Quantum phase transitions in the driven dissipative Jaynes-Cummings oscillator: from the dispersive regime to resonance

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by  Th. K. Mavrogordatos

Quantum phase transitions in the driven dissipative Jaynes-Cummings oscillator:
from the dispersive regime to resonance
Th. K. Mavrogordatos∗
Department of Physics and Astronomy, University College London,
Gower Street, London, WC1E 6BT, United Kingdom
(Dated: January 25, 2017)
We follow the passage from complex amplitude bistability to phase bistability in the driven dis-
sipativeJaynes-Cummingsoscillator. Quasidistributionfunctionsinthesteadystateareemployed,
for varying qubit-cavity detuning and drive parameters, in order to track a first-order dissipative
quantumphasetransitionuptothecriticalpointmarkingasecond-ordertransitionandspontaneous
symmetrybreaking. Wedemonstratethephotonblockadebreakdowninthedispersiveregime,and
find that the coexistence of cavity states in the regime of quantum bistability is accompanied by
pronounced qubit-cavity entanglement. Focusing on the roˆle of quantum fluctuations in the re-
7
sponse of both coupled quantum degrees of freedom (cavity and qubit), we move from a region of
1
minimal entanglement in the dispersive regime, where we derive analytical perturbative results, to
0
2 the threshold behaviour of spontaneous dressed-state polarization at resonance.

n
PACSnumbers: 42.50.Ct,42.50.Lc,42.50.Pq,03.65.Yz
a Keywords: dissipativequantumphasetransition,bistability,drivenJaynes-Cummingsoscillator
J

3
2 I. INTRODUCTION frequencies coincide, and in the dispersive regime, where
the cavity and qubit are strongly detuned in relation to

]
h TheJaynes-Cummings(JC)oscillatorisanarchetypal their dipole coupling strength [10, 13, 14].
p source of intricate quantum nonlinear dynamics arising At resonance, the mean-field nonlinearity diverges for
- from the coupling of a quantized electromagnetic mode zerophotonnumber,whileinthedispersiveregimebista-
t
n inside a resonator (cavity) to a two-level system (qubit) bility builds up in a perturbative fashion with no asso-
a [1]. The behaviour of quantum nonlinear oscillators has ciated threshold, unlike the laser. The perturbative ap-
u
beenasubjectofintensetheoreticalinvestigation(foran proach becomes inadequate when the qubit participates
q
[ overviewseeChapter7of[2])providingatthesametime activelyinthebistableswitchingforstrongerdriving[15].
the basis for numerous experiments in cavity and cir- In that regime, the system response comprises an aver-
1
cuitquantumelectrodynamics(seeforexample[3]where age over spontaneous switching between the metastable
v
the extended JC oscillator is driven out of equilibrium mean-fieldsteadystateswhereboththecavityandqubit
1
7 in the presence of dissipation). In addition, controlled are significantly excited. For complex amplitude bista-
6 light-matter interaction has shifted the center of interest bilityswitchingoccursbetweenadim(withlowern)and
6 in phase transitions from condensed matter to quantum a bright (with higher n) state, while in phase bistabil-
0 optics. ity both states have the same magnitude and opposite
1. Amongst the most discussed light-matter quantum phases following a transition from a discrete to a con-
0 phase transitions in the literature are the Dicke phase tinuous spectrum in the system quasienergies [9, 10, 16].
7 transition [4, 5], which is explicitly dissipative (with co- In first-order dissipative phase transitions for the cavity
1 operativeresonancefluorescenceasitsdrivenvariant[6]), field, weaker coupling implies a bigger photon number
v: and the laser which exhibits a second-order phase tran- required for the nonlinearity to manifest itself, yielding
i sition out of equilibrium [7]. In comparison to those, a response which is a non-analytic function of the drive
X
however, the driven JC model is fundamentally different [2, 10].
