PHYSICAL REVIEW A 78, 042323 共2008兲

Entanglement dynamics in a dispersively coupled qubit-oscillator system
D. Wahyu Utami and A. A. Clerk
Physics Department, McGill University, Montreal, Quebec, Canada H3A 2T8
共Received 12 March 2008; revised manuscript received 15 August 2008; published 23 October 2008兲
We study entanglement dynamics in a system consisting of a qubit dispersively coupled to a finitetemperature, dissipative, driven oscillator. The robustness against dissipation of two generic kinds of entanglement is studied: the qubit can be entangled with either the phase or amplitude of the oscillator’s motion. In thezero-temperature limit, an analytic expression is derived for the logarithmic negativity. We also discuss how
the generated entanglement may be detected via dephasing revivals, being mindful that revivals can occur evenin the absence of any useful entanglement.
DOI: 10.1103/PhysRevA.78.042323

PACS number共s兲: 03.67.Bg, 85.85.⫹j, 03.65.Yz

I. INTRODUCTION

There exists a long-standing interest in attempting to prepare and detect quantum states of macroscopic objects orcollective degrees of freedom. Such an experiment would be
more than a mere “proof of principle:” it would provide a
nontrivial test of our understanding of the quantum dissipative processes which cause such states to degrade with time,
and thus enforce the quantum to classical transition. Recent
advances suggest that submicrometer-scale mechanical resonators could be excellent candidate systems in which to pursue this goal 关1兴. Such resonators contain a truly macroscopic number of atoms; at the same time, they can be
fabricated to have high resonant frequencies and also highquality factors. This suggests that one has some hope ofcooling these systems to close to their quantum ground state,
and that decoherence due to the dissipative environment of
the resonator should be slow—in standard models, decoherence rates scale with the oscillator damping rate 关2兴. Nanomechanical resonators also have the advantage that they canbe strongly coupled to 共possibly coherent兲 electronic degrees
of freedom; this has recently been demonstrated to allow
sensitive position detection, approaching the fundamental
limits set by quantum back action 关3–5兴.
In this paper, we analyze the entanglement dynamics in anelectromechanical system where a dissipative mechanicalresonator is dispersively coupled to a superconducting qubit:
the state of the qubit simply shifts the frequency of the resonator 关6,7兴. Such a setup has the key advantage of being able
to work with qubit states which are first-order insensitive todephasing due to ever-present charge fluctuations 关8,9兴. As
such, the system is substantially different from the one analyzed in the seminal proposal of Armour et al. 关10兴, which
made use of quickly dephasing superpositions of charge
states in the qubit. While the dispersive coupling allows forlonger qubit coherence times, there is a price to pay: unlike
the proposal of Ref. 关10兴, the two energy eigenstates of thequbit do not yield different average forces on the oscillator.
As such, generating entanglement is a slightly more involved
affair.
We demonstrate that, with a dispersive coupling, there are
two generic ways to generate nonclassical, entangled
oscillator-qubit states: one can entangle the qubit either withthe amplitude of the oscillator’s motion, or with its phase.
1050-2947/2008/78共4兲/042323共7兲

We also study the robustness of these two kinds of entanglement against decoherence due to the dissipative environment, and discuss their detection using coherence revivals. A
fully analytical expression for the entanglement 共as measured
by the logarithmic negativity兲 is obtained for the zerotemperature case. In the finite-temperature case, we make useof an exact solution of the master equation to efficiently
calculate the time-dependent entanglement. We stress thatthe system studied here has already been realized in experiment, both with nanomechanical resonators 关11兴, and withsuperconducting stripline resonators in circuit QED experiments 关9兴. Our study also sheds light on general questions ofentanglement dynamics in the presence of dissipation, driving, and thermal noise, thus complementing other studies ofentanglement in qubit-plus-oscillator models 关12,13兴.
Note that issues of dispersive entanglement, while new inthe context of quantum electromechanics, have been studied
previously in atomic cavity QED systems. In particular, thegroundbreaking experiment of Brune et al. 关14兴 saw compelling evidence for entanglement between the state of an atomand the phase of a cavity coherent state. While theoreticalwork related to this experiment exists 关15–17兴, our study
differs in several key ways. We consider both phase andamplitude entanglement 共as defined above兲, and are explicitly interested in the effects of finite temperature, somethingthat is of crucial importance in nanomechanical systems. Unlike previous work, we also quantify the amount of entanglement using a rigorous entanglement monotone, the logarithmic negativity; we may thus rigorously distinguish a genericloss of qubit coherence from true qubit-oscillator entanglement.

