Rapid Steady State Convergence for Quantum Systems Using Time-Delayed Feedback Control

1.4 MB·

by  A L Grimsmo

Rapid Steady State Convergence for Quantum
Systems Using Time-Delayed Feedback Control
4
1
A.L. Grimsmo1, A.S. Parkins2 and B.-S. Skagerstam1
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2 1 DepartmentofPhysics,TheNorwegianUniversityofScienceandTechnology,
N-7491Trondheim,Norway
g 2 DepartmentofPhysics,UniversityofAuckland,PrivateBag92019,Auckland,
u
NewZealand
A
E-mail: [email protected]

6 E-mail: [email protected]
2
E-mail: [email protected]

]
h
Abstract. We propose a time-delayed feedback control scheme for open
p
quantumsystemsthatcandramaticallyreducethetimetoreachsteadystate. No
-
t measurementisperformedinthefeedbackloop,andwesuggestasimpleall-optical
n implementation for a cavity QED system. We demonstrate the potential of the
a schemebyapplyingittoadrivenanddissipativeDickemodel,asrecentlyrealized
u inaquantumgasexperiment. Thetimetoreachsteadystatecanthenreducedby
q twoordersofmagnitudeforparameterstakenfromexperiment,makingpreviously
[ inaccessiblelongtimeattractorsreachablewithintypicalexperimentalruntimes.

The scheme also offers the possibility of slowing down the dynamics, as well as
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qualitativelychangingthephasediagramofthesystem.
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2 PACSnumbers: 03.65.Yz 03.75.Gg 03.75.Kk 42.50.Dv
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1. Introduction
Steady states of open quantum systems, where driving forces and internal dynamics
are balanced by dissipation and/or other types of environmental noise, are often of
experimental interest. There are indeed a number of platforms currently available,
where the experimental control of the individual constituents of interacting quantum
systemsallowsprecisepreparationofinterestingandnon-trivialsteadystates,through
measurement and control of well-defined outputs and inputs of the system. These
include systemsusingtrappedionsorultra-coldatoms,opto-mechanicalsystems,and
systems based on cavity or circuit quantum electrodynamics (CQED) (see e.g. Refs.
[1–3]). Thiscanbeanalternativetoquantumstatepreparationbycoherent(unitary)
evolution, and is potentially of great interest to quantum information processing
technologies as a way of preparing computational resources, such as maximally
entangled states [4–7], or even providing for a route to quantum computation [8].
One great advantage of such an approach is that the steady state is robust against
variations of the initial state.
In practice, the time-scale on which such a steady state is reached is very
important; the generation of the desired state often requires a degree of control that
is hard to sustain over time, which can pose a challenge for finite-time experiments
[9–11]. In the present paper, we propose an all-optical feedback scheme relevant, for
example, to CQED systems, that can be used to i) change the stability of long time
attractors, so that one can switch between different behaviors, and ii) change the
characteristic time-scale for approaching a steady state, thus potentially speeding up
the convergence. The scheme we use is based on the time-delayed feedback control
method developed by Pyragas [12], and often referred to as the time-delay auto-
synchronization (TDAS) [13,14]. The control is based on coherent feedback, i.e., no
measurement is performed in the feedback loop, which can be advantageous, or even
necessary,forstabilizingthehighfrequencydynamicsofopticalsystemsorhighspeed
electrical circuits [13,14]. Another great strength of the approach is that it does not
require the steady state to be known a priori. We apply coherent TDAS, to our
knowledge, for the first time to a quantum system, with the feedback signal treated
quantum mechanically as well [15–17]. Delay-times have often been assumed to be
negligible in theoretical modelling of coherent quantum feedback, motivated by the
fact that a time-delay canintroduce undesirable instability to the system, andthat it
can often be made very small in practice [15,18]. In contrast, with TDAS, the delay
is the crucial ingredient for increasing the system’s stability. We note that delayed
coherentfeedbackhasalsobeenconsideredin[19,20]forasingle-atomsingle-excitation
system (and under the usual rotating wave approximation).
