Download Two Qubits Tavis-Cummings Model Beyond the Rotating Wave

Two Qubits Tavis-Cummings Model Beyond the Rotating Wave Approximation:
Degenerate Regime
S. Agarwal, S.M. Hashemi Rafsanjani and J.H. Eberly
arXiv:1106.0052v1 [quant-ph] 31 May 2011
Rochester Theory Center and the Department of Physics & Astronomy
University of Rochester, Rochester, New York 14627∗
(Dated: June 2, 2011)
We study the dynamics of two qubits interacting with a single mode of a harmonic oscillator
beyond the rotating wave approximation in the ideally degenerate regime. Exact analytic expressions
are obtained for state properties of interest, including qubit entanglement for a certain class of initial
states of the oscillator and the qubits. Qualitative differences and similarities in the evolution of the
qubits in the degenerate regime when the oscillator is treated quantum mechanically and classically
are discussed.
PACS numbers: ..........
I.
INTRODUCTION
Two level systems that interact with a harmonic oscillator can model many physical phenomena, such as
atoms interacting with an electromagnetic field [1, 2],
electrons coupled to a phonon mode of a crystal lattice
[3], or super-conducting qubits interacting with a nanomechanical resonator [4, 5], a transmission line resonator
[6, 7], or an LC circuit [8, 9]. In the area of quantum optics, the model describing the interaction of a single qubit
with a harmonic oscillator, called the Jaynes-Cummings
model (JCM) [1] is extensively studied in the literature
[10]. A generalization of the JCM to include many qubits
not interacting with each other, but interacting with a
single mode of a quantized harmonic oscillator, called the
Tavis-Cummings model (TCM) [11, 12], is also of great
interest as it provides a framework to study collective
properties of the qubits.
In physical situations where the qubits are nearly resonant with the oscillator and the coupling between the
qubits and the oscillator is weak, it is a good approximation to drop certain terms, called the counter-rotating
terms, from the Hamiltonian describing the evolution of
the qubits and the oscillator. Under this approximation,
called the rotating wave approximation (RWA), the dynamics of the system can be solved analytically [1].
In optical setups, the near resonance and the weak coupling conditions are usually satisfied under appropriate
conditions, justifying the use of the RWA to describe the
dynamics of the system [2]. With the recent advancements in the area of circuit QED, it is now possible to
engineer systems for which the near resonance and/or
the weak coupling conditions are not satisfied [13–15].
To understand the dynamics under such conditions, it is
necessary to study the evolution of the system beyond the
RWA. Analytical approximations and numerical methods
have been used to understand both qualitative and quantitative features of the dynamics beyond the RWA for the
∗
[email protected]
Jaynes-Cummings [16–29] and the Tavis-Cummings [30–
33] model.
In this report, we study quantum collective dynamics, in particular the entanglement properties of the two
qubits TCM, beyond the RWA, in the regime when both
the qubits are far from resonance with the oscillator.
We extend recent treatments [34–37] of the role of the
counter-rotating terms in the dynamics of entanglement
between the qubits.
In particular, we examine the idealized case of zerogap qubits, focusing on the entanglement dynamics of
the four Bell states with the oscillator initially in either
a thermal state, a coherent state or a number state. A
study of the dynamics of a degenerate qubit interacting
with a classical field, and a discussion of a physical system
which can be treated as a degenerate qubit, has been
given by Shakov and McGuire [38].
We find the eigenvalues and eigenfunctions of the
Hamiltonian of a two-qubit TCM in the degenerate
regime in section II. In section III, using the GlauberSudarshan P representation [39, 40] to describe the initial
state of the oscillator, we find the evolution of any initial
product state of the qubits and the oscillator. The entanglement properties between the qubits for a certain class
of initial states are studied in section IV. In section V, it
is shown that the dynamics of the qubits is qualitatively
different for the quantum mechanical and the classical
descriptions of the oscillator and we conclude in section
VI.
II.
THE MODEL
The Hamiltonian describing the dynamics of two
qubits interacting with a single harmonic oscillator is
given by [11]:
ω0 (1)
(σ̂ + σ̂z(2) ) + ~ω↠â + ~λ(â + ↠)(σ̂x(1) + σ̂x(2) ).
2 z
(1)
The two qubits are assumed to be similar but fundamentally distinguishable. The transition frequency of each
qubit (assumed to be the same for simplicity) is ω0 and
Ĥ = ~
2
can be written as:
 1

0√
0√
|1, 1i
0
1/
2
1/
 |1, 0i  
√2
√
 |0, 0i  = 
0 1/ 2 −1/ 2
|1, −1i
0
0
0
FIG. 1. Energy level diagrams for: (a) ω0 ≈ ω and (b) ω0 ≪
ω.
the oscillator frequency is ω. The coupling constant between oscillator and both the qubits (again assumed to be
the same for mathematical convenience) is λ. The raising and lowering operators for the harmonic oscillator are
(i)
(i)
conventional. The qubit operators σ̂z and σ̂x are the
usual Pauli matrices in the Hilbert space of the ith qubit.
The interaction part of the Hamiltonian that couples the
(1)
(2)
qubits to the oscillator is ĤI = ~λ(â + ↠)(σ̂x + σ̂x ).
While discussing the RWA, it is suggestive to write the
interaction Hamiltonian as:
(1)
(1)
(2)
(2)
ĤI = ~λ(â+↠)(σ̂+ + σ̂− )+~λ(â+↠)(σ̂+ + σ̂− ). (2)
If λ ≪ ω, ω0 and ω0 ≈ ω, one can neglect the counter ro(i)
(i)
tating terms: ↠σ̂+ and âσ̂− . This is called the rotating
wave approximation [1, 2]. The condition ω0 ≈ ω, when
the RWA is valid is shown in Fig. 1(a). The zero-gap
degenerate regime that we are interested in takes ω0 = 0,
an idealization of the case when ω0 ≪ ω and ω0 ≪ λ.
The condition ω0 ≪ ω is shown in Fig. 1(b).
