Heat and work along individual trajectories of a quantum bit
M. Naghiloo, D. Tan, P. M. Harrington, J. J. Alonso, E. Lutz, A. Romito, K. W. Murch
HHeat and work along individual trajectories of a quantum bit
M. Naghiloo, D. Tan,
1, 2
P. M. Harrington, J. J. Alonso, E. Lutz, A. Romito, and K. W. Murch
1, 6 Department of Physics, Washington University, St. Louis, Missouri 63130 Shenzhen Institute for Quantum Science and Engineering and Department of Physics,Southern University of Science and Technology, Shenzhen 518055, People’s Republic of China Department of Physics, Friedrich-Alexander-Universit¨at Erlangen-N¨urnberg, D-91058 Erlangen, Germany Institute for Theoretical Physics I, University of Stuttgart, D-70550 Stuttgart, Germany Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom Institute for Materials Science and Engineering, St. Louis, Missouri 63130 (Dated: April 30, 2019)We use a near quantum limited detector to experimentally track individual quantum state trajec-tories of a driven qubit formed by the hybridization of a waveguide cavity and a transmon circuit.For each measured quantum coherent trajectory, we separately identify energy changes of the qubitas heat and work, and verify the first law of thermodynamics for an open quantum system. Wefurther establish the consistency of these results by comparison with the master equation approachand the two-projective-measurement scheme, both for open and closed dynamics, with the help of aquantum feedback loop that compensates for the exchanged heat and effectively isolates the qubit.
Continuous measurement of a quantum bit can be usedto track individual trajectories of its state. Due to theintrinsic quantum fluctuations of a detector, measure-ment is an inherently stochastic process [1]. If a quan-tum system starts in a given state, then by accuratelymonitoring the fluctuations of the detector, it is pos-sible to reconstruct single quantum trajectories, whichdescribe the evolution of the quantum state conditionedto the measurement outcome [1]. The idea of quantumtrajectories made its transition from a theoretical tool(unraveling) to simulate open quantum systems [2] to aphysically accessible quantity with the experimental abil-ity of tracking these trajectories in optical [3, 4] and morerecently in solid state [5, 6] systems. Continuous monitor-ing of superconducting qubits has, for example, enabledcontinuous feedback control [7–9], the determination ofweak values [10–12], and the production of deterministicentanglement [13, 14]. In view of their ability to com-bine quantum trajectory monitoring with external uni-tary driving, these superconducting devices additionallyoffer a unique platform to explore energy exchanges andthermodynamics along single quantum trajectories.The laws of thermodynamics classify energy changesfor macroscopic systems as work performed by externaldriving and heat exchanged with the environment [15]. Inpast decades, these principles have been successfully ex-tended to the level of classical trajectories to account forthermal fluctuations [16]. By providing a theoretical andexperimental framework for determining work and heatalong individual trajectories, stochastic thermodynamicshas paved the way for the study of the energetics of micro-scopic systems, from colloidal particles to enzymes andmolecular motors [17, 18]. The further generalization ofthermodynamics to include quantum fluctuations facesunique challenges, ranging from the proper identificationof heat and work to the clarification of the role of co-herence [19–22]. Quantum heat is commonly associated with the nonunitary part of the dynamics [23–25], carry-ing over the classical notion of energy exchanged with thesurroundings. This definition has recently been extendedto the level of single discrete quantum jumps [26–31] andto individual continuous quantum trajectories [32, 33].Other definitions of quantum work and heat have beenput forward, for instance based on the single shot ap-proach [34, 35] or quantum resource theory [36, 37]. Thisdiversity of theoretical approaches emphasizes the crucialimportance of an experimental study.We here report the measurement of work and heat as-sociated with unitary and non-unitatry dynamics alongsingle quantum trajectories of a superconducting qubit.The qubit evolves under continuous unitary evolutionand is only weakly coupled to the detector. As a re-sult, information about its state may be inferred from themeasured signal without projecting it into eigenstates.This system might thus generically be in coherent su-perpositions of energy eigenstates. We show that themeasured heat and work are consistent with the first lawand prove the agreement with both the two-projective-measurement (TPM) scheme [38] and the master equa-tion approach [23–25]. We finally establish the corre-spondence with the TPM work in the unitary limit byemploying a phase-locking quantum feedback loop thateffectively compensates for the heat.
Heat and work along quantum trajectories.
In macro-scopic thermodynamics, work performed on a thermallyisolated system is defined as the variation of internal en-ergy, W = ∆ U [15]. According to the first law, heat isgiven by the difference, Q = ∆ U − W , for systems thatare not thermally isolated [15]. Thermal isolation is thusessential to distinguish heat from work. At the quantumlevel, identifying heat and work is more involved, becausequantum systems do not necessarily occupy definite en-ergy states. Energy changes are usually defined in termsof transition probabilities between energy eigenstates in a r X i v : . [ qu a n t - ph ] A p r the so-called two-projective-measurement (TPM) scheme[38]. For a driven quantum system described by theHamiltonian H t , the distribution of the total energy vari-ation ∆ U is thus given by [38], P (∆ U ) = (cid:88) m,n P τm,n P n δ [∆ U − ( E τm − E n )] , (1)where P n denote the initial occupation probabilities, P τm,n are the transition probabilities between initial andfinal eigenvalues E n and E τm of H t , and τ is the durationof the driving protocol. This relation has been used toexperimentally determine the work distribution in closedquantum systems such as NMR, trapped ion, and coldatom systems [39–41], for which ∆ U = W .However, in open quantum systems, the total energychange ∆ U cannot, in general, be uniquely separated intoheat and work [42] and several definitions have been pro-posed [23–37]. Open quantum systems can be describedwith density operator ρ t with evolution given by a quan-tum master equation [43], dρ t dt = − i (cid:126) [ H t , ρ t ] + L ρ t , (2)where L is a Lindblad dissipator. In this case, the firstlaw has been written in the usual form, ∆ ¯ U = ¯ Q + ¯ W ,with [23–25],¯ Q = (cid:90) τ dt tr (cid:20) dρ t dt H t (cid:21) , ¯ W = (cid:90) τ dt tr (cid:20) ρ t dH t dt (cid:21) . (3)As in classical thermodynamics, ¯ Q is the energy sup-plied to the system by the environment and ¯ W the workdone by external driving. The above definition of quan-tum work has been originally introduced by Pusz andWoronowicz in a C ∗ -algebraic context [44] and recentlyapplied to individual discrete quantum jumps [26–29].In our experiment, we examine how quantum heat andwork can be consistently identified for systems whose en-vironment consists of a continuously coupled quantumlimited detector, an effectively zero temperature reser-voir [1]. The ability to track quantum state trajectoriesenables energy changes to be decomposed separately intoheat and work components [32, 33]. The starting point ofour analysis is that the quantum state evolution consistsof both a unitary part and, because of the continuousmonitoring, an additional nonunitary component: theformer is again identified as work, the latter as heat, inanalogy to macroscopic thermodynamics [32, 33]. Specif-ically, for an infinitesimal time interval dt , a change of theconditional density operator for a single trajectory maybe written as d ˜ ρ t = δ W [˜ ρ t ] dt + δ Q [˜ ρ t ] dt , where δ W [˜ ρ t ]and δ Q [˜ ρ t ] are superoperators associated with the respec-tive unitary and nonunitary dynamics [32]. The tilde heremarks quantities that are evaluated in different realiza-tions of the experiment, as opposed to quantities aver-aged over the possible trajectories. The first law along a single quantum trajectory ˜ ρ t then reads d ˜ U = δ ˜ W + δ ˜ Q ,with δ ˜ W = tr[˜ ρ t − dt dH t ] and δ ˜ Q = tr[ H t d ˜ ρ t ] [45]. Whenintegrated over time, the first law takes the form,∆ ˜ U = (cid:90) τ d ˜ Udt dt = (cid:90) τ δ ˜ Wdt dt + (cid:90) τ δ ˜ Qdt dt, (4)for each quantum trajectory. Equation (4) is a quan-tum extension of the first law of stochastic thermody-namics. It relates the average change of energy ∆ ˜ U withthe path-dependent heat ˜ Q and work ˜ W . Similarly, wemay distinguish quantum heat and work contributions tochanges of the transition probabilities [32], d ˜ P m,n = δ ˜ P Wm,n + δ ˜ P Qm,n , (5)along single quantum trajectories [45].The consistency of the decompositions (4) and (5) maybe established in three independent ways: (i) the to-tal energy change along a trajectory, ∆ ˜ U = (cid:80) d ˜ U , andthe total transition probability, ˜ P nm = (cid:80) d ˜ P nm , may becompared to the TPM approach [38], (ii) the stochasticheat and work contributions (4) may be compared to themean quantities (3) after averaging over stochastic andquantum fluctuations, and (iii) finally, the work (4) alonga trajectory may be directly compared to the TPM result(1) in the unitary limit when heat vanishes. In that case,∆ ˜ U = ∆ U = E τm − E n = W [45]. Experimental set-up . The qubit is realized by the near-resonant interaction of a transmon circuit [46] and a threedimensional aluminum cavity [47] capacitivley coupled toa 50 Ω transmission line. Resonant coupling between thecircuit and cavity results in an effective qubit which isdescribed by the Hamiltonian, H q = − (cid:126) ω q σ z /
2, and de-picted in Figure 1a. The radiative interaction betweenthe qubit and transmission line is given by the inter-action Hamiltonian, H int = (cid:126) γ ( aσ + + a † σ − ), where γ is the coupling rate between the electromagnetic fieldmode corresponding to a ( a † ), the annihilation (creation)operator, and the qubit state transitions denoted by σ + ( σ − ), the raising (lowering) ladder operator for thequbit. By virtue of this interaction Hamiltonian, a ho-modyne measurement along an arbitrary quadrature ofthe quantized electromagnetic field of the transmissionline, ae − iϕ + a † e + iϕ , results in weak measurement alongthe corresponding dipole of the qubit, σ + e − iϕ + σ − e + iϕ [48]. In order to perform work on the qubit, we introducea classical time-dependent field described by the Hamil-tonian H R = (cid:126) Ω R σ y cos( ω q t + ϕ ), where ω q is the reso-nance frequency of the qubit and Ω R is the Rabi drivefrequency.Homodyne monitoring is performed with a Josephsonparametric amplifier [49, 50] operated in phase-sensitivemode. We adjust the homodyne detection quadraturesuch that the homodyne signal dV t obtained over the timeinterval ( t, t + dt ) provides an indirect signature [51] ofthe real part of σ − = ( σ x + iσ y ) /
2. The detector signal D r i v e D e c a y TransmoncircuitMicrowaveCavity Measurement Parametric amplifer DriveQuantum emitter Homodyne detection 3.02.01.0 P P , P + P (a) (b) (c)(d) (e) -3.0-2.0-1.01.00.50.0 E ne r g y ( h ω q ) E ne r g y ( h ω q ) ~ P , ~ ~ ~ Q W W Q δ Q~ δ W~ δ U~ Δ U~ Δ UTime ( μ s) Time ( μ s) N FIG. 1. Evaluating heat and work along single quantum trajectories. (a), Schematic of the qubit system, drive, and homodynedetection. (b), Work (blue), heat (red) and energy (green) along a single trajectory. The discrete timestep resolution is δt = 20ns, the smallest compatible with the detection bandwidth (c), A scatter plot of final work and heat contributions to the P , transition probability for an ensemble of ∼ experimental protocols of duration 2 µ s. Each experimental sequence terminateswith a projective measurement and the color of the points indicate the outcome of this measurement (orange: m = 1, purple: m = 0). The heat and work contributions are not necessarily bounded, but their sum is limited to [-1,0] as expected. (d),The total energy along a single quantum trajectory (green) compared to the total energy as determined from an ensemble ofprojective measurements at each time point (circles). The error bars indicate the standard error of the mean. (e), Projectivemeasurements binned and averaged according to the sum of the work and heat contributions ˜ P W , + ˜ P Q , . The error bars indicatethe standard error of the mean based on the number of occurrences ( N ) for each value of ˜ P W , + ˜ P Q , (inset). is given by dV t = √ ηγ (cid:104) σ x (cid:105) dt + √ γdX t , where η is thequantum efficiency of the