Main

Quantum thermodynamics should be more useful. The field has yielded fundamental insights, such as extensions of the second law of thermodynamics to small, coherent and far-from-equilibrium systems1,2,3,4,5,6,7,8. Additionally, quantum phenomena have been shown to enhance engines9,10,11,12, batteries13 and refrigerators14. These results are progressing gradually from theory to proof-of-principle experiments. However, quantum thermal technologies remain experimental curiosities, not practical everyday tools. Key challenges include control15 and cooling quantum thermal machines to temperatures that support quantum phenomena. Both challenges require substantial energy and effort but yield small returns. For example, one would expect a single-atom engine to perform only about an electronvolt of work16.

Autonomous quantum machines offer hope. First, they operate without external control. Second, they run on heat drawn from thermal baths, which are naturally abundant17. A quantum thermal machine would be useful in a context that met three criteria. (1) The machine fulfils a need. (2) The machine can access real-world different-temperature baths. (3) No or few extra resources are spent on maintaining whatever coherence is necessary for the machine’s operation.

We identify such a context: qubit reset. Consider a superconducting quantum computer starting a calculation. The computer requires qubits initialized to their ground states18. If left to thermalize with its environment as thoroughly as possible, though, a qubit could achieve only an excited-state population of ~0.01–0.03, or an effective temperature of 45–70 mK (refs. 19,20,21,22). Furthermore, such passive thermalization takes a few multiples of the qubit’s energy-relaxation time—hundreds of microseconds in state-of-the-art setups—delaying the next computation. A quantum machine cooling the qubits to their ground (minimal-entropy) states fulfils criterion (1). Moreover, superconducting qubits inhabit a dilution refrigerator formed from nested plates, whose temperatures decrease from the outermost to the innermost plate. These temperature plates can serve as heat baths, meeting criterion (2). Finally, the machine can retain its quantum nature if mounted on the coldest plate, next to the quantum processing unit, satisfying criterion (3). Such an autonomous machine would be a quantum absorption refrigerator.

Quantum absorption refrigerators have been widely studied theoretically14,17,23,24,25,26,27,28,29. A previous work30 reported a landmark proof-of-principle experiment performed with trapped ions. However, the heat baths were emulated with electric fields and lasers, rather than realized with physical heat reservoirs. Other quantum refrigerators, motivated by possible applications, have been proposed31 and tested32,33 but are not autonomous.

We report on a quantum absorption refrigerator realized with superconducting circuits. Our quantum refrigerator cools—and therefore resets—a target superconducting qubit autonomously. The target qubit’s energy-relaxation time is fully determined by the temperature of the hot bath that we can vary. Using this control, we can vary the energy-relaxation time by a factor of >70. The reset’s fidelity is competitive: the target’s excited-state population reaches below 3 × 10−4 ± 2 × 10−4 [effective temperatures as low as 22 (+2, −3) mK]. In comparison, state-of-the-art reset protocols achieve populations ranging from 8 × 10−4 to 2 × 10−3 (effective temperatures ranging from 40 mK to 49 mK)21,22. Our experiment demonstrates that quantum thermal machines can be not only useful, but also integrated with quantum information-processing units. Furthermore, such a practical autonomous quantum machine expends less control and thermodynamic work than its non-autonomous counterparts34,35,36.

Our absorption refrigerator consists of three qudits (d-level quantum systems; Fig. 1a). The auxiliary qudits Q1 and Q2 correspond to d = 2 and 3, respectively. Each of them couples directly to a waveguide that supports a continuum of electromagnetic modes. The waveguide can serve as a heat bath formed from photons of an arbitrary spectral profile. Here, nH and nC denote the average numbers of photons in the waveguides. The target of the refrigerator’s cooling is qubit Q3, which is undesirably coupled to an uncontrolled bath in its environment. This bath excites the target to an effective temperature TE. Nearest-neighbour qudits couple together with strengths g12 and g23. These couplings result in an effective three-body interaction37, a crucial ingredient in a quantum absorption refrigerator17,24,25,26. We engineer the three-body interaction such that one excitation in Q1 and one excitation in Q3 are simultaneously, coherently exchanged with a double excitation in Q2. Losing its excitation, Q3 is reset.

