New J. Phys. 20 (2018)013006 https://doi.org/10.1088/1367-2630/aa9cdd
PAPER
Superradiant to subradiant phase transition in the open system
Dicke model: dark state cascades
Michael Gegg, Alexander Carmele, Andreas Knorr and Marten Richter
1
Institut für Theoretische Physik, Nichtlineare Optik und Quantenelektronik, Technische Universität Berlin, Hardenbergstr. 36, EW 7-1,
D-10623 Berlin, Germany
1
Author to whom any correspondence should be addressed.
Keywords: Dicke model, collective effects in quantum optics, superradiance and subradiance, quantum entanglement, specific phase
transitions
Abstract
Collectivity in ensembles of atoms gives rise to effects like super- and subradiance. While
superradiance is well studied and experimentally accessible, subradiance remains elusive since it is
difficult to track experimentally as well as theoretically. Here we present a new type of phase transition
in the resonantly driven, open Dicke model that leads to a deterministic generation of subradiant
states. At the transition the system switches from a predominantly superradiant to a predominantly
subradiant state. Counterintuitively, the cavity decay is the crucial parameter for subradiant state
generation and not the individualizing process of spontaneous decay. The observed effect is thus a
cavity assisted generation of subradiant quantum coherences. Clear experimental signatures for the
effect are presented and entanglement properties are discussed. Letting the system relax into the
ground state generates a cascade of dark Dicke states, with dark state populations up to unity.
Furthermore we introduce a collectivity measure that allows to quantify collective behaviour.
1. Introduction
The open (and closed)system Dicke model has been a work horse in quantum optics and beyond for decades
[1–28]. Current research on Dicke model based systems includes novel laser-like systems [22], phase transitions
[19,26], quantum information and super/subradiance [14,17,23,24,27,29]. In recent years superradiance has
been investigated with respect to entanglement [23]and subradiance for its prospects to store quantum
information [24,29]. The Dicke model assumes Nidentical two-level systems, interacting with a bosonic
cavity mode.
Investigating subradiant effects in a consistent open system theory was not feasible for a long time since in a
straight forward approach the master equation scales exponentially in the number Nof two-level systems. This
renders full simulations even for small Nimpossible. Subradiance appears already for few particles, however the
behaviour towards more emitters cannot be modelled using the full exponential approach. Common limits and
approximations for both analytical and numerical treatments addressing this problem are also not suited to
study subradiance even for moderate N[4–10,14,17]. Usually for superradiance total spin conservation
(explained below)is assumed, entirely neglecting subradiant states. This reduces the numerical complexity to
∼N
2
[5]or sometimes even allows analytic solutions [8–10,17]. However ubiquitous phenomena in real systems
like decay processes and pure dephasing break the conservation of total spin symmetry. Therefore, both realistic
treatments and subradiant effects require a different methodology.
The formal permutation symmetry of the master equation itself allows, under some simple hypothesis on the
initial state, to reduce the complexity from an exponential scaling in Nto a polynomial scaling ∼N
3
, even
without total spin conservation [11,22,30–34]. This makes exact calculations for moderate emitter numbers
feasible and removes constraints imposed by assumptions and approximations. Furthermore the method can be
applied to any set of permutation symmetric multi-level systems [33,35].
OPEN ACCESS
RECEIVED
12 July 2017
REVISED
16 November 2017
ACCEPTED FOR PUBLICATION
23 November 2017
PUBLISHED
5 January 2018
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© 2018 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft
In this work we investigate the steady state population of subradiant states through decay and pure
dephasing processes—both do not conserve the total spin. The system is driven by an external laser and
increasing the driving strength results in a non-equilibrium phase transition in the steady state behaviour.
Counterintuitively, the cavity lifetime determines the population of the subradiant states: for short cavity
lifetimes (bad cavity limit)subradiant states are always suppressed by quantum coherence. Contrary, increasing
the cavity lifetime results in a collective, quantum coherent amplification of population in subradiant states.
Thus the quality of the collective behaviour of the two-level systems is drastically changed by tuning the cavity
lifetime, even though the cavity decay does not break the total spin symmetry and thus does not couple different
Dicke subspaces. The individual spontaneous emission process is a necessary requirement for the population of
subradiant Dicke states. However the associated decay rate has no influence on the population behaviour of the
subradiant states in typical parameter regimes for laboratory quantum optics and quantum information setups,
such as quantum dots, NV centres, Rydberg atoms, etc. The cavity quality is the only parameter determining the
quality of the collective effects in the present system. Therefore the coherences of the subradiant states are only
formed through the cavity degrees of freedom. The observed effect is thus a cavity assisted generation of
subradiant quantum coherences. The quantitative discussion of the collective behaviour of the system in the
steady state is enabled by the introduction of a collectivity measure.
In the bad cavity limit this setup resembles the scenario known as cooperative resonance fluorescence,
including individualization, which has been studied in the past using mean field theory and phase space
methods/large system size limits [4,6,9,10]. This removes an essential part of quantum coherent effects [36].