homodyne detection, γ is theradiative decay rate, and dX t is a zero-mean Gaussianrandom variable with variance dt .The qubit evolution, given both driven evolution H R and homodyne measurement results dV t , is described inthe rotating frame by the stochastic master equation [52], d ˜ ρ t = − i (cid:126) [ H R , ˜ ρ t ] dt + γ D [ σ − ]˜ ρ t dt + √ ηγ H [ σ − dX t ]˜ ρ t , (6)where D [ σ − ]˜ ρ = σ − ˜ ρσ + − ( σ + σ − ˜ ρ + ˜ ρσ + σ − ) and H [ O ]˜ ρ = O ˜ ρ + ˜ ρO † − tr[( O + O † )˜ ρ ]˜ ρ are the dissipationand jump superoperators, respectively. By taking theensemble average, Eq. (6) reduces to a master equationof the form (2) with dissipator L ρ t = γ D [ σ − ] ρ t , whichdescribes the coupling to a zero-temperature reservoir [1].We next introduce the experimental protocols to de-termine the stochastic heat and work contributions. Weidentify the instantaneous work contribution δ W [˜ ρ t ] withthe first (unitary) term in Eq. (6), while the instanta-neous heat contribution δ Q [˜ ρ t ] is associated with the lat-ter two (nonunitary) terms. Although the system could,in general, exchange energy with the detector in theform of heat or work, the homodyne measurement in ourexperiment only induces a zero-mean stochastic back-action, which guarantees no extra work is done by thedetection process. Having access to the instantaneous heat and work con-tributions from an individual quantum trajectory, wenow verify the first law in the form of Eqs. (4) and (5).For this, we initialize the qubit in the eigenstate n , andthen drive the qubit while collecting the homodyne mea-surement signal. Figure 1b shows the path-dependentinstantaneous heat and work contributions, δ ˜ Q and δ ˜ W ,and the corresponding changes in internal energy d ˜ U fora single trajectory originating in n = 0. After time τ ,we projectively measure [53] the qubit in state m andthen repeat the experiment several times. In Figure 1cwe show a scatter plot of the calculated heat and workcontributions to the transition probabilities, ˜ P Qm,n and˜ P Wm,n , for τ = 2 µ s. Each single quantum trajectory ex-hibits different heat and work contributions, highlightingthe stochastic nature of its quantum evolution. Using in-dividual heat and work trajectories we now address theconsistency of these decompositions in three independentways. (i) Total energy change —In order to establish the con-sistency of these results with the TPM scheme [38], wefirst show in Figure 1d the path-dependent total energyvariation ∆ ˜ U = (cid:80) δ ˜ U for a single trajectory and thepath-independent total energy change ∆ U = ( (cid:126) ω q ) P τ , obtained via projective measurements performed at var-ious intermediate times [45]. We find that the path-independent energy changes are in excellent agreementwith the energy changes along a single quantum tra-jectory. In Figure 1e we further compare the path- -10-50510 86420 Time( μ s) (a) (b) P ( Q ) ~ P ( W ) ~ Q ~ Q ~ W ~ W, ~ WQ__WQ~~WQ~~
FIG. 2. Comparison of stochastic and average heat and workquantities. (a), Individual heat and work trajectories ˜ Q , ˜ W are displayed as transparent red and blue traces. The meanof these individual trajectories (cid:104) ˜ Q (cid:105) , and (cid:104) ˜ W (cid:105) are displayedas dashed lines which are in good agreement with the meanvalues from the master equation, ¯ Q and ¯ W , Eq. (3), solid lines.(b), Distributions of ˜ Q and ˜ W at evolution time τ = 6 µ s. independent transition probability P , to the sum of thepath-dependent work and heat contributions, ˜ P W , + ˜ P Q , ,for experiments of variable duration τ = [0 , µ s. Weagain observe very good agreement. (ii) Correspondence with master equation definitions —Figure 2 displays the time evolution of the heat ˜ Q andwork ˜ W along single trajectories, as well as their respec-tive mean values. The ensemble average of the individualwork (cid:104) ˜ W (cid:105) and heat (cid:104) ˜ Q (cid:105) trajectories agrees well with thethe averaged values, ¯ Q and ¯ W , Eq. (3), thus recoveringthe expression by Pusz and Woronowicz [44] at the levelof unraveled quantum trajectories. In addition, the in-dividual trajectories allows examination of the heat andwork distributions (Fig. 2b) at each timestep. (iii) The unitary limit —We finally show correspon-dence of the quantum trajectory work ˜ W and the TPMwork, W = E τm − E n , for a single realization by experi-mentally isolating the system with a quantum feedbackloop [1]. The essence of feedback is to compensate forthe effect of the detector by adjusting the Hamiltonian ateach timestep, δ Q [˜ ρ t ], thus making the system effectivelyclosed. The dynamics of the system is then simply de-scribed by unitary evolution where only the work δ W [˜ ρ t ]contributes to changes in the state. In order to imple-ment feedback, we adapt the phase-locked loop protocolintroduced in Ref. [7]. This is achieved by multiplyingthe homodyne measurement signal with a reference oscil-lator of the form A [sin(Ω R t + φ ) + B ] yielding a feedbackcontrol, Ω F = √ η (cos(Ω t + φ ) − dV t /dt , that modu-lates the Rabi frequency of the qubit drive. The detectorheat exchange is eliminated by applying additional work, δ W F [˜ ρ t ] = ( i/ (cid:126) )[ (cid:126) Ω F σ y cos( ω q t + φ ) , ρ t ].Figure 3a shows the instantaneous feedback work, δ ˜ W F = (cid:126) ω q tr [Π m =1 δ W F [˜ ρ t ]] dt (with Π m the projec-tor onto eigenstate m ), together with the correspondinginstantaneous heat, δ ˜ Q = (cid:126) ω q tr [Π m =1 δ Q [˜ ρ t ]] dt , along (a) -0.10.00.10.2 86420 Time ( μ s) -0.100.1 δ Q -0.1 0.0 0.1 δ W F δ Q δ W F μ s) δ P -8-6-4-2 8642 ~~ ~ ~ E ne r g y ( h ω q ) δ P ( h ω q ) (h ω q ) F δ P δ P P , W ~ (b)(c) (d) , P ~P ~P FIG. 3. Quantum feedback loop. (a), Instantaneous heat andfeedback work along a single trajectory. The feedback workhas been time shifted by 20 ns to account for the time de-lay in the feedback circuit. The anti-correlation ( r = − . experimental it-erations. (d), Parametric plot of ˜ P W , versus ˜ P Q , (red) and˜ P F + Q , (blue) showing how the feedback cancels the heat, nar-rowing and shifting the distribution toward zero for τ = 6 µ s. a trajectory for a quantum efficiency of 35%. We ob-serve that the feedback partially cancels the heat at eachpoint in time. The anti-correlation between the instan-taneous feedback and heat contributions depicted in Fig-ure 3b confirms that the feedback loop compensates