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Non-Markovian Dynamics and Entanglement of Two-level Atoms in a Common Field
C. H. Fleming∗and N. I. Cummings†
Joint Quantum Institute, University of Maryland, College Park, Maryland 20742-4111, USA
Charis Anastopoulos‡
Department of Physics, University of Patras, 26500 Patras, Greece
B. L. Hu§
Joint Quantum Institute and Maryland Center for Fundamental Physics,
University of Maryland, College Park, Maryland 20742-4111, USA
(Dated: Jan 11, 2011)
We derive the stochastic equations and consider the non-Markovian dynamics of a system of
multiple two-level atoms in a common quantum field. We make only the dipole approximation for
the atoms and assume weak atom-field interaction, but no Born-Markov (BMA) or rotating-wave
approximation(RWA). These more accurate solutions are necessary if one wants to determine a)
whether late-time asymptotic entanglement exists and b) whether any initial state can avoid sudden
death, questions of practical importance for quantum information processing. We find that even
at zero temperature all initial states will undergo finite-time disentanglement (or eventually meet
with ‘sudden death’), in contrast to previous work. We also use our solution without invoking RWA
to fully characterize the necessary conditions for the sub-radiant dark state, which can be used to
preserve coherence and entanglement for long times. For sub-radiance and super-radiance to be
achieved, the atoms must be held close in relation to their resonant wavelength, and they must be
tuned closely in relation to the normal dissipation rate. This latter regime cannot be described
by Lindblad equations. Temperature does not alter the existence of such states. We discuss how
the phenomena of sub and super-radiance can be viewed as an interference phenomenon among
the noise processes and give a simple explanation of why the super-radiance emission rate scales
like the number of atoms squared. We also give an in-depth treatment of renormalization, which
takes into account the correlated influences between atoms and the importance of time dependent
renormalization in preserving causality.
I. INTRODUCTION
Atomic systems constitute an important setting for the
investigation of quantum decoherence and entanglement
phenomena essential for quantum information processing
considerations [1–4]. The physical principles underlying
these systems are quite well understood, and they can
be controlled and measured with great precision. One
aspect of quantum entanglement dynamics that has re-
ceived significant attention is the phenomenon of entan-
glement sudden death, or finite-time disentanglement,
while energy and local coherences only decay away expo-
nentially in time [5–7]. A common setting for theoretical
discussion of this phenomenon is atomic systems inter-
acting with the electromagnetic field [6, 8–12], serving
as an environment in the quantum open system (QOS)
perspective.
Much of the theoretical work on atom-field systems
is derived using the rotating-wave approximation (RWA)
[13–15]. When considering atomic dynamics with an
open-system approach, where there is a continuum of
∗hfleming@physics.umd.edu
†nickc@umd.edu
‡anastop@physics.upatras.gr
§blhu@umd.edu
field modes that are treated as a reservoir, the Born-
Markov approximation (BMA) is commonly invoked,
usually in combination with a RWA [13, 15, 16]. How-
ever, if such systems are central to many important in-
vestigations of broad relevance, both experimental and
theoretical, and one wishes to examine subtle features
such as entanglement dynamics, then great care must be
taken in the use of approximations and the accuracy of
results so derived. Both of these approximations require
an assumption of weak system-environment coupling to
be justified.
A fully non-Markovian treatment of multiple two-level
atoms in a common quantum field has yet to be carried
out in a manner which can predict entanglement evolu-
tion fully and address critical issues such as sudden death
of entanglement. There are several important reasons for
this. (1) Calculations have usually involved perturba-
tive master equations, either explicitly or by invoking the
BMA or RWA which both implicitly assume a perturba-
tive coupling to the environment. However, second-order
master equations fundamentally cannot rule out second-
order amounts of entanglement in many situations. This
is due to the little-known fact that second-order (non-
Markovian) master equations are not generally capable
of providing full second-order solutions except at early
times [17]. (2) Use of a RWA will lead to inaccurate pre-
dictions of late-time asymptotic entanglement differing
by second-order terms from the actual value, and at low
arXiv:1012.5067v2 [quant-ph] 12 Jan 2011
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2
temperatures this can potentially lead to entirely differ-
ent and erroneous qualitative features of entanglement
dynamics and sudden death. The RWA only captures
certain timescales to second order, while at any instance
density matrix elements can have second-order inaccu-
racies [18] (except at early times). However we would
note that a non-perturbative (or higher-order) treatment
of the model with an RWA system-environment interac-
tion is still qualitatively interesting. (3) Use of the RWA
also does not allow for consideration of near resonance
(as additional near-stationary terms are needed in the
Dirac picture)1. The existence of a sub-radiant dark
state generically requires the resonance condition, but
determining how critical this is requires some analysis of
the near-resonance regime.
In this work, rather than employing the BMA or RWA,
we use a conceptually straight-forward implementation
of perturbation theory, assuming only weak system-
environment coupling; We make careful and justified use
of the second-order master equation for the dynamics,
paying attention to the expected accuracy of the solu-
tions. We use an alternative (but compatible) means of
calculating the late-time asymptotics. In this way we are
able to show that the two atoms in a single field are not
asymptotically entangled, even when near resonance and
very close together — which is the criterion for a dark
state. This asymptotic behavior turns out to be rather
opposite to that of two oscillators in a field, which can be
asymptotically entangled [20]. In fact, we find that the
entanglement of any pair of atoms will always undergo
sudden death, regardless of the initial state. Further-
more we make a detailed analysis of how coherence can
be long lived amongst the ground state and dark states,
and we proceed to describe all relevant timescales of the
atom-field system. We explore what conditions are re-
quired for sub and super-radiance in terms of proximity,
tuning, and dissipation. In brief, to achieve dark and
bright states one requires proximity better than the res-
onant wavelength and tuning better than the ordinary
dissipation rates. Temperature only appears to change
the nature of these states and does not diminish their
existence (other than increasing any positive decay rates
linearly).
In physical terms the sub- and super-radiance of the
dark and bright states are ultimately a result of inter-
ference among the multiple noise processes provided by
the field modes evaluated at different locations. As such,
one cannot simply add the emission rates of two isolated
atoms. Some special mention should also be given to
our treatment of renormalization. Previous works have
only considered renormalization of the atoms individu-
ally, which is sufficient if the atoms are far apart, and
also simultaneous in time, which is sufficient in the late-
1Near resonant terms can be preserved in implementing the RWA,
but this will then lead to a master equation not of Lindblad-form
as in [19].
time regime.
entirety, which gives rise to an immersed dynamics more
similar to the free system and also more well behaved.
Our counter terms are also introduced along the light
cone, which keeps the full-time theory causal, and not
across all of space simultaneously.
We now describe the perturbative second-order mas-
ter equation in Sec. II. Then in Sec. III we explain our
method of solution, the resulting dynamics, and the ac-
curacy of these solutions. We also discuss the asymp-
totic state and entanglement dynamics specifically in
Sec. IIIC.
Here we “dress” the joint system in its
II.SECOND-ORDER MASTER EQUATION
A. System-environment coupling and correlations
We wish to investigate the properties of multiple atoms
interacting with a common electromagnetic field in free
space, which serves as the environment in the open quan-
tum system description. We will use the two-level ap-
proximation to describe the atoms, so that they are an
array of, otherwise non-interacting, qubits. Thus we can
write the free Hamiltonian of our system as
?
H0=
n
Ωnσ+nσ−n+
?
?k,s
ka†
?k,sa?k,s, (II.1)
where s indexes the two polarizations for each wave vec-
tor?k, and we have taken c = ? = 1. [The ground state
energy of the atoms is explicitly zeroed to make certain
expressions simpler.] We also make the dipole approx-
imation and assume that the atomic transition in each
atom will produce linearly polarized photons (i.e., both
ground and excited state are eigenstates of some compo-
nent of angular momentum with the same eigenvalue).
Under these assumptions the interaction of our system
with the environment can be represented with the bilin-
ear interaction Hamiltonian
?
where σxnis the x spin component of the nthqubit and
ln is its corresponding collective environment coupling.