These studies found bistable behaviour similar to absorptive optical bistability. This stems from a competition
between collective and individual behaviour. These older studies focused on mean excitations and two-level
system correlation functions—we place the focus on investigating superradiant and especially subradiant effects,
which requires a careful treatment of the quantum coherences in the system. Furthermore hyperradiance of two
individual non symmetric atoms, but also with an individual spontaneous decay as considered here, is
introduced and studied using a radiance witness in [28].
Experimentally accessible signatures of this effect and entanglement properties via spin squeezing are
discussed. Switching off the external driving once the steady state is reached, the subsequent relaxation into the
ground state forms a long-lived cascade of dark Dicke states. This results in a simple, deterministic protocol for
dark state preparation with populations close to unity under the influence of dephasing, with applications in
quantum information storage.
2. Model system
We consider the usual Dicke model with an additional classical optical, cw field Edriving all TLS identically.
Driving is necessary since subradiant states are excited states. In a frame rotating at the external laser frequency,
using the rotating wave approximation the system Hamiltonian reads
HbbJgJbJbEJJ,1
0 1 11 10 01 10 01
=D +D + + + +()() ()
††
where
0
D
,
1
D
are the mode and TLS detuning, gis the TLS-mode coupling, Eis the optical driving,
bb,
†
are
photonic operators and
J
ki
k
i
s=å,k11, 10, 01, 00=are the collective spin operators. Excited and ground
state of the individual TLS iare
1
i
ñ
∣
,
0
i
ñ
∣
. The spin operators, in the Bra and Ket representation, are 11
iii
11
s
=ñá∣∣
,
10
ii
i
10
s
=ñá∣∣
,01
ii
i
01
s
=ñá∣∣
and 00
iii
00
s
=ñá∣∣
. We assume resonant excitation field, cavity and TLS. Both cavity
and TLS are subject to loss and dephasing, using Lindblad formalism [37]. The master equation reads
H
i,.2
tde pd ph
rrr rrr¶= = + + +[ ] () () () ()
The Lindblad dissipators describe decay processes like individual radiative and non-radiative decay
22
i
ii i i
de 01 10 11 11
rg srs srrs=å--() ( )
, pure dephasing 2iz
iz
i
pd
rd srsr=å-() ( )
and cavity decay
b b bb bb22
ph
rk r rr=--() (
)
†† †
, see figure 1(a). We use z
ii i
11 0
0
s
ss=-
. All contributions to the master
equation except de
and
pd
are total spin preserving, (figure 1(b)). The total spin
l
l1+()
is the eigenvalue of
the
J
JJ JJ J2z
210 01 01 10
=+ +()
operator, with
J
12
ziz
i
s=å. The value of lvaries between
l
N2
max =for
the superradiant subspace and
l
0, 1 2
min =for the (most)subradiant subspace. The J
2
and J
z
eigenvalues
determine the coupling strength of the multi TLS (Dicke)state to the cavity mode and the coherent, external
drive. This coupling determines the rate of cavity photon generation as well as the pumping strength. The
magnitude of the coupling strength distinguishes between superradiance and subradiance. For superradiant
states the coupling strength scales superlinear in N, while for subradiant states the scaling is sublinear in Nand
some subradiant states are dark [38]. Dark means that the collective coupling to the cavity and the coherent,
external drive of these states vanishes, meaning these states cannot decay via collective interactions e.g. by
creating a cavity photon. However these states still decay into other states via the decay and dephasing processes
2
New J. Phys. 20 (2018)013006 M Gegg et al
de
and
pd
acting individually on the emitters, see figure 1(b). Generally the spin preserving contributions in
the master equation (like equation (1)) generate quantum correlations leading to collective TLS behaviour (both
super- and subradiance are collective effects)and the nonpreserving terms destroy correlations leading to
individualization (all properties scale exactly linear in N). However only the spin nonpreserving contributions
introduce coupling between superradiant and subradiant states, thus in order to prepare subradiant states an
interplay of collectivity and individualization is necessary.
Based on these considerations the distinction of the behaviour of the system in this work is twofold: we
distinguish between collective versus individual behaviour and superradiant versus subradiant behaviour. The
latter are special cases of collective behaviour. This twofold distinction seems crucial when investigating super-
and subradiance in the presence of dehasing and individual decay.
In the bad cavity limit (
g
k
)equation (2)corresponds to the cooperative resonance fluorescence setup
[4–6]. The system exhibits a non-equilibrium phase transition for increasing Efor both total spin preserving and
nonpreserving setups, where the nonpreserving setup was studied using mean field theory [4]. For longer cavity
lifetimes κthe system more and more resembles the absorptive optical bistability setup [39](instead of driving
the TLS, in optical bistability the cavity is driven, opposed to figure 1(a)). In the range investigated in this work
(
g
k~
)the clear distinction between cooperative resonance fluorescence and optical bistability breaks down,
thus combining these distinct fields of quantum optics. Besides the steady state, density matrix states with very
long lifetimes can exist in these systems, which lead to the observation of bistabilities in experiments with finite
measurement time [40]. In some limits these lifetimes go to infinity, resulting in a second steady state. For optical
bistability these long lifetimes are called tunnelling times [12,41], more generally this phenomenon is called
dissipative phase transition [42].
