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Entangling operations in nolinear two-atom Tavis-Cummings models
Roc´ıo G´omez-Rosas,1Carlos A. Gonz´alez-Guti´errez,2and Juan Mauricio Torres1, ∗
1Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico
2Instituto de Nanociencia y Materiales de Arag´on (INMA) and Departamento de F´ısica de la Materia Condensada,
CSIC-Universidad de Zaragoza, Zaragoza 50009, Spain
(Dated: August 9, 2021)
We derive an analytical approximate solution of the time-dependent state vector in terms of
material Bell states and coherent states of the field for a generalized two-atom Tavis-Cummings
model with nonlinear intensity dependent matter-field interaction. Using this solution, we obtain
simple expressions for the atomic concurrence and purity in order to study the entanglement in
the system at specific interaction times. We show how to implement entangling atomic operations
through measurement of the field. We illustrate how these operations can lead to a complete
Bell measurement. Furthermore, when considering two orthogonal states of the field as levels of a
third qubit, it is possible to implement a unitary three-qubit gate capable of generating authentic
tripartite entangled states such as the Greenberger–Horne–Zeilinger (GHZ) state and the W-state.
As an example of the generic model, we present an ion-trap setting employing the quantized mode
of the center of mass motion instead the photonic field, showing that the implementation of realistic
entangling operations from intrinsic nonlinear matter-field interactions is indeed possible.
I. INTRODUCTION
Entangling quantum gates are crucial in quantum in-
formation and quantum computation protocols such as
quantum teleportation, superdense coding, and Shor’s al-
gorithm [1,2]. For the implementation of these gates in
atomic qubits, cavity quantum electrodynamics (QED)
has played an important role, as generating and control-
ling entangled states has become an experimental real-
ity [3–10]. Some of these concepts and results have been
shared with other settings [11–13]. The celebrated Cirac-
Zoller controlled-NOT (CNOT) gate is an example in the
context of ion traps, where a Jaynes-Cummings interac-
tion [14] between electronic levels of the ions and its me-
chanical oscillatory mode has been exploited in order to
mediate the interaction between the ions [15,16]. Similar
applications have been found in the context of supercon-
ducting systems, where artificial atoms can be tailored to
specific needs [17–19]. While certain problems seem to be
solved, it is important to offer other advantageous alter-
natives for different experimental settings. For instance,
the Mølmer-Sørensen entangling gate in ion traps does
not require ground state cooling as the Cirac-Zoller gate
[20–22].
Recent proposals offer new perspectives exploiting the
multiphoton regime in cavity-QED such as the hybrid
quantum repeater utilizing dispersive and resonant inter-
actions of matter qubits and coherent light states [23–26].
It has been shown that using the Jaynes-Cummings inter-
action assisted with multiphoton states, it is possible to
implement a nonunitary entangling operation replacing
the CNOT gate in an entanglement purification protocol
[27,28]. Furthermore, exploiting a two-photon interac-
tion with multiphoton states, it is in principle possible to
∗jmtorres@ifuap.buap.mx
implement a complete Bell measurement (BM) by mea-
suring the state of the field [29]. An important feature to
achieve this BM is that the two-photon interaction model
presents perfect revivals of Rabi oscillations in the system
observables, in contrast to the Tavis-Cummings interac-
tion [30], where these revivals broaden in time [31,32].
A natural question is whether other models with perfect
revivals of Rabi oscillations could also be useful in this
type of protocols. This is relevant in an ion-trap im-
plementation of the model, as single phonon processes
are simpler to achieve than two-phonon ones [11]. Fur-
thermore, large coherent states in the motional degree
of trapped ions are nowadays accessible and controllable
[33–35], making it an interesting candidate to implement
multiphoton regime machinery from cavity QED as mul-
tiphonon ion-trap protocols.
In this paper, we study a generalized version of the two-
atom Tavis-Cummings model with a nonlinear matter-
field interaction. We derive an approximate analytical so-
lution of the time-dependent state vector given in terms
of material Bell states and coherent states of the field.
We find conditions where this approximation remains
valid and where perfect revivals of Rabi oscillations oc-
cur. The simple and general form of our solution allows
us to study the entanglement in the system and to gener-
ate entangling two-qubit and three qutbit quantum op-
erations. We present a viable realization in an ion-trap
setup, where the photonic oscillator is replaced by the
center of mass motion of the ions.
The manuscript is organized as follows. In Sec. II we
introduce a generic nonlinear two-atom Tavis-Cummings
model. We identify constants of motion, show its full
solvability, and derive a compact analytical approximate
solution. In Sec. III we present three examples of the
generic mode and we propose the implementation in an
ion-trap setting. In Sec. IV we study the dynamical fea-
tures and numerically test our approximate solution. We
study the entanglement in the system in Sec. V, where we
arXiv:2108.02813v1 [quant-ph] 5 Aug 2021
2
find approximate analytical expressions for concurrence
and purity of the atomic state. Based on our approxi-
mate solution, in Sec. VI, we present the implementa-
tion of entangling operations for the two and three-qubit
case, together with a Bell measurement protocol using a
second quantized mode.
II. GENERALIZED TWO-ATOM
TAVIS-CUMMINGS MODEL
In this section we present the Hamiltonian of a gener-
alized version of the Tavis-Cummings model [30] with a
nonlinear intensity dependent coupling. We identify con-
stants of motion that lead to an exact solvability. Similar
general models have already been considered in the past
and their exact solution is known [36–43]. However, here
we are interested in presenting a general approximate so-
lution for initial coherent states with large mean number
of quanta that is especially convenient to analyze the en-
tanglement in the system as it is expressed in terms of
material Bell states and coherent states of the field.
A. Hamiltonian and exact solvability
We consider the following Hamiltonian describing two
two-level atoms resonantly interacting with a quantized
harmonic oscillator
H=~ωI +V, I =a†a+Sz.(1)
The free Hamiltonian has been expressed in terms of the
operator Ithat represents the number of excitations in
the system. In the present case Icommutes with the
intensity dependent interaction operator
V≡Va=~Ωf(a†a)aS++a†f(a†a)S−.(2)
We have employed the creation and annihilation opera-
tors of the oscillator, aand a†. In Sec. III we will specify
the nature of the oscillator, that will be considered optical
or mechanical for different particular models. The inter-
action includes the intensity dependent function f(a†a)
leading to a nonlinear atom-field interaction. We have
also used the notation Vain order to stress the depen-
dence on the specific mode operator a, as it will probe
useful when we introduce a second mode and its opera-
tors band b†in Sec. VI.
As for the electronic degrees of freedom of the atoms,
we have introduced the following operators
S−=|gihe|1+|gihe|2, S+=S†
−
Sz=1
2|eihe|1+|eihe|2− |gihg|1− |gihg|2,(3)
where |ei1(|ei2) and |gi1(|gi2) are the excited and
ground states of the first (second) atom. The energy
difference between the atomic levels is given by ~ωand
coincides with a single quantum unit of energy of the
oscillator. Furthermore, ~Ω represents the coupling en-
ergy between the internal states of the atom and the os-
cillator degree of freedom. The resonance condition im-
plies a second constant of motion, namely S2= (S+S−+
S−S+)/2 + S2
z. If each two-level system is regarded as
a pseudospin, then the operator Splays the role of an
adimensional total pseudospin operator. The existence of
these two constants of motion implies that the eigenstates
of the Hamiltonian have to be simultaneous eigenstates
of Iand S2. Noting this fact, it is natural to work out
the problem in the following basis
|ϕni=|Ψ−i|ni,|ϕn
−1i=|ggi|n+ 1i,(4)
|ϕn
0i=|Ψ+i|ni,|ϕn
1i=|eei|n−1i,
where we have employed two of the Bell states
|Ψ±i=|gei±|egi
√2,|Φ±i=|ggi±|eei
√2.(5)
In the above definitions of the atomic states, we have
used the convention of labeling the first atom always to
the left, for instance |ei1|gi2=|egi.
