Anomalous heat flow and quantum Otto cycle with indefinite causal order

Abstract

The principle that heat spontaneously flows from higher temperature to lower temperature is a cornerstone of classical thermodynamics, often assumed to be independent of the sequence of interactions. While this holds true for macroscopic systems at equilibrium, here we show that, when the order of interactions between two identical thermalization channels is indefinite, an anomalous heat flow emerges, whereby heat can sometime flow from a colder entity to a hotter one. Taking advantage of this anomalous heat flow, we design a quantum Otto cycle with indefinite causal order, which not only achieves refrigeration but also generates work. The anomalous heat flow and the quantum Otto cycle are experimentally simulated in a photonic quantum setup, which provides a proof-of-principle demonstration of the theory.

Figures (12)

• Figure 1: Illustration of heat exchanges between the system (initially at temperature $T_{\texttt{S}}$, represented by a bottle with broken lines indicating the heat) and two identical thermalizing channels, $\mathcal{N}^{T_{\texttt{E}}}_{1}$ and $\mathcal{N}^{T_{\texttt{E}}}_{2}$, both at temperature $T_{\texttt{E}}$. We analyze two scenarios: $T_{\texttt{S}}<T_{\texttt{E}}$ (upper) and $T_{\texttt{S}}>T_{\texttt{E}}$ (lower). In both cases, the system interacts with an ICO by applying the quantum switch. The control qubit is initially prepared in a superposition state and subsequently measured in the basis $\left\{\left|+\right\rangle_{c},\left|-\right\rangle_{c}\right\}$ after the interaction. Anomalous heat flow occurs conditioned on the measurement outcomes of the control qubit: for $T_{\texttt{S}}<T_{\texttt{E}}$, if $\left|+\right\rangle_{c}$ is detected, heat is transferred from the cold system to the hot channels, whereas for $T_{\texttt{S}}>T_{\texttt{E}}$, if $\left|-\right\rangle_{c}$ is measured, heat flows from the cold channels to the hot system.
• Figure 2: Unfolded quantum switch of constant channels. The output of the quantum switch of two constant channels (top) can be reproduced by a quantum circuit where the state of the target system undergoes controlled ${\tt SWAP}$ operations with two fixed states (bottom). The two constant channels (in green) output fixed states $\tau_1$ and $\tau_2$, independently of their input. The target system (in red) is initially in the state $\rho$, while the control qubit (in blue) is initially in the state $\gamma$.
• Figure 3: Experimental setup. The quantum switch is realized by a Mach-Zehnder interferometer structure, which comprises two equivalent thermalizing channels $\mathcal{N}^{T_{\texttt{E}}}_{1}$ and $\mathcal{N}^{T_{\texttt{E}}}_{2}$ in the ICO. The two causal orders are characterized by red and blue optical paths, respectively. Polarization beam splitter (PBS), mirror (M), prism mirror (PM), half wave plate (HWP), quarter-wave plate (QWP), beam splitter (BS), interference filter (IF), fiber collimator (FC).
• Figure 4: (a) and (b) Heat changes in the system, i.e., $\Delta Q^{\pm}$, against $T_{\texttt{E}}/T_{\texttt{S}}$, when the control qubit is measured in the $\{|+\rangle_c,|-\rangle_c\}$ basis with its initial state set to $\ket{0}_{c}$ in (a) and $\ket{+}_{c}$ in (b), respectively. (c) and (d) The probabilities $P_{\pm}$ of measuring the control qubit in the $\ket \pm_{c}$ basis corresponding to scenarios (a) and (b), respectively. In all the figures, the curves denote theoretical predictions and the symbols represent the experimental data.
• Figure 5: (a) Schematic of work-heat conversion in an ICO-based thermal machine. The work cost for erasing the memory of Maxwell's demon allows the system to absorb heat from a cold source. Part of this heat is transferred to a hot source, and the remainder is converted into work on an external agent. (b) Illustration of the ICO-based Otto cycle. Strokes I (work input), III (work output), and IV (system cooling, combined with demon memory reset) are analogous to a standard Otto cycle. The cycle diverges in Stroke II, wherein the system undergoes an ICO process (green, "ICO"), followed by a measurement of the control qubit by the demon. A $|-\rangle_c$ result allows the cycle to continue; otherwise, the system is thermally reset via an auxiliary source (black dashed, "classical") to the state after Stroke I. This process iterates until a $|-\rangle_c$ measurement occurs.