II. SYSTEM

We consider a damped mechanical oscillator 共frequency␻M 兲 which is dispersively coupled to a qubit 共splitting frequency ␻qb兲. Setting ប = 1, the Hamiltonian takes the form
H = H0 + H␥ with

冉

H0 = 共␻ M + ␭␴ˆ z兲 â†â +

冊

1
␻qb
+
␴ˆ z + f共t兲共â + â†兲.
2
2

共1兲

Here, ␭ is the strength of the dispersive coupling, f共t兲 is aclassical external force applied to the resonator, and H␥ de-042323-1©2008 The American Physical Society

PHYSICAL REVIEW A 78, 042323 共2008兲

D. WAHYU UTAMI AND A. A. CLERK

scribes the damping of the oscillator 共damping rate ␥兲 by anequilibrium Ohmic bath at temperature T. We stress that sucha dispersive coupling can be easily realized in systems having nanomechanical resonators coupled to superconducting
qubits, as it emerges naturally from a Jaynes-Cumming-type
coupling in the relevant limit where ␻qb Ⰷ ␻ M 关8兴. Dispersive
couplings have also been achieved in the same way in recent
circuit QED experiments coupling superconducting qubits tostripline resonators 关18兴.
The dispersive coupling implies that the two qubit energy
eigenstates 兩 ↑ 典 , 兩 ↓ 典 each lead to different oscillator frequencies; equivalently, the effective frequency of the qubit depends on the energy of the oscillator. Unlike previous proposals 关10兴, the two qubit states do not yield different
oscillator forces, making entanglement generation somewhat
more subtle. One can easily show that if the oscillator starts
in a thermal state and is not driven, then there is never any
qubit-oscillator entanglement: the oscillator simply leads to astatistical uncertainty in the qubit’s frequency. Entanglement
generation thus necessarily requires that the oscillator start ina superposition of Fock states. The easiest experimental way
to achieve this is by driving the resonator, resulting in anonzero value of 具â共t兲典. We are thus led to consider entanglement in such 共possibly兲 nonequilibrium states.
In what follows, we will focus on the experimentally relevant regime of a high Q factor resonator, and where ␭Ⰶ ␻ M . We may thus make use of the high-Q form of the
Brownian-motion master equation for our system 关7兴:

exact solution of their time evolution for arbitrary temperature and driving force. Full details of this calculation areprovided in Ref. 关7兴.
One can write the resulting solution for the qubitoscillator density matrix in a physically transparent mannerthat is especially convenient for entanglement calculations.
We first let D̂关␣兴 denote the displacement operator exp共␣â†− ␣*â兲, ␴ = ↑ , ↓ the two qubit states, and define␳ˆ ␴␴⬘ = Trqb共␳ˆ 兩␴⬘典具␴兩兲,The solution for the qubit-diagonal parts of the density matrix may then be written as simple displaced thermal states:+ ␥neqD关â 兴␳ ,共2兲where for any operator Â we defineD关Â兴␳ˆ = Â␳ˆ Â† − 共Â†Â␳ˆ + ␳ˆ Â†Â兲/2.

共5a兲

␳ˆ ↓↓共t兲 = 21 D̂关␣↓共t兲兴␳ˆ eq关T兴D̂†关␣↓共t兲兴.

