Detection of atomic entanglement and electromagnetically induced transparency in velocity-selective coherent population trapping

0.26 MB·English

by  M. Kiffner

Detection of atomic entanglement and electromagnetically induced transparency in
velocity-selective coherent population trapping
M. Kiffner1 and K.-P. Marzlin2,1
1Fachbereich Physik der Universit¨at Konstanz, Fach M674, 78457 Konstanz, Germany∗
2Department of Physics and Astronomy, University of Calgary,
2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada
We investigate theoretically the optical properties of an atomic gas which has been cooled by
thelaser cooling method velocity-selective coherent population trapping. Wedemonstrate that the
application of a weak laser pulse gives rise to a backscattered pulse, which is a direct signal for
theentanglementintheatomicsystem,andwhichleadstosingle-particleentanglementonthefew-
photonlevel. Ifthepulseisappliedtogetherwiththepumplasers,italsodisplaysthephenomenon
of electromagnetically induced transparency. We suggest that the effect should be observable in a
5
gas of Rubidiumatoms.
0
0
PACSnumbers: 42.50.Gy, 03.67.Mn,03.75.-b,42.50.Ct
2

n
a I. INTRODUCTION We will show that the latter gives rise to a backscat-
J
teredbeam of lightwhen the VSCPT gas is probed with

8 Among a large variety of laser cooling schemes that a weak signal laser pulse (Sec. III). It will be demon-
1
have been developed, velocity-selective coherent popula- strated that this signal is absent for a mere mixture of

1 tiontrapping(VSCPT)belongstoasmallgroupofmeth- ground states and thus would provide a direct test of
v ods by which temperatures below the one-photon recoil the entanglement of the atomic system. This effect can
2 energy can be achieved. One-dimensional realizations of also be interpreted as a transfer of entanglement from
9 the VSCPT-method have been demonstrated for 4He [1] a single atom to a single photon (Sec. VI). There are
0 and 87Rb atoms [2]. In addition, VSCPT experiments two additional features of the VSCPT state which make
1 with Helium atoms have been carried out successfully in it a system with very special optical properties. First,
0
two [3] and three [4] dimensions. since it is prepared in a dark state one can expect that
5
0 ThefundamentalfeatureonwhichVSCPTreliesisthe a VSCPT gas also exhibits the phenomenon of electro-
/ preparation of an atomic dark state (see, e.g., Ref. [5]) magnetically induced transparency (EIT) [6, 7, 8]; this
h which does not couple to the pump lasers. However, will be examined in Sec. V. We remark here that EIT
p
thereisasignificantdifferencebetweenconventionaldark in a standing-wave geometry has also been studied by
-
t states and the state of a VSCPT gas: In the former case Affolderbach et al. [9]. Although the laser configuration
n
a itispreferabletoconsidersituationsinwhichtheatomic is similar to that considered here, the physical system
u center-of-mass motion can be neglected. This is for ex- is quite different since the experiments are conducted at
q ample the case if all laser fields are propagating in one roomtemperature. AsecondfeatureoftheVSCPTstate
v: direction, the corresponding dark state is then simply a is that it is a periodic state of matter in position space
i superposition of two hyperfine ground states. By con- ( z VSCPT exp( ipz/~)). While one may conjec-
X trast, the dark state of a VSCPT gas must depend on thur|e that thiis∝would ±cause the creation of band gaps in
r the center-of-mass motion to achieve the desired cooling thephotonicspectrumitwillbecomeclearinourderiva-
a
effect. Therefore, the atoms are exposed to two counter- tions that this is not the case.
propagatingpumplaserswithwavenumberk . Thedark
p
state of this laser configuration
1 II. THEORY OF THE VSCPT STATE
VSCPT := ~k + ~k (1)
p p
| i √2 |−i⊗| − i−| i⊗| i
(cid:2) (cid:3)
Our aim is to find an optical signature for the entan-
is then an entangled superpositionof states that contain
glement between the atomic internal and center-of-mass
both internal ( ) and center-of-mass degrees of free-
dom ( ~k )|.±i degreesoffreedominaVSCPTstate,aswellastoexplore
p
| ± i itspotentialforEITeffects. Todosowefirstdescribethe
It is this existence of entangled atomic coherences
features of an atomic VSCPT state, which has been ex-
which makes the optical properties of a VSCPT gas in-
perimentally realized with 4He [1] as well as 87Rb atoms
teresting. Ourgoalistoidentifyadistinctiveopticalsig-
[2]. The atoms are exposed to two counter-propagating
nal which is directly linked to the atomic entanglement.