The states in Eq. (4) are eigenstates of Iwith eigen-
value nthat takes values from −1 to ∞. These states
also fulfill the eigenvalue equations S2|ϕni= 0 and
S2|ϕn
li= 1, where l∈ {−1,0,1}. This implies that the
state |ϕniis an eigenstate of the Hamiltonian, as it is
the only one with eigenvalues nfor Iand 0 for S2. The
remaining three states, for fixed n, share the same eigen-
value for S2and therefore form a disconnected block of
the interaction Hamiltonian V. The matrix representa-
tion of each block with fixed ncan be expressed as
V(n)=~
0 Ωn0
Ωn0 Ωn−1
0 Ωn−10
,(6)
where Ωnis a real valued parameter dependent on nand
given by the follwowing expression
Ωn= Ω√2hn+ 1|a†f(a†a)|ni.(7)
The nonzero eigenvalues for each subspace can be
computed exactly and are simply given by En,±=
±~qΩ2
n+ Ω2
n−1. The eigenvectors can also be evaluated
exactly in closed-form, however, we will resort on approx-
imations that will probe useful, especially for analyzing
the atomic state when the field is initially prepared in a
coherent state with a large mean excitation value.
B. Time-dependent state vector
In order to simplify the calculations, we choose to work
in an interaction picture with respect to the free energy
3
~ωI that includes another time-independent transforma-
tion. In particular, the state vector in this frame is given
by
|Ψ(t)i=e−iIφ eiIω t|Ψ(t)iS,(8)
where |Ψ(t)iSis the state vector in the Schr¨odinger or
laboratory frame. The real parameter φis the phase of
the initial state of the field that is assumed to be prepared
in the Schr¨odinger picture in an arbitrary coherent state
|αeiφi=
∞
X
n=0
pneinφ|ni, pn=e−|α|2/2αn
√n!.(9)
In the interaction picture, however, we can restrict the
analysis to nonnegative values of α. In this way, we have
exploited the commutativity of the constant Iwith the
interaction Vin order to simplify the problem without
losing generality. As αis taken real in this work, the
mean number of quanta is given by N=ha†ai=α2.
For the total initial state of the system, we assume a
pure product state of the form |Ψ(0)i=|ψi|αi, where
the two atoms are allowed to start in an arbitrary pure
state |ψi, namely
|ψi=c−|Ψ−i+c+|Ψ+i+d−|Φ−i+d+|Φ+i.(10)
We have chosen to write the initial states in terms of Bell
states for later convenience. However, the basis of Eq.
(4), in which the Hamiltonain is block-diagonal, contains
two Bell states and two bare levels of the atoms. For this
reason and in order to keep track of the calculations, it
is useful to relate the following initial probability ampli-
tudes d−|Φ−i+d+|Φ+i=cg|ggi+ce|eei, where
d±=cg±ce
√2.(11)
Note that with the transformation in Eq. (8), the intial
atomic state is given in the laboratory frame as |ψiS=
eiSzφ|ψi.
The solution to the Schr¨odinger equation in the inter-
action picture defined in Eq. (8) is given by |Ψ(t)i=
e−iV t/~|Ψ(0)i. Using the basis states in Eq. (4) we can
formally expand the solution of the time-dependent state
vector as
|Ψ(t)i=c−|Ψ−i|αi+
∞
X
n=−1
Dn
X
l=−1
Cn,l(t)|ϕn
li(12)
with the limit in the second sum Dn= 1 −δn,0−2δn,−1
given in terms of the Kronecker delta. This limit depends
on the value of nand takes into account that for n=−1
there is only one state in the basis (Eq. (4) without |ϕni),
two states for n= 0, and three states for n≥2. These
states for low nwill have no significant contribution in
the limit of high number of excitation as e−α2/2'0.
Furthermore, we have used the fact that |ϕniis eigen-
state of Vwith zero eigenvalue and therefore its proba-
bility amplitude remains constant as c−pn. At t= 0 one
has the initial probability amplitudes Cn,−1(0) = pn+1cg,
Cn,0(0) = pnc+, and Cn,1(0) = pn−1ce. As the system is
exactly solvable, it is possible to obtain exact analytical
expressions for all the probability amplitudes in Eq. (12)
using the exact form of the evolution operator presented
in Appendix A.
In order to obtain manageable expressions we will re-
sort on three approximations. In the first one, we make
the replacements Ωn→Ωn−1/2and Ωn−1→Ωn−1/2in
Eq. (6). In this way, the eigenvectors of V(n)are in-
dependent of n. Provided that |Ωn−Ωn−1| Ωn, the
neglected part can be considered as a small perturba-
tion. This is indeed the case, for instance, when Ωn∝n
or Ωn∝√n. However, we will see later that Ωnmight
have a non monotonic dependence on n, but the condi-
tion might be fulfilled for a specific interval outside of
which the distribution pnin Eq. (9) presents vanishing
small contributions. With the first approximation, one
can find that the nonzero eigenenergies are given by
E(n)
±' ±~ωn, ωn=√2|Ωn−1/2|,(13)
where we have introduced the approximate eigenfrequen-
cies ωn. The second approximation is applied to the
Poissonian distribution in the coherent states, namely
pn−1'pn'pn+1, which relies on the condition of hav-
ing a large mean number of quanta N1. Using these
two approximations, one is lead to the following form of
the time-dependent probability amplitudes
Cn,l(t)'c+−d+
2(−1)leiωnt+d++c+
2e−iωnt−ld−pn−l
√2,
(14)
with l∈ {−1,0,1}. The third approximation is made to
the eigenfrequencies in (13) by Taylor expanding around
the mean photon number Nas
ωn'δN+ω0
Nn, δN=ωN−ω0
NN(15)
where we have used a prime to denote the first derivative,
namely ω0
n=dωn/dn. In the next section, we will show
that, despite the nonlinear form of the interaction, a lin-
ear behavior of the eigenfrequencies is indeed possible in
some models at least in an energy interval.
Substituting the expressions of Eq. (15) in Eq. (14)
and using the result in (12) one can approximate the
state vector |Ψ(t)i'|Ψap (t)iwith following expansion
in terms of coherent states of the field and material Bell
states
|Ψap(t)i= [|ζi|αi+|Υ(t)i]/N(t),(16)
where we have identified a time independent contribu-
tion, |ζi|αi, with the atomic stationary state
|ζi=c−|Ψ−i+d−|Φ−i.(17)
The time dependence is then present only in the following
atoms-oscillator state
|Υ(t)i=P±b±e∓i(δN+Szω0
N)t|φ±i|αe∓iω0
Nti,(18)
4
that is given in terms of two time dependent coherent
states accompanied by the following normalized material
states and their initial probability amplitudes
|φ±i=1
√2|Ψ+i±|Φ+i, b±=c+±d+
√2.(19)
Due to the performed approximations, one has to con-
sider the following normalization
N(t) = 1 + 2Re[b∗
+b−ei2h(t)]e−2Nsin2ω0
Ntsin2ω0
Nt(20)
−√2Im hb+e−ih(t)+b−eih(t)ie−2Nsin2ω0
Nt/2sin ω0
Nt,
with h(t) = δNt+Nsin ω0
Nt. This normalization will not
play a role in the forthcoming analysis for two reasons.
It approaches the unit value in the limit N→ ∞ as can
be seen from the behavior e−2Nsin2ω0
Nt/2sin ω0
Nt. The
second reason is that it attains unit value whenever ω0
Ntis
an integer multiple of 2πand those will be the interaction
times that will draw our attention.
The approximation in Eq. (15) is valid, as long as
higher orders in the Taylor expansion remain negligible.
This imposes a restriction on the maximum interaction
time ttb, i.e., when it is considerable less than a break-
down time tb, that can be obtained from the condition
|ω(j)
N(n−N)jtb|
j!= 1 ⇒tb=j!
(8N)j/2|ω(j)
N|,(21)
where ω(j)
Nis the first nonzero derivative of order j > 1.
In the previous expression we have taken into account the
standard deviation of the Poissonian distribution given
by α=√Nand therefore we have replaced |n−N|with
√8N. In this way, the sum of p2
nin the interval (N−
√8N , N +√8N) is larger than 0.995.
The result in Eq. (16) is the first important result of
this work, as it gives a general expression of the state
vector for an initial coherent state of the oscillator and
arbitrary atomic states. Similar expressions have been
found for the two-atom Tavis-Cummings model [31,44]
and first for the Jaynes-Cummings model [45]. However,
here we have presented a more general expression that
is valid for any model described by a Hamiltonian of the
form of Eq. (1). Furthermore, we will show that with this
expression in terms of material Bell sates, it is possible
to analyze in a more manageable way the entanglement
in the system.