3. Permutation symmetric method
The formal permutation symmetry of the master equation allows the incorporation of the individual TLS decay
and dephasing while having moderate numbers of TLS and photonic Fock states, since it reduces the number of
relevant TLS Liouville space states from
4
N
to NNN1236+++
(
)( )( ) . This method was first introduced by
Sarkar and Satchell in 1987 [11,12]as the few emitter analogue to the widely used phase space methods in
quantum optics, in particular the positive Prepresentation [31,43], which breaks down for 50two-level
systems [36]. The connection of this permutation symmetric method to the phase space methods can be seen
from the fact that there are exactly NNN1236+++
(
)( )( ) distinct, linearly independent ordered products
of the collective operators
J
JJ
p
z
qr
10 01
[31]. These operator products can also be used to expand the master equation
in actual calculations [27,31]. In recent years the permutation symmetric method has been independently
rediscovered by different groups using different approaches [22,26,32,33,44,45].
The states introduced in these methods may allow a more intuitive understanding of the processes in the
system depending on the mathematical formulation. In the following we will use the formulation developed in
[22,33,35]. For a permutation symmetric master equation the TLS density matrix is described by elements
nkl,,
[
]
with nklN
0
++ . These elements describe the full density matrix and their number scales
with ∝N
3
. For element nkl,,
[
]
nof the NTLS are in a
1
1
s
Liouville state, kare in a
10
s
state and lin a
0
1
s
state.
0
1
s
and
10
s
(
k0¹
and/or
l
0¹
)correspond to a quantum coherence/offdiagonal element in the individual
density matrix. The different elements can be interpreted as follows:
n,0,0
[
]
is the incoherent probability of
Figure 1. Illustrating the open Dicke model: (a)schematic representation of the system. (b)Dicke states for N=4. The lowest state in
each lsubspace is dark—the lowest state in the superradiant
lN2
max
=
subspace is the ground state. The interactions are depicted:
Hamiltonian part (purple,thick), dissipators
de
and pd
(black,thin)and dark state cascade (orange,curved). Dashed lines indicate
the additional states for N=5(with different values of
m
l,
).
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New J. Phys. 20 (2018)013006 M Gegg et al
finding the NTLS system with nexcited TLS. For instance preparing the system in a thermal state results in a
thermal distribution in the
n,0,0
[
]
densities, or preparing the system in the ground state is equivalent to
0, 0, 0 1
=[]
and zero for all other nkl,,
[
]
. The elements nkl,,
[
]
for kl,0¹describe quantum
correlations and thus are constructed from the offdiagonal elements of the full density matrix, including inter-
TLS coherences. For k=li.e.
nkk,,
[
]
these elements are real valued but still represent offdiagonal density
matrix elements/coherences between different TLS. The
nkk,,
[
]
are collective quantum contributions and
contribute to the collective Dicke state population of excited states in the system: more precise the
nkk,,
[
]
distinguish collective Dicke state populations from classical, individual excited state populations in the open
system, density matrix setting: the offdiagonal elements of the density matrix between different TLS are directly
connected to the collective effects in the many emitter setup. In the following we will explain this relation in
more detail and introduce a measure to distinguish collective Dicke behaviour from classical, individual
behaviour in the presence of dephasing. For more details on the permutation symmetric variant used here and
the density matrix elements nkl,,
[
]
please refer to appendix Aand [33,35,46].
The photonic degrees of freedom are treated using the usual bosonic number states, that are cutoff at
appropriate values in order to achieve convergence. Therefore the full calculations are carried out using the
quantities
nklm m,,; ,
lr
[
]
, including the photon degrees of freedom of the density matrix with an expansion
in the
mm
lr
ñá
∣∣
elements. This is explained more formally in appendix A.
4. Collectivity measure
Investigating super- and subradiant states requires a suitable measure. Unfortunately computing the respective
Dicke state populations is not sufficient for investigating collective effects and quantum coherence, if dephasing
is present: Dicke states
lm,ñ
∣
are eigenstates of J
2
and J
z
with corresponding quantum numbers
l
l1+()
,
lN
0
2 and ml
∣
∣.