When ω0 = 0, the non-RWA Hamiltonian takes the
form
Ĥ0 = ~ω↠â + ~λ(â + ↠)(σ̂x(1) + σ̂x(2) ).
(3)
In order to study the dynamics of the system in the degenerate regime, we first find the eigenstates, |Φi, and
eigenvalues, E, of Ĥ0 :
h
i
~ω↠â + ~λ(â + ↠)(σ̂x(1) + σ̂x(2) ) |Φi = E|Φi.
(4)
Eigenstates |Φi will be of the form |j, mi|φm i where |j, mi
(1)
(2)
are the eigenstates of (σ̂x + σ̂x ) and |φm i are the oscil(1)
(2)
lator eigenstates found from Ĥ0 by replacing (σ̂x + σ̂x )
by the eigenvalue corresponding to |j, mi [24].
(1)
(2)
The four eigenstates of (σ̂x + σ̂x ) are:
|j, mi = |1, ±1i, |1, 0i and |0, 0i,
(5)
with eigenvalues 2m. In terms of the simultaneous eigen(1)
(2)
(i)
states of σ̂x and σ̂x , σ̂x |±i = ±|±i, the states |j, mi

0 |+, +i
|+, −i
0

.

0 |−, +i
|−, −i
1
(6)
Having found |j, mi, let us now find |φm i that satisfy
the eigenvalue equation:
~ω↠â + ~2mλ(â + ↠) |φm i = E|φm i.
(7)
We denote λ/ω by β and mλ/ω by βm . Taking βm to be
real, and completing the square in equation (7), we get:
E
2
(↠+ 2βm )(â + 2βm )|φm i =
|φm i. (8)
+ 4βm
~ω
Using the displacement operator, D̂(x) = exp[x(↠− â)]
(for real x), we can write the expression on the left hand
of (8) as:
(↠+ 2βm )(â + 2βm )|φm i = D̂(−2βm )↠âD̂(2βm )|φm i.
(9)
†
As the operator D̂(−2βm )â âD̂(2βm ) is the number operator for a displaced harmonic oscillator with eigenstates
D̂(−2βm )|N i, we get from equations (8) and (9) that
|φm i = D̂(−2βm )|N i ≡ |Nm i.
(10)
Here |N i is the number state of the undisplaced oscillator, i.e. ↠â|N i = N |N i.
Corresponding to the state |Nm i, the energy, E, in
equation (7) takes value:
2
).
EN,m = ~ω(N − 4βm
(11)
In the discussion of two level systems interacting with a
harmonic oscillator beyond the RWA, the use of a displaced harmonic oscillator basis was first used by Crisp
[20]. The displaced oscillator states have the properties:
′
hNm |Nm
i = δN,N ′ ,
hNm |Mm′ i 6= 0.
(12)
The non-orthogonality condition, hNm |Mm′ i 6= 0, plays
an important role when one discusses the spectrum of the
system for ω0 6= 0.
III.
EVOLUTION OF ANY INITIAL PRODUCT
STATE OF OSCILLATOR AND QUBITS
Let the initial state of the system be given by the density matrix ρ̂(0) = Q̂ ⊗ F̂ , where Q̂ is the joint density
matrix of both the qubits and F̂ is the density matrix
of the oscillator. Writing Q̂ in the basis |j, mi, and the
oscillator density matrix in the diagonal coherent state
3
basis (Glauber-Sudarshan P representation), ρ̂(0) takes
the form:
Z
X
′
′
′
′
ρ̂(0) =
Qjm;j m |j, mihj , m | ⊗ d2 αP (α)|αihα|,
(13)
where Qjm;j ′ m′ = hj, m|Q̂|j ′ , m′ i. Evolving ρ̂(0) in time,
we get:
ρ̂(t) = e−iĤ0 t/~ ρ̂(0)eiĤ0 t/~ ,
XZ
=
d2 αP (α)Qjm;j ′ m′
×e−iĤ0 t/~ |j, mi|αihα|hj ′ , m′ |eiĤ0 t/~ .
It must be noted that for some oscillator states, the
function P (α) diverges, e.g. squeezed state. In such
cases, the integral in equation (19) can be ill-defined.
To deal with such states, instead of using the P (α) representation, one might find it convenient to write F̂ as:
Z
1
F̂ = 2
d2 αd2 α′ |αihα|F̂ |α′ ihα′ |
(20)
π
and then follow the evolution of the composite system.
(14)
IV.
DYNAMICS OF ENTANGLEMENT
BETWEEN THE QUBITS
From Appendix A, we get
e−iĤ0 t/~ |j, mi|αi = |j, mi|(α + 2βm )e−iωt − 2βm i
× exp (−2iβm f (ωt, α))
2
× exp 4iβm
(ωt − sin ωt) ,
(15)
where
f (ωt, α) = sin
ωt ∗ iωt/2
(α e
+ αe−iωt/2 ).
2
(16)
Using equation (15), ρ̂(t) takes the form:
XZ
ρ̂(t) =
d2 αP (α)Qjm;j ′ m′
× |j, mihj ′ , m′ |
⊗ |(α + 2βm )e−iωt − 2βm ih(α + 2βm′ )e−iωt − 2βm′ |
× exp (−2i(βm − βm′ )f (ωt, α))
2
2
× exp 4i(βm
− βm
(17)
′ )(ωt − sin ωt) .
The dynamics of the qubits can be followed by tracing
the oscillator degrees of freedom from equation (17) to
get the reduced density matrix for the qubits:
ρ̂q (t) = Tro {ρ̂(t)},
XZ
=
d2 αP (α)Qjm;j ′ m′
× |j, mihj ′ , m′ |
× h(α + 2βm′ )e−iωt − 2βm′ |(α + 2βm )e−iωt − 2βm i
× exp (−2i(βm − βm′ )f (ωt, α))
2
2
− βm
× exp 4i(βm
(18)
′ )(ωt − sin ωt) .