The environmental coupling for an atom at location ? rn
with dipole moment?dnis
HI=
n
σxnln
(II.2)
ln=
?
?k,s
ı
b
√k
??dn·? ??k,s
??
e+ı?k·? rna?k,s− e−ı?k·? rna†
?k,s
?
,
(II.3)
with b a constant and where ? ??k,sdenote the polarization
vectors perpendicular to?k for an electro-magnetic field
environment such as discussed in Ref. [14, 21, 22]. For a
scalar field environment one can simply neglect the dot
product.
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3
?30
?20
?1010 20 30
z
?0.5
0.5
1.0
FIG. 1: Comparison of sinc (bold), FS1, and FS0
(dashed). Sinc and FS1are extremely qualitatively
similar, both being unity at zero whereas FS0vanishes
at zero.
We calculate the resultant damping kernels corre-
sponding to the correlation functions of the scalar field
αnm(t,τ) = ?ln(t)lm(τ)?Eto be
˜ γnm(ω) = ˜ γ0sinc(rnmω), (II.4)
and for the electromagnetic field,
˜ γ0
?
FS1(rnmω)
?ˆdn·ˆdm
?
−1
2FS0(rnmω)
?ˆd?
n·ˆd?
m
??
(II.5)
,
in the Fourier domain. Here ? rnm= ? rn−? rmis the separa-
tion vector and?d?denotes the dipole component parallel
to ? rnm, with the entire functions
FS1(z) ≡3
FS0(z) ≡ 3(z2− 3)sin(z) + 3z cos(z)
2
(z2− 1)sin(z) + z cos(z)
z3
, (II.6)
z3
. (II.7)
In Fig. 1 we compare these functions. One can see that
the scalar-field correlations are very similar to that of the
electromagnetic field when?dn??dm⊥ ? rnm. Under this
condition, one can also see that the cross correlations,
which are very nonlocal, are maximized when the dipoles
are very close. Whereas when the dipoles are very far
apart, the cross correlations always vanish and thus all
noise can be treated independently. As we will wish to
maximize cross correlations, we will primarily work with
the scalar-field correlations, which one can think of as
being very similar to the parallel dipoles.
Our theory will be manifestly causal (as long as our
renormalization and state preparation is causal) given
that our field correlations are inherently causal. Note for
instance the temporal representation of the scalar-field
damping kernel
γnm(t) =˜ γ0
2δrnm(t), (II.8)
δr(t) ≡θ(r−|t|)
2r
,(II.9)
where θ is the Heaviside step function.
strictly adheres to the light cone.
The fluctuation-dissipation relation allows us to ex-
press the environmental correlations in terms of the
damping kernel as
This kernel
˜ α(ω) = ˜ γ(ω)
ω
sinh?ω
2T
?e−ω
2T,(II.10)
= 2 ˜ γ(ω)ω [¯ n(|ω|,T) + θ(−ω)] ,
and also noise kernel as
(II.11)
˜ ν(ω) = ˜ γ(ω)ω coth
?ω
2T
?
, (II.12)
= ˜ γ(ω)|ω|[¯ n(|ω|,T) + 1] ,(II.13)
where ¯ n(ω,T) is the thermal average photon number in
a mode of frequency ω. The damping kernel ˜ γ(ω) char-
acterizes dissipation, the noise kernel ˜ ν(ω) characterizes
diffusion, and the full quantum correlation ˜ α(ω) can be
thought to characterize decoherence [23]. For our model,
the near and far correlations are not ordered and there-
fore we cannot make any general statements regarding
one limit always providing more dissipation, diffusion,
and decoherence than the other. However, for two very
close and parallel dipoles the off-diagonal entries of the
kernels approach the diagonal values, and in doing so an
eigen-value must also vanish. At resonance this damping
and decoherence deficit can give rise to a “dark state” as
we explain more thoroughly in Sec. IIIB.
B.Master equation form and coefficients
The second-order master equation for the reduced den-
sity matrix of the dipoles can be expressed [24]
˙ ρ = (L0+ L2)ρ, (II.14)
in terms of the zeroth and second-order Liouville opera-
tors
L0ρ = [−ıH,ρ] ,
L2ρ =
nm
(II.15)
?
?σxn,ρ(Anm? σxm)†− (Anm? σxm)ρ?,
(II.16)
with the second-order operator most easily represented
by the ladder operators as
(Anm? σxm) = Anm(+Ωm) σ+m+ Anm(−Ωm) σ−m,
(II.17)
σ±≡1
2[σx+ ıσy] ,(II.18)
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4
and the second-order coefficients being related to the cor-
relation function as
Anm(ω) =1
2˜ αnm(ω) − ıP
?1
ω
?
∗ ˜ αnm(ω),(II.19)
here in the late-time limit (as compared to system and
cutoff timescales), where P denotes the Cauchy princi-
pal value. Higher-order master equation coefficients will
entail convolutions over several copies of the correlation
function combined with several products of the system
coupling operator.
The first portion of the second-order coefficient, or
Hermitian part (here real), is immediately given by
Eq. (II.11). Whereas the second term, or anti-Hermitian
part (here imaginary), must be evaluated via the convo-
lution
?+∞
and together they form a causal response function. These
are the coefficients which often require regularization and
renormalization. For now let us simply evaluate the bare
coefficients for non-vanishing r. For finite temperatures,
the coefficients exactly evaluate to
?
−
rnm
22T
Im[Anm(ω)] = −1
2π
−∞
dεP
?
1
ω − ε
?
˜ αnm(ε), (II.20)
Im[Anm(ω)] = +˜ γ0
?T
in terms of the Lerch Φ1function
rnm
?
1
πIm
?ω
Φ1
?
?ıω
− 1
2πT;2πTrnm
?
??
˜ γ0
ω−1
coth cos(rnmω)
?
, (II.21)
Φ1(z;λ) ≡
∞
?
k=1
e−kλ
k + z.
(II.22)
This functional representation is exact, though best for
positive temperature. Conversely, one also has the low-
temperature expansion
Im[Anm(ω)] =
˜ γ0
rnm
sign(ω)
π
∞
?
k=1
Sk
(II.23)
−
˜ γ0
rnm
1
π[sin(rnmω)ci(|rnmω|) − cos(rnmω)si(rnmω)] ,
in terms of the summand
Sk=Ei[(+kβ + ırnm)|ω|]
e(+kβ+ırnm)|ω|
+Ei[(−kβ + ırnm)|ω|] − ıπ
e(−kβ+ırnm)|ω|
,
(II.24)
and where the trigonometric integrals are defined
?∞
ci(z) ≡ −
?∞
si(z) ≡ −
z
dz?sin(z?)
?∞
z?
, (II.25)
z
dz?cos(z?)
?
z?
e−z?
z?
,
?
(II.26)
Ei(z) ≡ −
−z
dz?P
,(II.27)
however, for positive temperatures this expansion is not
well behaved for small energy differences. For zero tem-
perature, the exact relation (the second line in (II.23))
is well behaved and matches perfectly to the zero-
temperature limit of Eq. (II.21).
At resonance it may be useful to cast Eq. (II.16) in a
somewhat more familiar form as
˙ ρ = −ı[H + VRW+ VNRW,ρ] + DRW{ρ} + DNRW{ρ},
(II.28)
with the unitary generators
?
+
VRW≡ +
nm
?
?
?
Im[Anm(−Ω)]σ+nσ−m
nm
Im[Anm(+Ω)]σ−nσ+m, (II.29)
VNRW≡ +
nm
Anm(+Ω) − A∗
nm(−Ω)
2ı
σ+nσ+m
+
nm
Anm(−Ω) − A∗
nm(+Ω)
2ı
σ−nσ−m, (II.30)
and (pseudo) Lindblad-form dissipators
DRW{ρ} ≡
+
(II.31)
?
