Exchange of quantum states between coupled oscillators
Exchange of quantum states between coupled
oscillators
D. Portes Jr. and H. Rodrigues
Centro Federal de Educação Tecnológica do Rio de Janeiro
Departamento de Educação Superior-DEPES.
S. B. Duarte∗
Centro Brasileiro de Pesquisas F́ısicas /CNPq,
Rua Dr. Xavier Sigaud 150, CEP 22.290-180,
Rio de Janeiro (RJ), Brazil.
B. Baseia
Instituto de F́ısica, Universidade Federal de Goiás
PO-Box-131, 74.001-970, Goiania(GO), Brazil.
December 9, 2018
Abstract
Exchange of quantum states between two interacting harmonic oscilla-
tor along their evolution time is discussed. It is analyzed the conditions for
such exchange starting from a generic initial state and demonstrating that
the effect occurs exactly only for the particular states C0 |0 > +CN |N >,
which includes the interesting qubits components |0〉, |1〉. It is also deter-
mined the relation between the coupling constant and characteristic fre-
quencies of the oscillators to have the complete exchange.
1 Introduction
The engineering of quantum states of light fields and oscillators became an in-
teresting topic in the last years, due to its applications in : (i) fundamentals
of quantum mechanics (preparation of Schrodinger-cat states [1], their super-
position [2] and measurement of their decoherence [3], etc.); (ii) determination
of certain properties of a system (phase distribution P(θ) [4], Wigner [5] and
Husimi [6] functions, etc.); (iii) proposals for practical applications (quantum
lithography [7], quantum communication [8] - e.g., via hole-burning in Fock
∗corresponding author, e-mail : sbd@cbpf.br
http://arxiv.org/abs/0704.0815v2
space [9] - quantum teleportation [10], etc). However, a difficult situation ap-
pears when one wants to prepare a state of a system offering hard access [11].
In this case the difficulty may be circumvented by coupling the system having
hard access to a second system offering easy access, in which a desired state is
prepared with subsequent transfer to the first one. The success of this operation
depends on the model-Hamiltonian and on the initial state describing the whole
system.
Although the problem of two interacting harmonic oscillators has been ex-
haustively studied in the literature, the discussion about exchange of nonclas-
sical states between them is scarce. The coupled quantum oscillation problem
was considered earlier in [12, 13, 14], where the authors of those papers were
interested only in the energy of the system. Later on, in Ref [15] a full exchange
between quantum two-mode harmonic oscillators was presented, however the
issue was only concerned with the particular transfer of coherent states. In Ref.
[16] we have studied the transfer of certain properties (statistics and squeezing)
and in Ref. [17] we have studied the transfer of the most relevant part of
the state of a sub-system to another, through the simultaneous transfer of the
number and phase distributions, Pn and P (θ)
1 [17]; the solutions were found
numerically since the models were not exactly soluble.
In the present work we employ a distinct model-Hamiltonian, allowing us to
treat the problem analytically permitting us to analyze the transfer of generic
states. We show in which way one can get exact exchange of the states between
two interacting sub-systems. Exchange of states means simultaneous transfer
of states in two opposite directions ; so, it is more significant than the transfer
of states in one direction as studied in [17]. In the present case the transfer of a
state from the “easy-oscillator” to the “hard-oscillator” is observed by simply
monitoring the state of the easy-oscillator during the time evolution of the whole
system. For brevity, hereafter the easy- and the hard-oscillator will be referred
to as O1 and O2, respectively.
The Sect. II introduces the model-Hamiltonian allowing us to obtain the
evolution operator for this coupled system. In the Sect. III we consider differ-
ent types of initial states describing the entire system to study the mentioned
effect between the O1 and the O2 ( Sub-Sects. (A), (B),and (C) ), includ-
ing superpositions of states representing the qubits |0〉 and |1〉. The Sect. IV
contains the comments and conclusion.
