VOLUME 78, NUMBER 16 PHYSICAL REVIEW LETTERS 21APRIL 1997
Quantum State Transfer and Entanglement Distribution among Distant Nodes
in a Quantum Network
J. I. Cirac,
1,2
P. Zoller,
1,2
H. J. Kimble,
1,3
and H. Mabuchi
1,3
1
Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, California 93106-4030
2
Institut für Theoretische Physik, Universität Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria
3
Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, California 91125
(
Received 12 November 1996)
We propose a scheme to utilize photons for ideal quantum transmission between atoms located at
spatially separated nodes of a quantum network. The transmission protocol employs special laser
pulses that excite an atom inside an optical cavity at the sending node so that its state is mapped into
a time-symmetric photon wave packet that will enter a cavity at the receiving node and be absorbed by
an atom there with unit probability. Implementation of our scheme would enable reliable transfer or
sharing of entanglement among spatially distant atoms. [S0031-9007(97)02983-9]
PACS numbers: 89.70.+c, 03.65.Bz, 42.50.Lc
We consider a quantum network consisting of spatially
separated nodes connected by quantum communication
channels. Each node is a quantum system that stores quan-
tum information in quantum bits and processes this in-
formation locally using quantum gates [1]. Exchange of
information between the nodes of the network is accom-
plished via quantum channels. A physical implementa-
tion of such a network could consist, e.g., of clusters of
trapped atoms or ions representing the nodes, with opti-
cal fibers or similar photon “conduits” providing the quan-
tum channels. Atoms and ions are particularly well suited
for storing qubits in long-lived internal states, and recently
proposed schemes for performing quantum gates between
trapped atoms or ions provide an attractive method for lo-
cal processing within an atomyion node [2–4]. On the
other hand, photons clearly represent the best qubit carrier
for fast and reliable communication over long distances
[5,6], since fast and internal-state-preserving transportation
of atoms or ions seems to be technically intractable.
To date, no process has actually been identified for
using photons (or any other means) to achieve efficient
quantum transmission between spatially distant atoms [7].
In this Letter we outline a scheme to implement this basic
building block of communication in a distributed quantum
network. Our scheme allows quantum transmission with
(in principle) unit efficiency between distant atoms 1 and
2 (see Fig. 1). The possibility of combining local quan-
tum processing with quantum transmission between the
nodes of the network opens the possibility for a variety
of novel applications ranging from entangled-state cryp-
tography [8], teleportation [9], and purification [10], and
is interesting from the perspective of distributed quantum
computation [11].
The basic idea of our scheme is to utilize strong coupling
between a high-Q optical cavity and the atoms [5] forming
a given node of the quantum network. By applying laser
beams, one first transfers the internal state of an atom
at the first node to the optical state of the cavity mode.
The generated photons leak out of the cavity, propagate
as a wave packet along the transmission line, and enter
an optical cavity at the second node. Finally, the optical
state of the second cavity is transferred to the internal state
of an atom. Multiple-qubit transmissions can be achieved
by sequentially addressing pairs of atoms (one at each
node), as entanglements between arbitrarily located atoms
are preserved by the state-mapping process.
The distinguishing feature of our protocol is that by
controlling the atom-cavity interaction, one can absolutely
avoid the reflection of the wave packets from the second
cavity, effectively switching off the dominant loss channel
that would be responsible for decoherence in the commu-
nication process. For a physical picture of how this can
be accomplished, let us consider that a photon leaks out of
an optical cavity and propagates away as a wave packet.
Imagine that we were able to “time reverse” this wave
packet and send it back into the cavity; then this would
restore the original (unknown) superposition state of the
atom, provided we would also reverse the timing of the
laser pulses. If, on the other hand, we are able to drive
the atom in a transmitting cavity in such a way that the
outgoing pulse were already symmetric in time, the wave
packet entering a receiving cavity would “mimic” this time
reversed process, thus “restoring” the state of the first atom
in the second one.