r as it deals with the interaction of one field mode with
a Motivatedbythecurrentexperimentalandtheoretical
one quantum object, the qubit, necessitating the reap-
interestintheresponseofquantumnonlinearoscillators,
praisaloftherˆoleofquantumfluctuationsandadifferent
in this letter we track amplitude bistability, from its ori-
definition of the thermodynamic limit [8–11]. The open
gin in the dispersive regime, up to a critical point at res-
coherently-drivendissipativequbit-cavitysystemyieldsa
onance, where phase bistability takes over. Mean-field
bistableresponsewherequantumfluctuationsarerespon-
results guide us to extract the relevant scaling parame-
sible for switching between two metastable states that
ters used to define the “thermodynamic limit” for this
are long-lived in relation to the characteristic cavity and
√ driven resonator in which the number of photons is not
qubitdecaytimes[12]. The nsplittingoftheJCenergy
conserved. We present contour plots of quasidistribution
levelsisauniquefeaturedeterminingthenatureofbista-
functions for the cavity field, showing the passage from
bilitybothatresonance,wherethecavityandqubitbare
amplitude to phase bistability, and invoke the entangle-
ment entropy, calculated via the reduced qubit density
matrix, in order to demonstrate the active participation
∗ [email protected] of both quantum degrees of freedom in the emerging bi-
2
FIG.1. Dispersive amplitude bistability. (a)Steady-stateintracavityamplitudeasafunctionofthenormalizeddrivestrength
ε /γ in the semiclassical and the quantum description for ∆ω /γ = 340. The semiclassical bistability curve (solid line, with
d c
thesparselydashedpartcorrespondingtotheunstablebranch)depicting|α|issuperimposedontopofthequantumamplitude
(cid:112)
curves (cid:104)a†a(cid:105) (thinly dashed line) and |(cid:104)a(cid:105)| (dashed-dotted line). The latter exhibits the characteristic coherent cancellation
dip(pointA)oftheDuffingoscillator,andintersectsthesemiclassicalbistabilitycurveinthreepoints(B,C,D).Forthemarked
points A, B, C, D we plot the quasidistribution function Q(x+iy) for the corresponding intracavity field amplitude. (b)
Average cavity photon number (cid:104)a†a(cid:105) as a function of the drive parameters. The dashed line indicates the driving frequency
selected for (a). Parameters: g/δ=0.14, 2κ/γ =12, g/γ =3347, n =12.68.
scale
modality. We show further that, closer to resonance, en- ω −ω are the detunings of the cavity resonance fre-
d c,q
hanced multi-photon transitions appear for weak cavity quency ω and the qubit bare frequency ω from the
c q
excitation,followedbyabreakdownofthephotonblock- frequency of the drive, coupled to a resonant cavity
ade with stronger driving [10, 11]. mode with photon annihilation and creation operators
a and a†, respectively. The inversion operator σ is
z
related to the raising (lowering) operators σ (σ ) for
+ −
II. THE DISPERSIVE JC MODEL the qubit with two states |g(cid:105) (ground), |e(cid:105) (excited) via
σ = 2σ σ −1. The cavity mode is dipole-coupled to
z + −
TheJCHamiltoniandescribestheinteractionbetween the qubit with strength g, while the classical coherent
a single resonant cavity mode and a qubit; however, it field (with very high photon occupancy) is coupled to
does not account directly for the coupling to the envi- the resonant cavity mode with strength εd (also called
ronment which is included only in the formulation of the drive amplitude). The cavity field is also coupled to
Master Equation (ME) [17]. After adding dissipation in a Markovian thermal bath at zero temperature, which
a frame rotating with the frequency ω of the coherent induces a photon loss rate of 2κ. In addition to pho-
d
driving field, the Lindblad ME for the reduced system ton dissipation, there is also spontaneous emission to
density operator ρ (for a system Hamiltonian in the ro- modes different than the resonant cavity mode, with
tating wave approximation and setting (cid:126) = 1 for conve- rate γ. The strongly dispersive regime with weak spon-
nience) reads [9, 18] taneous emission is defined through a qubit-cavity de-
tuning such that δ ≡ |ω − ω | (cid:29) g (cid:29) 2κ (cid:29) γ. In
c q
ρ˙ =i∆ω [a†a,ρ]+i∆ω [σ σ ,ρ]+g[a†σ −aσ ,ρ] our results we show complex amplitude bistability for
c q + − − +
the intracavity field in the following region of the drive
+[ε a†−ε∗a,ρ]+κ(2aρa†−ρa†a−a†aρ)
d d phase space: 0 ≤ ∆ω ≤ g2/δ and ε < g2/δ < g (with
c d
+(γ/2)(2σ−ρσ+−ρσ+σ−−σ+σ−ρ). γ/(2κ) ≈ 0.1). At resonance, where δ = 0, phase bista-
(1) bility is associated with a particular point in the phase
space: (∆ω = 0, ε = g/2), and a threshold behaviour.