共5b兲

⬁

␳ˆ eq关T兴 = 共1 − e−⍀/共kBT兲兲 兺 e−n⍀/共kBT兲兩n典具n兩.

共6兲

n=0

Further, the displacement ␣␴共t兲 is simply the mean value of â
if the qubit state is frozen to be ␴. ␣↑共t兲 and ␣↓共t兲 obey the
expected classical equations of motion corresponding to adamped harmonic oscillator with frequency ␻ M ⫾ ␭:

␣␴共t兲 = Tr共â兩␴典具␴兩␳ˆ 共t兲兲,共7兲␣˙ ␴共t兲 = − i共␻M ⫾ ␭ − i␥/2兲␣␴共t兲 − if共t兲.

共8兲

In contrast to the above, the solutions to the qubit-offdiagonal parts of the density matrix are more complex, anddo not follow from some simple intuitive argument. We find␳ˆ ↑↓共t兲 = 关␳ˆ ↓↑共t兲兴† = 21 ei− ␻qbtY共t兲D̂关˜␣↑共t兲兴

共3兲

Here, ␥ Ⰶ ␻ M is the damping rate of the resonator due thebath, and neq is a Bose-Einstein factor evaluated at the bathtemperature T and energy ␻ M . ⌫␸ is the intrinsic dephasingrate of the qubit 共i.e., dephasing due to sources other than theoscillator兲. We consider the usual situation where the qubit
energy relaxation time is much longer than the dephasing
time, and ignore T1 processes.
We will further specialize to a class of experimentally
relevant initial conditions where at t = 0 there are no qubitoscillator correlations. The qubit will be taken to be initiallyin the superposition state 兩␺典 = 共兩 ↑ 典 + 兩 ↓ 典兲 / 冑2, while the oscillator state will correspond to a thermal equilibrium state thathas been displaced by the action of the classical driving force
f共t兲 共as yet unspecified兲. Such an uncorrelated state can berealized either by keeping the qubit-oscillator coupling off
until t = 0 共note that the coupling is tunable in electromechanical systems, e.g., Ref. 关4兴兲, or by quickly taking thequbit from its ground state 兩↓典 to 兩␺典 via a ␲ / 2 pulse. Notethat each member in this class of initial states has a Gaussian
form. As is discussed extensively in Ref. 关7兴, Eq. 共2兲 implies
that such states remain Gaussian for all times, allowing an␳ˆ ↑↑共t兲 = 21 D̂关␣↑共t兲兴␳ˆ eq关T兴D̂†关␣↑共t兲兴,Here, ␳ˆ eq共T兲 is the thermal equilibrium oscillator density matrix at temperature T:␳ˆ˙ = − i关H0, ␳ˆ 兴 + 共⌫␸/2兲D关␴ˆ z兴␳ˆ + ␥共neq + 1兲D关â兴␳ˆ
†共4兲⫻兵␳ˆ eq关T*共t兲兴e−i␾共t兲共n̂+1/2兲其D̂†关˜␣↓共t兲兴.

共9兲

We see that ␳ˆ ↑↓ also resembles a displaced thermal state.
However, neither the temperature T* nor the displacements
˜␣␴ are as expected: in fact, T* is time dependent. In addition,
there is an additional phase factor ␾共t兲 resulting from thedependence of the qubit frequency on the resonator energy.
Defining ␴共t兲 = coth关共␻ M / 2T*兲 + i␾ / 2兴, one finds␴˙ = − ␥关␴ − 共2neq + 1兲兴 − i␭共␴2 − 1兲

共10兲

with the initial condition T*共0兲 = T, ␾共0兲 = 0. In addition,
˜␣↑/↓ = 共a⫾兲 ⫾ 共1 − Re ␴ ⫿ i Im ␴兲共a+ − a−兲/共2 Re ␴兲,共11兲
where we have definedȧ⫾ = 关− i共␻ M ⫾ ␭ Re ␴兲 − ˜␥/2兴a⫾ − if共t兲

