Driving-Induced Resonance Narrowing in a Strongly Coupled Cavity-Qubit

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oration of a 190 nm thick aluminum layer and liftoff. In the second step, a bilayer resist is patterned by electron- beam lithography. Subsequently, shadow evaporation of two aluminum layers, 40 nm and 65 nm thick respectively, followed by liftoff define the qubit junctions. The chip is enclosed inside a copper package, which is cooled by a dilution refrigerator to a temperature of T = 23 mK. We employed both passive and active shielding methods to suppress magnetic field noise. While passive shielding is performed using a three-layer high permeability metal, an active magnetic field compensation system placed out- side the cryostat is used to actively reduce low-frequency magnetic field noise. We used a set of superconducting coils to apply DC magnetic flux. Qubit state control, which is employed in order to measure the qubit longitudinal T1 and transverse T2 relaxation times, is performed using shaped microwave pulses. Attenuators and filters FIG. 1: The device. (a) Electron micrograph of the flux qubit. are installed at different cooling stages along the trans- (b) Zoom out electron micrograph showing the qubit embed- mission lines for qubit control and readout. A detailed ded in the CPW resonator and its local flux control line. (c) description of sample fabrication and experimental setup Sketch of the experimental setup. The cavity transmission is can be found in [24, 25]. measured using a vector network analyzer (VNA). Monochro- matic flux qubit (FQ) driving is applied using a signal gen- erator (SG). (d) The measured cavity transmission (in dB III. THE DISPERSIVE REGION units) vs. ωf /2π (magnetic field detuning from the symmetry point) and ωdf /2π (cavity driving frequency). The power injected into the cavity is −112 dBm. For the device un- The circulating current states of the qubit are la- beled as x and y . The coupling between the der study ωc/2π = 6.6408 GHz, ω∆/2π = 1.12 GHz, g/2π = −5 | i | i 0.274 GHz and γc/ωc = 1.1 × 10 . The relaxation time T1 = cavity mode and the qubit is described by the term † x x y y 1.2 µs(1+0.45 ns × |ωf |) is obtained from energy relaxation g A + A ( ) in the system Hamil- −1 − | i h † | − | i h | measurements, and the rate T = 4.5 MHz (1 + 44 |ωf | /ωa) tonian, where A (A ) is a cavity mode annihilation (cre- 2 is obtained from Ramsey rate measurements [6]. The empiri- ation) operator, and g is the coupling coefficient. In the cal expressions for both T1 and T2 are obtained using approx- presence of an externally applied magnetic flux, the en- imate interpolation. ergy gap ~ωa between the qubit ground state and |−i first excited state + is approximately given by ~ωa = ~ 2 2 | i ~ ωf + ω∆, where ωf = (2IccΦ0/ )(Φe/Φ0 1/2), Icc ( I ) is the circulating current associated with− the state are compared with predictions of a semiclassical theory. cc y−p ( x ), Φ = h/2e is the flux quantum, Φ is the We find that good agreement can be obtained only in the 0 e externally| i | i applied magnetic flux and ~ω is the qubit limit of relatively small driving amplitudes. For higher ∆ energy gap for the case where Φ /Φ =1/2. driving amplitudes, better agreement is obtained with e 0 In the dispersive region, i.e. when g/ ∆ 1 where theoretical predictions derived by numerical integration ∆= ω ω and ω is the cavity mode angular| | ≪ frequency, of the master equation for the coupled system. c a c the coupling− between the cavity mode and the qubit gives rise to a resonance splitting. The steady state cavity mode response for the case where the qubit occupies the II. EXPERIMENTAL SETUP ground (first excited) states is found to be equivalent to the response of a mode having effective complex cavity angular resonance frequency Υ− (Υ+), where Υ± =Υc The investigated device [see Fig. 1(a) and (b)] contains ± ωBS Υba,Υc = ωc iγc is the cavity mode intrinsic a CPW cavity resonator weakly coupled to two ports that ± − are used for performing microwave transmission measure- complex angular resonance frequency, with ωc being the ments [see Fig. 1(c)]. A persistent current flux qubit [17], angular resonance frequency, γc the linear damping rate, 2 consisting of a superconducting loop interrupted by four and ωBS = g1/ (ωc + ωa) the Bloch-Siegert shift [21]. The Josephson junctions, is inductively coupled to the funda- term Υba is given by [35, 46–48] mental half-wavelength mode of the CPW resonator. We 2 1 i used a CPW line terminated by a low inductance shunt g1 ∆T2 Υba = − 2 , (1) for qubit driving [see Fig. 1(b) and (c)]. We fabricated ∆ 1 4g1 T1Ec − 1+ 2 2 + 2 the device on a high resistivity silicon substrate in a two- ∆ T2 ∆ T2 step process. In the first step, the resonator and the 2 control lines are defined using optical lithography, evap- where g1 = g/ 1 + (ωf /ω∆) is the flux dependent ef- q 3