C. Rabi oscillations and relevant time scales
Relevant timescales can be unveiled by evaluating ex-
pectation values of the system observables. It is not hard
to realize that these quantities depend on the overlaps
between coherent states of the form
hα|αeiω0
Nti=eiN sin ω0
Nte−2Nsin2ω0
Nt/2.(22)
Let us consider, as a figure of merit, the expectation value
of Sz, Eq. (3), with an initial state |eei|αi. Using the
overlap between coherent states and the solution to the
time-dependent state vector, Eq. (16), one can arrive to
the approximate expression
hSz(t)i ' e−2Nsin2ω0
Nt/2cos(δNt+Nsin ω0
Nt).(23)
From this expression, one can identify three different time
scales. The fastest one is given by the Rabi frequency ωN
determining fast oscillatory behavior. The oscillations
eventually vanish as they are modulated by a Gaussian
envelope; a phenomenon known as collapse of Rabi oscil-
lations [31,44–46]. This happens for times with vanish-
ing small values of the exponential in Eq. (23), when its
argument differs from integer (zero included) multiples
of 2π. The oscillations reappear when the argument of
the exponential in Eq. (23) vanishes, what is known as
revival of Rabi oscillations. These relevant times can be
evaluated from the previous expression and result in the
expressions for the Rabi time, collapse time, and revival
time that correspondingly are given by
tR= 2π/ωN, tc= 2/√N|ω0
N|, tr= 2π/|ω0
N|.(24)
One can note that the revival time always scales with
the collapse time as tr=π√Ntc, where Nis the mean
number of quanta in the oscillator. In Sec. IV we will
numerically study this behavior for the specific models
that will be presented in Sec. III. It is worth commenting
that the expression in Eq. (23) is only valid for times
where the linearization in Eq. (15) represents a faithful
approximation of the eigenfrequencies ωn.
D. State vector at fractional revival times
The revival time tr, as previously introduced in
Eq. (24), corresponds to the moment at which all com-
ponents of the oscillator state in Eq. (16) return to the
initial the condition |αi. At fractional multiples of this
revival time, the complete system attains interesting and
relevant states [31,32,44]. For instance, the state vector
at each odd integer multiple of a quarter of the revival
time, tr/4, is given as the completely separable state
|Υktr
4i=|ζ1,kiP±r−1b±e∓ikδNtr/4| ∓ iαi,(25)
|ζ1,ki=r|Ψ+i+ik|Φ−i
√2, r =p|c+|2+|d+|2,
with an odd integer k. We have arrived to this state using
Eq. (16) and the relation
e∓iSzπ/2|Φ+i=±i|Φ−i.(26)
It can be noted that in the state of Eq. (25), matter and
oscillator separate and that the atomic state is indepen-
dent from the initial condition. Perhaps not so evident
is the fact that the atomic state is a separable state for
5
any value of k, a property that can be simply tested with
any entanglement measure, such as the concurrence that
will later be used in this work. This means that even
if the atoms where initially entangled, no entanglement
remains at this time in any partition of the systems such
as: atom-atom or (any atom or both atoms)-field. This
phenomenon, with no entanglement in the system even if
it was initially entangled, has been refereed to as “basin
of attraction” in the Tavis-Cummings model [44]. It is
important to note that this only happens for the time-
dependent part of the state, |Υ(t)i, and therefore, this
feature applies only when the stationary part vanishes,
i.e., whenever c−=d−= 0.
At odd multiples of one half of the revival time, the
time-dependent part is given by
|Υ(k
2tr)i=|ζ2,ki| − αi,|ζ2,k i=ck|Ψ+i+dk|Φ+i(27)
with an odd integer kand the coefficients given by
ck=c+cos δN
k
2tr−id+sin δN
k
2tr(28)
dk=−i2k+1c+sin δN
k
2tr+i2kd+cos δN
k
2tr.(29)
In this case, one has again a product state of atoms and
oscillator. However, in this case, the atomic part might
be entangled. It is not hard to realize, as we will later
show, that |ζ2,kihas the same degree of entanglement as
the initial component c+|Ψ+i+d+|Φ+i. For this reason,
Eq. (27) will play an important role in identifying the
entanglement properties in the system and in order to de-
sign entangling operations that will be shown in Sec. VI.
III. SPECIFIC MODELS
In this section we present three examples of models
that can be described by the interaction Hamiltonian
in Eq. (2). We start with the Tavis-Cummings model
in order to compare our results with the most studied
example [31,44,47]. The Buck-Sukumar model [36] is
considered as it presents a particular nonlinear interac-
tion that induces an almost exact linear behavior of the
eigenfrequencies as required in Eq. (15). An ion-trap
nonlinear model [42] will be considered, as it represents
a viable experimental setting to this problem. We will
demonstrate that, despite the intrinsic nonlinear behav-
ior, a linearization of the eigenfrequencies is possible in
a restricted interval of the oscillator occupation number.
A. Two-atom Tavis-Cummings model
The Tavis-Cummings model describes the interaction
of an arbitrary number of two-level atoms interacting
with a single-mode of the quantized electromagnetic field
[30]. It can be viewed as an extension of the Jaynes-
Cummings model [14] for many atoms and it has there-
fore become a paradigm in cavity QED. The original
model was introduced in the same form as in Eq. (1) with
f(a†a) = 1 and with pseudo momentum operators S±
and Szfor arbitrary number of two level particles. Here,
however, we only consider the two-atom case that corre-
sponds to the atomic operators in Eq. (3) and whose in-
teraction Hamiltonian is diagonalizable in the block form
of Eq. (6).
As in this case f(a†a) = 1 in Eq. (2), the matrix ele-
ments in the blocks of the interaction potential, Eq. (6),
can be obtained from Ωn= Ω√2n+ 2. The eigenfre-
quencies or Rabi frequencies are obtained from Eq. (13)
and are ωn= Ω√4n+ 2. The relevant frequencies deter-
mining the total state in Eq. (16) can be found using Eq.
(15) as
ω0
N=2Ω
√4N+ 2, δN=2N+ 2
√4N+ 2.(30)
Therefore, in this model one can find that the relevant
time scales are given by
tR≈2π
Ω√N, tb≈√N
Ω, tr=2π√N
Ω.(31)
The shortest time scale corresponds to the Rabi oscilla-
tions period tR, followed by the time tbwhen the coherent
state approximation breaks down, see Eq. (21). Finally
one has the reappearance of Rabi oscillations at the re-
vival time tr. As tr> tb, the revival of Rabi oscillations
is not perfect in the Tavis-Cummings model and for this
reason the field components will deform leading to the
well known broadening of the revivals [46].
B. Two-atom Buck-Sukumar model
In 1980 Buck and Sukumar presented a simple the-
oretical model for the interaction of a two-level atom
with a single-mode electromagnetic field [36]. In this
model the atom-field coupling is assumed to be nonlin-
ear in the field variables and can be interpreted as an
intensity-dependent interaction. As the Buck-Sukumar
model (BS) is integrable and allows perfect revivals of
Rabi oscillations in the case of initial coherent fields, it
has drawn considerable theoretical attention in the past
[37,39,40]. A drawback of this model, however, is that
there is no obvious physical implementation.
Here we consider the Buck-Sukumar interaction for the
two-atom case, where f(a†a) = √a†ain Eq. (2). This
implies a linear dependence on nin the matrix elements
of the blocks of Vand its eigenfrequencies, namely Ωn=
Ω√2(n+1) and ωn= (2n+1)Ω. The relevant frequencies
in the time-dependent state vector in (16) are simply
given by
ω0
N= 2Ω, δN= Ω.(32)
The timescales are dictated in this case by the following
parameters
tR≈π
ΩN, tr=π
Ω, tb=N2
√2Ω.(33)
6
In contrast to the Tavis-Cummings model, here the
breakdown time of the coherent state approximation tb
scales as N2. In this case, the approximate value of
the eigenfrequencies are linear with nand therefore pre-
dict an infinite value of tb. Therefore, we have used
the exact dependence on nof the eigenvalues, which is
Ω√4n2+ 4n+ 2 '(2n+ 1)Ω. Another important differ-
ence is that here the revival time is independent of the
mean value of the oscillator N.