pump laser beams which share the same frequency ω
p
and Rabi frequency Ω . The electronic degrees of free-
p
dom of the atoms are modeled by a three-level system
∗Present address: Max-Planck-Institut fu¨r Kernphysik, in Λ-configuration (see Fig. 1). In addition, the atomic
Saupfercheckweg 1,69117Heidelberg,Germany center-of-mass motion is treated quantum mechanically.
2
In the one-dimensional cooling scheme considered here ground states can be pumped back to the excited state,
the momentum components perpendicular to the pump the atoms accumulate in VSCPT . The corresponding
| i
field are not observed and can therefore be traced out. dynamics is governed by the master equation
The atomic Hamiltonian is then given by
i
̺˙ = [H (t),̺]+ ̺ . (6)
pˆ2 −~ C Lγ
H := z + ~ω e e , (2)
0 0
2m | ih |
The term ̺ on the right hand side accounts for spon-
γ
L
where m is the mass of the atom and ω its resonance taneous emission and is given by [10, 11, 12]
0
frequency. Inthe rotatingwaveapproximationthe inter-
action Hamiltonian takes the form γ̺ := γ̺ee e e (7)
L − | ih |
γ
H := e eikpzˆ (3) [̺ e + ̺ e + + H.c.]
p e− e+
− | ih−| −2 | ih−| | ih |
(cid:0)
+ e + e−ikpzˆ e−iωpt~Ω + H.c., ~k0
p γ
| ih | (cid:1) + H(u)e~iuzˆ̺eee−~iuzˆdu
where Ω := dE /~ is the Rabi frequency of the pump 2 Z |−ih−|
p p −~k0
lasers (d := e dˆ ǫ(±) and dˆis the electric-dipole
moment operhato|r)·. E |d∓eniotes the electric-field ampli- ~k0
p γ
tude of the classical pump field Ep(z,t) = E(p+)(z,t)+ + 2 Z H(u)e~iuzˆ̺eee−~iuzˆdu|+ih+|
Ep(−)(z,t), with −~k0
E(p+)(z,t) := Epe−iωpt ǫ(+)eikpz +ǫ(−)e−ikpz (4) Jwit=h 1̺ǫǫ′ J:==h1ǫ|t̺r|aǫn′siitio(nǫ,cǫo′ns∈ider{eed, −he,r+e,}t)h.e fFuonrcttiohne
(cid:16) (cid:17) He(u) is↔givgen by H(u) := 3 (1+u2/(~k )2)/(8~k ). It
0 0
being thepositive-frequencypartofthe electricfieldand
has been shown in Ref. [10] that for a finite duration of
Ep(−) = E(p+)†. The circular polarization vectors are de- the pumping this process results in a finite atomic mo-
fined as ǫ(±) :=(e ie )/√2. mentum distribution. We approximate this state by a
x y
±
The special linear combination of ground states mixture of dark states with different momenta that is
described by the density matrix [13]
1
D(p) := ,p ~k +,p+~k (5)
| i √2 |− − pi − | pi ∞
(cid:2) (cid:3)
̺ := f(p) D(p) D(p) dp, (8)
f
is an eigenstate of the interaction Hamiltonian H and Z | ih |
p
−∞
thereforedecoupledfromthelightfield. However, D(p)
| i
is generally not an eigenstate of H . Only for p = 0 the
0 where f characterizes the momentum distribution. We
state VSCPT = D(p = 0) is an eigenstate of the assume here that f can be approximated by a Gaussian
| i | i
completeHamiltonianH (t):=H +H (t)andtherefore
C 0 p distribution centered around p=0,
stationary.
In principle, a VSCPT state can simply be created by f(p)= 1 e−p2/(2σp2) . (9)
pumping the atoms with laser light. Since spontaneous √2πσ
p
emissionincreasesthepopulationin VSCPT withacer-
| i
tain probability, and since all other combinations of the The momentum width is given by σ2 = p2f(p)dp. It
p
is a measure of the achieved final tempRerature of the
gas and varies with the coherent interaction time Θ as
z
σ 1/√Θ. The experiments presented in Refs. [1, 2]
p
|e,pi dem∼onstrate that a value of σp ~kp/2 is achievable.
