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Transfer of an unknown quantum state, quantum networks, and memory

27 Aug 2004-Physical Review A (American Physical Society)-Vol. 70, Iss: 2, pp 022323

Abstract: We present a protocol for transfer of an unknown quantum state. The protocol is based on a two-mode cavity interacting dispersively in a sequential manner with three-level atoms in the $\ensuremath{\Lambda}$ configuration. We propose a scheme for quantum networking using an atomic channel. We investigate the effect of cavity decoherence in the entire process. Further, we demonstrate the possibility of an efficient quantum memory for arbitrary superposition of two modes of a cavity containing one photon.

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Transfer of an unknown quantum state, quantum networks, and memory
Asoka Biswas and G. S. Agarwal
Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India
(Received 13 May 2003; revised manuscript received 19 May 2004; published 27 August 2004
)
We present a protocol for transfer of an unknown quantum state. The protocol is based on a two-mode cavity
interacting dispersively in a sequential manner with three-level atoms in the configuration. We propose a
scheme for quantum networking using an atomic channel. We investigate the effect of cavity decoherence in
the entire process. Further, we demonstrate the possibility of an efficient quantum memory for arbitrary
superposition of two modes of a cavity containing one photon.
DOI: 10.1103/PhysRevA.70.022323 PACS number(s): 03.67.Hk, 42.50.Pq
I. INTRODUCTION
In the quantum information theory [1], transfer of infor-
mation in the form of a coherently prepared quantum state is
essential. One can transfer a quantum state either by the
method of teleportation [2] or through quantum networking.
The basic idea behind a quantum network is to transfer a
quantum state from one node to another node with the help
of a career (a quantum channel) such that it arrives intact. In
between, one has to perform a process of quantum state
transfer (QST) to transfer the state from one node to the
career and again from the career to the destination node.
There have been some proposals [3] for quantum networking
using cavity-QED, where two atoms trapped inside two spa-
tially separated cavities serve the purpose of two nodes. In
Ref. [3], the task was to transfer the state of one atom into
the other via the process of QST between the atom and pho-
ton, where the latter is used as a career. The photon carries
the information through either free space or an optical fiber
between the cavities, and the success depends on the proba-
bilistic detection of photons or adiabatic passage through the
cavities. We note that, though it may be difficult to beat the
communication with photons, it is always interesting to ex-
plore the alternatives. In fact, very recently, quantum net-
work using a linear XY chain of N interacting qubits was
proposed. In this proposal, the quantum state can be trans-
ferred from the first qubit to the Nth qubit within micro-
scopic distance by preengineering interqubit interactions [4].
Further, storage of quantum states is also an important
issue. There have been several proposals for quantum
memory. For example, recent proposals [5,6] have shown
how to transfer the field state into atomic coherence by the
adiabatic technique and again retrieve the same through the
method of adiabatic following [5] or using the teleportation
technique [6]. Quantum memory of the individual polariza-
tion state into a collective atomic ensemble has been pro-
posed [7]. Initially, an entangled state of two pairs of atomic
ensembles is prepared, where the single-photon polarization
state is stored through a process similar to teleportation.
Though the information can be transferred back to the pho-
ton state, the protocol only succeeds with a probability 1/4.
Decoherence-free memory of one qubit in a pair of trapped
ions has also been experimentally demonstrated [8]. Maître
et al. [9] have proposed a quantum memory, where the quan-
tum information on the superposition state of a two-level
atom was stored in a cavity as a superposition of zero- and
one-photon Fock states. The holding time of such memories
is generally limited by the cavity decay time.
In this paper, we propose a scheme for QST to transfer the
unknown state of one atom to another atom where the atoms
are not directly interacting with each other. Note that by
direct spin interaction of the S
1
·S
2
kind, the quantum state
could be transferred from one atom to another within a mi-
croscopic range. In the present scheme, we show how a simi-
lar kind of interaction between two atoms can be mediated
via a cavity. Thus the atomic state can be transferred from
one atom to another in the mesoscopic range.