We will demonstrate the potential of our scheme by applying it to a highly
topical example, namely an open system version of the Dicke model. The Dicke
model is a paradigmatic model in quantum optics, describing the interaction of a
collection of two-level atoms with a single cavity mode [21]. This model has been
realized and studied in a number of recent experiments [9,22,23] based on a Bose-
Einsteincondensate(BEC)coupledtoafieldmodeofahighfinesseopticalcavity. The
cavity has a natural dissipative output channel, which has been used to monitor the
system in real time. In particular, the system undergoes a quantum phase transition
as an effective coupling strength between the BEC and the light field is increased
beyond a critical value, which can be observed through the intensity of the output
cavity field [9]. Spontaneous symmetry breaking has also been observed through a
3
heterodyne measurement scheme [22], and measuring the correlations of the density
fluctuations has been used to observe the diverging time-scale upon approaching the
criticalpoint[23]. This type ofmonitoringofa dissipativechannelis non-destructive,
andastheseexperimentshaveshown,offersaverypromisingroutefortheobservation
of complex many-body quantum dynamics.
Wewilltakeadvantageofthisdissipativechannelaswell,byusingitastheinput
to a non-invasive feedback loop that can alter the characteristic time-scale for the
relaxation of the system, as well as the stability of the long-time attractors. This is
particularly relevant for the BEC realization of the Dicke model where, as we will
discuss in more detail below, the approach to steady state can be slow compared
to typical experimental run times. Adverse effects such as spontaneous emission, or
atom loss, will eventually cause deviations from the desired, idealized behavior. This
isparticularlyproblematicclosetophaseboundaries,whereweexpectcriticalslowing
down. The system is therefore a very interesting test-case for our proposed feedback
scheme. We will treat the feedback in a semi-classical approach, where quantum
fluctuations are linearized. For the Dicke model, this approximation is valid in the
thermodynamiclimitofalargenumberofatoms. Theapproachwedevelopshouldbe
similarlyapplicabletoavarietyoftopicalquantum-opticalsystemsforwhichfeedback
can also have useful and interesting consequences (see e.g. Refs. [24–27]).
Thepaperisorganizedasfollows. InSection2weintroduceourfeedbackscheme
in a general setting, using the standard input-output theory for quantum optical
systems. Then, in Section 3, we introduce our primary system of study, the Dicke
model as recently experimentally realized, and apply our feedback scheme. We study
indetailthe effectoftheforcingdue tothe feedback,andfindoptimaldelay-timesfor
rapid convergence to steady state. We compare the performance of the system with
and without feedback, and demonstrate improvements in the relaxation time of two
orders of magnitude. In Section 4 we consider potential consequences for finite-time
experiments. In Section 5 we consider the influence of the feedback force on quantum
fluctuations. Finally, in Section 6, we give some concluding remarks.
2. Coherent time-delay auto-synchronization
The Pyragas’ time-delay auto-synchronization (TDAS) method is a continuous
feedback control method first developed to stabilize unstable periodic orbits and
equilibrium states embedded in a chaotic attractor [12–14,28]. We will also use the
method for manipulating the characteristictime-scale onwhicha stable fixedpoint is
approached. Briefly,the idea behind TDASforstabilizing adynamicalsystem,whose
classical state is given by x(t), is to apply one or more continuous feedback forces of
the formF(t)=k[x(t) x(t τ)]. The feedback force vanishes in steady state, or for
− −
a periodic orbit if the delay, τ, is a multiple of the period. This is referred to as non-
invasive feedback. The delay, and feedback strength, k, are parameters that should
be varied in experiment to achieve driving towards a particular long-time attractor.
We consider a CQED system, as illustrated in Fig. 1. The internal dynamics
of a cavity, consisting of a field mode, possibly interacting with other quantum and
classical degrees of freedom, is described by a Hamiltonian Hˆ. The cavity is assumed
tohavetwomirrorsbandc,correspondingtotwodistinctpairsofinput-outputports.
