Superharmonic Resonances in a Strongly Coupled Cavity-Atom System

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by  Eyal Buks

Superharmonic Resonances in a Strongly Coupled Cavity-Atom System
Eyal Buks,1 Chunqing Deng,2,3,4 Jean-Luc F.X. Orgazzi,2,5,4 Martin Otto,2,3,4 and Adrian Lupascu2,3,4
1Department of Electrical Engineering, Technion, Haifa 32000 Israel
2Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada N2L 3G1
3Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, Canada N2L 3G1
4Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, ON, Canada N2L 3G1
5Department of Electrical and Computer Engineering,
University of Waterloo, Waterloo, ON, Canada N2L 3G1
(Dated: February 1, 2016)
We study a system consisting of a superconducting flux qubit strongly coupled to a microwave
cavity. Thefundamentalcavitymodeisexternallydrivenandtheresponseisinvestigatedintheweak
6
nonlinear regime. We find that near the crossing point, at which the resonance frequencies of the
1
0 cavitymodeandqubitcoincide,thesignoftheKerrcoefficientchanges,andconsequentlythetypeof
2 nonlinear response changes from softening to hardening. Furthermore, thecavity response exhibits
superharmonic resonances when theratio between thequbit frequency and thecavity fundamental
n
mode frequency is tuned close to an integer value. The nonlinear response is characterized by
a
the method of intermodulation and both signal and idler gains are measured. The experimental
J
results are compared with theoretical predictions and good qualitative agreement is obtained. The
9
superharmonic resonances havepotential for applications in quantumamplification and generation
2
of entangled states of light.

]
h PACSnumbers: 85.25.Cp,03.65.Ge,42.50.Pq
p
-
t Cavity quantum electrodynamics (CQED) [1] is the
n
study of the interaction between photons confined in a
a
u cavity and atoms (natural or artificial). The interaction
q iscommonlydescribedbythe RabiorJaynes-Cummings
[ Hamiltonians[2],andithasbeenthesubjectofnumerous

1 theoretical and experimental investigations. An on-chip
v CQEDsystemcanberealizedbyintegratingaJosephson
9 qubit [3–5] (playing the role of an artificial atom) with
8 a superconducting microwave resonator (cavity) [6–8].
9 Superconducting CQED systems have generated a fast
7
growinginterestduetothepossibilitytoreachthestrong
0
[7] and ultra-strong [9, 10] coupling regimes, and due to
.
1 potentialapplicationsinquantuminformationprocessing
0
[4, 11–13].
6
1 In this study we investigate the driven dynamics of a
: stronglyinteractingsystemcomposedofasuperconduct-
v
i ing flux qubit [15, 16] and a coplanar waveguide (CPW)
X microwave cavity [9, 14, 17–21]. The nonlinear cavity FIG. 1: (color online) The CQED device. (a) Optical image
r response [22–36] is measured as a function of the mag- of a device used in the current experiment, which is nomi-
a nallyidenticaltotheonedescribedin[14],wheretheoverlaid
netic flux that is applied to the qubit. At weak driv-
dashed rectangle indicates the position of the qubit under
ing and when the ratio between the qubit frequency and
study. The resonator, which occupies the central part of the
the cavityfundamental mode frequency is tuned close to
chip, is coupled to the input/output ports using capacitors.
the value ω /ω =1 the common Jaynes-Cummings res-
a c (b) Electron micrograph showing a qubit embedded in the
onance, which henceforth is referred to as the primary
coplanar waveguide resonator and its local flux control line.
resonance, is observed. With stronger driving, however, (c) Electron micrograph of a fluxqubit.
and when the ratio ω /ω is tuned close to integer val-
a c
ues larger than unity, superharmonic resonances appear
in the measured response. Intermodulation (IMD) mea-
surementsareemployedto characterizethe nonlinearre-
coupled to two ports that are used for performing mi-
sponse [37–39]. The results are compared with the pre-
crowavetransmissionmeasurements[seeFig. 1(a)]. Two
dictions of a theoretical model, which is based on lin-
persistent current flux qubits [15], consisting of a super-
earizationoftheequationsofmotionthatgovernthe dy-
conducting loopinterruptedby four Josephsonjunctions
namics of the CQED system under study.