−1 −1 fective coupling coefficient, T1 = γ1 and T2 = γ2 are the qubit longitudinal and transverse relaxation times, respectively, and Ec is the averaged number of photons occupying the cavity mode. Note that the imaginary part of Υ± represents the effect of damping and the term proportional to Ec gives rise to nonlinearity. In the dispersive approximation this term is assumed to be 2 2 small, i.e. Ec ∆ T2/4g T1. Note also that when ≪ 1 ∆T2 1 the term Υba gives rise to a shift in the mode angular≫ frequency approximately given by χ, where 2 ∓ χ = g1/∆, and to a Kerr coefficient approximately given 4 3 by g1/∆ (4T1/T2). Network± analyzer measurements of the cavity transmission are shown in Fig. 1(d). In the region where 2 2 ∆ > 0 (i.e. ωf < ωc ω∆) two peaks are seen in the cavity transmission, the− upper one corresponds to the case where the qubitp mainly occupies the ground state, whereas the lower one, which is weaker, corresponds to the case where the qubit mainly occupies the first excited FIG. 2: The effect of qubit driving. Cavity transmission in state. dB units as a function of qubit driving frequency ωp/2π and cavity driving frequency ωdf /2π. The qubit driving amplitude ω1 in (b) is 100 times larger compared with the values used in (a). For both plots the qubit frequency is given by ωa/2π = IV. QUBIT DRIVING 5 GHz. The overlaid black dotted line in (b) is obtained by numerically calculating the transition frequencies between the The flux qubit is driven by injecting a signal having an- lowest-lying eigenvalues of the Hamiltonian (A1) using the gular frequency ωp and amplitude ω1 into the transmis- following parameters ω1/2π = 0.5 GHz, ωc/2π = 6.6408 GHz, sion line inductively coupled to the qubit [see Fig. 1(b) ω∆/2π = 1.12 GHz, ωf /2π = 4.9 GHz, ωa/2π = 5.0GHz and and (c)]. Network analyzer measurements of the cav- g1/2π = 0.150 GHz. ity transmission as a function of ωp for two fixed values of qubit driving amplitude ω1 are shown in Fig. 2(a) and (b). For both plots the qubit transition frequency is exhibit hardening and softening effects (corresponding to flux-tuned to the value ω /2π = 5 GHz. The frequency a positive and negative Kerr coefficient, respectively) in separation between the two resonances that are shown in some region of the qubit driving amplitude ω . Simi- Fig. 2 is consistent with what is expected from the above- 1 lar behavior is presented in Fig. 