Coherent oscillations inside a quantum manifold stabilized by dissipation
S. Touzard, A. Grimm, Z. Leghtas, S.O. Mundhada, P. Reinhold, R. Heeres, C. Axline, M. Reagor, K. Chou, J. Blumoff, K.M. Sliwa, S. Shankar, L. Frunzio, R.J. Schoelkopf, M. Mirrahimi, M.H. Devoret
CCoherent oscillations inside a quantum manifold stabilized by dissipation
S. Touzard, ∗ A. Grimm, Z. Leghtas, S.O. Mundhada, P. Reinhold, C.Axline, M. Reagor, K. Chou, J. Blumoff, K.M. Sliwa, S. Shankar, L. Frunzio, R.J. Schoelkopf, M. Mirrahimi,
2, 3 and M.H. Devoret Department of Applied Physics and Physics,Yale University, New Haven, CT 06520, USA Yale Quantum Institute, Yale University,New Haven, Connecticut 06520, USA QUANTIC team, INRIA de Paris, 2 Rue Simone Iff, 75012 Paris, France (Dated: November 15, 2017)
Abstract
Manipulating the state of a logical quantum bit usually comes at the expense of exposing it to decoher-ence. Fault-tolerant quantum computing tackles this problem by manipulating quantum information withina stable manifold of a larger Hilbert space, whose symmetries restrict the number of independent errors.The remaining errors do not affect the quantum computation and are correctable after the fact. Here weimplement the autonomous stabilization of an encoding manifold spanned by Schr¨odinger cat states in asuperconducting cavity. We show Zeno-driven coherent oscillations between these states analogous to theRabi rotation of a qubit protected against phase-flips. Such gates are compatible with quantum error correc-tion and hence are crucial for fault-tolerant logical qubits. a r X i v : . [ qu a n t - ph ] N ov he quantum Zeno effect (QZE) is the apparent freezing of a quantum system in one state un-der the influence of a continuous observation. This continuous observation can be performed bya dissipative environment [1–3]. It can be further generalized to the stabilization of a manifoldspanned by multiple quantum states, an operation which requires a dissipation that is blind to themanifold observables [4]. Harnessing this effect is crucial for the design of quantum computationschemes, since autonomous stabilization is a form of the feedback needed for quantum error cor-rection. When employing manifold QZE for correcting errors, motion inside the manifold can stillsubsist and can be driven by the combination of the dissipative stabilization and an external force[5–10]. Therefore manifold QZE offers a pathway towards the realization of logical gates compat-ible with quantum error correction. An example of such a system is provided by a superconductingmicrowave cavity, in which a dissipative process that annihilates photons in pairs at rate κ , actingtogether with a two-photon drive of strength (cid:15) projects the system onto the manifold spanned bySchr¨odinger cat states (cid:12)(cid:12) C ± α ∞ (cid:11) = N ( | α ∞ (cid:105) ± |− α ∞ (cid:105) ) , where | α ∞ (cid:105) is a coherent state of amplitude α ∞ = (cid:112) (cid:15) /κ and N is a normalization factor [11–13]. Each one of these states has a well de-fined photon number parity, which is conserved by the engineered dissipation. In this Schr¨odingercat states manifold, the displacement operator D ( α ) = exp( α a † − α ∗ a ) (where a is the annihilationoperator acting on the harmonic oscillator) has two effects: it changes the photon number parityand changes the amplitude of its component coherent states. The engineered dissipation leavesthe change in parity invariant and cancels the change in amplitude (Fig 1a). The net result of thisQuantum Zeno Dynamics (QZD) is to continuously vary the parity of Schr¨odinger cat states.These parity oscillations constitute the basis of an X-gate on a qubit encoded in the protectedmanifold | / (cid:105) P = N ( | α ∞ (cid:105) ± |− α ∞ (cid:105) ) . Encoding quantum information in superpositions ofSchr¨odinger cat states is compatible with quantum error correction (QEC) realized with quantumnon-demolition parity measurements [14–16]. Our gate is fundamentally different than previousmanipulations of Schr¨odinger cat states [17] as it operates while the manifold is stabilized. Thus,the quantum information is protected from out-of-manifold gate errors. Moreover the operationof the gate is not affected by the dominant source of errors: bit-flips. In fact, as the operationcommutes with them, it is compatible with a fault-tolerant scheme that would correct them afterthe operation.While related driven manifold dynamics have been proposed and observed [18–22], the non-linear dissipation specific to our experiment adds a crucial element: any drift out of the cat statemanifold