c d
In the right hand side of the above equation, ∆ω =
c,q
3
FIG.2. TheeffectiveKerrnonlinearity. JointquasidistributionfunctionW(x+iy)for∆ω /κ=72.50andfourdifferentvalues
c
of the drive strength: ε /κ=2.17,2.33,2.50,2.67 in (a)-(d) respectively, using Eq. (5) (Panel I) and the solution of Eq. (1)
d
for the reduced cavity density matrix (Panel II). Parameters: g/δ=0.14, 2κ/γ =12, g/γ =3347, n =12.68.
scale
Throughout our work, we solve numerically the Lind- classical intracavity amplitude α obeys the equation [16]
blad ME. We perform numerical simulations (employing
the open-source Matlab software Quantum Optics Tool-  −1
box) in a truncated Hilbert space for an initial product iε  2g2(κ−i∆ω )−1(γ−2i∆ω )−1
state with the qubit in the ground state and the cavity α=− d 1+ c q  ,
κ˜  8g2|α|2 
in a Fock state with zero photons [i.e. the initial ground  1+ 
(γ2+4∆ω2)
state ρ(0) = (|n=0(cid:105)(cid:104)n=0|) ⊗ (|g(cid:105)(cid:104)g|)]. The steady- q
state results are independent of the initial state, while (2)
convergence with respect to the number of photon levels with κ˜ =κ−i∆ωc, whichpredicts two metastablestates
has been ensured. The validity of the rotating wave ap- (dim and bright) and one unstable state that vanishes
proximationinthedispersiveregimeweareconsideringis in the presence of fluctuations. In contrast to the mean-
checkedagainstFig. 1ofRef. [19]andFig. 1ofRef. [20]. fieldprediction,thecurvedepicting|(cid:104)a(cid:105)|doesnotexhibit
The former shows that the counter-propagating terms any bistability. Quantum fluctuations out of equilibrium
can be omitted without affecting the physical picture manifest themselves through the absence of a Maxwell
even for the maximum cavity-qubit detuning considered, construction, since the line |(cid:104)a(cid:105)| does not cut the semi-
sincethecouplingstrengthg remainssufficientlysmaller classicalcurveintwoequalareas. Furthermore,thecurve
(cid:112)
than the bare cavity frequency (g/ω ≈ 0.03), while the for (cid:104)a†a(cid:105)doesnotexhibitthecoherentcancellationdip,
c
latter demonstrates good agreement between theory and whichishencesolelyaquantumphaseeffect. Thesecon-
experiment in a similar parameter regime. siderationsholdalsoforthedrivenDuffingoscillator(the
reader is referred to Fig. 1 of [18]) as a consequence of
non-constant diffusion coefficients in the corresponding
Fokker-Planck equation; here, however, we cannot for-
mulate such an equation due to the active participation
InFig. 1[seebothframes(a)and(b)]wedepictthethe of the qubit [8, 9].
intracavityphotonfield(inthecoherentstatespacewith
|α(cid:105) ≡ |x+iy(cid:105)) within a drive region where the quan-
tum fluctuations are responsible for the deviation from III. PERTURBATIVE EXPANSION FOR THE
the mean-field predictions. For low driving strengths, CAVITY BISTABILITY
(cid:112)
|α|, (cid:104)a†a(cid:105) and |(cid:104)a(cid:105)| coincide and the cavity is in the
dim state, resembling a vacuum state with Gaussian dis- At first we will examine the birth of dispersive am-
tribution. Withincreasingε /κ, thebrightstateaccrues plitude bistability for a driving frequency in the region
d
probability resulting in the coherent cancellation we ob- ∆ω (cid:39)g2/δandweakdrivestrength. Inthestronglydis-
c
serve at the point A. As we follow the curve for |(cid:104)a(cid:105)|, persivelimitthepresenceofthesmalltermg/δprecludes
probability transfers from the dim to the bright state thedivergenceofnonlinearityatlowintracavityfieldam-
crossing the boundary of a first-order dissipative quan- plitudes. When the length of the Bloch vector is con-
tum phase transition. The complex steady-state semi- served, in the absence of spontaneous emission (γ = 0),
4
FIG. 3. Boundary of the first-order phase transition in Panel I (a-c): Joint quasidistribution function Q(x+iy) for four
different points in the (∆ω /κ,ε /κ) phase space: (a) (56.83,16.67), (b) (47.33,33.33), (c) (39.83,50) and (d) (0,33.47).
c d
Parameters: g/δ=0.14,2κ/γ =12[inframes(a)-(c)]and2κ/γ =200(inframe(d)),g/γ =3347. Towards phase bistability at
the critical point (∆ω =0, ε =g/2) inPanel II:JointquasidistributionfunctionQ(x+iy)forfourdecreasingvaluesofthe
c d
qubit-cavitydetuningtocouplingstrengthratioδ/g: (a)7.12,(b)4.13,(c)1.14,(d)0. Parameters: g/γ =3347,2κ/γ =200.