3 of Ref. [47], which discussed dispersive shift χ, where χ = g2/∆. As can 1 exhibits cavity mode resonance line shapes of the same be seen from Fig. 2, the visibility∓ of the resonance cor- device in the absence of qubit driving. However, while responding to the qubit occupying the excited state at the nonlinearity observed in Ref. [47] is induced by cavity 6.637GHz is affected by both angular frequency ω and p driving, the one shown in Fig. 3(b) is induced by qubit amplitude ω of qubit driving. These dependencies are 1 driving. attributed to driving-induced qubit depolarization. The cavity driving induced nonlinearity reported in The comparison between Fig. 2(a) and Fig. 2(b), for Ref. [47] is well described by the above-discussed Kerr which the qubit driving amplitude ω is 100 times higher, 1 coefficients that can be calculated using Eq. (1) [see also reveals some nonlinear effects. One example is a non- Eqs. (4) and (A94) of Ref. [47]]. As is argued below, sim- linear process of frequency mixing of the externally ap- ilar nonlinearity can be obtained due to qubit driving. In plied driving tones, which gives rise to pronounced fea- the rotating wave approximation (RWA) the Hamiltonian tures in the data when ω is tuned close to the val- p of the closed system (consisting of a driven qubit and ues (ω + ω ) /2=2π 5.8 GHz and (3ω ω ) /2 = 0 a c a c Ha coupled cavity mode) in a frame rotated at the qubit 2π 4.2 GHz [see the overlaid× vertical white dotted− lines driving angular frequency ω is found to be given by [see in Fig.× 2(b)]. p Eqs. (A7), (A8) and (A9) of appendix A][49–55] The dependence of cavity transmission on qubit driving amplitude ω1 with a fixed driving frequency of ~−1 ′ † ′ ′ ′ † 0 = (ωR/2)Σz ∆pcA A + (g /2) AΣ− +Σ+A , ωp/2π = 5.16 GHz [ωp/2π = 5.52 GHz] is depicted in H − (2) Fig. 3 (a) and (b) [(c) and (d)]. Cross sections of the 2 2 color-coded plots shown in Fig. 3(a) and (c) (correspond- where ωR = ω1 + 4∆pa is the Rabi frequency, ∆pa = ′ ′ ing to different values of the qubit driving amplitude ω1) ωp ωa andq ∆pc = ωp ωc. The operators Σ± and Σz, are displayed in Fig. 3(b) and (d). which− are defined by Eq.− (A5), represent qubit opera- As can be seen from the cross sections shown in tors in the basis of dressed states. This Hamiltonian (2) Fig. 3(b) and (d), the cavity mode resonance line shapes has the same structure as the Hamiltonian in the RWA 4