l
N2
max =defines the superradiant subspace and
l
0, 1 2
min =defines the (most)
subradiant subspace, see figure 1(b). As an example consider the N=2 Dicke (or Bell)states: the superradiant
subspace consists of three states 1, 1-ñ
∣
,
1, 0ñ
∣
,1, 1+ñ
∣
while the subradiant subspace consists of a single dark
state
0, 0ñ
∣
. First we calculate the population of the Dicke states:
lm lm lm lm plm
t
r, , , , ,rñá = á ñá ñ =[∣ ∣ ] ∣ ∣ ( )
in the local basis. Using the permutation symmetric density matrix elements nkl,,
[
]
we can write these
populations as
plm a lm n alm n, , ,0,0 , 1,1,1 , 3
01
=+-¼() ()[ ] ()[ ] ()
with
n
mN2=+ . In the presence of dephasing the elements
nkk,,
[
]
for
k0¹
experience dephasing. The
nkk,,
[
]
represent quantum coherences between different TLS. If the dephasing is strong enough it will
completely suppress quantum correlations, i.e.
nkk,, 0
=[]
for
k0¹
. This represents a completely
incoherent mixture of TLS occupations. For varying numbers Nof TLS, different
n,0,0
[
]
distributions allow
a large variety of populations in super- and subradiant states even if quantum coherences between different TLS
are absent, since the relative size of the superradiant and subradiant Hilbert spaces vary for both Nand n.
Generally—when total spin non-conserving terms are included—the superradiant subspace population
decreases, since for large Nthe superradiant subspace is very small compared to the full Hilbert space (
N
1+
versus
2
N
). However without quantum coherences between different TLS (nkk k,, , 0
¹[] )the label super-
and subradiance becomes meaningless, since the inter-TLS quantum coherences are the signatures of the
collectivity of the Dicke states and reflect the redistribution of oscillator strength through collective effects (phase
locking). Thus—in the open Dicke model—n kkk,,
-[
]
are the key quantities that distinguish a super- or
subradiant state from a classical, incoherent mixture of TLS population (ijk,, 0
=[]for
j
k,0¹
). The decay
process de
and the pure dephasing
pd
act individually on every TLS and thus destroy the collectivity, resulting
in incoherent mixtures.
To quantify the effect of collectivity and distinguish between collective (super- and subradiance)and
individual (dipole moment scales linear in N)behaviour we introduce the ratio between the full Dicke subspace
population and its incoherent part
Rl pl m
alm m N
,
, 2,0,0 ,4
m
m0
å
å
=+
() ()
()[ ] ()
as a collectivity measure for the different Dicke subspaces l.
R
l1=() holds if the influence of quantum
correlations between the individual TLS on the subspace population is zero or negligible—the TLS act
individually.
R
l1<() /
R
l1>() holds if quantum correlations suppress/increase the respective subspace
occupation—the TLS act collectively. R(l)provides a reality check, since in any experiment dephasing is present
and isolated Dicke subspaces (or states)never occur.
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New J. Phys. 20 (2018)013006 M Gegg et al
5. Results and discussion
We solve equation (2)with our computer library PsiQuaSP [35,46]for master equations with reduced,
polynomial scaling (see appendix Afor a short introduction and [33]for more details). We use eigensolvers and
time integration from PETSc and SLEPc [47–50].
We use
g
5ps
1
=
-throughout this work and
1.0 ns
1
g
=
-
except for figures 5(c)and (d). Please note that
ultra-strong coupling effects are not present in the investigated parameter range. There are two types of
dephasing/individualization processes: spontaneous decay and pure dephasing. We first investigate the
spontaneous decay and investigate the effects of pure dephasing later. Including small pure dephasing preserves
all effects (see section 5.2 for a discussion).
5.1. Nature of the phase transition
In the steady state the most basic feature of the nonequilibrium phase transition is the change from the ground
state to a half excited TLS state with increasing external driving field (figure 2(a)). The mean field theory expects a
bistable behaviour [4], but in the full quantum treatment this is replaced by a slowing down in steady state
convergence [12,41](see figure 4(b)). Increasing the cavity quality (decreasing the ratio between cavity decay
rate and TLS-cavity coupling strength
g
k
)makes the transition sharper but overall the effect does not change
much. Contrary a drastic change is seen in the behaviour of the collectivity measure for the superradiant
subspace
R
lN2
max =()
,figure 2(b). While in the bad cavity limit the superradiant subspace population is
always increased by collective effects (
R
l1
max
()
), we observe an increased suppression (
R
l1
max
<()
)of the
superradiant subspace for increasing cavity lifetime/quality. This is accompanied by a drastic increase of
coherent cavity photons below and a pronounced bunching at moderate photon numbers above the phase
transition (figures 2(c)and (d)). The maximum in the second order photon correlation function indicates the
transition point from increased to suppressed superradiant subspace occupation. Please note that the cavity
decay does not lead to an effective dephasing/individualization contribution for the TLS, thus the population of
subradiant states through different cavity lifetimes is a highly nontrivial effect.