forexchanged heat at each timestep. In addition, by aver-aging the instantaneous heat and work contributions tothe transition probability over many iterations of the ex-periment (Fig. 3c), we clearly see how feedback workstoward canceling the heat on average. Similarly, at thelevel of single trajectories, the total transition probabilitymay be written as ˜ P τm,n = ˜ P Wm,n + ˜ P Qm,n + ˜ P Fm,n , with thework contribution from feedback ˜ P Fm,n . Figure 3d showsthe transition probabilities ˜ P W , versus ˜ P Q , + ˜ P F , . Bycomparing the transition probabilities with and withoutfeedback, we observe a significantly reduced heat contri-bution.In the presence of the quantum feedback loop we candecompose the instantaneous work along trajectories intowork imparted by the feedback and work associated withthe driving protocol, δ ˜ W . In the absence of the feed-back loop, the quantum dynamics of the qubit are givenby work δ W [˜ ρ t ] and heat δ Q [˜ ρ t ] superoperators; the heatchanges the state, causing the observed δ ˜ W to differ fromthe case of closed unitary evolution, δ ˜ W u . With the feed-back loop, the heat contribution is compensated at each -0.040.00.04 86420-0.08-0.040.00.040.08-0.08 0.080.0 Time( μ s) (a) δ W ∼ δ W Feedback ∼ δ
W Open ∼ ∼δ W u Unitary δ W u Unitary ∼ -0.08 0.080.0 ∼ δ W F eedba ck ∼ δ W O pen ∼ -0.08-0.040.00.040.08 (b) (c) δ W u Unitary
FIG. 4. Work along trajectories with and without feedback.(a), the instantaneous work δ ˜ W along a single trajectory inthe presence of feedback (blue) and the open loop configura-tion (no feedback) (red) is compared to the calculated instan-taneous work expected for pure unitary evolution (green). (b,c), The correlation between the instantaneous work δ ˜ W andthe work for a unitary evolution along a quantum trajectory.The linear regression fit (black lines) show correlation (slope0.437) when the feedback loop is employed, and no correlationin the open loop configuration (slope 0.006). timestep causing the instantaneous work δ ˜ W to matchthe expected unitary work δ ˜ W u . Figure 4 displays δ ˜ W for a single quantum trajectory in the presence of feed-back (blue) and for a different trajectory in the absenceof feedback (red) compared to the expected unitary work δ ˜ W u (green). Figure 4b,c show that in the presence offeedback the work is more closely correlated with theunitary work, with the correlation only limited by theefficiency of the feedback loop [45]. In the limit of unitquantum efficiency and null loop delay, a feedback loopcould exactly compensate for the exchanged heat [45]. Conclusion.
We have introduced experimental proto-cols for a continuously monitored driven qubit that allowto operationally define and individually measure quan-tum heat and work along single trajectories, account-ing for the presence of coherent superpositions of energyeigenstates. We have verified the first law of thermody-namics at the level of energy exchanges and of transi-tion probabilities. Moreover, we have demonstrated theconsistency of these results with the master equation ap-proach as well as with the TPM scheme, both for openand closed evolutions, with the help of feedback control.Our findings pave the way for future experimental andtheoretical studies in quantum thermodynamics [54] atthe single trajectory level.
Acknowledgements:
We acknowledge research supportfrom the NSF (Grants No. PHY-1607156 and No. PHY-1752844 (CAREER)), the ONR (Grant No. 12114811), the John Templeton Foundation, and the EPSRC (GrantNo. EP/P030815/1). This research used facilities at theInstitute of Materials Science and Engineering at Wash-ington University. K. W. M. acknowledges support fromthe Sloan Foundation. E. L. acknowledges support fromthe German Science Foundation (DFG) (Grant No. FOR2724). [1] K. Jacobs,
Quantum Measurement Theory (Cambridge,2014).[2] H. Carmichael,
An Open Systems Approach to QuantumOptics (Springer-Verlag, 1993).[3] C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes,S. Kuhr, M. Brune, J. Raimond, and S. Haroche, “Pro-gressive field-state collapse and quantum non-demolitionphoton counting,” Nature , 889 (2007).[4] C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, T. Ry-barczyk, G. Sebastien, P. Rouchon, M. Mirrahimi,H. Amini, and M. Brune, “Real-time quantum feedbackprepares and stabilizes photon number states,” Nature , 73 (2011).[5] K. W. Murch, S. J. Weber, K. M. Beck, E. Ginossar, andI. Siddiqi, “Reduction of the radiative decay of atomiccoherence in squeezed vacuum,” Nature , 62 (2013).[6] S. J. Weber, A. Chantasri, J. Dressel, A. N. Jordan,K. W. Murch, and I. Siddiqi, “Mapping the optimalroute between two quantum states,” Nature , 570(2014).[7] R. Vijay, C. Macklin, D. H. Slichter, S. J. Weber, K. W.Murch, R. Naik, A. N. Korotkov, and I. Siddiqi, “Stabi-lizing Rabi oscillations in a superconducting qubit usingquantum feedback,” Nature , 77 (2012).[8] M. S. Blok, C. Bonato, M. L. Markham, D. J. Twitchen,V. V. Dobrovitski, and R. Hanson, “Manipulating aqubit through the backaction of sequential partial mea-surements and real-time feedback,” Nature Physics ,189–193 (2014).[9] G. de Lange, D. Rist`e, M. J. Tiggelman, C. Eichler,L. Tornberg, G. Johansson, A. Wallraff, R. N. Schouten,and L. DiCarlo, “Reversing quantum trajectories withanalog feedback,” Phys. Rev. Lett. , 080501 (2014).[10] J. P. Groen, D. Rist`e, L. Tornberg, J. Cramer, P. C.de Groot, T. Picot, G. Johansson, and L. DiCarlo,“Partial-measurement backaction and nonclassical weakvalues in a superconducting circuit,” Phys. Rev. Lett. , 090506 (2013).[11] P. Campagne-Ibarcq, L. Bretheau, E. Flurin, A. Auff`eves,F. Mallet, and B. Huard, “Observing interferences be-tween past and future quantum states in resonance fluo-rescence,” Phys. Rev. Lett. , 180402 (2014).[12] D. Tan, S. J. Weber, I. Siddiqi, K. Mølmer, and K. W.Murch, “Prediction and retrodiction for a continuouslymonitored superconducting qubit,” Phys. Rev. Lett. ,090403 (2015).[13] D. Rist`e, M. Dukalski, C.A. Watson, G. de Lange,M. J. Tiggelman, Ya.M. Blanter, K.W. Lehnert, R. N.Schouten, and L. DiCarlo, “Deterministic entanglementof superconducting qubits by parity measurement andfeedback,” Nature , 350 (2013). [14] N. Roch, M. E. Schwartz, F. Motzoi, C. Macklin,R. Vijay, A. W. Eddins, A. N. Korotkov, K. B.Whaley, M. Sarovar, and I. Siddiqi, “Observation ofmeasurement-induced entanglement and quantum tra-jectories of remote superconducting qubits,” Phys. Rev.Lett. , 170501 (2014).