′2 ′2 2 where ξ = g / (4∆R) 1 g / 3∆R . Both hardening and softening effects− are attributed to the term † † ′ proportional to A AA AΣz in Eq. (3). The term proportional to A†A in the Hamiltonian (3) can be used to determine the shift in resonances that is induced by qubit driving. However, the comparison with the experimental results shown in Figs. 2 and 3 yields a moderate agreement. The inaccuracy is attributed to ′ throwing away counter-rotating terms of the form AΣ+ † ′ and A Σ− in the derivation of the Hamiltonian (2). Much better agreement is obtained by numerically calculating the eigenvalues of the Hamiltonian (A1). The results of this calculation are displayed in Figs. 2(b) and 3(a) by the overlaid black dotted lines. The parameters that have been assumed for the calculation are listed in the captions of Figs. 2 and 3. As can be seen from both Fig. 2(b) and the green- colored cross sections shown in Fig. 3(b) and (d), in some regions of qubit driving parameters the two dressed fundamental resonances overlap. In the overlap region, a pronounced linewidth narrowing is observed [see Fig. 3(b) and (d)]. In appendix A we show that the observed changes in linewidth of resonances can be qualitatively attributed to the Purcell effect for dressed states. In particular, FIG. 3: Dependence on qubit driving amplitude. The driving both narrowing and broadening are demonstrated by frequency is ωp/2π = 5.16GHz in (a) and (b) and ωp/2π = Eqs. (A14) and (A15) below. However, the analytical 5.52 GHz in (c) and (d). Cavity transmission in dB units as expressions given by Eqs. (A14) and (A15), which have a function of cavity driving frequency ωdf /2π and amplitude been obtained by assuming the limit of weak coupling (in logarithmic scale) P1 = 20 log10 (ω1/ω1,0) are shown in (a) and (c). Cross sections taken at values of P1 indicated by and by employing the RWA, are not applicable in the colored horizontal dotted lines in (a) and (c) are shown using region where the linewidth narrowing is experimentally the corresponding colors in (b) and (d). The overlaid black observed. Consequently, direct comparison between the- dotted line in (a) is obtained by numerically calculating the oretical predictions based on Eqs. (A14) and (A15) and transition frequencies between the eigenvalues of the Hamilto- data yields poor agreement. Moreover, the relatively in- nian (A1) using the following parameters ωp/2π = 5.16 GHz, tense driving in the region where the linewidth narrowing ω1,0/2π = 2.4 GHz, ω∆/2π = 1.12 GHz, ωf /2π = 4.873 GHz, is experimentally observed gives rise to stochastic transi- ωa/2π = 5.000 GHz and g1/2π = 0.150 GHz. tions between qubit states [56]. These stochastic transitions, which may give rise to the effect of motional narrowing [57, 58], cannot be adequately accounted for using of the same system in the absence of qubit driving [see the semicalssical approximation. Note that these phe- Eq. (A13) of Ref. [47]]. However, while for the case of nomena are closely related to the effct of driving-induced no qubit driving the so-called crossing point occurs when spin decoupling (e.g. Fig. 7.27 of Ref. [59]). the qubit angular frequency ωa coincides with the cavity Numerical analysis based on the stochastic Schrödinger mode angular frequency ωc, the condition ∆pc = ωR is equation is described below in appendix B. We find that satisfied at the crossing point of the driven system.± the effect of narrowing can be numerically reproduced † Applying the transformation T = U U [35] to the provided that both qubit and cavity driving amplitudes H H Hamiltonian 0 (2), where the unitary operator U is are sufficiently large. The analysis in this region is chal- H −1/2 −1 ′ 1/2 given by U = exp /2 tan g /∆R and lenging, since relatively long integration times are needed N S N where ∆ = ω + ∆ , = A†A +(1+Σ′ ) /2 and = to achieve conversion. As can be seen from Figs. 6 and 7, R R pc z AΣ′ A†Σ′ , yields (constantN terms are disregarded)S in the region were narrowing is numerically reproduced, − − + the system becomes multistable. ′4 † ′ −1 ′ g 1+ A AΣz † ~ T = ∆pc + ξΣ A A H − z − 12∆3 R ! V. CAVITY DRIVING ′ ′ 5 (ωR + ξ)Σz g + + O , The nonlinear response of a microwave cavity coupled 2 ∆R ! to a transmon superconducting qubit has recently been (3) studied in Ref. [60]. The experimental results, together 5

