Dynamical many-body phases of the parametrically driven, dissipative Dicke model

3.3 MB·English

by  R. Chitra

Dynamical many-body phases of the parametrically driven, dissipative Dicke model
R. Chitra and O. Zilberbeg
Institute for Theoretical Physics, ETH Zurich, 8093 Zu¨rich, Switzerland
ThedissipativeDickemodelexhibitsafascinatingout-of-equilibriummany-bodyphasetransition
as a function of a coupling between a driven photonic cavity and numerous two-level atoms. We
study the effect of a time-dependent parametric modulation of this coupling, and discover a rich
phasediagramasafunctionofthemodulationstrength. Wefindthatinadditiontotheestablished
normalandsuper-radiantphases,anewphasewithpulsedsuperradiancewhichwetermdynamical
normal phase appears when the system is parametrically driven. Employing different methods, we
5 characterizethedifferentphasesandthetransitionsbetweenthem. Specificheedispaidtotherole
1 ofdissipationindeterminingthephaseboundaries. Ouranalysispavestheroadfortheexperimental
0
study of dynamically stabilized phases of interacting light and matter.
2

n PACSnumbers: 42.50.Pq,05.30.Rt,32.80.Qk,42.65.Yj
a
J

8 Experimental progress in control and manipulation of of freedom of a BEC to a quantized mode of a laser-
2 light-matter quantum systems has generated a growing drivenopticalcavity,andthetheoreticallypredictednon-
interestinmany-bodyphenomenaoutofequilibrium[1]. equilibrium phase transition has been observed [7, 13].
]
s Well established examples of such systems include ultra- Moreover,theinevitablephotonleakageoutofthecavity
a cold atomic or ionic quantum gases in high finesse opti- aswellasdissipationoftheBEChasbeenshowntolead
g
calcavities[2],semiconductormicrocavitiesinthestrong to a considerable modification of the critical exponents
-
t couplingregime[1,3],andsuperconductingqubitsinmi- of the transition [14].
n
crowave resonators [4, 5]. The engineered interplay be-
a
u tween light and matter in these systems has led to the
q observation of a host of fascinating collective phases and
.
t quantum phase transitions including superfluidity in po-
a
laritons [6] and the super-radiant Dicke phase transition
m
in a Bose-Einstein condensate (BEC) coupled to an op-
-
d tical cavity [7].
n A defining feature of many of these systems is that
o
they are inherently driven and subject to dissipation.
c
Driven dissipative systems are usually treated within a
[
rotating frame formalism. This effectively renders the FIG.1. AsketchofaparametricallydrivenDickemodel. A
1
problem time-independent with the important feature single-modecavityisdrivenbyalaserbeamwithanoscillat-
v
that the asymptotic steady states are necessarily out of inglaser-fieldpowerP(t). Thecavityisnaturallyleakywitha
8
9 equilibrium. However, parametric driving of the system dissipationrateκ. Insidethecavity,anatomiccloudiscooled
and forms a Bose-Einstein condensate that is coupled to the
0 often does not allow the usual rotating frame simplifi- (cid:112)
driving laser with coupling λ(t) ∝ P(t) [cf. Eq. (1)]. The
7 cations. Consequently, the resulting interplay between
atoms are also coupled to an environment with a dissipation
0
interactions, dissipation, and parametric driving, could
. rateη. ThesystemisbestdescribedbyLiouvilliandynamics
1 lead to novel and exotic steady-state physics that has no [cf. Eq. (2)].