Theequationofmotionforthecavitymodeis,accordingtothestandardinput-output
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Figure 1. Aschematicillustrationofourfeedbackcontrolscheme. Theinternal
dynamics of the cavity isdescribed byaHamiltonian, Hˆ. Thecavity consists of
two mirrors,b (left) and c (right), with decay rates κb and κc respectively. The
light-blue tilted bars denote beam splitters BS1 and BS2, with transformation
properties defined by the unitary matrices S1 and S2, respectively. The circle
denotes a delay-time of τ and a φ phase shift to one of the feedback arms. We
assumethatFaradayisolators(not shown)separatethe cavityinputandoutput
fields.
theory for optical quantum systems [29],
daˆ
=i[Hˆ,aˆ] (κ +κ )aˆ √2κ ˆb (t) √2κ cˆ (t). (1)
b c b in c in
dt − − −
Here aˆ is the cavitymode annihilationoperator, κ and κ are the decay ratesfor the
b c
two mirrors, and ˆb (t), cˆ (t) the annihilation operators of the input fields incident
in in
on the respective mirrors. Here, and in the following, we will, when convenient,
suppress the time argument for any system operator evaluated at time t, but keep
the time argument for input and output fields for clarity. We will assume that cˆ (t)
in
correspondstoavacuumfield,althoughitwouldalsobeofinteresttoconsideradrive
here. The input fields obey the commutation relations
[ξˆ (t),ξˆ† (t′)]=δ(t t′), (2)
in in −
whereξˆ denotesanyoftheinputmode operatorsˆb orcˆ . Inaddition,anyvacuum
in in in
input field ξˆ (t) correspondsto a Gaussianwhite noise operatorwith zeromean, and
in
its only non-vanishing correlationfunction is
ξˆ (t)ξˆ† (t′) =δ(t t′). (3)
in in −
D E
The corresponding output fields are given as
ˆb (t)=√2κ aˆ(t)+ˆb (t), (4a)
out b in
and
cˆ (t)=√2κ aˆ(t)+cˆ (t). (4b)
out c in
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The output from mirror c is now split into two feedback arms, as illustrated in Fig.
1. One of the feedback arms has a time-delay τ and a phase shift φ, while we set
the time-delay and phase shift of the other arm to zero for simplicity. The input and
output fields for the two beam-splitters BS1 and BS2 shown in Fig. 1 are related
through
fˆ(t) νˆ (t)
1 =S 1 , (5a)
(cid:18) fˆ2(t) (cid:19) 1(cid:18) cˆout(t) (cid:19)
and
ˆb (t) eiφfˆ(t τ)
νˆin(t) =S2 fˆ2(t−) , (5b)
(cid:18) 2 (cid:19) (cid:18) 1 (cid:19)
where S and S are unitary matrices, νˆ (t) is a vacuum input field to beam splitter
1 2 1
BS1,andνˆ (t)isthe(unused)otheroutputfieldfrombeamsplitterBS2. Wewantthe
2
time-delayed and time-undelayed fields to be incident on mirror b in opposite phase,
so as to give the desireddestructive interference in steady state. For this purpose, we
choose the beam splitter transformations to be given by
e−iφ/2 s reiφ/2
S1 = √2 re−iφ/2 − s , (6a)
(cid:18) (cid:19)
and
e−iφ/2 r seiφ/2
S2 = √2 se−−iφ/2 − r , (6b)
(cid:18) − (cid:19)
where r,s 0 (real) and r2 +s2 = 2. In passing, we remark that these are not the
≥
most general choice of beam splitter transformations but sufficient for our purposes.
We now find forˆb (t):
in
ˆb (t)=
in
rs s2 r2
[cˆ (t) cˆ (t τ)] e−iφ/2νˆ (t) e−iφ/2νˆ (t τ)
out out 1 1
2 − − − 2 − 2 −
rs
= √2κ [aˆ(t) aˆ(t τ)]+˜b (t), (7)
c in
2 − −
where we have defined
˜b (t)
in
≡
rs s2 r2
[cˆ (t) cˆ (t τ)] e−iφ/2νˆ (t) e−iφ/2νˆ (t τ). (8)
in in 1 1
2 − − − 2 − 2 −
The mode operator˜b (t) satisfies Eq. (2) andEq. (3) andis thus the “vacuum part”
in
of the field incident on mirror b.