is projected back into it. 2n our experiment, schematically shown in Fig 1, the drive-dissipation is implemented usingtwo-photon transitions between two electromagnetic modes. The first one has a high quality factorand stores the Schr¨odinger cat states. We refer to it as the storage (subscript S). The second one isused as an engineered cold bath that removes rapidly the entropy from the storage. We refer to it asthe reservoir (subscript R). We employ the four-wave mixing capability of a Josephson junction,together with two microwave pumps, to stimulate those transitions. In order to make resonant theconversion from one reservoir photon into two storage photons and vice-versa, the first pump isset at frequency f S − f R . The second pump, set at frequency f R , combines with the first one tocreate pairs of photons in the storage. When the dynamics of the reservoir mode is eliminated, thedensity matrix of the storage mode ρ is given by the Lindblad equation dρdt = − i (cid:126) [ H S , ρ ] + κ D [ a S − α ∞ ] ρ, where H S is a Hamiltonian acting on the storage and D [ L ] ρ = 2 L ρ L † − L † L ρ − ρ L † L is the Lind-blad superoperator. As the Lindblad superoperator is engineered to be the dominant term in the dy-namics, the dynamical steady states of the system are given by the coherent states |± α ∞ (cid:105) . The mi-crowave pumps set the phase and amplitude of the complex amplitude α ∞ . The Hamiltonian partof the equation contains the self Kerr effect of the storage mode induced by the Josephson junctionand the linear drive that induces the coherent oscillations: H S / (cid:126) = − χ SS / a † S a S ) + ( (cid:15) a † S + h . c . ) .The frequency of the coherent oscillations is maximum when the phase of the linear drive (cid:15) is per-pendicular to the phase of the stabilized Schr ¨odinger cat states. Thus, this linear drive displaces theSchr¨odinger cat state perpendicularly to the stabilization axis while the dissipation continuouslyprojects the system back to the stabilized manifold. If the drive respects the adiabaticity condition | (cid:15) | (cid:28) | α ∞ | κ then the net effect of the linear drive is to induce parity oscillations within thestabilized manifold, at frequency Ω = 2 (cid:15) | α ∞ | [12].The adiabaticity condition [6–10] sets an upper-bound on the frequency of these oscillations,fixed by the maximum κ that we can engineer. In order to observe this dynamics, we also need thecoherence time of the storage mode to be larger than the period of the oscillations. The architecturewe designed was key to engineer both a highly coherent storage mode and a large coupling to theenvironment. We implement the storage mode into a long-lived post cavity made of aluminium[23](Fig 1b). Its finite lifetime induces two types of errors on a protected qubit encoded in thestabilized manifold. First, in absence of stabilization, the amplitude of the Schr¨odinger cat statesdecays until eventually the two coherent states are no longer distinguishable. This error happens at3ate κ / π =1 . . Second, when the environment is observed to have absorbed a photon, theprojected density matrix of the storage mode suffers a parity jump, which corresponds to a bit-fliperror in our encoded qubit. For a stabilized cat state containing ¯ n = | α ∞ | photons on average,they happen at a rate ¯ nκ [24]. Additionally, the above mentioned Kerr effect would distort thecoherent states at rate χ SS / π ∼ in absence of stabilization. In order to achieve a fast non-linear dissipation, the storage cavity is coupled to a transmon qubit embedded into a coaxial tunnel[25], whose lifetime is engineered to be much less than that of the storage (
317 ns ), and we use itas the entropy reservoir to induce the QZE. While the reservoir is efficient to dispose of the entropyof the storage mode, its low lifetime has two impacts on the coherence of the storage. First, weassociate the lifetime of our storage cavity to the Purcell effect (usually κ / π is of order
100 Hz [23]). Second the finite temperature of the reservoir causes additionnal dephasing of the storagemode (see supplement [26]). However, the direct coupling between the storage and the reservoirmodes led to a non-linear dissipation rate of κ / π =176 kHz . The two orders of magnitudeseparating κ and κ are enough to observe the parity oscillations while respecting the adiabaticitycondition. The storage cavity is also coupled to a very coherent transmon whose coherence times( T = 70 µ s , T ∗ = 30 µ s ) are large compared to the time it takes to perform a parity measurementof the storage cavity using the dispersive coupling (