共12兲

and ˜␥ = ␥ + ␭␻ M Im ␴. Note that the coherent state displacements ˜␣␴ coincide with the simple classical displacements
␣␴ only in the limit of zero temperature.
Finally, the prefactor Y共t兲 in the expression for ␳ˆ ↑↓共t兲 in
Eq. 共9兲 describes the lack of purity of the qubit-oscillator

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ENTANGLEMENT DYNAMICS IN A DISPERSIVELY…

state: the qubit-oscillator system will in general not be in apure state. This can result either from unwanted entanglement with the oscillator’s dissipative bath, or from the intrinsic qubit dephasing described by ⌫␸ in Eq. 共2兲. Defining
pn共T兲 = 具n兩␳ˆ eq共T兲兩n典 we find冉 冕te−⌫␸t exp − i␭
Y共t兲 =

0dt⬘共␴ + 2a+a−*兲

冊

兺n pn共T*兲e−i␾共n+1/2兲具n兩D̂†关˜␣↓兴D̂关˜␣↑兴兩n典

.

共13兲

III. ENTANGLEMENT

Having solved for the evolution of the density matrix inthe previous section, we now turn to characterizing theamount of qubit-oscillator entanglement it possesses. This is
somewhat involved, as in general our system will be in amixed state. Similar to many recent works, we will use thelogarithmic negativity EN to quantify the amount of qubitoscillator entanglement 关19,20兴. EN is a rigorous entanglement monotone applicable to mixed state systems, and isstrictly zero for unentangled systems. One has EN= log2共2N + 1兲, where the negativity N is the absolute valueof the sum of the negative eigenvalues of the partially transposed density matrix ␳ˆ PT. It is worth emphasizing that thelogarithmic negativity is a complex function of the entiredensity matrix, not just, e.g., the qubit coherence Tr␳ˆ ↑↓. As
such, even in the case of zero intrinsic qubit dephasing 共i.e.,⌫␸ = 0兲, a loss of qubit coherence does not in itself imply theexistence of any qubit-oscillator entanglement; such loss ofcoherence could instead be due to the effects of the oscillator’s dissipative bath.
As discussed in the previous section, we always start froma state at t = 0 where the qubit is uncorrelated with the oscillator, and is in an equal superposition of its two energy
eigenstates. The two qubit states in this superposition eachlead 共via the dispersive coupling兲 to different oscillator frequencies; this difference will be exploited to yield entangledstates which, in the absence of dissipation or thermal noise,
would have the simple form
There are thus two generic ways to generate entanglement
共see diagram in Fig. 1兲. The first is to entangle the qubit withthe phase of the resonator, i.e., arg具â典. The second genericapproach is to entangle the qubit with the amplitude of theoscillator’s motion, i.e., 兩具â典兩. Note that entanglement withphase was studied in the seminal experiments of Brune et al.
关14兴, and in related works of theory 关15–17兴. However, thelogarithmic negativity associated with this kind of entanglement has not been studied before, nor its robustness against
finite temperatures.
In what follows, we will discuss phase and amplitude entanglement at both zero and nonzero bath temperature T. In
the zero-temperature case, the solution of the master equation in Eqs. 共5兲 and 共9兲 yields an exact expression for EN共t兲.
Letting cos关␪共t兲兴 = 円具␣↑共t兲兩␣↓共t兲典円, one finds

FIG. 1. 共Color online兲 Illustration of amplitude entanglement
共left兲 and phase entanglement 共right兲. The two displaced states corresponding to the qubit up and down are drawn. In the case of
amplitude entanglement, ␣↑ gets driven to a higher oscillation amplitude compared to ␣↓, as only ␣↑ is on resonance. In the case ofphase entanglement, there is no driving force; however, we start ata large coherent state amplitude. Both of the states gradually decay,
but while this occurs, ␣↑ oscillates faster than ␣↓.

