Resonance fluorescence from an artificial atom in squeezed vacuum

D. M. Toyli,1, ∗ A. W. Eddins,1, ∗ S. Boutin,2 S. Puri,2 D. Hover,3

V. Bolkhovsky,3 W. D. Oliver,3, 4 A. Blais,2, 5 and I. Siddiqi1

1Quantum Nanoelectronics Laboratory, Department of Physics, University of California, Berkeley CA 947202Department de Physique, Universite de Sherbrooke,

2500 boulevard de l’Universite, Sherbrooke, Quebec J1K 2R1, Canada3Massachusetts Institute of Technology Lincoln Laboratory, Lexington, MA 02420, USA

4Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA5Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada

(Dated: July 19, 2016)

We present an experimental realization of resonance fluorescence in squeezed vacuum. We stronglycouple microwave-frequency squeezed light to a superconducting artificial atom and detect the result-ing fluorescence with high resolution enabled by a broadband traveling-wave parametric amplifier.We investigate the fluorescence spectra in the weak and strong driving regimes, observing up to3.1 dB of reduction of the fluorescence linewidth below the ordinary vacuum level and a dramaticdependence of the Mollow triplet spectrum on the relative phase of the driving and squeezed vacuumfields. Our results are in excellent agreement with predictions for spectra produced by a two-levelatom in squeezed vacuum [Phys. Rev. Lett. 58, 2539-2542 (1987)], demonstrating that resonancefluorescence offers a resource-efficient means to characterize squeezing in cryogenic environments.

Keywords: Quantum Physics, Atomic and Molecular Physics, Quantum Information

I. INTRODUCTION

The accurate prediction of the fluorescence spectrumof a single atom under coherent excitation, comprisingcanonical phenomena such as the Mollow triplet [1, 2], isa foundational success of quantum optics. As the Mol-low triplet provides a clear signature of coherent light-matter coupling, in recent years such spectra have beenwidely applied to probe quantum coherence in artificialatoms based on quantum dots [3] and superconductingqubits [4, 5]. Resonance fluorescence is predicted to of-fer analogous spectroscopic means to identify and char-acterize atomic interactions with squeezed light [6–10],which is known for its potential to enhance measurementprecision in applications ranging from gravitational wavedetectors [11] to biological imaging [12]. However, it re-mains a challenge to demonstrate that exciting atomicsystems with squeezed light strongly modifies resonancefluorescence, in part due to the stringent requirementthat nearly all the modes in the atom’s electromagneticenvironment must be squeezed [13].

Recently, the circuit quantum electrodynamics archi-tecture has emerged as a compelling platform for study-ing squeezed light-matter interactions. The low dimen-sionality of microwave-frequency electrical circuits natu-rally limits the number of modes involved in atomic inter-actions [14], such that squeezing a single mode can havea significant effect. Experiments have thus demonstratedthat microwave-frequency squeezed light can modify thetemporal radiative properties of an artificial atom, lead-ing to phase-dependent radiative decay [15]. However,these experiments lacked a means to probe the resulting

∗ Equal contributors.

fluorescence, and predictions for the spectrum of reso-nance fluorescence in both the weak [6] and strong [7, 8]driving regimes remain unexplored. Here, we employ twosuperconducting parametric amplifiers, one to generatesqueezing and one to detect the resulting atomic fluores-cence, to perform a systematic study of the dependenceof resonance fluorescence on the properties of squeezedvacuum. We observe subnatural fluorescence linewidthsand a strong dependence of the Mollow triplet spectrumon the relative phase of the driving and squeezed vac-uum fields, in excellent agreement with theoretical pre-dictions. These results enable experimental access to themany theoretical studies on atomic spectra in squeezedreservoirs that have followed [9, 10], and demonstrate theutility of resonance fluorescence for the characterizationof itinerant squeezed states, an important capability forthe development of proposed schemes to enhance qubitreadout with microwave-frequency squeezed light [16, 17].

Our experimental implementation integrates recentadvances in parametric amplifiers and superconduct-ing artificial atoms (Fig. 1(a)). We produce itiner-ant squeezed microwaves using a Josephson paramet-ric amplifier (JPA) [18], which can be understood as afrequency-tunable LC resonator (Fig. 1(b,c)). A dilu-tion refrigerator cools the experiment to below 30 mKsuch that the microwave field incident to the JPA is verynearly in its vacuum state. Modulation of the JPA’s res-onant frequency by a pump tone at twice that frequencysqueezes the vacuum fluctuations in one quadrature ofthe resonator field while amplifying those in the other.Ideally, the JPA’s output field is an ellipse in phase spacedescribed by the parameters N and M , with quadraturephase variances V (θ) given by

V (θ) =1

2

[N +M cos(2(θ − ϕ)) +

1

2

], (1)

arX

iv:1

602.

0324

0v2

[qu

ant-

ph]

15

Jul 2

016

2

where M2 = N(N+1). The squeezing axis lies along theangle θ = ϕ+π/2 in phase space, at which the minimumquadrature variance occurs (see Appendix B for defini-tions). In practice, N and M are diminished from theirideal values by a factor η < 1 due to dilution of the itin-erant squeezed state with vacuum noise via microwavecomponent loss or other imperfections.