Using the relation of scalar product between two coherent
2
∗
∗
states, hν2 |ν1 i = e−|ν1 −ν2 | /2 e(ν2 ν1 −ν2 ν1 )/2 , we get:
XZ
ρ̂q (t) =
d2 αP (α)Qjm;j ′ m′
× |j, mihj ′ , m′ |
× exp −8(βm − βm′ )2 sin2 ωt/2
× exp (−4i(βm − βm′ )f (ωt, α))
2
2
× exp 4i(βm
− βm
′ )(ωt − sin ωt) .
(19)
The reduced density matrix for the qubits, equation (19),
will be used in the next section for studying the dynamics
of entanglement between the two non-interacting qubits.
Using a Hamiltonian similar to (3), it was shown in [41]
that two non-interacting, initially unentangled qubits can
get entangled due to their interaction with a common
bath (harmonic oscillator) initially in the ground state.
In this section, we study the dynamics of entanglement
between the two qubits for various different initial states
of the oscillator and the qubits. For all the initial states
that we study, the qubits and the oscillator are chosen
to be initially separable, and are taken to be of the form
of equation (13). Of all the possible initial states of the
form (13), we choose to analyze the states for which the
qubits are initially maximally entangled and are in one
of the four possible Bell states:
1
|Ψ± i = √ (|e, gi ± |g, ei),
2
1
|Φ± i = √ (|e, ei ± |g, gi),
2
(i)
(21)
(i)
where σ̂z |ei = |ei and σ̂z |gi = −|gi.
From equation (6), we know that for any initial state of
the oscillator, the states |Ψ− i and |Φ− i are eigenstates of
Ĥ0 . So, the reduced density matrices for the qubits will
not change if they start in the state |Ψ− i or |Φ− i and the
qubits will remain maximally entangled no matter what
the initial state of the oscillatoris.
(1)
The local unitary operator, σ̂x ⊗ 1̂(2) , commutes
with the Hamiltonian and connects the states |Ψ+ i and
|Φ+ i:
|Ψ+ i = σ̂x(1) ⊗ 1̂(2) |Φ+ i,
i
h
(22)
σ̂x(1) ⊗ 1̂(2) , Ĥ0 = 0.
Since any measure of entanglement calculated for two
different states that are connected by a local unitary
transformation has the same value [42], the entanglement
dynamics between the qubits for the initial state |Ψ+ i
will be the same as entanglement dynamics between the
qubits for the initial state |Φ+ i. One should note that
this result is valid for any initial state of the field. Because of this, we explicitly study only the states for which
the initial state of the qubits is |Ψ+ i.
4
When the qubits start in the state |Ψ+ i, it turns out
that the expression for concurrence [43], a measure that
quantifies entanglement between two qubits, has an analytic general form that is valid for all the initial states
of the oscillator. We first derive that general expression
and then study explicitly the cases for which the initial
state of the oscillator is either a thermal state, a coherent
state or a number state.
For the state |Ψ+ i, the initial density matrix for the
qubits, Q̂, takes the form:


1 0 0 −1
10 0 0 0 
Q̂ = 
,
(23)
0 0 0 0
2
−1 0 0 1
where the rows and columns are defined in the basis:
|1, 1i, |1, 0i, |0, 0i and |1, −1i. We denote the initial
state of the oscillator by F̂ , which is characterized by
the Glauber-Sudarshan function P. As time evolves, by
using equation (19), we calculate the reduced density matrix for the qubits in the |j, mi basis and get:


1
0 0 −I(t)
1 0
0 0
0 
,
(24)
ρ̂q (t) = 
0
0 0
0 
2
∗
−I(t) 0 0
1
where
Z
d2 αP (α) exp (−8iβf (ωt, α))
× exp −32β 2 sin2 (ωt/2) .
I(t) =
(25)
In order to calculate the concurrence in the usual form
between the two qubits, we need to write ρ̂q (t) in the
|e, ei, |e, gi, |g, ei and |g, gi basis:


1 − u(t) iv(t)
iv(t) 1 − u(t)
1  −iv(t) 1 + u(t) 1 + u(t) −iv(t) 
ρ̂q (t) = 
.
4 −iv(t) 1 + u(t) 1 + u(t) −iv(t)
1 − u(t) iv(t)
iv(t) 1 − u(t)
(26)
The terms u(t) and v(t) are the real and imaginary parts
of I(t) respectively. For this density matrix, the concurrence is given by:
C(t) = |I(t)| ,
Z
2
= d αP (α) exp (−8iβf (ωt, α))
× exp −32β 2 sin2 (ωt/2) ,
= P̃ (8β sin(ωt), 8β(cos(ωt) − 1))
× exp −32β 2 sin2 (ωt/2) .
(27)
The function P̃ , defined in equation (27), is the Fourier
transform of P:
Z
P̃ (k1 , k2 ) = dxdyP (x, y) exp (−i(k1 x + k2 y)), (28)
with x and y being respectively the real and imaginary
parts of α in the integral (25). From equation (27), we
see that knowing P (α) or its Fourier transform, P̃ , that
characterizes the initial state of the oscillator, we can
follow the evolution of entanglement between the qubits.
Qubits initially in state |Ψ+ i and oscillator in
the thermal state: The dynamics of a system interacting with a thermal reservoir is generally associated with
decoherence of the system. However, following the work
of Bose et al. [44], it was shown in [45] that entanglement (which is a manifestation of coherent superposition
of states of a multi-partite system), can be created between the two qubits of the Tavis-Cummings model with
the oscillator initially prepared in the thermal state. Environment induced entanglement is an area of active research [41, 46].
Entanglement dynamics of the two-qubits TavisCummings model with the oscillator initially in a thermal state is investigated in [47–50]. In particular, Roa et
al. [50], studied the entanglement dynamics between the
qubits in the dispersive coupling regime within the rotating wave approximation, where the qubits were taken to
be far off resonance from the oscillator and the coupling
between the qubits and the oscillator taken to be weak.