[15] A. B. Pippard, Elements of Classical Thermodynamics (Cambridge, 1966).[16] C. Jarzynski, “Equalities and inequalities: Irreversibilityand the second law of thermodynamics at the nanoscale,”Annual Review of Condensed Matter Physics , 329–351(2011).[17] U. Seifert, “Stochastic thermodynamics, fluctuation the-orems and molecular machines,” Rep. Prog. Phys. ,126001 (2012).[18] S. Ciliberto, R. Gomez-Solano, and A. Petrosyan,“Fluctuations, linear response, and currents in out-of-equilibrium systems,” Annual Review of Condensed Mat-ter Physics , 235–261 (2013).[19] R. Gallego, J. Eisert, and H. Wilming, “Thermody-namic work from operational principles,” New Journalof Physics , 103017 (2016).[20] S. Deffner, J. P. Paz, and W. H. Zurek, “Quantum workand the thermodynamic cost of quantum measurements,”Phys. Rev. E , 010103 (2016).[21] P. Kammerlander and J. Anders, “Coherence and mea-surement in quantum thermodynamics,” Scientific Re-ports , 22174 (2016).[22] N. Cottet, S. Jezouin, L. Bretheau, P. Campagne-Ibarcq,Q. Ficheux, J. Anders, A. Auff`eves, R. Azouit, P. Rou-chon, and B. Huard, “Observing a quantum Maxwelldemon at work,” Proceedings of the National Academyof Sciences , 7561–7564 (2017).[23] R. Alicki, “The quantum open system as a model of theheat engine,” Journal of Physics A: Mathematical andGeneral , L103–L107 (1979).[24] Herbert Spohn and Joel L. Lebowitz, “Irreversible ther-modynamics for quantum systems weakly coupled tothermal reservoirs,” in Advances in Chemical Physics (John Wiley & Sons, Inc., 2007) pp. 109–142.[25] R. Kosloff, “A quantum mechanical open system as amodel of a heat engine,” The Journal of Chemical Physics , 1625–1631 (1984).[26] H.-P. Breuer, “Quantum jumps and entropy production,”Phys. Rev. A , 032105 (2003).[27] B. Leggio, A. Napoli, A. Messina, and H.-P. Breuer,“Entropy production and information fluctuations alongquantum trajectories,” Phys. Rev. A , 042111 (2013).[28] F. W. J. Hekking and J. P. Pekola, “Quantum jump ap-proach for work and dissipation in a two-level system,”Phys. Rev. Lett. , 093602 (2013).[29] Z. Gong, Y. Ashida, and M. Ueda, “Quantum-trajectorythermodynamics with discrete feedback control,” Phys.Rev. A , 012107 (2016).[30] J. M. Horowitz, “Quantum-trajectory approach to thestochastic thermodynamics of a forced harmonic oscilla-tor,” Phys. Rev. E , 031110 (2012).[31] J. M. Horowitz and J. M R Parrondo, “Entropy produc-tion along nonequilibrium quantum jump trajectories,”New Journal of Physics , 085028 (2013).[32] J. J. Alonso, E. Lutz, and A. Romito, “Thermodynamicsof weakly measured quantum systems,” Phys. Rev. Lett. , 080403 (2016).[33] C. Elouard, D. A. Herrera-Mart`ı, M. Clusel, and A. Auff´eves, “The role of quantum measurement instochastic thermodynamics,” npj Quantum Information , 9 (2017).[34] J. Aberg, “Truly work-like work extraction via a single-shot analysis,” Nature Communications (2013).[35] N. Y. Halpern, A. J. P. Garner, O. C. O. Dahlsten, andV. Vedral, “Introducing one-shot work into fluctuationrelations,” New Journal of Physics , 095003 (2015).[36] F. G. S. L. Brand˜ao, M. Horodecki, J. Oppenheim, J. M.Renes, and R. W. Spekkens, “Resource theory of quan-tum states out of thermal equilibrium,” Phys. Rev. Lett. , 250404 (2013).[37] F. G. S. L. Brand˜ao and G. Gour, “Reversible frameworkfor quantum resource theories,” Phys. Rev. Lett. ,070503 (2015).[38] P. Talkner, E. Lutz, and P. H¨anggi, “Fluctuation the-orems: Work is not an observable,” Phys. Rev. E ,050102 (2007).[39] T. B. Batalh˜ao, A. M. Souza, L. Mazzola, R. Auccaise,R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara,M. Paternostro, and R. M. Serra, “Experimental recon-struction of work distribution and study of fluctuationrelations in a closed quantum system,” Phys. Rev. Lett. , 140601 (2014).[40] S. An, J.-N. Zhang, M. Um, D. Lv, Y. Lu, J. Zhang, Z.-Q.Yin, H. T. Quan, and K. Kim, “Experimental test of thequantum Jarzynski equality with a trapped-ion system,”Nature Physics , 193199 (2015).[41] F. Cerisola, Y. Margalit, S. Machluf, A. J. Roncaglia,J. P. Paz, and R. Folman, “Using a quantum work me-ter to test non-equilibrium fluctuation theorems,” NatureCommunications (2017).[42] M. Campisi, P. H¨anggi, and P. Talkner, “Colloquium:Quantum fluctuation relations: Foundations and appli-cations,” Rev. Mod. Phys. , 771–791 (2011).[43] Heinz-Peter Breuer and Francesco Petruccione, The The-ory of Open Quantum Systems (Oxford University Press,2002).[44] W. Pusz and S. L. Woronowicz, “Passive states and KMSstates for general quantum systems,” Communications inMathematical Physics , 273–290 (1978).[45] Further details are given in supplemental material.[46] Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck,D. I. Schuster, J. Majer, Alexandre Blais, M. H. Devoret,S. M. Girvin, and R. J. Schoelkopf, “Charge-insensitivequbit design derived from the Cooper pair box,” Phys.Rev. A , 042319 (2007).[47] H. Paik, D. I. Schuster, L. S. Bishop, G. Kirchmair,G. Catelani, A. P. Sears, B. R. Johnson, M. J. Reagor,L. Frunzio, L. I. Glazman, S. M. Girvin, M. H. De-voret, and R. J. Schoelkopf, “Observation of high coher-ence in Josephson junction qubits measured in a three-dimensional circuit QED architecture,” Phys. Rev. Lett. , 240501 (2011).[48] M. Naghiloo, N. Foroozani, D. Tan, A. Jadbabaie, andK. W. Murch, “Mapping quantum state dynamics inspontaneous emission,” Nature Communications , 11527(2016).[49] M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton,L. R. Vale, and K. W. Lehnert, “Amplification andsqueezing of quantum noise with a tunable Josephsonmetamaterial,” Nature Physics , 929–931 (2008).[50] M. Hatridge, R. Vijay, D. H. Slichter, J. Clarke, andI. Siddiqi, “Dispersive magnetometry with a quantum limited SQUID parametric amplifier,” Phys. Rev. B ,134501 (2011).[51] A. N. Jordan, A. Chantasri, P. Rouchon, and B. Huard,“Anatomy of fluorescence: Quantum trajectory statis-tics from continuously measuring spontaneous emission,”Quantum Stud.: Math. Found. , 237 (2016).[52] A. Bolund and K. Mølmer, “Stochastic excitation duringthe decay of a two-level emitter subject to homodyne andheterodyne detection,” Phys. Rev. A , 023827 (2014).[53] M. D. Reed, L. DiCarlo, B. R. Johnson, L. Sun, D. I.Schuster, L. Frunzio, and R. J. Schoelkopf, “High-fidelityreadout in circuit quantum electrodynamics using theJaynes-Cummings nonlinearity,” Phys. Rev. Lett. ,173601 (2010).[54] J. Gemmer, M. Michel, and G. Mahler, Quantum Ther-modynamics (Springer, 2004). [55] K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi,“Observing single quantum trajectories of a supercon-ducting qubit,” Nature , 211 (2013).[56] M. Naghiloo, D. Tan, P. M. Harrington, P. Lewalle,A. N. Jordan, and K. W. Murch, “Quantum causticsin resonance-fluorescence trajectories,” Phys. Rev. A ,053807 (2017).[57] P. Campagne-Ibarcq, E. Flurin, N. Roch, D. Darson,P. Morfin, M. Mirrahimi, M. H. Devoret, F. Mallet, andB. Huard, “Persistent control of a superconducting qubitby stroboscopic measurement feedback,” Phys. Rev. X ,021008 (2013).[58] D. Rist`e, C. C. Bultink, K. W. Lehnert, and L. Di-Carlo, “Feedback control of a solid-state qubit using high-fidelity projective measurement,” Phys. Rev. Lett. ,240502 (2012). SUPPLEMENTAL MATERIALHeat and work definitions and contributions to transition probabilities