This is because, despite accurately modeling the ﬁxed

points, which henceforth are referred to as the bright and dim metastable states (see Fig. 5), the semiclassical equations of motion give no information regarding the occupation probabilities of the two metastable states in T the overall state of the system, which can be written as

66 66 66 66 66 66 66 66 66 ρ = pbρb + pdρd , (4) where ρb(ρd) and pb(pd) represent the bright (dim) state and its probability respectively.

The experimental results shown in Fig. 4 exhibit a T sharp dip in cavity transmission TNA at drive powers above 109dBm. A very similar feature has been ex- 66 66 66 66 66 66 66 66 66 − perimentally observed before in [60] and theoretically discussed in Refs. [46, 61], for which the full quantum theory of the single nonlinear oscillator has been developed in FIG. 4: Nonlinear response to cavity driving. The cavity [62]. The origin of this dip is the destructive interference transmission TNA is measured as a function of cavity driv- between the two metastable states. Since the system is ing frequency ωdf /2π for different values of the cavity driv- coupled to an external reservoir, fluctuations in the quan- ing power Pda. These data are compared with a numerical tum state ensue and occasionally cause major switching calculation of the steady state of the Lindblad master equation (orange line), which is detailed in appendix C. In (a-c) events between the bright and dim states. When the complex amplitude of the cavity state is averaged over the frequency ωf /2π is flux-tuned to 5.5GHz, and in (d-f) to 7.8 GHz. Different values of γc are used at each drive power an ensemble of many such switching events, there is typ- to account for the increase in the quality factor of the cavity ically a narrow region in the frequency-power space where with occupation. We use γc = (a) 0.314, (b) 0.251, (c) 0.126, the two complex amplitudes partially cancel each other. (d) 0.314, (e) 0.251 and (f) 0.126 MHz. By using the Lindblad master equation to model the system, we are able to take account of these fluctuations which cause these switching events and we produce the numerical fits seen in Fig. 4. Comparison between the with theoretical analysis [46, 61], indicate that the re- predictions derived from the numerical integration of the sponse to strong cavity driving is affected by the sig- master equation and the ones analytically derived from nificant coherent driving of the qubit as well as by the the semiclassical equations of motion is shown in Fig. 5. stochastic transitions between qubit states. The effect of cavity driving can be characterized by a dephasing rate and by a measurement rate. Both rates have been numerically calculated and analytically estimated in Ref. VI. SUMMARY [56]. Measurements of the cavity transmission TNA of our Our main finding is the linewidth narrowing that is device as a function of cavity driving frequency ωdf /2π obtained by applying intense qubit driving. The effect and power Pda are shown in Fig. 4. No qubit driving is experimentally robust, however, its theoretical mod- is applied during these measurements. We demonstrate eling is quite challenging. Further study is needed to nonlinearity of the softening type in Fig. 4(a-c), whereas explore the possibility of exploiting this effect for long- hardening is demonstrated in Fig. 4(d-f). We obtained time storage of quantum information. We also find that the data shown in Fig. 4 by sweeping the cavity driv- bistability, which is predicted by the semiclassical model ing frequency ωdf /2π upwards. Almost no hysteresis is for monochromatic cavity driving, is experimentally in- observed when the sweeping direction is flipped. accessible. This effect and related observations can be The measured cavity transmission TNA can be com- satisfactorily explained using numerical integration of the pared with theoretical predictions based on the semiclas- master equation for the coupled system. sical approximation. Such a comparison has been performed in Ref. [47] based on data that has been obtained from the same device. Good quantitative agreement was found in the region of relatively small cavity driving am- VII. ACKNOWLEDGMENTS plitudes [47]. However, when the cavity is strongly driven, the non- EB and PB contributed equally to this work. The work linearity introduced to the system by the qubit causes of the Waterloo group is supported by NSERC and CMC the onset of bistability and the semiclassical approxima- and the work of the Technion group by the Israeli Science tion alone is unable to reproduce the cavity transmission. Foundation. 6

Appendix A: Dressed states

In this appendix the semiclassical dynamics of a driven qubit coupled to a cavity mode is discussed [49–55]. In the RWA the Hamiltonian 0 of the closed system in a H frame rotated at the qubit driving angular frequency ωp is given by [see Eq. (A13) of Ref. [47]]

−1 ω1 (Σ+ +Σ−) ~ 0 = ∆paΣ + H − z 4 † † g1 AΣ− +Σ+A ∆pcA A + , − 2 (A1)

where ∆pa = ωp ωa, ~ωa is the qubit energy, Σ is the − z qubit longitudinal operator, ω1 is the driving amplitude, † Σ+ and Σ− =Σ+ are qubit rotated transverse operators, ∆pc = ωp ωc, ωc is the cavity mode angular frequency, † − A A is the cavity mode number operator and g1 is the coupling coeﬃcient. The Bloch equations of motion for the expectation values a = A , p+ = Σ+ and pz = Σz are obtained from theh Heisenbergi h equationsi of motionh i and the com- † mutation relations A, A = 1, [Σ , Σ±] = Σ± and z ± [Σ+, Σ−] = 2Σz by adding ﬂuctuation and dissipation terms and by averaging