218 ns ). We use this transmon qubit to measurethe Wigner function of the storage mode.Our experimental protocol follows a fixed sequence of pulses which contains three parts(Fig 2a). The first step is the initialization of the system in the encoding manifold, which isdone with pulses generated by an optimal control algorithm [17]. As it involves transient statesthat are not Schr¨odinger cat states and that are entangled to the transmon qubit, this method in-duces errors on the protected encoding that are not corrected by the stabilization. However, it iscurrently the fastest method available. The second part is the stabilization of the manifold, whichis done with or without the rotation drive. Finally, in the third part, the Wigner function of thestorage cavity at a given point in phase space is measured [14, 27].We characterized the initialization and the quality of the measurement by taking a full Wignertomography of the storage cavity initialized in | (cid:105) P and | (cid:105) P (Fig 2b). The raw data consisted ofsingle shot parity measurements realized with a parametric amplifier and averaged without anyfurther normalization. The phase locking of the different drives ensured that the stabilization axisof the Schr¨odinger cat states was aligned with the Wigner representation axis, and that the rotationdrive was perpendicular to the stabilization axis [26]. The right column illustrates our ability to go4rom an even/odd parity Schr¨odinger cat state to a ”Yurke-Stoler” cat state [28]. The zero value ofthe Wigner function at the center of phase space shows that these states had no parity. They weregenerated by a rotation of π/ in the encoding manifold. It is important to note that the cat is notpushed sideway. The fringes are moving, but the ”blobs” remain in place (Fig 2b).In order to investigate the parity oscillations more closely, we restricted the measurement of theWigner function to the center of phase space (photon number parity measurement). In Fig 3a wepresent the time evolution of cat states, initially in the even state. We measured their parity over µ s while they were stabilized (Fig 3a). For ¯ n =
2, 3 and 5 we observed decay time constantsof respectively µ s , µ s and µ s . This behaviour arises from the natural single-photon jumpsof the cavity. They correspond to bit-flips within the encoding manifold which eventually destroythe coherence of the encoded qubit. The coherence of the encoded qubit is lost at a rate nκ [24]. This is close to what was found in the experiment and thus shows that the decoherence ismainly due to bit-flips happening during the stabilization. With the rotation drive turned on, theoscillations of the parity over time are similar to Rabi oscillations for a two-level system. For adrive with strength (cid:15) the equivalent Rabi frequency [12] is given by Ω = 2 (cid:15) | α ∞ | . We chose afirst drive strength (cid:15) that gave a single oscillation in parity within the decay time of a Schr¨odingercat state with amplitude ¯ n =
2. We then repeated the experiment for drive strengths that weremultiples of (cid:15) and for different amplitudes of the initial cat states. On each panel the frequency ofthe oscillations increases with the drive strength. By looking at curves that correspond to the samedrive strength over different panels (same colour) we see that the frequency of oscillation alsoincreases with the amplitude of the initial state. We obtained theory predictions by numericallyintegrating the evolution of the density matrix and superimposed them on the data. The parametersof the theory were all provided by the results of independent experiments [26].We also present in Fig 3b the frequency of the oscillations as a function of the normalized drivestrength (Fig 3b). According to theory, the oscillation frequency should depend linearly on thedrive strength. The linear fit for ¯ n = 2 gives (cid:15) / π =7 kHz . This value, when compared to ¯ nκ ,means that we respected the adiabaticity condition for this drive strength. However, when the drivestrength increases, this condition is no longer fulfilled. Subsequently, we predict the oscillationfrequencies for ¯ n = ¯ n = (cid:15)/(cid:15) increases,the oscillations decay faster. This is explained by the fact that the gate does not perfectly respectthe adiabaticity condition | (cid:15) | (cid:28) | α ∞ | κ . Nevertheless, when the number of photons in the initialstate is larger, the decay constant of the oscillations gets closer to the ideal limit: the adiabaticitycondition is easier to fulfill for higher number of photons. Although encoding with a