To study the entanglement dynamics between the
qubits in the degenerate regime, beyond the RWA and
for arbitrary coupling strengths, for the initial state of
the qubits being |Ψ+ i and that of the oscillator being
a thermal state using equation (27), we note that for a
thermal state P (α) is given by [51]:
Pth (α) =
1
exp (−|α|2 / hni).
π hni
(29)
In the above equation, hni is the average number of excitations for a given temperature which characterizes the
thermal
pstate. We note that as P (α) is a gaussian of
width hni,
pits Fourier transform, P̃ , is also a gaussian
of width 1/ hni.
Using Pth (α) in equation (27), we get:
Cth (t) = exp −32β 2 sin2 (ωt/2)(1 + 2 hni) .
(30)
In Fig. 2, Cth (t) is plotted for various values of hni and
β = 0.05 and β = 0.1. We notice that entanglement
between the qubits oscillates periodically with time period 2π/ω. Within each time period, the entanglement
between the qubits decreases more rapidly from its maximum value as one increases hni or β. The minimum en
tanglement between the qubits, exp −32β 2 (1 + 2 hni) ,
reduces as hni or β is increased. These are consequences
of P̃ becoming narrower as hni or β increases.
Qubits initially in state |Ψ+ i and oscillator in a
coherent state: The dynamics of a single qubit interacting with a coherent state of the oscillator has been studied extensively in the literature and has revealed many
important phenomena that are unique features of quantum mechanics such as collapse and revival of atomic
5
<n>=1
<n>=3
<n>=10
Cth(t)
0.8
0.6
β=0.1
β=0.2
β=0.5
0.5
0.4
0.2
0
0
0
0
0.5
1
1.5
ωt/2π
2
2.5
<n>=1
<n>=3
<n>=10
0.5
1
1.5
ωt/2π
1
1.5
ω t/2π
2
2.5
3
3
0.5
0
0
0.5
FIG. 3. Time dependence of entanglement between the two
qubits in the degenerate regime given by Cco (t) for various
value of β.
1
Cth(t)
1
Cco(t)
1
2
2.5
3
FIG. 2. Time dependence of entanglement between the two
qubits in the degenerate regime given by Cth (t) for various
value of hni. Top: β = 0.05 and bottom: β = 0.1.
coherence [52], entanglement and disentanglement of the
qubit with the oscillator and in process preparation of
the Schrödinger Cat state [53], etc. Within the RWA,
the collapse and revival of a two-qubit TCM was studied
in [54], and the entanglement dynamics for various different initial states of the oscillator, including coherent
state, was done in [55].
For the oscillator initially in the coherent state |α0 i,
P (α) is given by [51]:
Pcoh (α) = δ 2 (α − α0 ).
(31)
With Pcoh (α), I(t) in equation (25) is evaluated to be:
Icoh (t) = exp (−8iβf (ωt, α0 ))
× exp −32β 2 sin2 (ωt/2) .
(32)
From equations (27) and (32), and the fact that f (ωt, α0 )
is real, the concurrence for the coherent state case is
found to be:
Ccoh (t) = exp −32β 2 sin2 (ωt/2) .
(33)
We see from equation (33) that the entanglement between
the qubits does not depend on the oscillator strength, α0 ,
and only depends upon β and ω. The fact that Ccoh (t)
does not depend upon α0 is a consequence of P̃ being
just a phase as Pcoh (α) is a delta function. In Fig. 3, we
plot Ccoh (t) for various values of β. We see from Fig. 3
that Ccoh (t) is periodic with time period 2π/ω. As in the
thermal state case, the entanglement between the qubits
decreases more rapidly from its maximum value as one
increases β.
Qubits initially in state |Ψ+ i and oscillator in
a number state: To study the entanglement dynamics
between the qubits in the degenerate regime when the
oscillator starts in the Fock state, we note that P (α) for
a Fock state |N i is given by [51]:
exp (αα∗ )
∂ 2N
2
PN (α) =
δ
(α)
.
(34)
N!
∂αN ∂α∗N
With PN (α) in equation (25), I(t) becomes (see Appendix B):
N X
N
1
2(N −m)
IN (t) =
(8iβ sin (ωt/2))
m
(N
−
m)!
m=0
× exp −32β 2 sin2 (ωt/2) ,
= LN (8β sin (ωt/2))2
× exp −32β 2 sin2 (ωt/2) ,
(35)
where LN (x) is a Laguerre polynomial. Using equations
(35) and (27), we calculate the concurrence for the number state case:
CN (t) = LN (8β sin (ωt/2))2 × exp −32β 2 sin2 (ωt/2) .
(36)
The concurrence, CN (t), is plotted in Fig. 4 for various values of N and β = 0.05 and β = 0.1. We see
from the figure that CN (t) is periodic with time period
2π/ω. Within each time period, for certain values of N
and β, the qubits momentarily become disentangled before getting entangled again. As N or β is increases,
entanglement between the qubits gets sharply peaked at
times that are integer multiples of 2π/ω.
Dependence of entanglement on coupling
strength: As discussed in section II, for the RWA to
be valid, one has to ensure that λ ≪ ω, ω0 . So, given ω
and ω0 , there is an upper limit of λ for which RWA is
valid. In contrast to the RWA case, the results in this report are correct for all values of λ. We see from equations
6
1.0
(a) Thermal State
th
N=1
N=3
N=10
C
CN(t)
1
<n>=1
<n>=3
<n>=10
0.5
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.4
0.5
1.0
(b) Coherent State
0.5
1
1.5
ωt/2π
2
2.5
1
0.5
N=1
N=3
N=10
0
0
0.1
0.2
0.3
1
0.5
N=1
N=3
N=10
(c) Number State
CN
CN(t)
3
Ccoh
0
0
0.5
0
0
0.5
1
1.5
ω t/2π
2
2.5
3
FIG. 4. Time dependence of entanglement between the two
qubits in the degenerate regime given by CN (t) for various
value of N . Top: β = 0.05 and bottom: β = 0.1.