We extend our idea of QST to a quantum network, where
we transfer the state of one cavity to another spatially sepa-
rated cavity. For this we use long-lived atoms as career, and
make use of the QST process to transfer the state of the
cavity to an atom and again to the target cavity. Our protocol
for quantum networking provides a deterministic way to
transfer the quantum state between the cavities. This protocol
does not require any kind of probability arguments based on
the outcome of a measurement. Further, we propose the re-
alization of a quantum memory of arbitrary superposition of
two modes of a cavity which contains only one photon. This
superposition state can be stored in the long-lived states of
the neutral atoms and retrieved in another two-mode cavity
later, deterministically. Our proposal relies on the techno-
logical advances and realizations as described in Ref. [10].
The structure of the paper is as follows. In Sec. II, we
describe the model and provide the relevant equations. In
Sec. III, we discuss how transfer of an unknown quantum
state can be performed between two atoms. We provide an
estimate of possible decoherence in this process due to cavity
decay. In Sec. IV, we extend our scheme to quantum net-
works and quantum memory.
II. MODEL CONFIGURATION
To describe how the QST protocol works, we consider a
three-level atom in the configuration interacting with a
two-mode cavity (see Fig. 1). The modes with annihilation
operators a and b interact with the e g and ef
transitions, respectively. The Hamiltonian under the rotating
wave approximation can be written as
PHYSICAL REVIEW A 70, 022323 (2004)
1050-2947/2004/70(2)/022323(5)/$22.50 ©2004 The American Physical Society70 022323-1

H =
eg
e典具e +
fg
f典具f +
1
a
a +
2
b
b + g
1
e典具ga
+ g
2
e典具fb + H.c.其兴, 1
where
lg
l e, f is the atomic transition frequency,
i
i
1,2 is the frequency of the cavity modes a and b, and g
i
is the atom-cavity coupling constant. We assume g
i
to be
real.
We work under the two-photon resonance condition and
assume large single-photon detuning. After adiabatically
eliminating the excited level e in the large detuning do-
main, we derive an effective Hamiltonian describing the sys-
tem of Fig. 1,
H
eff
=−
g
2
关兩g典具ga
a + f典具fb
b
g
2
关兩g典具fa
b + f典具gab
, 2
where =
eg,f
1,2
is the common one-photon detuning of
the cavity modes and g
1
=g
2
=gⰆ⌬. The condition g
1
=g
2
can be satisfied by proper choice as we can choose appropri-
ate transitions in atomic systems, frequencies, etc. Note that
if one considers the levels g and f as Zeeman sublevels,
then these conditions are automatically satisfied. In that case,
we may consider the two modes of the cavity as two or-
thogonal polarization states of a photon. Now note that the
first two terms in Eq. (2) represent the self-energy terms and
the last two terms give the interaction leading to a transition
from the initial state to the final state. The probability ampli-
tudes of relevant basis states g典兩n,
and f典兩n−1,
+1 in
the state vector
t兲典 = d
g
t兲兩g,n,
+ d
f
t兲兩f,n −1,
+1典共3
are given by
d
g
t =
nXY
n +
+1
+ d
g
0,
d
f
t =
+1XY
n +
+1
+ d
f
0, 4
where X=
nd
g
0+
+1d
f
0, Y =expig
2
n+
+1t/
1, and n and
are the respective photon numbers in the
modes a and b. We note that the effective interaction (2) can
be seen as an interaction between two qubits defined via the
atomic variables and field variables
S
+
= f典具g, S
= g典具f, S
z
=
1
2
共兩f典具f g典具g兩兲;
R
+
= a
b, R
= ab
, R
z
=
1
2
a
a b
b. 5
In the single-photon space, the field operators R
±
, R
z
satisfy
spin-1/2 algebra and thus the interaction (2) can be written
as an interaction between two qubits,
H
eff
g
2
R
+
S
+ R
S
+
−2R
z
S
z
. 6
In view of the above form of the effective interaction, we
conclude that our system of Fig. 1 can be used for a number
of quantum logic operations.