Schr¨odingercat state of larger amplitude increases the bit-flip rate, given by nκ , it increases the quality ofour manipulation.Finally, we measured the effect of this protected Rabi rotation on an arbitrary state of theencoding manifold. We represent a state of the protected qubit by a vector in the Bloch sphere. Itscoordinates were found by measuring the equivalent Pauli operators for this encoding [29]. Theeffect of the gate is accurately described by its effect on the 6 cardinal points of an octahedronwithin the Bloch sphere. We present the results for a manifold encoding using | α ∞ | = 3 (Fig 4).The octahedron on the left shows the initial state. To illustrate the effect of the gate we chose aspecific rotation of π/ using a drive strength (cid:15) = 6 (cid:15) . This corresponded to a gate time of . µ s .We compared the results with those obtained by waiting for the same amount of time withoutapplying a drive, which corresponded to applying the identity.The next step after manipulating an encoded and protected qubit contained in the stabilizedmanifold of cat states is to address the fault-tolerance of logical operations. A future versionof our experiment, which should be accessible with current techniques, will increase κ aboveany coupling to other modes and thus achieve two goals: first, it will improve the gate qualityto make it better than a gate on a physical qubit. Second, it will suppress the dephasing dueto finite temperature in other modes and thus suppress one remaining decoherence channel. Allpossible remaining errors would then be equivalent to bit-flip errors, which can be corrected byfault-tolerant joint parity measurements [30, 31] on several cavities.We acknowledge Victor Albert and Liang Jiang for helpful discussions, Kyle Serniak and LukeBurkhart for their work on the fabrication process, R´emi Bisognin and Renan Goupil for their par-ticipation in the experiment. Facilities use was supported by the Yale SEAS clean room, YINQE,and NSF MRSEC DMR-1119826. This research was supported by the Army Research Office(ARO) under Grant No.W911NF-14-1-0011. J.B. and K.C. acknowledge partial support from theARO Grant No. W911NF-16-1-0349. P.R. acknowledges partial support from the Air Force Office6cientific Research under Grant No. FA9550-15-1-0015. S. S. acknowledges partial support fromthe ARO Grant No. W911NF-14-1-0563. C.A. acknowledges support from the NSF GraduateResearch Fellowship under Grant No. DGE-1122492. ∗ [email protected][1] B. Misra and E. C. G. Sudarshan, J. Math. Phys. , 756.[2] W. M. Itano, D. J. Heinzen, J. J. Bollinger, and D. J. Wineland, Phys. Rev. A , 2295.[3] S. R. Wilkinson, C. F. Bharucha, M. C. Fischer, K. W. Madison, P. R. Morrow, Q. Niu, B. Sundaram,and M. G. Raizen, Nature , 575.[4] P. Facchi, V. Gorini, G. Marmo, S. Pascazio, and E. C. G. Sudarshan, Phys. Lett. A , 12.[5] A. Beige, D. Braun, B. Tregenna, and P. L. Knight, Phys. Rev. Lett. , 1762 (2000).[6] A. Carollo, M. F. m. c. Santos, and V. Vedral, Phys. Rev. Lett. , 020403 (2006).[7] O. Oreshkov and J. Calsamiglia, Phys. Rev. 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D(β)
Reservoir @ f R Reservoir @ 2f S - f R W( α ) Δt Figure 2. Experimental protocol and Wigner tomography result. (a) Sequence of different drives outlinedin fig 1. The storage is either initialized in a cat state N ( | α ∞ (cid:105) ± |− α ∞ (cid:105) ) with optimal control pulses orin a coherent state |± α ∞ (cid:105) with a displacement D( ± α ∞ ). The drives stabilizing the manifold are turnedon in
24 ns (purple and yellow). They are on for a duration ∆ t during which the storage drive (cyan) canbe turned on to induce the parity oscillations. The drives are left on for another
500 ns and then turned offin
24 ns after which the Wigner function is measured. (b) In the left column is the Wigner functions ofthe storage cavity after initialization in an even or odd cat state ( | α ∞ | = 3) . The right column shows thecorresponding Wigner functions after a quarter of an oscillation. The colormaps are averaged raw data ofthe Wigner function measurement (see text) and the orange circles are cuts along Re( α )=0. The grey solidlines are theoretical curves corresponding to even or odd cats (left column, lines 1 and 2 respectively) andparityless cats (right column, lines 1 and 2). The only fit parameter in the theory is the renormalization ofthe amplitude by a factor 0.87 on the left, and 0.65 on the right. These factors account for the fidelity of theparity measurement and the decay of the fringes of the cat states during the stabilization. t (μs)
10 20 30 400.0-0.50.5 W ( ) Fit