Our artificial atom consists of a transmon supercon-ducting qubit coupled to an aluminum microwave waveg-uide cavity [19]. As the qubit is nearly resonant with thecavity’s fundamental mode, their strong coupling pro-duces well-separated polariton states described by theJaynes-Cummings Hamiltonian (Fig. 1(d)) [14, 15, 20].When only two of these states are coupled to squeezedradiation, the system is expected to exhibit the samedynamics as a single two-level atom in squeezed vac-uum [21]. We drive the system via two ports (antennas)coupled to the cavity mode, where one port is coupledmore than four times as strongly as the other. Squeezedvacuum from the JPA is directed by a circulator to thecavity’s strongly-coupled port, while the weakly-coupledport allows for application of a coherent tone to drive po-lariton Rabi oscillations. Relatively weak coupling at thelatter limits dilution of the squeezing by vacuum fluctu-ations in the connecting transmission line, such that thetwo inputs sum to a squeezed vacuum displaced in phasespace by the coherent drive. Most of the resulting fluores-cence exits the cavity through the strongly coupled port.This fluorescence, in combination with the squeezed ra-diation reflecting off of the strongly coupled port, thenpasses through the circulator towards the first stage ofamplification.

To detect the fluorescence with high precision weutilize the recently-developed Josephson traveling waveparametric amplifier (JTWPA) [22]. Previous polaritonMollow triplet studies have relied on cross-correlationof two measurement chains and data processing using afield-programmable gate array (FPGA) to overcome theadded noise of following semiconducting amplifiers thatwould otherwise necessitate prohibitively long averagingtimes [20]. Here, the JTWPA acts as a superconduct-ing preamplifier with near-quantum-limited noise perfor-mance that mitigates the effect of this added noise andthus facilitates direct fluorescence measurements with amicrowave spectrum analyzer. Moreover, the high dy-namic range of the JTWPA ensures that squeezed vac-uum power reflecting off of the cavity does not causecompression of the amplified fluorescence spectra. Fig-ure 1(e) plots resonance fluorescence spectra measured inordinary vacuum over a 50 MHz span as a function of thecoherent drive amplitude applied at the upper polaritonfrequency (ω1,+). For large drive amplitudes, we see athree-Lorentzian Mollow triplet spectrum, reflecting thepolariton Rabi oscillations.

flux pump

180°hybrid

∑

Δ

(a)

(b)

(c)

cavity

0

ωc

+ qubit + rabi

ωc

1

2

1,+

1,–

2,+

2,–0,f

~2g

ω1,+

ω1,++Ωω1,+–Ω

(d)

(e)

Driv

eam

plitu

de (a

.u.)

Fluorescence spectrum

Frequency (GHz)

Rel

. pow

er (a

.u.)

0

1

0.5

span5 MHz

100 μm

2ω ω

7.28 7.29 7.3 7.31 7.32

FIG. 1. (a) Simplified experimental setup. A qubit-cavitysystem is driven by a coherent drive combined with squeezedvacuum from a Josephson Parametric Amplifier (JPA). Theresulting fluorescence is amplified by a Josephson Travel-ing Wave Parametric Amplifier (JTWPA). (b,c) False-coloredphotomicrograph and schematic of the JPA. The port (red) atleft flux-couples a pump tone to the JPA, producing quadra-ture squeezing. (d) Coupling to the qubit splits the cavitystates into polariton states. Driving Rabi oscillations between|0〉 and |1,+〉 splits each state into a doublet. The three dis-tinct transition frequencies produce the Mollow triplet spec-trum. (e) Mollow triplet spectra in ordinary vacuum, normal-ized by the power spectrum with no coherent drive to accountfor the JTWPA’s gain ripple.

II. RESONANCE FLUORESCENCE INSQUEEZED VACUUM

We next examine the artificial atom’s fluorescencewhen the polariton is excited only by squeezed vacuumwith no coherent drive. In the reflection geometry, non-classical correlations must exist in the driving field to ob-serve any fluorescence peak amidst the broadband noise.To quantitatively analyze the power spectra, we adaptthe prediction of Gardiner [6, 23] to the case of a two-port

3

cavity measured in reflection, leading to the expression

2πSR(ω) =N

ηc+ γ[M − (1− ηc)N

2N + 1

(γy

ω2 + γ2y

)− M + (1− ηc)N

2N + 1

(γx

ω2 + γ2x

)], (2)

Here N and M describe the squeezed state inside thecavity and γ is the radiative linewidth in ordinary vac-uum. As predicted for a single-port cavity [6, 23], thespectrum is the sum of a broad background representingreflected noise power, a broad negative Lorentzian withhalf width γx = γ( 1

2 + N + M), and a narrow positive

Lorentzian with half width γy = γ( 12 +N−M). The neg-

ative Lorentzian represents absorption of correlated pho-ton pairs at ±ω which are then reemitted at the atomicresonance frequency to produce the positive Lorentzian.Thus if all the incident power is reflected, the areas ofthe two Lorentzians are equal. However, in practice somepower escapes through the second cavity port, which en-hances the relative spectral weight of the negative dipand couples in unsqueezed vacuum modes that dilute thesqueezing inside the cavity by the factor ηc. Experimen-tally, we determine the total linewidth γ = 304 kHz fromfits to the Mollow triplet spectra in ordinary vacuum andthe factor ηc = 0.81 through reflection measurements ofthe polariton resonance (see Appendix A 1 for details).

We observe reflection spectra in good agreement withEq. 2—a representative spectrum for 1.4 dB of JPA sig-nal gain (GJPA, see Appendix A 2) and a correspondingfit are shown in Fig. 2(a). From the fit we determineγy, indicating 2.4 dB of squeezing below the standardvacuum limit for these conditions. While the spectraexhibit negative dips as expected, the small amplitudesand broad spectral range of these features limit the reli-able determination of γx through this measurement. No-tably, all of these spectral features rapidly diminish whenthe squeezer pump is detuned by an amount compara-ble to γ (Fig. 2(b)). Although much smaller than thesqueezer bandwidth, the detuning causes the squeezedstate to appear as thermal noise in the frame of the ar-tificial atom. By removing the symmetry of two-modecorrelations about the atomic transition, the detuningsuppresses the two-photon process by which off-resonantpower can be absorbed and then resonantly emitted,thereby suppressing the squeezing induced fluorescence.