0 counterpart in the undriven case. Parametric driving is
5
increasingly used as an experimental tool in diverse con- Inthiswork,weanalyzetheimpactofparametricdriv-
1
: texts,e.g.,inthegenerationofFloquettopologicalinsu- ing on the phase diagram of the dissipative Dicke model.
v lators [8], improved measurement fidelity with squeezed Specifically,weconsideramodulationoftheatom-cavity
i
X quantumstates[9,10]andunconventionalphenomenain coupling, whichiseasilyrealizableincurrentexperimen-
r cavity QED [11]. tal setups, see Fig. 1. Using a combination of mean field
a
A prime example of a system exhibiting light-matter theoryandeffectiveHamiltonians,weobtainarichphase
collective phenomenon is the Dicke model [12]. Here, a diagramcomprising: (i)theNPwithparametricamplifi-
bosonic/cavitymodeiscoupledtoalargenumberoftwo- cation,(ii)theSPphase,and(iii)anoveldynamicalnor-
levelatoms. ItexhibitsaZ symmetrybreakingquantum malphase(D-NP),whichappearstobeadynamicallyro-
2
phasetransitionfromanormalphase(NP),whereallthe tating NP with pulsed super-radiance. We elucidate the
atomsareintheirgroundstateandthecavityisempty,to vital role that dissipation plays in modifying the com-
a super-radiant phase (SP), where the atoms are excited plex phase topography of this nonequilibrium system.
and the cavity is in a coherent state. This model has Our analysis presents parametric driving as a promis-
recently been realized by coupling the external degree ing frontier in the search for exotic collective phases in
2
(a) α¯ x¯ y¯ z¯
2.0 | | 2.0 | | 0.5 2.0 | | 0.5 2.0 | | 0.5
1.5 2.0 1.5 1.5 1.5
D-NP
Ω Ω 0.3Ω 0.3Ω 0.3
/ 1.0 / 1.0 NP / 1.0 / 1.0
ω0 1.0ω0 ω0 ω0
0.5 0.5 0.1 0.5 0.1 0.5 0.1
SP
0.0 0.0 0.0 0.0 0.0
0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0
(cid:15)/Ω (cid:15)/Ω (cid:15)/Ω (cid:15)/Ω
(b) max α(ω=0) max x(ω=0) max y(ω=0) max z(ω=0)
2.0 { 6 } 2.0 { 6 } 0.5 2.0 { 6 } 0.5 2.0 { 6 } 0.5
1.5 2.0 1.5 1.5 1.5
Ω Ω 0.3Ω 0.3Ω 0.3
/ 1.0 / 1.0 / 1.0 / 1.0
ω0 1.0ω0 ω0 ω0
0.5 0.5 0.1 0.5 0.1 0.5 0.1
0.0 0.0 0.0 0.0 0.0
0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0
(cid:15)/Ω (cid:15)/Ω (cid:15)/Ω (cid:15)/Ω
FIG.2. CharacteristicsofthedynamicalphasediagramoftheparametricallydrivenDickemodelasafunctionofcavity/atoms
frequencyω =ω =ω andparametricmodulationstrength(cid:15)forbarecouplingλ =0.4Ωanddissipationratesκ=η=0.1Ω.
0 c a 0
We obtain these characteristics from the numerical steady-state solutions of the mean field equations [cf. Eqs. (6)-(8)]. Each
point, with a 10−3 resolution, is a result of a different numerical integration. Superimposed (striped overlay) is the normal
modestabilityzone[cf.Eq.(5)]. (a)Densityplotsoftheabsolutevalueofthesteady-statetime-averagedorderparameters. (b)
Insteady-statetheorderparametersareoscillatingwithacomplexbeatstructure[15]. Thedensityplotsareoftheamplitude
of the maximal frequency ω contributing to this oscillation. We see three distinct phases appearing as a function of (cid:15), i.e. ,
extensionsofthenormalandsuper-radiantphases(NPandSP),aswellasanoveldynamicalphase(D-NP).Thesethreephases
meet at a shared multicritical point. The properties of each phase are summarized in Table I.