[see Fig. 1(c)], are inductively coupled to the CPW res-
TheinvestigateddevicecontainsaCPWcavityweakly onator [see Fig. 1(b)]. In the current experiment, how-
2
ever,only one qubit significantly affects the cavity mode which is given by
response, whereas the other one is made effectively de-
coupled by detuning its energy gap away from the mode Υc =ωc−iγc+(Kc−iγc4)Ec , (2)
frequency. A CPW line terminated by a low inductance
where E is the averaged number of photons occupying
shuntisusedtosendmicrowavepulsesforcoherentqubit c
the cavity mode. The imaginary part of Υ represents
control[seeFig. 1(b)]. Thedeviceisfabricatedonahigh c
the effect of damping and the terms proportional to E
resistivitysiliconsubstrate,in a two-stepprocess. Inthe c
determinetheweaknonlinearresponse. Theeffectofthe
first step, the resonator and the controllines are defined
flux qubit on the cavity response in the weak nonlinear
using optical lithography, evaporation of a 190nm thick
regime is theoretically evaluated in sections I and II of
aluminum layer and liftoff. In the second step, a bilayer
the SM for the case where g/|ω −ω | ≪ 1. The cou-
resist is patterned by electron-beam lithography. Sub- c a
pling between the cavity mode and the qubit gives rise
sequently, shadow evaporation of two aluminum layers,
to a resonance splitting. The steady state cavity mode
40nmand65nmthickrespectively,followedbyliftoffde-
response for the case where the qubit mainly occupies
fine the qubit junctions.
thestate|±i(groundandfirstexcitedstates)isfoundto
The chip is enclosed inside a copper package, which
be equivalent to the response of a mode having effective
is cooled by a dilution refrigerator to a temperature of
complex cavity angular resonance frequency Υ given
eff
T = 23mK. Both passive and active shielding methods
by
are employed to suppress magnetic field noise. While
passive shielding is performed using a three-layer high Υ =Υ ±ω ±Υ , (3)
eff c BS ba
permeability metal, an active magnetic field compensa-
tionsystemplacedoutsidethecryostatisusedtoactively where ωBS =g12/(ωc+ωa) is the Bloch-Siegertshift [10]
reduce low-frequency magnetic field noise. A set of su- (see SM section II). The term Υba is given by (see SM
perconducting coils is used to apply DC magnetic flux. section I.I)
Qubitstatecontrolisperformedusingshapedmicrowave
pulses. Attenuators and filters are installed at different Υ =−g12 1− ∆1iT2 , (4)
coolingstagesalongthe transmissionlines for qubitcon- ba ∆11+ 1 + 4g12T1Ec
trol and readout. A detailed description of sample fabri- ∆21T22 ∆21T2
cation and experimental setup can be found in [14, 18]. g =g/β is the flux dependent effective coupling coeffi-
1 f
The main theoretical results needed for analyzing the cient, where the coefficient β is given by
f
experimentalfindingsarebrieflydescribedbelow[deriva-
tions are given in the supplemental material (SM)]. ω 2
f
The circulating current states of the qubit are la- βf = 1+ , (5)
beled as |xi and |yi. The coupling between the s (cid:18)ω∆(cid:19)
cavity mode and the qubit is described by the term ∆ = ω −ω is the detuning between the angular fre-
1 p a
−g A+A† (|xihx|−|yihy|) in the system Hamil- quency of the externally injected pump tone ωp and the
tonian, where A (A ) is a cavity mode annihilation (cre- qubit angular resonance frequency ω , and T (T ) is
(cid:0) (cid:1) † a 1 2
ation) operator, and g is the coupling coefficient. In the thequbitlongitudinal(transverse)relaxationtime. Note
presence of an externally applied magnetic flux, the en- thatwhen∆ T ≫1andwhenthequbitmainlyoccupies
1 2
ergy gap ~ωa between the qubit ground state |−i and thestate|±ithetermΥbagivesrisetoashiftinthemode
first excited state |+i is taken to be given by ~ωa = angular frequency approximately given by ∓g12/∆1 and
~ ω2+ω2 (see SM section I.A), where a shift in the value of the Kerr coefficient approximately
f ∆
given by ± g4/∆3 (4T /T ). Similar theoretical results
1 1 1 2
p
have been obtained in Ref. [28], in which the unitary
2I Φ Φ 1
cc 0 e (cid:0) (cid:1)
ωf = ~ Φ − 2 , (1) transformation that diagonalizes the Hamiltonian of the
(cid:18) 0 (cid:19) closed system has been applied to the system’s master
equation.