C. Ion-trap nonlinear model
The last and most important model that will be con-
sidered consists on two ions trapped in a linear har-
monic potential driven by a classical monochromatic ra-
diation field. In this case aand a†represent the an-
nihilation and creation operators of the ions center of
mass motion [22,42]. The free Hamiltonian is given by
H0=~ωSz+~νa†a, i.e., the frequency νof the me-
chanical oscillator differs from the transition frequency
of the atoms. The coupling with the electronic levels is
mediated by the external monochromatic field whose fre-
quency is tuned to the first vibrational sideband and is
given by ωL=ω−ν. With these conditions, the interac-
tion Hamiltonian is time independent in the interaction
picture and is also well described by Eq. (2) with the
following intensity-dependent function
f(a†a) = ηe−η2/2
∞
X
m=0
(−η2)m
m!(m+ 1)!a†mam.(34)
In this case, the nonzero matrix elements of the interac-
tion potential can be expressed in terms of a Laguerre
polynomial, namely
Ωn= Ωηr2
n+ 1e−η2/2L(1)
n(η2).(35)
This polynomial will clearly display nonlinear behavior
that will be inherited by the eigenfrequencies ωn. How-
ever, for a given value of the Lamb-Dicke paramenter η,
it is possible to find an interval around a certain value of
Ndisplaying approximately linear behavior with n. In
principle, it is possible to find the most suitable value of
the mean phonon number for a given value of the Lamb-
Dicke paramenter ηby analyzing the form of the La-
guerre polynomial as a function of n. However, the task
greatly simplifies by expressing the Laguerre polynomi-
als in terms of Bessel functions [35,48,49] which, in our
case, is a good approximation whenever η24n+4. Do-
ing so one can find the following approximate expression
Ωn'√2ΩJ1(2η√n+ 1) and therefore the eigenfrequen-
cies become
ωn'2Ω J12ηqn+1
2,(36)
where J1(√x) is the Bessel function of first kind
and order one. The eigenfrequency ωnis plotted in
Fig. 1for two different values of the Lamb-Dicke para-
menter. Relating the argument of the Bessel function
as 2ηpn+ 1/2 = √x, it is possible to analyze the func-
tion for arbitrary values of η. One can then note that
there is an approximate linear behavior in the interval
x∈(7.25,12.65). Indeed, one can realize that a linear
approximation in this interval differs on average from the
original function in less than 1%. For this estimation, we
have performed a Taylor expansion around x0, the zero
of the function d2J1(√x)/dx2, which is the point where
the slope of J1(√x) changes behavior. In this way, one
is able to find a relation between the mean number of
quanta Nand the Lamb-Dicke paramenter ηas
N=x0
4η2−1
2, x0= 9.95161.(37)
The value of x0is written to six digits precision and it
was obtained using the Newton-Raphson method. The
value of Ndecreases as ηincreases. Therefore, in order
to fit a Poissonian distribution with standard deviation
√Nin the linear interval, one has to fulfill the condition
η≤2.7/√32x0≈0.156905. For this reason, large val-
ues of the Lamb-Dicke parameter cannot be used in this
scheme. In Fig. 1we have also plotted the probability
amplitude of each number state in the coherent state of
Eq. (9) for two different values of the mean number of
quanta N. It is to be noted that for a smaller value of the
Lamb-Dicke paramenter, the mean number Nincreases
and also the number of states lying in the linear part of
the function. For this reason, in the limit of large N,
one does not require to perfectly fit the optimal value of
Nin Eq. (37). The generation of large motional coher-
ent states in trapped ions is nowadays possible [33,35]
offering interesting perspective to implement this model.
Using the results in Eqs. (36) and (37) one can obtain
the relevant frequencies for the state vector (16) as
ωN= 2ΩJ1(√x0)≈0.558924Ω
ω0
N=Ω√x0
2N+ 1(J0(√x0)−J2(√x0)) ≈ −2.50163Ω
2N+ 1 (38)
δN=ωN−Nω0
N≈0.558924 + 2.50163N
2N+ 1 .Ω
An important feature to note here is that this quantities
are given only in terms of the optimal value of N, there-
fore, indirectly depending on η. In this form, a similar
analysis as for the previous two models is also possible in
this case. As for the time scales, it is no difficult to find
that the relevant values are given by
tR≈2π
0.56Ω, tr=π4N+ 2
2.5Ω , tb≈N3/2
10Ω .(39)
In this model, the period of the Rabi oscillations is inde-
pendent of the mean phonon number Nand the revival
time trscales linearly with N. The breakdown time of
the coherent state approximation roughly relates to the
revival time as tb≈√N/50. Therefore, in order to have
7
0 50 100
0.
0.5
1.
0.
0.089
0.18
0 1000 2000 3000
0.
0.5
1.
0.
0.041
0.081
FIG. 1. Eigenfrequency ωnas a function of the oscillator
quantum number nfor two different values of the mean num-
ber of quanta and the Lamb-Dicke parameter: N= 85,
η= 0.170582 (left), and N= 2000, η= 0.0352653 (right).
Approximately linear behavior can be appreciated around N.
In orange, we present the probability amplitude pnof a num-
ber state in a coherent state |αi, with N=α2. In both cases,
vanishing small contributions of pnlie outside of the apparent
linear interval of ωn.
a faithful description, one has, in principle, to achieve
large mean phonon numbers. For instance, for an accu-
rate description up to an interaction time tr/2 one re-
quires values of N > 625. In the next section, however,
we will show that even with moderate values of N, the
model offers a reasonable description.
In order to present a clear comparison between the
models, in Table Iwe present a summary of the depen-
dence on Nof the different times for the three cases pre-
sented.
Model ΩtRΩtcΩtrΩtb
Tavis-Cummings 2π/√N2 2π√N√N
Buck-Sukumar π/N 1/√N π N 2/√2
ion-trap 11.2 1.6√N5N0.1N3/2
TABLE I. Relevant time scales for three different models in
terms of the mean number of photons N: Rabi oscillations
period, collapse time, revival time, and breakdown time of
the coherent state approximation.
IV. DYNAMICAL FEATURES
In this section we present the results and comparison of
numerical calculations of dynamical features of the three
specific models introduced in Sec. III. We focus on the
collapse and revival of Rabi oscillations and we test our
analytical result with numerically exact calculations.
A. Rabi oscillations and phase space representation
As mentioned in Sec. II C, the relevant time scales of
the system can be obtained from evaluating the expec-
tation value of observables in the system. As figure of
merit, in this work we have chosen to evaluate the mean
FIG. 2. Expectation value of the operator Sz, Eq. (3), for
two different values of the mean number of quanta: N= 85
(top plots), N= 2000 (bottom plots). In the left column, the
initial Rabi oscillations and its collapse is presented. In the
right column the first revival of Rabi oscillations is displayed
around a time tr. Black, blue, and red curves, respectively
correspond to the Tavis-Cummings model, the Buck-Sukumar
model, and the ion-trap model.
value of Sz, which can be analytically evaluated from our
approximate expression in (16) with the result given in
(23). In Fig. 2we have plotted the numerically exact
result of hSz(t)ifor the different models with an initial
atomic state |eeiand mean number of photons N= 85
and N= 2000. Black, blue and red curve correspond,
respectively, to the Tavis-Cummings, Buck-Sukumar and
ion-trap models. We present the Rabi oscillations close
to t= 0 (left column) and around t=tr(right column).
The first evident feature is that for all three models the
collapse of the Rabi oscillations occurs at the same frac-
tion of the revival time tr, i.e., the Gaussian envelope is
the same in terms of the adimensional time t/tr. This
is in complete agreement with the analytical approxi-
mation given in Eq. (23). For the reappearance of the
Rabi oscillations around tr, only the Tavis-Cummings
model presents a broadening of the oscillatory region.
The Buck-Sukumar model presents perfect revivals for
the two values of N. The ion-trap model presents no ap-
parent enhancement, however, for N= 85 the oscillations
display asymmetries. The revival seems to be perfect in
this model for N= 2000.
The collapse and revival of Rabi oscillations can be elu-
cidated by visualizing the state of the oscillator in phase
space with the aid of some quasiprobability distribution.
In this work we rely on the Husimi function that can
be regarded as the expectation value of the oscillator re-
duced density matrix ρos with respect to a coherent state
|βi, namely
Q(β) = hβ|ρos(t)|βi/π, (40)
ρos(t) = Trat {|Ψ(t)ihΨ(t)|}.
We have used the notation Trat for the partial trace with
respect to the atomic electronic degrees of freedom and
8
FIG. 3. Husimi function of the reduced density matrix for the
oscillator in an initial coherent state |αiand for interaction
times t=rr/4 (left column) and t=tr/2 (right column). The
results corresponds to the Tavis-Cumming model, the Buck-
Sukumar model, and the ion-trap model for the first, second,
and third row respectively.
we have considered βas a complex parameter. Recon-
struction of a Husimi Q-function has been experimentally
achieved on single 171Yb+ions in a harmonic potential
by using Raman laser beams [50].
