Dissipative stabilization of squeezing beyond 3 dB in a microwave mode
R. Dassonneville, R. Assouly, T. Peronnin, A. A. Clerk, A. Bienfait, B. Huard
DDissipative stabilization of squeezing beyond 3 dB in a microwave mode
R. Dassonneville, R. Assouly, T. Peronnin, A. A. Clerk, A. Bienfait, and B. Huard Univ Lyon, ENS de Lyon, Univ Claude Bernard,CNRS, Laboratoire de Physique, F-69342 Lyon, France Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA (Dated: February 9, 2021)While a propagating state of light can be generated with arbitrary squeezing by pumping aparametric resonator, the intra-resonator state is limited to 3 dB of squeezing. Here, we implement areservoir engineering method to surpass this limit using superconducting circuits. Two-tone pumpingof a three-wave-mixing element implements an effective coupling to a squeezed bath which stabilizesa squeezed state inside the resonator. Using an ancillary superconducting qubit as a probe allows usto perform a direct Wigner tomography of the intra-resonator state. The raw measurement providesa lower bound on the squeezing at about (6 . ± .
2) dB below the zero-point level. Further, we showhow to correct for resonator evolution during the Wigner tomography and obtain a squeezing ashigh as (8 . ± .
8) dB. Moreover, this level of squeezing is achieved with a purity of ( − . ± .
4) dB.
I. INTRODUCTION
One of the most striking predictions of quantum me-chanics is that even in the ground state of an harmonicoscillator, any quadrature measurement is noisy. Zeropoint fluctuations can however be engineered and low-ered for one quadrature of the field at the expense ofthe other. These squeezed states have become a cen-tral resource for quantum information processing. Theycan be used to boost the sensitivity of many measure-ments including gravitational wave detection [1–4], per-form quantum secure communication [5, 6] and used formeasurement-based continuous-variable quantum com-puting [6, 7]. Squeezing is usually generated by para-metrically pumping a resonator. This process gener-ates squeezing of both the intra-resonator and outgoingfields. While any amount of squeezing can theoreticallybe obtained for the outgoing field, the steady-state intra-resonator squeezing is limited to 3 dB below the zeropoint fluctuations.Intra-resonator squeezing beyond 3 dB can in principlebe attained by injecting squeezed light into the resonatorinput using an external source of squeezed radiation [8–10]. In practice however, the achievable squeezing in suchschemes is limited by losses associated with transportingand injecting the extremely fragile squeezed state into theresonator. A more attractive approach is to use reservoir-engineering techniques [11], where tailored driving re-sults in the cavity being coupled to effective squeezeddissipation [12–14]. These methods can also surpass the3 dB limit, and do not involve transporting an externally-prepared squeezed state. Reservoir-engineering intracav-ity squeezing beyond 3 dB has recently been achieved formechanical modes, both in optomechanical systems [15–18] as well as in a trapped ion platform [19].In this work, we experimentally demonstrate thatreservoir-engineering squeezing beyond 3 dB can also beachieved for purely electromagnetic intracavity modes,namely a microwave-frequency mode in a superconduct-ing quantum circuit. Using the well developed circuit- QED toolbox, we also perform a direct tomography ofthe intra-resonator squeezed state instead of inferring theresonator state from the measured output mode. This isachieved through the use of an ancillary superconduct-ing qubit, which enables in-situ
Wigner tomography ofthe squeezed intracavity microwave mode. The intra-cavity squeezing factor reaches at least ( − . ± .
2) dB,going well beyond the 3 dB limit. We also probe thenon-classicality of the squeezed state by investigating itsphoton number statistics [20], and use our tomographicmethod to carefully study the full dynamics of the dissi-pative generation of squeezing. This work thus presentsan interesting platform to stabilize, manipulate and char-acterize Gaussian states in-situ. Our stabilization tech-nique could also be extended beyond simple squeezedstates to other continuous variable states such as cat orgrid states [21–26] by taking advantage of the large non-linearities that can be engineered in circuit-QED.
II. SYSTEM AND MODEL
Our device consists in a Josephson Ring Modulator(JRM) [30] coupling one mode (the cavity ) which wewould like to stabilize in a squeezed state, and a secondauxiliary mode strongly coupled to a transmission line(the dump ). The cavity and dump have resonant frequen-cies ω c / π = 3 .
741 55 GHz and ω d / π = 11 .