Above the phase transition collectivity favours the most subradiant subspace l
min
: the dependence of
R
lmax
()
on the number of TLS N,figure 3(a), shows a growing collective change in population of the superradiant
subspace for increasing N.Infigure 3(b)the ratio
R
l2
max -()
is plotted—it switches from collective
suppression below to collective increase above the transition (this subspace only exists for
N4
). However the
collective increase in population decreases for increasing N. For N=4, 5 there are three different lsubspaces:
l
max
,
l
1
max
-
and
l
2
max
-
. Thus for N=4, 5 the subspace
l
2
max
-
corresponds to the most subradiant
subspace i.e. N=4:
l
4220
min =-=
and N=5:
l
12
min =. In these two cases the collective increase in
Figure 2. Leaving the bad cavity limit: variation of the external pumping strength for different ratios
g
k
:(a)the normalized TLS
excitation number
n
NJN
11
=á ñ ,(b)the relative superradiant subspace occupation
R
lN2
max
=(
)
,(c)the cavity output rate
mbbkk=á
ñ
†and (d)the photonic second order correlation function
g
0
2(
)
() : drastic qualitative change for
g
k
approaching unity.
5
New J. Phys. 20 (2018)013006 M Gegg et al
population is strongest. For larger Nsubspaces with smaller lexist, e.g.
N
ll6: 3
min max
==-
. Looking at R(0)
(only defined for even N, always corresponds to the most subradiant subspace),figure 3(c), we see that the
increase due to collective effects increases with N. Hence the collective increase is always most pronounced in the
most subradiant subspace (l
min
)above the phase transition. Remarkably, below the phase transition the
subradiant subspaces are completely suppressed, see figures 3(b),(c).
The total occupation in the superradiant subspace goes to zero above the phase transition for
N
¥
,
figure 3(d). Naively we could associate this with subradiance. However for
E
¥
the TLS are in a completely
incoherent, equipartitioned state [51]and the superradiant subspace is only depopulated since this subspace
becomes very small compared to the full Hilbert (Liouville)space for large N. This is clearly not a collective
effect. This illustrates that (in the steady state)it is impossible to distinguish between collective and individual
behaviour by using Dicke state occupations alone.
However by looking at both the absolute and relative populations we conclude that in the good cavity and
large Nlimit the system changes from a predominantly superradiant to a predominantly subradiant state at the
phase transition. This constitutes the main result of this work.
In figure 4the scaling of experimentally more accessible quantities with the number of individual TLS Nis
presented: the normalized TLS excitation develops a kink for increasing N, indicating a second-order transition,
figure 4(a). The smallest magnitude nonzero eigenvalue
1
l
of the Liouville operator
(see equation (2)), which
corresponds to the slowest time scale in the system to reach steady state, decreases around the phase transition
for increasing N,figure 4(b). It might even vanish for
N
¥
, creating a second steady state. This could be
measured for instance in a hysteresis cycle typical for optical bistability experiments [19,40,52]. The intracavity
mean photon number shows the formation of a local minimum at the transition and an increase in the peak
intensity, figure 4(c). Also bunching (
g
01
2
>()
()
)increases for increasing N,figure 4(d). Overall the transition
becomes sharper and more pronounced for increasing Nand decreasing
g
k
, since these parameters increase
the system size. This displays a typical property of phase transitions, which are well defined only in the
thermodynamic limit (infinite system size)and blur for small system sizes [4,53,54].
5.2. Robustness test, entanglement and the spontaneous decay time
So far all results were presented without including pure dephasing. Now we investigate the robustness of the
collective effects at the phase transition against pure dephasing: in figure 5(a)we see that the collective behaviour
of the relative Dicke subspace population is reduced for increasing δ. However the effect of clear distinction of
superradiant state below and subradiant state above phase transition is preserved for
d
g~. The general trend of
total Dicke subspace occupation is not affected by pure dephasing, as in figure 3(d).
Figure 3. Increasing the system size: relative Dicke subspace occupation for varying N:(a)the superradiant subspace
lN2=
,(b)
lN22=-
,(c)l=0. These states have no interactions due to the Hamiltonian. They only couple to states with l0>through
decay and dephasing. (d)Absolute occupation in the superradiant subspace: approaching zero above the phase transition for
N
¥
, even without correlations.
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New J. Phys. 20 (2018)013006 M Gegg et al
In the spin preserving setup the TLS are entangled via spin squeezing below the phase transition [17]. Spin
squeezing is a concept originating from quantum metrology, where it was developed around the idea that
squeezed atomic coherent states could be used for measurement precision below the shot noise limit, but also
has attracted a lot of attention as an entanglement witness [55–58]. Here we employ the spin squeezing
inequalities (SSI)introduced by Tóth et al that are explicitly derived as an entanglement witness for many two-
(and multi-)level system setups [59,60]. The spin preserving case does not contain any subradiant states/effects
and cannot model the effects of pure dephasing. The spin preserving and nonpreserving scenarios are two limits
Figure 4. Experimental signatures for varying N:(a)the normalized TLS excitation number
n
NJN
11
=á ñ ,(b)the renormalized
Liouvillian gap
1
lg
∣
∣
,(c)the rescaled intracavity photon number
m
NbbN=á ñ
†and (d)the second order correlation
g
0
2(
)
() .