Icc (−Icc) is the circulating current associated with the The effect of the qubit on cavity response is experi-
state |yi (|xi), Φ0 = h/2e is the flux quantum, Φe is mentally investigated using transmission measurements.
theexternallyappliedmagneticfluxand~ω∆isthequbit The color coded plots in Fig. 2 exhibit the measured
energy gap for the case where Φe/Φ0 =1/2. (panels a and c) and calculated (panels b and d) cav-
The decoupled cavity mode is characterizedby an an- ity transmission (in dB units) vs. ω /2π, for the case
f
gular resonance frequency ω , Kerr coefficient K , lin- where the power injected into the cavity is −127 dBm
c c
eardamping rate γ andcubic damping (two-photonab- (panels a and b) and −112 dBm (panels c and d). In
c
sorption) rate γ . The response of the decoupled cavity the first step of the theoretical calculation, which has
c4
in the weak nonlinear regime (in which, nonlinearity is generated the theoretical predictions plotted in panels b
takenintoaccounttolowestnon-vanishingorder)canbe and d, fixed points are found by calculating steady state
described by introducing the complex and mode ampli- solutions of the equations of motion that govern the dy-
tude dependent cavity angular resonance frequency Υ , namics of the system (see SM section I.I). Then in the
c
3
FIG. 2: (color online) The measured (panels a and c) and calculated (panels b and d) cavity transmission (in dB units) vs.
ωf/2π. Forthepanelsontheleft(panelsaandb)thepowerinjectedintothecavityis−127dBm,whereasforthepanelsonthe
right (panels c and d) the power is −112 dBm. The following parameters have been assumed in the calculation: T =23mK,
ωc/2π = 6.6408GHz, ω∆/2π = 1.12GHz, g/2π = 0.274GHz, γc1/ωc = 5×10−6 and γc2 = 1.1×γc1. The relaxation time
T1 = 1.2µs(1+0.45ns×|ωf|) is obtained from energy relaxation measurements (in the range of qubit frequencies well below
ωc/2π),andtherateT2−1 =4.5MHz(1+44|ωf|/ωa)isobtainedfromRamseyratemeasurements[14]. Inpanelathemeasured
off-resonancetransmission issignificantlyhigherthanthecalculatedone(seepanelb)duetoinstrumentalnoise,whichhasnot
been taken into account in the theoretical modeling. In the region where ∆ = ωc−ωa > 0 two peaks are seen in the cavity
transmission, theupperonecorrespondstothecase wherethequbitmainly occupies thegroundstate, whereas thelower one,
which is weaker, corresponds to thecase wherethequbitmainly occupies thefirst excited state. Thelower peak is less visible
in the data seen in panel a, which was obtained at lower input power, due to reduced signal to noise ratio. In panels c and d
theprimary (labeled by ±1) and superharmonic (labeled by ±2, ±3 and 4) resonances are indicated byarrows.
secondstepinput-outputrelationsareemployedinorder behavioris obtainedwhenthe qubitmainly occupies the
to calculate the cavity transmission(see SM section I.E) first excited state. The probability for this to happen,
[40]. The assumed device parameters are listed in the which depends on the ratio between thermal energy and
caption of Fig. 2. qubit energy gap,is non-negligible in the currentexperi-
ment. The comparisonbetween the experimental results
WhilethecavityresponseseeninpanelsaandbofFig.
(panels a and c) and the theoretical predictions (panels
2 is nearly linear, nonlinearity is observed in the results
b and d, respectively) yields an acceptable agreement.
depicted in panels c and d, which are obtained at higher
input power. The measured response exhibits hardening It is well known that the flux qubit is expected to
(softening) when ∆ < 0 (∆ > 0), for the case where strongly affect the response of the cavity mode near the
1 1
thequbitmainlyoccupiesitsgroundstate. Theopposite primary resonance, i.e. when the ratio ω /ω is tuned
a c
4
Ingeneral,nonlinearcavity responseis commonlyem-
ployed for frequency mixing, which in turn can be used
for signal amplification [37, 38, 41–43] and noise squeez-
ing [39, 41]. An amplifier based on flux qubits has been
recently demonstrated in Ref. [44]. Here we employ the
methodofIMDtocharacterizefrequencymixing. Inthis
method, two monochromatictones are combinedand in-
jected into the cavity: an intense pump tone at angu-
lar frequency ω and amplitude bin, and a weaker signal
p c1
tone at angular frequency ω = ω +ω and amplitude
s p
cin. The cavity transmission is measured and the spec-
c1
tralamplitude ofthe output signaltone atfrequency ω ,
s
which is labeled by cout(ω), and the spectral amplitude
c2
oftheso-calledidlertoneatfrequency2ω −ω =ω −ω,
p s p
which is labeled by cout(−ω), are recorded. The corre-
c2
sponding signal gain G = cout(ω)/cin 2 and idler gain
s c2 c1
G = cout(−ω)/cin 2 are determined, and the experi-
i c2 c1 (cid:12) (cid:12)
mental findings are compar(cid:12)ed with the(cid:12)theoretical pre-
(cid:12) (cid:12)
diction(cid:12)s, which are b(cid:12)ased on the linearized equations of
motion of the system (see SM section I.L).