Fig. 1: Quantum-absorption-refrigerator scheme and level diagram.
figure 1

a, Conceptual scheme with three qudits. Qubit Q1 couples directly to a waveguide at a rate Γ1; and qudit Q2, to another waveguide at a rate Γ2. Qubit Q3 couples undesirably to an uncontrolled bath in its environment. This bath keeps Q3 at an effective temperature TE. The coupling rate ΓE determines the natural energy-relaxation time of Q3. The waveguides can operate as heat baths containing photons of average numbers nH and nC. The interqudit couplings (g12, g23) engender a process in which one excitation in Q1 and one excitation in Q3 are simultaneously exchanged with a double excitation in Q2. This exchange helps reset Q3. When heat baths drive this process, the system operates as an autonomous quantum refrigerator. The average heat currents are depicted by wide arrows from the baths of the hot (\({{\mathcal{J}}}_{{\rm{H}}}\)), cold (\({{\mathcal{J}}}_{{\rm{C}}}\)) and target (\({{\mathcal{J}}}_{{\rm{T}}}\)) systems. By energy conservation, \({{\mathcal{J}}}_{{\rm{H}}}+{{\mathcal{J}}}_{{\rm{T}}}={{\mathcal{J}}}_{{\rm{C}}}\). b, False-colour micrograph of the device implemented with superconducting circuits. Q2 is frequency-tunable due to a flux current line and two parallel Josephson junctions, as magnified in the inset. c, Level diagram showing tensor products |q1q2q3〉 of the qudits’ energy eigenstates. |101〉 and |020〉 are resonant if ω1 + ω3 = 2ω2 + α2. At resonance, a three-body interaction couples the states at a rate A. d, Distributions over the qudits’ experimentally observed transition frequencies. The Lorentzian distributions’ widths represent spectral widths. The red-shaded box depicts the spectral density nH of photons injected into the waveguide coupled to Q1. This synthesized noise realizes the refrigerator’s hot thermal bath. Analogous statements concern the blue box, nC, Q2 and the cold bath.

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As the heat baths drive the resetting, the system operates autonomously as a quantum absorption refrigerator17. A generic thermodynamic model describes such a refrigerator as follows. Heat flows from a hot bath (coupled to Q1) into an intermediate-temperature bath (coupled to Q2). A heat current of \({{\mathcal{J}}}_{{\rm{H}}}\) (\({{\mathcal{J}}}_{{\rm{T}}}\)) flows out of the bath of Q1 (Q3). A net heat current \({{\mathcal{J}}}_{{\rm{C}}}={{\mathcal{J}}}_{{\rm{H}}}+{{\mathcal{J}}}_{{\rm{T}}}\) enters the bath of Q2 (Fig. 1a). That is, a temperature gradient, rather than work, coaxes heat out of the target qubit.

The qudits are Al-based superconducting transmons that have Al/AlOx/Al Josephson junctions38. We arrange the qudits spatially in a linear configuration (Fig. 1b). The capacitances between the transmons couple the qudits mutually. Qudit Q1 has a transition frequency of ω1/(2π) = 5.327 GHz; and qudit Q2, a variable frequency of ω2/(2π). Q1 directly couples capacitively to a microwave waveguide at dissipation rate Γ1/(2π) = 70 kHz; and Q2 couples to another waveguide at Γ2/(2π) = 7.2 MHz. The third qubit, Q3 has transition frequency ω3/(2π) = 3.725 GHz. Q3 couples dispersively to a coplanar waveguide resonator. Via the resonator, we read out the state of Q3 and drive Q3 coherently. In addition, Q3 couples to the uncontrolled bath in its environment at a rate ΓE. In our proof-of-concept demonstration, Q3 stands for a computational qubit that is being reset and that may participate in a larger processing unit. In the present design, Q3 has a natural energy-relaxation time Trelax = 1/ΓE = 16.8 μs, limited largely by Purcell decay into the nearest waveguide, and a residual excited-state population of Pres = 0.028.