?
nm
Γnm(¯ n(Ω,T) + 1)(2σ−nρσ+m−{σ+mσ−n,ρ})
+
nm
Γnm¯ n(Ω,T)(2σ+nρσ−m−{σ−mσ+n,ρ}) ,
DNRW{ρ} ≡
+
(II.32)
?
?
nm
A∗
nm(−Ω)+Anm(+Ω)
2
(2σ+nρσ+m−{σ+mσ+n,ρ})
+
nm
A∗
nm(+Ω)+Anm(−Ω)
2
(2σ−nρσ−m−{σ−mσ−n,ρ}) ,
where Γnm= Ω ˜ γnm(Ω) is the zero-temperature value of
Re[Anm(−Ω)]. The RW terms are among those preserved
in the rotating-wave approximation (RWA), which results
in a Lindblad master equation even outside of the Marko-
vian regime [18]. At zero temperature these coincide with
the form of the master equation in Ref. [10] and their ex-
pression for the distances dependence of Γnm. The NRW
terms are the so-called “counter-rotating” terms that are
neglected in the RWA (though not necessarily VNRW).
1.Asymptotic regularization and renormalization
Note that sinc(ω/Λ) is a high frequency regulator:
sinc(z) : [0,∞) → [1,0) sufficiently fast for all of our
integrals to converge. Therefore we don’t need to con-
sider any additional regularization in our damping kernel
if we do not evaluate sinc(rω) for vanishing r. Instead
of allowing r to vanish for self-correlations, we impose a
high frequency cutoff r0 = Λ−1, perhaps motivated by
the non-vanishing physical size of the dipole. The more
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5
common alternative is to introduce cutoff regularization
directly into the field coupling lnin Eq. (II.3), often by
treating the coupling strength b as a form factor with
some gradual?k-dependence. Different choices of cutoff
regulators will yield the same results to highest order
in Λ, and the theory should be somewhat insensitive to
these details in the end.
Given some form of regularization, the coefficients are
then bounded yet cutoff sensitive. The remaining cutoff
sensitivity is reduced through a renormalization scheme.
The typical scheme in quantum open systems is to sub-
tract off the zero-energy correction
Im[Anm(0)] = −
˜ γ0
2rnm
.(II.33)
This is equivalent to the quadratic σxnσxmcounter-
term which arises from moving the bilinear system-
environment interaction into the square of the environ-
ment’s harmonic potential (c.f.
are numerous reasons for this choice of renormaliza-
tion. Most importantly it is the minimal renormalization
which ensures a lower bound in the energy spectrum of
the interacting system + environment Hamiltonian for
all strengths of interaction [24].
A second reason for the choice of renormalizing (II.33)
is that in the quantum Langevin equation [25, 26] this
term appears as a system-Hamiltonian shift after trans-
forming from the (somewhat pathological) dissipation-
kernel representation to the more well-behaved and
positive-definite damping-kernel representation.
nomenologically, (II.33) is precisely the term to renor-
malize if one desires to keep the homogeneous dynamics
of the interacting system most closely resembling that of
the free theory.
A final general (though abstract) argument for renor-
malization of specifically (II.33) is that it is equivalent to
the enforcement of a gauge invariance between the system
and environment. This is most physical in the Brownian
motion Hamiltonian for an oscillator bilinearly coupled
to a bath of harmonic oscillators
Eq. (II.34)).There
Phe-
p2
2m+mω2
?
where the gauge invariance enforced is specifically trans-
lation invariance. Renormalized in this way, the full sys-
tem + environment is translationally invariant to global
coordinate shifts.
The bare coefficients contain linear cutoff sensitivity
while the renormalized coefficients are only logarithmi-
cally cutoff sensitive. As depicted in Fig. 2, this cutoff
sensitivity only appears for small r as sinc(rω) acts as a
natural cutoff regulator regardless of any cutoff regula-
tion we might put in by hand. Small r divergence appears
in the coefficients, and thus the induced frequency shifts,
of other works [10, 11, 27, 28] because they must not have
2
(x−xeq)2
???
system
+1
2
?
k
p2
mk
?
k
+ mkω2
k
?
??
xk−
ckx
mkω2
k
?2
?
environment+interaction
,
(II.34)
considered regularization of the full influence of the envi-
ronment, including all cross terms. Here we have chosen
to regularize and renormalize all coefficients Anm(ω) and
not simply the auto-coefficients Ann(ω).
Putting aside the previous arguments, one might con-
sider renormalization to be a choice of model. However
one is not free to choose any form of joint regularization.
In previous works, when the cross terms were left un-
regularized, the implication is that the environment cor-
relation function α(ω) (II.11) and related influence ker-
nels are not positive definite for large ω. Their environ-
ment correlation therefore does not strictly correspond
to any microscopic model and there can potentially be
some pathology associated with this.
Finally we question the physical implications of renor-
malization or the lack thereof in its entirety. Without full
renormalization of cross terms, one has a theory where
neutral atoms have 1/r potential interactions at close
range. This does not appeal to physical expectations.
2.Full-time regularization and renormalization
For the full-time evolution of initially uncorrelated
states, one must apply the full-time coefficients
?t
which must exhibit causal behavior as the field correla-
tions are causal. At zero temperature there is a (r−t)−1
pole in the integrand which can be encapsulated by con-
tour integrals for t > r. The encapsulation of this pole
produces an activation jolt in the master equation coef-
ficients precisely at t = r which we plot in Fig. 3. Prior
to the jolt, the master equation coefficients are roughly
zero; whereas after the jolt, the coefficients are roughly
their asymptotic value. For positive temperatures there
is an infinite series of poles increasingly attenuated by
the rising temperature.
With the consideration of renormalization, one be-
comes even more directly confronted with causality. If
renormalization is applied to the entire system simulta-
neously, e.g.
Anm(ω;t) =
0
dτ e−ıωταnm(τ), (II.35)
Im[Anm(ω;t)] → Im[Anm(ω;t)] − Im[Anm(0;∞)],
(II.36)
then the renormalization will be felt instantaneously over
finite distances. Effectively such an acausal renormaliza-
tion is introducing a counter term into the Hamiltonian
at t = 0 for which there is nothing to counter until t > r.
Whereas if renormalization is applied at a retarded time,
e.g.
Im[Anm(ω;t)] → Im[Anm(ω;t)](II.37)
− θ(t−rnm)Im[Anm(0;∞)],
where θ(z) denotes the unit-step function, then the
renormalized theory will be as causal in its behavior as
Page 6
6
0.51.01.52.0
? r
?3.0
?2.5
?2.0
?1.5
?1.0
?0.5
?Γ?
0???1Im?A?
0.51.01.5 2.0
? r
?3
?2
?1
1
2
3
?Γ?
0???1Im?A?ren
FIG. 2: Separation dependence of asymptotic coefficients Im[Ar(±Ω)] for (left) the bare coefficients and (right) the
renormalized coefficients, where the bold curve denotes evaluation at +Ω and the other −Ω.
5101520
? t
?1.2
?1.0
?0.8
?0.6
?0.4
?0.2
r
Γ?
0
Re?A?
5 10 1520
? t
0.5
1.0
r
Γ?
0
Im?A?
FIG. 3: Re[Anm(−Ωm;t)] (left) and renormalized Im[Anm(−Ωm;t)] (right) for a zero temperature reservoir at
rnm= 10/Ωm. The bold line denotes the asymptotic coefficients. For the latter plot, the dashed curve is the result
of simultaneous renormalization and is acausal.
the non-renormalized theory. Renormalization (and any
state preparation [29]) must be performed in a causal
manner (along the light cone) if one desires causal evo-
lution. Improper renormalization, in the context of a
factorized initial state of the system and environment,
will create (apparently) mediated interactions between
the atoms which are activated before mediation can ac-
tually occur. Such a theory is Hamiltonian, but not rel-
ativistic.
III. SECOND-ORDER SOLUTIONS
From an analysis of the full-time coefficients (see
Fig. 3), one can see that each coefficient jolts on at t = r.