2 Model-Hamiltonian: evolution operator
We start from the Hamiltonian
H/h̄ = ω1a
1 a1 + ω2a
2 a2 + λ
a+1 a2 + a1a
, (1)
1Since the number and phase are canonically conjugate operators they are complementary,
in the sense that simultaneous transfer of number and phase distributions, Pn and P (θ),
concerns the transfer of the major part of the state describing a system.
where a+i (ai) stands for the raising (lowering) operator of the i− th oscillator,
i = 1, 2; ωi and λ are real parameters standing for the i-th oscillator frequency
and coupling constant, respectively. The equations of motion for the operators
a1(t) and a2(t) can be solved analytically,
a1(t) =
c2e−iω
t + s2e−iω
a1(0) + cs
t − e−iω
a2(0), (2)
a2(t) =
c2e−iω
t + s2e−iω
a2(0) + cs
t − e−iω
a1(0),
where,
ω′1 = ω1 + λ
, (3)
ω′2 = ω2 − λ
x2 + 1
x2 + 1
ω1 − ω2
. (5)
The parameter s and c satisfy the condition c2+s2 = 1, they define the auxiliary
operators
a′1 = c a1 + s a2 , (6)
a′2 = −s a1 + c a2 ,
which decouple the above Hamiltonian. The following relations also hold:
ω′1 + ω
2 = ω1 + ω2 , (7)
ω′1 − ω′2 =
It is convenient for our purposes to find the time dependent state vector or
density operator in the Schrodinger picture. One formal prescription is to work
with Wigner representation of the state and obtain the time-dependent density
operator from the Wigner function[19], for which the time evolution is easily
obtained. However, it is a hard task to restore analytical or numerical values
for the density matrix ρ(t) in the Fock basis from the time dependent Wigner
function. To overcome this difficult we will show that for the Hamiltonian given
by Eq.(1) there is an analytical expression for the evolution operator U(t), which
defines the solution of the Schrodinger equation, allowing us to get directly the
matrix ρ(t) in the Fock basis. This kind of approach was already used in Ref
[18], but only treating the system in the resonant case (ω1 = ω2). In [18] the
author studied the transfer of state starting from the particular one photon
state. Our results permit one to obtain an analytical expression for the matrix
element U(t), for the Hamiltonian (1) not restricted to the resonant case and
permitting easy application to a generic initial state. Consequently, the problem
of transfer of states can be more comfortably discussed using the present results.
To obtain the operator U(t), we define the (auxiliary) unitary operator
Us(t) which is associated to a rotation and decouples the Hamiltonian,
U−1s ai Us = a
i . (8)
We have,
U−1s = U−s , (9)
in view of the reverse transformation
a1 = c a
1 − s a′2 , (10)
a2 = s a
1 + c a
We denote {|n1, n2〉0} as representing the Fock′s basis, eigenvectors of the
(old) number operator Ni = a
i ai , whereas {|n1, n2〉s} is the same for the
(new) number operator Ni(s) = a
i. We have,
Us|n1, n2〉s = |n1, n2〉0, (11)
|n1, n2〉s = U−s|n1, n2〉0.
If we represent Us in the Fock
′s basis {|n1, n2〉0}, we obtain
n1, n2
m1, m2
= 0〈n1, n2|Us|m1,m2〉0 (12)
= s〈n1, n2|m1,m2〉0.
Next, to reconstruct the operator Us in the Fock’s basis, we start from
s〈n1, n2| a′1|m1,m2〉0 = s〈n1, n2| (c a1 + s a2) |m1,m2〉0, (13)
Since the operators a′i act on the basis {|n1, n2〉s} whereas the ai act on the
basis {|n1, n2〉0}, we get
n1 + 1s〈n1 + 1, n2|m1,m2〉0 = c
m1 s〈n1, n2|m1 − 1,m2〉0 (14)
m2 s〈n1, n2|m1,m2 − 1〉0,
which, after using the Eq.(12), leads to
n1, n2
m1, m2
n1−1,n2
m1−1,m2
n1−1,n2
m1,m2−1
, (15)
and similarly, repeating the procedure for the operator a′2, we find
n1, n2
m1, m2
n1,n2−1
m1−1,m2
n1,n2−1
m1,m2−1
. (16)
Using the Eqs. (15), (16) plus the unitary condition U †sUs = UsU
s = 1 we
obtain, after a lengthy calculation, the expression
n1, n2
m1, m2
= δn1+n2, m1+m2
n1!n2!
m1!m2!
(−1)n2 cm1−n2 sm2+n2 (17)
min(n2,m2)
k=max(0,m2−n1)
(−1)−k
n2 − k
(U−s)
n1, n2
m1, m2
= (−1)m2−n2 (Us)n1, n2m1, m2 . (18)
The time evolution operator U(t) may be written in the basis {|n1, n2〉s} as
U(t) =
k1,k2
|k1, k2〉s e−i(k1ω
+ k2ω
s〈k1, k2| , (19)
for H is diagonal in this basis. Finally from the Eqs.(12) and (19) we obtain
the expression
n1, n2
m1, m2
k1,k2
e−i(k1ω
+ k2ω
)t (U−s)
n1, n2
k1, k2
(U−s)
m1, m2
k1, k2
, (20)
restricted to n1 + n2 = k1 + k2 = m1 +m2 , whereas U
n1, n2
m1, m2
= 0 otherwise.