The interqudit couplings hybridize the qudit modes. The hybridization, together with the Josephson junctions’ nonlinearity, results in a three body interaction (Supplementary Information). For this interaction to be resonant, the qudit frequencies must meet the condition \({\omega }_{1}+{\omega }_{3}=2{\omega }_{2}^{{\rm{res}}}+{\alpha }_{2}\). Here, \({\omega }_{2}^{{\rm{res}}}\) denotes the Q2 frequency that satisfies the equality, and α2 denotes the anharmonicity of Q2. The interaction arises from a four-wave-mixing process: one excitation in Q1 and one excitation in Q3 are simultaneously exchanged with a double excitation in Q2 (Fig. 1a)37. To satisfy the resonance condition in situ, we make Q2 frequency-tunable38. We control the frequency with a magnetic flux induced by a nearby current line. The device is mounted in a dilution refrigerator that reaches 10 mK.

To describe the resonance condition, we introduce further notation. Let |0〉 and |1〉 denote the ground and first excited states of any qudit. Let |2〉 denote the second excited state of Q2. We represent a three-qudit state by |q1q2q3〉 ≔ |q1〉1 ⊗ |q2〉2 ⊗ |q3〉3. The resonance condition leads to coherence between the states |101〉 and |020〉. This coherence is a key quantum feature of our refrigerator. Two processes, operating in conjunction, reset Q3. (1) Levels |101〉 and |020〉 coherently couple with an effective strength A (Fig. 1c). (2) Q2 dissipates into its waveguide at a rate Γ2. The combined action of (1) and (2) brings |101〉 rapidly to |010〉 (and then to |000〉), thereby resetting Q3.

We engineer the heat baths of Q1 and Q2 as follows. First, we synthesize radiation using room-temperature electronics (Supplementary Fig. 1). This radiation has a white-noise spectral profile over a selected frequency range. The radiation is injected into microwave coaxial cables, which are interlinked by dissipative microwave attenuators thermalized at different temperatures of the cryostat. The attenuators reduce the incoming radiation’s power and simultaneously introduce quantum thermal noise39,40. The last attenuator, at 10 mK, contributes noise that is predominantly quantum vacuum noise. This resulting radiation finally reaches the waveguides of Q1 and Q2. Quantum noise is generally characterized by a non-symmetrical spectral density, resulting in different emission and absorption rates of a qudit41. Here the predominance of the spontaneous emission of Q2 into the cold bath (as opposed to absorption) is critical to the refrigerator’s operation.

The synthesized radiation’s bandwidth is selected to include the transition frequencies of Q1 (|0〉↔|1〉) and Q2 (|0〉↔|1〉 and |1〉↔|2〉) (Fig. 1d). Within this bandwidth, the radiation can be approximated as a thermal field. Outside the bandwidth, however, the radiation deviates from thermality. Therefore, we designate this field as quasithermal. Its effective temperature, TH,C, depends on the average number of photons at the transition frequencies of qudits Q1,2: nH,C = 1/[exp(ℏω1,2/kBTH,C) – 1], where kB is Boltzmann’s constant. We can vary nH,C by regulating the synthesized noise’s power. This setup enables the whole system to function as a quantum thermal machine. The quasithermal baths induce transitions in Q1 and Q2, autonomously driving the reset via the three-body interaction.

Having specified the setup, we demonstrate the three-body interaction. We verify that Q3 can be reset via resonant driving of Q1 if and only if Q2 meets the resonance condition. The qudits begin in |000〉; accordingly, we issue two microwave drive pulses (Fig. 2a). The first is a Gaussian π pulse that excites Q3 to state |1〉: |000〉→|001〉.

Fig. 2: Three-body interaction.
figure 2

a, Pulse scheme (see the main text for description). b, Two-dimensional plot of the excited-state population of Q3 (\([\langle {\sigma }_{z}\rangle +{\mathbb{1}}]/2\)), as a function of (1) the flux bias voltage (left axis) modulating the frequency of Q2 and (2) the detuning between the Q1 drive frequency \({\omega }_{1}^{{\rm{d}}}/(2\pi )\) and ω1/(2π) (bottom axis). Q1 is driven for Δt = 2 μs during the pulse scheme, after which we read out (RO) the state of Q3 via the resonator. The left axis translates directly into the right axis—the detuning of the Q2 frequency, ω2/(2π), from the resonant value, \({\omega }_{2}^{{\rm{res}}}/(2\uppi )\). The white patch evidences an avoided crossing, where |101〉 and |020〉 become resonant (Fig. 1c). c, Excited-state readout of Q3 as a function of the duration Δt of the Q1 drive, at select drive rates Ω/(2π). The solid black lines are fits based on the model shown in Supplementary Section II.