We next drive the polariton to generate a Mollowtriplet spectrum (Fig. 3(a)). The sideband splitting iskept small so that when the JPA is pumped for gainthe squeezing spectrum can be approximated as constantover the Mollow triplet spectral range. As the polari-ton’s coherent drive and the JPA’s flux pump are gen-erated from the same microwave source, their relativephase can be controllably varied (Fig. 3 insets). This con-figuration facilitates observation of the Mollow triplet’sstrong dependence on this relative phase as predicted byCarmichael, Lane, and Walls [7, 8]: at modest JPA gainsall three peaks can be resolved with the center peak oscil-

600

400

200

0

-200

-400

-600

0 0.5 1-0.5-1-1.5f - f1,+ (MHz)

½ f pu

mp -

f 1,+ (

kHz)

-0.25

0

0.25

0.50

0.75

1.00

GJPA = 1.4 dB

M - N = 0.21(1)

Rel

. pow

er (a

.u.)

γY = 89(3) kHz

1.5

Rel

. pow

er (a

.u.)

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

(a)

(b)

FIG. 2. (a) Reflected power as a function of frequency forexcitation with only squeezed vacuum (GJPA = 1.4 dB). Thespectrum is normalized by a measurement in ordinary vac-uum to account for the JTWPA’s gain ripple. Zero relativepower corresponds to the background level of the broadbandsqueezed power. The red ellipses are phase space representa-tions of the squeezed states inside and outside the cavity asinferred from the fit to Eq. (2). (b) Reflected power mea-sured over a constant frequency span as the frequency of theJPA pump is varied. The peak in the spectrum diminishesand broadens as the JPA pump is detuned from the polaritonresonance.

lating between subnatural (Fig. 3(b)) and supernatural(Fig. 3(c)) half linewidths of γy and γx, respectively. Tofurther demonstrate the phase dependence, we first fitthe data with an approximate model of three Lorentzianpeaks; example fits appear as black dashed curves in Fig.3(a-c). The results of these fits exhibit the expectedout-of-phase oscillations of the center peak and sidebandlinewidths as a function of this relative phase (Fig. 3(d)).

III. SQUEEZING CHARACTERIZATION

While the approximate Lorentzian model clearly illus-trates the phase dependence of the spectra, we quan-titatively characterize the squeezing levels through fits

4

Pow

er (a

.u.)

Relative squeezing angle Ф (deg)

(d)1.0

0.8

0.6

0.4

0.2

0.0

Full

linew

idth

(MH

z)

270180900

Central peak FWHM Lower peak FWHM Upper peak FWHM

Sine fit of central peak widths Sine fit of sideband widths

1.3

1.2

1.1

1.0

(a)

1.4

1.3

1.2

1.1

1.0

1.5

1.2

1.1

1.0Pow

er (a

.u.)

Pow

er (a

.u.)

Frequency (GHz)

(c)

7.298 7.3 7.302 7.304

Ф = 0

(b)

Ф = π/2

FIG. 3. Reflection spectra for excitation with a coherentdrive (a) in ordinary vacuum, (b) with Φ = π/2 (GJPA = 1.5dB), and (c) with Φ = 0 (GJPA = 1.5 dB), where Φ + π/2 isthe phase angle between the coherent drive and the squeez-ing axis (see Appendix B for definitions). The spectra arenormalized to account for the JTWPA’s gain ripple such thatthe background level in ordinary vacuum is one; the parabolicbackgrounds (orange, dashed) in (b) and (c) result from thereflected JPA noise power. (d) The black dashed curves arefit using an approximate three-Lorentzian model; linewidthsof this fit are plotted in d as a function of Φ. Error bars re-flect fit uncertainties. The two dashed curves are out-of-phasesine functions fit to the central-peak and sideband linewidthsinferred from measurements in squeezed vacuum, while thedashed horizontal lines indicate the corresponding linewidthsinferred from measurements in ordinary vacuum.

Gain (dB)

M –

N

Squeezing (dB

)

0

1

2

3

4

76543210

0.30

0.25

0.20

0.15

0.10

0.05

Squeezing + Rabi

Squeezing only

Theory fit (η = 0.55)

√1/2 – (M – N)2

FIG. 4. The squeezing level in the artificial atom’s envi-ronment as a function of gain GJPA. The quantity M − Ndetermines the minimal quadrature variance of a squeezedstate as indicated in the inset. The values of M − N areinferred from measurements of the fluorescence spectrum insqueezed vacuum with (red) and without (blue) a coherentRabi drive. The error bars reflect fit uncertainties. The mea-surements with no Rabi drive give a more reliable indicationof the squeezing level as they are less sensitive to backgroundeffects and experimental drift. The dashed orange curve is aone-parameter fit to the blue data points yielding η = 0.55.

to analytic expressions for the power spectra havingboth Lorentzian and small non-Lorentzian dispersiveconstituents (Appendix B, Eq. (B20)). These expres-sions extend the results of Refs. [7, 8] to arbitrary rela-tive phases and do not assume observation through un-squeezed vacuum modes. We find that these analyticalexpressions are able to accurately model the measuredpower spectra even at large GJPA when the sidebands aresignificantly broadened and the frequency-dependence ofthe squeezer background becomes prominent.

