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arXiv:quant-ph/0611128v1 11 Nov 2006

Dual species matter qubit entangled with light

S.-Y. Lan, S. D. Jenkins,∗T. Chaneli` ere,⋆D. N. Matsukevich,†

C. J. Campbell, R. Zhao, T. A. B. Kennedy, and A. Kuzmich

School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430

(Dated: February 1, 2008)

We propose and demonstrate an atomic qubit based on a cold85Rb-87Rb isotopic mixture, entan-

gled with a frequency-encoded optical qubit. The interface of an atomic qubit with a single spatial

light mode, and the ability to independently address the two atomic qubit states, should provide

the basic element of an interferometrically robust quantum network.

PACS numbers: 42.50.Dv,03.65.Ud,03.67.Mn

Quantum mechanics permits the secure communica-

tion of information between remote parties [1, 2, 3]. How-

ever, direct optical fiber based quantum communication

over distances greater than about 100 km is challenging

due to intrinsic fiber losses. To overcome this limitation it

is necessary to take advantage of quantum state storage

at intermediate locations on the transmission channel.

Interconversion of the information from light to matter

to light is therefore essential.

interface photonic communication channels and storage

elements that lead to the proposal of the quantum re-

peater as an architecture for long-distance distribution

of quantum information via qubits [4, 5].

Recently there has been rapid progress in interfacing

photonic and stored atomic qubits. Two-ensemble encod-

ing of matter qubits was used to achieve entanglement of

photonic and atomic rubidium qubits and quantum state

transfer from matter to light [6]. This was followed by

a more robust single-ensemble qubit encoding [7], which

led to full light-matter-light qubit interconversion and

entanglement of two remote atomic qubits [8].

recently, both two-ensemble and single-ensemble atomic

qubits were reported using cesium gas [9, 10].

To realize scalable long distance qubit distribution

telecommunication-wavelength photons and long-lived

quantum memory elements are required [11]. Although

multiplexing of atomic memory elements vastly improves

the dependence of entanglement distribution on storage

lifetime [12], there remains, the problem of robust atomic

and photonic qubits for long-distance communication.

Two-ensemble encoding suffers from the problem of long-

term interferometric phase stability, while qubit states

encoded in a single ensemble are hard to individually ad-

dress.

A protocol for implementing entanglement distribution

with an atomic ensemble-based quantum repeater has

been proposed [5]. It involves generating and transmit-

ting each of the qubit basis states individually, in prac-

tice via two interferometrically separate channels. Under

prevailing conditions of low overall efficiencies it provides

improved scaling compared to direct qubit entanglement

distribution [4]. Its disadvantage is the necessity to sta-

bilize the length of both transmission channels to a small

It was the necessity to

More

FIG. 1: (a) Schematic shows two orthogonal qubit states (ar-

rows) encoded in two atomic ensembles coupled to distinct

spatial light modes [5, 6], (b) the proposed architecture en-

codes the states in a two species atomic mixture, coupled to

a single light mode.

fraction of the optical wavelength, as the distribution of

qubit entanglement is sensitive to the relative phase fluc-

tuations in the two arms.

In this Letter we propose an interferometrically robust

quantum repeater element based on entangled mixed

species atomic, and frequency-encoded photonic, qubits,

Fig. 1. This avoids the use of two interferometrically

separate paths for qubit entanglement distribution. The

qubit basis states are encoded as single spin wave excita-

tions in each one of the two atomic species co-trapped in

the same region of space. The spectroscopically resolved

transitions enable individual addressing of the atomic

species. Hence one may perform independent manipu-

lations in the two repeater arms which share a single

mode transmission channel. Phase stability is achieved

by eliminating the relative ground state energy shifts of

the co-trapped atomic species, as is in any case essential

to successfully read out an atomic excitation [13].

We consider a co-trapped isotope mixture of85Rb and

87Rb, containing, respectively, N85and N87atoms cooled

in a magneto optical trap, as shown in Fig. 2. Unpo-

larized atoms of isotope ν (ν ∈ {85,87}) are prepared

in the ground hyperfine level

???5S1/2,F(85)

??a(ν)?, where

??a(85)?