light-matter systems. Stability A1 Stability A2 |α¯| |x¯| |y¯| |z¯|
NP yes yes 0 0 0 1/2
The single mode parametrically driven Dicke model is
SP no yes (cid:54)=0 (cid:54)=0 (cid:54)=0 (cid:54)=0
described by the Hamiltonian
D-NP no ((cid:63)) no ((cid:63)) 0 0 0 <1/2
N 2(cid:126)λ(t) N TABLE I. Summary of normal mode [cf. Eq. (5)] and mean
H(t)=(cid:126)ω a†a+(cid:126)ω si + si(a+a†), (1) field [cf. Eqs. (6)-(8)] analyses. In Fig. 2(a), we observe
c a z √N x threemainregions,dubbednormalphase(NP),super-radiant
i=1 i=1
(cid:88) (cid:88) phase(SP),anddynamicalnormalphase(D-NP).Eachregion
manifests a different behavior summarized here, i.e. , which
where siα with α=x,y,z are the spin operators describ- normal mode is stable in each region [15], and what are the
ing the ith two level atom, and a,a† represent the cavity valuesofthedifferentorderparameters. IntheD-NPregion,
creation and annihilation operators. The cavity’s reso- we denote by “no ((cid:63))” that at least one normal mode is un-
nance frequency is ω , whereas the atoms are considered stable.
c
to be identical with level spacing (cid:126)ω . We consider a
a
time-dependentcouplingbetweentheatomsandthecav- ρ of the system
sys
ity of the form λ(t)=λ +(cid:15)cos(2Ωt). Such a coupling is
0
dρ i
easily generated by a modulation of the laser power that sys = [H(t),ρ ]+κ[2aρ a† a†a,ρ ]
drives the cavity [16]. For (cid:15)=0, the system exhibits the dt −(cid:126) sys sys −{ sys}
well known continuous phase transition from a NP to a η N
+ [2si ρ sj si sj ,ρ ], (2)
SP when the coupling λ0 √ωcωa/2 [12]. For (cid:15) = 0, N − sys +−{ + − sys}
we reiterate that the param≥etric driving described(cid:54)here i(cid:88),j=1
cannot be rotated away by a suitable choice of frame. where sj = sj +isj are ladder operators. The first
± x y
Indeed, recent treatments of similar modulations of the
term on the r.h.s. describes the standard Hamiltonian
Dicke model were addressed using a mapping to para-
evolution and the last two terms represent the Marko-
metric oscillators, and a partial phase diagram for the
vian dissipation for both cavity and a global dissipation
NP was obtained [17, 18]. Here, we explicitly include for the atoms in Lindblad form with rates κ and η, re-
dissipationforboththecavityandtheatoms(seeFig.1)
spectively [19]. Note that this approach is valid in the
and analyze the impact of parametric driving on the full
Born-Markov limit of weak dissipation.
phase diagram of the dissipative Dicke model, see Fig. 2. In the absence of paramteric driving, (cid:15)=0, the NP is
The driven and dissipative nature of the system is de- well described by considering the collection of two-level
scribed by a Liouvillian equation for the density matrix atoms as constituting a giant spin S = si aligned
i
(cid:80)
3
along the z axis [20]. The deviations of this giant spin Mathieu oscillators where dissipation results in a modifi-
awayfromthisquantizationaxiscanbecharacterizedby cationofthestabilitycriterionfortheparametricoscilla-
the standard Holstein Primakoff representation for the tor. Inparticular, weakMarkoviandissipationleadstoa
spin operators, S = b†b N,S = √N b†bb, and pronouncedstabilizationoftheNPinthevicinityofthese
z − 2 − −
S+ =b†√N −b†b,whereb,b† arestandardbosonicoper- resonantfrequenciesforsmalldriving(cid:15)(cid:28)λ0,andbarely
ators[20]. Thisapproachcanbeextendedtoaddressthe affects the stability at higher drive amplitudes [24, 25].
stabilityoftheNPinthepresenceofparametricdriving, Indeed in Fig. 2, we see substantial stabilization of NP
(cid:15)=0. SincedeviationsofSfromthez axisareexpected at the resonant frequency f2,1. Such stabilization of the
(cid:54)
to be small, as N we can map the Dicke Hamil- NP in the many-body context of the Dicke model is a
→ ∞
tonian [cf. Eq. (1)] onto the problem of two harmonic manifestationoftheexplicitlydissipationdependentnon-
oscillators whose coupling is parametrically driven, equilibrium asymptotic state.
