We propose a scheme for teleporting an atomic center-of-mass wave function
between distant locations. The scheme uses interactions in cavity quantum
electrodynamics to facilitate a coupling between the motion of an atom trapped
inside a cavity and external propagating light fields. This enables the
distribution of quantum entanglement and the realization of the required
motional Bell-state analysis.Comment: 4 pages, 3 figure

Kimble, H. J.

Parkins, A. S.

English

arXiv

We propose a scheme for teleporting an atomic center-of-mass wave function between distant locations. The scheme uses interactions in cavity quantum electrodynamics to facilitate a coupling between the motion of an atom trapped inside a cavity and external propagating light fields. This enables the distribution of quantum entanglement and the realization of the required motional Bell-state analysis

Caltech Authors - Main

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Proposal for teleportation of the wave function of a massive particle
A.S. Parkins1 and H.J. Kimble2
1Department of Physics, University of Auckland, Auckland, New Zealand
2Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, CA 91125, U.S.A.
(August 9, 2018)
We propose a scheme for teleporting an atomic center-of-
mass wave function between distant locations. The scheme
uses interactions in cavity quantum electrodynamics to facili-
tate a coupling between the motion of an atom trapped inside
a cavity and external propagating light fields. This enables
the distribution of quantum entanglement and the realization
of the required motional Bell-state analysis.
PACS numbers: 03.67.Hk, 42.50.-p, 42.50.Vk
In a landmark work of 1993, Bennett et al. [1] discov-
ered a procedure for teleporting an unknown quantum
state from one location to another. The essential ingre-
dient in their protocol is quantum entanglement of a bi-
partite system shared by the sender, Alice, and receiver,
Bob. This shared entanglement, in unison with suitable
measurements performed by Alice and communicated via
classical channels to Bob, ‘mediates’ the state transfer.
Since the work of Bennett et al., a variety of possible
experimental schemes for the teleportation of quantum
states of two-state systems have been proposed, in large
part by the quantum optics community (see, e.g., [2–5]).
In an exciting recent development, the first experimental
investigations of teleportation of such states have been
performed [6,7] with, in particular, the polarization state
of a photon providing the two-state system of interest.
Complementing this work on two-state systems has
been research into the teleportation of states of infinite-
dimensional systems [8,9], culminating last year in the
experimental demonstration of quantum teleportation of
optical coherent states [10]. This experiment was based
on the specific proposal of [9], utilizing squeezed-state
entanglement and balanced homodyne measurements of
the light fields. Given that the experiment employed only
standard optical elements and measurement techniques,
it offers significant promise of further intriguing possibili-
ties for quantum information processing with continuous
quantum variables, including quantum dense coding [11],
and universal quantum computation [12].
Another burgeoning field of research in quantum infor-
mation science is the implementation of quantum logic
with trapped atoms or ions. Inspired by the proposal of
Cirac and Zoller [13] for a quantum computer based on
the motional and internal degrees of freedom of a collec-
tion of trapped ions, impressive experimental progress
has been made towards controlling quantum proper-
ties in such systems [14,15]. Particular advantages of
trapped atom systems include long coherence times and
the exquisite control of transformations between mo-
tional and internal states.
In view of this tremendous potential of both light- and
motion-based schemes for quantum information process-
ing, it is sensible to investigate possibilities for combining
the two approaches and their distinct advantages. One
particular application would be to quantum networks
for distributed quantum computing and communication,
where, e.g., the ‘distribution’ is accomplished with light
fields [16], while local processing is performed on mo-
tional states of a collection of trapped atoms. Indeed,
with the protocols of [16] in mind, we have recently pro-
posed and analyzed a cavity-QED-based system that en-
ables the transfer of quantum states between the motion
