Autonomous quantum heat engine

Abstract

Quantum heat engines provide attractive means in quantum thermodynamics for studying the fundamentals of the flow of heat and work. Previous experimental implementations of heat engines operating at the level of a few excitation quanta have utilized external driving, which has made the observation of the produced work challenging. Conversely, autonomous quantum heat engines only require a flow of heat to operate and generate work. However, autonomous quantum heat engines have not yet been experimentally demonstrated in any system despite numerous theoretical investigations. Here, we experimentally realize an autonomous quantum heat engine based on superconducting circuits. We construct the engine circuit implementing an approximate Otto cycle by coupling two superconducting resonators with a superconducting quantum interference device, and coupling this system to spectrally filtered hot and cold reservoirs. By varying the experimental conditions, we observe coherent microwave power generation arising from the internal dynamics of the system driven only by the thermal reservoirs. Our results validate previous theoretical predictions for this circuit and pave the way for detailed studies of quantum effects in heat engines and for using heat-generated coherent microwaves in circuit quantum electrodynamics.

Figures (8)

• Figure 1: Working principle and experimental setup of the autonomous quantum heat engine.a, Illustration of an ideal quantum Otto cycle. The quantum working body is depicted as a quadratic potential energy in an eigenfrequency--photon-number plane. The red and blue arrows illustrate the heat coming in and out of the working body, respectively, and the yellow arrows similarly denote the related work. b, Circuit-element model of the autonomous quantum heat engine, comprised of five subcircuits: cold filter resonator + resistor (dark blue, c), hot filter resonator + resistor (red, h), working-body resonator + SQUID loop (orange, a), driving resonator (deep purple, b), and a probing feedline (light blue). For $x \in \{\rm a,b,c,h\}$, each resonator has impedance $Z_x$ and bare frequency $f_x$, and each resistor has resistance $R_x$ at temperature $T_x$. $C_{x,\rm out}$ and $C_{x,\rm cpl}$ denote output and coupling capacitors, $C_{\rm PPC}$ is the parallel plate capacitance, and $L_{\rm cpl}$ is the galvanic inductive coupling to the SQUID. The electric current with amplitude $A_\mathrm{b}$ of the driving resonator (cyan) modulates the effective flux $\Phi(t)$ through the SQUID with critical currents $I_1$ and $I_2$. Dashed boxes indicate the circuit components contributing to effective frequencies $f_\mathrm{a}'$ and $f_\mathrm{b}'$ of the working-body and driving resonators, respectively. c, Evolution of the working-body resonance frequency on top of the Lorentzian noise spectra of the hot and cold heat reservoirs as a function of frequency. d, Simplified experimental setup together with a false-color image of the heat engine circuit. The sample consists of hot (red) and cold (blue) filtering CPW resonators, the working-body resonator (orange), and the driving resonator (deep purple). The driving resonator is galvanically coupled to the working-body resonator through a SQUID (inset). A DC-driven magnetic coil controls the external flux outside the sample holder. We probe the driving resonator using a capacitively coupled feedline connected to selected measurement instruments at room temperature. For thermal-noise injection, we use both a 100-$\Omega$ heated resistor and an RF noise drive connected via a switch on the hot filter feedline.
• Figure 2: Engine characterization and calibration.a, b, Magnitude of the microwave transmission coefficient as a function of normalized external flux and probe frequency for (a) the filtering and working-body resonators and (b) the driving resonator. The red and blue dashed lines in (a) depict the avoided crossings between the working-body resonator and the filtering resonators with matching colors to Fig. \ref{['fig:setup']}d. The optimal flux point for the heat engine is marked with white dashed lines in (a) and (b). c, As (b) but measured in the vicinity of the optimal flux point with RF noise on at $-132$ dBm power and $-125$ dBm probe power. The white dashed line displays the flux value for which the resonance dip of the driving resonator is most enhanced. d, Trace of the transmission magnitude along the white dashed line in (c) for RF noise on and off. e, Transmission magnitude as a function of the RF noise power and probe frequency at the optimal flux point for $-129$ dBm probe power. f, Total quality factor of the driving resonator extracted from (e) as a function of the RF noise power (bottom axis) and calculated noise temperature (top axis). The rolling average window size is 10 samples. The default parameters of the device are listed in Table \ref{['tab:exp_parameters']}.
• Figure 3: Heat engine characteristics in transmission measurements.a, Transmission magnitude as a function of RF noise power and probe frequency at $-140$ dBm probe power. b, Transmission magnitude as a function of probe frequency for $-150$ dBm probe power and $-128$ dBm noise power around the minimum point of (a). The obtained quality factors and the average photon number of the driving resonator from the fit are displayed in the inset. c, Measurement scheme for the time-domain measurements. The top panel illustrates the RF noise continuously fed through the drive line of the hot filter, and the bottom panel shows the kick pulse and the averaged integration window of the outgoing signal in the sample feedline. d, Average signal amplitude as a function of idle time and kick pulse frequency at $-128$ dBm noise power, measured using the protocol in (c). The red dashed line marks the dip of the resonance feature at 316.3 MHz probed in (e), and the purple dashed line marks the peak of the resonance feature at 316.4 MHz probed in (f). e, Evolution of the resonator state as a function of idle time in the normalized IQ plane sliced from the purple dashed line in (d). The lines between points highlight the direction of temporal evolution with 10 ns idle-time steps between points. f, Decay of the resonator state excitation as average signal amplitude as a function of the idle time for noise on and off for up to 5 µ s evolution. The rolling average window is 30 samples. The default parameters of the device are listed in Table \ref{['tab:exp_parameters']}.
• Figure 4: Autonomous photon generation and power distribution.a, Power spectrum as a function of RF noise power and spectrum analyzer frequency. The direction of the noise power sweep is from low to high in the top panel and high to low in the bottom panel, indicated by the white arrows. The white dashed line marks the spontaneous emergence point of the emission. b, Power spectrum as a function of spectrum analyzer frequency for $-126$ dBm noise power. The linewidth of the spectrum is 30 kHz, and the integrated power from the spectrum is $P\approx 5$ aW. c, Histogram of normalized single shots from the incoming signal power of the heat engine for on and off peak of the spectrum in (a) for $-126$ dBm noise power. d, Distribution of single shots from the signal generated by the heat engine in the normalized IQ plane for $-126$ dBm noise power. e, Difference between the distribution in (d) and a Gaussian fitted to that distribution in the normalized IQ plane. The length of the integration windows for (c), (d), and (e) is 20 µ s. The default parameters of the device are listed in Table \ref{['tab:exp_parameters']}.
• Figure 5: Blackbody heater as a thermal noise source.a, Transmission magnitude as a function of normalized external flux and probe frequency with blackbody heater on. The temperature of the blackbody heater is approximately 12.3 K, and the base temperature is 187 mK. b, Transmission magnitude as a function of heater temperature (bottom axis), base temperature (top axis), and probe frequency. c, Power spectrum as a function of heater temperature (bottom axis), base temperature (top axis), and spectrum analyzer frequency. The temperature for the emergence point of the emission is marked with white dashed lines in (b) and (c). The default parameters of the device are listed in Table \ref{['tab:exp_parameters']}.