Figure 4 plots M − N , which indicates the squeezinglevel, as a function of GJPA, comparing results from fitsof the Mollow triplet spectra (red points) to correspond-ing results from fits of spectra measured with no Rabidrive (blue points). Both measurements of M − N arewell-described by a one-parameter fit with overall effi-ciency η = 0.55 (orange dashed line), comprising lossesdue to both ηc and component loss (see Appendix A 1).For the range of GJPA measured here, we observe cavitysqueezing levels as high as 3.1 dB below the standardvacuum limit; factoring out cavity losses correspondingto ηc suggests up to 4.4 dB of squeezing in the itinerantsqueezed state incident to the cavity. While the resultsof the measurements with and without a Rabi drive arein reasonable agreement, the simplicity, speed, and rela-tive insensitivity to background features of the measure-ments with no Rabi drive recommend that technique asa resource-efficient detector of squeezing, which could beimplemented in a single-port cavity to further improveprecision.

Both generating and characterizing the squeezing in asuperconducting qubit’s environment are of central im-portance to proposed schemes for enhancing qubit mea-surement via interferometric readout [16, 24]. The ra-

5

tiometric methods presented here achieve this charac-terization through spectral measurements that are con-siderably simpler to implement than alternative de-tection methods that require time-domain qubit con-trol [15]. Moreover, as subnatural fluorescence linewidthsare also expected to occur in two-mode squeezed vac-uum [25], these metrological techniques are similarlyrelevant to measurement schemes employing two-modesqueezed light [17].

More generally, our results validate the canonical pre-dictions for resonance fluorescence in squeezed vacuumand exemplify the JTWPA’s broad utility in microwavequantum optics. These experimental techniques directlyenable investigations of numerous predicted phenomenaincluding fluorescence in non-Markovian squeezed reser-voirs [26] and in parameter regimes expected to producequalitatively distinct spectra [27]. Finally, while thisarea of experimental study has long been led by theo-retical prediction, combining the fluorescence detectiontechniques demonstrated here with the wide range ofatomic environments realizable in circuit quantum elec-trodynamics enables the study of squeezing’s impact onresonance fluorescence in previously unforeseen settingssuch as multimode strong coupling [28] and the ultra-strong coupling regime [29].

ACKNOWLEDGMENTS

The authors thank E. Flurin, C. Macklin, and L.Martin for useful discussions. Work was supportedin part by the Army Research Office (W911NF-14-1-0078); the Office of Naval Research (N00014-13-1-0150);NSERC; the Office of the Director of National Intelli-gence (ODNI), Intelligence Advanced Research ProjectsActivity (IARPA), via MIT Lincoln Laboratory underAir Force Contract FA8721-05-C-0002; and a Multidisci-plinary University Research Initiative from the Air ForceOffice of Scientific Research MURI grant no. FA9550-12-1-0488. The views and conclusions contained herein arethose of the authors and should not be interpreted as nec-essarily representing the official policies or endorsements,either expressed or implied, of ODNI, IARPA, or theU.S. government. The U.S. government is authorized toreproduce and distribute reprints for governmental pur-pose notwithstanding any copyright annotation thereon.A.W.E. acknowledges support from the Department ofDefense through the NDSEG fellowship program.

Appendix A: Experimental Methods

1. Qubit-cavity implementation

The qubit was composed of an aluminum transmoncircuit fabricated on double side polished silicon charac-terized by EC/h = 200 MHz and EJ/h = 33.1 GHz. Thetransmon circuit was coupled to a 3D aluminum cavity

with resonance frequency ωc/2π = 7.1051 GHz at rateg/2π = 202 MHz. We calculate the frequency of thebare qubit to be 7.091 GHz. To bring the cavity close toresonance with the qubit, we placed extra silicon chips inthe cavity as a low-loss dielectric based on finite-elementsimulation. Most of the final detuning of 14 MHz wasdue to aging of the qubit between experimental cycles.

As mentioned in the main text, coupling to unsqueezedvacuum modes dilutes the squeezing in the artificialatom’s environment from that which is produced atthe JPA output. We probe the cavity contributionto this dilution by measuring the |1,+〉 polariton reso-nance in reflection through the strongly coupled cavityport. Through this port, we measure external and in-ternal quality factors of Q1+,ext = 26, 500 and Q1+,int =110, 000, the latter characterizing the total loss rate dueto the extra silicon substrates and the weak coupling ofthe cavity’s second port. This coupling to unsqueezedmodes limits the achievable squeezing, corresponding toηc = Q1+,int/(Q1+,ext + Q1+,int) = 0.81. Given thatwe experimentally observe η = ηlossηc = 0.55, we in-fer ηloss = 0.68. This number is in reasonable agree-ment with expectations for the combined effect of internalparamp loss and component loss in the circulator, hybrid,and cables. In future experiments, ηc could be optimizedby reducing the coupling of the weakly coupled port, withηloss limiting the minimum achievable linewidth.

Approximating our multilevel qubit-cavity system as atwo-level state requires the use of squeezed light with suf-ficiently narrow bandwidth to avoid exciting the system’shigher levels. For a resonant Jaynes-Cummings Hamil-tonian this imposes the restriction that κJPA < 0.6g [21].However, it is important to account for the fact that thehigher levels of the transmon qubit can perturb the fre-quencies of the dressed states. We directly characterizedthe relevant transition frequencies using two-tone spec-troscopy [30]. When resonantly driving the |0, g〉 to |1,+〉transition with a coherent tone we find that the closesttransition is the |1,+〉 to |2,+〉 transition at 7.262 GHz,corresponding to a detuning of -38 MHz. The JPA wasdesigned to have a single-side bandwidth smaller thanthis detuning.

To confirm that our qubit-cavity system was well-thermalized to the 30 mK base stage of our dilutionrefrigerator and that thermal photons were not con-taminating the vacuum environment, we measured theexcited-state population of the polariton states using aRabi population measurement [31, 32]. These measure-ments indicate the excited state population was less than1.2%, corresponding to Nth < 0.01, thus negligibly im-pacting the squeezing levels quoted in the main text.