of a trapped atom and propagating light fields [17,18],
which should lead to new capabilities for the synthesis
and control of quantum states for both motion and light.
A particular example from [18] is the possibility of
creating an EPR (Einstein-Podolsky-Rosen) state in
position-momentum for distantly separated atoms. The
creation of such an entangled state between remote par-
ticles suggests an avenue for achieving teleportation of
an unknown wave function of a trapped particle between
the EPR sites. In this letter, we propose and analyze
one such protocol that enables the teleportation of an
unknown one-dimensional atomic center-of-mass wave
function and that should be attainable within the con-
text of emerging experimental capabilities for trapping
atoms in cavity QED [19,20].
Our proposed teleportation scheme is shown schemat-
ically in Fig. 1. Each of Alice (A), Bob (B), and Vic-
tor (V) possess an atom trapped inside an optical cav-
ity. The aim is to teleport the (x-dimension) motional
state of Victor’s atom to Bob’s atom. This is achieved
via the three usual stages for continuous quantum vari-
ables [9,10] – (i) preparation of quantum entanglement
between Alice and Bob’s atoms, (ii) Bell-state (homo-
dyne) measurement by Alice, and (iii) phase-space dis-
placement [D(α∗)] by Bob (given the classical result α of
Alice’s measurement). Finally, the state of Bob’s atom
may be examined by Victor for verification of the qual-
ity of the teleportation (or indeed, physically delivered
to Victor). Stages (i) and (ii), as illustrated in Fig. 1,
employ the cavity-mediated motion-light state transfer
scheme of [17], to which we now turn our attention.
Briefly, a single two-level atom (or ion) is tightly con-
fined in a harmonic trap located inside a high-finesse op-
tical cavity. The atomic transition of frequency ωa is cou-
pled to a single mode of the cavity field of frequency ωc
1
and also to an external (classical) laser field of frequency
ωL and strength EL. The physical setup and excitation
scheme are depicted in Fig. 2(a). The cavity is aligned
along the x-axis, while the field EL is incident from a di-
rection in the y-z plane. Both the cavity field and EL are
far from resonance with the atomic transition, but their
difference frequency is chosen so that they drive Raman
transitions between neighboring motional number states
(i.e., ωc − ωL = νx, with νx the x-axis trap frequency).
A number of assumptions are made in order to achieve
the desired motion-light coupling: (i) Atomic sponta-
neous emission is neglected and the internal atomic dy-
namics adiabatically eliminated. (ii) The size of the har-
monic trap, located at a node of the cavity field, is taken
to be small compared to the optical wavelength (Lamb-
Dicke regime), enabling the approximations sin(kx̂) ≃
ηx(b̂x + b̂
†
x) and EL(ŷ, ẑ, t) ≃ EL(t)e−iφL , where ηx (≪ 1)
is the Lamb-Dicke parameter and x̂ = (h̄/2mνx)
1/2(b̂x +
b̂†x). (iii) The trap frequency νx and cavity field decay
rate κ are assumed to satisfy νx ≫ κ≫ |(g0ηx/∆)EL(t)|,
where g0 is the single-photon atom-cavity mode coupling
strength, and ∆ = ωa − ωL. The first inequality allows a
rotating-wave approximation to be made with respect to
the trap oscillation frequency, while the second inequality
enables an adiabatic elimination of the cavity field mode.
Under these conditions, the motional mode dynamics
in the x direction is well described by the simple quantum
Langevin equation [17]
˙̃bx ≃ −Γ(t)b̃x +
√
2Γ(t) ãin(t) , (1)
where b̃x = e
iνxtb̂x and Γ(t) = [g0ηx|EL(t)|/∆]2/κ.
The operator ãin(t) obeys the commutation relation
[ãin(t), ã
†
in
(t′)] = δ(t−t′) and describes the quantum noise
input to the cavity field (in a frame rotating at the cav-
ity frequency). From the linear nature of (1), it follows
that the statistics of a (continuous) light field incident
upon the cavity can be ‘written onto’ the state of the
atomic motion. This also means that entanglement be-
tween separate light fields can be transferred to entan-
glement between separate motional states, as we discuss
below. From a consideration of the input-output theory
of optical cavities [21], it also follows that measurements
on the cavity output field amount to measurements on
the motion of the atom. In particular, one can show that
ãout(t) ≃ −ãin(t) +
√
2Γ(t) b̃x(t) . (2)
So, for a vacuum input field, homodyne measurements
on the cavity output field realize position or momentum
measurements (or some mixture, depending on the local