Theory n = 2n = 3n = 5 ε / ε τ / τ Ω ( M H z ) (a) (b) ε = ε ε = 2ε ε = 4ε ε = 8ε NO DRIVE
NO QZDε = ε ε = 2ε ε = 4ε ε = 8ε W ( ) W ( ) Figure 3. Characterization of the oscillations. (a) Evolution of the measured parity as a function of time.The initial cat states are even, with ¯ n = | α ∞ | = 2 , , (circles, squares, diamonds). The storage drive iseither off (black markers) or on (coloured markers) with various strengths given in units of a chosen basestrength (cid:15) . Simulations are shown as solid lines. The minimum of each experimental curve is emphasizedfor each drive strength (full marker with black contour). (b) A fit of the data gives the frequency Ω andthe time constant τ of the decaying oscillations. The former is plotted as a function of the relative drivestrength (cid:15)/(cid:15) (top panel). The case ¯ n =2 is fitted with a linear function (dashed line). Based on this, we makepredictions for ¯ n =3, 5 (solid lines). The bottom panel shows the characteristic decay time of the oscillations τ , normalized by the decay time of the non-driven case τ X Y Ζ X Y Ζ X Y π/2
Figure 4. Gate on cardinal points of the Bloch sphere of the protected manifold. An arbitrary cat state N (cid:0) cos( θ ) (cid:12)(cid:12) C + α ∞ (cid:11) + sin( θ ) e iφ (cid:12)(cid:12) C − α ∞ (cid:11)(cid:1) is represented by a point on a sphere. Six initial states are chosen,corresponding to the cardinal points ( θ, φ ) = (0 , , ( π, , ( ± π/ , , ( π/ , ± π/ , with | α ∞ | = 3 , andtheir equivalent Pauli operators are measured. The markers corresponding to each initial state are respec-tively red and blue circles, grey up/down triangles, and black up/down triangles. The initial octahedronformed by those points (left) is either transformed under the action of the identity (upper right) or a rotationof π/ around the X axis (lower right). UPPLEMENTARY METHODSExperimental setup
The detailed setup is shown on Fig. SS1. The top of this figure (above the 300K dashedline) shows the relevant room temperature electronics while the bottom half shows the wiring ofthe dilution refrigerator. The measurement setup is on the right hand side of the figure. Twogenerators (Readout and local oscillator (LO)) were used to perform a heterodyne measurementon the Wigner transmon. We performed single shot measurements of the state of the transmonqubit using a Josephson Parametric Converter as a parametric amplifier. The two middle brancheswere used to control the two quadratures of the signal addressing the transmon qubit and thestorage (crimson and cyan). The storage branch served 3 different purposes. First, it was usedto perform fast displacements on the storage cavity (both for state preparation and to measurethe Wigner function). Second we used it for the slow drive that performs the gate. To this end,we used directional couplers to create an additional, strongly attenuated, path on the right sideof the storage branch. Finally the left side of the storage branch was part of an interferometerthat created the reservoir drive by mixing the frequency-doubled storage tone with the pump tone.This way, the reservoir drive was phase-locked to the pump tone and to the storage tone, such thatdrifts in the phase of a generator were not affecting the experiment. We also had control over thequadratures of the pump tone and the reservoir drive. Those 4 controls made it possible to sweepthe phase and the frequency of the relevant modes of the experiment and thus simplified the waywe tuned it (as shown in next sections). Inside the dilution refrigerator, on the side of the pump,we used a combination of non-dissipative elements that combined drives of very different powersand attenuated the pump without warming up the base plate. We used two directional couplersthat combined the storage and reservoir drives with the pump tone and attenuated the pump toneby sending 99% of its power back to the 4K stage of the dilution refrigerator on another line. Thissetup also provided a way to diagnose the side of the reservoir by measuring in reflection.
Hamiltonian and parameters
The Hamiltonian of our system, when it is not driven, is well described by the usual circuit QEDHamiltonian [32] containing Kerr effect χ ii / a † i a i ) and cross-Kerr χ ij ( a † i a i )( a † j a j ) . The couplingbetween the reservoir and the transmon qubit has been designed to be small. Our measurements did13ot reveal a coupling that would be bigger than the linewidth of either mode. We thus neglectedany coupling between those two modes in our model. The coupling between the storage andthe transmon qubit was large enough ( .