ig1p+ a˙ = (γc i∆pc) a , (A2) − − − 2 iω1pz p˙+ = (γ2 + i∆pa) p+ ig1ap , (A3) − − 2 − z FIG. 5: The bistable regime. The cavity drive power is given ∗ ∗ ∗ iω1 p+ p+ ig1 ap+ p+a by Pda = 20 log10 (ω2/ω2,0) where ω2,0/2π = 340 GHz. In (a) p˙z = γ1 (pz p0)+ − + − , we plot the Wigner function of the cavity state in the bistable − − 4 2 regime, which is obtained by solving for the steady state of (A4) the master equation at a drive power of Pda = −109dBm and where overdot denotes time derivative, γc is the cav- a drive frequency of ωda/2π = 6.6445GHz. These parameters are marked by the red crosses in panel (b) and in Fig. ity mode damping rate, γ1 and γ2 are the qubit longi- 4(b). Two metastable states can be seen: a bright state at tudinal and transverse damping rates, respectively, the ~ Cb = 4.64 − 3.73i and a dim state at Cd = −1.88 − 0.39i. coefficient p0 = tanh (( ωa) / (2kBT )) is the value − These two states correspond to the fixed points produced us- of pz in thermal equilibrium (when ω1 = g1 = 0), ing the semiclassical equations of motion, marked by blue kB is the Boltzmann’s constant and T is the temper- crosses. Next in (b) we examine the boundaries of the bistable ature. In the absence of coupling. i.e. when g1 = regime. By examining the cavity Wigner function over a range 0, the steady state solution of Eqs. (A3) and (A4) is of drive powers and frequencies we map the region in which given by p+,ss = (iω1pz,ss) / (2γ2 +2i∆pa) and pz,ss = we find two peaks corresponding to the bright and dim states. 2 − 2 2 −1 When two peaks can be identified we calculate the metric p0 1+ γ2ω1 / 4γ1γ2 +4γ1∆pa . B = 1 − |pb − pd| as a measure of bistability. This is plot- Consider the transformation ted in the colormap above. Meanwhile the dashed black lines ′ Σ+ Σ+ mark the boundaries of the region in which the semiclassi- ′ cal equations of motion have two fixed points. These meth- Σ− = Ma Σ− , (A5)  ′    ods produce significant overlap and both predict the onset of Σz Σz bistability around Pda = −117dB. We also see that the re-     gion of maximum bistability predicted by the master equation where (yellow strip) lies either close to or within the semiclassical sin α+1 sin α−1 bistable region at all powers. However there are significant 2 2 cos α sin α−1 sin α+1 − differences in the limits of the bistable region, particularly at Ma = 2 2 cos α , (A6)  cos α cos α −  the upper freuqency limit. The master equation predicts this 2 2 sin α limit should increase with drive power, whereas the semiclas-   sical equations predict the opposite. and where tan α = ( 2∆pa/ω1). Note that the trans- ′ − ′ formed operators Σ± and Σz satisfy the commutation 7 relations Σ′ , Σ′ = Σ′ and Σ′ , Σ′ = 2Σ′ pro- z ± ± ± + − z vided that the original operators Σ± and Σz satisfy 30 [Σz, Σ±]= Σ± and [Σ+, Σ−]=2Σz. Under this± transformation the first two terms 2 20 of the Hamiltonian 0 (A1) become ∆paΣz + )| H − ′ 2 2 10 α 125 kHz ω1 (Σ+ +Σ−) /4 = (ωR/2)Σz, where ωR = ω1 + 4∆pa | is the Rabi frequency. In the RWA, in whichq counter- 0 rotating terms are disregarded, the equations of motion ⟨⟩ (A2), (A3) and (A4) are transformed into . ′ ′