oscillator phase) on the trapped atom. This enables the
necessary Bell-state analysis to be performed.
To begin the teleportation procedure, Victor’s atom
is prepared in a particular motional state |φ〉Vx in the
x dimension (e.g., by the techniques of [15]). With the
motion-light coupling switched off in Victor’s cavity (i.e.,
ELV = 0), this state is assumed to remain unchanged until
required by Alice for the Bell-state analysis.
Next, a position-momentum EPR state of Alice and
Bob’s atoms is prepared by using the motion-light cou-
pling described in (1), with input light fields from a non-
degenerate optical parametric amplifier (NOPA). This
preparation, described in detail in [18], is depicted in
Fig. 2(b). The two quantum-correlated output light fields
from a NOPA (operating below threshold) are separated
and made to impinge on Alice and Bob’s cavities, respec-
tively. Assuming ΓA = ΓB = Γ, after a time t ≫ Γ−1,
the following pure entangled motional state is prepared,
|ψ〉AB = SAB(r)|0〉Ax|0〉Bx
= [cosh(r)]
−1
∞
∑
m=0
[− tanh(r)]m |m〉Ax|m〉Bx , (3)
where |m〉A,Bx are Fock states of the motional modes and
SAB(r) = exp[r(b̃Ax b̃Bx − b̃†Axb̃
†
Bx)], with r the ‘entangle-
ment’ parameter. Once this state has been prepared, the
atom-cavity couplings are turned off (ΓA,ΓB → 0), as is
the NOPA pump field. Again, we assume that the en-
tangled state (3) remains unchanged until the next step
in the procedure.
At this stage in the protocol, the total system state is
|ΨI〉 = |φ〉Vx |ψ〉AB . (4)
The Bell-state analysis performed by Alice is depicted
in Fig. 3. At a predetermined time, Victor switches on
his atom-cavity coupling ΓV via ELV(t), thus converting
the state |φ〉Vx to that of a freely propagating field de-
livered to input beam-splitter BS of Alice’s sending sta-
tion. With due accounting for propagation delay, Al-
ice has likewise switched on the coupling ΓA from her
cavity, where, for simplicity, ΓV = ΓA = Γ. Note that
Victor and Alice’s cavities both have vacuum inputs at
this stage. The two cavity output fields are combined by
Alice at the 50/50 beamsplitter BS, the two outputs of
which are incident on homodyne detectors D±. Through
the input-output relation (2), and through the mixing
of the cavity output fields at the beamsplitter, these
detectors effect homodyne measurements on the modes
c̃± = 2
−1/2 (b̃Vx±b̃Ax). The effect of these measurements
is to project the system state onto quadrature eigenstates
of the modes c̃±, given by |χ±〉± = Q†±(χ±)|0〉±, where
Q†±(χ±) = (2π)
−1/4 exp[−(1/2)(c̃†±eiθ± − χ±)2 + χ2±/4],
with θ± the local oscillator (LO) phases [22–24]. In
[23,24], this projection is proved with the assumption
that the local oscillator photon flux matches the tem-
poral shape of the signal flux [which in our case is set by
Γ(t)], while the variable χ is shown to be equivalent to
the integrated homodyne photocurrent.
For the two homodyne measurements we choose LO
phases θ+ = 0 and θ− = π/2. With these choices one
can show that, in terms of the original mode operators,
Q+(χ+)Q−(χ−) = (2π)
−1/2 exp
(
−|α|2/2
)
2
· exp
(
−b̃Vxb̃Ax + αb̃Vx + α∗b̃Ax
)
, (5)
where α = (χ+ + iχ−)/
√
2 . The motional state of Bob’s
atom following the homodyne measurements, with results
χ±, can thus be written
|ϕ〉Bx ∝ Vx〈0|Ax〈0|Q+(χ+)Q−(χ−)|ΨI〉
∝ Vx〈α∗|Ax〈α| exp
(
−b̃Vxb̃Ax
)
|ΨI〉. (6)
Using properties of the squeezing operator SAB(r) [21],
one can further reduce this to
|ϕ〉Bx ∝ Vx〈α∗| exp
[
Λ
(
b̃Vx − α∗
)
b̃†Bx
]
|φ〉Vx|0〉Bx , (7)
where Λ = tanh(r). Expanding |φ〉Vx in terms of the
coherent states, i.e., |φ〉Vx = π−1
∫
d2β Vx〈β|φ〉Vx |β〉Vx,
the right-hand-side of (7) becomes
1
π
∫
d2β Vx〈β|φ〉Vx Vx
〈α∗|β〉Vx
Vx〈Λα∗|Λβ〉Vx
· DB(−Λα∗)DB(Λβ) |0〉Bx , (8)
where DB(β) = exp(βb̃
†
Bx − β∗b̃Bx) is the coherent dis-
placement operator for Bob’s atom. In the limit of strong
squeezing and entanglement (Λ → 1), (8) approaches
DB(−α∗)
1
π
∫
d2β Vx〈β|φ〉Vx |β〉Bx . (9)
That is, |ϕ〉Bx approaches a state which, apart from a
coherent displacement by −α∗, is identical to the initial
motional state (in the x dimension) of Victor’s atom.
Given the measurement results χ±, transmitted to Bob
via a classical channel, the final step in the teleportation
procedure is for Bob to apply a coherent displacement
α∗ (assuming Λ ≃ 1) to the motional state of his atom,
i.e., DB(α
∗)|ϕ〉Bx → |φ〉Bx. In practice, this might be
achieved by applying an electric field (in the case of a
trapped ion) along the x-axis which oscillates at the trap
frequency νx, or, alternatively, by applying off-resonant