(30), (33) and (36) that the entanglement depends on the
coupling strength and the oscillator frequency through
the parameter β 2 sin2 (ωt/2). This parameter is periodic
with time period T = 2π/ω. At times that are integer
multiples of T , the qubits are maximally entangled for all
coupling strengths. This can be seen from Figs. 2, 3 and
4. To understand the role of the coupling strength, λ, in
the entanglement between the qubits, we plot in Fig. 5
the concurrence for the thermal state, the coherent state
and the number state as a function of the dimension-less
coupling strength β evaluated at half the time period, i.e.
t = π/ω. We have:
Cth (π/ω) = exp −32β 2 (1 + 2 hni) ,
Ccoh (π/ω) = exp −32β 2 ,
CN (π/ω) = LN 64β 2 exp −32β 2 .
(37)
(38)
(39)
We see from the above set of equations that for all
the three initial states of the oscillator, concurrence decreases exponentially as β increases. Equation (37), for
various values of hni, is plotted in Fig. 5(a) from which
we see that the concurrence decreases faster as the mean
thermal excitation number hni is increased. For the coherent state, equation (38) is plotted in Fig. 5(b) from
which the exponential decrease of concurrence is evident.
In Fig. 5(c), CN (π/ω) is plotted for various values of N
where we see that the exponential decrease of concurrence is modulated by the Laguerre polynomial.
0
0
0.1
0.2
β
0.3
0.4
0.5
FIG. 5. Entanglement between the qubits as a function of β
for (a) thermal state, (b) coherent state, and (c) number state
evaluated at ωt/2 = π/2.
V.
CLASSICAL-QUANTUM COMPARISON
It is well established within the rotating wave approximation that the predictions about the dynamical
properties of the qubits interacting with a classical harmonic oscillator are qualitatively different from the predictions when the oscillator is treated quantum mechanically. Contrary to the classical case involving a prescribed driving field, the quantum mechanical treatment
for the corresponding coherent field predicts the qubit
population inversion to show collapse and revival [52].
These arise from the infinitely many, but discrete, Rabi
frequencies for qubit evolution associated with the infinite set of Fock states defining a coherent state. The
quantum mechanical predictions have been confirmed experimentally [56]. In view of this, it is a question whether
there are qualitative differences between the predictions
that can be made on the basis of the non-RWA calculations in Sec. IV and a set of corresponding non-RWA
predictions to be made when the oscillator is treated as
classical, while the qubits remain quantum mechanical.
A mixed quantum-classical theory of this kind is usually called semi-classical and the oscillator is taken as an
external influence. Here we want something a bit different because in Sec. IV the oscillator has not been treated
as an external influence, but rather as open to back reaction from the qubits. Jaynes first addressed this issue of
back reaction in the cavity QED context with a so-called
“neo-classical” theory [1, 57] in which the oscillator (cav-
7
ity electric field in his case) was driven by the qubit via
its quantum average values. In this way, although influenced by the quantized qubits, the oscillator remains
unquantized for all time and takes only numerical, not
operator, values.
We will adopt the Jaynes prescription in order to include back reaction. The simplest approach is to derive
equations of motion for the various dynamical quantities
using the original Hamiltonian (3). For the oscillator, instead of working with non-hermitian operators â and ↠,
we work with the real dynamical quantities, momentum
p̂ and co-ordinate q̂ defined in Appendix C. We then replace operators by numerical variables, which we indicate
simply by removing the carets. These variables evolve
from classically specifiable initial values p(0) and q(0), as
driven by expectation values of the qubit σ̂x variables:
p
q̇ = p + ~λ 2/~ωhσx(1) + σ̂x(2) i,
(40)
2
ṗ = −ω q,
(41)
˙σ̂ (i) = 0,
(42)
x
p
˙σ̂ (i) = −2λ 2/~ω pσ̂ (i) ,
(43)
y
z
p
˙σ̂ (i) = 2λ 2/~ω pσ̂ (i) .
(44)
y
z
As it happens, the states we have worked with have
zero expectation of the σ̂x operators, which don’t evolve
themselves, so the oscillator evolves completely independently. For the qubits starting in the state |Ψ+ i,
there are only four non-zero density matrix elements:
h1, ±1|ρ̂q |1, ±1i and h1, ±1|ρ̂q |1, ∓1i. The diagonal density matrix elements do not change in time because
(1)
(2)
σ̂˙ x + σ̂˙ x = 0. To find the evolution of the off-diagonal
elements, we note that:
h1, 1|ρ̂q (t)|1, −1i = T r{ρ̂q (t)|1, −1ih1, 1|}
(1)
(2)
= T r{ρ̂q (0)Ŝ+ (t)Ŝ+ (t)},
D
E
(1)
(2)
= Ŝ+ (t)Ŝ+ (t) ,
(45)
1 (i)
(σ̂ (t) + iσ̂y(i) (t)).
2 z
(46)
where
(i)
Ŝ+ (t) =
From equations (43) and (44), we get the following equation for the required expectation values
r
d (1)
2
(2)
(1)
(2)
hŜ+ (t)Ŝ+ (t)i = −4iλ
phŜ+ (t)Ŝ+ (t)i. (47)
dt
~ω
Solving equations (40), (41) and (47) we get
(1)
(2)
(1)
(2)
hŜ+ (t)Ŝ+ (t)i = hŜ+ (0)Ŝ+ (0)i
p
× exp 8iβ 2ω/~ q(0) sin2 (ωt/2)
p
× exp −4iβ 2/~ω p(0) sin(ωt) ,(48)
where q(0) and p(0) are the initial position and momentum of the oscillator respectively. For the initial state
(1)
(2)
of the qubits being |Ψ+ i, hŜ+ (0)Ŝ+ (0)i = −1/2. The
entanglement between the qubits is then given by (equations (27) and (45)):
(1)
(2)
C(t) = 2|hŜ+ (t)Ŝ+ (t)i|.
(49)
Since the time dependent terms in equation (48) enter
only as phases, the entanglement between the qubits do
not change and the qubits remain maximally entangled.
A crucial point used in deriving equation (48) is the
fact that the oscillator is assumed to have well defined
initial position and momentum. Because of the uncertainty principle, within quantum mechanics, no state can
be a simultaneous eigenstate of position and momentum.