≈
σ+ σ−
ωp, kp ωp, kp III. COHERENT BACKSCATTERING OF A
−
WEAK SIGNAL BEAM
,p ~k +,p+~k
p p
|− − i | i
Theopticalresponseofanatomicgascanbedescribed
FIG. 1: Λ-type pumping scheme including translational de-
grees of freedom. A σ+ polarized beam travels along the bytheMaxwell-Blochequations,whichincludeanatomic
positivez-directionandcouplesthegroundstate|−,p−~kpi master equation of the form (6) and the wave equation
for the electric field,
to the excited state |e,pi. The counter-propagating beam is
σ− polarizedandcouplesthegroundstate|+,p+~kpitothe
1 1
excited state. ∂2 ∆ E = P¨ . (10)
(cid:18)c2 t − (cid:19) −c2ε
0
3
perturbation theory. Expanding the density matrix as
I II III
̺=̺ +̺ we then find
0 1
E ̺˙ = ̺ (15)
1 VSCPT-gas 0 L0 0
i
k , ω
s s ∂ ̺ = [H ,̺ ] . (16)
t − L0 1 −~ s 0
(cid:2) (cid:3)
z The Liouville operator is defined as
0
z =0 z =a L
i
( ):= [H , ] + ( ) (17)
FIG. 2: Idealized set-up after the pump laser beams have L0 · −~ 0 · Lγ ·
been switched off. The VSCPT gas stretches from z = 0 to
and governs the time evolution of the free atom.
z =a (region II) and is surrounded by vacuum (region I and
III). The signal beam, a plane-wave with frequency ωs, σ+ Wefirstconsidertheidealcasewhere̺0 isgivenbythe
stationarystate ̺ VSCPT VSCPT. For brevity we
tphoelalreifzta.tion and amplitude E0 is shone into the probe from will only discuss tsh≡e i|ncomingihsignal be|am E1 since the
second beam can be treated in an analogous way [14].
The commutator on the right hand side of eq. (16) is
comprised of two time-dependent parts
Foradilutegas,atom-atominteractionscanbeneglected
so thatthe macroscopicpolarizationP canbe expressed i [H ,̺ ]= iΩ1 e,~∆k VSCPTe−iωst + H.c. (18)
as the local mean value of the single-atom dipole opera- −~ s s √2| ih |
tor,
that vary with e−iωst and eiωst, respectively (∆k :=
P(z,t):=ρ¯Trint ̺(z,t)dˆ . (11) ks −kp). This inhomogeneity gives rise to the follow-
ing steady-state solution
(cid:0) (cid:1)
Here, ρ¯ denotes the mean atomic density and the trace
1
runs over the internal states (̺(z,t)= z ̺(t) z ). ̺ (t) = χ (ω )Ω e−iωst e,~∆k , ~k (19)
1 0 s 1 p
h | | i −2 | ih− − |
To investigate the optical properties of a gas that has
been cooled by means of the VSCPT method, we gen- 1
+ χ (ω )Ω e−iωst e,~∆k +,~k + H.c.
erally consider the behavior of a weak signal laser beam 2 0 s 1 | ih p|
interacting with a homogeneous distribution of cooled
with
atoms of finite width a. In this section we examine the
casewherethesignalbeamisswitchedonafterthepump ~∆k2 γ −1
lasersareswitchedoff. WewillseeinSec.Vthattheopti- χ0(ωs):= Er +i +∆s . (20)
(cid:20)(cid:18) − 2m (cid:19) 2 (cid:21)
calresponsewillbe quite differentwhenthe pump lasers
are not switched off. E := ~k2/(2m) defines the recoil frequency and ∆ :=
r p s
The setup to investigate the optical response is shown ω ω denotes the detuning ofthe signalfieldfromres-
s 0
inFig.2. Attime t=0,the pump fieldis turnedoff and ona−nce. In the following we will use this steady-state
a σ+ polarized signal field solution instead of a full solution that fulfills the correct
initial condition ̺ (t) = 0. This is justified if the in-
1
E(+)(z,t)=E (z)ǫ(+)ei(ksz−ωst) (12) teraction time between the laser (pulses) and the atoms
1 1
is long compared to the natural lifetime of the excited
withslowlyvaryingamplitudeE1(z)andfrequencyωs is state, since in this case all non-stationary contributions
appliedtotheprobefromtheleft. Theassociatedatomic (which are solutions of the homogeneous equations) are
evolution can be derived from Eq. (6) with HC replaced damped away.