1
N共t兲 = 关冑共1 − 兩Y兩兲2 + 4兩Y兩sin2 ␪ − 共1 − 兩Y兩兲兴,4

共15兲

where for T = 0 Eq. 共13兲 for Y yields
兩Y共t兲兩 =e−⌫␸t exp共− 2␭兰t0dt⬘共兩␣↑␣↓兩sin ␾↑↓兲兲cos ␪共t兲

共16兲

with ␾↑↓共t兲 = −arg关␣↑共t兲␣↓*共t兲兴. As expected, N 共and hence
the entanglement兲 is an increasing function of the distinguishability sin2 ␪ of the two oscillator states. It is also amonotonically decreasing function of the purity Y共t兲: as time
progresses, the bath can distinguish the two oscillator states兩␣␴典, thus reducing both Y共t兲 and the amount of oscillatorqubit entanglement. One might expect that 兩Y共t兲兩 should depend only on the history of the overlap cos ␪共t⬘兲; this is not
the case. Instead, the decay of 兩Y共t兲兩 is sensitive to the historyof sin ␾↑↓共t兲, i.e., the sine of the relative phase between thetwo oscillator coherent state amplitudes. Somewhat surprisingly, times when ␾↑↓ = ␲ do not contribute to the decay of兩Y共t兲兩, even though the overlap between the two oscillator
coherent states is maximally small at such times.
Turning to the more general case where the bath temperature T ⬎ 0, we again use our exact Eqs. 共5兲–共12兲 to solve forthe system dynamics. Unfortunately, the resulting solution
for the density matrix 关as given in Eqs. 共5兲兴 does not allow
for an exact evaluation of the logarithmic negativity. It does,
however, allow for a simple numerical evaluation of N: one
simply converts ␳ˆ into a matrix in a basis of displaced Fockstates, and then numerically finds the partial transpose andthe corresponding negative eigenvalues. For T ⬎ 0, EN共t兲 is
not simply a function of the overlap 具␣↑ 兩 ␣↓典共t兲 and the purity
Y共t兲 关as given by the full expression Eq. 共13兲兴—the entire
matrix structure of ␳ˆ ↑↓ is relevant.
Finally, in what follows we will be especially interested in
how the oscillator dissipation affects the dynamics of entanglement. We will thus focus on the case of small intrinsic
qubit dephasing, ⌫␸ → 0; we will comment on the effects of anonzero ⌫␸ in Sec. IV.

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D. WAHYU UTAMI AND A. A. CLERK

E

N

0.6
0.4

-2

10

-3

10

-4

1

2

0
0

-5

10

0

1

-1

neq

3

4

2

3

5

t(γ )
FIG. 2. 共Color online兲 Logarithmic negativity EN versus time.
EN is a non-monotonic function of time in the amplitudeentanglement setup, while it decays monotonically in the phaseentanglement setup. Blue lines correspond to the amplitudeentanglement setup; red lines correspond to the envelope of theoscillating EN in the phase-entanglement setup. Solid lines are for
neq = 0, dot-dashed for neq = 0.5, and dashed for neq = 1.0. For thephase-entanglement curves, we took ␣0 = 0.76 to maximize the total
integral of EN共t兲 at neq = 0. We also chose ␣ f = 3.74 so that the integral of EN共t兲 for amplitude entanglement at neq = 0 is the same as thephase case. The inset shows the total time integral of EN for bothphase and amplitude entanglement as a function of neq. In thephase-entanglement case, the integrated entanglement tends to zeronear neq = 1. In all cases, ␭ = 0.01␻ M , ␥ = 10−5␻ M , ⌫␸ = 0.
A. Amplitude entanglement