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The second pulse is flat and coherently drives Q1 (effecting |001〉↔|101〉) at a frequency \({\omega }_{1}^{{\rm{d}}}\) with a rate Ω for a duration Δt. Subsequently, we perform qubit-state readout on Q3 (we measure \([\langle {\sigma }_{z}\rangle +{\mathbb{1}}]/2\)) via the resonator of Q3.

We investigate the readout’s dependencies on \({\omega }_{1}^{{\rm{d}}}\) and on the flux bias voltage (proportional to flux current) that modulates the tunable Q2 frequency. We have fixed Δt = 2 μs and Ω/(2π) = 200 kHz. The microwave drives that we observe deplete the excited-state population of Q3 (Fig. 2a). The depletion is the greatest when \({\omega }_{2}={\omega }_{2}^{{\rm{res}}}\) and the drive is resonant (\({\omega }_{1}^{{\rm{d}}}={\omega }_{1}\))—when the resonant coupling A between |101〉 and |020〉 is the strongest.

The excited state of Q3 is depleted by the cascaded processes |001〉↔|101〉↔|020〉→|010〉. The combined effect of these processes resembles optical pumping—used to achieve population inversion in atomic physics—that enables qubit reset. Away from the resonance condition, the resonant coupling A decreases. Consequently, the excited-state population of Q3 drops less as the |101〉–|020〉 detuning grows.

Furthermore, we study the effect of increasing the drive rate Ω (Fig. 2c). When Ω = 0 MHz, Q3 decays to its ground state (resets) at its natural energy-relaxation time (16.8 μs). As Ω increases, the reset happens increasingly quickly. By fitting a model based on a Lindblad master equation (Supplementary Section II), we determine that the three-body interaction has a strength of A/(2π) = 3.2 MHz.

Having demonstrated the three-body interaction, we operate the three-qubit system as a quantum thermal machine. To measure the system’s performance, we implement a three-step pulse sequence (Fig. 3a). (1) Excite Q3 to near |1〉 (to an excited-state population of 0.95). (2) Fill the waveguides with quasithermal photons, as described above, for a variable time interval Δt. (3) Measure the excited-state population of Q3 (the first two excited states combined), Pexc, using a Rabi population-measurement scheme19,20. This scheme allows for a more accurate population measurement than standard qubit-state readout. This scheme functions optimally when the second-excited-state population of Q3 is negligible compared with the first-excited-state population. However, this condition may not always be met when the latter is extremely small (≲0.004). Nonetheless, we account for the second-excited-state population of Q3, determined theoretically from a comprehensively fitted model, in all the population measurements, which are recalibrated accordingly (Supplementary Section III). We assume that the second-excited-state population is exponentially suppressed, arising from the same uncontrolled bath causing the residual first-excited-state population. This recalibration is negligible, except in some narrow subsets of our experimental data, which lie outside the regime in which we evaluate our refrigerator’s performance (Fig. 4).

Fig. 3: Autonomous refrigeration enabled by a hot bath.
figure 3

a, Three-step pulse scheme. Initialization brings the state of Q3 close to |1〉. During refrigeration, Q1 and Q2 interact with the synthesized quasithermal fields for a duration Δt. Finally, the excited-state population of Q3, Pexc, is measured via a Rabi population-measurement scheme. Pexc represents the combined populations of the first and second excited states; the latter is calculated based on a fitted theoretical model and is negligible except at intermediate values of Δt (Supplementary Section III). b, Pexc as a function of Δt, at select values of nH, the average number of photons in the hot bath. The x axis is split into two different regimes of Δt values: a low regime Δt ∈ [0, 10] μs and a high regime Δt ∈ [10, 80] μs. Q2 experiences no synthesized quasithermal field. We estimate that nC ≈ 0.007 due to the residual thermal field. The dashed green line shows the residual excited-state population of Q3 (defined in the main text) as Pres = 0.028. The dotted blue line shows the excited-state population that Q3 would have at the cold bath’s temperature (45 mK), PC = 0.020. The grey area represents our estimate of the noise floor (Supplementary Section IV). Near the noise floor, some measurements yield small negative values, represented by data points at the bottom axis. The solid lines represent global fits to the experimental curves. The fits are calculated from the model shown in Supplementary Section II. nH is the sole free-fitting parameter.