The evolution operator obtained in Eq.(20) allows us to study the time evo-
lution of the whole state describing our bipartite system composed by coupled
oscillators, represented by the Hamiltonian in the Eq.(1). In the next section
we will study the exchange of states between these oscillators and, as a natural
assumption, we will suppose the O2 initially in its ground state |0〉. The O1 is
assumed to be previously prepared in various initial states, firstly starting from
an arbitrary state |φ〉.
3 Exchange of generic state
Let us consider that the whole (bipartite) system is initially in the state
|Ψ(0)〉 = |φ〉 ⊗ |0〉 , (21)
whose components in the Fock’s basis are given by,
|Ψ(0)〉 =
Cn, 0(0)|n, 0〉 , (22)
since Cn1, n2(0) = 0 for n2 6= 0. In the Schrodinger representation, the coeffi-
cients Cn1,n2(t) are obtained from Cn1,n2(t) = 〈n1, n2|U(t)|Ψ(0)〉, which, using
Eq. (22) and the constraint n1 + n2 = n, results in the form
Cn1,n2(t) = Cn1+n2,0(0)U(t)
n1,n2
n1+n2,0
. (23)
In particular, we have that
Cn,0(t) = Cn,0(0)U(t)
n,0 , (24)
C0,n(t) = Cn,0(0)U(t)
n,0 . (25)
The exchange of states between the oscillators will occur after an instant τ ,when
C0,n(τ ) = Cn,0(0) and
|Ψ(τ)〉 =
C0, n(τ )|0, n〉 , (26)
or, |Ψ(τ )〉 = |0〉 ⊗ |φ〉. This shows that exchange of states allows us to verify
the transfer of states to the O2 by monitoring the time evolution of the O1.
From the Eqs. (17) and (18) we have,
n−l,l =
(n− l)!l!
cn−l sl , (27)
n−l,l =
(n− l)!l! (−1)
cl sn−l . (28)
The substitution of the Eqs. (27) and (28) in the Eq. (20) results
n,0 = (−1)
(n− l)!l!
(−1)n−l cnsne−i (n−l) ω
t e−i l ω
t . (29)
where we recognize the Newton’s binomial expression,
n,0 = (−1)
e−i ω
t − e−i ω
or, replacing the auxiliary parameters ω′1, ω
2 by ω1, ω2 and λ (cf. Eq. (7)),
n,0 = e
ω1+ω2
−2 i s c sin( λ
, (31)
and, consequently,
C0,n(t) = Cn,0(0) e−i
ω1+ω2
−2 i s c sin( λ
. (32)
In a similar way we get,
Cn,0(t) = Cn,0(0) e−i
ω1+ω2
c2e−i
t + s2ei
. (33)
From Eq. (32) we see that a partial exchange of states will occur when
λt/sc = (2k + 1)π, i.e., in the time intervals τk = (sc/λ) (2k + 1)π. The effect
attains the highest efficiency when the product sc is maximum, i.e., when s =
c = 1/
2 and τk = (k + 1/2)π/λ. According to the Eq. (4) this implies x = 0
and the resonance condition ω1 = ω2 = ω (cf. Eq. (5)),
C0,n(τk) = (−i)n Cn,0(0) e−i ω n τk . (34)
However, we note that even at resonance we obtain no exchange of states, due to
the presence of the phase factor exp
ωτk +
affecting the coefficients
of the state describing both oscillators in the Fock’s representation. In this gen-
eral case we obtain
∣C0,n(τk)
∣Cn,0(0)
∣, which means exchange of statistics
between the two oscillators. This can also be seen comparing both reduced
density matrix, ρ
m1, m2(τk) and ρ
m1, m2(0), in the Fock’s representation,
ρ(2)m1, m2(τk) = e
−i (ωτk+π2 ) (m1−m2) ρ(1)m1, m2(0) , (35)
which exhibits the distinction between their off-diagonal elements. As well
known, while the state of a system offers its complete description, the same
is not true for the statistics, which contains only partial informations of the
system.