To entangle the qubit with the amplitude of the oscillator’s motion, we start at t = 0 with the qubit in a superpositionof its eigenstates, and the oscillator in a thermal state with
具â典 = 0. The oscillator is then driven with a force f共t兲= ␥␣ f cos关共␻ M + ␭兲t兴 共␣ f ⬎ 0兲. In the relevant limit of ahigh-Q oscillator where ␥ Ⰶ ␭, f共t兲 will cause 兩␣↑共t兲兩 to growto a large value ␣ f , while 兩␣↓兩 will be smaller by a large
factor ␻ M / ␥: hence, the amplitude of the oscillator’s motion
will become entangled with state of the qubit, and we would
expect the qubit-oscillator entanglement to grow with time.
However, at long enough times, the dissipative bath coupled
to the oscillator will destroy this entanglement: the bath candistinguish the two states 兩␣␴典, as described by Y共t兲. These
two competing tendencies lead to EN being a nonmonotonic
function of time; this is shown in Fig. 2.
At T = 0, one can use Eqs. 共5兲 and 共15兲 to derive simple,
exact expressions for EN共t兲 for the amplitude entanglement
setup. Focusing on the relevant limit ␥ Ⰶ ␭ Ⰶ ␻ M , and taking⌫␸ = 0, one finds that the negativity N共t兲 共and hence the entanglement兲 is independent of ␭, and determined by冉

Y = exp

0.5

10

0.2
0
0

5

*

-1

10

neq

0.8

1

10

0

10

|α↑ - α↓|2-1Integrated EN (γ )1冊

共␣ f 兲2关共2 − e−␥t/2兲2 − 1 − ␥t兴 ,2

cos2 ␪ = exp关− 共␣ f 兲2共1 − e−␥t/2兲2兴.

共17a兲
共17b兲

Note that in the long time limit the overlap cos ␪ saturates,
while Y共t兲 decays exponentially to zero. This exponential
decay is easily understood from Eq. 共16兲: at long times, the2t (γ-1)

4

0
0

0.5

-1

t (γ )

1

FIG. 3. 共Color online兲 Left: Distance between effective oscillator displacements ˜␣↑共t兲, ˜␣↓共t兲 entering the expression for ␳ˆ ↑↓. The
apparent width of some curves is due to rapid oscillations withfrequency ␭ / ␲. Right: Bose-Einstein factor associated with the effective temperature T*共t兲 appearing in the expression for ␳ˆ ↑↓. Bothcases are for the amplitude entanglement setup with ␣ f = 3.74. On
the left, the highest curves correspond to the lowest neq = 0; on theright, the value of the curve at t = 0 gives neq. Temperatures are
共black兲, 0.1 共blue兲, 0.5 共green兲, and 0.1 共red兲. ␭, ␥, and ⌫␸ as in Fig.
1. For T ⬎ 0, one finds ˜␣␴ ⫽ ␣␴; temperature reduces the distinguishability of the ˜␣␴.

phase difference ␾↑↓ is nonzero and independent of time.
Nonzero temperatures dramatically suppress amplitudeentanglement, as is shown in Fig. 2. Even though the two“classical⬙ amplitudes ␣↑ , ␣↓ continue to have very different
magnitudes at T ⬎ 0 共they are of course independent of T兲,
the displacements ˜␣↑ , ˜␣↓ which determine ␳ˆ ↑↓ 关cf. Eq. 共9兲兴
become less distinguishable as T is increased. This behavior
is shown in Fig. 3, along with the time dependence of theeffective temperature T*共t兲 appearing in the expression for␳ˆ ↑↓.
B. Phase entanglement

To entangle the qubit with the resonator phase, we prepare
the system at t = 0 so that the qubit is again in a superpositionof its two eigenstates, and the oscillator is in a state of motion characterized by the coherent state amplitude ␣0 共e.g.,
one could drive the oscillator at ␻m, keeping the coupling to
the qubit off until t = 0兲. All driving forces on the oscillator
are turned off at t = 0, and the coupled system is allowed to
evolve. The magnitudes of both coherent states ␣↑ , ␣↓ will beidentical, and will decay at a rate ␥. In contrast, the phase of
the oscillator coherent state will wind at a frequency determined by the qubit. We have thus prepared a state where thephase, not the amplitude, of the oscillator’s motion is entangled with the qubit.
Shown in the top panel of Fig. 4 is EN共t兲 for the phaseentanglement setup, again in the case of zero intrinsic qubit
dephasing. The entanglement drops to zero periodically at afrequency ␭ / ␲: at these times, the phases of the two oscillator states ␣↑ , ␣↓ are aligned, implying cos ␪ = 1. This alignment also implies that the coherence of the qubit is partiallyrestored, as is shown in the bottom panel of Fig. 4, where weplot 2兩Tr␳ˆ ↑↓兩. These revivals of coherence are not perfect, as
the dissipative bath leads the qubit-oscillator system to lose
purity: one finds that the factor Y共t兲 decays in a steplikefashion. This loss of purity also causes a gradually decaying