A control force is generated from the difference between the current cavity field,
aˆ(t), and the field at some point in the past, aˆ(t τ). This forcing is then fed back
−
into the system. By making use of Eq. (7) in Eq. (1) we obtain
daˆ(t)
=i[Hˆ,aˆ(t)] (κ +κ )aˆ(t) √2κ ˜b (t) √2κ cˆ (t)
b c b in c in
dt − − −
+k(aˆ(t τ) aˆ(t)) , (9)
− −
where k rs√κ κ satisfies 0 k √κ κ . We remark that the vacuum input fields
b c b c
≡ ≤ ≤
˜b (t) and cˆ (t) are not independent, as can be seen from Eq. (8).
in in
Inthe followingsectionwewillapply this schemeto anopenversionofthe Dicke
model, and find optimal delay-times for parameters motivated by recent experiment.
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3. Application to the open Dicke model dynamics
We will first introduce the generalized Dicke model that will be the main object of
our study. It describes the interaction of N two-level atoms with a single mode of
the electro-magnetic field. The Hamiltonian describing the internal dynamics of the
system is
2g U
Hˆ =ω Jˆ +ωaˆ†aˆ+ Jˆ aˆ+aˆ† + Jˆaˆ†aˆ, (10)
0 z x z
√N N
(cid:0) (cid:1)
where parameters ω and ω are atomic and cavity frequencies, respectively, g is the
0
linear interaction strength, and U is a non-linear coupling constant. The operators
aˆ and aˆ† are, again, the annihilation and creation operators for the cavity mode,
and Jˆ ,Jˆ,Jˆ are collective atomic operators satisfying the conventional angular
x y z
{ }
momentum commutation relations. The Hamiltonian in Eq. (10) is identical to the
conventional Dicke model [21] Hamiltonian if U is set to zero.
We will not enter into the details on the underlying physics of the BEC
experiments realizing Eq. (10), but refer the reader to Ref. [9]. Briefly, the two-level
atoms of the Dicke model are realized through pairs of discrete momentum states
of a BEC. The linear coupling to the cavity field, (2g/√N)Jˆ (aˆ† + aˆ), effectively
x
describes Rayleigh scattering of photons between the cavity mode and an auxiliary
pump laser. Importantly, the effective coupling strength, g √P, can be tuned via
∼
the laser intensity P. The parameter ω is determined by the detuning of the cavity
mode frequency from the pump laser frequency, and is therefore controllable. The
collective atomic frequency ω is fixed by the optical wave-vector and atomic mass
0
(i.e., it is set by the recoil energy). It is therefore not readily tunable like the other
parameters. The non-linear coupling constant U is given by a dispersive light-shift.
Wewillconsiderparameterswherethisnon-linearcouplingdoesnotplayamajorrole,
but refer to Refs. [10,11,30] for discussions on the very interesting dynamics that can
result from this term.
We will assume that the cavity consists of two mirrors, both with decay rates
κ = κ κ/2 (for simplicity), and implement the feedback scheme introduced in
b c
≡
the previous section and illustrated in Fig. 1. We will, furthermore, assume the
“thermodynamic” limit N , where we can employ a semi-classical approach to
→ ∞
findexpectationvaluesofsystemoperators,asdescribedbelow. LaterinSection5,we
examinequantumfluctuationsbylinearizingthe operatorequationsofmotionaround
these expectation values.