[The jolting (here logarithmic divergence) is a result of
the factorized initial conditions and would become a more
smooth activation upon considering properly correlated
initial states or switching on the interaction gradually.]
So for t < r the atoms roughly evolve independently
(equivalent to r → ∞) and then for t > r the atoms be-
come aware of each other’s presence and evolve jointly.
If there is any acausal behavior, such as creation of en-
tanglement outside of the light cone, then it would have
to be very small.
Because the master equation coefficients mostly
asymptote to constant values quite quickly here in the
weak-coupling regime, as can be seen in Fig. 3, it is suffi-
cient to consider a sequence of constant Liouvillians [29].
E.g. for two atoms
?L[∞](∞) t < r
and therefore the full-time propagator can be sufficiently
approximated by a chain of exponential propagators, here
?etL[∞](∞)
L[r](t) ≈
L[r](∞) t > r,
(III.1)
G[r](t) ≈
t < r
etL[r](∞)er L[∞](∞)t > r.
(III.2)
Page 7
7
A more accurate full-time treatment would be sensitive to
initial conditions, and our factorized initial conditions are
not reasonable enough to warrant that level of scrutiny
for any physical applications. For the remainder of the
paper, we will be interested in the t ? r regime. The
challenge then lies in calculating the evolution due to
etL[r](∞).
A.Dynamics
The open-system dynamics are described approxi-
mately by the time-independent Liouvillian L[r](∞),
which we will now write simply as L. The time evolution
is then approximately etL, and it can be computed (anal-
ogously to the time-independent Schr¨ odinger equation)
simply from the solutions of the eigen-value problem
L{o} = f o,(III.3)
where f is an eigen-frequency and o a right eigen-
operator (super-vector)2. In principle this can be per-
formed numerically with the super-matrix operators, but
when confronted with numerical instability we resorted
to a careful application of canonical perturbation theory,
as can found in Ref. [24]. Because the master equation it-
self is perturbative, there is no loss in accuracy by finding
the solutions perturbatively.
At zeroth-order our eigen-value problem corresponds
to the energy-level differences and outer-products of en-
ergy states
L0{|ωi??ωj|} = −ıωij|ωi??ωj| ,
where ωij= ωi−ωj. The environment induces frequency
shifts (including decay) and basis corrections such that
the eigen-operators are no longer dyadic in any basis of
Hilbert-space vectors. Some degree of degeneracy is also
inescapable as ωii= ωjj= 0.
As our system coupling is non-stationary, with no addi-
tional degeneracies the cross-coupling will have no effect
upon the second-order frequencies of the perturbed off-
diagonal operators, and the fijcorresponding to |ωi??ωj|
for i ?= j are given by
fij= −ıωij+ ?ωi|L2{|ωi??ωj|}|ωj? ,
which reference no cross-correlations. Second-order cor-
rections to the eigen-operators o (and thus states) can
then be found by perturbative consistency with the mas-
ter equation. Dynamics of the diagonal operators and
any other degenerate (and near-degenerate) subspaces
must be treated much more carefully with degenerate
perturbation theory. For the energy states, their second-
order dynamics are encapsulated by a Pauli master equa-
tion.This gives rise to their second-order relaxation
(III.4)
(III.5)
2These eigen-operators are often referred to as the damping basis
of the master equation.
5101520 25
? r
1
2
3
4
?
Γ
?0,0?
????
?0,1?
????
?1,1?
FIG. 4: Decay rates of the (zeroth-order) stationary
operators for two resonant atoms in a zero-temperature
environment at varying separation distance. The legend
indicates the pure states they approximately correspond
to in the order they occur at the vertical axis.
510 1520 25
? r
1
2
3
4
?
Γ
?0,0?????
?0,0?????
?1,1??0,0?
?1,1?????
?1,1?????
FIG. 5: Decoherence rates of the (zeroth-order)
non-stationary operators for two resonant atoms in a
zero-temperature environment at varying separation
distance. The legend indicates the matrix elements they
correspond to in the order they occur at the vertical
axis.
rates and zeroth-order eigen-operators.
ent degeneracy, ωii = ωjj = 0 and any resonant fre-
quencies, their second-order operator perturbations re-
quire the fourth-order Pauli master equation [17, 24]. In
Sec. IIIC we use an alternative means to calculate cor-
rections to the asymptotic or reduced thermal state using
only the second-order coefficients. To summarize, in gen-
eral the matrix elements of the solution ρ(t) expressed
in the (free) energy basis will be accurate to O(γ) off
the diagonal but only to O(1) on the diagonal (though
timescales are known to O(γ)). This inaccuracy in the
diagonals is an inherent limitation of any perturbative
master equation, including those derived under the RWA
or the BMA [17]. With the RWA, however, all matrix
Due to inher-
Page 8
8
2468 10
∆?
Γ
1
2
3
4
?
Γ
?0,0?????
?0,0?????
?1,1??0,0?
?1,1?????
?1,1?????
FIG. 6: Decoherence rates of the (zeroth-order)
non-stationary operators for two atoms in a
zero-temperature environment at varying detuning and
vanishing separation, r12? Ω1,Ω2, with γ = ?Ω?/100.
The legend indicates the matrix elements they
approximately correspond to for small detunings (in the
order they occur at the vertical axis) as to compare
with Fig. 5.
elements are only good to O(1)3.
In Figs. 4–5 we plot all relaxation rates associated
with the two-atom system as a function of proximity,
where γ is specifically the decoherence rate of a single
isolated atom. For large separation the decay rates for
|Ψ±? ≡ (|0,1? ± |1,0?)/√2 are 1+1 times γ (which would
be Nγ for N atoms), as the noise processes are indepen-
dent and the decay rates are additive. Whereas at prox-
imity they become 0 and N2times γ for |Ψ−? and |Ψ+?
respectively, as the noise processes are maximally corre-
lated and display destructive and constructive interfer-
ence. In Fig. 6 we plot all non-stationary decoherence
rates associated with the two-atom system as a function
of detuning. To achieve a dark state, the tuning of the
two atoms must be much better than the dissipation,
δΩ ? γ, which counter-intuitively implies that weak-
dissipation is not always desirable to preserve coherence.
However, this condition makes more sense if thought of
in another way: The dark state arises from the destruc-
tive interference of the emission from the two atoms. If
the emission from each atom is characterized by center
frequency Ωnand an emission line width γ, then the con-
dition δΩ ? γ simply specifies that the emission lines of
the atoms must overlap enough that their emissions are
not distinguishable from one another. This allows the
required destructive interference.
3When looking only at observables time-averaged over many sys-
tem periods 2π/Ω some of these additional discrepancies gener-
ated by the RWA can be greatly reduced.
1. The Atomic Seesaw
One behavior which is qualitatively different from the
closed-system evolution is the damped oscillations be-
tween the singly-excited states. More specifically for any
initial state of the form
?a + δ
with all positive coefficients, then in addition to the Bell
state decay one will also have damped oscillations of the
form
ρ0=|0,1?
|1,0?
+ıb
−ıb a − δ
?
?0,1|
?1,0|,(III.6)
[δ cos(f1t) − bsin(f1t)]e−γ1t(|0,1??0,1| − |1,0??1,0|)
+ı[bcos(f1t) + δ sin(f1t)]e−γ1t(|0,1??1,0| − |1,0??0,1|)
(III.7)
which can oscillate from one excited state to the other
excited state. But this happens very slowly, with the
frequency
f1= 2˜ γ01 − cos(Ωr)
r
, (III.8)
for all temperatures. The oscillation arises from the mas-
ter equation term defined in Eq. (II.29), and should be
present in conventional calculations using the RWA. This
particular frequency vanishes for small separation; with-
out our choice of regularization and renormalization, as
detailed in Sec. IIB1, it would diverge.