ig1p+ − a˙ = (γc i∆pc) a , (A7) − − − 2 ′ ′ iωR ′ ′ ′ p˙ = γ p ig ap , (A8) + − 2 − 2 + − 1 z ⟨⟩ ′ ′∗ ′ ∗ ′ ′ ′ ′ ig1 ap+ p+a p˙ = γ (p p )+ − , (A9) z − 1 z − 0 2 where the effective coupling coefficient g′ is given by 1 ′ g1 = (1/2) (sin α + 1) g1, the transformed damping rates ′ ′ ′ 2 γ1 and γ2 are given by γ1 = γ2 + (γ1 γ2) sin α and ′ 2− γ2 = (1/2) (γ1 + γ2)+(1/2) (γ2 γ1) sin α, respectively, −′ and the polarization coefficient p0 is related to p0 by FIG. 6: Simulation of the cavity spectrum in the nonlinear ′ sin α regime. We use a cavity drive amplitude of ω2/2π = 1 MHz, p = p0 . (A10) 0 a qubit drive frequency of ωp = 5.5 GHz and a qubit drive γ2 1+ γ1−γ2 sin2 α γ1 γ2 amplitude of ω1 = 0.863 GHz. The system displays multistability and three distinct metastable states can be identified, Note that the equations of motion (A7), (A8) and (A9) which are labelled by d↓, d↑ and b and assigned the colours become unstable when [63–67] orange, green and red respectively. We plot the square cavity amplitude (a), qubit polarization (b) and occupation proba- ′ 2 bility (d) of each of these three states against the cavity drive ′2 2γcγ2 ∆L g 1+ , (A11) frequency ωdf /2π. In panel (a) we see that the cavity ampli- 1 ≥− p′ ′ 2 0 (γc + γ2) ! tude of state b is significantly larger than the amplitudes of states d↓ and d↑. Hence we refer to b as bright and d↓ and d↑ where ∆L = ∆pc ωR/2. as dim. The black line is produced by averaging the cavity − ′ In the limit where the coupling coefficient g1 is suffi- amplitude over 9.6 ms of evolution before taking the square ′ ciently small, at and near steady state the term pz in Eq. of the absolute value. It displays a narrow resonance at the (A8) can be approximately treated as a constant, and bare cavity frequency. The full width at half maximum is only consequently Eqs. (A7) and (A8) can be expressed in a 125 kHz, which is 33% of the natural linewidth of 377 kHz. In matrix form as panel (b) we see that states d↓ and d↑ occur when the qubit is polarized up and down respectively whereas the qubit polar- d a a ization associated with state b varies with the drive frequency. ′ + MP ′ =0 , (A12) dt p+ p+ Finally in panel (c) we see the occupation probabilities of the three states. Away from the cavity resonance the stability of where the matrix MP is given by state b falls. This causes the narrowing observed in panel (a).

′ ig γc i∆pc 2 MP = − R . (A13) ig′p′ γ′ iω z 2 − 2 To lowest nonvanishing order in g′ the eigenvalues of the matrix MP are given by

′ ′2 pz g 2 ′4 Γc = γc i∆pc + ′ + O g , (A14) − γ2 γc + i∆L As can be seen from Eqs. (A14) and (A15), in this limit − ′ ′2 pz g the coupling gives rise to repulsion-like behavior of the ′ iωR 2 ′4 ′ ′ Γ2 = γ2 ′ + O g . (A15) damping rates, i.e. γc,eff γ2,eff > γc γ2 (it is as- − 2 − γ2 γc + i∆L − | − | − sumed that p′ < 0). This behavior can be considered as z The real parts of Γc (Γ2) represents the effective damp- a generalization of the Purcell effect [10] for the case of ′ ing rates γc,eff (γ2,eff ) of the cavity-like (qubit-like) mode. dressed states. 8

Δ αΔ α Δ

⟨ α Δ

−

Δ α ⟨ ⟨ μ ⟩

⟩ ⟨ ⟨ αΔ

FIG. 7: Here we examine a quantum state trajectory produced at ωdf /2π = 6.6409 GHz. In panel (a) we plot the real and imaginary parts of the cavity amplitude. The cavity is observed to jump between three metastable states. Examples of states d↑, d↓ and b are highlighted between the vertical dashed lines in green, orange and red respectively. By referring to panel (b) we see that state d↑ occurs when the qubit has positive polarization, state d↓ occurs when it has negative polarization and state b occurs when the qubit freely varies over the range −1 < hσzi < 1. In panel (c) we plot a histogram of the cavity amplitude throughout 9.6 ms of evolution. The three metastable states are clearly identiﬁed as three clusters in the plane. Switching pathways leading between these clusters can also be observed.