2. Josephson parametric amplifier

The JPA used in this work was designed to providea high degree of squeezing despite operating over a nar-row bandwidth. The JPA’s squeezing performance is ex-

6

pected to inversely correlate with the dimensionless non-linearity Λ/κJPA [33], where Λ is the Kerr nonlinearityand κJPA is the resonator bandwidth. Here κJPA was in-tentionally kept small (κJPA,external/2π = 21 MHz whenωJPA = ω1,+) through the use of input coupling capac-itors to avoid exciting higher transitions in the qubit-cavity system. To produce a weak nonlinearity, Λ wasreduced through the incorporation of geometric induc-tance, which reduces the Josephson junction participa-tion ratio (p = LJ/Ltotal), and through the use of a twoSQUID array. The former reduces Λ by p3, whereas thelatter reduces Λ by 1/N2

SQUID for fixed LJ, where NSQUID

is the number of SQUIDs. With these modifications, weestimate Λ/κJPA ∼ 5 × 10−5 at the polariton frequency.For comparison, Λ/κJPA ∼ 6×10−3 for an ideal lumped-element JPA consisting of a capacitance shunted by asingle SQUID with a 100 Ohm input impedance (set bythe 180 hybrid launch). In addition, the amplifier waspumped for gain by modulating the flux through the twoSQUID loops at twice the resonator frequency. Relativeto alternative methods such as double-pumping [15], flux-pumping allowed us to minimize the number of passivecomponents between the JPA and the artificial atom. Allof the JPA circuitry, including the flux pump input andthe hybrid, was shielded by an aluminum box and cryop-erm at the base stage of the dilution refrigerator.

For each JPA pump amplitude, the JPA power gain(GJPA) was characterized using a vector network ana-lyzer. These GJPA values refer to phase preserving gainof the amplifier, as the gain was characterized for a sig-nal slightly detuned such that the measurement band-width was small compared to the detuning. For an idealsqueezed state, GJPA is related to the power gain in theamplified and squeezed quadratures by the expression

2

(N ±M +

1

2

)=(√

GJPA ±√GJPA − 1

)2

. (A1)

3. Data acquisition and analysis

Fig. 5 displays a full schematic of the experimentalsetup. The polariton drive and the JPA flux pumpwere phase locked by generating them from the same mi-crowave source and doubling the flux pump frequency.This allowed the relative phase between the two tonesto be controllably varied by adjusting the bias voltageson an IQ mixer. All spectra were directly acquired us-ing a microwave spectrum analyzer. As our analysis fo-cuses on the incoherent portion of the fluorescence spec-trum, we kept the spectrum analyzer resolution band-width much smaller than any of the relevant spectrallinewidths such that the coherent portion of the spec-trum could be omitted by dropping a small number ofpoints around the polariton frequency. In addition, eachspectrum was normalized by an equivalent measurementwith both the JPA pump and coherent drive off in or-der to eliminate background trends originating from the

gain ripple of the JTWPA [22] and following amplifiers.We note that, as we are working in the photon blockaderegime, we expect the fluorescence intensity in the Mol-low triplet measurements (not including the backgroundsqueezed noise power) to be approximately hωγ = −140dBm. For all measurements the JTWPA was pumped toprovide roughly 20 dB of power gain.

After normalizing the data, we determined the squeez-ing levels using the analytic fit expressions outlined inthe main text. Here we describe additional details of thefitting procedures. For Fig. 2(a), the squeezing level wasinferred by fitting the data to Eq. 2 from the main textincluding an overall scaling and offset. In addition, the fitincludes a parabolic background term to account for thefrequency-dependence of the squeezed noise power pro-duced by the JPA. Although small in Fig. 2(a), this back-ground becomes important when analyzing such spectraat large gains as is done to determine the squeezing levelspresented in Fig. 4. To determine the total linewidth (γ)used to constrain the inputs to Eq. 2 in the main text, wefit the Mollow triplet spectra in ordinary vacuum usingthe methodology described in Appendix B to account fordispersive spectral features and the weakly coupled cav-ity port. In Fig. 3, to illustrate the phase dependence ofthe Mollow triplet, the spectra were approximated as thesum of three Lorentzians and a parabolic background.Approximating the peaks neglects the aforementioneddispersive features. Therefore, to more rigorously charac-terize the squeezing level through this measurement, weperform nonlinear regression using the analytic expres-sions in Appendix B. In Fig. 4 we focus on characteriz-ing M −N by measuring spectra at four relative phasesbetween the coherent drive and squeezed vacuum fieldsnear the phase that minimizes the center peak linewidth.For each gain point, the four spectra were fit with jointparameters; only background and scaling terms were al-lowed to vary among the four traces to compensate forsmall experimental drift. As in Fig. 2, for these fits γ wasindependently determined by fitting the Mollow tripletspectrum in ordinary vacuum.

Appendix B: Numerical modeling of fluorescencespectra

In the papers of Carmichael, Lane and Walls [7, 8],an expression for the resonance fluorescence spectrum ofa two-level atom driven by broadband squeezed light isgiven for the limiting cases 2Φ = 0, π, where Φ is therelative phase between the Rabi drive and the squeez-ing angle. Note that here, as depicted in Fig. 6, anangular factor of two has been introduced compared toRefs. [7, 8] to simplify the phase space picture. In or-der to analyze the experimental data for an arbitraryphase Φ, we generalize these analytical expressions. InSec. B 1, we define the notation used and give the opticalBloch equations. In Sec. B 2, we obtain the resonancefluorescence spectrum using the Bloch equations and the

7

BP

F

JPA(see fig. 1)

2x

7.3003 GHz(f1,+)

-5dB

I Q

LPF

-20d

B7.0217 GHz

-30d

B-2

0dB

LPF2

LPF

-20d

BLP

F2-1

0dBJTWPA

LPF

HEMT(4K)