B.The dark state
All stationary (and thus decoherence-free) states ρDof
the open-system must satisfy the relation
LρD= 0,(III.9)
and are thus right eigen-supervectors of the Liouvillian
with eigenvalue 0. As the Liouvillian is not Hermitian,
there is no trivial correspondence between the left and
right eigen-supervectors. The super-adjoint of the mas-
ter equation [16] time-evolves system observables and for
closed systems can be contrasted
L0ρ = −ı[H,ρ],
L†
(III.10)
0S = +ı[H,S].(III.11)
The left eigen-supervector S†
therefore satisfy
Dcorresponding to ρDmust
L†SD= 0. (III.12)
So for every stationary or decoherence-free state ρDthere
is a symmetry operator SDwhose expectation value is a
constant of the motion. The thermal state or reduced
thermal state is such a state. In the limit of vanishing
coupling strength, this state is the familiar Boltzmann
Page 9
9
thermal state. One can check that the symmetry op-
erator in this case is proportional to the identity and
corresponds to Tr[ρ] being a constant of the motion.
For two resonant dipoles, with Ωn = Ω, there is an-
other stationary state in the limit of vanishing separa-
tion r12= r. Because of degeneracy, any superposition
of states
|Ψ? = a1|1,0? + a2|0,1? ,(III.13)
is also an energy state and therefore annihilated by both
L0and L†
the noise processes ln(t) become exactly correlated and
identical. Their contributions to the interaction Hamil-
tonian can then be collected into
0. Further note that for vanishing separation,
HI1+ HI2= (σx1+ σx2)ln = Σxln. (III.14)
Next we note the equality
Σx|1,0? = Σx|0,1? ,(III.15)
so that for the Bell states
|Ψ±? ≡
1
√2{|1,0? ± |0,1?} ,(III.16)
the noise adds destructively for |Ψ−? and constructively
for |Ψ+?. Therefore |Ψ−? is a decoherence-free state (dark
state) of the open system for vanishing separation and
at resonance, regardless of coupling strength or temper-
ature. And whereas |Ψ−? appears dark (sub-radiant),
|Ψ+? appears bright (super-radiant). [Note that for anti-
parallel dipoles, these roles will be reversed due to the
anti-correlated noise.]
In this particular case the left and right eigen-
supervectors are equivalent, and so it is the dark-state
component ?Ψ−|ρ|Ψ−? which is a constant of the mo-
tion. However, unlike the thermal state, if the separa-
tion is no longer vanishing then this is not some pertur-
bative limit of a stationary state but of a very long-lived
state. The final constant of motion, which we have val-
idated by analyzing the eigen-system of L, corresponds
to the coherence between the ground state and the dark
state or ?0,0|ρ|Ψ−?. Using these constants of motion, for
two very close dipoles in a zero-temperature environment
with initial state ρ0, the system will relax into the state
ρ1= (1 − b)|0,0??0,0| + b|Ψ−??Ψ−|
+ c|0,0??Ψ−| + c∗|Ψ−??0,0| ,
b ≡ ?Ψ−|ρ0|Ψ−? ,
c ≡ ?0,0|ρ0|Ψ−? ,
to zeroth order in the system-environment coupling,
whereupon the system has bipartite entanglement b.
While our (regularized) model is well behaved in the
mathematical limit r → 0, it is important to remember
that physically the model is no longer valid for sufficiently
small r. At small enough r other terms would come into
play, including electrostatic interaction, and eventually
(III.17)
(III.18)
(III.19)
the atoms would cease to even be distinct. We are assum-
ing that this scale is much smaller than all other scales in
our model (except perhaps the cutoff). This means that
we can sensibly consider cases where r is small compared
to the other parameters, but r cannot vanish completely.
Since the coefficients of our master equation are contin-
uous in r, it is useful to consider r = 0 to understand the
limiting behavior as r becomes small. The existence of
the dark state we’ve discussed at r = 0 means that this
state will be almost completely dark when r is small;
thus, any initial state ρ0will first relax approximately
into the state given in Eq. (III.17) within the ordinary
relaxation timescale γ, and then on a much longer relax-
ation timescale τ, where roughly 1/τ ≈ γ(Ωr)2for small
r, the system will fully thermalize. However, this ex-
pression for the dark state is only to zeroth-order in the
system-environment coupling.
the subsequent final state of decay one needs the second-
order asymptotics that we discuss in Sec. IIIC.
Finally we would note that this “dark state” is a very
general feature of resonant multipartite systems with
similar linear couplings to a shared environment. One
can rather easily work out that for a pair of resonant
linear oscillators with these same noise correlations the
sum mode is thermalized, and the difference mode is de-
coherence free for vanishing separation. The separation
dependence of the entanglement dynamics of two reso-
nant oscillators was considered in Ref. [20], while that of
(effectively) two very close oscillators was considered in
Ref. [30, 31].
In order to understand
1.N-Atom dark and bright states
The sub-radiant dark state achieves destructive inter-
ference in the environmental noise (and thus little-to-no
emission) while the bright state achieves constructive in-
terference in the noise (and thus near-maximal emission).
For the super-radiant bright state one essentially couples
the system to N copies of the same noise process ln(t) and
therefore the super-radiant emission rate can be propor-
tional to N2. An N2dependence does appear the case as
we demonstrate in Fig. 7. The emission rate is (pertur-
batively) determined by the noise correlation (the square
of the noise process). Both results differ having from N
independent noise processes where one can simply add
the N independent noise correlations which results in an
emission rate at most proportional to N.
Following the previous approach, we define a proper
dark state as an atomic state annihilated by L0and HI
regardless of the state of the environment. Let us con-
sider an assembly of N resonant dipoles at close proxim-
ity. We first note that the superposition
?
of energy states with the same total excitement S is also
an energy state and therefore annihilated by L0. Defining
|Ψ? =
?sn=S
as1,s2,···,sN|s1,s2,··· ,sN? ,(III.20)
Page 10
10
012345
N
5
10
15
?max
Γ
FIG. 7: Maximal (over states) second-order decay rates
as a function of the number of atoms N, at zero
temperature and in close proximity. The solid curve
denotes the best quadratic fit and has a corresponding
p-value of 2.4%, which is fairly significant in
corroborating an N2dependence.
the collective spin operator
Σx=
?
n
σxn, (III.21)
such that the interaction Hamiltonian can be expressed
HI= Σxln;(III.22)
a proper dark state must then satisfy Σx|Ψ−? = 0 and
will be decoherence free. For N = 2 this is the familiar
Bell state that we’ve already labeled |Ψ−?.
In considering large N the structure is essentially just
what was studied by Dicke [32], so following that ap-
proach we define collective y and z spin operators Σyand
Σzas well as raising and lowering operators Σ+and Σ−,
analogously to Eq. (III.21), as well as Σ2= Σ2
And we can note that the free Hamiltonian for the atoms
only differs from Σz by a multiple of the identity, so
all the eigenstates of that Hamiltonian are also eigen-
states of Σz. A basis for the Hilbert space of the sys-
tem can be specified by the eigenstates of Σ2with eigen-
values j(j + 1) and Σz with eigenvalues m (though for
N > 2 there will be degeneracy, so that additional quan-
tum numbers are needed to identify a specific state). The
dark state we seek must then satisfy Σz|Ψ−? = m|Ψ−?
and Σx|Ψ−? = 0.
only states with j = 0 and m = 0 can satisfy these
requirements simultaneously.
when N is even, and that set of states has dimension
N!/[(N/2 + 1)!(N/2)!]. These are also the dark states in
the RWA, as they are in the null space of both Σ+and
x+Σ2
y+Σ2
z.
As the discussion in [32] implies,
Such states only occur
Σ−. For N = 4 these states take the form
|Ψ−? = a1(|0,0,1,1? + |1,1,0,0?) + a2(|0,1,0,1? + |1,0,1,0?)
+ a3(|0,1,1,0? + |1,0,0,1?),
0 =an,
(III.23)
?
n
(III.24)
where every pair in parenthesis is spin-flip symmetric.
One can easily check that any such state is annihilated
by Σx.