Appendix B: Simulating line narrowing the cavity drive frequency. Therefore in order to model the transmitted power TNA we must examine the cavity amplitude A in a frame rotating with the drive. This Experimentally we have observed narrowing in the cav- h i ity spectrum which occurs when a drive is applied to the is given by qubit. In an attempt to model this narrowing we per- α(∆dp,t)= Tr ρ(t)A exp i∆dpt . (B3) form simulations of the cavity response by unravelling − the Lindblad master equation using a quantum jump (Monte-Carlo) stochastic Schroödinger equation. In a Input-output relations can then be used to calculate TNA frame rotating with the qubit drive at angular frequency from this amplitude. ωp we use the rotating wave approximation (RWA) to We now attempt to reproduce the spectrum seen in write down the Hamiltonian as: Fig. 3(b). In order to observe narrowing we must drive the cavity in the nonlinear regime. We take a cavity −1 −1 † ~ (t)= ~ 0 + ω2 A exp(i∆dpt)+ A exp( i∆dpt) drive amplitude of ω /2π = 1.00 MHz. The remaining H H − 2 (B1) parameters are set to ω /2π = 1.726 MHz, ω /2π = 1 p where the time independent part of the Hamiltonian 0 5.50 GHz, g /2π =0.150 GHz, ω /2π = 5 GHz, γ /2π = H 1 a c is given in Eq. (A1) and the time dependent cavity drive 377 kHz and γ1/2π = 40.7 kHz. Using these parameters oscillates at the frequency ∆dp = ωdf ωp. In order to we produce the spectrum in Fig. 6 by evolving the state − describe dissipation due to loss of photons from the cavity of our system over 9.6 ms for a range of cavity drive we use the Lindblad operator √γc A, while to describe frequencies. The long time average α(∆dp) displays a dissipation in the qubit we use √γ Σ−. After combining full width at half maximum of 125 kHz, significantly less these elements the evolution of the state of the system is than the natural linewidth of 377 kHz. described by: This narrowing can be explained when we realise that in the presence of a strong cavity drive the system dis- i + ∂tρ = ~[ (t),ρ]+ γc [A]ρ + γ1 [Σ ]ρ. (B2) plays multistability and the line narrowing is due to a − H D D bright cavity state (b) which is most stable over a narrow We use this equation to numerically evolve the state range of frequencies close to the bare cavity resonance. ρ(t) over time and study the cavity amplitude A = Close to the cavity resonance the system occupies the Tr(ρ(t)A). We find that the time-dependence of Ah con-i bright state and the transmitted power is high. However h i tains two main frequencies: ∆dp and ∆cp, due to the away from this point the system may also occupy two drive and cavity frequency respectively. The experimen- other dim states (d↓ and d↑), which causes a sharp drop tal data presented in Fig. 3 were measured by mixing the in the transmitted power and a narrow linewidth. signal transmitted through the cavity with a reference at In Fig. 7 we examine these metastable states more 9 closely. We plot the cavity amplitude and qubit polar- In the above the detuning between the cavity drive and ization over 170 µs of evolution at ω /2π =6.6409 GHz. the cavity resonance is given by ∆dc = ωdf ωc, while d − The two dim states, labelled d↑ and d↓, occur when the detuning between the cavity drive and the qubit fre- the qubit is polarized in the up and down directions quency is given by ∆da = ωdf ωa. Since the relax- respectively. Meanwhile the bright state occurs when ation rate of the qubit depends− on the magnetic field the qubit is depolarised and varies widely over the range detuning from the symmetry point we must take account 1 < σz < 1. of this in our calculations. For Figs. 4(a-c) we have − h i ωf /2π =5.5GHz and γ1/2π =6.29kHz, whereas for Figs. 4(d-f) we have ωf /2π =7.8GHz and γ1/2π =4.02kHz. Appendix C: Spectra calculations The master equation above does not include a Lindblad operator to describe pure dephasing of the flux qubit. In order to calculate the response of our system to cav- Since we are operating the qubit far from its symmetry ity driving (without qubit driving) we use the following point, pure dephasing will be dominated by flux noise, master equation: and in [25] the power spectral density (PSD) of this noise was found to have a 1/f 0.9 form. Unfortunatley we can- ∂tρ = i[ /~,ρ]+ γc [A]ρ + γ1 [Σ+]ρ , (C1) not account for this noise in the master equation, because − H D D the Markovian approximation requires that the PSD is which consists of non-unitary components calculated ac- well-behaved at zero frequency. However, even without 1 cording to (L)ρ = L†ρL (L†Lρ+ρL†L) and a unitary the inclusion of pure dephasing, the master equation is D − 2 component which obeys the Hamiltonian given by still able to explain the major features of the spectra

−1 † measured in Fig. 4. ~ = ∆daΣ + ω2 A + A H − z † g1 † ∆dcA A + Σ−A +Σ +A . (C2) − 2

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