In view of this, it is a question whether with a classical
description of the oscillator one can or cannot get modulations in entanglement of the qubits if the initial position
and momentum of the oscillator are not well defined. We
now show that by averaging equation (48) over appropriate joint probability distributions of q(0) and p(0), say
P (q0 , p0 ), we get expressions for concurrence that match
with the quantum mechanically derived results for thermal states (equation (30)) and the coherent states (equation (33)).
The classical probability distribution P (q0 , p0 ) of a
harmonic oscillator in thermal equilibrium is gaussian.
For an appropriate probability distribution of the oscillator in a coherent state, |α0 i, we use the Wigner distribution of the coherent state [51] which is also a gaussian.
So, for the purpose of thermal states and coherent states,
we take P (q0 , p0 ) to be of the general form:
(p0 − p̄0 )2
(q0 − q̄0 )2
1
,
−
exp −
P (q0 , p0 ) =
π∆q∆p
(∆q)2
(∆p)2
(50)
where ∆q and ∆p are the initial uncertainties in position
and momentum, centered at q̄0 and p̄0 respectively. Taking the absolute value after averaging equation (48) over
P (q0 , p0 ), we get
Z
(1)
(2)
C̄(t) = 2 dq0 dp0 P (q0 , p0 )hŜ+ (t)Ŝ+ (t)i ,
ω(∆q)2
4
2
= exp −32β sin (ωt/2)
~
2
(∆p)
2
2
× exp −8β sin (ωt)
.
(51)
~ω
From the above equation, we see that the concurrence
does not depend on the average values q̄0 and p̄0 and
only depends on the uncertainties, ∆q and ∆p. If both
the uncertainties, ∆q and ∆p, are not simultaneously
zero, the concurrence does change in time.
For a p
thermal state characterized
√ by temperature T ,
∆qth = 2KT /ω 2 and ∆pth = 2KT where K is the
Boltzmann constant. Using this in equation (51), we get:
KT
2
2
.
(52)
C̄th (t) = exp −64β sin (ωt/2)
~ω
8
In order to compare equation (52) with the quantum result (equation(30)), we note that in the high temperature
limit, i.e. when KT ≫ ~ω, the average excitation number hni = KT /~ω. Now, replacing the factor (1 + 2hni)
in the exponential of equation (30) by 2KT /~ω, we get
exactly the same expression for concurrence as derived
with the classical description of the oscillator, equation
(52).
A coherent state
state for
p is a minimum uncertainty
√
which ∆qcoh = ~/ω and ∆pcoh = ~ω. Putting this
in equation (51), we get:
C̄coh (t) = exp −32β 2 sin2 (ωt/2) ,
(53)
which exactly matches with the corresponding quantum
result, equation (33).
The above results show that the modulations in the entanglement between the qubits is a consequence of initial
uncertainty in q0 and p0 , and not an intrisically quantum
property. However, the uncertainty in the simultaneous
assignment of values to canonically conjugate variables is
inherent in quantum mechanics and cannot be avoided.
This fundamental uncertainty is reflected in the general
equation for concurrence that is valid for all quantum
states, equation (27), from which we see that for all states
there is a time dependent factor, exp −32β 2 sin2 (ωt/2) ,
which corresponds to the modulation of entanglement for
the minimum uncertainty state, i.e. the coherent state.
VI.
CONCLUSION
In this report, we studied the dynamics of the twoqubit Tavis-Cummings model in the degenerate regime
beyond the RWA, obtaining exact analytic expressions
for state properties of interest, including qubit entanglement. The general form of the time evolved density matrix of the composite system for any initial product state
of the qubits and the oscillator was derived. The qubits
initially prepared in the |Ψ− i and |Φ− i states were shown
to remain maximally entangled throughout the dynamical evolution. The dynamics of entanglement between
the qubits initially prepared in the |Ψ+ i state was shown
to be the same as that for the qubits initially prepared
in the state |Φ+ i for any initial state of the oscillator. A
general expression for concurrence, quantifying the entanglement between the qubits initially prepared in the
state |Ψ+ i and the oscillator in any state characterized
by the Glauber-Sudarshan function P (α) was derived.
When the initial state of the oscillator is either the
thermal state, coherent state or number state, the entanglement dynamics between the qubits was shown to be
periodic with time period 2π/ω. This periodic behavior
is a consequence of the fact that our system consists of
displaced harmonic oscillators with frequency ω. A general feature of entanglement for all initial states of the
oscillator was a more sharply peaked entanglement as the
dimension-less coupling strength β increased. This feature can be understood as a manifestation of the common
factor exp −32β 2 sin2 (ωt/2) in equation (27), which is
present for all initial states of the oscillator.
When the initial state of the oscillator is a coherent
state, it was shown that the entanglement between the
qubits does not depend on the oscillator strength α0 . The
periodicity of entanglement between the qubits with a
period independent of the strength of the coherent state
was reported in [50] where the dynamics between the
qubits and the oscillator was studied in the dispersive
regime within the RWA. Although in [50] the periodicity
was found not to depend on the strength of the coherent
state, the amplitude of oscillations of entanglement depended on α0 in contrast to the results found here in the
degenerate regime.
In section V, we studied the dynamics of the state of
the qubits by treating the oscillator classically. The back
action of the qubits on the oscillator evolution equations
was included in the spirit of Jaynes’ original neoclassical
treatment [57]. It was shown that temporal entanglement
modulation is also present classically, but only if there is
uncertainty in the initial position and/or momentum of
the oscillator.
In all the calculations in the current paper, the qubits
were assumed to be perfectly degenerate. This is obviously an idealization. A more realistic scenario would
have quasi-degenerate qubits with ω0 6= 0 but still satisfying the condition ω0 ≪ ω, λ. The existence of comparable analytic expressions for the collective properties
of such qubits with non-zero ω0 is an important question
for future work.
ACKNOWLEDGEMENT
Partial financial support was received from DARPA
HR0011-09-1-0008, ARO W911NF-09-1-0385, and NSF
PHY-0855701.