Thedrivingfieldhasaphasedifferenceof−π/2withrespecttothedriveinFig. 1(a),leadingtoarotationofthedistribution
by that angle in the x-y plane, as expected from Eq. 2.
the steady-state complex field amplitude is given by the agreement with the semi-classical prediction of Eq. 4 for
relation [10] σ = −1). The series expansion also renormalizes pro-
z
gressively the driving phase space such that the effective
(cid:40) (cid:34) g2 (cid:18) 4g2 (cid:19)−1/2(cid:35)(cid:41)−1 drive strength and frequency are functions of the system
α=−iε κ−i ∆ω − 1+ |α|2 .
d c δ δ2 operators [14].
We can then derive the Wigner function for the ef-
(3)
According to Eq. 3, we can identify n =δ2/(4g2) as fective dressed Duffing oscillator [18], calculated via the
scale
generalized P-representation [21–23]
the dispersive scale parameter. This number approaches
infinity for g → 0 (at constant δ) and in that sense
2 | F (c,2ε˜ α∗)|2
the “thermodynamic limit”, where fluctuations vanish, W(α,α∗)= e−2|α|2 0 1 d , (5)
is a weak-coupling limit (for a constant co-operativity π 0F2(c,c∗,2|ε˜d|2)
parameter C = g2/(κγ)). Here, the displayed nonlin-
where F (λ;x) and F (λ,µ;x) are generalized hyper-
earity presents similarities to absorptive optical bista- 0 1 0 2
geometric functions of the variable x, with parameters
bility where, setting γ = 0 in the Maxwell-Bloch equa-
λ,µ. Here, c = (κ − i∆ω(cid:48))/(iχ) and ε˜ = ε /(iχ)
tion solutions a posteriori, we find the scaling parame- c d d
with χ = (g4/δ3)σ . The effective detuning ∆ω(cid:48) =
ter ∆ω2/(2g2) [10] (note also that Eq. (2) with ∆ω = z c
c c ∆ω +(g2/δ)σ −(g4/δ3)(2σ +1) accounts for the cor-
∆ω ≡∆ω is identical to Eq. 28 of [10]). c z z
q rection by the dispersive shift and higher-order terms.
We apply the dispersive transformation to diag-
We note that the perturbative expression (5) repro-
onalize the JC Hamiltonian, generating the term
(cid:112) ducestheGaussianformofthedistributionfunctioncor-
δ 1+n /n , where n =a†a+σ σ is the operator
s scale s + − respondingtoavacuumstate,W =(2/π)e−2|α|2,forvery
of system excitations (see [14] and [20] for more details).
lowdrivingstrengthsε /κ,andallowsustotrackthepro-
Expanding Eq. (3) to the lowest order in n /n (with d
s scale gressiveparticipationofthevariousnonlineartermsaris-
|α|2 the semiclassical analogue of n ) yields
s ingfromthehypergeometricfunction F . Itistherefore
0 1
(cid:26) (cid:20) g2 (cid:18) 2g2 (cid:19)(cid:21)(cid:27)−1 moreinstructivetowriteaperturbationseriesexpansion
α=−iε κ−i ∆ω − 1− |α|2 , (4)
d c δ δ2 for the numerator:
icnavaitgyreDemuffienntgwaipthproExqi.ma(t3io2n) wofe[1re0t]a.inInontlyhetedrrmessseudp- W(α,α∗)= F(2(/cπ,c)e∗−,22||εα˜|2|2)(cid:12)(cid:12)(cid:12)(cid:12)1+ Dz + 2!zD2 +···(cid:12)(cid:12)(cid:12)(cid:12)2,
0 2 d 1 2
to the second-order in ns/nscale, and the reduced Hamil- √ (6)
tonianacquiresthequarticcorrection(g4/δ3)σ a†2a2 (in with z = −8ε˜ α∗ and D =[(c+m−1)!]/[(c−1)!]=
z d m
5
c·(c+1)·...·(c+m−1), showing explicitly the devel-
opment of nonlinearity for increasing drive strength. In
the regime where n /n ≈ 1, the Duffing approxima-
s scale
tion breaks down as the qubit vector becomes increas-
ingly entangled to the cavity mode, moving towards the
equatorial plane in the Bloch sphere representation [16].