From Fig. 2, we see that the NP occupies only a small
H (t)=(cid:126)ω a†a+(cid:126)ω b†b+(cid:126)λ(t)(b+b†)(a+a†). (3) part of the phase diagram when the system is paramet-
NP c a
rically driven. However, what lies beyond these stable
Focusing on the case ω ω = ω , Eq. (3) can be NP regions cannot be accessed by the current approach
0 a c
≡
diagonalized in terms of normal modes, and requires another method, such as mean field theory,
which is well justified for the Dicke model in the limit of
H (t)=(cid:126)Ω (t)A†A +(cid:126)Ω (t)A†A , (4) N 1. This method was successfully used for studying
NP 1 1 1 2 2 2 (cid:29)
theSPwhichhasbrokenZ symmetry,forthenon-driven
2
where for m = 1,2, Ω2 (t) = ω2 ( 1)m2λ(t)ω are case (cid:15) = 0 [12, 20, 26]. The mean field ansatz that we
m 0 − − 0
time-dependent normal mode frequencies, and A = use states that the total density matrix in the steady
m
√1 [ +(a ( 1)mb)+ −(a† ( 1)mb†)] are the corre- state is a product state of the individual density matri-
s2Ωp√mo2n±Sdωmi0ng. nF−oorrm−caolmmpoudtaetSoiompnearlats−oimrs−pwliictihtyc,oweffie cailesnotsasSsum±m=e cmeas,trρicsyesso=ftρhce⊗caviNit=y1a⊗nρdit,hwehiethreatρocma,nrdesρpiecatrievedlyen[s1i9t]y.
2 ω0Ωm Furthermore, sinc(cid:81)e all atoms are identical, we assume all
κ=η γ[21].
≡ ρ tobeequivalent. SubstitutingthisansatzintoEq.(2),
Each normal mode is a quantum Mathieu paramet- i
we obtain a set of coupled non-linear equations for the
ric oscillator, i.e. , its fundamental harmonic frequency
mean-fieldorderparametersoftheparametricallydriven
varies sinusoidally in time [22, 23]. The stability of
and dissipative Dicke model
eachquantumparametricoscillatorcanbededucedfrom
its displacement Tr ρ(t)(A +A† ) . It results in a
m m α˙ = iω α 2iλ(t)x κα, (6)
c
complex stability diagram comprising “Arnold tongues” − − −
(cid:8) (cid:9) x˙ = ω y ηx, (7)
which delineate regions where the displacement, though − a −
parametrically amplified, remains bounded (stable), and y˙ =ωax 2λ(t)[α+α∗]z ηy, (8)
− −
those where the displacement grows exponentially with
where we have defined the order parameters α =
time (unstable) [15, 24, 25]. Incidentally, the result-
a /√N, x = sx /N, y = sy /N, z =
ing stability diagram for the quantum oscillator is the (cid:104) (cid:105) (cid:104) i i(cid:105) (cid:104) i i(cid:105)
same as that of the classical damped Mathieu oscilla- (cid:104) iszi(cid:105)/N, and ha(cid:80)ve assumed z = (1(cid:80)/2)2−|x|2−|y|2
tor obeying the classical equations of motion for the since the mean field equations satisfy the constraint
displacement[15, 24, 25], x(cid:80)2+y2+z2 =1/4. (cid:112)
Equations (6)-(8) form a set of coupled non-
x¨ +γx˙ +Ω2 (t)x =0, (5) autonomous differential equations. Using numerical stiff
m m m m
ordinary differential equations solvers, we integrate this
The combination of the stability diagrams of the two set of equations to a long time limit, where we ob-
normalmodesyieldsthestabilityoftheNP,i.e.,theNP tain a convergent behavior. We find that, generically,
isstableonlyifbothdissipativenormalmodesarestable, thesteady-statemean-fieldsolutionsshowoscillatorybe-
see shaded area in Fig. 2. In the absence of dissipation, havior around a zero or non-zero mean value [15]. In
κ = η = γ = 0, each normal mode A is unstable in Figs. 2(a), we plot the absolute time-averaged order pa-
m
thelimitofinfinitesimalparametricdriving(cid:15) 0atres- rameters, α¯ , x¯, y¯ and z¯, averaged over a sufficiently
→ | | | | | | | |
onant frequencies f = (λ )2+n2Ω2 + ( 1)m(λ ) long time window in the steady-state[15]. Note, that the
m,n 0 0
−
where n = 1,2,3... [22, 23]. We find that the lower absolute value is taken for presentation reasons only, we
(cid:112)
boundary of the NP is dictated by the lowest Arnold always have α¯,y¯,z¯>0 and x¯<0. Based on these mean
tongue of A , i.e. , where the time-independent part of values, we find that the SP, characterized by α¯ = 0, ex-
2
(cid:54)
Ω (t) becomes negative. The remaining stability bound- tends from its zero drive region of ω 2λ onto a large
2 0 0
≤
aries of NP are determined by frequencies where either regime spanning both small and large drive amplitudes.