III. QUANTUM STATE TRANSFER PROTOCOL
We next demonstrate how the dynamics of an atom in a
two-mode cavity can be used to implement the QST proto-
col. Hereafter, we will use the term
pulse” to denote an
equivalent traversal time T of the atom through the cavity
such that 2g
2
T/=
. The time T could be controlled by
selecting the atomic velocity.
We assume that the atom A is initially in an unknown
state,
i
A
=
g
A
+
f
A
, 7
where
and
are unknown arbitrary coefficients. The state
i
A
of atom A is to be transferred to another atom B which is
elsewhere. Preparing the cavity in a state 0,1 (i.e., initially
one photon in the b mode), we send the atom A through the
cavity for a certain time which is equivalent to a
pulse.
After atom A comes out of the cavity, atom B in state
i
=
g +
f典共8
is sent through the cavity. Here
and
are arbitrary co-
efficients and need not be known. Atom B also experiences a
pulse during the interaction with the cavity. The entire
process can be described as follows:
i
A
0,1
pulse on atom A
g
A
0,1
1,0典兲
B atom enters
g
A
i
B
0,1
1,0典兲
pulse on atom B
g
A
i
B
0,1
1,0典兲. 9
FIG. 1. Three-level atomic configuration with levels g, e, and
f interacting with two orthogonal modes of the cavity, described
by operators a and b. Here g
1
and g
2
represent the atom-cavity
coupling of the a and b modes with the corresponding transitions
and is the common one-photon detuning.
A. BISWAS AND G. S. AGARWAL PHYSICAL REVIEW A 70, 022323 (2004)
022323-2

If one prepares the cavity initially in state 1,0, then
following a similar sequence to the above, the final state will
be f
A
i
B
0,1
1,0典兲. Note that atom B has already
acquired the state i of atom A, i.e., the state i is transferred
from atom A to atom B.
More generally, our QST protocol can be written as
i
A
i
B
0,1 +
1,0典兲
cav
U
g
f典兲
A
i
B
cav
,
10
where U
=U
A
U
B
, U
k
(k A,B) =exp
iH
eff
T/其兴 denotes the
-pulse operation on the atom k,
and
cav
=
0,1
cav
1,0
cav
. 11
Our protocol has interesting features: (a) the initial states of
the atoms can be arbitrary, and (b) the field state can also be
an arbitrary superposition of 0,1 and 1,0. Note that in the
case of a two-level atom interacting with a resonant single-
mode cavity, the QST protocol from one atom to another
atom has difficulties associated with a relative phase which
can be changed either by using a conditional phase shift
which is essentially a two-qubit operation [see Eq. (3.8) of
Ref. [10]] or by applying a resonant microwave field to the
atomic qubit.
We note that if the initial state of the atom B is g (or f)
and the cavity is initially in state 0,1 (or 1,0), then we
can not only transfer the state of atom A to B, but we also
can interchange the states between them. However, the QST
protocol described here cannot be interpreted as a SWAP
gate. As in the usual version of a quantum gate, atoms A and
B must interact with the field simultaneously. We also note
that, in the process of coherence transfer between two atoms
using, for example, the scheme of Ref. [11], the atoms must
be addressed by the pulses simultaneously, which is basically
a local interaction. In the present protocol, the atoms interact
with the
pulse in a sequential manner. This is essentially a
nonlocal process.
Extending the idea of QST described above to a number
of atoms, we can transfer the state of any atom to the con-
secutive atom. This means that if we consider a sequel of
atoms, then the state of any atom can be transferred to the
consecutive atom which will pass the cavity after the former
leaves the cavity. The procedure of transfer of atomic states
to consecutive atoms has been shown schematically in Fig. 2.