The simplest possible configuration of quantum trans-
mission between two nodes consists of two atoms 1 and
2 which are strongly coupled to their respective cavity
modes (see Fig. 1). The Hamiltonian describing the inter-
action of each atom with the corresponding cavity mode
FIG. 1. Schematic representation of unidirectional quantum
transmission between two atoms in optical cavities connected
by a quantized transmission line (see text for explanation).
0031-9007y97y78(16)y3221(4)$10.00 © 1997 The American Physical Society 3221
VOLUME 78, NUMBER 16 PHYSICAL REVIEW LETTERS 21APRIL 1997
is ( ¯h 1)
ˆ
H
i
v
c
ˆa
y
i
ˆa
i
1v
0
jrl
ii
krj 1 gsjrl
ii
kgjˆa
i
1 H.c.d 1
1
2
3V
i
stdfe
2ifv
L
t1f
i
stdg
jrl
ii
kej 1 H.c.gsi1, 2d .
(1)
Here, ˆa
i
is the destruction operator for cavity mode i with
frequency v
c
, jgl, jrl, and jel form a three-level system
of excitation frequency v
0
(Fig. 1), and the qubit is stored
in a superposition of the two degenerate ground states.
The states jel and jgl are coupled by a Raman transition
[3,4,12], where a laser of frequency v
L
excites the atom
from jel to jrl with a time-dependent Rabi frequency
V
i
std and phase f
i
std, followed by a transition jrl !
jel which is accompanied by emission of a photon into
the corresponding cavity mode, with coupling constant
g. In order to suppress spontaneous emission from the
excited state during the Raman process, we assume that
the laser is strongly detuned from the atomic transition
jDj¿V
1,2
std, g, j
Ù
f
1,2
j (with D v
L
2v
0
). In such
a case, one can eliminate adiabatically the excited states
jrl
i
. The new Hamiltonian for the dynamics of the two
ground states becomes, in a rotating frame for the cavity
modes at the laser frequency,
ˆ
H
i
2dˆa
y
i
ˆa
i
1
g
2
D
ˆa
y
i
ˆa
i
jgl
ii
kgj 1dv
i
stdjel
ii
kej
2 ig
i
stdfe
if
i
std
jel
ii
kgja
i
2 H.c.gsi1, 2d .
(2)
The first term involves the Raman detuning d v
L
2
v
c
. The next two terms are ac-Stark shifts of the ground
states jgl and jel due to the cavity mode and laser
field, respectively, with dv
i
std V
i
std
2
ys4Dd. The last
term is the familiar Jaynes-Cummings interaction, with
an effective coupling constant g
i
std gV
i
stdys2Dd [13].
The notation jel as “excited” and jgl as “ground” state is
motivated by this analogy.
Our goal is to select the time-dependent Rabi frequen-
cies and laser phases [14] to accomplish the ideal quantum
transmission
sc
g
jgl
1
1 c
e
jel
1
d jgl
2
≠j0l
1
j0l
2
jvacl
!jgl
1
sc
g
jgl
2
1c
e
jel
2
d≠j0l
1
j0l
2
jvacl , (3)
where c
g,e
are complex numbers; in general, they have
to be replaced by unnormalized states of other “specta-
tor” atoms in the network. In (3), j0l
i
and jvacl represent
the vacuum state of the cavity modes and the free elec-
tromagnetic modes connecting the cavities. Transmission
will occur by photon exchange via these modes.
In a quantum stochastic description employing the
input-output formalism the cavity mode operators obey
the quantum Langevin equations [15]:
d ˆa
i
dt
2ifˆa
i
,
ˆ
H
i
stdg 2kˆa
i
2
p
2kˆa
sid
in
st dsi1, 2d .
(4)
The first term on the right-hand side (RHS) of this equa-
tion gives the systematic evolution due to the interaction
with the atom, while the last two terms correspond to pho-
ton transmission through the mirror with loss rate k, and
(white) quantum noise of the vacuum field incident on the
cavity i, respectively. The output of each cavity is given
by the equation [15]
ˆa
sid
out
std ˆa
sid
in
st d 1
p
2k ˆa
i
st d , (5)
which expresses the outgoing field at the mirror as a sum
of the incident field plus the field radiated from the cavity.