A becomes unstable. The impact of disspation on the At the critical frequency ω (ω ,(cid:15)), we observe a tran-
m crit 0
stability of NP can be understood from the physics of sition to a region with α¯ =x¯=y¯=0. Contrasting these
4
results with those obtained through the study of normal derstood from Floquet analysis [27]. In Figs. 2(b), we
modes, we see that for (cid:15) (cid:46) 0.3Ω, the NP lies above the plot the amplitude of the largest ω =0 peak in the FFT
(cid:54)
line defined by ω with z¯ = 1/2. The NP SP tran- landscape in order to quantify the overall extent of the
crit
↔
sition in this regime is thus an extension of the usual oscillation. We find that, SP has weak oscillations in all
continuous Dicke transition at zero drive to finite para- of the order parameters, whereas the D-NP is strongly
metric driving. Note that the details of the transition oscillatory in the x y plane. These aspects are better
−
may still differ from the standard Dicke transition as the highlighted in Fig. 3, by plotting the steady-state time-
parametrically-driven NP accommodates a large number dependent trajectory of the total spin si (t) on the
(cid:104) i | (cid:105)
of photons in the cavity. Bloch sphere for different parameter values. It appears
(cid:80)
Interestingly, at (cid:15) 0.3Ω, we see a sudden change thatadistinguishingcriterionbetweentheSPandD-NP
∼
in the curvature of ω . This exactly signals the point is whether the trajectory encircles the z-axis. Our re-
crit
wheretheNPendsandanoveldynamicalphase,dubbed sults seem to indicate that the most plausible candidate
dynamical-NP (D-NP), starts. As opposed to the NP, fortheD-NPisa”normalphase”inadynamicalrotating
thoughα¯ =0,thisphasehasoscillatoryα(t)anddoesnot frame.
haveitsspinalignedalongthez-axis,i.e.,z¯<1/2. Addi- Combining the results from the normal modes and
tionally,thisregioncorrespondsexactlytotheparametri- mean field analyses, we see that periodic modulation of
callyunstableArnoldtonguesoftheaforementionednor- the atom-light coupling results in a rich phase diagram,
mal modes. As a result, the point ((cid:15) 0.3Ω,ωcrit), ap- characterized by a multitude of dynamical phase bound-
∼
pears to be a multicritical point where the three phases: aries between the NP, SP and the intriguing new phase,
NP, SP, and the new D-NP intersect. The principal fea- D-NP. All three phases meet at a multicritical point. In
tures of the three phases are summarized in Table I. the D-NP, the cavity periodically emits pulses of pho-
tons with opposing phases, which should be detectable
experimentally. The NP SP boundary is principally
→
dictated by where the normal mode A becomes unsta-
2
ble, whereas the NP D-NP boundary is fixed by the
→
instabilityofanyofthemodesA [15]. Withinourmean
m
fieldapproach,wefindthetransitionsSP NP,D-NPto
→
be continuous, though the latter is rather sharp. How-
ever, the nature of the NP D-NP transition cannot
→
be studied within our approach. The topography of the
phase diagram is expected to vary with the choice of λ
0
and the strength of dissipation. Consequently, though
the three phases would exist, the highly sensitive multi-
critical point may disappear.