≡

a

= 3

?

,

??a(87)?

≡

???5S1/2,F(87)

a

= 2

?

, and

F(ν)

f

is the total atomic angular momentum for level

Page 2

2

FIG. 2: Schematic of the experimental set-up showing the

geometry of the addressing and scattered fields from the co-

trapped isotope mixture of

laser fields generate signal and idler fields, respectively de-

tected at D1 and D2; E1, E2 are optical frequency filters.

PM1-4 are light phase modulators, φs and φi are relative

phases of the driving rf fields, see text for details.

85Rb-87Rb. The write and read

??f(ν)?.

with energies ?ω(ν)

??b(ν)?

??5P1/2

polarization eH = ˆ z and temporal profile ϕ(t) (normal-

ized to unity

?dt |ϕ(t)|2= 1) impinges on an electro-

quencies ck(85)

w

= ckw+ δωw and ck(87)

(δωw = 531.5 MHz) nearly resonant on the respective

isotopic D1(??a(ν)?↔

Raman scattering of the write fields results in signal pho-

tons with frequencies ck(ν)

s

= ck(ν)

??b(ν)?↔

ν with vertical polarization eV is given by

We consider the Raman configuration with

ground levels

??a(ν)?

corresponds to the ground hyperfine level with

smaller angular momentum, while level

?hyperfine level with F(ν)

and

??b(ν)?

and excited level

??c(ν)?

a , ?ω(ν)

b, and ?ω(ν)

c

respectively. Level

??c(ν)?

is the

c

= F(ν)

a . A 150 ns long

write laser pulse of wave vector kw = kwˆ y, horizontal

optic modulator (EOM), producing sidebands with fre-

w

= ckw− δωw

??c(ν)?) transitions with detunings

w +(ω(ν)

b

??c(ν)?transitions. The positive frequency com-

∆ν = ck(ν)

w − (ω(ν)

c

− ω(ν)

a ) ≈ −10 MHz. Spontaneous

−ω(ν)

a ) on the

ponent of the detected signal electric field from isotope

ˆE

(ν)(+)(r,t) =

?

×us(r)ˆψ(ν)

?k(ν)

s

2ǫ0

e−ick(ν)

s

(t−ˆk(ν)

s

·r)

s (t −ˆks· r)eV, (1)

where us(r) is the transverse spatial profile of the

signal field (normalized to unity in its transverse

plane), and

ˆψ(ν)

s (t) is the annihilation operator for

the signal field.These operators obey the usual free

field, narrow bandwidth bosonic commutation relations

[ˆψ(ν)

s

(t′)] = δν,ν′δ(t − t′). The emission of V -

polarized signal photons creates correlated atomic spin-

wave excitations with annihilation operators given by

s (t),ˆψ(ν′)†

ˆ s(ν)= cosθνˆ s(ν)

−1− sinθνˆ s(ν)

+1,(2)

where

cos2θν=

F(ν)

a ?

m=−F(ν)

a

X(ν)2

m,−1/

?

α=±1

F(ν)

a ?

m=−F(ν)

a

X(ν)2

m,α,

X(ν)

Gordan coefficients, and the spherical vector components

of the spin wave are given by

m,α ≡ CF(ν)

a

1 F(ν)

c

m 0 m

CF(ν)

m−α α m

b

1 F(ν)

c

is a product of Clebsch-

ˆ s(ν)

α

=

F(ν)

a ?

m=−F(ν)

a

X(ν)

m,α

??F(ν)

a

m=−F(ν)

a

???X(ν)

m,α

???

2ˆ s(ν)

m,α. (3)

The spin wave Zeeman components of isotope ν are

given in terms of the µ-thνRb atom transition opera-

tors σa(ν),m; b(ν),m′ and the write uw(r) and signal us(r)

field spatial profiles

ˆ s(ν)

m,α= iA(ν)

?

(2F(ν)

a

+ 1)

Nν

Nν

?