JPA

pum

p am

plitu

de c

ontro

l

BP

F-1

3dB

-10d

BB

PF

V

-20d

B

-30d

B-6

dB

3D-transmon

(absorbtive)

BPF 11-18 GHzbandpass filter

2x amplifyingdoubler

LPF 12 GHz lowpass filter

switch

LPF2 lossy filter

isolator,various bands

amplifier

spectrumanalyzer to computer

voltage variableattenuatorV

-20 dBdirectionalcoupler

I Q IQ mixer

splitter/combiner

circulator

room temperature

< 30 mK

(100mK,4K) (100mK,4K) (4K) (4K)

cryoperm, Al

cryoperm, Cu

VNAinout

13

2

cryoperm, Cu

FIG. 5. Experimental setup diagram. The JPA pump tone, which enters the cryostat at (1), is generated through a frequencydoubler followed by a voltage variable attenuator that allows programmatic control of the JPA gain. The gain can be measuredby the VNA at (2). Power from the same generator is split off before doubling to create the phase-locked coherent drive at(3). The relative phase of this drive is controlled through DC voltages on the IQ mixer. Each time the phase is stepped, thetwo switches just before the spectrum analyzer are thrown such that the analyzer samples the drive power, which is fed backon to ensure power flatness versus phase. These switches are toggled back before measurement of fluorescence spectra on thespectrum analyzer.

8

quantum regression theorem. Finally, in Sec. B 3, we dis-cuss the calculation of the reflection spectrum and thebroadband squeezing approximation.

1. Definitions and steady-state solutions of theoptical Bloch equations

When an atom is strongly coupled to a cavity withdipole coupling g, the eigenstates of the system are thedressed-states called polaritons formed by a superposi-tion of the bare atom, cavity states: |+, n〉 = [|g, n〉 +

|e, n− 1〉]/√

2, |−, n〉 = [|g, n〉 − |e, n− 1〉]/√

2, with vac-uum Rabi splitting 2g

√n. It has been shown that for a

strong g each vacuum-Rabi resonance behaves as a two-level system [34]. As described in the main text, thesystem is driven at the upper polariton |+, 1〉 = [|g, 1〉+

|e, 0〉]/√

2 frequency. As long as no other dressed-statesare excited, the ground state |g, 0〉 and |+, 1〉 behave asan effective two-level system. The next closest transitionfrequency corresponds to the |+, 1〉 to |+, 2〉 transition,

with ω+,2 − ω+,1 = (2 −√

2)g ∼ 0.6g. As a result, thetwo-level approximation holds for κJPA,Ω 0.6g, whereκJPA is the linewidth of the JPA acting as the source ofsqueezing and Ω the coherent drive amplitude.

Working in this limit, we only consider the two-levelsystem |g, 0〉, |+, 1〉, driven in resonance in a broadbandsqueezed vacuum environment, κJPA Ω, γ with γ thelinewidth of the two-level system. With this approxima-tion and working in the rotating frame of the drive, themaster equation describing the two-port system is [35]

ρ =iΩ

2[σ+ + σ−, ρ] + γintD[σ−]ρ

+ γext(N + 1)D[σ−]ρ+ γextND[σ+]ρ

− γextMS[σ+]ρ− γextM∗S[σ−]ρ,

(B1)

where N and M are the second-order moments of the in-coming squeezed field, D[A]ρ = AρA†− 1

2 (A†Aρ+ρA†A)

and S[A]ρ = AρA− 12 (A2ρ+ ρA2) are superoperators.

In order to simplify the notation, we use the quantumefficiency ηc = γext/(γint + γext) to rewrite the masterequation as a two-level system coupled to a single port

ρ =iΩ

2[σ+ + σ−, ρ] + γ(N + 1)D[σ−]ρ

+ γND[σ+]ρ− γMS[σ+]ρ− γM∗S[σ−]ρ,(B2)

with effective parameters N = ηcN and M = ηcM suchthat γextN = γN .

The optical Bloch equations in the presence of squeez-ing for the effective single port system can be obtainedfrom Eq. (B2), leading to

〈σx〉 = −γ+〈σx〉+ γM 〈σy〉〈σy〉 = γM 〈σx〉 − γ−〈σy〉 − Ω〈σz〉〈σz〉 = Ω〈σy〉 − γN 〈σz〉 − γ,

(B3)

where σx,y,z are the standard Pauli matrices, γ± =γ(N ± |M | cos 2Φ + 1

2

), γM = γ |M | sin 2Φ and γN =

γ (2N + 1). The steady-state solutions of these equationsare

〈σx〉ss =ΩγγM

γNγ2NM + Ω2γ+

,

〈σy〉ss =Ωγγ+

γNγ2NM + Ω2γ+

,

〈σz〉ss = − γγ2NM

γNγ2NM + Ω2γ+

,

(B4)

with the rate γ2

NM = γ2[(N + 1

2

)2 −M2].

2. Resonance fluorescence of a driven two-levelsystem in a squeezed environment

The resonance fluorescence spectrum is obtained fromthe Fourier transform of a time-domain correlation func-tion [35]

S(ω) =1

πRe

∫ ∞0

dt 〈σ+(t)σ−(0)〉ss eiωt. (B5)

Noting that the correlation function of interest can beexpressed as [8]

〈σ+(t)σ−(0)〉ss = [〈σx(t)σx(0)〉ss + 〈σy(t)σy(0)〉ss+ i 〈σy(t)σx(0)〉ss − i 〈σx(t)σy(0)〉ss], (B6)

we use Eq. (B3) and the quantum regression formula [35]to obtain the linear system of equations,

∂t

〈σx(t)σx,y(0)〉ss〈σy(t)σx,y(0)〉ss〈σz(t)σx,y(0)〉ss

=

B

〈σx(t)σx,y(0)〉ss〈σy(t)σx,y(0)〉ss〈σz(t)σx,y(0)〉ss

− γ 0

0〈σx,y〉ss

, (B7)

with the matrix

B =

−γ+ γM 0γM −γ− −Ω0 Ω −γN

. (B8)

The results of Ref. [7, 8] are obtained for γM = 0. Here,we generalize these results.