3.1 The complete exchange of state
It is shown in the last section that it is not possible to have a complete exchange
of states for a generic initial state because the phases are not transferred (see
Eq.35). Here we show that when the state of oscillator O1 is given by the super-
position C0|0〉+ CN |N〉 whereas O2 is in the vacuum state, complete exchange
of states occurs. Note that this state includes in particular the important case
C0|0〉+ C1|1〉 using the qubits |0〉, |1〉 having potential applications in quantum
communication [20] and in quantum computation [21]. It was shown that this
state exhibits squeezed fluctuations [22].
Next, let us consider the whole system initially in the superposed state
|Ψ(0)〉 = C0,0(0)|0, 0〉+ CN,0(0)|N, 0〉 . (36)
In this case we verify perfect exchange of states between the oscillators for a
convenient choice of the parameters involved. Assuming the resonance condition
in the Eq.(32) we have, for C0,0(t) = C0,0(0),
C0,N (t) = CN,0(0) e−i (ω t+π/2)N sinN (λ t) . (37)
Partial exchange of states will occur when t = τ0 = π/(2λ),which results in
C0,N (τk) = C
N,0(0) e−i π/2(ω/λ+1)N , (38)
whose meaning is the exchange of statistics. The exchange of states becomes
complete (exact) when C0,N (τk) = C
N,0(0), namely, when
, (39)
with m integers. Taking m = 1 and ω in the microwave domain (ω ∼ 109Hz)
the time spent to transfer the state C0|0 > + C1|1 > from the O1 to the O2
results τ0 = π/(2λ) ∼ 10−9s, since λ = ω/3 (cf. Eq.(39)), which is smaller than
the typical decoherence time for such systems (τd ∼ 10−3s), as it should.
Note that the previous initial state C0|0〉+CN |N〉 describing the O1 includes
the Fock states |N〉, obtained from C0 = 0 and CN = 1. In this case exact
exchange of states no longer requires the Eq. (39). The reason comes from the
phase factor appearing in the Eq. (39), now becoming a global phase with no
physical relevance. In this case the exchange of states is exact for any instant
tk = τ0 + 2πk/λ.
4 Comments and Conclusion
An analytical procedure applied to a convenient model-Hamiltonian describing
two coupled oscillators allows us to get the exact evolution operator for the
entire system (Sect. II). This approach, through the use of distinct initial states
and parameters (Sub-Sects. (A), (B) of Sect. III), makes easy the study of
exchange of states between such sub-systems. In all cases we have shown that
the fidelity of the process is maximum when the resonance condition, ω1 =
ω2, is attained. Assuming the O2 always in the vacuum state we find, sub-
Section by sub-Section, that: (A) partial exchange of states is achieved when
the initial state of the O1 is arbitrary, for the time intervals t = τk = (k +
1/2)π/λ; the efficiency of partial exchange is maximum when the product sc
is maximum (sc = 1/2); however, while the occurrence of exchange of states is
partial, exchange of statistics is obtained exactly, as shown in the Eqs. (34),
(35); (B) exact exchange of states occurs when the O1 starts from the initial
superposed state C0|0〉 + CN |N〉, in the time intervals tk = τ0 + 2πk/λ, with
the requirement in Eq. (39). If the Eq.(39) is not obeyed, exchange of states
will occur at the same time intervals, but now the effect is only partial; Exact
exchange of states is also found in the particular case of (B), setting C0 = 0
and CN = 1, which means the O1 starting from a Fock state |N〉. In this case
the exchange of states occurs exactly at the same time intervals found in (B),
no matter the Eq. (39) is obeyed or not.
As final remarks we mention that exchange of states and its efficiency could
be investigated for other model-Hamiltonians and, as explained before, the ef-
fect goes beyond those studied in [16] and [17]. To our knowledge, exchange of
states in coupled systems and even exchange of certain properties, are subjects
receiving little attention in the literature [23] - with the remarkable exception
of quantum teleportation [21], an effect having a very distinct nature (requiring
the presence of quantum channels and entangled states), which occurs in the
absence of coupling between the two sub-systems. In the context of teleporta-
tion, exchange of states appears with the name ”identity interchange” [24] and
”two-way teleportation” [25].
4.1 Acknowledgements
The authors thank the CNPq (SBD, BB) and FAPERJ (DPJ) for the partial
supports.
4.2 References
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	 Introduction
	Model-Hamiltonian: evolution operator
	Exchange of generic state
	The complete exchange of state
	Comments and Conclusion
	Acknowledgements
	References