In Fig. 3we have plotted the Husimi function Q(β) for
the three models described in Sec. III and for two differ-
ent interaction times: tr/4 and tr/2. We have used two
excited atoms as initial state and a coherent state for the
oscillator with α=√85. The initial state |αiremains
as a stationary component of the mode for all time as
evidenced in the plots. It can be noted that the time-
evolving field components of the Tavis-Cummings model
(top plots) suffer from a distortion already for a time tr/4
and this feature is more notorious at tr/2. In contrast, all
mode components in the Buck-Sukumar model (middle
plots) retain their shape. This an evidence of their evolu-
tion as coherent states. In the case of the ion-trap model,
the field components follow the same trajectory slightly
distorting their shape. This behavior corroborates a good
agreement with the coherent state approximation, even
with the moderate value N= 85.
FIG. 4. Average fidelity as a function of time of the ap-
proximated state vector in (16) with respect the numerically
exact state vector for two different values of the mean num-
ber of quanta: the solid (dashed) line correspond to N= 85
(N= 2000). The average has been preformed over 1000 ran-
dom initial conditions.
B. Fidelity of the approximate state vector
In the previous subsection we have briefly analyzed the
collapse and revival phenomenon. We have observed that
the approximations given in Sec. II seem plausible given
the fact that the revival of the oscillations and the mode
components in phase space do not broaden for the Buck-
Sukumar and the ion-trap models. Let us now turn our
attention to the numerical analysis of the validity of our
analytical calculation. In order to test the approximation
in Eq. (16), we consider the fidelity between the exact
state vector |Ψ(t)iand its approximation |Ψap(t)ias a
function of time that is given by
F(t) = |hΨap(t)|Ψ(t)i|2/N.(41)
The normalization Nof |Ψap(t)iis given in (20) and,
as mentioned before, it gives only as small contribution
close the revivals of oscillations. In Fig. 4we have plotted
the fidelity F(t) averaged over 1000 random initial condi-
tions uniformly distributed according to the correspond-
ing Haar measure. For the three cases we have chosen two
different values of the mean number of quanta: N= 85
presented in full line and N= 2000 in dashed line. For
the Tavis-Cummings model (black curves) the fidelity
drops well before the first revival. The Buck-Sukumar
model (blue curves) displays very good fidelity for the
complete time interval. This is expected as the coherent
state approximation is predicted to hold for longer time,
as in this case tb/tr∝N2. For the ion-trap model (red
curves), the fidelity is maintained above 0.9 for N= 85
and greatly improves for N= 2000, corroborating the
expected agreement given by the time scales in Table I.
In the context of quantum computation and quan-
tum information tasks with atomic qubits, the oscillator
might be considered as an auxiliary degree of freedom.
In this situation, the state of the mode does not play
an important role, and one is mainly concerned with the
9
FIG. 5. Fidelity of the approximate atomic reduced den-
sity matrix with respect to its numerically exact counterpart.
averaged over 1000 initial conditions. Two values of Nare
considered and indicated in the plot for the three different
models as indicated in the legend.
atomic state. Therefore, the most important state to test
is the reduced density matrix of the atoms whose fidelity
with respect to the exact reduced state can be evaluated
as
Fat(t) = Trqpρat(t)ρap
at (t)pρat(t)2
.(42)
The reduced atomic density matrices are taken from
the exact and approximated total state vector as
ρap
at (t) = Trosc|Ψap (t)ihΨap(t)|and similarly for ρat (t) =
Trosc|Ψ(t)ihΨ(t)|. In Fig. 5we plot the atomic fidelity
Fat(t) as function of time and averaged over 1000 random
initial conditions. Remarkably, the fidelity is extremely
good for all the models, including the Tavis-Cummings
model, for times where the field components separate,
i.e., for times different to trand tr/2. Around these
times, the Tavis-Cummings model fails to achieve a good
fidelity, however, the Buck-Sukumar model displays good
fidelity for any value of N. In the case of the ion-trap
model, the fidelity increases with N. This result also
corroborates that the coherent state approximation ac-
curately describes the Buck-Sukumar model and the ion-
trap model for large values of N. In the case of the Tavis-
Cummings model, although the coherent state approxi-
mation fails to describe the complete state, the atomic
state is well described for times that are not close to the
revival time and half the revival time. This happens be-
cause the field components follow the trajectory of the
coherent states in the approximation.
V. ENTANGLEMENT ANALYSIS
Entanglement is an important feature of the system in
the context of quantum information and quantum com-
putation, especially the atomic entanglement when the
atoms are regarded as qubits. This quantity has been
previously studied for the Tavis-Cummings model [47,51]
and some interesting properties have been introduced
Refs. [31,44]. However, the quantitative study has been
limited to specific initial conditions and numerical calcu-
lations. As the system is exactly solvable, one could, in
principle, calculate in closed form certain entanglement
measures for any bipartition of the system. However,
the resulting expressions will surely be complicated and
difficult to analyze. Here, we take advantage of our ap-
proximation in order to evaluate remarkable simple an-
alytical expressions for any initial condition at specific
times. In order to carry out this study, we evaluate the
reduced density matrix for the two-qubit system given
by ρat(t) = Trosc|Ψ(t)ihΨ(t)|. Furthermore, we concen-
trate our attention to specific interations times given by
jktr/4, namely at odd multiples kof a quarter (half) of
the revival time for j= 1 (j= 2). At these times the
density matrix assumes the following simple form
ρat jk
4tr=|ζihζ|+|ζj,kihζj,k |(43)
with kand odd positive integer. For j= 1 one has to
use the state in Eq. (25), and the state in Eq. (27)
for j= 2. The ket |ζiis the stationary atomic state
given in Eq. (16). Using ρat it is possible to evaluate the
entanglement between the atoms and the entanglement
between atoms and the oscillator.
A. Two-atom entanglement
We rely on the concurrence [52] as a measure of the en-
tanglement between the atoms. For a general two-qubit
sate ρ, it is defined as
C(ρ) = max(0, λ1−λ2−λ3−λ4),(44)
where the four λi’s are the square roots of the eigenvalues
of the positive non-Hermitian operator ρeρin decreasing
order. We have also introduced the Pauli operator σyand
eρ=σ⊗2
yρ∗σ⊗2
ywhere ρ∗is obtained from ρafter complex
conjugation in the computational basis. For a pure state,
the concurrence reduces to C(|ψi) = |hψ|e
ψi| with |e
ψi=
σy⊗σy|ψi∗and the complex conjugated vector |ψi∗in
the computational basis. By noting that the Bell states
fulfill the relations |e
Ψ±i=±|Ψ±i, and |e
Φ±i=∓|Φ±i, it
is not hard to realize that the concurrence of the initial
state in Eq. (10) is given by
C(|ψi) = c2
−−d2
−−c2
++d2
+.(45)
This result will serve as guidance for the concurrence of
the atomic mixed states after the interaction with the
oscillator.
Having introduced the entanglement measure and its
initial form, it is now appropriate to evaluate this quan-
tity for the mixed state of the atoms after the interac-
tion with the mode. At odd quarters of the revival time,
10
FIG. 6. Average concurrence as a function of time for the
three models as indicated in the legend for N= 2000. The
average was taken from 1000 random initial conditions. The
red dots indicate the average analytical value at fixed times
given in Eqs. (46) and (46). The legend indicates the curve
for each model, that present a considerable overlap.
the expression for the concurrence can be evaluated in
closed form using Eqs. (43) and (44). The calculations
are somehow tedious as hζ|e
ζ1,ki 6= 0, however, one can
find that the only two nonzero values of λiare given
by (q|c2
−−d2
−|2+ 2|dr|2± |c2
−−d2
−|)/2. Therefore, the
concurrence reduces to the simple expression
C(ρat(k
4tr)) = |hζ|e
ζi| =c2
−−d2
−.(46)
This results shows that the atomic entanglement at this
point has contribution only from the stationary part of
the state vector. This is in line with what is expected
from the basin of attraction [44] that no longer touches
a minimum for nonzero c−and d−.
At half revival time, the reduced density matrix of
the atoms is given by a rank two operator with its con-
stituents fulfilling the property hζ|ζ2,ki=hζ|e
ζ2,ki= 0.