More generally we define an improper dark state as
one only annihilated by L and not HI (i.e., stationary
in the coarse-grained open-system dynamics but not in
the full closed system dynamics), thus being dependent
upon the state of the environment and even the coupling
strength. In the simplest case we can consider the zero-
temperature environment. For the second-order dynam-
ics, upward transitions are automatically ruled out from
the lack of thermal activation. The only term that could
lead to population of higher excitation states is the sec-
ond term in Eq. (II.31), which vanishes at T = 0. Rather
than investigating the master equation, we can then sim-
ply demand that the lowest-order decay transitions are
vanishing, meaning that if
then ?S?|Σx
excited states. We can also state this in terms of the col-
lective spin operators we have defined, by saying that we
demand that |Ψ−? is an eigenstate of Σzwith eigenvalue
m, and that all matrix elements onto states with lower m?
values must vanish. Since Σx=1
that there will be non-vanishing matrix elements onto
states with m?= m−1 unless m = −j. So any state with
m = −j is an imperfect dark state at zero temperature,
and there are N!(2j+1)/[(N/2 + j + 1)!(N/2 − j)!] such
states [32]. Interestingly, in the RWA such states (when
combined with a vacuum field) are also stationary states
but of the closed-system dynamics. For N = 3 and at
zero temperature, all such dark states can be expressed
??ΨS
−
?has total excitation S,
??ΨS
−
?= 0 for all S?≤ S lesser and equally
2(Σ++ Σ−), we know
|Ψ−? = a1|1,0,0? + a2|0,1,0? + a3|0,0,1? ,
0 =
an,
(III.25)
?
n
(III.26)
for weak coupling to the field. These dark states also exist
for positive temperature, but they take on a different
form.
C.Asymptotics
To zeroth order in the system-environment interaction,
the asymptotic steady state is Boltzmann, which can be
expressed
?
ρTn≡1
2
ρT=
n
ρTn,(III.27)
?
1 − tanh
?Ωn
2T
?
σzn
?
,(III.28)
Page 11
11
in terms of Pauli matrices. The asymptotic state of the
second-order master equation is consistent with this re-
sult and can additionally provide some of the second-
order corrections δρTvia the constraint
L0{δρT} + L2{ρT} = 0. (III.29)
These will specifically be the off-diagonal or non-
stationary perturbations. In general, to find the second-
order corrections to the diagonal elements of the den-
sity matrix one needs to compute contributions from the
fourth-order Liouvillian [17].
It has been shown [24, 33] that for non-vanishing in-
teraction with the environment the off-diagonal elements
of the asymptotic state match reduced thermal state
?
where ZC(β) is the partition function of the system
and environment with non-vanishing interaction.
will refer to G(β) as the thermal Green’s function; this
function can be expanded perturbatively in the system-
environment coupling as
G(β) ≡
1
ZC(β)TrE
e−β(H+HE+HI)?
,(III.30)
We
G(β) =
1
Z0(β)e−β H+ G2(β) + ··· ,(III.31)
where Z0(β) is the partition function of the free system.
The second-order corrections are given by
?ωi|G2(β)|ωj? =
?
nmk
Rnm
ijk
Z0(β)?ωi|σxm|ωk??ωk|σxn|ωj? .
(III.32)
All terms with ωi= ωjare zero, so that this expression
gives no correction to the diagonal elements of the den-
sity matrix. Otherwise, the (non-resonant) off-diagonal
coefficients are given by
?
?e−βωiAmn(ωki)−e−βωjAmn(ωkj)
with the free ground-state energy set to zero. These coef-
ficients agree perturbatively with those from Eq. (III.29).
Because such an expansion is inherently secular in β, it is
valid only at a sufficiently high temperature such that the
perturbations are small compared to the smallest Boltz-
mann weight,
?
Rnm
ijk
??
ωi?=ωj≡ Ime−βωkAnm(ωik)−Anm(ωjk)
ωi− ωj
?
+ Im
ωi− ωj
?
,(III.33)
γ
Ω? e−β(Ωn+Ωm)=
¯ n(Ωn,T)
¯ n(Ωn,T)+1
??
¯ n(Ωm,T)
¯ n(Ωm,T)+1
?
.
(III.34)
The expansion does not apply at lower temperatures. Re-
liability of the expansion at higher temperature suggests
that the diagonal corrections to the asymptotic state
must be suppressed there.
Since neither the second-order master equation nor the
perturbative expansion of the thermal Green’s function
can give the full low-temperature solution, including di-
agonal corrections, it appears that in general this will re-
quire the fourth-order master equation coefficients. How-
ever, at zero temperature the thermal state is simply the
ground state of the total system-environment Hamilto-
nian. This ground state can be calculated perturbatively
from the Hamiltonian as usual in a closed system, and the
zero-temperature reduced thermal state follows directly.
All three of these formalisms are fully consistent as shown
in Ref. [24]. At zero temperature the off-diagonal second-
order corrections to the asymptotic state are still of the
form given in Eqs. (III.32) and (III.33), with the coeffi-
cients evaluated in the limit β → ∞. The diagonal (and
resonant) perturbations are given by
??
lim
β→∞Im
lim
β→∞Rnm
ijk
?
ωi=ωj=(III.35)
?
e−βωkd
dωiAnm(ωik) + e−βωid
dωiAmn(ωki),
where only a handful of terms are non-vanishing. We
note that the expression inside the limit in Eq. (III.35)
has both the correct low and high-temperature limits,
so it may be roughly correct for all temperatures, but
we have yet to fully investigate the fourth-order master
equation.
For most regimes the second-order thermal state can
now be expressed entirely in terms of the second-order
master equation coefficients and limits thereof, therefore
we can say that the environmentally induced correlations
do vanish for large separations with a power-law decay
like 1/r and 1/r2.
1.Entanglement of Two Atoms
Now we will consider the bipartite entanglement be-
tween any two atoms, labeled n and m in a common
quantum field. We begin with some remarks that apply
to any system of two qubits. We focus on the late-time
dynamics of this system; we will compute the reduced
density matrix for their asymptotic state ρnmand derive
the asymptotic value of entanglement between these two
atoms. We will see that this computation will also al-
low us to show that all entangled initial states become
disentangled at a finite time.
To quantify the bipartite entanglement we will use
Wootters’ concurrence function [34], which is a mono-
tone with a one-to-one relationship to the entanglement
of formation for two qubits. The concurrence is defined
as
C(ρnm) =max{0,C(ρnm)}
C(ρnm) =λ1−
where λ1 ≥ λ2 ≥ λ3 ≥ λ4 are the eigenvalues of the
matrix
(III.36)
??
λ2−
?
λ3−
?
λ4
(III.37)
ρnm˜ ρnm≡ ρnm(σynσymρ∗
nmσynσym) ,(III.38)
Page 12
12
which are always non-negative. A two-qubit state is en-
tangled if and only if C > 0. It is important to note that
C (ρ) is a continuous function of the matrix elements of ρ
(since the eigenvalues of a matrix can be written as a con-
tinuous function of the matrix elements [35]); this then
implies that any density matrix with C < 0 lies in the in-
terior of the set of separable states (with every sufficiently
nearby state also separable), while states with C > 0 lie
in the interior of the set of entangled states. States with
C = 0 are separable but include states that lie on the
boundary between the two sets, infinitesimally close to
both entangled states and the interior of the separable
states. Any separable pure state lies on this boundary
[36].
Given the late-time asymptotic state of two atoms
ρnm, one can easily compute the asymptotic entangle-
ment from C (ρnm). Based on the preceding paragraph,
however, we know that this will also tell us something
qualitatively about the late-time entanglement dynam-
ics. If C (ρnm) < 0 then (assuming only continuous evo-
lution in state space) every initial state must become
separable at some finite time as it crosses into the set of
separable states. Likewise, if C (ρnm) > 0 then all ini-
tial states lead to entanglement at sufficiently late time
and any sudden death of entanglement must be followed
by revival. In models such as ours which have a unique
asymptotic state, it is only when C (ρnm) = 0 that this
qualitative feature of the late-time behavior will depend
on the initial state, with some entangled states remaining
separable after some finite time and others becoming dis-
tentangled only asymptotically in the limit t → ∞ as in
[6, 10]. Previous work has pointed out that the late-time
entanglement dynamics can be determined by the asymp-
totic state in this way [37, 38], with Yu and Eberly [39]
discussing the role of C in predicting sudden death. In
Refs. [38, 39] the authors consider models with multiple
steady states, which introduces additional dependence on
initial conditions.
