Amplifying microwave pulses with a single qubit engine fueled by quantum measurements

TL;DR

The paper addresses how quantum measurement backaction can serve as an energy source for driving a quantum engine. It implements a measurement-powered engine using a superconducting transmon to repeatedly measure $sigma_x$ and to feed back the drive phase, enabling amplification of an incoming microwave signal and direct work extraction. By comparing the measured gain $G(t)$ of the outgoing field with work inferred from qubit tomography, it validates direct work probing and demonstrates cyclic, sustained operation under feedback while showing how open-loop operation leads to engine decay. The results highlight the practical potential of measurement backaction as an energetic resource and emphasize the importance of qubit coherence and frequency stability for robust quantum thermodynamic tasks.

Abstract

Recent progress in manipulating individual quantum systems enables the exploration of engines exploiting non-classical resources. One of the most appealing is the energy provided by the inherent backaction of quantum measurements. While a handful of experiments have investigated the inner dynamics of engines fueled by measurement backaction, powering a task by such an engine is missing. Here we demonstrate the amplification of microwave signals by an engine fueled by repeated quantum measurements of a superconducting transmon qubit. Using feedback, the engine acts as a quantum Maxwell demon operating without a hot thermal source. Measuring the gain of this amplification constitutes a direct probing of the work output of the engine, in contrast with inferring the work by measuring the qubit state along its evolution. Observing a good agreement between both work estimation methods, our experiment validates the accuracy of the indirect method. We characterize the long-term stability of the engine as well as its robustness to transmon decoherence, loss and drifts. Our experiment exemplifies the use of energy brought by quantum measurement backaction.

Figures (12)

• Figure 1: a) A measurement-powered quantum engine amplifying microwave pulses. b) Simplified schematic of the setup with an optical image of the superconducting circuit in false colors. A transmon qubit (green) can be directly driven and its outgoing field is measured via heterodyne detection. The qubit state is read out via its dispersive interaction with a resonator (blue). A Purcell filter (red) inhibits the qubit decay through the readout resonator. c) Principle and pulse sequence of the engine : in /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $1$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $1$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $1$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $1$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) 1 , the qubit is initialized in state $\ket{+x}$ via a $\pi/2$-pulse after a measurement-based active reset in $\ket{-z}$. In /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $2$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $2$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $2$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $2$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) 2 , an incoming resonant microwave signal drives the qubit and is amplified by stimulated emission. Measuring the gain reveals the work output. We measure the output power $P_\mathrm{out}$ during the Rabi drive of amplitude $\Omega$ and duration $t_R$. The useful work is given by $W = (P_\mathrm{out} - P_\mathrm{in}) t_R$ where $P_\mathrm{in}$ (dash lines) is calibrated by detuning the qubit using the AC Stark effect (see \ref{['app:calib_P']}). The gain is obtained with $G(t) = P_\mathrm{out}/P_\mathrm{in}$. In /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $3$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $3$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $3$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $3$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) 3 , the qubit is projectively measured in its $\hat{\sigma}_x$ basis. In /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $4$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $4$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $4$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $4$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) 4 , depending on the outcome of the readout, the qubit is actively reinitialized in $\ket{+x}$ and the thermodynamic cycle restarts at /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $2$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $2$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $2$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) $2$ /csteps/inner xsep /csteps/inner ysep /csteps/inner ysep (0,) (0,0)*(0,0)(0,) (-.50,0) 2 . The qubit-readout dispersive interaction gives a QND-measurement of $\hat{\sigma}_z$. It is transformed into a QND $\hat{\sigma}_x$ measurement by applying a $-\pi/2$ pulse before and a $+\pi/2$ pulse after the readout pulse. If the qubit is measured in the $\ket{-x}$ state, a $-\pi/2$ pulse is applied instead of the $+\pi/2$ thanks to FPGA-based real-time feedback.
• Figure 2: a) Dots: measured excess gain $G-1$ (left) and excess power $P/\hbar \omega_\mathrm{q} \Omega$ (right) as a function of time during the first three cycles of the normal engine (orange) or in open-loop configuration (blue). Here, $\Omega/2\pi = 14.2kHz$ and $t_\mathrm{R} = 8µs$. Error bars combine the standard deviation divided by the square root of sampling number and a 2% relative uncertainty on the calibrated gain $G_\mathrm{meas}$. Shadows: inferred excess gain and power using the tomography of the qubit state in b) with Eqs. (\ref{['eq:expval_pow']}),(\ref{['eq:gain']}), and assuming a 1% uncertainty due to gate and readout infidelities. Solid red (green) lines: case of infinite $T_1$ and $T_2$ and resonant drive of the (open-loop) engine. Dashed lines: result of numerical simulations assuming resonant drive, $T_1=25\mu s$ and $T_2=32\mu s$. Solid orange/blue lines: result of simulation assuming a distribution of qubit frequencies and coherence times $T_2$ (see \ref{['app:stability']} and \ref{['app:simulations']}). The visible dead times of 536ns correspond to the duration $t_\mathrm{meas}$ of steps 3 and 4 in the engine cycle. b) Crosses: measured Bloch coordinates $\expval{\hat{\sigma}_x(t)}$ as a function of $\expval{\hat{\sigma}_z(t)}$ during the same first three cycles of the engine from left to right (top, orange) or in the open-loop configuration without feedback (bottom, blue). Other lines correspond to their counterpart in a).
• Figure 3: a) Dots: the mean measured excess gain $\overline{G_c}-1$ averaged over 40 cycles as a function of Rabi frequency $\Omega$ in log-scale representation. Note that the log scale is cut around zero (gray area). Colored dashed lines : prediction for an ideal qubit with infinite lifetime and coherence times and resonant driving. Solid gray line : small angle limit $2\Gamma_\mathrm{c}/\Omega$. b) Dots: normalized extracted work per cycle $\overline{W}/\hbar\omega_\mathrm{q}t_R\Omega$ averaged over 40 cycles as a function of Rabi angle $\theta = \Omega t_\mathrm{R}$. Black dashed line: prediction for an ideal qubit with infinite lifetime and coherence times and resonant driving. a-b) Error bars combine the standard deviation divided by the square root of sampling number and a 2% relative uncertainty on the calibrated gain $G_\mathrm{meas}$. The different colors correspond to different durations $t_\mathrm{R}$ (see legend). Solid lines are numerical simulations assuming a distribution of qubit frequencies and coherence times $T_2$ (see \ref{['app:stability']} and \ref{['app:simulations']}). The star indicates the parameters used in \ref{['fig:cyclic_evol']}.
• Figure 4: Schematic of the microwave wiring. The Josephson Traveling Wave Parametric Amplifier (TWPA) was provided by the Lincoln Lab. The RF source colors refer to the frequency of the matching element in the device up to a modulation frequency. Identically colored sources represent a single instrument with a split output.
• Figure 5: Qubit state readout: histograms of the readout quadratures when the qubit has been prepared in its thermal equilibrium state (close to its ground state $\ket{-z}$ in the left panel) or after a $\pi$-pulse from its thermal equilibrium state (close to its excited state $\ket{+z}$).