Figure 5. Robustness, entanglement and decay time: (a)the ratio R(l)for N=5 for ll l,
min max
=and varying δ: the clear switching at
the phase transition survives for
d
g~
.(b)Entanglement via spin squeezing inequalities: entanglement below the transition for
d
g<
.(c)and (d)The influence of the spontaneous decay time on the population of subradiant states is negligible in realistic
parameter ranges. This is especially surprising since this parameter is the only parameter that couples the different Dicke subspaces
(for
0
d
=
). The corresponding ratio
g
g
correspond to the interval 210210
52
´´
--
–.
7
New J. Phys. 20 (2018)013006 M Gegg et al
of the same physical system [31,61]. Thus an investigation of entanglement in our setup and its preservation
under dephasing is desirable: we find that the SSI by Tóth et al detect entanglement below the phase transition
for
d
g<, see figure 5(b)(see appendix Bfor the SSI and a definition of the quantity in figure 5(b)). Hence the
entanglement detected in the spin preserving setup is still present for spin nonpreserving setups and even for
moderate pure dephasing times.
In figures 5(c)and (d)we vary the spontaneous decay time over typical parameter ranges for quantum optics
and quantum information setups, such as quantum dots, NV centres and Rydberg atoms. The quantity shown is
the relative subradiant subspace occupation for N=5. There is hardly any effect at all, only in the limit of
unrealisticly short decay times there is a visible dependence. The qualitative behaviour, that subradiant states are
amplified in a good cavity and are not amplified in a bad cavity, is not influenced at all by this parameter.
However setting this parameter to zero would result in a decoupling of the different Dicke subspaces and the
R
12()
curves in figures 5(c)and (d)would be fixed at zero. This seems contradictory that the behaviour of the
system for decreasing γdoes not converge towards the 0
g
=scenario. This contradiction can be removed by
remembering that these results correspond to the steady state.Infigure 4(b)the renormalized Liouvillian gap
1
l
gis shown. This gap scales with γ, meaning that with decreasing γthe steady state convergence time increases
and becomes infinite for
0
g
. Thus this particular steady state will never be reached. This corresponds to a
dissipative phase transition, the Liouvillian gap closes, and the different Dicke subspaces form non-interacting
subspaces in Hilbert/Liouville space. Thus in this limit the setup is truly multistable with the dimension of the
Liouvillian null space being equal to the number of distinct lvalues.
In summary the spontaneous decay time is a necessary condition for the population of dark states in the
system but has no influence on the actual steady state in realistic parameter ranges—γjust scales the time
necessary to reach this steady state. Therefore we conclude that—for a finite γ—the only parameter responsible
for the dark state coherences and the superradiant to subradiant switching is in fact the cavity quality, which
however does not lead to the population of dark states by itself. Therefore the presented effect is a highly
nontrivial emergent property, arsing as an interplay of dissipative, individual and coherent, collective, cavity
processes.
5.3. Dark state cascades
Super- and subradiance are concepts related to time evolution and so far we have only discussed the steady state:
now, we drive the system to the steady state with maximum
R
l
min
()
(see figures 3(b)and (c)) and then,
afterwards, we switch off the driving field. The system relaxes into the ground state and we observe that a cascade
of dark states is generated, figure 6(a):
p12, 12-()
and
p32, 32-()
are the populations in the lowest states
of the smallest
l
l
mi
n
=and intermediate
l
ll
max min
>>
subspace for N=5 TLS (see also figure 1(b)). Both
states are dark. They are populated on time scales of the inverse TLS-photon coupling constant
g
1
-
, because the
higher energy, bright states of the associated lsubspaces decay via the emission of cavity photons. The cavity
photons subsequently leave the cavity through the cavity decay. After the initial fast population of the
ll,-ñ
∣
states due to the TLS cavity interaction the dynamics are governed by spontaneous emission. At this point the
only states populated are the lowermost states in each lsubspace and the relative population of these states is
determined by the relative population of the respective subspaces before switching off the drive. The overall dark
state population subsequently decays on the slower time scale g5000
11
g
=
--
towards the ground state of the
TLS (
52, 52-ñ
∣
). The decay follows the Dicke state cascade
ppp12, 12 32, 32 52, 52- - -()()()
. In general for different N: all
m
l>-
states relax to the
m
l=-
states on time scales of the inverse TLS-photon coupling constant
g
1
-
which is orders of magnitude
faster than the decay time
1
g
-
. Subsequently the dark states
ll,-ñ
∣
relax in a cascade to the lower energy, dark
states ll1, 1+--ñ
∣
with minimal lon time scales of
1
g
-
towards the ground state ll,
max max
-ñ
∣
,figure 1(b).
Please note that the overall occupation in subradiant dark states reaches values close to unity. In figure 6(b)we
see that increasing the number of TLS also increases the total dark state occupation during ground state
relaxation. Also the single dark state of the most subradiant state does not experience any initial fast population,
since there are no higher energy, bright states in this case.