The results are exhibited in Fig. 3, in which the cav-
FIG. 3: (color online) Cavity transmission (panels on the
ity transmission S (left panels) and the signal G and
left) and IMD gain (panels on the right). Experimental data 21 s
idler G gains (right panels) are plotted vs. pump fre-
points are labeled by crosses whereas the solid lines repre- i
sent the theoretical predictions (see SM section I.L) for the quency fp = ωp/2π for different values of the pump
cavity transmission S21, for thesignal gain Gs (blue)and for input power Pp. The magnetic flux for these measure-
the idler gain Gi (green). The parameters that have been ments is set to a value for which ωf/2π = 8.1GHz and
employed for the calculation are listed in the caption of Fig. ∆ /2π =−1.5GHz. Relatively good agreementbetween
1
2. The detuning between the signal and pump frequencies is data and theory (see SM section I.L) is found for the re-
(ωs−ωp)/2π =50kHz. sultsseeninFig. 3,however,thedeviationbetweendata
and theory becomes larger at higher powers. Further
study is needed in order to identify the sources of dis-
crepancy,and to improve the accuracy of the theoretical
close to unity (see the points labeled by ±1 in panels c
predictions accordingly.
and d of Fig. 2). With sufficiently large driving ampli-
tude,however,higherordernonlinearprocessesmaygive In summary, superharmonic resonances in the device
rise to superharmonic resonances, which occur near the understudyhavebeenexperimentallyobserved. Wethe-
points at which the ratio ω /ω is an integer larger than oretically show that a relatively simple CQED model of
a c
unity (see the points labeled by ±2, ±3 and 4 in panels asystemcomposedoftwocoupledelements,asinglecav-
c and d of Fig. 2). The cavity response near a superhar- ity mode havingno intrinsicnonlinearityanda two-level
monic resonance is theoretically evaluated in section III system, can account for the main experimental findings.
ofthe SM. It is found that the sameEq. (4) canbe used Further study will aim at expanding the range of valid-
todescribe the effectofthe flux qubitoncavityresponse ity of the theoretical predictions in order to account for
near a superharmonic resonance, provided that the cou- the experimental results at higher levels of input power.
pling coefficient g is replaced by g , which is given by Futureexperimentswillexplorethepossibilityofexploit-
1 n
(see SM section III) ing nonlinearity for improving the fidelity of qubit read-
outandemploysuperharmonicresonancesforgenerating
4g ω E1/2 highlycorrelatedstatesofthemicrowavecavityfield(e.g.
1 f c
gn =g1J1 n , (6) thecreationofentangledpairsofmicrowavephotonsnear
− ωpω∆ !
the n=2superharmonicresonanceviatwo-photonstim-
ulated emission events).
where J is the l’th Bessel function of the first kind, and
l
thedetuning∆ =ω −ω isreplacedby∆ =nω −ω , We thank Feyruz Kitapli and Pol Forn-D´ıaz for use-
1 p a n p a
where at the superharmonic resonance ω /ω = n. As fuldiscussions. Thisresearchissupportedbythe Gerald
a c
can be seen by comparing panels c and d of Fig. 2, the SchwartzandHeatherReismanFoundation,NSERC,In-
calculatedand measuredcavityresponse near the super- dustry Canada,CMC, CFI, OntarioMRI andthe Israeli
harmonic resonances exhibit an acceptable agreement. Science Foundation.
5
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Superharmonic Resonances in a Strongly Coupled Cavity-Atom System -
Supplemental Material
Eyal Buks,1 Chunqing Deng,2,3,4 Jean-Luc F.X. Orgazzi,2,5,4 Martin Otto,2,3,4 and Adrian Lupascu2,3,4
1Department of Electrical Engineering, Technion, Haifa 32000 Israel
2Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada N2L 3G1
3Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, Canada N2L 3G1
4Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, ON, Canada N2L 3G1
5Department of Electrical and Computer Engineering,
University of Waterloo, Waterloo, ON, Canada N2L 3G1
(Dated: February 1, 2016)
6
1
0 In the first section of the supplemental material the B. Qubit Energy Eigenstates
2 equations of motion are derived and linearized and the
n response is evaluated in the weak nonlinear regime. The The energy eigenstates of the decoupled qubit are
a second section is devoted to the Bloch-Siegert shift, and given by |±i
J the third one discusses the superharmonic resonances.

