal energy and the corresponding engine's output for different values of the measurement efficiency $\eta$ and the feedback delay $\tau_{\textnormal{delay}}$. The engine's output is calculated by subtracting the system's energy from the energy the system would have if no feedback took place. Under ideal measurement and feedback ($\eta = 1$ and $\tau_{\textnormal{delay}} = 0$, blue curves with circles), the system is pinned to its initial state. In such case, all the energy that would have been pumped into the system is extracted. Under feedback delay and efficient measurements ($\eta = 1$ and $\tau_{\textnormal{delay}} = 0.1T$, orange curves with squares), the engine has null output until feedback starts acting. Without delay but inefficient measurements ($\eta = 0.7$ and $\tau_{\textnormal{delay}} = 0$, yellow curves with rhombi), the engine's work output diminishes. Finally, under inefficient measurements and feedback delay ($\eta = 0.5$ and $\tau_{\textnormal{delay}} = 0.2T$, purple curves with triangle), the engine has negative output for a period of time, but eventually reaches positive output. Note that the regimes considered are overly conservative. Experimental measurement efficiencies on superconducting qubits typically fall within the range $\eta_{\textnormal{meas}} \in [0.7,0,8]$PhysRevLett.126.020502PhysRevX.9.011004, and feedback delay is typically orders of magnitude below qubit coherence times PhysRevX.3.021008.