µ

σµ

a(ν),m; b(ν),m

×ei(k(ν)

s

−k(ν)

w )·rµus(rµ)u∗

w(rµ).(4)

The effective overlap of the write beam and the detected

signal mode [14] is given by

A(ν)=

??

d3r|us(r)u∗

w(r)|2n(ν)(r)

Nν

?−1/2

, (5)

where n(ν)(r) is the number density of isotope ν. The

interaction responsible for scattering into the collected

signal mode is given by

ˆHs(t) = i?χϕ(t)

?

cosηˆψ(85)†

s

(t)ˆ s(85)†

+sinηˆψ(87)†

s

(t)ˆ s(87)†?

+ h.c., (6)

where χ ≡

rameter,

?χ2

85+ χ2

87is a dimensionless interaction pa-

χν≡

√2d(ν)

A(ν)∆ν

cbd(ν)

ca

k(ν)

(2F(ν)

s k(ν)

w n(ν)

w Nν

+ 1)?ǫ0

a

?

?

?

?

?

?

α=±1

F(ν)

a ?

m=−F(ν)

a

???X(ν)

is the

m,α

???

2

,

(7)

d(ν)

average number of photons in the write pulse sideband

with frequency ck(ν)

w , and the parametric mixing angle

η is given by cos2η = χ2

tion picture Hamiltonian also includes terms represent-

ing Rayleigh scattering and Raman scattering into unde-

tected modes. One can show, however, that these terms

commute with the signal Hamiltonian (Eq. (6)) and with

the operatorsˆψ(ν)

result, the interaction picture density operator for the

ca and d(ν)

cb

are reduced matrix elements, n(ν)

w

85/(χ2

85+ χ2

87). The interac-

s (t) and ˆ s(ν)to order O(1/√N). As a

Page 3

3

signal-spin wave system (tracing over undetected field

modes) is given byˆUˆ ρ0ˆU†, where ˆ ρ0 is the initial den-

sity matrix of the unpolarized ensemble and the vacuum

electromagnetic field, and the unitary operatorˆU is given

by

lnˆU = χ(cosηˆ a(85)†ˆ s(85)†+ sinηˆ a(87)†ˆ s(87)†− h.c.), (8)

where ˆ a(ν)

=

?dtϕ∗(t)ˆψ(ν)

ciently weak we may writeˆU − 1 = χ(cosηˆ a(85)†ˆ s(85)†+

sinηˆ a(87)†ˆ s(87)†)+O(χ2), i.e., the Raman scattering pro-

duces entanglement between a two-mode field (frequency

qubit) and the isotopic spin wave (dual species matter

qubit). Although we explicitly treat isotopically distinct

species, it is clear that the analysis is easily generalized

to chemically distinct atoms and/or molecules.

To characterize the nonclassical correlations of this sys-

tem, the signal field is sent to an electro-optic phase mod-

ulator (PM2 in Fig. 2) driven at a frequency δωs =

δωw−??ω(87)

ponents into a central frequency cks= c(k(85)

with a relative phase φs. A photoelectric detector pre-

ceded by a filter (an optical cavity, E1 in Fig. 2) which

reflects all but the central signal frequency is used to mea-

sure the statistics of the signal. We describe the detected

signal field using the bosonic field operator,

s (t) is the discrete signal

mode bosonic operator. When the write pulse is suffi-

a

−ω(87)

b

?−?ω(85)

a

−ω(85)

b

??/2 = 1368 MHz.

s

+ k(87)

The modulator combines the two signal frequency com-

s

)/2

ˆψs(t,φs) =

?

ǫ(85)

s

2

e−iφs/2ˆψ(85)

s

(t) +

?

ǫ(87)

s

2

eiφs/2ˆψ(87)

s

(t)

+

?

1 − ǫ(85)

2

s

e−iφs/2ˆξ(85)

s

(t) +

?

1 − ǫ(87)

2

s

eiφs/2ˆξ(87)

s

(t)

where ǫ(ν)

agation losses and losses to other frequency sidebands

within PM2, andˆξ(ν)

s (t) represents concomitant vacuum

noise. While quantum memory times in excess of 30 µs

have been demonstrated [15], here the spin wave qubit

is retrieved after 150 ns by shining a vertically polar-

ized read pulse into a third electro-optic phase modulator

(PM3 in Fig. 2), producing two sidebands with frequen-

cies ck(85)

r

and ck(87)

r

resonant on the

and

??b(87)?

izontally polarized idler photons emitted in the phase

matched directions k(ν)

i

the retrieval dynamics using the effective beam split-

ter relationsˆb(ν)=

ǫ(ν)

ǫ(ν)

r

is the retrieval efficiency of the spin wave stored in

the isotopeνRb,ˆb(ν)=

?dtϕ(ν)∗

quency ck(ν)

i

photon emitted from the

s

∈ [0,1] is the signal efficiency including prop-

??b(85)?