It can be seen that none of the foregoing discussion
is specific to the concurrence; it would apply to any
quantity that is a continuous function of the density ma-
trix, takes on negative values for some separable states,
and is an entanglement monotone when non-negative. If
we have such an unmaximized entanglement function E
from which an entanglement monotone can be defined
by E = max{0,E}, then we can use it just as we have
discussed using C above. As illustrated qualitatively in
Fig. 8, entanglement sudden death occurs because the un-
maximized entanglement function asymptotes towards a
negative value, whereas any entanglement monotone (de-
rived from E or otherwise) cannot go below zero, leading
to an abrupt sudden death of entanglement when E be-
comes negative.
An important point arises from the facts we have noted
about C and separability: At sufficiently low temper-
ature the O(γ) corrections to the asymptotic state are
required to calculate the sign of C (ρnm) and, therefore,
even the qualitative features of late-time entanglement
t
????
??t?
FIG. 8: Qualitative plot of an (unmaximized)
entanglement function showing dynamics including
entanglement sudden death, revival, and asymptotic
separability.
dynamics. At zero temperature, the zeroth-order asymp-
totic state is simply the ground state of the system, as-
suming no degeneracy at the ground energy, according
to Eq. (III.27). So the zeroth-order asymptotic state is
a pure separable state. This means that it lies on the
boundary between the entangled and separable states,
and in general some initial states will suffer sudden death
while others will not, as depicted in Fig. 9a. But any non-
zero perturbation, however small, can lead to asymptotic
entanglement or can place the asymptotic state in the
interior of the separable states, implying sudden death
for all initial conditions. Fig. 9b shows each of these
situations. Thus, knowing only the zeroth-order asymp-
totic state one can make no meaningful prediction about
late-time entanglement dynamics, and this will always be
the case when using the rotating-wave approximation,
because it neglects the second-order corrections to the
asymptotic state [18]. This makes calculations such as
[10] incapable of correctly predicting these features.
At positive temperature the zeroth-order asymptotic
state is simply the Boltzmann state ρT, which lies in the
interior of the set of separable states [37], and
ρT˜ ρT=e−(Ωn+Ωm)/T
Z0(T)2
1,(III.39)
so that C (ρT) = −2e−(Ωn+Ωm)/(2T)/Z0(T). The O(γ)
corrections to ρnmwill yield order O(γ) corrections to
ρnm˜ ρnm.Then simply from the definition of C we
know that so long as the temperature is sufficiently high
that Eq. (III.34) is satisfied the corrections to ρnmwill
cause at most O(γ) corrections to C (ρnm) so that it
must remain negative. Consequently, the second-order
asymptotic state still lies in the interior of the separa-
ble states, and all initial states will suffer entanglement
sudden death at sufficiently late times. For lower tem-
peratures it does not appear that the sign of C (ρnm)
can be generically predicted, and one must find the late-
Page 13
13
(a) Pure Asymptotic State
(b) Mixed Asymptotic State
FIG. 9: A schematic representation of the evolution in
state space. The white area represents entangled states
(C > 0), while the gray areas represent separable states
C ≤ 0 with the dark gray representing states with
C = 0. The asymptotic state is represented by ?, while
initial states are represented by ?. In (a) we have the
asymptotic state on the boundary as in the zeroth-order
at T = 0. In (b) two scenarios are shown that can arise
from a small perturbation moving the asymptotic state
off the boundary, into the interior of one of the two sets.
This illustrates how such a perturbation qualitatively
changes the late-time entanglement dynamics.
time asymptotic state for the specific system in ques-
tion which generally requires terms from the fourth-order
master equation.
Returning to the specifics of the particular model ex-
amined in this paper, from Eq. (III.32) it can be read-
ily seen that the atoms are correlated in the asymptotic
state at all temperatures, and from our second-order co-
efficients these correlations experience power-law decay
with separation. However, we find based on Eqs. (III.31),
(III.32), and (III.33) that when the high-temperature ex-
pansion is valid (according to Eq. (III.34)) the asymp-
totic state has C (ρnm) < 0.
Eqs. (III.33) and (III.35) also give C (ρnm) < 0.
both cases the asymptotic state lies in the interior of
the separable states, and all initial states become separa-
ble permanently after some finite time. With this prop-
erty upheld for zero and high temperatures, we suspect
this to be true at all temperatures, making entanglement
sudden death a generic feature which happens in every
case in this model. Of course, as discussed in Sec. IIIB,
for closely spaced atoms there can be a dark state, so
that entanglement persists over a long timescale before
eventually succumbing to sudden death. It should also
be noted that, while this examination of the asymptotic
behavior tells us that entanglement always remains zero
after some finite time, we do find O(1) sudden death and
revival of entanglement at earlier times for some initial
states (similar to [10]).
In Fig. 10 we plot C as it varies with separation dis-
tance and frequency detuning. As a consistency check
we also calculated the logarithmic negativity and found
it to be consistent with the concurrence to second order.
The behavior of the entanglement is markedly different
from that of two oscillators in a field. The separation
dependence of two resonant oscillators was considered in
Ref. [20] and the more general solution will be given in
Ref. [40]. For two oscillators, there can be asymptotic en-
tanglement if they are held very close and near enough to
resonance with each other. Separation and detuning then
causes the entanglement monotones to decay away. For
the two-atom case studied here asymptotic entanglement
does not exist, and resonant tuning with proximity will
only exacerbate the problem. Permanent sudden death
of entanglement occurs because the unmaximized entan-
glement functions can trend below zero within a finite
amount of time and without the need of any asymptotic
limit. We would finally note that while the concurrence
function does appear to be increasing for large detuning,
the parameters drift outside of the weak-coupling regime
as one of the frequencies becomes very small.
At zero temperature,
In
IV.DISCUSSION
In this paper we have derived the dynamics of a col-
lection of two-level atoms under a dipole approximation
interacting with a common quantized electromagnetic
field assuming only weak coupling and not the Born-
Markov approximation (BMA) or rotating-wave approxi-
mation (RWA). The solution we have derived here there-
fore yields greater accuracy than those derived using the
RWA, which is assumed in most prior studies of such
systems. We have also presented a method of finding
the zero-temperature asymptotic state to higher accu-
racy than is possible directly with a second-order master
Page 14
14
510 1520 25
? r
?0.6
?0.5
?0.4
?0.3
?0.2
?0.1
?
Γ
?
0.2 0.40.60.8
∆?
???
?0.60
?0.55
?0.50
?0.45
?0.40
?0.35
?
Γ
?
FIG. 10: Unmaximized concurrence for two resonant atoms at various separation distance (left) and two close atoms
at various frequency detunings (right) at zero temperature and for γ = ?Ω?/100.
equation. We have used this to show that even at zero
temperature the bipartite entanglement between any pair
of atoms will undergo sudden death for all initial atomic
states, in contrast to the predictions of previous theo-
retical treatments [10] under BMA or RWA. (We will
point out specific deficiencies of [11] in a later communi-
cation.) Finally, we have characterized the various decay
rates that are present in this solution without the RWA
and the sub- and super-radiant states that exist.
We have argued that in the RWA there can be inac-
curacies in all entries of the density matrix that are of
the order of the damping rate γ. By contrast, when rep-
resented in the (free) energy basis the solution we have
derived here will have off-diagonal elements that are ac-
curate at second-order, having O(γ2) errors.
this solution diagonal matrix elements (and matrix el-
ements between any two degenerate energy states) can
still have O(γ) errors, due to a fundamental limitation
of any weak-coupling master equation. However, the ex-
pectation of any operator that has vanishing diagonals
in the energy basis (including atomic dipole operators),
will have only O(γ2) errors. Moreover, unlike some other
methods of solution, our solution can be applied when
the atoms have distinct frequencies.