Overall subradiant correlations are clearly dominant in this cascade, since without these correlations the
excitation in the TLS would still decay via the TLS–cavity interaction Hamiltonian. The superradiant to
subradiant phase transition and the dark state cascade could be exploited for a controlled generation of
subradiant states with dark state occupations up to unity.
The sole requirement for the dark state cascade to occur is individualization: the cascade also occurs in the
bad cavity limit and strong pure dephasing (also for
d
g
)limit. The only difference is that the total transient
populations are lower but still approach unity for
N
¥
: subradiant states are always populated in the
presence of external driving as long as individualization is present and the superradiant subspace becomes very
small to the total Hilbert/Liouville space for large N. Thus the system will have an increasing population in
subradiant states for large Nalso in the bad cavity and strong dephasing limit. In the case of strong pure
8
New J. Phys. 20 (2018)013006 M Gegg et al
dephasing the lifetime of the dark state cascade drops (coherence time). However the effect of favoring
subradiant states and the distinction between incoherent/thermal/individual versus quantum coherent/
collective two-level system behaviour in the steady state relies on the moderate cavity quality and the low pure
dephasing.
6. Conclusion
Experimental systems for observing the effects presented in this paper have to meet certain requirements: the
pure dephasing of the TLS coherences should be small compared to the decay rate, i.e.
d
g~. In experimental
settings this is usually referred to as lifetime limited coherence time, since then the coherence time in the
emitters is essentially limited by the spontaneous radiative lifetime. This can be realized with e.g. Rydberg
ensembles [29,62,63]or with NV centres [64]and quantum dots [65]at low temperatures. For quantum dots
lifetime limited coherence times of 0.63 ns were reported [65]. Also a small inhomogeneous broadening is
required, since it would likely blur the presented effect. For quantum dots this is more challenging than for NV
centres and Rydberg ensembles. Generally, the decay rate γis not a crucial parameter but the ratio between decay
and pure dephasing. If pure dephasing is too large the steady state effects are blurred, in the ground state
relaxation subradiant state occupation is decreased and coherence times are shorter. However the dark state
cascade effect is stable even against larger pure dephasing
d
g>.
The parameters used in this study are realistic for NV centres, quantum dots and Rydberg atoms and the
behaviour is stable over a wide parameter range. Especially varying the spontaneous decay rate γover realistic
parameter ranges has no influence on the discussed effect. For steady state coherences and entanglement
properties the relative strength of the pure dephasing
d
gis crucial, not the absolute value of δ.
In summary we have shown that the nonequilibrium phase transition of cooperative resonance fluorescence
changes drastically when leaving the bad cavity limit: subradiant Dicke states are amplified through cavity
assisted coherences and clear experimental signatures of this effect emerge. Letting the system relax into the
ground state generates a dark state cascade that can be utilized to store quantum information.
Figure 6. Ground state relaxation and the dark state cascade: (a)driving the system to the maximum subradiance point with
subsequent relaxation to the ground state N=5,
0
d
=
: a cascade of dark states is generated. Total dark state occupation close to
unity. (b)Same cascade for N=6: the single dark state of the most subradiant subspace does not experience initial population, since
there are no higher energy Dicke states in this subspace. The overall dark state population is higher than for N=5. A single
prototypical bright state shows initial Rabi oscillations but decays very fast and remains unpopulated throughout the relaxation
process.
9
New J. Phys. 20 (2018)013006 M Gegg et al
Acknowledgments
We thank Nicolas Naumann for useful discussions, and gratefully acknowledge support from the Deutsche
Forschungsgemeinschaft (DFG)through SFB 951 (MG, MR, AK)and through the School of Nanophotonics of
the SFB 787 (MG)and BR 1528/8-2 (AC, AK).
Appendix A. Details to the permutation symmetric method
The permutation symmetry of the master equation equation (2)confines the dynamics of the density matrix
onto the subspace of symmetrized Liouville space states [32,33,35,46]:
nnn,, , A.1
nnnn
11 10 01 11 10 01 00
11 10 01 00
ssss=
ÄÄÄÄ
ˆ[] ()
with
n
Nn n n
00 11 10 0
1
=- - -
. The symmetrization operator is defined as P
P
=åˆ, where
P
ˆ
is the
permutation operator and the sum is over all possible permutations Pof two-level systems. This expression is
not normalized since the method is numerically more stable without normalization [33]. The density matrix can
be expanded in the symmetric states using the Hilbert–Schmidt inner product, tracing over both the photonic
and TLS degrees of freedom
nnn nnn,, tr ,, . A.2
11 10 01 11 10 01
r=[][
ˆ[]] ()
This corresponds to the pure TLS density matrix. The full degrees of freedom of the present system are given by
both TLS and cavity degrees of freedom, therfore for the actual calculations we use the quantities
nnnmm nnn mm,,;, tr ,, , A.3
lr l r11 10 01 11 10 01
r=Äñá[][
ˆ[]∣∣] ()
including the photonic degrees of freedom with normal bosonic Fock states. Equations of motion can be derived
from this expression by taking the time derivative and inserting the quantum master equation. In the PsiQuaSP
library this is greatly facilitated by the use of a sketch representation for the symmetric basis states and the action
of the Liouville space operators, there no derivation of equations of motion is required [35,46]. The population
in all states outside the symmetric Liouville subspace equation (A.1)is zero, if it is zero in the initial state.