Remarkably, we see that dissipation leads to a sizable
FIG. 3. Trajectories on the Bloch sphere of the atoms
stabilization of the NP. Due to the parametric nature of
order parameters as a function of time in steady-state,
thenormalmodes,theNPinthedrivencasecanmanifest
f[x(t),y(t),z(t),t]. The trajectories are for (cid:15) = 0.5Ω, λ =
0
a dissipation-assisted generation of substantial entangle-
0.4Ω, and κ = η = 0.1Ω. The trajectory that does not en-
circle the z-axis (magenta) is in SP with ω = 0.6Ω. The ment/squeezingbetweentheatomsofthecondensateand
0
trajectories that encircle the z-axis (red and yellow) are for the cavity [21]. The physical signature of such entangle-
ω0 =0.65Ω and ω0 =1.5Ω, respectively. ment as well as the impact of parametric driving and
dissipation on the critical exponents defining the differ-
To better understand the nature of the D-NP, as well ent phase transitions merit in-depth studies. It would
as the effects of the drive (cid:15) = 0 on the NP and SP, also be interesting to extend the present work to other
(cid:54)
we analyze the oscillatory behaviour around the steady- parameter regimes like ωa << ωc realized in current ex-
state mean-field solutions of Eqs. (6)-(8). Typically, we perimental setups [7].
find that each phase has a different oscillatory behav- Our work shows that parametric driving is a powerful
ior: (i)inNP,theorderparametersconvergetozeroand tool in the quest for new physics, which exists exclu-
do not oscillate, (ii) in SP, the oscillations are small but sively in the realm far from equilibrium. The richness
mildly grow with (cid:15), and (iii) in D-NP, the order param- of the physics seen in the simple Dicke model presages
eters oscillate strongly around zero [15]. We, then, Fast intriguing phenomena in time-dependent systems, which
Fourier Transform (FFT) the steady-state solutions for requires the development of new theoretical methodolo-
eachω and(cid:15). InbothSPandD-NPregimes,theoscilla- gies. This frontier is potentially best explored using ex-
0
tionshaveacomplexbeatstructurethatcorrespondstoa perimental light-matter systems.
combofpeakedfrequenciesatω =0,aswellasω =Ω,ω We would like to thank T. Donner, R. Mottl, R.
0
(cid:54) (cid:54)
[15]. Note that the frequencies that appear can be un- Landig, T.Esslinger, L.Papariello, and E. van Nieuwen-
5
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0
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6
SUPPLEMENTAL MATERIAL
Tr ρ(t) A1+A†1 Tr ρ(t) A2+A†2
2.0 n (cid:16) (cid:17)o 2.0 n (cid:16) (cid:17)o
I. NORMAL PHASE 1.5 1.5 unstable
stable
Ω Ω
/ 1.0 / 1.0 stable
ω0 ω0
A normal mode with time-dependent frequency mod- unstable
0.5 0.5
ulation [cf. Eq. (4) in the main text] can be generically unstable
describedbytheHamiltonianforaparametricoscillator: 0.0 0.0
0.0 0.5 1.0 0.0 0.5 1.0
p2 1 (cid:15)/Ω (cid:15)/Ω
H = + [ω2+µ(t)]x2. (I.1)
2m 2 0
FIG. I.2. The numerical stability diagram of the normal
For a classical oscillator with sinusoidal modulation, the
modes A [cf. Eq. (4) in the main text] for λ = 0.4Ω and
m 0
above Hamiltonian describes the well known Mathieu
κ = η = 0.1Ω. As the drive in the Dicke model affects each
parametric oscillator [22, 23]. The quantum parametric modedifferently,weobtaintwo“Arnoldtongue”stabilitydi-
Hamiltonian can be rewritten as agramsthatareshiftedandscaledwithrespecttoeachother.
µ(t) µ(t) Superposing the zones where both modes are stable leads to
H =[ω0+ 2ω ]a†a+ 4ω (a2+a†2) thestabilityofthenormalphase(NP)[cf.Fig.2inthemain
0 0 text].