The output field of the first cavity constitutes the input
for the second cavity with an appropriate time delay, i.e.,
ˆa
s2d
in
st d ˆa
s1d
out
st 2td, where t is a constant related to
retardation in the propagation between the mirrors. The
output field of the second cavity is, therefore,
ˆa
s2d
out
std ˆa
s1d
in
st 2td1
p
2kfˆa
1
st2td1 ˆa
2
stdg . (6)
Introducing this relation in Eq. (4) we obtain
d ˆa
1
dt
2 ifˆa
1
,
ˆ
H
1
stdg 2kˆa
1
2
p
2kˆa
s1d
in
st d , (7a)
d ˆa
2
dt
2 ifˆa
2
,
ˆ
H
2
stdg 2kˆa
2
22kˆa
1
st2td
2
p
2kˆa
s1d
in
st 2td. (7b)
Note that the first equation is decoupled from the sec-
ond one; i.e., we consider here only a unidirectional cou-
pling between the cavities (see Fig. 1) [16]. The present
model is a particular example of a cascaded quantum sys-
tem and can be described within the formalism developed
by Gardiner and Carmichael [17,18]. We can eliminate
the time delay t in these equations by defining “time
delayed” operators for the first system (atom 1 cavity),
e.g., ˜astd; ˆast2td, etc.; in a similar way we rede-
fine the Rabi frequency
˜
V
1
std V
1
st 2td, and phase
˜
f
1
std f
1
st 2td. In the following we will assume that
we have performed these transformations, and for simplic-
ity of notation we will omit the tilde. This amounts to
setting t ! 0 in all these equations. Equations (7a) and
(7b) have to be solved with the corresponding equations
for the atomic operators and with the condition that the
field incident on the first cavity is in the vacuum state,
i.e., ˆa
s1d
in
st d jC
0
l 0 ;t .
In the present context, it is convenient to adopt the
language of quantum trajectories [18,19]. Let us consider
a fictitious experiment where the output field of the second
cavity is continuously monitored by a photodetector
(see Fig. 1). The evolution of the quantum system
under continuous observation, conditional to observing a
particular trajectory of counts, can be described by a pure
state wave function jC
c
stdl in the system Hilbert space
(where the radiation modes outside the cavity have been
eliminated). During the time intervals when no count
is detected, this wave function evolves according to a
Schrödinger equation with the non-Hermitian effective
Hamiltonian
ˆ
H
eff
st d
ˆ
H
1
st d 1
ˆ
H
2
st d 2 iks ˆa
y
1
ˆa
1
1 ˆa
y
2
ˆa
2
1 2ˆa
y
2
ˆa
1
d.
(8)
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VOLUME 78, NUMBER 16 PHYSICAL REVIEW LETTERS 21APRIL 1997
The detection of a count at time t
r
is associated with a
quantum jump according to jC
c
st
r
1 dtdl ~ ˆcjC
c
st
r
dl,
where ˆc ˆa
1
1 ˆa
2
[17,19]. The probability density for
a jump (detector click) to occur during the time interval
from t to t 1 dt is kC
c
stdjˆc
y
ˆcjC
c
stdldt [17,19].
We wish to design the laser pulses in both cavities in
such a way that ideal quantum transmission condition (3)
is satisfied. A necessary condition for the time evolution is
that a quantum jump (detector click; see Fig. 1) never oc-
curs, i.e., ˆcjC
c
stdl 0 ;t, and thus the effective Hamil-
tonian will become a Hermitian operator. In other words,
the system will remain in a dark state of the cascaded quan-
tum system. Physically, this means that the wave packet is
not reflected from the second cavity. We expand the state
of the system as
jC
c
stdl c
g
jggl j00l 1 c
e
fa
1
stde
2if
1
std
jegl j00l 1a
2
stde
2if
2
std
jgel j00l 1b
1
stdjggl j10l 1b
2
stdjggl j01lg . (9)
Ideal quantum transmission (3) will occur for
a
1
s2`d a
2
s1`d 1, f
1
s2`d f
2
s1`d 0.