Here the atoms A, B, C, etc. are sent through another iden-
tical bimodal cavity in initial state 0,1. After passing
through this cavity, atom C is again prepared in state i.
Thus, using a second cavity in this way, we can transfer the
state of the first atom A to a third atom C. Clearly, if we used
n number of cavities in this sequence, we could transfer the
state of atom A to the n+1th atom in the sequence.
Effects of decoherence: Fidelity of the QST protocol
Decoherence is a strong limiting factor in the realization
of any quantum computational protocol. The interaction of
the atom and the cavity with the environment causes them to
decay and results in decoherence. Thus, one has to consider
the effect of decoherence to examine with how much effi-
ciency the desired outcome can be produced. These calcula-
tions can be done in the density-matrix framework using the
following Liouville equation:
˙
=−
i
H
eff
,
a
a
a
−2a
a
+
a
a
b
b
b
−2b
b
+
b
b, 12
where
a
and
b
are the decay constants of the two modes
and H
eff
is given by Eq. (2).
In the present case, to investigate the effect of decoher-
ence, let us consider a possible scheme. We consider g and
f to be the Rydberg levels as in Haroche’s experiments. In
that case, we can use a bimodal microwave cavity like the
one used by Haroche’s group. We use parameters similar to
those in the experiments by Haroche and his co-workers. If
the cavity coupling constant g is 2
50 kHz and the cavity
decay constant
a
=
b
=
for each mode is 2
100 Hz,
then
/g=0.002. Further, for =10g, we calculate the cavity
interaction time to be 50
s for a
pulse, which is consis-
tent with the interaction time possible to achieve in a micro-
wave experiment. One sends the atoms with a velocity
10
2
cm s
−1
through a few-cm-long cavity to achieve this
interaction time. Using these parameters, we calculate the
fidelity F that the first step of the evolution (9) occurs. The
variation of FT with the decay constant
is shown in Fig.
3(a), where T is the interaction time of the atom with the
cavity. Note that the probability that the state of atom A is
transferred to the cavity remains more than 90% for
=0.002g. We next show [see Fig. 3(b)] the variation of the
fidelity F2T+
of the entire process (9) to occur with the
time delay
between the atoms A and B for
=0.002g.Itis
clear that the probability that the atom B acquires the desired
state remains above 80% even at g
=20共⬅
63
s.
IV. EXTENSIONS OF QUANTUM STATE
TRANSFER PROTOCOL
A. Quantum networks
Now we show how the above QST protocol can be made
useful in preparing a quantum network, in which long-lived
atomic states are used to communicate between the two
nodes of the network. We assume that there are two identical
two-mode cavities C
1
and C
2
, which are considered as two
nodes of the network. Let us consider that the cavity C
1
is
initially in a state 0,1. To prepare this cavity in a superpo-
sition state,
FIG. 2. Schematic diagram for the QST protocol for a number of
atoms interacting with the two-mode cavity in a sequential manner
for a time T=
/2g
2
.
TRANSFER OF AN UNKNOWN QUANTUM STATE, PHYSICAL REVIEW A 70, 022323 (2004)
022323-3

E
cav
=
0,1
cav
1,0
cav
, 13
we send an atom A in state i through the cavity (see Fig. 4)
such that the atom A experiences a
pulse. Now our goal is
to transfer this cavity state E
cav
to the other node C
2
. For
that we send a second atom B through the cavity C
1
after A
comes out of it. We see that the atom B is prepared in state i
through the evolution (9). This atom is now sent through the
second node C
2
which is initially in state 0,1. In this way,
the state E
cav
of node C
1
is transferred to the node C
2
.
Extending the above idea to a number of distant nodes
(cavities), we thus can transfer the state E
cav
from one node
to another node of the proposed quantum network via a
quantum channel (atom). For example, to send this state
E
cav
from C
2
to another node (say, C
3
), we can send a third
atom C through these two nodes subsequently.