APPENDIX A: EVOLUTION TERM
e−iĤ0 t/~ |j, mi|αi =
= e−i(ωâ
=e
†
1
2
â+λ(â+↠)(σ̂x
+σ̂x
))t
−i(ω↠â+λ(â+↠)2m)t
|j, mi|αi,
|j, mi|αi,
†
2
= |j, miD̂(−2βm )e−iωtâ â D̂(2βm )|αie4iβm ωt ,
= |j, mi|(α + 2βm )e−iωt − 2βm i
ωt
× exp (−2iβm sin (α∗ eiωt/2 + αe−iωt/2 ))
2
2
× exp (4iβm
(ωt − sin ωt)).
(54)
APPENDIX B: CALCULATION OF IN (t)
We define
I˜N (t) = IN (t) exp −32β 2 sin2 (ωt/2) ,
(55)
9
and evaluate I˜N (t) for PN (α) given by equation (34):
Z
I˜N (t) = d2 αPN (α)
ωt
× exp (−8iβ sin (α∗ eiωt/2 + αe−iωt/2 ))
2
Z
1
∂ 2N
2
=
δ
(α)
d2 α exp (|α|2 )
N!
∂αN ∂α∗N
ωt
× exp (−8iβ sin (α∗ eiωt/2 + αe−iωt/2 ))
2
Z
1
∂ 2N
2
=
δ
(α)
d2 α exp (|α|2 )
N!
∂αN ∂α∗N
∗
× exp (A(α a + αb)).
(56)
APPENDIX C: HAMILTONIAN CONTEXT
For an explicit example of context for the generic
Hamiltonian (3) we provide a version of the original treatment by Jaynes [57]. In this case we are describing a version of cavity or circuit QED. The electromagnetic field
mode is interpreted as an oscillator, and the associated
field Hamiltonian is written in terms of real dynamical
variables, momentum p̂ and co-ordinate q̂ as:
1X 2
2 2
(p̂ + ωm
q̂m ).
(60)
2 m m
where p̂m and q̂m are the expansion coefficients of the
vector potential and the transverse electric field:
Ĥ =
where
A = −8iβ sin
a = eiωt/2 ,
ωt
,
2
and b = e−iωt/2 .
Using the identity
Z
∂ 2N
2
d2 α
δ
(α)
f (α, α∗ )
∂αN ∂α∗N
∂ 2N
∗
f
(α,
α
)
|α=α∗ =0 ,
=
∂αN ∂α∗N
(57)
X
~ˆ r , t) = −√ǫ0
A(~
~um (~rm )q̂m (t),
m
X
~ˆ r , t) = √ǫ0
~um (~rm )p̂m (t),
E(~
(61)
m
(58)
we get
or their equivalents in a resonant circuit. In QED these
are operators obeying canonical commutation relations.
The function ~um (~rm ) is the orthonormal vector function (i.e., incorporating polarization) that defines the mth
mode of the field, which has frequency ωm .
The interaction Hamiltonian is derived in the familiar
~ˆ E.
~ˆ For a
way from the dipole -field coupling: Ĥint = −d.
single excited mode labeled a, and a qubit with transition
dipole matrix element d~12 , this takes the explicit form
2
∗
1
∂ 2N
I˜N (t) =
e|α| eA(α a+αb) |α=α∗ =0 ,
N
∗N
N ! ∂α ∂α
∗
1 ∂N
[(α∗ + Ab)N eAaα ]|α∗ =0 ,
=
∗N
N ! ∂α
N 1 X N
∂m
∗
N
=
(α
+
Ab)
N ! m=0 m
∂α∗m
N −m
∂
Aaα∗
×
|α∗ =0 ,
e
∂α∗N −m
N N!
1 X N
A2(N −m) (ab)N −m ,
=
N ! m=0 m (N − m)!
N N!
1 X N
ωt
=
(−8iβ sin )2(N −m) .
N ! m=0 m (N − m)!
2
(59)
where we have introduced the coupling constant λ used
throughout,
as well as the oscillator
amplitudes via p̂a =
p
p
~ω/2(â + ↠) and ωa q̂a = i ~ωa /2(â − ↠). As usual,
the Pauli matrix expresses the operator character of the
two-level dipole moment. In Sec V, we removed the hats
and regarded p̂a , q̂a , â, and ↠as classical numerical variables.
[1] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51, 89
(1963).
[2] L. Allen and J.H. Eberly in Optical Resonance and TwoLevel Atoms (Dover Publications, 1987).
[3] T. Holstein, Ann. Phys. (N.Y.) 8, 325 (1959).
[4] E.K. Irish and K. Schwab, Phys. Rev. B 68, 155311
(2003).
[5] K.C. Schwab and M.L. Roukes, Phys. Today 58, 36
(2005).
[6] A. Blais, R.S. Huang, A. Wallraff, S.M. Girvin, and R.J.
Schoelkopf, Phys. Rev. A 69, 062320 (2004).
[7] A. Wallraff, D.I. Schuster, A. Blais, L. Frunzio, R.-S.
Huang, J. Majer, S. Kumar, S.M. Girvin, and R. J.
Schoelkopf, Nature (London) 431, 162 (2004).
[8] I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura,
C.J.P.M. Harmans, and J.E. Mooij, Nature (London)
431, 159 (2004).
[9] J. Johansson, S. Saito, T. Meno, H. Nakano, M. Ueda, K.
√
Ĥint = −(d~12 .~ua (0)) ǫ0 p̂a (t)σ̂x ,
≡ ~λ(â + ↠)σ̂x ,
(62)
10
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
Semba, and H. Takayanagi, Phys. Rev. Lett. 96, 127006
(2006).
B.W. Shore and P.L. Knight, J. Mod. Opt. 40, 1195
(1993) and references in there.
M. Tavis and F.W. Cummings, Phys. Rev. 170, 379
(1968).
M. Tavis and F.W. Cummings, Phys. Rev. 188, 692
(1969).