laser fields which drive stimulated Raman transitions be-
tween neighboring trap levels [15]. After this, control of
Bob’s atom can be passed to Victor, who is free to con-
firm the overall quality of the teleportation protocol, e.g.,
along the lines analyzed in [25].
Issues of practicality associated with the motion-light
state transfer procedure central to our teleportation
scheme have been discussed elsewhere [17,18]. In brief,
desired conditions are of (i) strong coupling optical cav-
ity QED, such that g20/(κγ) ≫ 1, where γ is the
atomic spontaneous decay rate, and (ii) strong confine-
ment of the atoms with minimal motional-state deco-
herence. Both of these conditions have been achieved
separately [14,15,19,20], and we expect that future ex-
periments trapping single atoms inside optical cavities
will be able to meet these criteria simultaneously. Note
that timescales for motional-state decoherence of trapped
ions can be of the order of milliseconds [15]; the typi-
cal timescale involved in our teleportation scheme, Γ−1,
would likely be of the order of microseconds [17,18]. Fi-
nally, calculations in [9] suggest reasonable (nonclassical)
teleportation fidelities to be possible with values of the
squeezing parameter r > 1.
To conclude, we note that the scheme given here is just
one of a number of possibilities that we have analyzed.
One could, e.g., eliminate Victor’s atom and cavity and,
as the state to be teleported, choose the motional state of
Alice’s atom along an axis orthogonal to the x-axis. Af-
ter preparing the entangled (x-dimension) motional state
|ψ〉AB of Alice and Bob’s atoms, the orthogonal motional
modes of Alice’s atom could be linearly mixed within the
trap itself, in the fashion of a beamsplitter, using suitable
interactions with auxiliary laser fields [26,27], after which
coupling to the cavity field and homodyne measurement
of the output light field would again provide the Bell-
state analysis. In addition, as we will discuss elsewhere
[27], it is possible to eliminate the NOPA from the scheme
and use only trapped atoms interacting with cavity and
laser fields both to produce and distribute the quantum
entanglement required for the teleportation protocol.
ASP gratefully acknowledges support from the Mars-
den Fund of the Royal Society of New Zealand. HJK
is supported by the National Science Foundation, by
DARPA via the QUIC Institute which is administered
by ARO, and by the Office of Naval Research.
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869 (1998).
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3
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tion.
FIG. 1. Schematic of proposed teleportation scheme for
atomic wavepackets. Preparation of motional-state entangle-
ment between Alice and Bob and Bell-state analysis by Alice
are facilitated by cavity-mediated motion-light couplings.
FIG. 2. (a) Proposed setup and excitation scheme for cou-
pling between the motion of a trapped atom and a quantized
optical cavity mode, and thence to a freely propagating ex-
ternal field. The cavity is assumed to be one-sided, i.e., one
mirror is taken to be perfectly reflecting. (b) Preparation of
a position-momentum EPR state of Alice and Bob’s atoms.
The two output fields from a nondegenerate parametric ampli-
fier (NOPA) impinge on Alice and Bob’s cavities, respectively.
Faraday isolators (F) facilitate a unidirectional coupling be-
tween the entangled light source and the atom-cavity systems.
FIG. 3. Schematic of Alice’s Bell-state analysis. The out-
put field representing Victor’s unknown state is combined by
Alice at a 50/50 beamsplitter (BS) with the output field from
her cavity. The resulting output fields from the BS are in-
cident on homodyne detectors D±. The cavity output fields
follow the motional modes, which decay on a timescale Γ−1.
The local oscillator fields (LO±) are pulsed, with temporal
profiles chosen to match that of the cavity output fields.
4
Figure 1
φ
 EPR
 light 
source
Bell-state
 analysis
ALICE
BOB
VICTOR Vx
VICTOR
D(α  )
C
lassical inform
ation (α)
✗✓
*
F
pump
NOPA
Figure 2
F
mode 1
mode 2
ψ
AB
laser
field
cavity
 field
atom trap
|e>
|g>
∆
ν
laser cavity
x
z y
κ
F
aout
x
ALICE
BOB
(a)
(b)
ain
εL
 motional 
excitations
time
integrated
  signals
(2Γ)–1
–
χ+
LO+
D–
aout
V
Figure 3
BS
LO–
D+
φVICTOR Vx
ALICE
aout
A


Proposal for teleportation of the wave function of a massive particle

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