We emphasize that our protocol of quantum networking is
distinct from the teleportation protocol of Davidovich et al.
[12]. Their protocol depends on the Bell state measurements,
whereas in our protocol no Bell measurement is ever made.
We further note that the present scheme can be used to
spread entanglement between two distant cavities. For this,
one first sends an atom A in state g through the first cavity
C
1
prepared initially in the state 1,0 such that the atom
experiences a
/2 pulse 2g
2
T/=
/2. This would prepare
the atom and the cavity in the following entangled state:
AC
1
=
1
2
e
i
/2
共兩g
A
1,0
1
+ f
A
0,1
1
. 14
Next the atom passes through a second cavity C
2
initially in
the state 0,1 and experiences a
pulse. Thus, at the end of
this process, the two cavities are prepared in an entangled
state of two modes as
C
1
C
2
=
1
2
e
i
/2
关兩1,0
1
0,1
2
0,1
1
1,0
2
. 15
Clearly one can spread entanglement between the atom and
the cavity to another distant cavity. Note that in our proposal,
entanglement is created between the modes of the two dif-
ferent cavities. The entanglement between two modes of a
single cavity has been produced in [13].
B. Storage and retrieval of an arbitrary superposition state
of two modes of a cavity
We now discuss how the present
-pulse technique can be
used to prepare an efficient quantum memory for arbitrary
superposition of two cavity modes, where there is only one
photon present in either mode. Let us consider a two-mode
cavity which is in a superposition state of two modes [see
Eq. (13)],
E
cav
=
0,1
cav
1,0
cav
, 16
where
and
are known coefficients. Now we send an atom
in state (8) through the cavity. Applying a
pulse on it, we
can map the superposition of E
cav
into the state of the atom.
This procedure can be written as
i
典兩E
cav
i典兩
cav
, 17
where i=
g+
f and
cav
is given by Eq. (11). Be-
cause, the states g and f of the atom are radiatively long-
lived, information about the state of the cavity can be stored
inside the atom for a sufficiently long time. To retrieve this
information into the cavity, we prepare a second cavity in
either of the states 0,1 or 1,0 and send the atom in state
i through the cavity. Upon applying a
pulse, the cavity
can again be prepared in the superposition state as before.
The retrieval of superposition can be shown as
FIG. 4. Schematic diagram for the quantum network between
distant cavities via the atomic channel. Description of the figure is
in the text.
FIG. 3. (a) Variation of the fidelity FT of mapping the state of
the atom A in the cavity C
1
with
/g. We have assumed that the
cavity decay rates are the same for both the modes and =10g. (b)
Variation of the fidelity F calculated at time 2T+
, with the time
delay
between the atoms for
=0.002g and =10g.
A. BISWAS AND G. S. AGARWAL PHYSICAL REVIEW A 70, 022323 (2004)
022323-4

i典兩0,1
cav
g典兩E
cav
, i典兩1,0
cav
f典兩E
cav
. 18
We should mention here that the quantum memory pro-
posed here for the cavity state is expected to work better
since the information is being stored inside the long-lived
atomic states g and f. However, the transfer time of the
cavity state to the atom is limited by the cavity holding time
and the atom must stop interacting with the cavity before it
decays. We also note that if the two modes are degenerate
and correspond to two states of circular polarizations, then
Eq. (16) can be viewed as a superposition of two polarization
states of a photon. In such a case, our proposal corresponds
to storage and retrieval of the polarization states of a photon.
V. CONCLUSION
In conclusion, we have presented a protocol for the trans-
fer of a quantum state from one atom to another atom. This
protocol can be extended to a number of atoms passing
through sequential cavities and thus one can set up a quan-
tum network. We have further shown how an efficient quan-
tum memory of arbitrary superposition of two cavity modes
can be built up. Our proposals have certain advantages as we
work with long-lived states of atoms. We provide a proper
estimate of the efficiency of the state transfer protocol
against cavity decoherence.
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022323-5
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