T. Niemczyk et al., Nat. Phys. 6, 772 (2010).
P. Forn-Dı́az, J. Lisenfeld, D. Marcos, J.J. Garcı́a-Ripoll,
E. Solano, C.J.P.M. Harmans, and J.E. Mooij, Phys. Rev.
Lett. 105, 237001 (2010).
A. Fedorov, A.K. Feofanov, P. Macha, P. Forn-Dı́az,
C.J.P.M. Harmans, and J.E. Mooij, Phys. Rev. Lett. 105,
060503 (2010).
S. Swain, J. Phys. A: Math. Gen. 6, 192 (1973).
K. Zaheer and M. S. Zubairy, Opt. Comm. 73, 325
(1989).
S.J.D. Phoenix, J. Mod. Opt. 36, 1163 (1989).
B.W. Shore, Chapter 4 of The Theory of Coherent
Atomic Excitation: Vol. 1, Simple Atoms and Fields
(Wiley-Interscience Publication, 1990).
M.D. Crisp, Phys. Rev. A 46, 4138 (1992).
G.A. Finney and J. Gea-Banacloche, Phys. Rev. A 50,
2040 (1994).
E. A. Tur, Opt. Spectro. 89, 574 (2000).
E. A. Tur, Opt. Spectro. 91, 899 (2001); E. A. Tur, arXiv:
math-ph/0211055.
E.K. Irish, J. Gea-Banacloche, I. Martin, and K.C.
Schwab, Phys. Rev. B 72, 195410 (2005).
E.K. Irish, Phys. Rev. Lett. 99, 173601 (2007).
J. Larson, Phys. Scr. 76, 146 (2007).
J. Hausinger and M. Grifoni, New J. Phys. 10, 115015
(2008).
J. Hausinger and M. Grifoni, Phys. Rev. A 82, 062320
(2010).
J. Casanova, G. Romero, I. Lizuain, J.J. Garcı́a-Ripoll,
and E. Solano, Phys. Rev. Lett. 105, 263603 (2010).
J. Seke, Quantum Opt. 3, 127 (1991).
J. Seke, Il Nuovo Cimento D, 13, 1429 (1991).
A.B. Klimov and S.M. Chumakov, A Group-Theoretical
Aproach to Quantum Optics: Models of Atom-Field Interactions (Wiley-VCH 2009).
D. Zueco, G.M. Reuther, S. Kohler, and P. Hänggi, Phys.
Rev. A 80, 033846 (2009).
J. Jing, Z.G. Lü, and Z. Ficek, Phys. Rev. A 79, 044305
(2009).
Z. Ficek, J. Jing, and Z.G. Lü, Phys. Scr. T140, 014005
(2010).
Q.H. Chen, Y. Yang, T. Liu, and K.L. Wang, Phys. Rev.
A 82, 052306 (2010).
[37] S.M. Hashemi Rafsanjani, S. Agarwal, and J.H. Eberly,
arXiv:1105.2835v1 [quant-ph].
[38] K.K. Shakov and J.H. McGuire, Phys. Rev. A 67,
033405 (2003).
[39] R.J. Glauber, Phys. Rev. 131, 2766 (1963).
[40] E.C.G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
[41] S. Oh and J. Kim, Phys. Rev. A 73, 062306 (2006).
[42] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki,
Rev. Mod. Phys. 81, 865 (2009).
[43] W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[44] S. Bose, I. Fuentes-Guridi, P.L. Knight, and V. Vedral,
Phys. Rev. Lett. 87, 050401 (2001).
[45] M.S. Kim, J. Lee, D. Ahn and, P.L. Knight, Phys. Rev.
A 65, 040101(R) (2002).
[46] D. Braun, Phys. Rev. Lett. 89, 277901 (2002); F. Benatti, R. Floreanini, and M. Piani, Phys. Rev. Lett. 91,
070402 (2003); Z. Ficek, Opt. and Spect. 108, 347 (2010);
[47] L.S. Aguiar, P.P. Munhoz, A.V. Barranco, and J.A.
Roversi, J. Opt. B: Quantum Semiclass. Opt. 7, S769
(2005).
[48] L. Zhou, X.X. Yi, H.S. Song, and Y.Q. Guo, J. Opt. B:
Quantum Semiclass. Opt. 6 378 (2004).
[49] L. Jin, X.S. Hua, C.H. Ping, and L. Jian, Comm. in Theo.
Phys. 51, 637 (2009).
[50] L. Roa, R. Pozo-González, M. Schaefer and P. Utreras,
Phys. Rev. A 75, 062316 (2007).
[51] M.O. Scully and M.S. Zubairy in Quantum Optics (Cambridge University Press, 1997).
[52] J.H. Eberly, N.B. Narozhny, and J.J. SanchezMondragon, Phys. Rev. Lett. 44, 1323 (1980); N.B.
Narozhny, J.J. Sanchez-Mondragon, and J.H. Eberly,
Phys. Rev. A 23, 236 (1981); H.I. Yoo, J.J. SanchezMondragon, and J.H. Eberly, J. Phys. A 14, 1383 (1981).
[53] S.J.D. Phoenix and P.L. Knight, Phys. Rev. A 44, 6023
(1991); S.J.D. Phoenix and P.L. Knight, Phys. Rev. Lett.
66, 2833 (1991); J. Gea-Banacloche, Phys. Rev. Lett. 65,
3385 (1990); J. Gea-Banacloche, Phys. Rev. A 44, 5913
(1991); V. Buzek, H.M. Cessa, P.L. Knight, and S.J.D.
Phoenix, Phys. Rev. A 45, 8190 (1992);
[54] Z. Deng, Opt. Commun. 54, 222 (1985).
[55] T.E. Tessier, I.H. Deutsch, and A. Delgado, Phys. Rev A
68, 062316 (2003).
[56] M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J.M. Raimond, and S. Haroche, Phys. Rev. Lett.
76, 1800 (1996).
[57] E.T. Jaynes, Stanford University Microwave Laboratory
Report 502 (1958).