By making use of Eq. (9) and Eq. (10), we can now write down the Heisenberg
equations of motion for the cavity mode and the spin operators:
daˆ(t) U 2g
= i ω+ Jˆ(t) aˆ(t) i Jˆ (t) κaˆ(t)
z x
dt − N − √N −
(cid:18) (cid:19)
+k(aˆ(t τ) aˆ(t)) √2κaˆ (t),
in
− − −
dJˆ (t) U
x = ω + aˆ(t)†aˆ(t) Jˆ(t),
0 y
dt − N
(cid:18) (cid:19)
dJˆ(t) U 2g
y = ω + aˆ(t)†aˆ(t) Jˆ (t) aˆ(t)+aˆ(t)† Jˆ(t),
0 x z
dt N − √N
(cid:18) (cid:19)
dJˆ(t) 2g (cid:0) (cid:1)
z = aˆ(t)+aˆ(t)† Jˆ(t), (11)
y
dt √N
(cid:0) (cid:1)
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where now 0 k κ/2, and aˆ (t) = 1/√2 ˜b (t)+cˆ (t) is used to denote the
in in in
≤ ≤
sum of the vacuum input fields through the t(cid:16)wo mirrors and(cid:17)where, for clarity, the
time-dependence hasbeenmadeexplicit. Since˜b (t),whichcorrespondstoavacuum
in
part of the field incident on mirror b, is not independent of the input field cˆ (t) on
in
mirror c, aˆ (t) does not obey the usual relations Eq. (2) and Eq. (3), but instead we
in
have that
[aˆ (t),aˆ† (t′)]= aˆ (t)aˆ† (t′) =δ(t t′)
in in in in −
k 1D E 1
+ δ(t t′) δ(t t′+τ) δ(t t′ τ) , (12)
κ − − 2 − − 2 − −
(cid:20) (cid:21)
which can be verified by making use of Eq. (8).
The non-linearoperatorequations,Eq. (11), cannotbe solveddirectly,andwith
delayedfeedback(τ =0),noequivalentmasterequationcanbederived. Byneglecting
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fluctuations andfactorizingoperatorproducts,we can,however,derivea closedsetof
equations of motion for the five real-valued variables:
aˆ aˆ
x Re h i , x Im h i ,
1 2
≡ √N ≡ √N
Jˆ Jˆ Jˆ
x y z
j h i, j h i, j h i, (13)
x y z
≡ N ≡ N ≡ N
which take the form
dx (t)
1
= κx (t)+ ω+Uj (t) x (t)+k (x (t τ) x (t) ,
1 z 2 1 1
dt − − −
dx (t) (cid:0) (cid:1) (cid:0) (cid:1)
2
= κx (t) ω+Uj (t) x (t) 2gj (t)+k x (t τ) x (t) ,
2 z 1 x 2 2
dt − − − − −
dj (t) (cid:0) (cid:1) (cid:0) (cid:1)
x = ω +U(x2(t)+x2(t)) j ,
dt − 0 1 2 y
djy(t) = ω(cid:0)+U x2(t)+x2(t) j(cid:1)(t) 4gx (t)j (t),
dt 0 1 2 x − 1 z
dj (t) (cid:0) (cid:0) (cid:1)(cid:1)
z
= 4gx (t)j (t). (14)
1 y
dt
Here, again, the time-dependence has been made explicit for clarity. These equations
of motion conserve the total length of the spin j2+j2+j2, and we will restrict our
x y z
study to states on the Bloch sphere and thus always assume the constraint
j2+j2+j2 =1/4. (15)
x y z
Below we will also make use the rescaled complex variable α aˆ /√N = x +ix ,
1 2
≡ h i
and, for notational convenience, we also write x (x ,x ,j ,j ,j ).
1 2 x y z
≡
In the absence of feedback, i.e. k=0, this model has been explored theoretically
in great detail, both in the thermodynamic limit N in Refs. [10,11], and for
→ ∞
finite N and including all quantum effects in Ref. [30]. In Refs. [10,11], steady states
were found analytically by setting the left-hand sides in Eq. (14) equal to zero, and
a further stability analysis was performed by linearizing around the fixed points. A
surprisingly rich phase diagram was uncovered, with the appearance of several new
phasesduetothepresenceofthenon-linearcouplingU andthecavitydecayparameter
κ when comparedto the conventionalDicke model[31,32]. One of the key findings in
Ref.[11]wasthattheemergenttime-scalesofthecollectivedynamicsvarysignificantly
8
throughout the phase diagram, and that the run-times of current experiments may
not be sufficient to reach the long time attractor in all cases.