This feature enables a simple calculation of the concur-
rence that is given by
C(ρat(k
2tr)) = |hζ|e
ζi| − |hζ2,k|e
ζ2,ki|(47)
=c2
−−d2
−−d2
+−c2
+≤C(|ψi),
where we have used c2
k,φ −d2
k,φ =d+
φ
2−c2
+. The last in-
equality follows from the reverse triangle inequality and
indicates that the entanglement at odd multiples of one
half of the revival time is always smaller than the initial
entanglement. The result can be interpreted as if there
was a sort of competition between the entanglement in
the two components leading to a maximum possible en-
tanglement if either c−=d−= 0 or if d+=c+= 0.
In order to test our analytical prediction, in Fig. 6we
have plotted the concurrence averaged over 1000 random
initial conditions. Our analytical prediction is indicated
with a red dot and displays an accurate prediction to the
numerical calculation. In all the chosen values of time,
we note that there is a critical point in the behavior of the
concurrence; in this case all of them are maximums. This
behavior can change, however, depending on the initial
probability amplitudes.
B. Atoms-oscillator entanglement
In order to measure the entanglement between both
atoms and the oscillator one can use the purity of any of
the two density matrices. As we have already evaluated
it for the atomic system in Eq. (43), we will use it to
evaluate the purity of the reduced atomic state as
P(t) = Tr ρ2
at(t).(48)
Unit value of the purity corresponds to a pure reduced
state and therefore no entanglement between atoms and
oscillator. The minimum value of the purity is 1/4 and
corresponds to a maximally mixed state of the atoms and
correspondingly maximum entanglement in the atoms-
oscillator bipartition.
Using Eq. (43) it is not difficult to calculate the atomic
purity. For odd multiples of one quarter of the revival
time the result is given by
Pktr
4=p2+ (1 −p)(1 − |c−|2).(49)
p=|c−|2+|d−|2.
Taking odd multiples of one half of the revival time, one
arrives to the following result
Pktr
2=p2+ (1 −p)2≤Pktr
4.(50)
Note that entanglement between the mode and the atoms
at odd multiples of tr/4 and tr/2 depends entirely on the
the initial probabilities of the states |Ψ−iand |Φ−i. If
non of these states are initially populated, the purity of
the atomic reduced density matrix is one and therefore no
entanglement is present in the atoms-oscillator partition
at this specific times. Also when p= 1, the purity takes
unit value at these two times. From the previous expres-
sions one can find that the minimum value of the purity
is 1/2, attainable for p2= 1/2. Therefore it is impossible
to maximally entangle the two atoms with the oscillator.
Nevertheless, the amount of achievable degree of entan-
glement is good enough to generate authentic tripartite
entangled states as will be shown in the next section.
VI. ENTANGLING OPERATIONS
In this section we introduce entangling operations that
can be implemented with the aforementioned system and
that can be exploited in quantum information protocols.
The results rely on the approximate solution of the state
vector in Eq. (16) for an interaction time tr/2 where the
time dependent part takes the simple form in Eq. (27).
As we will be concentrated only in this interaction time,
11
it is convenient to introduce the following shorthands to
be used in this section
U=e−iV tr/~, θ =δNtr/2.(51)
We anticipate that some the resulting two-qubit opera-
tions are not unitary, however, they can be of importance
in quantum information tasks. For instance, it has been
shown that one of them can replace the CNOT gate in
a recurrence entanglement purification protocol [27,28].
This does not impose a major loss in the protocol, as
recurrence purification protocols are probabilistic in na-
ture as the implementation of unitary gates is followed
by measurement in the computational basis where half of
the results have to be discarded. Here, we will introduce
a new entangling operation that can also assist in such a
protocol.
A. Two-qubit operations
Let us first consider a scheme to implement quantum
operations based on Bell state projectors. By inspecting
Eqs. (16) and (27) one can rewrite the approximate so-
lution to the state vector at time tr/2 in the following
convenient form
U|ψi|αi 'c−|Ψ−i+d−|Φ−i|αi
+ (c+|Ψθi+d+|Φθi)| − αi,(52)
with the orthogonal and maximally entangled states
|Ψθi= cos θ|Ψ+i+isin θ|Φ+i,
|Φθi=−isin θ|Ψ+i − cos θ|Φ+i.(53)
The coefficients c1and d1in Eq. (28) respectively repre-
sent the initial probability amplitudes of these two states.
We note that a measurement of the oscillator state |αi
or | − αipostselects the atoms in one of two orthogonal
states. This would correspond to two different two-qubit
operations. However, projecting on | − αipostselects the
atoms in a state that depends on the parameters of the
system and not merely on the initial atomic state. This
can be overcome by initially applying a quantum gate
that transforms the symmetric Bell states, while leaving
the antisymmetric ones invariant. For this purpose, we
introduce the following unitary gate
Gθ=|ΨθihΨ+|−|ΦθihΦ+|+|Ψ−ihΨ−|+|Φ−ihΦ−|.(54)
The minus sign in the second element is crucial, as in this
way the required quantum gate is separable and can be
expressed in terms of separable (single atom) gates as
Gθ=gθ⊗gθ, gθ= cos θ
2I+isin θ
2σx.(55)
Using this gate before the interaction, one can obtain the
state at half the revival time given by
U Gθ|ψi|αi 'c−|Ψ−i+d−|Φ−i|αi
+c+|Ψ+i − d+|Φ+i| − αi.(56)
With this result, measuring the state of the oscillator in
|αior | − αiwould respectively correspond to the follow-
ing quantum operations
M=|Ψ−ihΨ−|+|Φ−ihΦ−|(57)
L=|Ψ+ihΨ+|−|Φ+ihΦ+|.(58)
These Hermitian operators can be regarded as the mea-
surement operators of a positive operator valued measure
(POVM) [1]. The operators Mand Lfulfill M2+L2=I.
Furthermore, the sum of the two of them M+Lis a uni-
tary operator. The gate Mhas already been used in
place of the usual CNOT gate in purification protocols
[27]. The operation Lcan also be considered in a setting
of this type in order to improve the efficiency of these
purification protocols. A manuscript with these results
is also in preparation by two of the authors.
It is important to comment that for the measurement
of the photonic field a projection onto coherent states is
not strictly necessary. A projection onto position eigen-
states or weighted sum of position eigenstates close to
the coherent state would lead to the same atomic postse-
lection. This can be implemented using a balanced hom-
dyne measurement as explained in [31]. In ion traps, one
would require a measurement of the position of the ions
[11]. As for the single qubit gates gθ, these can, in prin-
ciple, be implemented by driving the atomic transition
with laser pulses and properly controlling their duration
[4,53]. In the case of trapped ions, the carrier resonance
has to be chosen in order to avoid excitation of the me-
chanical mode [11].
B. Three-qubit operations
As we have seen in Sec. V, the dynamics of this model
is able to generate simultaneous entanglement between
the two atoms and also between these two and the oscil-
lator. For this reason, it is natural to expect the possibil-
ity of tripartite entanglement in the system. It is known
that there are two inequivalent types of tripartite entan-
gled states of three qubits [54] that can be represented
by the following states
|GHZi= (|000i+|111i)/√2,(59)
|Wi= (|001i+|010i+|100i)/√3.(60)
These two state, W-state and the GHZ state, posses en-
tanglement among any bipartition of the three qubits.
We will show that it is possible to generate both of theses
paradigmatic tripartite entangled states from an initial
separable state and using as entangling gate the evolu-
tion operator Uin Eq. (51).
Let us first consider the generation of GHZ states, as
it follows directly from the solution of the state vector.
Examining Eq. (56), it is possible to note that in order to
generate a GHZ state, it suffices to initialize both atoms
in the ground state, where d−=d+= 1/√2 and c−=
12
c+= 0, and apply the separable atomic gate Gθfollowed
by the evolution operator U. With these conditions one
can obtain a GHZ state in the following way
U Gθ|ggi|αi=|ggi|α, −i +|eei|α, +i
√2,(61)
|α, ±i =−(| − αi±|αi)/√2.
We have introduced the symmetric and antisymmetric
Schr¨odinger cat states |α, ±i that can be considered as
the two states of a two level system. This is plausi-
ble provided that αis large enough in order to have
hα, +|α, −i ' 0.
The generation of a W-state from a separable state
is more involved, but also not difficult to achieve using
the unitary evolution. For this case, we will first intro-
duce an effective form of the evolution operator Uwhere
its action as a three-qubit gate is more evident. In this
case, we will use for the third qubit the nearly orthogo-
nal states | ± αi. As the solution of the state vector in
Eq. (52) at the specific time tr/2 involves only these two
coherent states, one can identify that the evolution op-
erator connects the coherent state |αiwith | ± αi. The
same is true for | − αithat is only connected to | ± αi,
as one can note by evaluating the action of the evolu-
tion operator on |ψi| − αiusing the interaction picture
as defined in Eq. (8). This means that Uis closed in
the subspace spanned by these two coherent states that
can be regarded as states of a third qubit. With this in
mind and by analyzing Eq. (52), it is possible to find the
following form of the evolution operator
U'M⊗Ia+K⊗|−αihα|+K∗⊗ |αih−α|,(62)
K=|ΨθihΨ+|+|ΦθihΦ+|.