(10)
The first term on the RHS of (9) does not change
under the time evolution generated by H
eff
. Defining
symmetric and antisymmetric coefficients b
1,2
sb
s
7
b
a
dy
p
2, we find the following evolution equations
Ù
a
1
std g
1
stdb
a
stdy
p
2, (11a)
Ù
a
2
std 2g
2
stdb
a
stdy
p
2, (11b)
Ù
b
a
std 2g
1
stda
1
stdy
p
2 1 g
2
stda
2
stdy
p
2, (11c)
where we have chosen the laser frequencies v
L
1
Ù
f
1,2
std
so that d g
2
yD and
Ù
f
1,2
std dv
i
std (12)
in order to compensate the ac-Stark shifts; thus Eqs. (11a),
(11b), and (11c) are decoupled from the phases. The dark
state condition implies b
s
std 0, and therefore
Ù
b
s
std g
1
stda
1
stdy
p
2 1 g
2
stda
2
stdy
p
2 1kb
a
std;0,
(13)
as well as the normalization condition
ja
1
stdj
2
1 ja
2
stdj
2
1 jb
a
stdj
2
1. (14)
We note that the coefficients a
1,2
std and b
s
std are real.
The mathematical problem is now to find pulse shapes
V
1,2
std ~ g
1,2
std such that the conditions (10), (11a),
(11b), (11c), and (13) are fulfilled. In general this is a
difficult problem, as imposing conditions (10) and (13) on
the solutions of the differential equations [(11a), (11b),
and (11c)] give functional relations for the pulse shape
whose solution are not obvious. We shall construct a
class of solutions guided by the physical expectation that
the time evolution in the second cavity should reverse the
time evolution in the first one. Thus, we look for solutions
satisfying the symmetric pulse condition
g
2
std g
1
s2tds;td. (15)
This implies a
1
std a
2
s2td, and b
a
std b
a
s2td. The
latter relation leads to a symmetric shape of the photon
wave packet propagating between the cavities.
Suppose that we specify a pulse shape V
1
std ~ g
1
std for
the second half of the pulse in the first cavity (t $ 0) [20].
We wish to determine the first half V
1
s2td ~ g
1
s2td (for
t . 0), such that the conditions for ideal transmission (3)
are satisfied. From (13) and (10) we have
g
1
s2td 2
p
2 kb
a
std 1 g
1
stda
1
std
a
2
std
st . 0d .
(16)
Thus, the pulse shape is completely determined provided
we know the system evolution for t $ 0. However, a
difficulty arises when we try to find this evolution, since
it depends on the yet unknown g
2
std g
1
s2td for t . 0
[see Eqs. (11a), (11b), and (11c)]. In order to circumvent
this problem, we use (13) to eliminate this dependence in
Eqs. (11a) and (11c). This gives
Ù
a
1
std g
1
stdb
a
stdy
p
2, (17a)
Ù
b
a
std 2kb
a
std 2
p
2 g
1
stda
1
std (17b)
for t $ 0. These equations have to be integrated with the
initial conditions
a
1
s0d
∑
2k
2
g
1
s0d
2
1k
2
∏1
2
, (18a)
b
a
s0d f1 2 2a
1
s0d
2
g
1
2
, (18b)
which follow immediately from a
1
s0d a
2
s0d, and (14)
and (13) at t 0. Given the solution of Eqs. (17a) and
(17b), we can determine a
2
std from the normalization (14).
In this way, the problem is solved since all the quantities
appearing on the RHS of Eq. (16) are known for t $ 0.
It is straightforward to find analytical expressions for the
pulse shapes, for example, by specifying V
1
std const
for t . 0, as will be done in the following.