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Fig. 4: Performance metrics of the quantum absorption refrigerator.
figure 4

a, After a 105 μs reset protocol, the excited state of Q3 reaches a steady-state population PSS. Left: PSS as a function of the hot bath’s average photon number, nH. The corresponding temperature TH is translated along the top axis. Two experimental curves are at two values of the cold bath’s average photon number, nC. Right: PSS as a function of nC (translated into temperature TC on the top axis), at two nH values. Some measurements yield small negative values, represented by data points at the bottom axis. The dashed green and dotted blue lines are the same as in Fig. 3b. The error bars represent standard deviations about the mean values represented by symbols (Supplementary Section IV). b, Reset time (time required for Pexc of Q3 to reach 0.01) as a function of nH. All the solid lines are theoretical predictions calculated from the model shown in Supplementary Section II.

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We raise the effective temperature of the hot bath and investigate how Pexc responds. To do so, we elevate the average number nH of quasithermal photons in the hot bath by increasing the spectral power of the synthesized noise in Q1’s waveguide. We perform this study in the absence of synthesized noise in the cold bath (coupled to Q2), which contains the minimal average number nC of photons. We infer the minimal nC from an independent measurement, using Q2 as a thermometer40: nC = 0.007, associated with a temperature TC = 45 mK. The greater the nH value, the more quickly Pexc decays as we increase Δt (Fig. 3b). At the low value nH = 0.16, Pexc drops below the residual excited-state population Pres = 0.028 (Fig. 3b, green dashed line) that Q3 would achieve if left alone for a long time. From this value, we infer that the effective temperature of Q3’s environment bath TE = 50 mK (Fig. 1a). If thermalized at the cold bath’s temperature (45 mK), Q3 would have an excited-state population of PC = 0.020 (Fig. 3b, blue dotted line). If the hot bath is excited, Pexc reaches a value that is at least an order of magnitude lower than Pres and PC. Our refrigeration scheme clearly outperforms passive thermalization with either the intrinsic bath of Q3 or the coldest bath available. At nH = 19.38 (TH = 5.1 K), refrigeration reduces the effective energy-relaxation time of Q3, Trelax, from 16.8 μs to 230 ns. This reduction is by a factor of >70. Q3’s population declines below 2 × 10−3 over 1.8 μs, before approaching a steady-state value below 0.0008—our measurement protocol’s noise floor (Supplementary Section IV).

In an independent measurement, we study the steady-state population PSS as a function of nH or nC, keeping the other quantity fixed (Fig. 3a). We define PSS as the Pexc value achieved after Δt = 105 μs. This definition stems from the observation that, when the refrigerator is inactive (nH = 0.003), Q3 naturally relaxes to its steady-state residual population Pres by Δt = 105 μs. PSS decreases rapidly as nH increases. Furthermore, PSS reaches its lowest values when nC minimizes at 0.007, such that Q2 is not excited. We overestimate the lowest-reached PSS value and its error margin by computing the mean and standard deviation of all the measured PSS values that lie below 0.0008, the noise floor (Supplementary Section IV shows the methodology). PSS reaches a minimum of <3 × 10−4 ± 2 × 10−4, equivalent to a temperature TSS = 22(+2, −3) mK. This result is remarkably close to the prediction from a general theory of a quantum absorption refrigerator17: \({T}_{{\rm{SS}}}=\frac{2{\omega }_{2}+{\alpha }_{2}-{\omega }_{1}}{(2{\omega }_{2}+{\alpha }_{2})/{T}_{{\rm{C}}}-{\omega }_{1}/{T}_{{\rm{H}}}}\) = 18.6 mK, equivalent to PSS = 6.7 × 10−5. In the limit as nH→∞, TSS decreases marginally to 18.5 mK. TSS does not depend on temperature TE of the target’s effective bath, if ΓE is very small (≪1/Trelax), as in our system during refrigeration.