382 GHz anddecay rates κ c / π = 40 kHz and κ d / π = 8 MHz. Oursetup also has an ancillary transmon qubit coupled to thecavity; its only role is to perform intra-resonator Wignertomography (Fig. 1.a).When applying a pump at frequency ω − = ω d − ω c , andwithin the rotating-wave approximation (RWA) and stiffpump condition, the JRM leads to a beam-splitter inter-action Hamiltonian ˆ H − / (cid:126) = g − ˆ d † ˆ c + g ∗− ˆ d ˆ c † , where thepump amplitude controls the coupling strength g − be-tween the cavity and dump modes described by bosonicoperators ˆ c and ˆ d . It mediates coherent exchange of pho-tons between the cavity and the dump and thus lossless a r X i v : . [ qu a n t - ph ] F e b pumpsqubitreadoutcavity Wigner tomography a)b)c) cavity dumpWigner tomograph FIG. 1. a) Principle of the experiment. A cavity mode (green)at frequency ω c is coupled to a dump mode at frequency ω d (orange) via a Josephson Ring Modulator (JRM, in purple).The dump mode is strongly coupled to a cold transmissionline through which the JRM is pumped at both frequencies ω + = ω c + ω d (two-mode squeezing) and ω − = ω d − ω c (photonconversion). A squeezed vacuum state is stabilized into thecavity as a result. An ancillary qubit with an ancillary read-out resonator is used as a Wigner tomograph. The contoursof the Wigner functions of each mode are shown as coloredregions in the quadrature phase space, while a dashed circlerepresents the vacuum state. b) Frequencies of the involvedmodes and drives. c) Pulse sequence. The sum pump at ω + with amplitude g + and the difference pump at ω − with am-plitude g − are applied for a time t s . After a waiting time t w ,the Wigner function of the cavity W ( α ) is measured usinga cavity displacement by − α followed by a parity measure-ment [27–29]. frequency conversion [31, 32]. In contrast, a pump ap-plied at frequency ω + = ω d + ω c mediates a paramet-ric down conversion process involving cavity and dump,ˆ H + / (cid:126) = g + ˆ d † ˆ c † + g ∗ + ˆ d ˆ c . The pump amplitude controlsthe coupling strength g + . On its own, this kind of pump-ing leads to phase-preserving amplification [30, 33] andgeneration of two-mode squeezed states [34]. Note thatin order to avoid parasitic nonlinear effects, we operatethe JRM at a flux point which maximizes these three-wave mixing terms while cancelling the four-wave mixingterms [35, 36].Simultaneously pumping at these two frequencies en-ables various interesting phenomena such as effectiveultrastrong coupling [37, 38] or directional amplifica-tion [39–41]. Here, using a long-lived cavity mode, weshow that this double pumping scheme can stabilize a squeezed state [12, 13]. Indeed, in the rotating frame,and setting the phase references such that g ± are posi-tive, the total Hamiltonian readsˆ H/ (cid:126) = ˆ d ( g + ˆ c + g − ˆ c † ) + h.c. (1)In the case where g + < g − , this Hamiltonian can bereinterpreted as a beam splitter interaction between thedump mode and a Bogoliubov mode ˆ β = cosh( r )ˆ c +sinh( r )ˆ c † with r = tanh − ( g + /g − ). It readsˆ H/ (cid:126) = G ˆ d ˆ β † + h.c., (2)where the coupling strength is G = (cid:113) g − − g . In theideal case where the coupling rate κ d of the dump modeto a reservoir at zero temperature is much larger thanany other rates, and where the cavity lifetime κ − c is un-limited, the Hamiltonian leads to the relaxation of theBogoliubov mode into its ground state. In that state,the cavity mode is a vacuum squeezed state with squeez-ing parameter r = tanh − ( g + /g − ).The signature of this squeezing is best seen in thequadrature phase space of the cavity mode. We de-note X − and X + the quadratures of the cavity modethat have the smallest and largest variances in a givenstate. In the vacuum state of the cavity ( r = 0),the variance of the quadratures corresponds to the zeropoint fluctuations (cid:10) X ± (cid:11) | (cid:105) = X . The squeezing factorone can generate in the ground state of the Bogoliubovmode is simply a scaling of the variances by the factor S ± = (cid:10) X ± (cid:11) /X = e ± r . In the general case, where theBogoliubov mode is not cooled down to its ground state,these factors become [13] S ± = e ± r (cid:10) ( β ∓ β † ) (cid:11) . (3)We thus see that in principle, the 3 dB squeezing limitcan be surpassed arbitrarily by having g + approach g − from below (as this causes the squeezing parameter r to diverge). However, in this limit the effective cou-pling rate G of the Bogoliubov mode to the dump goesdown to zero. As a result, the competition between thisengineered decay channel and the intrinsic cavity loss(rate κ c ) prevents the Bogoliubov mode from reaching itsground state. This both degrades the effective squeezingof the steady state, as well as its purity. Thus, for anyvalue of g − there exists an optimum value of g + that min-imizes the variance (cid:10) X − (cid:11) . This minimum increases withthe value of g − and is finally expected to saturate to alevel set by the damping rates S − ≥ κ c / ( κ c + κ d ), whichreflects the fact that the damping rate of the dump κ d sets an upper limit to the coupling of the bosonic modeto the effective squeezed reservoir.We thus see that a prerequisite for achieving squeezingwell beyond 3 dB is to engineer a large ratio κ d /κ c . Forour sample parameters, we have κ d (cid:39) κ c , leading toa lower bound of S − ≥ −
23 dB [13]. Further, taking a) b)c)
FIG. 2. Characterization of the stabilized squeezed state. a) Top panels: measured steady-state squeezing S − = (cid:10) X − (cid:11) /X (left)and anti-squeezing S + = (cid:10) X (cid:11) /X (right) factors. Bottom panels: theoretical prediction for S ± using Eq. (4). Green dashedlines correspond to the value g opt − as a function of g + that minimizes the squeezing S − according to Eq. (4) (i.e. calculatedby neglecting Kerr nonlinearities). b) Purity P (top, green), squeezing S − (bottom, blue) and anti-squeezing S + (bottom,orange) factors as a function of g + /g − for a fixed value g − / π = 1 .
85 MHz (cut along the arrow in Fig. a). Circles are thenormalized eigenvalues of the covariance matrix of the measured Wigner functions at each pump amplitudes as shown in (c)and reach a squeezing factor as low as S − = ( − . ± .
2) dB. Points with error bars are the values obtained when correctingfor cavity evolution during Wigner tomography (see Appendix D 1), which reveals a stabilized squeezing reaching as low as S − = ( − . ± .