ω2+µ(t)a¯†a¯, (I.2)
≡ 0
(cid:113)
where the ladder operators a,a† are defined with respect where the dissipative kernel K(t) = dωJ(ω)e−iωt and
to the time independent model (i.e., µ(t)=0). J(ω) is the spectral density of the dissipative bath.
(cid:82)
These coupled first order integro-differential equations
3 3 are rather difficult to solve and numerical solutions are
o2 o2 needed. However, for standard Markovian dissipation,
(cid:17) (cid:17)
A†1 1 A†2 1 induced by cavity leakage in the rotating frame or cou-
+ +
A1 0 A2 0 pling to an ohmic bath, K(t) = γδ(t) where γ is the
ρt() n(cid:16)−1 ρt() n(cid:16)−1 ωω00==00..57 dwaemspeeintgharattteh.eySubbesctoitmuetinagsethtiosfinlinEeaqrs.O(ID.5E)s.anOdb(sIe.r6v)-,
Tr −2 Tr −2 ω0=0.9 ables and correlation functions can easily be obtained
3 3
− 0 20 40 60 80 100 − 0 20 40 60 80 100 from these solutions. For example,
tΩ tΩ
a(t) =G(t) a(0) +L∗(t) a†(0) + F(t) , (I.7)
(cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105) (cid:104) (cid:105)
FIG. I.1. Characteristic numerical time-integration plots of
Eq.(I.8)indifferentregionsofthephasediagram. Allcurves where the expectation values are with respect to the ini-
are with λ0 = 0.4Ω, (cid:15) = 0.5Ω, and κ = η = 0.1Ω. We see tialdensitymatrix. Assuminginitialconditionssuchthat
that for (i) ω0 =0.5Ω, mode A1 is stable whereas A2 is not, F(t) =0, which are expected for baths with no partic-
(ii)ω =0.7Ω,modeA isunstablewhereasA isstable,and (cid:104) (cid:105)
0 1 2 ular ordering, we obtain
(iii) ω =0.9Ω, both modes A are stable.
0 m
x(t) =Tr ρ(t) a+a† (I.8)
Weconsideracouplingtoanexternalbathwhichisin (cid:104) (cid:105)
p(0)
the rotating wave approximation. Hence, the solutions =Re[(cid:8)G(t)(cid:0)+L(t)](cid:1)x(cid:9)(0) Im[G(t)+L(t)](cid:104) (cid:105).
totheHeisenbergequationsofmotion,inthepresenceof (cid:104) (cid:105)− mω0
dissipation, for the operators a and a† take the form [28]
For arbitrary initial conditions, the stability of the oscil-
a(t)=G(t)a(0)+L∗(t)a†(0)+F(t), (I.3) lator is dictated by whether the pre-factors [G(t)+L(t)]
a†(t)=G∗(t)a†(0)+L(t)a(0)+F†(t), (I.4) grow with time as one approaches the asymptotic state.
For the parametric oscillator, we expect it to become
whereGandLaretimedependentfunctionsobeyingthe
exponentially unstable as the strength of the driving is
initial conditions G(0)=G∗(0)=1 and L(0)=L∗(0)=
increased [24]. Choosing µ(t) = gcos(2Ωt), the zones of
0. F isanoperatortermwhichstemsfromthedissipation
stability can be traced in the ω g plane. The result-
and also depends on the functions G and L. It satisfies 0−
ing stability diagram is the same as that for the classical
the condition F(0) = F†(0) = 0. The functions G,L
Mathieu oscillators, which can also be extracted from
obey the integro-differential equations
theclassicalequationsofmotions[cf.Eq.(5)inthemain
G˙(t)=−i(ω0+µ2ω(t0))G−iµ2ω(t0)L−(cid:90)0tdsK(t−s)G(s), (I.5) texTto].obtain the full NP stability diagram of the Dicke
L˙(t)=i(ω0+ µ2ω(t0))L+iµ2ω(t0)G−(cid:90)0tdsK(t−s)L(s), (I.6) mofobdoetlh[seneorFmigal2minotdheesmAamin, tmext=], w1,e2s,tuwdityhthfreeqstuaebnicliiteys
7
(a) (c) tion studied here.