In order to solve this set of linear equations for ar-bitrary time t, we make use of the Laplace transform:Lf(t) =

∫∞0

e−stf(t) dt. Performing this transforma-tion on Eq. (B7) we obtain

L

〈σx(t)σx,y(0)〉ss〈σy(t)σx,y(0)〉ss〈σz(t)σx,y(0)〉ss

=

(s1−B)−1

〈σxσx,y〉ss〈σyσx,y〉ss

〈σzσx,y〉ss −γs 〈σx,y〉ss

, (B9)

9

Φ

X1

X2

φc

X1

X2

φV(φ)

V(φ+π/2)

(a) (b) (c)

X1

X2

1/2

FIG. 6. States of light can be fully described by quasiprobability distributions (here Wigner functions) in phase space, whereX1 and X2 are the two quadratures of the light field. a, The vacuum state is described by a circular Gaussian distribution ofwidth 1/2 centered at the origin. The red disk indicates the area in which the distribution is greater than its half-maximumvalue. b, The distribution for a squeezed vacuum state has an increased width along the amplification axis at angle ϕ and adecreased width along the squeezing axis at angle ϕ+ π/2. A slice of this distribution taken through the origin at angle θ hasa variance given by Eq. 1 of the main text, with the amplified variance given by 1/2(N +M + 1/2) and the squeezed variancegiven by 1/2(N −M +1/2). Note an angular factor of 2 has been introduced in Eq. 1 in the main text compared to Refs. [7, 8]to simplify the phase space picture. c, Similarly, Φ, as used in Fig. 3 of the main text, indicates the angle of the amplificationaxis of a displaced squeezed state relative to the displacement angle, ϕc. Experimentally, ϕc is set by the phase of the coherentRabi drive.

with the relations

(s1−B)−1

=1

D(s)

(s+ γ−) (s+ γN ) + Ω2 γM (s+ γN ) −γMΩγM (s+ γN ) (s+ γN ) (s+ γ+) − (s+ γ+) Ω

γMΩ (s+ γ+) Ω (s+ γ−) (s+ γ+)− γ2M

, (B10)

D(s) = (s+ γN )[(s+ γ−) (s+ γ+)− γ2

M

]+ (s+ γ+) Ω2. (B11)

In order to obtain the time-domain solution, an inverseLaplace transform must be applied, which requires theroots λ0,1,2 of the cubic polynomial D(s). In the limitingcases 2Φ = 0, π, the rate γM is zero and the roots ofD(s) are easily obtained as [7, 8]

λ0 = −γ+,

λ1,2 = −γ− + γN2

± 1

2

√(γ− − γN )

2 − 4Ω2.(B12)

However, for an arbitary phase Φ, finding analytical solu-tions requires the cubic roots formula. As these analyti-cal expressions are rather cumbersome, but easily found,they are not reproduced here. With these expressions,D(s) can be expressed as

D(s) = (s− λ0) (s− λ1) (s− λ2) . (B13)

Using this factorization, the inverse Laplace transform

reads

fn(t) = L−1

sn

D(s)

=

C0λn0 eλ0t + C1λ

n1 eλ1t + C2λ

n2 eλ2t − δn,−1

λ0λ1λ2(B14)

with n ∈ −1, 0, 1, 2 and where we have defined thecoefficients

C0 =1

(λ0 − λ1) (λ0 − λ2),

C1 =1

(λ1 − λ0) (λ1 − λ2),

C2 =1

(λ2 − λ0) (λ2 − λ1).

(B15)

10

Hence time-domain solutions are〈σx(t)σx,y(0)〉ss〈σy(t)σx,y(0)〉ss〈σz(t)σx,y(0)〉ss

= W

〈σxσx,y〉ss〈σyσx,y〉ss〈σzσx,y〉ss

− γ 〈σx,y〉ss v,(B16)

with the matrix W = L−1

(s1−B)−1

, and vector v

given by

W =

f2 + (γ− + γN ) f1 +(γ−γN + Ω2

)f0 γMf1 + γNγMf0 −γMΩf0

γMf1 + γNγMf0 f2 + (γN + γ+) f1 + γNγ+f0 −Ωf1 − Ωγ+f0

γMΩf0 Ωf1 + Ωγ+f0 f2 + γNf1 + γ2NMf0

, (B17)

v = L−1

1

s(s1−B)

−1

001

=

−γMΩf−1

−Ωf0 − Ωγ+f−1

f1 + γNf0 + γ2NMf−1

. (B18)

Using properties of the Pauli matrices and Eq. (B16), wecan write the correlation function of Eq. (B6) as

〈σ+(t)σ−(0)〉ss = K0eλ0t+K1eλ1t+K2eλ2t+K, (B19)

with

K = −Ωγ (γM + iγ+)

λ0λ1λ2〈σ−〉ss ,

Kj = Cj[

2λ2j + γN (3λj + γN ) + Ω2

](1 + 〈σz〉ss)

+ Ω [γM + i (γ+ + λj)]

(1 +

γ

λj

)〈σ−〉ss

.