8) dB. Solid lines come from the model Eq. (4). c) Selected measured Wigner functions along the same axis g − / π = 1 .
85 MHz, for various g + /g − ratios as indicated in the labels. The star indicates the Wigner function at optimumsqueezing. into account the thermal equilibrium occupancies n thc and n thd of the cavity and dump modes, and in the limit ofthe experiment where κ c , G (cid:28) κ d , Eq. (3) leads to (seeAppendix C or [18] for formula without approximation) S ± (cid:39) κ c (2 n thc + 1) + Γ ± (2 n thd + 1) κ eff (4) where we introduce κ eff = κ c + 4 G /κ d and Γ ± = 4( g − ± g + ) /κ d . Eq. (4) also makes it clear that the intrinsic lossrate κ c and non-zero environmental temperatures alsolower the purity of the steady state P = Tr (cid:0) ρ (cid:1) below 1,where ρ is the steady state cavity density matrix. Thisfollows from the fact that P = 1 / (cid:112) S − S + for a Gaussianstate.To measure the squeezing and anti-squeezing factors S ± , we perform a full in-situ Wigner tomography [27–29]using an ancillary transmon qubit at frequency ω q / π =4 .
327 31 GHz (Fig. 1.a). It couples dispersively to thecavity with a dispersive shift χ/ π = − .
28 MHz. Athird resonator, at frequency ω r / π = 6 .
293 GHz, is usedto perform single-shot readout of the qubit state witha fidelity of 96 % in a 380 ns integration time. Fromthe Wigner function, we compute the covariance ma-trix of the cavity mode quadratures and diagonalize itto extract the minimum and maximum cavity quadra-ture variances (cid:104) X ± (cid:105) . Due to its coupling to the qubit,the cavity acquires an induced parasitic self-Kerr non-linearity − K ˆ c † ˆ c and a qubit-state-dependent self-Kerr − K e ˆ c † ˆ c | e (cid:105)(cid:104) e | with K/ π = 20 kHz and K e / π =70 kHz (measured in a previous run of the experiment).These non-linearities distort the squeezed state and thusreduce the effective squeezing factor, similarly to whatoccurs for Josephson parametric amplifiers (JPA) [42].While no analytical solution taking into account the Kerreffects exists, Eq. (3) and Eq. (4) still provide a gooddescription when G (cid:29) K . In the future, these non-linearities could be harnessed as a resource to stabilizemore complex non-Gaussian states [21, 22, 43, 44]. III. STEADY-STATE SQUEEZING
The key advantage of reservoir engineering is that thedesired target state is prepared in the steady state, inde-pendent of the initial cavity state: one can simply turnon the pumps and wait. We thus turn on g + and g − for a duration t s = 4 µ s (cf Fig. 1.c) that is long enoughto establish a steady state, and immediately afterwardsmeasure the Wigner function W ( α ). To perform the mea-surement at each amplitude α , we start by applying a cal-ibrated displacement D ( − α ) to the cavity state using acavity drive at ω c with a pulse shape chosen to be a 13 nswide hyperbolic secant and whose complex amplitude isproportional to − α . We then measure the cavity parityoperator by reading out the qubit state after performingtwo π/ π/χ = 152 ns. We perform phase-cycling,running each sequence twice with an opposite phase forthe second π/ ×
25 pixels approxi-mately aligned to the squeezing axis. Due to the finitewindow size (cf Appendix E), we could only resolve anti-squeezing up to 11 dB (dotted dash line in Fig. 2.b). EachWigner tomogram is averaged over 5000 realizations. Toincrease the repetition rate and limit the low-frequencydrifts, the cavity is first emptied by applying a differencepump g − , cooling it down to a thermal vacuum statewith residual population n thc = n thd = 0 . ± .
003 (seeAppendix B 3). The qubit is also reset to its groundstate using measurement-based feedback. Furthermore, to minimize the low-frequency noise as much as possi-ble, we interleave pump-on-measurements with pump-off-measurement. We thus obtain experimental squeezingfactors S ± = (cid:0)(cid:10) X ± (cid:11) / (cid:10) X (cid:11) off (cid:1) · (cid:0)(cid:10) X (cid:11) off /X (cid:1) by firstnormalizing the measured variances (cid:10) X ± (cid:11) with the mea-sured pumps-off variances (cid:10) X (cid:11) off and then correcting forthe thermal occupancy (cid:10) X (cid:11) off /X = (0 . ± .
03) dB.The pumping strengths g + and g − are calibrated usingindependent measurements (Appendix B 1). We estimatea statistical uncertainty of ± . g + and g − . We observe a maximum squeezingof ( − . ± .
2) dB well below the − . ± .
2) dB and thus a statepurity of ( − . ± .
2) dB. For each value of the rate g + ,the largest squeezing we observe occurs for g − close to g + . This trend is expected from the analytical Kerr-freemodel Eq. (4) of the system, which predicts the work-ing point of largest squeezing as a function of g + (greendashed line in Fig. 2.a) [13]. However, contrary to the ex-pected monotonic increase of the optimal squeezing fac-tor S − with g + (Kerr-free prediction in bottom panels ofFig. 2.a), we find a global maximum squeezing at a finitevalue of ( g − , g + ). Note that for g + > g − the system be-comes unstable: the qubit gets ionized [46], preventing usfrom measuring the Wigner functions (grey shade area).In Fig. 2.b-c), we show the squeezing and anti-squeezing as a function of g + /g − as well as some mea-sured Wigner tomograms, for g − / π = 1 .