At sufficiently low temperature, the zeroth-order
asymptotic state (given by the RWA) is near the bound-
ary between the separable and entangled states, and
the small perturbation induced by the environment at
O(γ) can push it into either set. Depending on which
set the perturbed asymptotic states fall into, all states
may experience entanglement sudden death or all may
become entangled asymptotically. We have presented a
second-order solution for the asymptotic state of any two
atoms, which allows us to say decisively that the zero-
temperature asymptotic state of those atoms is separa-
ble, and pairwise entanglement of all atoms experiences
sudden death regardless of the initial state.
It should be noted that, for example, in some optical-
frequency atomic systems the O(γ) corrections we discuss
Even in
can be quite small, with γ/Ω being perhaps something on
the order of 10−9. Though lowest order corrections to the
timescales cannot be ignored (as they are responsible for
the presence of dissipation), corrections of this size to the
values of the density matrix elements at any instant can
easily be considered negligible. However, in the case of
a theoretical study of entanglement sudden death, where
one wishes to distinguish asymptotic decay to zero from
vanishing in finite time, small perturbations can become
vitally important, as they do at low temperature. And in
optical frequency atomic systems at room temperature
the thermal-average photon number will be far smaller
than 10−9, placing the system deep into what we are
considering the low-temperature regime for entanglement
dynamics.
We have characterized the sub- and super-radiant
states that exist in this model when the RWA is not
used.We have shown that there is still a long-lived,
highly-entangled dark state when the atoms have small
enough separation, and sudden death of entanglement
occurs only on the much longer timescale of decay of
this state (assuming it had some population in the ini-
tial state). In this simple model, decoherence-free dark
states are achievable for arbitrary temperature and dis-
sipation, whereas typically these factors together are the
primary cause of decoherence. This result is achieved
through interference phenomena in the noise processes
themselves. Both destructive and constructive interfer-
ence occur, producing dark states and bright states. In
this model the number of such states can be fairly large,
which is a favorable condition for QIP.
We close with a few remarks: 1) With the knowledge
of distance dependence, to preserve entanglement in time
one should place the atoms very close to each other in
the field, so as to produce strong cross correlations in
the noise. But at some proximity one must also consider
further atom-atom interactions, perhaps self-consistently
within the confines of field theory. 2) Qualitative differ-
ences between systems under the two-level and dipole-
Page 15
15
interaction approximations and harmonic systems sug-
gests a degree of model dependence in some of the phe-
nomena considered; this merits further investigation into
the consequences of these approximations. 3) Many other
sorts of level structures are relevant to experimental sys-
tems, both in terms of the number of levels involved and
the angular momentum exchange with the field.
methodology and conceptions developed in this work can
The
be applied for the analysis of the non-Markovian dynam-
ics of more general systems, from which one can perhaps
better understand how model-dependent the entangle-
ment behavior considered herewith is.
Acknowledgment This work is supported partially by
NSF Grants PHY-0426696, PHY-0801368, DARPA grant
DARPAHR0011-09-1-0008 and the Laboratory of Physi-
cal Sciences.
[1] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995).
[2] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod.
Phys. 73, 565 (2001).
[3] D. L. Moehring, P. Maunz, S. Olmschenk, K. C. Younge,
D. N. Matsukevich, L. Duan,
449, 68 (2007).
[4] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura,
C. Monroe, and J. L. O’Brien, Nature 464, 45 (2010).
[5] K.
˙Zyczkowski, P. Horodecki, M. Horodecki,
R. Horodecki, Phys. Rev. A 65, 012101 (2001).
[6] T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404
(2004).
[7] T. Yu and J. H. Eberly, Science 323, 598 (2009).
[8] M. Y¨ ona¸ c, T. Yu, and J. H. Eberly, J. Phys. B 39, S621
(2006).
[9] M. Y¨ ona¸ c, T. Yu, and J. H. Eberly, J. Phys. B 40, S45
(2007).
[10] Z. Ficek and R. Tanas, Phys. Rev. A 74, 024304 (2006).
[11] C. Anastopoulos, S. Shresta,
quant-ph/0610007.
[12] C. Anastopoulos, S. Shresta, and B. L. Hu, Quantum.
Inf. Process. 8, 549 (2009).
[13] G. S. Agarwal, Quantum Statistical Theories of Sponta-
neous Emission and Their Relation to Other Approaches
(Springer-Verlag, Berlin, 1974).
[14] D. F. Walls and G. J. Milburn, Quantum Optics
(Springer-Verlag, Berlin Heidelberg, 1995).
[15] H. J. Carmichael, Statistical Methods in Quantum Optics
I (Springer, New York, 1999).
[16] H. P. Breuer and F. Petruccione, The Theory of Open
Quantum Systems (Oxford University Press, Oxford,
2002).
[17] C. H. Fleming and N. I. Cummings, “On the accuracy
of perturbative master equations,” (2010), submitted to
Phys. Rev. E, arXiv:1010.5025 [quant-ph].
[18] C. H. Fleming, N. I. Cummings, C. Anastopoulos, and
B. L. Hu, J. Phys. A 43, 405304 (2010), arXiv:1003.1749
[quant-ph].
[19] M. Scala, B. Militello, A. Messina, S. Maniscalco, J. Piilo,
and K. Suominen, J. of Phys. A 40, 14527 (2007).
[20] S.-Y. Lin and B. L. Hu, Phys. Rev. D 79, 085020 (2009).
and C. Monroe, Nature
and
and B. L. Hu, (2006),
[21] G. S. Agarwal, Phys. Rev. A 4, 1778 (1971).
[22] C. Anastopoulos and B. L. Hu, Phys. Rev. A 62, 033821
(2000).
[23] C. H. Fleming, B. L. Hu,
ence strength of multiple non-markovian environments,”
(2010), submitted to Phys. Rev. E, arXiv:1011.3286
[quant-ph].
[24] C. H. Fleming and B. L. Hu, “The evolution of gen-
eral systems in non-markovian environments,”
in preparation.
[25] G. W. Ford, J. T. Lewis,
Rev. A 37, 4419 (1988).
[26] C. H. Fleming, B. L. Hu,
equilibrium fluctuation-dissipation inequality, and non-
equilibrium uncertainty principle,” (2010), submitted to
Eur. Phys. Lett., arXiv:1012.0681 [quant-ph].
[27] R. H. Lehmberg, Phys. Rev. A 2, 883 (1970).
[28] Z. Ficek and R. Tanas, Phys. Rep. 372, 369 (2003).
[29] C. H. Fleming, B. L. Hu,
generation of system-environment correlations with state
preparation,” (2011), in preparation.
[30] J. P. Paz and C. A. Roncaglia, Phys. Rev. Lett. 100,
220401 (2008).
[31] J. P. Paz and C. A. Roncaglia, Phys. Rev. A 79, 032102
(2009).
[32] R. H. Dicke, Phys. Rev. 93, 99 (1954).
[33] T. Mori and S. Miyashita, J. Phys. Soc. Japan 77, 124005
(2008).
[34] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[35] G. W. Stewart, Matrix Algorithms:
(SIAM, 2001).
[36] R. Horodecki,P. Horodecki,
K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
[37] T. Yu and J. H. Eberly, (2007), arXiv:0707.3215 [quant-
ph].
[38] M. O. Terra Cunha, New J. Phys. 9, 237 (2007).
[39] T. Yu and J. H. Eberly, J. Mod. Opt. 54, 2289 (2007).
[40] C. H. Fleming, A. Roura, and B. L. Hu, “Exact analyt-
ical solutions to the master equation of quantum brow-
nian motion for a general environment ii,”
preparation.
and A. Roura, “Decoher-
(2011),
and R. F. O’Connell, Phys.
and A. Roura, “Non-
and A. Roura, “Dynamical
Eigensystems
M. Horodecki,and
(2011), in
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