As an illustration, we have numerically integrated the
full time-dependent Schrödinger equation with the ef-
fective Hamiltonian (8). The results are displayed in
Fig. 2(a). We have used a pulse shape calculated using
the above procedure, with g
1
std 2dv
1
std k ; const
for t . 0 [see Fig. 2(b)]. As Fig. 2 shows, the quantum
transmission is ideal.
In practice there will be several sources of imperfec-
tions. First, there is the possibility of spontaneous emis-
sion from the excited state during the Raman pulses. Its
effects can be accounted for in the effective Hamiltonian
(8) by the replacement D ! D1iGy2, where G is the
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VOLUME 78, NUMBER 16 PHYSICAL REVIEW LETTERS 21APRIL 1997
FIG. 2. Populations a
1,2
std
2
and b
a
std
2
for the ideal trans-
mission pulse g
1
std g
2
s2td given in the inset, specified by
g
1
st $ 0d 2dv
1
st $ 0d k const.
decay rate from level jrl. If we denote by t [ømax 3
s1yk,1yg
1,2
d] the effective pulse duration, the probability
for a spontaneous emission is of the order of GsV
2
1,2
1
4g
2
dys8D
2
dt ø 1. For g
1
ø k this probability scales like
1yD, so that the effects of spontaneous emission are sup-
pressed for sufficiently large detunings. A second source
of decoherence will be losses in the mirror and during
propagation. They can be taken into account by adding a
term 2ik
0
s ˆa
y
1
ˆa
1
1 ˆa
y
2
ˆa
2
d in H
eff
(8), where k
0
is the ad-
ditional loss rate. Typically, we expect k
0
ø k. Never-
theless, one can overcome the effects of photon losses by
error correction [21]. We have included these imperfec-
tions in our numerical simulations. Figure 3 shows the
probability of a faithful transmission F as a function of
k
0
yk for different values of GyD for the same parameters
and pulse shapes as in Fig. 2.
In conclusion, we have proposed for the first time a pro-
tocol to accomplish ideal quantum transmission between
FIG. 3. Fidelity of transmission F including the effects of
mirror losses and spontaneous emission as a function of k
0
yk
for GyD 0, 0.01, and 0.05 (solid, dashed, and dot-dashed
lines, respectively). Other parameters are as in Fig. 2.
two nodes of a quantum network. Our scheme has been
tailored to a potential network implementation in which
trapped atoms or ions constitute the nodes, and photon
transmission lines provide communication channels be-
tween them. Extensions of the present scheme will be
presented elsewhere [11], including error correction and
new quantum gates in cavity quantum electrodynamics.
We thank the members of the ITP program Quantum
Computers and Quantum Coherence for discussions. This
work was supported in part by the Österreichischer Fonds
zur Förderung der wissenschaftlichen Forschung, by the
European TMR network ERB4061PL95-1412, by NSF
PHY94-07194 and PHY-93-13668, by DARPAyARO
through the QUIC program, and by the ONR.
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transmission of (correlated) photon states is achieved
(see Ref. [8]). In contrast, here we are interested in the
interface between atoms and photons.
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[13] We ignore for the moment the small effects produced
by spontaneous emission during the Raman process. Its
effects will be studied in the context of Fig. 3.
[14] One could also modulate the cavity transmission, but this
is technically more difficult.
[15] C. W. Gardiner, Quantum Noise (Springer-Verlag, Berlin,
1991).
[16] In a perfect realization of the present scheme no light field
will be reflected from the second mirror, and therefore the
assumption of unidirectional propagation is not needed.
[17] C. W. Gardiner, Phys. Rev. Lett. 70, 2269 (1993).
[18] H. J. Carmichael, Phys. Rev. Lett. 70, 2273 (1993).
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Summer School, edited by E. Giacobino et al. (Elsevier,
New York, to be published).
[20] V
1
std has to be such that a
1
s`d 0. This is fulfilled if
V
1
s`d . 0, which also guarantees that the denominator in
(16) does not vanish for t . 0.
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