85 MHz. For g + < . g − , the measured variances are well captured byEq. (4) with exponentially increasing squeezing factors.For g + > . g − , the measured variances start deviatingfrom the theory (solid lines in Fig. 2.b). As can be seen inFig. 2.c), the squeezed states are not Gaussian anymorein this parameter region. The Wigner functions developan S-shape, a typical signature of the cavity self-Kerr.We attribute this effect to higher order terms we have sofar neglected: the self-Kerr rate induced by the qubit onthe cavity, as well as a residual four-wave mixing term inthe JRM Hamiltonian (see Appendix D 2).While the raw measurement of the Wigner functionprovides a good estimate of the steady-state squeezingparameter (circles in Fig. 2.b), the finite measurementtime needed to perform tomography leads to a systematicerror. During this finite measurement time, the pumptones g ± are off, implying that the cavity is no longercoupled to an effective squeezed reservoir. The squeezedstate thus degrades due to the intrinsic cavity loss. Afurther error is caused by evolution under the cavity-selfKerr nonlinearity during this time. Both these effectscause our Wigner function methods to underestimate thetrue value of the steady-state squeezing.It is possible to correct for this measurement error andretro-predict via numerical simulation the squeezing fac-tors S ± associated with the state prepared at the endof the stabilization period [47]. To that end, we con-sider a series of input model Gaussian states for whichwe numerically implement our experimental Wigner to-mography measurement. At the end of these simulations,we obtain a mapping from Gaussian states to measuredsqueezing and anti-squeezing factors that we are able toinvert in order to retro-predict the stabilized state (Ap-pendix D 1). Using this correction improves the bestsqueezing estimate to ( − . ± .
8) dB (dots with errorbars in Fig. 2.b) with purity ( − . ± .
4) dB.It is interesting to compare our stabilization techniqueto other intra-resonator microwave squeezing generationschemes. One possibility consists in driving a cavity witha squeezed input state that is externally generated bya Josephson parametric amplifier (JPA) [10, 48]. Highsqueezing factors [42, 48–50] ( (cid:39) −
10 dB) can be achievedin the amplifier output field. However, transferring thisstate into a cavity is challenging as it is extremely sen-sitive to microwave losses, resulting in degraded squeez-ing and purity. For comparison, we consider a resonatordriven by a pure squeezing source (in practice a JPA). Toachieve the same intracavity squeezing and purity as oursetup ( S − = − . P = − . ∼− . .
15 dB. This levelof loss is smaller than the typical insertion loss of com-mon microwave components. It is hard to achieve, evenif all elements are fabricated in a single-chip architecture.We can also compare against another approach for gen-erating (but not stabilizing) a squeezed state, based onthe use of arbitrary state preparation techniques (e.g. theSNAP gate protocol [51]). In Ref. [52], the authors usedsuch an approach to obtain a squeezing factor of − .
71 dBfor a purity of − .
86 dB with a non-deterministic successrate of 15 %.
IV. NONCLASSICAL PHOTON DISTRIBUTION
One of the hallmarks of vacuum squeezed states is thatthey are quantum superpositions involving only even-number photon Fock states. Such ideal states have theform | ψ (cid:105) = (cid:80) ∞ k =0 tanh( r ) k √ (2 k )!2 k k ! | k (cid:105) . Our device allowsus to directly verify this unique, non-classical aspect ofthe squeezed states we stabilize in our cavity [20]. This isbecause our coupling to the ancilla qubit is strong enoughto place us in photon-number-resolved regime where dis-tinct cavity photon numbers can be resolved by mea-suring the effective qubit frequency, i.e. χ (cid:29) Γ withΓ = (11 µ s) − the qubit coherence rate.After preparing the squeezed state, we perform spec-troscopy of the qubit using a narrow-bandwidth π -pulseat a varying probe frequency ω followed by qubit readout(green curve in Fig. 3.a). The observed peak heights ateach frequency ω − ω q ≈ nχ allow us to determine thecavity photon-number distribution P ( n ) [53]. We correct a) b) FIG. 3. a) Number photon distribution measurement usingqubit spectroscopy. Solid lines: measured probability P | e (cid:105) that the qubit gets excited by a 200 ns wide hyperbolic secant π -pulse of frequency ω after a cavity state is stabilized. Blue:near vacuum state when no pumps are applied. Orange: ther-mal state when only a pump g + is applied. Green: squeezedvacuum state when both pumps g + and g − are applied. Ver-tical lines indicate the qubit resonance frequency conditionedon the cavity having n photons. Filled circles: numerical sim-ulations. b) Dots with error bars: Klyshko number K n (seemain text) calculated from the qubit spectroscopy. Orangebars: expected Klyshko number for a thermal state at anytemperature. Green bars: Klyshko numbers predicted withthe model described in the text. this dataset for the qubit residual thermal population(1 %), the finite fidelity of the π -pulse and readout er-rors. Interestingly, the peaks are not evenly spaced in fre-quency due to the higher nonlinear term − K e ˆ c † ˆ c | e (cid:105)(cid:104) e | with K e / π = 70 kHz. The photon number distribution P ( n ) are then obtained from the qubit excitation proba-bility at ω q − n ( χ + 2 K e n ) (vertical lines). For compar-ison, we also measure the photon number distribution P ( n ) for two other cavity states: a thermal equilibriumstate when no pumps are applied (blue in Fig. 3) anda thermal state that we create by only applying a sumpump g + / π = 0 .