9 + cosθ sinθ x
2 | i = sin2θ cos2θ |yi , (4)
(cid:18)|−i(cid:19) (cid:18)− 2 2 (cid:19)(cid:18)| i(cid:19)
] I. WEAK NONLINEAR RESPONSE
h where
p
- A. The Closed System ω∆
t tanθ = , (5)
n ω
f
a
The Hamiltonian of the closed system, formed by
u H0 and the corresponding eigenenergies are
the flux qubit and the cavity mode, is taken to be given
q
[ by ~ω
ε = a , (6)
1 ± ± 2
1 K
v ~ 1 =ω A A+ + cA A AA
− 0 c † † †
9 H 2 2 where
(cid:18) (cid:19)
8 ω
+ f (x x y y)
9 2 | ih |−| ih | ω = ω2+ω2 . (7)
7 ω a f ∆
0 + ∆ (x y + y x) q
2 | ih | | ih |
. The following relations
1 g A+A (x x y y) .
†
0 − | ih |−| ih |
6 (cid:0) (cid:1) (1) |xihx|−|yihy|=cosθ Σz −sinθ(Σ++Σ−) , (8)
1
v: The cavity mode angular resonance frequency is labeled and
Xi cbayvωitcy,mKocdisetahnencihaviliattyiomnoodpeeKraetrorr.coTehffieccieonetffiacnidenAt~isωtfhies |xihy|+|yihx|=sinθ Σz+cosθ(Σ++Σ−) , (9)
r related to the externally applied magnetic flux Φ by
a e
hold, where
~ω I Φ
f cc 0
2 = 2π φe , (2) Σz =|+ih+|−|−ih−| , (10)
Σ = + , (11)
+
whereIcc = y ∂ 0/∂Φe y ( Icc)isthecirculating Σ =| ih+−| , (12)
currentassoc−iahted|wHiththest|atei y− (x ),Φ =h/2eis − |−ih |
0
| i | i
the flux quantum, and the normalized applied magnetic and thus the Hamiltonian can be expressed as
0
flux φ is given by H
e
1 K
~ 1 =ω A A+ + cA A AA
Φ 1 − 0 c † † †
φe =2π e . (3) H (cid:18) 2(cid:19) 2
(cid:18)Φ0 − 2(cid:19) + ωaΣ
z
2
The coefficient ~ω is the qubit energy gap, and g is
∆ g A+A† [cosθ Σz sinθ(Σ++Σ )] .
the coupling constant. The frequencies ωf, ω∆ and g are − − −
(13)
assumed to be time independent. (cid:0) (cid:1)
2
C. Damping by
~−1 =~−1 0
H H
Damping is takeninto accountusing a model contain- 4
ing reservoirshaving dense spectrum of oscillator modes + dω ωa†cn(ω)acn(ω)
interacting with both the cavity mode and the qubit. n=1Z
X
The cavity mode is assumed to be coupled to 4 semi- 3 γ
infinite transmission lines. The first two, denoted as c1 + πcn dω eiφcnA†acn(ω)+e−iφcna†cn(ω)A
and c2, are feedlines (or ports), which are linearly cou- n=1r Z
X (cid:2) (cid:3)
pled to the cavity mode with coupling magnitudes γ γ
and γc2 and coupling phases φc1 and φc2, respectivelcy1, + 2cπ4 dω eiφc4A†A†ac4(ω)+e−iφc4a†c4(ω)AA
and which are employed to deliver the input and output r Z h i
2
signals. The third, denoted as c3, is linearly coupled to
+ dω ωa†qn(ω)aqn(ω)
the cavity mode with a coupling magnitude γ and a
c3 n=1Z
coupling phase φ , and it is used to model linear dissi- X
c3 γ
pdaentiootned(dausec4to, iisnnteornnlainlesaorulyrcceos)u,pwlehdetroeatshethceavfoitrythmoondee, +r2qπ1 Z dω(cid:16)eiφq1Σ+aq1(ω)+e−iφq1a†q1(ω)Σ−(cid:17)
withacouplingmagnitudeγ andacouplingphaseφ , γ
and is employed to model noc4nlinear dissipation (duect4o + 4qπ2 dω eiφq2Σzaq2(ω)+e−iφq2a†q2(ω)Σz .
internal sources). The qubit is assumed to be coupled r Z (cid:16) (cid:17)
(22)
to 2 semi-infinite transmission lines, with coupling mag-
nitudes γ and γ and coupling phases φ and φ ,
q1 q2 q1 q2
respectively. Whilethefirstisemployedtomodelenergy
relaxation, the second is employed to model dephasing.