Compatible initial states are e.g. the ground state and the thermal equilibrium. The number of different
symmetric basis states and thus the overall scaling of the method is
NNN N1236
3
+++~
(
)( )( )
. For
N=2 we retrieve 10 basis states. The N=2 states that occur in the Dicke state expansion are the classical
occupation probabilities
0, 0, 0
00
100
2
ss=á ñ[]
(TLS ground state),1, 0, 0 11
100
200
111
2
ss ss=á + ñ[] (one TLS
excited),
2, 0, 0
11
111
2
ss=á ñ[]
(both TLS excited)and the quantum correlation
0, 1, 1 10
101
201
110
2
ss ss=á + ñ[] , with
tr... ... r
á
ñ= [
]
. Exchanging the indices
1
2«
leaves these states
invariant—they are permutation symmetric.
Continuing the example for N=2 the expectation values for the Dicke state projectors can be expanded in
the symmetrized basis states: p1, 1 0,0,0-=()[
]
,p1, 0 1 2 1, 0, 0 0, 1, 1=+() ([ ] [ ])
,
p1,1 2,0,0=() [
]
and p0, 0 1 2 1, 0, 0 0, 1, 1=-() ([ ] [ ])
, using the trace condition of [34].
Appendix B. Spin sqeezing inequalities
We employ the SSI introduced by Tóth et al [59,60]as entanglement measure. Tóth et al derived seven
inequalities that are satisfied by any separable N-qubit state, hence the violation of any of these inequalities
implies entanglement. Four of the seven inequalities detect entanglement in our setup, but the violation of two
equations is equivalent: the coherent driving field introduces a time dependent phase factor caused by local
unitary transformations which do not affect entanglement [66]but cause the violation of the SSI to oscillate back
and forth between the two associated inequalities (between (B.1)and (B.2)and between (B.3)and (B.4)). The
four SSI that detect entanglement in our setup are
JJ
NNJ
210, B.1
yz x
22 2áñ+áñ- - - D()() ()
JJ
NNJ
210, B.2
xz y
22 2
áñ+áñ- - - D()() ()
JNN NJJ
2
410, B.3
xyz
222
áñ+ --- D+D
()
()[()()] ()
JNN NJJ
2
410, B.4
yxz
222
áñ+ --- D+D
()
()[()()] ()
where the variances are defined as
AAA
22 2
D=áñ-áñ
(
)
. In order to simplify the discussion we only show one
SSI in our plot:
10
New J. Phys. 20 (2018)013006 M Gegg et al
JJ
NNJ
210, B.5
yz x
A
22 2
áñ+áñ- - - D
()() ()
≕
hence Ais the quantity plotted in figure 5(b). Since strictly speaking the quantities
J
y
2
á
ñ
and
J
x
2
á
ñ
do not have a
defined steady state, but oscillate with the phase factor mentioned above, we set t=0 and thus set the phase
factor to unity throughout the plot in figure 5(d). Since, as stated above, the local unitary transformations
causing the oscillation do not affect the entanglement, this is a valid approach. In the following the local unitary
transformation is explained:
On resonance the Hamiltonian of the system in a frame rotating at the external laser frequency l
w
reads
HgJbJb EJ J.B.6
10 01 10 01
=+++()() ()
†
The corresponding master equation for the setup considered in this work is
H
i,, B.7
tde pd ph
rr r¶= = + + +[] ()
where ρis the rotating frame density matrix. The transformation between normal frame and rotating frame is
given by
ee, B.8
nHt Ht
irot irot
rr=-()
with the normal frame density matrix
n
r
and the Hamiltonian
HbbJ.B.9
lrot 11
w=+() ()
†
The Hamiltonian acts locally on the density matrix, in the sense that each TLS experiences an individual unitary
transformation, i.e.
ee. B.10
J
i
N
1
i
11 11
=s
=
()
Such a transformation leaves the quantum correlations invariant [66]. Nonetheless some quantities arising in
the SSI experience a time dependency through this transformation. In fact only the rotating frame density matrix
has a stationary steady state, the normal frame density matrix
n
r
exhibits an oscillating steady state, where
diagonal entries are stationary and offdiagonal entries oscillate with a phase of multiples of l
w
.
The quantities
J
xy,
2
á
ñ
and Jxy,
2
D
(
)are explicitly time dependent in the normal frame. By adding
equations (B.1)–(B.4)respectively, one can derive time independent inequalities, which however do not detect
entanglement in our setup.
ORCID iDs
Marten Richter https://orcid.org/0000-0003-4160-1008
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