43 MHz (orange curve). Note that thisthermal state is obtained by tracing out the dump modefor the vacuum two mode squeezed state that is stabilizedbetween cavity and dump [54].For the squeezed state (green curve with g − / π =2 . g + / π = 1 .
42 MHz), we observe a non-monotonic behavior: the weight of even photon num-bers is enhanced, whereas that of odd photon numbers issuppressed (note the log scale here). The non-zero butsmall population of odd Fock states indicates a deviationfrom an ideal squeezed vacuum state. The measured dataclosely fits to our numerical simulation (dots in Fig. 3.a).The measurement done with the pumps off gives the ther-mal population of the cavity n thc = 0 .
017 (blue dots) andalso indicates the measurement noise floor. For the ther-mal state, we observe a Bose-Einstein distribution witha population n thc = 1 . P repre-sentations [55]. Equivalently, it also manifests itselfin the behaviour of so-called Klyshko numbers K n =( n + 1) P ( n − P ( n + 1) /n P ( n ) . A state is non-classicalif for one or more integers n , K n < P function cannot be well behaved). For example, aperfect squeezed vacuum state, as it only includes evenphoton numbers, exhibits infinite odd Klyshko numbersand zero even Klyshko numbers. Ref. [20] computed theKlyshko numbers for a squeezed state generated by an ex-ternal JPA and observed a Klyshko number smaller than1, even though they only observed monotonic behaviorin the photon distribution P ( n ).For a thermal state, the Klyshko number are givenby K th n = ( n + 1) /n independently of temperature (or-ange bars in Fig. 3.b). We observe this universal relationwith the prepared thermal state (orange points with er-rorbar). Interestingly, it is a striking demonstration ofthe fact that a two mode squeezed state generates a ther-mal distribution when tracing out one of the modes. It isexpected from the maximally entangled state at a givenaverage energy. We do not show the Klyshko numberswhen the pumps are off because P ( n ) is below the noisefloor.For the squeezed state, we observe ample oscillationsin the Klyshko numbers (notice the log scale again). Wemeasure K = 0 .
23 and K = 0 . P ( n ), our numerical model (green bars)reproduces the observed Klyshko numbers. V. STABILIZATION DYNAMICS AND DECAYOF SQUEEZING
Our measurements establish that, as expected, thereservoir engineering scheme we implement is able tostabilize a squeezed state in the cavity. In addition to a)b)
FIG. 4. Dynamics of the squeezing factors. a) Measuredsqueezing (dots) and anti-squeezing (diamonds) factors nor-malized by their steady state values S ss ± as a function of thestabilization time t s for g + / π = 1 .
16 MHz and various g − .The data are shifted by 1 for each value of g − ranging in g − / π = [1.48, 1.85, 2.22, 2.59, 2.96, 3.33] MHz (from lightgreen to dark blue). Solid lines: results of the numericalsimulation. Rectangles indicate the predicted characteristicstabilization times κ d / G assuming 2 % relative uncertaintyon g + and g − . b) Selected measured Wigner tomograms for g − / π = 1 .
85 MHz and g + / π = 1 .
16 MHz after various sta-bilization times t s . characterizing the steady state, it is also interesting toask how long the scheme takes to prepare the steadystate. For the ideal (Kerr-free) system, and in the limitof a large dump-mode damping, one can use adiabaticelimination to show that this preparation timescale is κ d / G [13].We can directly test this prediction in our experiment.The measured squeezing and anti-squeezing factors areshown in Fig. 4.a) as a function of the time t s duringwhich the pumps are turned on for g + / π = 1 .
16 MHzand for various values of g − . By normalizing the squeez-ing and anti-squeezing factors S ± by their steady-statevalues S ss ± , we observe, as expected, that the steady-stateis reached in a typical time of κ d / G that decreases with g − (rectangles in Fig. 4.a). As a consequence, the stabi-lization time increases with squeezing when consideringa fixed g + value, as long as G dominates both the cav-ity loss rate κ c and its self-Kerr rate K . This is well- a)b) FIG. 5. Decay of the squeezed state towards thermal equi-librium. a) Dots: measured squeezing factor S − (blue) andanti-squeezing factor S + (orange) as a function of the wait-ing time t w during which the pumps are turned off after theywere at g + / π = 1 .
42 MHz and g − / π = 2 . K/κ c = 0 .
5. Dashed lines:same simulations but without self-Kerr ( K = 0). b) MeasuredWigner functions after various pump off times t w from 20 nsto 15 . µ s as indicated on each label. understood from the cooling dynamics of the Bogoliubovmode: larger squeezing parameters r = tanh − ( g + /g − )are obtained for smaller values of G = (cid:113) g − − g butthey lead to a longer relaxation time. The evolution ofthe squeezing factors is reproduced using numerical sim-ulation of the master equation (solid lines in Fig. 4.a).It is also interesting to examine experimentally thetime-evolution of the full cavity Wigner functions. InFig. 4.b), the evolution at g + / π = 1 .
16 MHz and g − / π = 1 .
85 MHz (global minimum of the squeezingfactor) shows how the squeezing establishes with some ro-tation and distortion of the Gaussian distribution due toKerr effect as the average number of photons gets larger.The steady-state is thus reached in about κ d / G buthow fast does it disappear once the pumps are turnedoff? Operating at g + / π = 1 .