This form is only valid for an interaction time tr/2 and
initial coherent states | ± αi. In the previous expression
we have used Iaas the identity operator in the oscillator
space. Although not evident at first glance, the operator
Kis Hermitian and fulfills the relation K2=L2. An
important feature of this effective evolution operator is
its evident three-qubit gate character that is suitable to
analyze three qubits states.
In order to generate a W-state, we choose the specific
value θ=δNtr/2 = π/4 of the angle in |Ψθiand |Φθi
given in Eqs. (51) and (53). This restricts the value of
the mean number Nand it is not possible to achieve in
every model. For instance, in the Buck-Sukumar model
δNand the revival time are constant. In the ion-trap
model, however, Ncan be chosen according to Eq. (38)
in order to achieve θ=πδN/|ω0
N|=π/4. Note that for
large enough value of the mean number of quanta N, its
value can slightly differ from the optimal one in Eq. (37),
as the coherent state amplitude are narrowly centered in
the linear region of ωn(see Fig. 1for N= 2000). With
this in mind, and using the following initial separable
condition
|ψ1i=|gi|gi+i|ei
√2|αi+√2| − αi
√3,(63)
it is not hard to realize that applying the evolution op-
erator results in
U|ψ1i=1
√3|ψ2i|αi+r2
3|Ψ+i| − αi,(64)
|ψ2i= (|Ψ−i+i|Φ−i − i√2|Φ+i)/2.
We have introduced the separable state |ψ2ithat is or-
thogonal to |Ψ+i. In order to bring this state to a more
obvious form of a Wstate, the next task is to find a
separable unitary gate that fulfills T|ψ2i=|eeiand
T|Ψ+i=|Ψ+i. The problem can be solved with the
aid of the following separable gate
T=γg†
π/4⊗γgπ/4, γ =i|eihe|+|gihg|.(65)
In this way, using the gate Tone can immediately find
that starting from a separable state |ψ1i, one can obtain
T U|ψ1i=|eei|αi+|gei| − αi+|egi| − αi
√3,(66)
which is a more evident form of a W-state as introduced
in Eq. (59).
We have shown the potential to generate authentic tri-
partite entangled states using the unitary evolution of the
model as entangling operation. Furthermore, it is also
relevant to note that for the generation of the W-state,
a state similar to a Schr¨odinger cat is needed. Although
this might be considered as a drawback, it is evident from
the generated GHZ state in Eq. (61) that this model also
offers a direct form of generating Schr¨odinger cat states
by measuring the atoms in the computational basis.
C. Bell measurement
Let us briefly sketch a procedure to implement a Bell
measurement by taking advantage of the interaction dy-
namics of the general model introduced above. It is not
hard to conceive such a protocol by inspecting the solu-
tion for the state vector in terms of atomic Bell states and
coherent state states of the oscillator. First let us come
back to the state in Eq. (56) after applying the gate Gon
the initial state followed by the atoms-oscillator interac-
tion U. This procedure can be considered as an atomic
state splitter, in the sense that one of the material com-
ponents remains invariant accompanying the oscillator
state |αi, while another material component follows an-
other mode state | − αithat is orthogonal to the first one.
This interesting feature can be further applied in order to
separate again the two components into four components
in terms of Bell states. In order to do so, one requires
to interchange the Bell states |Φ±iusing a rotation as in
Eq. (26), followed by an interaction with an additional
oscillator with operators band b†described by Vbas in
Eq. (2). With all these considerations, one can come up
with the unitary gate
U=e−iSz
π
2G†
θUbeiSz
π
2UaGθ,(67)
13
FIG. 7. Quantum circuit representation of the Bell measure-
ment protocol with coherent states | ± αia,b used as auxiliary
qubits and with eiSzπ/2=iZ ⊗iZ. Here we use the notation
Z=|eihe|−|gihg|as the Pauli-Zquantum gate commonly
used in quantum computing. At the final stage of the cir-
cuit, modes aand bare measured leading to four possible
outcomes that postselect the atoms in one of the four Bell
states as indicated by the state in Eq. (68).
where we have distinguished between different modes us-
ing their annihilation operator as subscript. Applying
this unitary gate to an initial arbitrary atomic state with
two coherent states, one obtains the following state vec-
tor with four components
U|ψi|αia|αib'd−|Φ−i|αia| − αib+c−|Ψ−i|αia|αib
+d+|Ψ+i| − αia|αib+c+|Φ+i| − αia| − αib.(68)
In this final state, a different combination of coherent
states is accompanied by a specific Bell state multiplied
by its initial probability amplitude. Therefore, by dis-
criminating the four coherent states in the two oscilla-
tors, one is able to postselect the atomic state in one
of the four Bell states. For instance, measuring oscilla-
tor states close to |αi| − αicorresponds to a projection
onto |Φ−i, as this would happen with probability |d−|2.
Analogue procedures apply to all four Bell states. The
process can be visualized using the useful circuit repre-
sentation shown in Fig. 7. It is worth commenting, that
the discrimination of oscillator states need not be a pro-
jection onto coherent states. It suffices a measurement
of the oscillator in a localized state close to a specific
coherent state.
VII. CONCLUSIONS
We have presented a theoretical analytical study
of a nonlinear intensity dependent two-atom Tavis-
Cummings model. The exact solvability of the model has
been shown by identifying two constant of motion. By in-
troducing a convenient interaction picture, we have been
able to solve the time dependent problem for initial arbi-
trary coherent states, using only coherent states that lie
in the positive axis in phase space. By considering large
mean number of quanta in the oscillator, we have derived
an analytical approximate expression given in terms of
atomic Bell states and oscillator coherent states that has
been numerically tested using its fidelity with respect to
the exact expression. As particular cases of this model,
we have revised in detail three particular models: the
Tavis-Cummings model, the Buck-Sukumar model, and
the nonlinear ion-trap model. We have shown that in
the experimentally feasible ion-trap model, the coherent
state approximation can accurately describe the dynam-
ics, when carefully choosing the mean number of quanta
for a given Lamb-Dicke parameter. The approximate so-
lution of the time-dependent state vector has proven to
be very useful in analyzing the dynamical features, and
more specifically, the entanglement in the system. The
most important result is that, with the approximate form
of the state vector, we have been able to introduce en-
tangling operations for two-qubtis and three qubits in a
compact form. The results in this work show that the
physical implementation of entangling operations relying
on nonlinear Tavis-Cummings models can be realized in
current ion-trap experiments, opening new avenues for
the implementation of basic quantum protocols assisted
by multi-phonon states.
ACKNOWLEDGMENTS
R. G.-R. is grateful to CONACYT for financial sup-
port under a Doctoral Fellowship. C. A. G.-G. acknowl-
edges funding from the Spanish MICINN through the
project MAT2017-88358-C3-1-R, and the program Ac-
ciones de Dinamizaci´on “Europa Excelencia” EUR2019-
103823. The authors would also like to thank Ralf Bet-
zholz for his useful comments on this manuscript.
Appendix A: Exact form of the evolution operator
For the sake of completeness, in this appendix we
briefly present the exact solution of the evolution opera-
tor U(t) = e−iV t/~. The exact expression of each block
of U(t) for n > 1 is given by
U(n)(t) =
Ω2
n−1+Ω2
nC(t)
ν2
n
Ω2
nS(t)
iνn
Ωn−1Ωn(C(t)−1)
ν2
n
Ω2
nS(t)
iνnC(t)Ω2
n−1S(t)
iνn
Ωn−1Ωn(C(t)−1)
ν2
n
Ω2
n−1S(t)
iνn
Ω2
n+Ω2
n−1C(t)
ν2
n
with the shorthands C(t) = cos νntand S(t) = sin νnt,
and the exact form of the eigenfrequencies νn=
qΩ2
n+ Ω2
n−1. For n= 0 the blocks of the interaction
operator Vand of the evolution operator U(t) are two
dimensional and they are given by
V(0) = 0~Ω0
~Ω00!, U(0)(t) = cos Ω0t−isin Ω0t
−isin Ω0tcos Ω0t!,
with the basis states |ggi|1iand |Ψ+i|0i. For n=−1,
we have only one state, |ggi|0i, and therefore the blocks
14
are one-dimensional with V(−1)=0 and U(−1) =1. Us-
ing these blocks, the time-dependent amplitudes in Eq.