Here we will now focus our attention on the following set of parameters as taken
from the recent experiment in Ref. [23]:
ω ,ω,U,κ = 8.3 10−3,14.0, 8.0,1.25 2πMHz. (16)
0
{ } { · − }·
With this choice of parameters, a phase transition from the normal phase, x⇓
≡
(0,0,0,0, 1/2), to a super-radiant phase with x ,x ,j = 0, happens at a critical
1 2 x
− { }6
coupling strength [11]
ω [(ω U/2)2+κ2]
0
g = − =0.19 2πMHz. (17)
c
s 4(ω U/2) ·
−
Below this critical coupling there are two fixed points: the normal phase
x⇓ (0,0,0,0, 1/2), (18)
≡ −
and the inverted phase
x⇑ (0,0,0,0,1/2), (19)
≡
where only the former is stable in the absence of feedback (k =0). Above the critical
coupling, both of these phases are unstable in the absence of feedback, and two new
stable fixed points, xSR, come into existence, given by [11]
±
ω g2(4ω2 U2) ω Uκ2
0
− − , ifU =0,
jSR = −U −s U2(ω0U +4g2) 6
z  gc2 , ifU =0,
−2g2
jSR = 1/4 (jSR)2,
x ± − z
jSR =0,q
y
2gjSR
αSR xSR+ixSR = x . (20)
≡ 1 2 −ω+UjSR iκ
z −
The two fixed points xSR differ only in the choice of sign for j and α, related to
± x
a duality emerging from the invariance of Eq. (10) under the parity transformation
aˆ aˆ,Jˆ Jˆ. We can treat the two solutions simultaneously, and we will
x x
→ − → −
refer to both as “the super-radiant phase”, and denote them both by xSR, when the
difference between them is of no importance. The super-radiant phase transition (in
the absence of feedback) is qualitatively illustrated in Fig. 2.
The treatment in Refs. [10,11] showed that the approach to steady state can be
exceedingly slow. This is partly related to the relatively small value of ω , and canin
0
fact be interpreted as critical slowing down, due to closeness to phase boundaries in
an extended parameter space [10,11]. By careful adiabatic elimination of the cavity
mode, under the condition ω,κ ω , one can find an effective rate describing the
0
{ }≫
incoherentdynamicsofthe atoms. Inthe normalphase,forexample,itis foundtobe
4κg2(ω U/2)ω / (ω U/2)2+κ2 2 [11] (see also equation Eq. (21b) below). At
0
≃ − −
g =g thisisroughly0.3 2πHz,whichindicatesaremarkablyslowdecaytakingonthe
c
(cid:2) · (cid:3)
orderofseconds. Theparameterω ,describingthecollectiveatomicfrequency,isfixed
0
by the opticalwavevectorandthe atomicmass,andcanthereforenoteasilybe made
larger in practice. In Fig. 3, we show two typical examples of the atomic inversion,
9
Figure2. Qualitativeillustrationoftheorderparametersjz(leftfigure)andjx
(rightfigure),asfunctionsofg,intheabsenceoffeedback(k=0). Belowcritical
coupling,g<gc,therearetwofixedpointsx⇑ (green)andx⇓ (blue),whereonly
the latter is stable. Above critical coupling, g > gc, they are both unstable,
and two new stable fixed points, xSR (red), come into existence, satisfying Eq.
±
(20). NotethatthepositionsoffixedpointsstaythesameunderTDASfeedback
control,onlytheirstabilitymightchange.
j (t), as a function of time, below critical coupling, with g/g =0.74 and above,with
z c
g/g = 1.1. The other parameters are as in Eq. (16). In both cases the initial state
c
is taken to be x = (0,0,1/√12,1/√12,1/√12). In Fig. 4, we similarly show the
time-evolutionof the normalizedphoton number α(t)2 for the same initial state and
| |
parameters. These figures clearly show the exceedingly slow approach towards the
stable steady state, in agreement with the predictions from Ref. [11].
