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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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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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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.

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