0.6
eters 0.3 eters0.15 αx FigW.eI.s1o,lvewetheprceosernretspcohnadriancgterEisqtsi.c (In.u5)m-(eIr.i6c)alantdimien-
aram 0.0 aram0.10 y integration plots of Eq. (I.8) for the normal modes Am.
erp 0.3 erp0.05 z Repeating this procedure for different ω0 and (cid:15), and by
ord− ord checking for converging/diverging solutions [cf. Fig. I.1]
0.6 0.00
− 0 500 1000 1500 2000 0.0 0.5 1.0 1.5 2.0 we find the stability diagram for both modes Am, see
tΩ ω/Ω Fig. I.2. The superposition of the two stability diagrams
(b) 0.6 (d)
then yields the stability of the normal phase shown in
eters 0.3 eters00..45 αx Fig. 2.
m m
ara 0.0 ara0.3 y
p p0.2 z
order−0.3 order0.1 II. MEAN-FIELD ANALYSIS
0.6 0.0
− 0 500 1000 1500 2000 0.0 0.5 1.0 1.5 2.0
tΩ ω/Ω In the main text, we obtained a set of coupled non-
autonomous mean-field equations [cf. Eqs. (6)-(8) in the
main text] for the mean field parameters α,x and y.
FIG.II.1. Characteristicnumericalanalysisofthemean-field
equations[cf.Eqs.(6)-(8)inthemaintext]leadingtothedis- Theseequationsweresolvednumericallywithavarietyof
played phase diagram [cf. Fig. 2 in the main text]. All plots ODE solvers. Characteristic numerical time-integration
are with λ0 = 0.4Ω, (cid:15) = 0.5Ω, and κ = η = 0.1Ω. (a) and plots of the solutions to these equations are shown in
(c) are characteristic numerical time-integration plots of the Figs.II.1(a)and(b). Notethatthesolutionsconvergeto
mean-fieldequationswithω =0.6Ωandω =0.65Ω,respec-
0 0 theasymptoticregimefortimestΩ 500andareoscilla-
tively. (b) and (d) are the corresponding Fast Fourier Trans- ∼
tory. The time scale for reaching the asymptotic regime
form (FFT). We see that in the super-radiant phase (SP)
varies with the parameters. Repeating this procedure
region [Figs. (a) and (b)], all order parameters are oscillat-
ingaroundanon-zeromeanwithrelativelysmalloscillations as a function of ω0 and (cid:15), the time-average of the order
amplitudes. In the dynamical normal phase (D-NP) region parametersoverthesteady-statebehaviour(thelastone-
[Figs. (c) and (d)], apart from z, all order parameters have third of the integrated time) is presented in Figs. 2(a) in
a zero mean value, but their oscillations are large taking the the main text.
full length of the central spin in its x and y components.
We find that typically the solutions show sinusoidal
oscillations characterized by the frequency of the para-
metric drive. However, in certain parameter regimes,
[cf. Eq. (4) in the main text]
the order parameters oscillate strongly with a complex
Ω2(t)=ω2+2λ ω +2ω (cid:15)cos(2Ωt), (I.9) beat structure involving multiple frequencies. To ana-
1 0 0 0 0
Ω2(t)=ω2 2λ ω 2ω (cid:15)cos(2Ωt). (I.10) lyze all these solutions in a systematic manner, we Fast
2 0 − 0 0− 0 Fourier Transform (FFT) the steady-state signals, see
To simplify our calculation, we also assume that the Figs. II.1(c) and (d). The largest amplitude of a finite
baths the two modes couple to, have identical spectral frequency in such plots serves as a measure for the ex-
densities[21]. Thoughrelaxingthisconditionwouldlead tentoftheoscillationthattheorderparamatersundergo.
tomoretechnicalcomplexity,itshouldnothaveanynon- This amplitude is plotted as a function of ω and (cid:15) in
0
trivial physical consequence in the limit of weak dissipa- Figs. 2(b) in the main text.