42 MHz and g − / π =2 . t w (Fig. 5.a). A fast decrease of the squeezing factor is ob-served in a characteristic time shorter than the cavityrelaxation time κ − . We attribute this deviation fromthe behavior expected of a perfectly harmonic oscillator(dashed lines) to the self-Kerr effect induced by the trans- mon qubit onto the cavity. The corresponding predictedevolution of squeezing factors is shown with K/κ c = 0 . S − as K/κ c increasesbeyond about 1 (Appendix D 3). Since K (cid:39) κ c in theexperiment, we are close to a critical damping regime. VI. CONCLUSION
Using dissipation engineering, we have shown thestabilization of a squeezed state in a microwave res-onator with a squeezing factor greatly exceeding the stan-dard 3 dB limit for coherent in-situ parametric pump-ing. We directly measure the squeezing factor by per-forming a direct Wigner tomography using an ancillaryqubit. Correcting for state evolution during measure-ment, we infer that we achieve a squeezing factor of( − . ± .
8) dB. While reservoir-engineered squeezingof mechanical modes has previously been demonstrated,this is the first demonstration of this method (to ourknowledge) in an electromagnetic system. The reser-voir engineering technique used here thus extends thestate-of-the-art for intra-resonator microwave squeezing.Moreover, the produced squeezed state is close to a purestate with purity of ( − . ± .
4) dB. A displaced vacuumsqueezed state could also be stabilized in our system byadding a coherent drive on the dump.Beyond the stabilization of Gaussian squeezed states,the techniques presented here could be useful for the sta-bilization of far more complex states. As discussed, Kerrnonlinearities already play an appreciable role in our ex-periment. Future work could use this nonlinearity di-rectly as a resource for non-Gaussian state preparation.Recent work has demonstrated that the combination ofsqueezing-via-parametric driving with Kerr interactionscan be used to generate cat states [21, 43] and even en-tangled cat states [44]. The combination of dissipativesqueezing (as realized here) with Kerr interactions couldsimilarly yield complex cat-like states. Our techniquescould also be used to generate squeezed Fock states [56],squeezed Schr¨odinger’s cat states [57] or for the prepara-tion of grid states without the need for measurement [23–26]. These engineered squeezed states could find manyapplications. Indeed, used to erase which-path informa-tion, they can increase gate fidelity [58]; used to increasedistinguishability, they can improve qubit state read-out [9, 14, 59, 60]. Squeezing can also be used in spindetection to enhance the light-matter coupling [61, 62].Finally, dissipative squeezing techniques employed ona single site of a lattice of microwave resonators (seee.g. Ref. [63]) can serve as a shortcut for effectively gen-erating highly-entangled many-body states [64, 65].
ACKNOWLEDGMENTS
We are grateful to Olivier Arcizet and Alexandre Blaisfor discussions. This work was initiated during a discus-sion that happened during Les Houches Summer Schoolin July 2019. We acknowledge IARPA and Lincoln Labsfor providing a Josephson Traveling-Wave ParametricAmplifier. The device was fabricated in the cleanroomsof Coll`ege de France, ENS Paris, CEA Saclay, and Ob-servatoire de Paris. This work is part of a project thathas received funding from the European Union’s Horizon2020 research and innovation program under grant agree-ment No 820505. AC acknowledges support from the AirForce Office of Scientific Research MURI program, underGrant No. FA9550-19-1-0399.
Appendix A: Steady-state Wigners tomograms
The Wigners tomograms of all the points of Fig. 2 areavailable on [66].
Appendix B: Sample and setup
The sample is the same as in Ref.[36] albeit for a dif-ferent cool-down. The measurement setup is also simi-lar with the addition of the pump at the sum frequency(Fig. 6). The two local oscillators for the pumps aregenerated by mixing the output of the two microwavesources that are used to generate the dump and cavitydrives. Intermediate frequency (IF) signals – tens of MHz– generated by the Quantum Machines’ OPX hardwareare upconverted by these local oscillators. Finally, wecombine and amplify the two pumps before combiningthem to the dump port inside of the dilution refrigera-tor.To successfully stabilize and measure a squeezed stateon a well-defined squeezing axis ( g ± real), a good phasecoherence is required between the pumps and cavitydrives. Our setup ensures this condition by deriving thepumps from the dump and cavity local oscillators. Onedifficulty of our experiment is the large power requiredfor the pumps to reach maximal squeezing factor. Thisrequires the use of a room-temperature amplifier after themixers (Fig. 6). This amplifier has a slow temperature-induced drift in gain, leading to a relative error of 2 % onthe pump amplitudes (corresponding to the horizontalerrorbars in Fig. 2.b).
1. Calibration of the pumps
This section shows how to relate the IF amplitudes A − and A + to the rates g − and g + .To calibrate g − , we measure the mean photon numberin the cavity after applying the pump when the cavity is initially populated with a coherent state α = √
6. De-pending on the amplitude A − and duration 4 σ of thepump pulse, the rate at which the cavity coherently ex-changes excitations with the dump varies (Fig. 7). Dueto the large dissipation rate of the dump, the oscillationsof the cavity mean photon number (cid:104) n (cid:105) are damped. Byfitting the oscillations using a master equation, we find,as expected, a linear dependence of g − as a function of A − that we use as calibration.To calibrate g + , we measure the mean photon numberin the cavity n thc after applying a square pulse with am-plitude A + for 100 ns when the cavity is initially in vac-uum. The mean photon number is measured via cavity-induced Ramsey oscillations [36]. The only differencewith the former reference is that the distribution of pho-ton numbers is thermal instead of Poissonian. Hence, thephase acquired by the qubit during the waiting time ofthe Ramsey sequence differs and leads to a final qubitexcitation probability of P e ( t ) = n thc (1 − cos χt ) + 12(1 − cos χt )( n thc + 1) n thc + 1 e − Γ t . Using a time dependent master equation, the mean pho-ton number is converted into a two-mode squeezing rate g + . The curve g + as function of A + is non linear (Fig. 8),likely due to higher order non-linearities in the Hamilto-nian. The calibration g + ( A + ) is then obtained by inter-polating the measurement.