(B20)

Finally, putting all of these results together, we obtainfrom Eq. (B5) the resonance fluorescence spectrum

S(ω) =1

π

2∑j=0

−[KRj λ

Rj +KI

j

(ω + λIj

)](λRj)2

+(ω + λIj

)2

+Kδ(ω),

(B21)where we have defined real and imaginary parts such thatλj = λRj + iλIj , and Kj = KR

j + iKIj . Neglecting the

last term, we find that the resonance fluorescence spec-trum is the sum of three Lorentzians, each multipliedby a function containing corrections that are linear infrequency. Figure 7 displays experimental spectra ex-hibiting such dispersive features. As we show below, seeEq. (B26), in the large Rabi drive limit, the spectrum ispurely Lorentzian (Im Ki ∼ 0).

3. Reflection spectrum

The results of the previous section assume that theabove spectrum is directly measured by looking at the

fluorescence of the qubit, i.e. ignoring the squeezed bath.However, in the experiment described here, the measure-ment is performed in reflection implying that the signalfluorescence is combined with radiation from the squeezerreflected by the cavity. Following Ref. [6], with u(t) = 0for t < 0 and u(t) = 1 for t > 0, the full reflection spec-trum from the strongly coupled port (coupling rate γext)is related to the time-domain correlation function,

⟨b†out(t)bout(0)

⟩= Nδ(t)

+ γextu(t)⟨[σ+(t), Nσ−(0)− Mσ+(0)

]⟩+ γextu(−t)

⟨[Nσ+(t)− Mσ−(t), σ−(0)

]⟩+ γext 〈σ+(t)σ−(0)〉 ,

(B22)

where M , N characterize the field in the transmissionline. Hence the reflection spectrum is given by

SR(ω) =1

π

∫ ∞0

⟨b†out(t)bout(0)

⟩dt

=N

2π+ γextS(ω)

+γext

πRe

∫ ∞0

dt eiωt⟨[σ+(t), Nσ−(0)− Mσ+(0)

]⟩,

(B23)

where S(ω) is the resonance fluorescence spectrum de-fined in Eq. (B21). Following the standard conventions,the output field operator bout has units of root frequencyand thus the reflection spectrum SR(ω) is unitless whileS(ω) has units of inverse frequency. Eq. (B23) assumesM real and a single decay channel γ. In the more gen-eral case where M = |M | e2iΦ, and rewriting the equation

11

−4 −3 −2 −1 0 1 2 3 4Frequency (MHz)

1.4

1.5

1.6

1.7

1.8

1.9

−3 −2 −1 0 1 2 3Frequency (MHz)

−4 −3 −2 −1 0 1 2 3 4Frequency (MHz)

S(ω

)Ф = 1.35 Ф = 1.58 Ф = 1.81(a) (b) (c)

FIG. 7. Mollow triplet spectra measured at GJPA = 6.6 dB. The spectra are measured near Φ = π/2, the relative phasebetween the squeezed vacuum and coherent drive fields that minimizes the center peak linewidth. Even though the Rabifrequency (1.2 MHz as in Fig. 3) is small compared to the measured frequency range, the sidebands are broadened such thatthey are no longer apparent. The dashed black line represents the best fit to the power spectra, taking into account both thefluorescence spectrum and the frequency-dependent squeezer background. The spectra are simultaneously fit with γ fixed bymeasurements of the Mollow triplet in ordinary vacuum and also constrained by the relative phase difference between eachmeasurement. From these measurements, we infer M −N = 0.24 for these conditions.

in terms of the total damping rate γ = γint + γext and effective field moments M and N,

SR(ω) =N

2πηc+ γ (N + ηc)S(ω)

+γ

πRe

∫ ∞0

dt eiωt[|M | e2iΦ 〈σ+(0)σ+(t)〉 − |M | e2iΦ 〈σ+(t)σ+(0)〉 −N 〈σ−(0)σ+(t)〉

].

(B24)

It is important to recall that the above expression isobtained using the broadband squeezing approximation.Since the two-level atom probes the field only in a smallfrequency range, this approximation is valid for the res-onance fluorescence spectrum. The squeezer backgroundis, however, frequency dependent. We take this into ac-count in analyzing the experimental data by replacingthe first term of Eq. (B24) by the frequency dependentspectrum N(ω) for the output field of a parametric am-

plifier [36]. With this modification, the analytical expres-sion is found to be in excellent agreement with cascadedmaster equation simulations (not shown).

In order to relate Eq. (B24) to the results of the maintext, we now examine useful limiting cases. First, weconsider the reflection spectrum in the absence of thecoherent Rabi drive (Ω = 0). In this case, we obtain(Eq. (2) of the main text)

SR(ω) =N

2πηc+

1

2π

γ

2N + 1

[γy(M − (1− ηc)N)

ω2 + γ2y

− γx(M + (1− ηc)N)

ω2 + γ2x

], (B25)

where, γx,y = γ(N ±M + 1/2). From this equation wesee that for a lossless single-sided cavity ηc = 1, a fluo-rescence peak is observed in the reflection spectrum onlyif M 6= 0. The spectrum is a sum of narrow and broadlinewidth Lorentzians. However for ηc 6= 1, some of theincident intensity is lost which enhances the relative spec-tral weight of the negative dip. Finally, in the large Rabidrive limit Ω γ, γ+, γ−, γN , we obtain

SR(ω) =N

2πηc+ηc2π

γγ+

ω2 + γ+2

+1

8π

γ(γN + γ−)

(ω − Ω)2 + (γN/2 + γ−/2)2

+1

8π

γ(γN + γ−)

(ω + Ω)2 + (γN/2 + γ−/2)2

(B26)

12

This equation indicates that for a strong Rabi drive thereflection spectrum is a sum of three Lorentzians centeredat 0,±Ω and of full widths 2γ+ = γ(2N +2M cos 2Φ+1)

and γN+γ− = γ(3N−M cos 2Φ+3/2). This explains thecosine dependence with opposite phases of the linewidthsas plotted in Fig. 3 of the main text.

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