For a continuously monitored driven quantum system, one can consistently identify unitary and nonunitary contri-butions to (i) the evolution of the conditional density operator ˜ ρ t along a single trajectory, (ii) to the energy of thesystem ˜ U t (averaged over quantum fluctuations), and (iii) to changes δ ˜ P m,n of the transition probabilities [32]. Thedefinition of heat and work in the manuscript stems from the association of work with the deterministic driving of thequantum system, and of heat with the stochastic evolution due to the detection process, d ˜ ρ t = δ W [˜ ρ t ] dt + δ Q [˜ ρ t ] dt ,where δ W [˜ ρ t ] and δ Q [˜ ρ t ] are superoperators associated with the respective unitary and nonunitary dynamics. In fact,the change in the internal energy between times t and t + dt along a single quantum trajectory may be expressed as, d ˜ U t =tr[ H t (˜ ρ t − dt + d ˜ ρ t )] − tr[ H t − dt ˜ ρ t − dt ]=tr[˜ ρ t − dt dH t ] + tr[ H t δ W [˜ ρ t ] dt ] + tr[ H t δ Q [˜ ρ t ] dt ]= δ ˜ W t + δ ˜ Q t , (7)where in the second line dH t = H t − H t − dt and tr[ H t δ W [˜ ρ t ] dt ] = − ( i/ (cid:126) )tr[ H t [ H t , ρ t ] dt ] = 0. In the last line, δ ˜ W t = tr[˜ ρ t − dt dH t ] and δ ˜ Q = tr[ H t d ˜ ρ t ], indicating that work is related to a change of the Hamiltonian and heat tothe nonunitary changes in the state. Equation (7) shows that there are actually two contributions to the averagedwork δ ˜ W t : one coming from the variation dH t of the Hamiltonian and one coming from the superoperator δ W [˜ ρ t ].However, the average of the latter vanishes due to the properties of the trace. The superoperator δ W [˜ ρ t ], by contrast,directly contributes to the (unaveraged) density operator ˜ ρ t and transition probabilities δ ˜ P m,n . We note that theassociation of δ W [˜ ρ t ] with work at this level is limited to the case of driven unitary dynamics.The different contributions of heat and work to the quantum evolution are reflected in different contributions tothe transition probabilities. The changes to the transition probabilities due to heat and work are defined as δ ˜ P Qm,n =tr [Π m δ Q [˜ ρ t ]], and δ ˜ P Wm,n = tr [Π m δ W [ ρ t ]], where Π m is the projective measurement operator of the eigenstate m at time t and the trajectory ˜ ρ ( t ) originates in eigenstate n . Correspondingly, we define the path-dependent totaltransition probabilities, ˜ P Qm,n = (cid:90) τ dt δ ˜ P Qm,n , ˜ P Wm,n = (cid:90) τ dt δ ˜ P Wm,n . (8)The definition of heat and work contributions to the evolution of the density matrix, d ˜ ρ t , imply that, starting from adensity matrix ρ corresponding to an eigenstate n , the total transition probabilities ˜ P τm,n are given by˜ P τm,n = tr[Π m ˜ ρ t ] = P m,n + (cid:90) τ dt tr[Π m δ Q [˜ ρ t ]] dt + (cid:90) τ dt tr[Π m δ Q [˜ ρ t ]] dt = ˜ P Qm,n + ˜ P Wm,n , (9)The instantaneous heat, work, and feedback are also expressed in terms of energy by δ ˜ Q = (cid:126) ω q tr [Π m =1 δ Q [˜ ρ t ]], δ ˜ W = (cid:126) ω q tr [Π m =1 δ W [˜ ρ t ]], and δ ˜ W F = (cid:126) ω q tr [Π m =1 δ W F [˜ ρ t ]], respectively. These changes reflect only the instantaneouschanges in energy and do not depend on the initial state of the system.Transition probabilities can be experimentally obtained by preparing the qubit in a specific initial state n andterminating the experiment at time τ with a projective measurement [53] in the energy basis. These projectivemeasurements allow us to determine the total energy change the qubit and compare to the total energy calculatedfrom the heat and work along individual quantum trajectories, in a manner similar to the tomographic validationfor trajectories in previous work [6, 55]. In Figure 1d we show the total energy changes of the qubit ∆ U obtainedfrom the transition probability P , which was determined from an ensemble of 10 experiments of variable duration τ where the path-dependent energy ˜ U was within ± . (cid:126) ω q of the black curve shown in Figure 1d at time τ . Thetransition probability is corrected for the finite readout fidelity of 65%. In Figure 1e, we compare the measured,path-independent, transition probability P , , to the path-dependent transition probability ˜ P W , + ˜ P Q , , by binningexperiments of variable duration according to the final path-dependent transition probability and determining thetransition probability P , at each point from the outcomes of the projective readout. The close agreement between P , and ˜ P W , + ˜ P Q , indicates that the two independent measures of the qubit energy are in agreement, therebyconfirming the first law of thermodynamics. The unitary limit