↔

??c(85)?

↔

??c(87)?

transitions, respectively. This re-

sults in the transfer of the spin wave excitations to hor-

= k(ν)

w − k(ν)

s

+ k(ν)

r . We treat

?

r ˆ s(ν)+

?

(t)ˆψ(ν)

1 − ǫ(ν)

r

ˆξ(ν)

r , where

ii

(t) is the dis-

crete idler bosonic operator for an idler photon of fre-

, ϕ(ν)

i(t) is the temporal profile of an idler

νRb spin wave (normalized

to unity), andˆψ(ν)

an idler photon emitted at time t. As with the signal

operators, the idler field operators obey the usual free

field, narrow bandwidth bosonic commutation relations

[ˆψ(ν)

ii

(t′)] = δν,ν′δ(t −t′). A fourth EOM, PM4,

driven at a frequency δωi= δωw−(∆85+∆87)/2 = 531.5

MHz combines the idler frequency components into a

sideband with frequency cki = c(k(85)

a relative phase φi. The combined idler field is measured

by a photon counter preceded by a frequency filter (an

optical cavity, E2 in Fig. 2) which only transmits fields

of the central frequency cki. The detected idler field is

described by the bosonic field operator,

i

(t) is the annihilation operator for

(t),ˆψ(ν′)†

i

+ k(87)

i

)/2 with

ˆψi(t,φi) =

?

ǫ(85)

i

2

eiφi/2ˆψ(85)

i

(t) +

?

ǫ(87)

i

2

e−iφi/2ˆψ(87)

i

(t)

+

?

1 − ǫ(85)

2

i

eiφi/2ˆξ(85)

i

(t) +

?

1 − ǫ(87)

2

i

e−iφi/2ˆξ(87)

i

(t)

where ǫ(ν)

agation losses and losses to other frequency sidebands

within PM4, andˆξ(ν)

i

(t) represents associated vacuum

noise. The write-read protocol in our experiment is re-

peated 2 · 105times per second.

Thesignal-idlercorrelations

dependentcoincidencerates

tectionefficiencyfactors,

?dts

From the state of the atom-signal system after the write

process, ˆUˆ ρ0ˆU†, (Eq.8), we calculate the coincidence

rates to second order in χ,

i

∈ [0,1] is the idler efficiency including prop-

result

given,

by

inphase-

toupde-

=Csi(φs,φi)

?dti

?ˆψ†

s(ts,φs)ˆψ†

i(ti,φi)ˆψi(ti,φi)ˆψs(ts,φs)

?

.

Csi(φs,φi) =χ2

4

?

µ(85)cos2η + µ(87)sin2η

+Υ

?

µ(85)µ(87)sin2ηcos(φi− φs+ φ0)

?

(9)

where µ(ν)≡ ǫ(ν)

amplitude and phase, respectively, such that

r ǫ(ν)

iǫ(ν)

s , and Υ and φ0represent a real

Υe−iφ0= e−(δφ2

s+δφ2

i)/2

?

dtϕ(85)∗

i

(t)ϕ(87)

i

(t),(10)

and we account for classical phase noise in the rf driving

of the EOM pairs PM1,4 and PM2,3, by treating φsand

φias Gaussian random variables with variances δφ2

δφ2

irespectively, see Fig. 2. When the write fields are

detuned such that the rates of correlated signal-idler coin-

cidences are equal (i.e., when µ(85)cos2η = µ(87)sin2η),

the fringe visibility is maximized, and Eq. (9) reduces to

sand

Csi(φs,φi) =χ2

2µ(85)cos2η[1+Υcos(φi−φs+φ0)]. (11)

Page 4

4

0

200

400

600

800

1000

50150250 350

counts in 5 minutes

φi,degrees

FIG. 3: Measured Csi(φs,φi) as a function of φi for φs =

0, diamonds and for φs = −π/2, circles.

absorbed into the arbitrary definition of the origin, i.e., φ0 is

defined to be zero. Solid lines are sinusoidal fringes based on

Eq. (11) with Υ = 0.86. Single channel counts of D1 and D2

show no dependence on the phases.