Note thatallcouplingparametersareassumedto be fre- D. The equations of motion
quency independent. The following Bose
The Heisenberg equations of motion are generated ac-
cording to
A,A† =1, (14) dO = i O,~ 1 , (23)
−
dt − H
acn(ω),a†c(cid:2)m(ω′)(cid:3)=δn,mδ(ω−ω′) , (15) (cid:2) (cid:3)
(cid:2)aqn(ω),a†qm(ω′)(cid:3)=δn,mδ(ω−ω′) , (16) where O is an operator and is the total Hamiltonian,
H
[a (ω),a (ω )]=0, (17) hence
(cid:2) cn cm ′(cid:3)
[a (ω),a (ω )]=0, (18)
qn qm ′
dA
= iωcA iKcA†AA
dt − −
+ig[cosθ Σ sinθ(Σ +Σ )]
z +
− −
3
γ
and qubit i cneiφcn dω a (ω)
cn
− π
n=1r Z
X
2γ
i c4eiφc4 dω A†ac4(ω) ,
− π
r Z
(24)
[Σ ,Σ ]=2Σ , (19)
z + +
[Σ ,Σ ]= 2Σ , (20)
z
− − −
[Σ ,Σ ]=Σ , (21)
+ z
− dΣ
z
= 2igsinθ A+A (Σ Σ )
† +
dt − − −
γ (cid:0) (cid:1)
q1
2i dω
− 2π
commutation relations are assumed to hold. r Z
× eiφq1Σ+aq1(ω)−e−iφq1a†q1(ω)Σ− ,
(cid:16) (cid:17) (25)
The Hamiltonian of the system is takento be given
H
3
dΣ
+ and
=iωaΣ+ ig A+A† (2cosθ Σ++sinθ Σz)
dt −
t 1
−irγ2qπ1 Z dω(cid:0)e−iφq1a†q(cid:1)1(ω)Σz Ztc dt′ δ(t−t′)f(t′)= 2sgn(t−tc)f(t) , (33)
γq2 where sgn(x) is the sign function
+i dω
π
r Z
+1 if x>0
× eiφq2Σ+aq2(ω)+e−iφq2a†q2(ω)Σ+ , sgn(x)= 1 if x<0. , (34)
(cid:26)−
(cid:16) (cid:17) (26)
lead to
dacn(ω) 1 ∞ dω a (ω)
cn
dt √2π
=(cid:26)−−iiωωaacc4n((ωω))−−iipγ2γcππ4cnee−−iφiφc4cnAAA n=n=1,24,3 , =(cid:26)Za−icna4∞ic(nnt)(t−)−i√iγ2pc4γe2c−nieφ−c4iAφcn(tA)A(t)(t) n=n=1,42,3 ,
(27)
p (35)
da (ω) γ and
q1 = iωaq1(ω) i q1e−iφq1Σ , (28)
dt − − r2π − 1 ∞ dω a (ω)
cn
and √2π
Z−∞
daqd2t(ω) =−iωaq2(ω)−irγ4qπ2e−iφq2Σz . (29) =(cid:26) aoca4uoctnu(tt)(t+)+i√iγ2pc4γe2c−nieφ−c4iAφcn(tA)A(t)(t) n=n=1,24,3 ,
(36)
E. Input-Output Relations where the input operators are given by
latTedhebyfiealdssuopmeirnagtoeritahcenr(itn,iωti)alatvatliumeefotrcathnebfieelcdalocpu-- aicnn(t)= √12π ∞ dω e−iω(t−t0)acn(t0,ω) , (37)
Z−∞
erator a (t ,ω) at time t or final value for the field
cn 0 0
and the output operators by
operator a (t ,ω) at time t . The time t is assumed
cn 1 1 0
to be in the distant past whereas t is assumed to be in
the distant future, i.e. t0 ≪t≪t11. Time integration of aocnut(t)= √12π ∞ dω e−iω(t−t1)acn(t1,ω) . (38)
(27) using initial values at time t0 <t yields Z−∞
a (ω) Equations(35)and(36)yieldthe followinginput-output
cn
relations
e−iω(t−t0)acn(t0,ω)
= −i γπcne−iφcn tt0dt′ e−iω(t−t′)A(t′) n=1,2,3 , aocnut(t)−aicnn(t)
 −i γp2cπ4e−iφec−4iωt(t0t−Rdtt0′)ae−c4iω(t(0t−,ωt′))A(t′)A(t′) n=4 =(cid:26)−−i√i√γc24γe−cnieφ−c4iAφc(ntA)A(t)(t) n=n=1,42,3 .