In the absence of a detector, the system under consideration reduces to an isolated system and our definition of workagrees with the standard definition in Eq. (1) of the manuscript. In order to see this for a two energy measurementprotocol, we note that our distinction between heat and work amounts to the separation of work-like and heat-likecomponents of the transition probabilities, ˜ P Wm,n and ˜ P Qm,n . When the system detector coupling is vanishing, one has˜ P Wm,n = P τm,n . In fact, for an isolated system initially prepared in the E n energy eigenstate, the trajectory is onlydictated by the unitary evolution, with the nonunitary part being zero along the entire trajectory, δ Q [˜ ρ t ] /dt = 0.Hence ˜ P Qm,n = 0 and d ˜ ρ t = δ W [˜ ρ t ] dt . Together with Eq. (9), this implies the work-like component of the transitionprobability is reduced to that of an isolated system,˜ P Wm,n = ˜ P τm,n = tr[Π m ˜ ρ τ ] = P τm,n . (10)In addition, our formalism also reproduces the physics of an isolated system at the level of a single two-energy-measurement realization. In that case, one identifies four possible trajectories corresponding to the transitions fromthe initial states of energy E n to the final states with energy E τm . The probability of such a trajectory is P n P τm,n .Each of these trajectories consists of a unitary evolution from 0 to t f ending with a density matrix ρ f and a finalextra nonunitary step from t f to t f + ∆ t M = τ determined by the measurement, during which the Hamiltonian isunchanged and ρ f → ρ m = | m (cid:105)(cid:104) m | . The work contribution from the unitary evolution is, (cid:90) τ δ ˜ Wdt dt = (cid:90) t f δ ˜ Wdt dt = tr[˜ ρ f H τ − ˜ ρ H ]= tr[˜ ρ f H τ ] − E n . (11)The contribution from the last step does not involve exchange of energy with the detector and is therefore regardedas work, although it arises from nonunitary evolution [30]. This yields (cid:90) τ δ ˜ Wdt dt = (cid:90) t f +∆ t M t f δ ˜ Wdt dt = tr[˜ ρ m H τ − ˜ ρ f H τ ] = E τm − tr[˜ ρ f H τ ] . (12)This correctly reproduces the change of internal energy, ∆ ˜ U = ∆ U = E τm − E n = W associated with the trajectoryfrom energies E n to E τm . We therefore recover the full probability distribution in Eq. (1) of the manuscript. Heat and work tracking
The stochastic master equation (Eq. 3) is used to update state of the qubit conditioned on the collected homodynesignal which is digitized in 20 ns steps, and scaled such that its variance is γdt [48]. Our identification of work andheat as the respective unitary and nonunitary changes of the state applies in the laboratory frame. However, it is -0.4-0.20.00.20.4 δ Q -0.4 0.0 0.4 δ W f -0.100.000.10 86420 Time( μ s) δ Q δ W f -1.0-0.50.00.51.0 z a v g -1.0 0.0 1.0x avg θ R xx z A BCB’ θ Q (a) (b) (c) (d) 500ns0 ns ~ ~ ~ ~ E ne r g y ( h ω q ) z Supplemental Figure 1.
Simulation of optimal unitary feedback. (a), Schematic of the feedback operation, the qubitevolution from A at time t to C at time t + dt comprises of two different types of evolution; unitary evolution due to theRabi drive ( A → B ) and stochastic evolution due to coupling to the environment. Because the environment is monitored withnonideal quantum efficiency we effectively average over some of the stochastic evolution reducing the state purity ( B → C ). Theideal unitary feedback maintains the phase relation with unitary evolution by application of a rotation − θ Q to state B (cid:48) . (b),Instantaneous contributions of heat and feedback to the transition probability P , for a single run of experiment. (c), Scatterplot of the instantaneous heat versus feedback for 100 runs of experiment which shows an anti-correlation ( r = − .
9) betweenfeedback and heat contributions to the transition probability P . (d), Ensemble behavior of the trajectories in presence offeedback with no delay (red) and a 500 ns loop delay (blue) showing a persistent Rabi oscillations. convenient to calculate the state trajectories in the frame rotating with the qubit drive and identify energy changesin the rotating frame. We break the evolution into discrete timesteps. Each timestep i is divided into two substeps.The first substep updates the ˜ ρ [ i ] by the unitary terms. The second substep updates ˜ ρ [ i ] with the non-unitary termsgiven by the discretized stochastic master equation (in Itˆo form) [32, 56]. d ˜ ρ [ i ] = γ (1 − ˜ ρ [ i ]) dt + √ η ( dV [ i ] − √ ηγ ρ [ i ] dt )2˜ ρ [ i ](1 − ˜ ρ [ i ]) , (13) d ˜ ρ [ i ] = γ ˜ ρ [ i ] / dt + √ η ( dV [ i ] − √ ηγ ρ [ i ] dt )(1 − ˜ ρ [ i ] − ρ [ i ]) . (14)Therefore, in each timestep, we accordingly distinguish between instantaneous work and instantaneous heat in therotating frame. Since the transformation to the lab frame is set by the deterministic driving, it is straightforward todetermine these energy changes in the lab frame. Feedback
In this section, we analyze in detail the feedback protocol used in the experiment and compare it with optimalfeedback protocols in the presence of finite efficiency and feedback loop delay.If η = 1 and given a pure initial state, the state of the system is pure at all times, and is described by a vector onthe surface of the Bloch sphere. An ideal feedback loop would then exactly and immediately compensate for the heatexchanged with the environment resulting in completely unitary evolution of the state. This is not possible at finiteinefficiency, where the evolution of the system is no longer constrained to the surface of the Bloch sphere. Since theevolution due to the feedback protocol is unitary, it preserves the length of the Bloch vector, and cannot maintainpure evolution once purity has been lost. Therefore, it is impossible to exactly compensate for the exchanged heatwith a unitary operation.Given access to the trajectory in real time, the best possible feedback is to maintain the phase of the oscillationas if the qubit state were to undergo closed unitary evolution (Supplemental Fig. 1a). In this case, the feedbackoutput would be Ω F dt = − θ Q where θ Q can be calculated by the state of the qubit at t and t + dt . In this case wewould not have control over the purity of the state and it will change by measurement backaction from point to point.Supplemental Figure 1 shows simulation results for this type of feedback for 35% quantum efficiency. As depicted inSupplemental Figure 1b, the exchanged heat contribution is compensated by the feedback contribution. The scatterplot in Supplemental Figure 1(c) shows the anti-correlation of these contributions. Therefore, as we expect, thesystem will behave more like a closed system and we observe persistent Rabi oscillations with 70% of full contrast(Supplemental Fig. 1d). However, a realistic feedback loop would also have a finite loop delay, given by the time ittakes for the measurement signal to travel to a detector and for the feedback output to be calculated. Considering afeedback loop delay of ∼
500 ns [57, 58], Supplemental Figure 1(d) (blue curve) shows that the feedback performancewould be reduced.0 -1.0-0.50.00.51.0 Z a v g -1.0 0.0 1.0X avg (a) (b) (c) (d)100ns0 ns-0.4-0.20.00.20.4 δ Q -0.4 0.0 0.4 δ W f -0.10.00.1 86420 Time ( μ s) -2 0 20.30.20.10.0 F eedba ck sc a l e Offset δ Q ~ δ W f ~ ~ ~ Supplemental Figure 2.