The angle φ0 is

Fig. 3 shows coincidence fringes as a function of φi

taken for two different values of φs. The detection rates

measured separately for85Rb and87Rb were (a) 53 Hz

and 62 Hz on D1 and (b) 95 Hz and 107 Hz on D2, respec-

tively. These rates correspond to a level of random back-

ground counts about 2.5 times lower than the minima of

the interference fringes. This implies that the observed

value of visibility Υ = 0.86 cannot be accounted for by

random photoelectric coincidences alone. The additional

reduction of visibility may be due to variations in the

idler phases caused by temporal variations in the cloud

densities during data accumulation, while the effects of

rf phase noise are believed to be negligible.

Following Ref. [16] we calculate the correlation func-

tion E(φs,φi), given by

Csi(φs,φi) − Csi(φs,φ⊥

Csi(φs,φi) + Csi(φs,φ⊥

i) − Csi(φ⊥

i) + Csi(φ⊥

s,φi) + Csi(φ⊥

s,φi) + Csi(φ⊥

s,φ⊥

s,φ⊥

i)

i),

(12)

where φ⊥

ogy with polarization correlations, the detected signal

[idler] fieldˆψs[i](t,φ⊥

?ˆψs[i](t,φs[i]),ˆψ†

cal local hidden variable theory yields the Bell inequality

|S| ≤ 2, where S ≡ E(φs,φi) − E(φ′

E(φ′

i) [17]. Using Eq.(11), the correlation function is

given by

s[i]

= φs[i]+ π.We note that, by anal-

s[i]) is orthogonal toˆψs[i](t,φs[i]), i.e.,

?

s,φi) − E(φs,φ′

s[i](t′,φ⊥

s[i])= 0. One finds that a classi-

i) −

s,φ′

E(φs,φi) = Υcos(φs− φi+ φ0).

Choosing, e.g., the angles φs = −φ0, φi = π/4, φ′

−φ0− π/2, and φ′

S = 2√2Υ.

Table 1 presents measured values for the correlation

function E (φs,φi) using the canonical set of angles

φs,φi. We find Sexp= 2.44±0.04 ? 2 - a clear violation

(13)

s=

i= 3π/4, we find the Bell parameter

TABLE I: Measured correlation function E(φs,φi) and S for

∆t = 150 ns delay between write and read pulses; all the errors

are based on the statistics of the photon counting events.

φs

0

0

φi

π/4

3π/4

π/4

3π/4

E(φs,φi)

0.629 ± 0.018

−0.591 ± 0.018

−0.614 ± 0.018

−0.608 ± 0.018

Sexp = 2.44 ± 0.04

−π/2

−π/2

of the Bell inequality. This value of Sexp is consistent

with the visibility of the fringes Υ ≈ 0.86 shown in Fig.

3. This agreement supports our observation that system-

atic phase drifts are negligible. We emphasize that no

active phase stabilization of any optical frequency field is

employed.

In conclusion, we report the first realization of a

dual species matter qubit and its entanglement with a

frequency-encoded photonic qubit.

ployed two different isotopes, our scheme should work for

chemically different atoms (e.g., rubidium and cesium)

and/or molecules.

This work was supported by NSF, ONR, NASA, Al-

fred P. Sloan and Cullen-Peck Foundations. Present ad-

dresses:

sit` a dell’ Insubria, 22100 Como, Italy;⋆Laboratoire Aim´ e

Cotton, CNRS-UPR 3321, Bˆ atiment 505, Campus Uni-

versitaire, 91405 Orsay Cedex, France;†Department of

Physics, University of Michigan, Ann Arbor, Michigan

48109.

Although we em-

∗Dipartimento di Fisica e Matematica, Univer-

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