 p R (30) (39)
and using finite values at time t1 >t yields Similarly for the bath operators that are coupled to
the qubit one has [see Eqs. (28) and (29)]
a (ω)
cn
= −i γπcnee−−iiφωc(nt−ttt11)datc′ne(−t1iω,ω(t)−t′)A(t′) n=1,2,3 √. 12π Z−∞∞dω aq1(ω)=aiqn1(t)−irγ4q1e−iφq1Σ− , (40)
 −ipγp2cπ4e−iφec−4iRωt(t1t−Rdtt1′)ae−c4iω(t(1t−,ωt′))A(t′)A(t′) (n31=)4 wheraeiqn1(t)= √12π ∞ dω e−iω(t−t0)aq1(t0,ω) , (41)
Z−∞
Integrating a (ω) over ω and using the following rela-
cn
and
tions
∞ dω e−iω(t−t′) =2πδ(t−t′) , (32) √12π ∞ dω aq2(ω)=aiqn2(t)−i γ8q2e−iφq2Σz , (42)
Z−∞ Z−∞ r
4
where the input qubit operators as
aiqn2(t)= √12π ∞ dω e−iω(t−t0)aq2(t0,ω) . (43) the output cavity opaeiqnrna=torcsiqnnaes−iωpt , (53)
Z−∞
Thus, the equation of motion for A becomes [see Eqs. aocnut = bocnut+cocnut e−iωpt , (54)
(24) and (35)]
the cavity mode annih(cid:0)ilation ope(cid:1)rator as
dA + iω +γ +(iK +γ )A A A A=ARe−iωpt , (55)
c c c c4 †
dt
and the qubit operator Σ as
=ig[c(cid:2)osθ Σ sinθ(Σ +Σ )] (cid:3) +
z +
− − Σ =Σ eiωpt . (56)
3 + +R
−i 2γcneiφcnaicnn−2i√γc4eiφc4A†aicn4 , In terms of these notations Eq. (44) becomes
n=1
Xp dA
(44) dtR + −i∆pc+γc+(iKc+γc4)A†RAR AR
where +i 2γh eiφc1bin i
c1 c1
γc =γc1+γc2+γc3 . (45) −igpcosθ Σzeiωpt−sinθ Σ+Re2iωpt+Σ†+R
= h, (cid:16) (cid:17)i
Furthermore, by making use of the following relations A
V
(57)
1
Σ+Σ = (1+Σz) , (46) where
− 2
1 ∆ =ω ω , (58)
Σ Σ+ = (1 Σz) , (47) pc p− c
− 2 −
and where
Σ Σ = Σ Σ =Σ , (48)
z + + z +
− 3
Σ−Σz =−ΣzΣ− =Σ− , (49) VA =−i 2γcneiφcncicnn−2i√γc4ei(φc4+ωpt)A†Rcicn4 ,
onefindsthattheequationofmotionforΣz becomes[see nX=1p (59)
Eqs. (25) and (40)]
Eq. (50) becomes
ddΣtz +γq1(1+Σz)+2igsinθ A+A† (Σ+−Σ−) ddΣtz +γq1(1+Σz)
=2i√γq1 −Σ+eiφq1aiqn1+e−i(cid:0)φq1aiqn1†Σ(cid:1)− , +2igsinθ ARe−iωpt+A†Reiωpt Σ+Reiωpt−Σ†+Re−iωpt
(cid:16) (cid:17) (50) = , (cid:16) (cid:17)(cid:16) (cid:17)
z
V
(60)
andtheequationofmotionforΣ becomes[seeEqs. (26)
+
and (42)] where
dΣ+ iω Σ + γq1 +γ Σ Vz =2i√γq1 −eiφq1Σ+Rciqn1+e−iφq1ciqn1†Σ†+R , (61)
a + q2 +
dt − 2 (cid:16) (cid:17)
and Eq. (51) becomes
(cid:16) (cid:17)
+ig A+A (2cosθ Σ +sinθ Σ )
† + z
dΣ γ
=−i(cid:0)√γq1e−i(cid:1)φq1aiqn1†Σz d+tR +i∆1Σ+R+ 2q1 +γq2 Σ+R
+i 2γq2 Σ+eiφq2aiqn2+e−iφq2aiqn2†Σ+ . +ig ARe−iωpt+A(cid:16)†Reiωpt 2c(cid:17)osθ Σ+R+sinθ Σze−iωpt
p (cid:16) (cid:17) (51) = (cid:16), (cid:17)(cid:0) (cid:1)
+
V
(62)
F. Cavity External Drive where
∆ =ω ω , (63)
1 p a
Consider the case where a monochromatic pump tone −
havingamplitudebin andangularfrequencyω isinjected and where
c1 p
tinhteoinppourtt1c.avInityaofrpaemraetorrostaatriengexaptreasnsgeudlaarsfrequency ωp V+ =−i√γq1e−iφq1ciqn1†Σz
+i 2γq2 eiφq2Σ+Rciqn2e−iωpt+e−iφq2ciqn2†Σ+Reiωpt .