In Fig. 2(a) we depict the cavity field distribution in
the absence of entanglement with the qubit. The exci-
tation pathways ‘flow’ around the nodes of the Wigner
function in a spiral-like fashion, as the departure from
the Gaussian form becomes more apparent. These per-
turbative distributions approximate very well the exact
ME results, in which the qubit is included as an inde-
pendent degree of freedom [see Fig. 2(b)]. Hence, the
agreement verifies the fact that the qubit participates
only in dressing the cavity with a Kerr term depending
on σ =(cid:104)σ (cid:105)=−1. Treating σ as a constant of motion
z z z
when solving Hamilton’s equations (for arbitrary exci-
tation but for a short time scale in comparison to γ−1)
FIG. 4. Amplitude bistability for both cavity and qubit. (a)
underliesthemethodfollowedbytheauthorsofRef. [20], Entanglement entropy S as a function of the driving fre-
q
whoprovideasemiclassicalexpressionfor|α|2 inthedis- quency and strength. (b) Entanglement entropy (x20, red
persive regime. The cavity response is described therein curve), |(cid:104)σ (cid:105)| (green curve), |(cid:104)a(cid:105)| (orange curve) and auto-
−
by a skewed Lorentzian curve with an amplitude depen- correlationfunctiong(2)(0)(bluecurve)forvaryingdrivefre-
dent frequency shift, approaching the value σzg2/δ for quency and εd/κ = 16.67 (corresponding to the top level of
|α|→0, and restoring the linear regime limit [24]. Con- the phase space diagram in (a)). Parameters: g/δ = 0.14,
versely, for large intracavity amplitudes and δ =0, their 2κ/γ =12, g/γ =3347, nscale =12.68.
expressionyieldsthedrivedetuningcorrespondingtothe
tworesonantpaths,asrevealedbythenonlinearequation
metastable states [12, 28]. The resulting critical slow-
ε2
|α|2 = d , (7) ing down is a direct consequence of nonlinear dynamics
κ2+[∆ω ∓g/(2|α|)]2
c (see Chapter 5 of [8]) and the departure from a Gaus-
that explains the split Lorentzian response at resonance. sian probability distribution. As the authors of Ref. [12]
This distinct split has a direct relation to the origin of note, “bistability is a macroscopic phenomenon reached
phase bistability, namely the formation of two quasi- inthelimitnscale →∞”. Inourcase,thebimodaldistri-
independent excitation ladders with a vanishing connec- butions identify distinct states that are long-lived on the
tion between them [10]. The region of high powers in time scale γ−1 (and consequently on the scale (2κ)−1)
the strongly dispersive regime can be accessed for non- even for nscale =12.68. Bimodality is depicted in Fig. 3
demolition qubit readout with ∆ω ∼ κ [20, 25, 26]. [Panel I, frames (a)-(c)], associated with maximal qubit-
c
Dynamical mapping of the qubit to the photon states cavity entanglement, as we will see later on. The bright
has been also proposed in [27] as an optimized proto- state is quadrature-squeezed along the mean-field direc-
col, exploiting the first-order phase transition by means tion (in a similar way to resonance fluorescence [8, 29]),
of which the photon blockade breaks down (see also the another display of the JC nonlinearity in this regime.
following section). InthePanelIofFig. 3weplotquasidistributionfunc-
tions showing coexistent metastable states in a region
where the qubit is significantly excited and approaches
IV. PHASE TRANSITION CROSS-OVER progressively the equator in the Bloch sphere. For in-
creasing drive amplitude ε /κ we observe a growing sep-
d
Wewillnowdelineatethedefiningfeaturesofcomplex arationofthetwometastablestatedistributionsfollowed
amplitude bistability with the aim of approaching phase by a change in their orientation [frames (a)-(c)]. The
bistability, the occurrence of which signals the ultimate quasidistribution function in Fig. 3 (d) of Panel I il-
difference between the effective Duffing and the full JC lustrates a precursor of phase bistability for ε = g/2,
d
nonlinearity. For that purpose we plot the Q function lacking nevertheless complete symmetry with respect to
in the steady state, Q(x+iy)=(1/π)(cid:104)x+iy|ρ |x+iy(cid:105), the horizontal axis (and hence having peaks of unequal
c
for the reduced cavity matrix ρ and the coherent state height) because δ (cid:54)= 0. We build upon this theme in
c
|x+iy(cid:105). The region of coexisting states with probabili- the Panel II of Fig. 3, where we track the emergence
ties of the same order of magnitude marks the bound- of phase bistability for decreasing values of δ/g, and
ary in the phase space of the drive where quantum ∆ω = 0, ε = g/2. As δ/g → 0, nonlinearity is trig-
c d
fluctuations induce equiprobable transitions between the gered by lower photon numbers and the two peaks ap-
6
proach each other (for the same values of ε /κ), while
d
complete symmetry is restored only when δ =0.