2. Cavity displacement calibration
The calibration of the displacement of the cavity un-der a pulsed coherent drive is performed by counting themean photon number. The method chosen to count themean photon number is to use the ancillary qubit andreadout as a vacuum detector [36]. This method alsoallows us to extract the cavity decay rate.
3. Cavity thermal population
The cavity thermal population is extracted fromthe cavity-induced Ramsey oscillations of the ancillaryqubit [36]. With the reset protocol, consisting of a swappulse ( g − ) between cavity and dump modes followed bya measurement-based feedback initialisation of the qubitin its ground state, we measured a mean photon num-ber n thc = 1 . ± . · − corresponding to an effectivetemperature of (44 ±
2) mK for the cavity.
4. Correction and uncertainty on the quadraturevariances
We wish to extract the squeezing and anti-squeezingfactors by normalizing the measured variances to the
Ecco4 K300 K EccoTWPA Cryoperm shielddump cavity qubitreadout-30 -20 -20-20-20-20-20 -10 -10 -10Band-pass filterEcco Low-pass filterHigh-pass filterRF source -X X dB attenuatorIsolatorMixerSingle sidebandmixerIQ-mixer AmplifierChannel of OPX DAC Eccorsorb filter Splitter or combinerDirectional couplerDiplexer Channel of OPX ADC SSB HEMT0.1 K17 mK0.8 K >7GHz<7GHz<7GHz >7GHz
SSB <10GHz>10GHz
FIG. 6. Schematic of the measurement setup. The rf sources color refers to the frequency of the matching element in the deviceup to a modulation frequency. Multiple instances of a microwave source with the same color represent a single instrument withsplit outputs. The sum (blue) and difference (red) pumps are obtained by mixing the cavity and dump rf sources to ensurephase stability. The TWPA [67] was provided by Lincoln Labs. FIG. 7. Calibration of the rate g − . Dot: measured meanphoton number in the cavity as a function of pump pulsewidth σ with a hyperbolic secant shape for three amplitudes A − . Solid lines: prediction of the photon number using amaster equation using g − / A − = 74 MHz / V.FIG. 8. Calibration of the rate g + . Dots: measured meanphoton number n thc as function of the amplitude A + . Numer-ical simulations allow us to extract the rate g + (top axis) thatleads to a given n thc (see text). Solid line: third order polyno-mial fit of g + as function of A + that is used as an empiricalcalibration. zero-point fluctuations. However, the residual thermalpopulation offsets the measured value of the zero-point-fluctuations by a factor 2 n th + 1. This also means thatall the measured squeezing factors have to be offset by(0 . ± .
03) dB. Due to other sources of uncertainty,such as fluctuations on the cavity displacement pulses, wemeasure a higher statistical uncertainty for the pump-offvariances of ± . Appendix C: Kerr-free analytical model
This derivation, which can be found in Ref. [18], isgiven here for completeness. When continuously pump-ing at the difference and sum of the resonance frequencieswith rates g − and g + , the Langevin equations read˙ˆ d = − κ d d + i ( g − ˆ c + g + ˆ c † ) + √ κ d ˆ d in ˙ˆ c = − κ c c + i ( g − ˆ d + g + ˆ d † ) + √ κ c ˆ c in , (C1)where the cavity (dump) input field operators ˆ c in ( ˆ d in )verify [ˆ b in ( t ) , ˆ b in ( t (cid:48) )] = δ ( t − t (cid:48) ) and (cid:68) ˆ b † in ( t )ˆ b in ( t (cid:48) ) (cid:69) = n thb δ ( t − t (cid:48) ) for b = c, d . Solving the Langevin equationsfor the steady-state, the squeezing S − and anti-squeezing S + factors are given by S ± = 4( g − ∓ g + ) κ d (2 n thd + 1)( κ d + κ c )(4 G + κ d κ c )+ [4 G + κ d ( κ d + κ c )] κ c (2 n thc + 1)( κ d + κ c )(4 G + κ d κ c ) . (C2)Assuming G , κ c (cid:28) κ d , Eq. (C2) gives the simplifiedEq. (4) given in the main text. Appendix D: Modeling the Kerr effect
The Kerr effect is not included in the analytical modeldescribed in Appendix C. It induces spurious effects,which reduce the maximal squeezing factor and accel-erate the relaxation of squeezing. In this section, weshow how to take these effects into account. We simulateour system using the QuantumOptics.jl library [68]. Thesteady-state simulations are run on an Nvidia Geforce1080Ti GPU, which allows us to reach Hilbert space di-mensions of about 1800. All of the other simulationsare run on the CPU. Except for the Wigner tomographyretro-prediction, the qubit is not simulated but we takeinto account the Kerr effect it induces on the cavity. Inthe case of the Wigner tomography retro-prediction, thedump is adiabatically eliminated.