(12) can be evaluated as Cn(t) = U(n)Cn(0) with the
column vector Cn(t) containing the coefficients Cn,l(t),
l∈ {−1,0,1}. The approximate expressions in Eq. (14)
can also be obtained directly from the exact expressions
using the approximations explained in Sec. II B.
[1] M. A. Nielsen and I. L. Chuang. Quantum computa-
tion and quantum information, (Cambridge University
Press2000).
[2] J. I. Cirac, W. D¨ur, B. Kraus, and M. Lewenstein. Phys.
Rev. Lett., 86, 544 (2001).
[3] S.-B. Zheng and G.-C. Guo. Phys. Rev. Lett., 85, 2392
(2000).
[4] J. M. Raimond, M. Brune, and S. Haroche. Rev. Mod.
Phys., 73, 565 (2001).
[5] S. Osnaghi, P. Bertet, A. Auffeves, P. Maioli, M. Brune,
J. M. Raimond, and S. Haroche. Phys. Rev. Lett., 87,
037902 (2001).
[6] S. Ritter, C. N¨olleke, C. Hahn, A. Reiserer, A. Neuzner,
M. Uphoff, M. M¨ucke, E. Figueroa, J. Bochmann, and
G. Rempe. Nature, 484, 195 (2012).
[7] C. N¨olleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe,
and S. Ritter. Phys. Rev. Lett., 110, 140403 (2013).
[8] B. Casabone, A. Stute, K. Friebe, B. Brandst¨atter,
K. Sch¨uppert, R. Blatt, and T. E. Northup. Phys. Rev.
Lett., 111, 100505 (2013).
[9] A. Reiserer and G. Rempe. Rev. Mod. Phys., 87, 1379
(2015).
[10] H. Walther, B. T. H. Varcoe, B.-G. Englert, and
T. Becker. Rep. Prog. Phys., 69, 1325 (2006).
[11] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland. Rev.
Mod. Phys., 75, 281 (2003).
[12] J. S. Pedernales, I. Lizuain, S. Felicetti, G. Romero,
L. Lamata, and E. Solano. Sci. Rep., 5, 1 (2015).
[13] A. Blais, A. L. Grimsmo, S. M. Girvin, and A. Wallraff.
Rev. Mod. Phys., 93, 025005 (2021).
[14] E. T. Jaynes and F. W. Cummings. Proc. IEEE, 51, 89
(1963).
[15] J. I. Cirac and P. Zoller. Phys. Rev. Lett., 74, 4091
(1995).
[16] F. Schmidt-Kaler, H. H¨affner, M. Riebe, S. Gulde, G. P.
Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Es-
chner, and R. Blatt. Nature, 422, 408 (2003).
[17] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R.
Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A.
Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M.
Girvin, and R. J. Schoelkopf. Nature, 449, 443 (2007).
[18] J. M. Fink, R. Bianchetti, M. Baur, M. G¨oppl, L. Steffen,
S. Filipp, P. J. Leek, A. Blais, and A. Wallraff. Phys. Rev.
Lett., 103, 083601 (2009).
[19] N. M. Linke, D. Maslov, M. Roetteler, S. Debnath,
C. Figgatt, K. A. Landsman, K. Wright, and C. Mon-
roe. Proc. Natl. Acad. Sci. U.S.A., 114, 3305 (2017).
[20] K. Mølmer and A. Sørensen. Phys. Rev. Lett., 82, 1835
(1999).
[21] H. H¨affner, C. F. Roos, and R. Blatt. Phys. Rep., 469,
155 (2008).
[22] C. F. Roos. New J. Phys., 10, 013002 (2008).
[23] P. van Loock, T. D. Ladd, K. Sanaka, F. Yamaguchi,
K. Nemoto, W. J. Munro, and Y. Yamamoto. Phys. Rev.
Lett., 96, 240501 (2006).
[24] T. D. Ladd, P. van Loock, K. Nemoto, W. J. Munro, and
Y. Yamamoto. New J. Phys., 8, 184 (2006).
[25] J. Z. Bern´ad and G. Alber. Phys. Rev. A, 87, 012311
(2013).
[26] J. Z. Bern´ad. Phys. Rev. A, 96, 052329 (2017).
[27] J. Z. Bern´ad, J. M. Torres, L. Kunz, and G. Alber. Phys.
Rev. A, 93, 032317 (2016).
[28] J. M. Torres and J. Z. Bern´ad. Phys. Rev. A, 94, 052329
(2016).
[29] C. A. Gonz´alez-Guti´errez and J. M. Torres. Phys. Rev.
A, 99, 023854 (2018).
[30] M. Tavis and F. W. Cummings. Phys. Rev., 170, 379
(1968).
[31] J. M. Torres, J. Z. Bern´ad, and G. Alber. Phys. Rev. A,
90, 012304 (2014).
[32] J. M. Torres, J. Z. Bern´ad, and G. Alber. Appl. Phys.
B, 122, 117 (2016).
[33] K. G. Johnson, J. D. Wong-Campos, B. Neyenhuis,
J. Mizrahi, and C. Monroe. Nat. Commun., 8, 1 (2017).
[34] M. J. McDonnell, J. P. Home, D. M. Lucas, G. Imreh,
B. C. Keitch, D. J. Szwer, N. R. Thomas, S. C. Webster,
D. N. Stacey, and A. M. Steane. Phys. Rev. Lett., 98,
063603 (2007).
[35] J. Alonso, F. M. Leupold, Z. U. Sol`er, M. Fadel,
M. Marinelli, B. C. Keitch, V. Negnevitsky, and J. P.
Home. Nat. Commun., 7, 11243 (2016).
[36] B. Buck and C. V. Sukumar. Phys. Lett. A, 81, 132
(1981).
[37] E. A. Kochetov. J. Phys. A, 20, 2433 (1987).
[38] M. Chaichian, D. Ellinas, and P. Kulish. Phys. Rev.
Lett., 65, 980 (1990).
[39] D. Bonatsos, C. Daskaloyannis, and G. A. Lalazissis.
Phys. Rev. A, 47, 3448 (1993).
[40] A. Rybin, G. Miroshnichenko, I. Vadeiko, and J. Timo-
nen. J. Phys. A, 32, 8739 (1999).
[41] O. de los Santos-S´anchez, C. Gonz´alez-Guti´errez, and
J. R´ecamier. J. Phys. B, 49, 165503 (2016).
[42] W. Vogel and R. L. de Matos Filho. Phys. Rev. A, 52,
4214 (1995).
[43] F. H. Maldonado-Villamizar, C. A. Gonz´alez-Guti´errez,
L. Villanueva-Vergara, and B. M. Rodr´ıguez-Lara. arXiv
preprint arXiv:2010.13867 (2020).
[44] C. E. Jarvis, D. A. Rodrigues, B. L. Gy¨orffy, T. P. Spiller,
A. J. Short, and J. F. Annett. New J. Phys., 11, 103047
(2009).
[45] J. Gea-Banacloche. Phys. Rev. A, 44, 5913 (1991).
[46] J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-
Mondragon. Phys. Rev. Lett., 44, 1323 (1980).
[47] T. E. Tessier, I. H. Deutsch, A. Delgado, and I. Fuentes-
Guridi. Phys. Rev. A, 68, 062316 (2003).
[48] S. G. Orthogonal Polynomials, (American Mathematical
Society1975), 4 edition.
[49] B. Muckenhoupt. Proc. Am. Math. Soc., 24, 288 (1970).
[50] D. Lv, S. An, M. Um, J. Zhang, J.-N. Zhang, M. S. Kim,
and K. Kim. Phys. Rev. A, 95, 043813 (2017).
15
[51] C. A. Gonz´alez-Guti´errez, R. Rom´an-Ancheyta, D. Espi-
tia, and R. Lo Franco. Int. J. Quantum Inf., 14, 1650031
(2016).
[52] W. K. Wootters. Phys. Rev. Lett., 80, 2245 (1998). En-
tanglement.
[53] D. Meschede and A. Rauschenbeutel. Adv. At. Mol. Opt.
Phys., 53, 75 (2006).
[54] W. D¨ur, G. Vidal, and J. I. Cirac. Phys. Rev. A, 62,
062314 (2000).