aicnn =(cid:26)(cid:0)bicnnc+icnnceic−nni(cid:1)ωep−t iωpt n=n=2,31,4 , (52) p (cid:16) (64(cid:17))
5
G. Rotating Wave Approximation where
In the rotating wave approximation (RWA), in which ΘR AR,A†R,Σz,Σ+R,Σ†+R
rapidly oscillating terms are disregarded, the equations (cid:16) (cid:17)
of motion (57), (60) and (62) become = −i∆pc+γc+(iKc+γc4)A†RAR AR
+ih 2γc1eiφc1bicn1+ig1Σ†+R , i
dA
dtR + −i∆pc+γc+(iKc+γc4)A†RAR AR p (74)
+ip2γhc1eiφc1bicn1+ig1Σ†+R =VA , i (65) Θz AR,A†R,Σz,Σ+R,Σ†+R
(cid:16)Σ P (cid:17)
z 0
= T− +2ig1 ARΣ+R−Σ†+RA†R ,
1
dΣ (cid:16) (cid:17)
z (75)
+γ (1+Σ )
q1 z
dt
and
+2ig1 ARΣ+R−Σ†+RA†R =Vz ,
(cid:16) (cid:17) (66) Θ+ AR,A†R,Σz,Σ+R,Σ†+R
Σ(cid:16) (cid:17)
+R
and = T +i∆1Σ+R+ig1A†RΣz .
2
(76)
dΣ γ
+R q1
+i∆ Σ + +γ Σ
1 +R q2 +R
dt 2 The forcing terms = , = and
A A A z z z
+ig1A†RΣz =V+ , (cid:16) (cid:17) vFa+lu=e. VT+he−cohVeffi+iciheFnatvePa,Vvwahn−iicshhhViinsgigtivFheenrmbayVl e−xpheVctiation
(67) 0
1
P = , (77)
where 0 −2n +1
0
represents the expectation value Σ in thermal equi-
g =gsinθ . (68) z
1 h i
librium in the absent of external driving and when the
coupling between the qubit and the cavity can be disre-
garded. The time T , which is given by
H. Linearization 1
P
0
T = , (78)
1
Expectationvaluesoftheoperators , and are −γ
A z + q1
V V V
evaluatedbyassumingthatbathmodesareallinthermal
is the qubit longitudinal relaxation time, and the time
equilibrium[1]. Tofirstorderinthedampingcoefficients
T , which is given by
one finds that vanishes [see Eq. (59)] and that [see 2
A
Eqs. (61) and (h6V4)i] P0 2T1
T = = , (79)
= 2γ n Σ , (69) 2 −γ2q1 +γq2 1+ 2γγqq12
z q1 0 z
hV i − h i
is the qubit transverse relaxation time.
γ
q1
= 2 +γ n Σ , (70)
+ q2 0 +R
hV i − 2 h i I. Fixed Points
(cid:16) (cid:17)
where n is the Bosonic thermal occupation number.
0
The solution is expressed as
With the help of Eqs. (65), (66), (67), (69) and (70)
the equations of motion become A =α +a , (80a)
R R R
Σ =P +σ , (80b)
dA z z z
R
dt +ΘR =FA , (71) Σ+R =P+R+σ+R , (80c)
where both α and P are complex numbers, P is a
R +R z
real number, and the operators a , σ and σ are con-
dΣ R z +R
z
+Θ = , (72) sidered as small. Fixed points are found by solving
z z
dt F
Θ α ,α ,P ,P ,P =0, (81a)
and R R ∗R z +R +∗R
Θz(cid:0)αR,α∗R,Pz,P+R,P+∗R(cid:1)=0, (81b)
dΣ
d+tR +Θ+ =F+ , (73) Θ+(cid:0)αR,α∗R,Pz,P+R,P+∗R(cid:1)=0. (81c)
(cid:0) (cid:1)