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

ជ

1

·S

ជ

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 兩e典↔ 兩g典 and 兩e典↔兩f典

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 = ប关

eg

兩e典具e兩 +

fg

兩f典具f兩 +

1

a

†

a +

2

b

†

b + 兵g

1

兩e典具g兩a

+ g

2

兩e典具f兩b + H.c.其兴, 共1兲

where

lg

共l苸 e, f兲 is the atomic transition frequency,

i

共i

苸 1,2兲 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 兩e典 in the large detuning do-

main, we derive an effective Hamiltonian describing the sys-

tem of Fig. 1,

H

eff

=−

បg

2

⌬

关兩g典具g兩a

†

a + 兩f典具f兩b

†

b兴

−

បg

2

⌬

关兩g典具f兩a

†

b + 兩f典具g兩ab

†

兴, 共2兲

where ⌬ =

eg,f

−

1,2

is the common one-photon detuning of

the cavity modes and g

1

=g

2

=g共Ⰶ⌬兲. 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 兩g典 and 兩f典 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 兩g典兩n,

典 and 兩f典兩n−1,

+1典 in

the state vector

兩

共t兲典 = d

g

共t兲兩g,n,

典 + d

f

共t兲兩f,n −1,

+1典共3兲

are given by

d

g

共t兲 =

冑

nXY

n +

+1

+ d

g

共0兲,

d

f

共t兲 =

冑

+1XY

n +

+1

+ d

f

共0兲, 共4兲

where X=

冑

nd

g

共0兲+

冑

+1d

f

共0兲, Y =exp关ig

2

共n+

+1兲t/⌬兴

−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

+

= 兩f典具g兩, S

−

= 兩g典具f兩, S

z

=

1

2

共兩f典具f兩 − 兩g典具g兩兲;

R

+

= a

†

b, R

−

= ab

†

, R

z

=

1

2

共a

†

a − b

†

b兲. 共5兲

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

⬅ −

បg

2

⌬

共R

+

S

−

+ R

−

S

+

−2R

z

S

z

兲. 共6兲

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,

兩i典

A

=

␣

兩g典

A

+

兩f典

A

, 共7兲

where

␣

and

are unknown arbitrary coefﬁcients. The state

兩i典

A

of atom A is to be transferred to another atom B which is

elsewhere. Preparing the cavity in a state 兩0,1典 (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

兩i

⬘

典 =

␣

⬘

兩g典 +

⬘

兩f典共8兲

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:

兩i典

A

兩0,1典

↓

pulse on atom A

兩g典

A

共

␣

兩0,1典 −

兩1,0典兲

↓ B atom enters

兩g典

A

兩i

⬘

典

B

共

␣

兩0,1典 −

兩1,0典兲

↓

pulse on atom B

兩g典

A

兩i典

B

共

␣

⬘

兩0,1典 −

⬘

兩1,0典兲. 共9兲

FIG. 1. Three-level atomic conﬁguration with levels 兩g典, 兩e典, and

兩f典 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 兩1,0典, then

following a similar sequence to the above, the ﬁnal state will

be −兩f典

A

兩i典

B

共

␣

⬘

兩0,1典−

⬘

兩1,0典兲. Note that atom B has already

acquired the state 兩i典 of atom A, i.e., the state 兩i典 is transferred

from atom A to atom B.

More generally, our QST protocol can be written as

兩i典

A

兩i

⬘

典

B

共

␥

兩0,1典 +

␦

兩1,0典兲

cav

→

U共

兲

共

␥

兩g典 −

␦

兩f典兲

A

兩i典

B

兩

典

cav

,

共10兲

where U共

兲=U

A

共

兲U

B

共

兲, U

k

共

兲 (k苸 A,B) 关=exp

兵−iH

eff

T/ប其兴 denotes the

-pulse operation on the atom k,

and

兩

典

cav

=

␣

⬘

兩0,1典

cav

−

⬘

兩1,0典

cav

. 共11兲

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 兩0,1典 and 兩1,0典. 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 兩g典 (or 兩f典)

and the cavity is initially in state 兩0,1典 (or 兩1,0典), 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 兩0,1典. After passing

through this cavity, atom C is again prepared in state 兩i典.

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 共n+1兲th 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

ប

关H

eff

,

兴 −

a

共a

†

a

−2a

a

†

+

a

†

a兲

−

b

共b

†

b

−2b

b

†

+

b

†

b兲, 共12兲

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 兩g典 and

兩f典 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 F共T兲 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 F共2T+

兲 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兲.

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 兩0,1典. 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

兩E典

cav

=

␣

兩0,1典

cav

−

兩1,0典

cav

, 共13兲

we send an atom A in state 兩i典 through the cavity (see Fig. 4)

such that the atom A experiences a

pulse. Now our goal is

to transfer this cavity state 兩E典

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 兩i典

through the evolution (9). This atom is now sent through the

second node C

2

which is initially in state 兩0,1典. In this way,

the state 兩E典

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 兩E典

cav

from one node

to another node of the proposed quantum network via a

quantum channel (atom). For example, to send this state

兩E典

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 兩g典 through the ﬁrst cavity

C

1

prepared initially in the state 兩1,0典 such that the atom

experiences a

/2 pulse 共2g

2

T/⌬=

/2兲. This would prepare

the atom and the cavity in the following entangled state:

兩⌿典

AC

1

=

1

冑

2

e

i

/2

共兩g典

A

兩1,0典

1

+ 兩f典

A

兩0,1典

1

兲. 共14兲

Next the atom passes through a second cavity C

2

initially in

the state 兩0,1典 and experiences a

pulse. Thus, at the end of

this process, the two cavities are prepared in an entangled

state of two modes as

兩⌿典

C

1

C

2

=

1

冑

2

e

i

/2

关兩1,0典

1

兩0,1典

2

− 兩0,1典

1

兩1,0典

2

兴. 共15兲

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)],

兩E典

cav

=

␣

兩0,1典

cav

−

兩1,0典

cav

, 共16兲

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 兩E典

cav

into the state of the atom.

This procedure can be written as

兩i

⬘

典兩E典

cav

→ − 兩i典兩

典

cav

, 共17兲

where 兩i典=

␣

兩g典+

兩f典 and 兩

典

cav

is given by Eq. (11). Be-

cause, the states 兩g典 and 兩f典 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 兩0,1典 or 兩1,0典 and send the atom in state

兩i典 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 F共T兲 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

兩i典兩0,1典

cav

→ 兩g典兩E典

cav

, 兩i典兩1,0典

cav

→ − 兩f典兩E典

cav

. 共18兲

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 兩g典 and 兩f典. 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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