We proceed now to the study of entanglement as a
measure of the joint participation of both quantum de-
grees of freedom, employing the von Neumann entropy
for the reduced qubit density matrix ρ = Tr ρ in the
q c
steady-state (where Tr denotes the partial trace over
c
the cavity field states), defined as S = −Tr[ρ lnρ ] =
q q q
(cid:80)
− λ lnλ . Theeigenvaluesλ ofthereducedqubit
i=1,2 i i i
matrix ρ = (ρ ,ρ ;ρ∗ ,ρ ) are given by the expres-
q gg ge ge ee
sion [30, 31]:
1(cid:20) (cid:113) (cid:21)
λ = 1± (ρ −ρ )2+4|ρ |2 . (8)
1,2 2 gg ee eg
TheentropyS quantifiestheentanglementbetweenthe
q
two quantum oscillators, assessing the purity of the re-
duced quantum state for the qubit in the steady state
(following the evolution of the open system from an ini-
tial pure state). In the dispersive regime there is still
appreciable entanglement between the cavity and qubit
despitetheirstrongdetuning. Ithasrecentlybeenshown
thatentanglementisalsopresentinthelinearregion[32],
which we have neglected when setting (cid:104)σ (cid:105) = −1 in our
z
analytical mapping to the Duffing oscillator. The entan-
glement entropy tracing a first-order phase transition in
the drive phase space is shown in Fig. 4(a). From the
linearregion,whereentanglementisveryweak[lightblue
region in Fig. 4(a) appearing at ∆ω = g2/δ], we move
c
tothenonlinearregimewherethemaximumshiftstothe
left with a very steep drop, in a similar manner to the
average photon number (cid:104)n(cid:105)=(cid:104)a†a(cid:105), due to the presence
ofgrowingamplitudebistability[comparetoFigs. 1and
3(a) of [10]]. In Fig. 4(b) we plot the second-order cor-
relation function for zero time delay, g(2)(τ =0), defined
via the relation g(2)(0) = (cid:104)n(n−1)(cid:105)/((cid:104)n(cid:105)2), in order to
reveal the effect of quantum fluctuations. The peak of
quantum correlations is shifted relatively to the entan-
glement entropy maximum, with the two curves (blue
and red, respectively) intersecting closer to the position
of the coherent cancellation dip in the cavity amplitude
|(cid:104)a(cid:105)| [similar to the point A in Fig. 1(a)] and the pseu-
dospin projection |(cid:104)σ (cid:105)|. The aforementioned dip has a
−
purely quantum origin at zero temperature, which ex-
plains the amplification of quantum fluctuations in that
region [18, 33]. On the other hand, the maximum of FIG. 5. Nonlinearity for varying nscale. (a) Growing entan-
thevonNeumannentropyoccursatthefrequencywhere glemententropySq asafunctionofthedrivingfrequencyfor
g/δ = 0.14 (with n = 12.68) and five equispaced drive
the dim and bright state distributions attain peaks of scale
amplitudes in the range ε /γ =[20,100]. The red curve cor-
equal height, as we can observe in Fig. 5(a). When d
responds to ε /γ = 100 at the peak of which we plot the
δ/g → 0 the system response becomes highly nonlinear d
function Q(x+iy) for the intracavity amplitude distribution
for low drive strengths, as n decreases. We observe
scale (inset). The green curve depicts the relative difference of the
enhancedresonantmulti-photontransitions[insetofFig. Q function peak values h , h (with h > h ), defined as
1 2 1 2
5(b)] gradually disappearing in the region of high drive r = (h −h )/h . (b) Growing entanglement entropy S
1 2 1 q
strengths [main panel of Fig. 5(b)]. This phenomenon is as a function of the driving frequency for g/δ = 0.87 (with
referredtoasbreakdown of the photon blockade[seeFigs. n =0.33) and five equispaced drive amplitudes (for each
scale
2(a) and 5(a) of [11], and [10] for an extensive discussion differentcolour,increasinginthedirection: lightblue,green,
at resonance – δ = 0] accompanied by the appearance orange, red, deep blue) in the range εd/γ =[200,1000]. The
upper-right inset depicts successive multi-photon resonances
of amplitude bistability [see the Q function plot in the
forthesamefivedriveamplitudesasin(a). Thebottom-right
bottom inset of Fig. 5(b)].