29 MHz ) to perform a fast parity measurement [14][27](
218 ns (cid:28) T , T ). The coupling between the reservoir and the storage was such that we reacheda large value for κ compared to κ while having a Purcell limit on the lifetime of the storage thatwas high enough (here µ s ) to see oscillations in the parity before the decoherence of cat states.In Table SI we give the coherence times of each mode. The lifetime of an electromagnetic modestored in an aluminium post cavity is usually of order [23]. We attribute our lower lifetimeto the Purcell effect from the reservoir to the storage cavity. We attribute the finite dephasingtime of the storage cavity to the finite temperature of the modes it was coupled to, particularly thereservoir (see next section). Table SII shows the measured system parameters. The only parameternot accessible via standard techniques (see supplement of [13]) was the coupling between thestorage and the reservoir. As we were not in the photon number splitting regime, we could notdeduce this quantity using a spectroscopy experiment. Instead, we used a known quantity, theanharmonicity of the reservoir, in order to measure it. When we applied the pump tone to thereservoir, the frequencies of both the reservoir and the storage underwent a Stark-shift ∆ R and ∆ S respectively. From the full system Hamiltonian we derived χ RS = 2 χ RR ∆ S / ∆ R [13]. As χ RR was known by inducing two-photon transition between | g (cid:105) R to | f (cid:105) R , we deduced χ RS (see TableSII). Phase-flips characterization
In our encoding, phase-flips correspond to leakage between the coherent states |± α ∞ (cid:105) . Theywere measured by looking at the difference of the Wigner functions at points ± α ∞ after initializingthe storage cavity in a coherent state | α ∞ (cid:105) . On Fig. SS2 we show that if the temperature ofour system had been 0K, our manifold stabilization would have protected the encoding againstphase-flips exponentially with the average number of photons. The actual experiment revealed amuch faster phase decay that did not seem to depend on the number of photons of the cat state.We reproduced this decay by introducing a finite thermal population in the reservoir mode in oursimulations (see section on simulations). We used this dependence as a way to evaluate the thermalpopulation of the reservoir mode for simulations of our experiment such as on Fig. 3 of the mainpaper. 14 uning the frequency matching condition The experiment required a precise frequency matching condition for the frequency of the pumptone: f P = 2 f S − f R . The Hamiltonian of 2 modes coupled through a Josephson junction predictsthat when a pump tone is applied, the frequency of each mode is renormalized due to a Stark-shift,which complicated the tuning phase of our experiment. We proceeded by fixing the frequency ofthe pump tone a little bit above the bare frequency matching condition (to account for the Stark-shift) and then swept its amplitude. We see on Fig. SS3 that the frequency of the storage cavitymoved (and so did the frequency of the reservoir) until the frequency matching condition was met.This condition was obtained when we observed an anti-crossing in the storage spectroscopy. OnFig. SS4 we looked at the overlap of the state of the storage with Fock state | (cid:105) while we weresweeping the amplitude of the pump tone (x-axis) and the frequency of the reservoir drive (y-axis), without sending any signal at the frequency of the storage. When the two-photon conversionhappened, the average photon number in the storage was not 0. We see that at the same amplitudeas the previous anti-crossing there was an efficient conversion of reservoir photons into pairs ofstorage photons over a wide range of frequencies for the reservoir drive. Simulations
In Fig. 3 of the main paper, we used Python and the open source library QuTip [33] to simulatethe time evolution of our system. The Hilbert space that we considered was composed of thereservoir (as a three-level system) and the storage (harmonic oscillator with a truncation at 30levels). We decided to neglect the effect of the transmon qubit for two reasons. First the experimentwas performed after checking that the transmon qubit was in its ground state. Second, repeatedmeasurements of the transmon qubit in its equilibrium state showed a thermal population of only5% while its T was µ s . The timescale corresponding to jumps towards the first excited statethus was [24] T / ¯ n th = 1 . which was enough to consider that the transmon qubit would stayin its ground state. We simulated the following Hamiltonian:15 RS (cid:126) = (cid:16) g