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ENTANGLEMENT DYNAMICS IN A DISPERSIVELY…

1

0.2

N

neq=0.1

Integrated EN (γ-1)

2 |Tr ρ

↑↓

|

E

0.5

n =0.5
eq

0.15

0
1

0.5

n =1
eq

n =2
eq

0.1

0.05

0
1

|Y|

0
0

2

4

6

8

10

α02/neq

0.5
0

0.01

0.02

0.03
−1

0.04

0.05

t(γ )

FIG. 4. 共Color online兲 Phase entanglement. 共a兲 Time-dependent
logarithmic negativity EN, ␣0 = 0.76. Solid line is for neq = 0, dashedfor neq = 1.0, and dash-dotted for neq = 5.0. 共b兲 Qubit recoherences
associated with phase entanglement; same parameters. 共c兲 Purity 兩Y兩of qubit-resonator system; same parameters. All with ␭ , ␥ , ⌫␸ as inFig. 1.

envelope for the entanglement oscillations, as can be seen in
Fig. 4. At zero temperature, and for ␥ Ⰶ ␭ Ⰶ ␻ M , the decay of
Y共t兲 is given by冉

Y共t兲 = exp −␣20
兵4␭2 − e−␥t关4␭2 + 2␥␭ sin共2␭t兲␥2 + 4␭2冊+ 2␥2 sin2共␭t兲兴其 .

共18兲

Note that Y共t兲 does not tend to zero in the long time limit.
It is interesting to note that for a fixed bath temperature
the total amount of phase entanglement 关as measured by thetime integral of EN共t兲兴 has a maximum as a function of theinitial coherent state amplitude ␣0; this is shown in Fig. 5.
For ␣0 too small, the two oscillator states are not sufficiently
distinguishable, and EN共t兲 = 0 remains small for all times,
while for too large an ␣0, the bath very rapidly distinguishes
the two oscillator states in the superposition, and EN共t兲 decays rapidly to zero.
Finally, it is interesting to compare the phase and amplitude entanglement setups at finite temperature; this is done in
Fig. 2, where we plot only the envelope of the oscillating
entanglement in the phase case. As can be seen from theinset, phase entanglement is more robust against nonzero T
than amplitude entanglement; the total time integral of EN共t兲
decays far more slowly as a function of neq in the phase case,for parameters that yield the same entanglement at neq = 0.
Further, while both kinds of entanglement are suppressed bynonzero T, entanglement in the phase case can remain large
for short times 共i.e., for times t such that 1 / ␭ Ⰶ t Ⰶ 1 / ␥兲.
Thus, phase entanglement shows a certain increased resilience against thermal dissipation compared to amplitude entanglement.

FIG. 5. 共Color online兲 Total entanglement 共as measured by thetime integral of EN兲 for the phase-entanglement setup, as a functionof the initial coherent state amplitude ␣0 and neq. For a finite neq,
there is an optimal value of ␣0 which maximizes the total entanglement. In all cases, ␭ = 0.01␻ M , ␥ = 10−5␻ M , ⌫␸ = 0.