1. Retro-prediction of the Wigner tomography
In order to correct for the error introduced by the cav-ity evolution during Wigner tomography, we resort tosimulations of the cavity and qubit alone. Indeed, in theabsence of pumps, the effect of the JRM on the cavityis negligible. We numerically implement our experimen-tal Wigner tomography pulse sequence on a truncatedHilbert with up to 50 excitations for the cavity and thetwo qubit states. Starting from a range of initial squeezedstates for the cavity, with variances ( S i − , S i+ ), we simulate1 c)d) f)e)a)b) FIG. 9. a-b) Representation of the map f − W . Pre-measurement squeezing S i − and anti-squeezing S i+ as a function of themeasured squeezing and anti-squeezing factors S W ± . Circles: simulated values. Colors: linear interpolation of f − W . c-d) Versionof Fig. 2.a corrected for the measurement error during Wigner tomography. e-f) Color: retro-predicted uncertainty on thesqueezing factors ∆ S i − and ∆ S i+ owing to a measurement uncertainty of ± . S W ± . the outcome of the faulty Wigner tomography by com-puting the variances ( S W − , S W+ ) of the simulated Wignertomograms.This data-set provides a function f W that maps actualvariances ( S i − , S i+ ) of the pre-measured quantum stateto the variances ( S W − , S W+ ) extracted from the measuredWigner tomograms. As this function empirically appearsbijective, the retro-prediction is performed by interpolat-ing its inverse f − W . The interpolated f − W for the initialsqueezing and anti-squeezing, as well as the simulatedpoints, are shown in Fig. 9.a and b respectively. Theretro-predicted initial squeezing and anti-squeezing cor-responding to Fig. 2.a-b are shown in Fig. 9.c-d respec-tively.Assuming the ± . S W ± , and retro-predicting the evolution during Wigner to-mography, we obtain an uncertainty ∆ S ± on the retro-predicted squeezing and anti-squeezing factors that de-pends on the value of the measured squeezing and anti-squeezing (Fig. 9.e-f).
2. Steady-state simulations
As seen in Fig. 2.b, the analytical Kerr-free modelfails to quantitatively describe the squeezing factor at g + > . g − (Fig. 10.a). Here, we compute how higher or- der terms in the Hamiltonian may explain this difference.The first term we consider is the Kerr effect − Kc † c induced by the qubit on the cavity (Fig. 10.b). Thissimulation accurately predicts the optimal g + but stillfails to reproduce the measured squeezing factors above g + /g − = 0 .
7. Experimentally, we aim for a JRM fluxbias that maximizes the three-wave mixing term whilecancelling the four-wave mixing term. However, smalldeviations from this sweet spot create four-wave mix-ing terms between the cavity, dump and pumps. Con-trary to the retroprediction simulations which modela situation where the pumps are turned off, these ex-tra terms may have a significant impact on the squeez-ing factor where the pumps are turned on. In theRWA, the four wave-mixing term leads to three kindsof interactions, a cross-Kerr between cavity and dump K cd c † cd † d , an AC-Stark frequency shift due to the pumps2( | p − | + | p + | )( K pc c † c + K pd d † d ) and parametric squeez-ing drive due to pump inter-modulation K pc p + p ∗− c † + K pd p ∗ + p ∗− d † + h.c. . The JRM also induces a self-Kerrinteraction for the dump, but we neglected it as it isone order of magnitude smaller than K pd p ∗ + p ∗− in ourcase [69] and much smaller than the dissipation rate κ d anyway. The rates K cd , K pc and K pd are not mea-sured in this run. Realistic values K cd / π = 250 kHz, K pc | p − | / π = 172 kHz and K pd | p − | / π = 172 kHz can2 b)a) c) FIG. 10. a, b and c) Crosses, retro-predicted squeezing factors for g − / π = 1 .
85 MHz as in Fig. 2.b. Shaded areas correspondto different models with 2 % uncertainty on g + and g − ; in a), Kerr-free analytical model, in b), steady-state simulations with K/ π = (20 ±
2) kHz, in c), steady state simulations including in addition to the Kerr effect some JRM four-wave mixing terms, K cd / π = 250 kHz, K pc | p − | / π = (172 ±
4) kHz and K pd | p − | / π = (172 ±
4) kHz. change the squeezing factors at the large g + , which com-forts the assumption that higher order nonlinearities mayexplain the deviations we observe between our analyticalmodel and the measured squeezing factors.To numerically compute the steady-state squeezingand anti-squeezing as a function of g − and g + , we use aniterative method to find the Liouvillian eigenvalues on atruncated Hilbert space comprising up to 60 excitationsfor the cavity and 30 excitations for the dump.
3. Simulations of the squeezing dynamics
The dynamics of stabilization and decay of squeezingare computed by solving the master equation on a trun-cated Hilbert space comprising up to 20 excitations forthe cavity and 16 excitations for the dump.To understand the effect of the self-Kerr term on thesqueezing decay, we simulate the evolution of squeezingfor varying waiting time t w and various self-Kerr rates K (Fig. 11). We initialize the cavity state at t w = 0in a Gaussian state with the measured S − = − . S + = 6 . K = 0), we observe an exponential damping of squeez-ing due to the cavity relaxation. For nonzero K , thesqueezing factor also oscillates in time. Our experimentalvalue K/ π = 20 kHz is closed to the critically dampedregime where the effective decay time is maximally re-duced. This observation highlights the crucial role ofKerr effect in the imperfections of our Wigner tomogra-phy technique used to estimate the variances. Appendix E: Effect of finite size Wigner tomograms
Due to experimental constraints, the probed quadra-ture phase space must be finite. A rectangular window − x ≤ Im( α ) ≤ x and − y ≤ Re( α ) ≤ y is chosen,where x = 1 . y = 2 .
7. This induces a systematic