a S a † R + g ∗ a † S a R (cid:17) + (cid:16) (cid:15) R a † R + (cid:15) ∗ R a R (cid:17) + (cid:16) (cid:15) a † S + (cid:15) ∗ a S (cid:17) − χ RR a † R a R − χ SS a † S a S − χ RS ( a † R a R )( a † S a S )+ ∆ R a † R a R + ∆ R + ∆ P a † S a S . As described in [13], this Hamiltonian is reducible to the one detailed in the main paper. Thevalue of g and of (cid:15) R were deduced from measuring the frequency Stark-shift of the reservoir for agiven pump strength and from the size ¯ n of the stabilized cat states [13]. The χ ij were all measuredfrom independent experiments and their values are given in Table SII. ∆ R and ∆ P were accountingfor possible detunings of the microwave pumps from the perfect frequency matching condition.The simulations for Fig. 3 were done at the frequency matching condition ∆ R = ∆ P = 0 . Thevalue of (cid:15) was measured from the frequency of the parity oscillations on Fig. 3. The full Lindbladequation is then dρ RS dt = − i (cid:126) [ H RS , ρ RS ] + κ D [ a S ] ρ RS + 1 + ¯ n th T R D [ a R ] ρ RS + ¯ n th T R D (cid:104) a † R (cid:105) ρ RS , where the dissipation timescales are given in Table SI and ¯ n th corresponds to the thermal popu-lation of the reservoir mode. We chose a tensor product between a cat state in the storage cavityand a thermal state in the reservoir as initial state. The thermal population of the reservoir was de-duced from the phase-flip measurement on Fig. SS2. The numerical integration of the differentialequation gave a final density matrix for the entire system. In order to compare the result to thedata, we took a partial trace over the reservoir mode and then calculated the Wigner function forzero displacement in phase space. We scaled it by the amplitude of our parity measurement for agiven cat state and superimposed the resulting curve to the data (see Fig. 3).16 toragePump Qubit Readout LO LR Ix2LR II QI QL IRL IRI QI Q
JPC Pump -30 dBEE E E E E E EE to diagnosis to diagnosis to diagnoseto diagnosis to VNAREF to ADC SIG to ADCto ADC for tuning -20 dB -20 dB -20 dB -20 dB-10 dB -10 dB -10 dB-10 dB-6 dB -1 dB-3 dB
DC block
50 Ω
DirectionalCouplerBandpass FilterLowpass EccosorbLowpass filterIsolator -20 dB-13 dB
Circulator
50 Ω
Figure S1. Experimental setup. See text of first section. = 2 (T = 0K)n = 3 (T = 0K)n = 5 (T = 0K)n = 2 (observed)simulations (finite T)n = 3 (observed)n = 5 (observed) Time (μs)
20 40 60 80 10000.51.0 ( W ( β ) - W (- β )) / ( W ( β ) + W (- β )) Figure S2. Temperature dependent phase-flip error. Normalized leakage from the initial coherent state | α ∞ (cid:105) to the state of opposite phase as a function of time. The dashed red, yellow and blue curves give thetheoretical behavior at zero temperature for ¯ n =
2, 3 and 5 respectively. The corresponding markers showthe experimentally observed values. Each of the latter curves is well reproduced by simulation (solid blacklines) for a thermal population of 1.5% for ¯ n = ¯ n = ¯ n , the theory curve for ¯ n = 3 has a lower leakagerate than ¯ n = 5. P r obe de t un i ng ( k H z ) Figure S3. Cavity Stark shift. We do a spectroscopy of the storage cavity while applying the pump withdifferent strengths. We send a µ s square pulse and measure the overlap of the resulting storage statewith the Fock state | (cid:105) . The frequency of the storage pulse is modulated around the value corresponding tothe bare frequency of the storage cavity (given by zero probe detuning). The x-axis shows the amplitudeof the pump. The zero means that no tone is sent while 1 corresponds to the maximum amplitude that ourcontrol electronics can deliver. The right panel is a zoom of the anti-crossing corresponding to the frequencymatching condition being met. R e s e r v o i r d r i v e de t un i ng ( k H z ) Figure S4. Frequency of the reservoir drive versus amplitude of the pump tone. We play the pump tone andthe reservoir drive during µ s and plot the overlap of the resulting storage state with | (cid:105) for different fre-quencies of the reservoir drive and amplitudes for the pump tone. The reservoir drive frequency is expressedby its detuning from the Stark-shifted frequency of the reservoir at the frequency matching condition. Theright panel is a zoom on the anti-crossing taking place at that particular point. UPPLEMENTARY TABLES
Table I. Frequencies and coherence times of the experimental design.Mode Frequency (
GHz ) T (µ s ) T (µ s )Qubit 5.89064 70 30Storage 8.10451 92 40Reservoir 6.6373 0.317 -Table II. Parameters of the Hamiltonian of the experimental device. χ ij / π ( MHz ) Qubit Storage ReservoirQubit 268 - -Storage 2.29 ∼0.003 -Reservoir - 0.471 86