C. Detecting entanglement

We now discuss how entanglement may be detected using
revivals in the coherence of the qubit. Consider the case ofphase entanglement: as shown in Fig. 4, such entanglement
leads to qubit recoherences 关i.e., the magnitude of the qubit’s
off-diagonal density matrix element 兩Tr␳ˆ ↑↓共t兲兩 is nonmonotonic in time兴. This quantity represents the time-dependentdephasing of the qubit, and is measurable via either a standard Ramsey interference experiment 关21兴, or via state tomography. While such revivals of coherence have been used
to detect nonclassical states in other situations 关22兴, and havebeen proposed as a way to detect entanglement in nanoelectromechanical system 共NEMS兲 关10兴, one must be careful:
it is possible to have coherence revivals without any qubitoscillator entanglement. In our system, a purely thermal stateoscillator state with 具â共t兲典 = 0 yields dephasing revivals, but
zero qubit-oscillator entanglement.
Despite this caveat, one can still use dephasing revivals as
a proxy for detecting entanglement. We focus on phase entanglement, as the amplitude-entanglement setup does notlead to any dephasing revivals, but rather a monotonic 共but
nonexponential兲 decay of 兩Tr␳ˆ ↑↓共t兲兩. The basic idea in using
coherence revivals to detect phase entanglement is that theFourier spectrum of the time-dependent dephasing lets one
unambiguously distinguish an initial thermal state with ␣= 0 and EN共t兲 = 0, from a phase-entangled state where ␣ ⫽ 0.
This difference is essentially the number splitting effect discussed in Refs. 关7,23,24兴, and measured in Ref. 关18兴: thepeaks in the Fourier spectra are directly related to the number
distribution in the oscillator. Thus, in a thermal state, one
expects the peaks to follow a simple geometric series. In
contrast, when ␣0 is nonzero, the distribution begins to resemble more the Poisson distribution associated with a number state. Observing this difference, and comparing it totheory, would be a convincing way to detect the phase entanglement we have described. The difference between thetwo spectra is shown in Fig. 6.

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D. WAHYU UTAMI AND A. A. CLERK

−4

x 10

0.4

Area

Re[S(ω)]

3
2

0

1
0

0.2

0

0

1

n
1

ω (2 λ)22FIG. 6. 共Color online兲 Phase-entanglement spectrum. S共␻兲, thereal part of the Fourier transform of the qubit’s time-dependent
coherence 2兩Tr␳ˆ ↑↓共t兲兩. The red dashed curve is for neq = 0.5, ␣0 = 0.
There are revivals of coherence, but strictly zero qubit-oscillator
entanglement; S共␻兲 shows peaks whose area follows a geometricseries. In contrast, the solid blue curve corresponds to ␣0 = 1.23,
neq = 0.5. Here, one gets both coherence revivals and entanglement.
The dephasing spectrum is markedly different: the peak areas do
not form a geometric series. The inset shows the integrated area foreach spectrum in the two cases. All with ␭ , ␥ , ⌫␸ as in Fig. 1.
IV. EXPERIMENTAL CONSIDERATIONS AND
CONCLUSIONSTo realize these ideas in a quantum electromechanicalsystem, there are many challenges to be addressed. One musthave a sufficiently strong qubit-resonator coupling. The resulting dispersive coupling ␭ must be much larger than both
the resonator damping ␥, as well as the intrinsic qubit
dephasing rate ⌫␸. As indicated by Eq. 共13兲, the qubit
dephasing rate causes the purity Y共t兲 of the qubit-oscillator
state to decay over and above the decay causes by the oscil-

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ground state, one can still obtain appreciable entanglement.
In conclusion, we have studied entanglement in a dispersive qubit-oscillator system. We have identified two generickinds of entanglement 共phase and amplitude entanglement兲,and have shown that in general, phase entanglement is morerobust against the effects of dissipation. We have also discussed how one can detect phase entanglement by using revivals in the qubit’s coherence, and how one can distinguish
this from revivals occurring without any qubit-oscillator entanglement.
ACKNOWLEDGMENTS

This work was supported by the Alfred P. Sloan Foundation, the Canadian Institute for Advanced Research, and
NSERC.

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