inset depicts the quasiprobability function Q(x+iy) corre-
sponding to the marked point M. Parameters: 2κ/γ = 12,
g/γ =3347.
7
(cid:18) g (cid:19)−1
α=−iε κ±i , (9)
d 2|α|
as well as a bistable qubit vector lying on the equatorial
plane (ζ ≡ (cid:104)σ (cid:105) = 0) with ν ≡ (cid:104)σ (cid:105) = ±α/(2|α|). In
z −
that regard, phase bistability corresponds to maximally
entangled states of the two coupled quantum degrees of
freedom, in which the qubit polarization and the cavity
field are not enslaved to the external drive, as already
predicted by the mean-field analysis of Ref. [17].
VI. CONCLUSION
In this letter we have examined the interplay of
qubit-cavity entanglement and cavity bimodality when
connecting the dispersive and the resonance regimes
in the driven dissipative Jaynes-Cummings model for
varying qubit-cavity detuning. For the assessment of
the cavity nonlinearity we have employed both the
FIG. 6. Changing excitation paths at resonance. Entangle-
mean-fieldandtheMasterEquationtreatmentincluding
mententropyS intheregionofphasebistability(δ=0)asa
q quantum fluctuations. We have followed the change of
functionofthedrivingfrequencypastthethresholdε =g/2
d the intracavity field quasidistribution functions from the
(compare with Fig. 1 of Ref. [10] and see Eq. (7), in the
strongly dispersive regime to the gates of a critical point
absenceofspontaneousemission). Theinsetsdepictthejoint
related to a second-order quantum phase transition at
quasidistributionfunctionQ(x+iy)forthemarkedpointsA,
BandC.Parameters: 2ε /g=1.06,2κ/γ =500,g/γ =3347. resonance. We have also included in our discussion the
d
complex amplitude bistability encountered in the driven
dissipative Duffing oscillator, adopting a perturbative
approach for weak driving fields. This is a region of
minimal entanglement and very weak qubit involvement
V. PHASE BISTABILITY
in the formation of the system nonlinearity, for the
quantum description of which we have employed an
Letusfinallylinktheincreasingentanglemententropy analyticalformoftheWignerquasidistributionfunction.
to the appearance of phase bistability past the thresh- The growing participation of both coupled quantum
old set by the critical point of the second-order quantum degrees of freedom marks the passage from a first-order
dissipative phase transition: (∆ω = δ = 0, ε = g/2). to a second-order dissipative quantum phase transition.
c d
At resonance, the nonlinearity can be triggered by low
photon numbers with a different scaling parameter, as- Acknowledgments. The author wishes to thank
sociated with a strong-coupling limit [10], as opposed to H. J. Carmichael and E. Ginossar for inspiring discus-
the strongly dispersive regime. Fig. 6 shows the devel- sions. He acknowledges support from the Engineering
opment of a phase-bimodal distribution as we cross the and Physical Sciences Research Council (EPSRC) under
line ∆ω = 0, where the entanglement entropy has a lo- grants EP/I028900/2 and EP/K003623/2.
c
cal maximum. For growing drive strength, the entropy
atpointBincreasesandthetwopeaksoftheQfunction
movefurtherapartcomparedtotheirthresholdposition,
always remaining symmetrical with respect to the hori-
zontal axis.
Atthisstage,itisinstructivetoinvokeforafinaltime
the solution above threshold of the so-called neoclassical
equations, i.e. the semiclassical equations that conserve
the length of the Bloch vector [10] (in the absence of
spontaneousemission)whicharealsocombinedtoderive
the steady-state expression of Eq. 3 in the dispersive
regime. Neoclassical theory predicts a parity-breaking
transition at resonance, according to the equation
8
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