Simulation of phase-lock feedback. (a), Instantaneous contribution of heat and feedback tothe transition probability P for a single run of experiment. (b), Scatter plot of instantaneous heat versus feedback for 100runs of the experiment which shows anti-correlation ( r = − .
81) between the feedback and heat contributions to transitionprobability P , . (c), Ensemble behavior of the trajectories in presence of feedback with no delay (red) and 100 ns loop delay(blue) showing a persistent Rabi oscillations. (d), Simulated feedback efficiency versus feedback scale ( A ) and offset ( B ). Thegreen dot indicates the expected parameters for optimal feedback. The feedback loop implemented in our experiment differs from the optimal feedback in that it does not requirereal-time state tracking and error processing. This feedback takes a copy of homodyne signal, dV t , and multiplies itby a sinusoidal reference signal, A [sin(Ω R t + φ ) + B ] resulting in a feedback signal Ω F , which is used to modulate thedrive amplitude.This feedback loop essentially implements a phase-locked loop, and in order to clarify how the loop works we maycast the stochastic master equation (3) in terms of the Bloch components x and z , dz = +Ω xdt + γ (1 − z ) dt + √ ηx (1 − z )( dV t − γ √ ηxdt ) (15) dx = − Ω zdt − γ xdt + √ η (1 − z − x )( dV t − γ √ ηxdt ) . (16)It is apparent that by canceling the last two terms, the evolution would be unitary as we expect for a closed system.However, with only unitary rotations we can change dz and dx in the following way, dz = Ω F xdt, dx = − Ω F zdt, (17)where, we wish to cancel all the stochastic terms in (15) with the unitary terms (17). Regardless how complicatedΩ F is, with finite efficiency, it is impossible to compensate for all terms as mentioned earlier and the best choicerecovers about 70% of purity for 35% quantum efficiency. To understand how the phase-locked loop approximates theoptimal feedback, we consider just the z ≡ (cid:104) σ z (cid:105) component of the state. This is reasonable since all thermodynamicsparameters e.g. work, heat and transition probabilities directly relate to the z component. This requires Ω F xdt = −√ ηx (1 − z )( dV t − γ √ ηxdt ), where for weak measurement γ √ ηxdt is negligible compared to dV t . Thus, we haveΩ F = −√ η (1 − z ) dV t /dt . The essence of the phase-locked loop is to replace z with cos(Ω t + φ ), which is the “target” z that would be obtained for closed evolution. Here φ = 0 ( π ) for an initial ground (excited) state. This choice for z has a two-fold effect: not only is this a reasonable approximation for z in presence of feedback but it also locks theoscillation phase which addresses the damping term in (15). Note that the choice of phase φ only affects the transientbehavior and appears as a overall phase shift in the persistent Rabi oscillations without affecting the contrast. Thusthe feedback signal is, Ω F = √ η (cos(Ω t + φ ) − dV t /dt, (18a)This equation suggests that the scale for feedback should be around A = √ η/dt ∼
30 which is in agreement withoptimal value found empirically in the experiment of A = 34. Experimentally, this factor may be set by pre-amplification of the homodyne signal. Note that this result also suggests the offset term of B = − B ∼ − A ∼
35 as we expect.1
Digitizer - d B - d B
3D transmonCu + Cryoperm shields(3, 0.8, 0.1 K) 9 mK H E M T ( K ) LJPA Δ A l + C r y o p e r m - d B - d B L P F L P F LPF φ IQ mixerAmplifierVariable attenuatorPhase shifterSplitter/CombinerDirectionalcoupler -10 dB
IsolatorAttenuatorSwitchLossy filter Circulator Δ
180 hybrid xyz x * y = z Analogmultiplier φ φ IsolationswitchStaterotations7.257 GHz 360 MHzLOLO selection RF f φφ xyz x * y = z Supplemental Figure 3.
Experimental setup
The qubit and Josephson parametric amplifier share a signal generator tomaintain the phase relation defining the amplification quadrature. We use a double sideband technique to pump the parametricamplifier. The homodyne signal is split for the purpose of feedback and state tracking. Both the feedback signal and homodynesignal are digitized for state tracking in a post-processing step.
Experimental setup and parameters
The transmon circuit was fabricated by double angle evaporation of aluminum on a high resistivity silicon substrate.The circuit was placed at the center of a 3D aluminum waveguide cavity machined from 6061 aluminum. Thebare cavity frequency is ω c / π = 7 .
257 GHz. The near-resonant interaction between the circuit and the cavity(characterized by coupling rate g/ π = 136 MHz) results in hybrid states, as described by the Jaynes-CummingsHamiltonian. The lowest energy transition of hybrid states ( ω q / π = 6 .
541 GHz) can therefore be considered a “one-dimensional” artificial atom because the radiative decay of the system is dominated by the cavity’s coupling to a 50Ω transmission line. This radiative decay was characterized by a decay of rate γ = 1 . µ s − . Resonance fluorescencefrom the artificial atom is amplified by a near-quantum-limited Josephson parametric amplifier, consisting of a 1 . I = 1 µ AJosephson junctions. The amplifier produces 20 dB of gain with an instantaneous 3-dB-bandwidth of 50 MHz. Thequantum efficiency was measured to be 35%. We drive the qubit by sending a resonant coherent signal via a weaklycoupled transmission line, and the strength of the drive is characterized by a Rabi frequency of Ω / π (cid:39) Statistical Analysis