Also, raising the cold bath’s temperature impedes the reset. Increasing nC to 0.07—exciting Q2 more—leads PSS (as a function of nH) to saturate at a higher value. Finally, consider fixing nH and increasing nC. PSS increases rapidly and then saturates near 0.36. This saturation occurs largely independently of nH. The greater the nH, though, the greater the initial (low-nC) PSS.

A standard figure of merit in the thermodynamic analysis of refrigerators is the coefficient of performance (COP)26. The COP is to refrigerators as efficiency is to heat engines. The steady-state COP is defined as \({{\mathcal{J}}}_{{\rm{T}}}/{{\mathcal{J}}}_{{\rm{H}}}\) (Fig. 1a), which we numerically calculate from the theoretical model shown in Supplementary Section II. The steady-state COP is 0.7 when TH = 5.1 K and TSS = 22 mK. In terms of COP, our quantum refrigerator performs comparably to a macroscopic absorption refrigerator—namely, a common air conditioner [COP ≈ 0.7 (ref. 42)]. In the quasistatic limit (as \({{\mathcal{J}}}_{{\rm{T}}}\to 0\)), the COP can reach its theoretical upper bound—the Carnot limit: \(\frac{{T}_{{\rm{E}}}({T}_{{\rm{H}}}-{T}_{{\rm{C}}})}{{T}_{{\rm{H}}}({T}_{{\rm{C}}}-{T}_{{\rm{E}}})}\). Our quantum refrigerator has a Carnot bound of 0.95 (>0.7), satisfying the second law of thermodynamics.

Another important performance metric is the time required to reset Q3. We define the reset time as the time required for Pexc to reach 0.01 (corresponding to 38.5 mK). The reset time reaches as low as 970 ns before rising slowly with nH (Fig. 3b). We attribute the observed upturn to an excessive dephasing of the coherent process |101〉↔|020〉, which is critical for refrigeration.

In summary, we have demonstrated the first quantum thermal machine being deployed to accomplish a useful task. The task—the reset of a superconducting qubit—is crucial to quantum information processing. The machine—a quantum absorption refrigerator formed from superconducting circuits—cools and resets the target qubit to an excited-state population lower than that achieved with state-of-the-art active reset protocols, without requiring external control. Nevertheless, the refrigeration can be turned off when the target qubit serves in a computation: one can either change the hot bath’s temperature or detune a qudit out of resonance, using an on-chip magnetic flux.

Our refrigerator has two main quantum features—discrete energy levels and a coherent exchange coupling between states |101〉 and |020〉. Another salient feature of our quantum thermal machine is its use of waveguides as physical heat baths. In contrast, other experiments have emulated heat baths30,35. Our heat baths consist of quasithermal fields—syntheses of quantum thermal fields and finite-bandwidth artificial microwave noise. Our approach allows control over the baths’ temperatures, the ability to tailor spectral properties of the heat baths and the selection of the level transitions to be heated. Thus, this method can facilitate a rigorous study of quantum thermal machines. Our experimental setup can be modified to exploit real-world thermal baths, such as different-temperature plates of a dilution refrigerator. We have already demonstrated that our quantum refrigerator can reset a qubit effectively if it has access to a hot bath at a temperature of a few kelvins, without the need for tuning. Superconducting coaxial cables, together with infrared-blocking filters43, can expose the qudits to thermal radiation emitted by hot resistors anchored to a suitable plate of the dilution refrigerator44,45. The modification adds no significant heat load to the base-temperature plate; nor does it compromise the performance of the quantum information-processing unit. One can activate the thermal reset on demand in two different ways: (1) by using a microwave switch46 to toggle Q1’s bath between hot and cold or (2) by dynamically detuning Q2 in and out of the resonance condition that enables the reset process.

Our quantum refrigerator initiates a path towards the experimental studies of quantum thermodynamics with superconducting circuits coupled to propagating thermal microwave fields. Superconducting circuits may also offer an avenue towards scaling quantum thermal machines similarly to quantum information processors. Our experiment may inspire the further development of useful, real-world applications of quantum thermodynamics47 to quantum information processing48,49,50, thermometry40,45,51, algorithmic cooling32,52, timekeeping53 and entanglement generation54. This work marks a significant step in quantum thermodynamics towards practical applications.