In order to obtain the characteristic time-scales governing the approach to a
fixed point, x¯(t) x¯, when feedback is applied, we consider a small perturbation,
≡
y(t) = x(t) x¯, and linearize the equations of motion around the solution. By
−
using an Ansatz y(t)=exp(λt)y , we can then derive a characteristic equationfor λ.
0
Each solution λ corresponds to a characteristic (inverse) time-scale for the dynamics
close to the steady state. In the presence of feedback, k,τ > 0, there are in fact
an infinite number of solutions λ . However, a crucial result in the analysis of delay
k
differential equations is that there are only a finite number in any real half-plane
Reλ>σ, σ R [33]. Thus it becomes feasible to find the slowest λ that ultimately
k
∈
governs the time-scale for approaching or leaving a fixed point. The details of a
stability analysis for Eq. (14) are given in Appendix A.
A steady state solution x¯ is stable only if all λ have negative real parts, i.e.,
k
Reλ < 0. The time-scale for the approach to a stable steady state solution is thus
k
governedby the eigenvalue with realpart closest to zero, which we will denote by λ .
1
The key to our control scheme is that the eigenvalues can be manipulated through
the variation of k and τ. In particular, both the magnitude and sign of Reλ can be
1
changed. This opens up the possibility of changing the emergent time-scales, as well
as qualitatively changing the phase diagram of the system, by changing the stability
of a steady state.
We can find λ numerically by solving a transcendental characteristic equation,
1
giveninEq. (A.5). InFig. 5,weplotReλ asafunctionofτ fork =κ/2,g/g =0.74
1 c
and 1.1, while the other parameters are kept as in Eq. (16). For g/g = 0.74, we
c
linearize around the normal phase, x⇓ Eq. (18), and the inverted phase x⇑ Eq. (19),
10
0.4
0.2
z 0.0
j
−0.2
−0.4
0.4
0.2
z 0.0
j
−0.2
−0.4
0 20 40 60 80 100 120 140 156.50 156.75 157.00
t(ms) t(ms)
Figure 3. Time-evolution of the collective inversion, jz(t), for an initial state
x = (0,0,1/√12,1/√12,1/√12). The Hamiltonian coupling of the spin and
cavitydegreesoffreedominducesoscillationsthatareveryrapidcomparedtothe
relaxationtime. Theshadedregionsintheleftpanelsshowtheoscillatingsolution,
whilethelinesshowthesamesignalafteralow-passfilterhasbeenapplied. The
right panels show a zoom of the oscillating solutions for late times. The dashed
lines show the exact steady state values for k = 0. Top panels: g/gc = 0.74;
thenormalphase,x⇓ asinEq. (18),istheonlystablefixedpointintheabsence
of feedback. The blue line shows the time-evolution without feedback. The red
line is with k = κ/2 and τ = 50µs. The green line shows the time-evolution
for k = κ/2 and τ = 100µs, for which the normal phase is unstable, and the
system exhibits persistent oscillations in the long time limit. Bottom panels:
g/gc =1.1; the super-radiantphase, xSR as inEq. (20), is the only stable fixed
pointintheabsenceoffeedback. Thebluelineshowsthetime-evolutionwithout
feedback,andtheredlineisfork=κ/2andτ =50µs. Forbothpanels,allother
parametersaregiveninEq. (16).
whicharealwaysvalidfixedpoints. Forg/g =1.1wealsolinearizearoundthesuper-
c
radiant fixed point, xSR Eq. (20). This analysis can be used to find optimal values
for τ closeto the steady state. We observethata minimum value for Reλ is reached
1
for a delay of around τ 50µs, for both the normal phase when g/g = 0.74 and
c
≃
the super-radiant phase when g/g = 1.1. The value of Reλ is, without feedback
c 1
(τ = 0), roughly 0.14 2π Hz and 0.35 2π Hz, respectively, for the two cases. In
− · − ·
comparison,atthe firstminima,with τ =52µsandτ =50µs,the valuesare 26 2π
− ·
and 58 2π Hz, i.e., two orders of magnitude larger.
− ·
In Fig. 3 and Fig. 4 the time-evolution of the collective inversion,
j (t), and the normalized photon number, α(t)2, from an initial state x =
z
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