by H0+Hs(t), with Weremarkthat,asaconsequenceoffirst-orderpertur-
bationtheory,thepopulationoftheexcitedstateremains
Hs(t) = ~Ω1eikszˆe−iωst e zero. This explains the simple form of solution (19) and
− | ih−|
~Ω eik2zˆe−iωst e + + H.c. (13) effectively allows us to replace the full decoherence term
2
− | ih | (7) by
and Ω :=dE (z)/~. The second electric field
i i γ
coh
̺:= [̺ e +̺ e + +H.c.] .
Lγ −2 e−| ih−| e+| ih |
E(+)(z,t):=E (z)ǫ(−)ei(k2z−ωst) (14)
2 2
An important feature of Eq. (19) is the non-vanishing
(tkh2e a:=nsaktsz−con2skipste≈nt−(skese)bhealoswb)e.eTnoinsotrlvoeduthceedMtaoxwkeeellp- pcoehaerriefntchee[̺s1ta]et+e ̺:=0 =he|V̺1S|C+PiT. TVhSisCePffTectisdroeepslancoetdabpy-
| ih |
Blochequationswefirstconsidertheatomicmasterequa- an incoherent mixture of the form
tion. Since the signal beam is assumed to be weak, 1
its influence on the atoms can be treated in first-order ̺mix := , ~kp , ~kp + +,~kp +,~kp ,
2 |− − ih− − | | ih |
(cid:2) (cid:3)
4
and therefore is a signal for the coherence between the solution of Eq. (24) is found to be
atomic ground states.
E (z) = (27)
We will now show that this coherence creates a 1
backscatteredlightbeamandthereforegeneratesasignal E δcosh[δ(a−z)]+i(n0−∆k)sinh[δ(a−z)]e−i∆kz
ofthe entanglementbetweenatomic internalandcenter- 0 δcosh[δa]+i(n ∆k)sinh[δa]
0
−
of-massdegreesoffreedom. Todothiswesolvethewave
equation (10) in paraxialapproximation, i.e., we neglect E (z) = E in0sinh[δ(a−z)] e−i∆kz .
the terms ∂2E (z) (i 1,2 ). This is justified if the 2 0 δcosh[δa]+i(n ∆k)sinh[δa]
z i ∈ { } 0−
envelopes E (z) are slowly varying over one wavelength,
i
The envelopes E containa periodic partwhose period
so that 1,2
λis givenby the inverseimaginarypartofδ. For atomic
∂2E (z) k ∂ E (z) . (21) densities lower than 1012cm−3 the parameter λ is usu-
z i ≪ s | z i | ally orders of magnitudes larger than the typical length
(cid:12) (cid:12)
(cid:12) (cid:12) of a VSCPT gas(i.e., a few centimeters); the periodicity
Inserting ̺ of Eq. (19) and the corresponding contri-
1
bution induced by E into Eq. (11) leads to would then not be observable. Generally, for aδ 1
2 | | ≪
and ∆k 0 Eq. (27) simplifies considerably and be-
→
ρ¯d∗ comes
P(+)(z,t) = χ (ω ) Ω Ω eikszǫ(+) (22)
0 s 2 1
2 (cid:16)(cid:0) − (cid:1) E (z) = E 1 n0z (28)
+ Ω1−Ω2 eik2zǫ(−)(cid:17)e−iωst . 1 0 (cid:18) − n0a−i(cid:19)
(cid:0) (cid:1)
n (a z)
Theterm Ω ǫ(−) isadirectconsequenceof[̺ ] =0. E (z) = E 0 − ,
1 1 e+ 2 0
∼ 6 (cid:18) n a i (cid:19)
In the paraxial wave equation [15], 0 −
which is an excellent approximation for realistic param-
ks∂zE1(z)ǫ(+)eiksz (23) eters.
k ∂ E (z)+2i∆kE (z) ǫ(−)eik2z = eiωst P¨(+) , Fig. 3showsaplotoftheintensityoftheincomingand
− s z 2 2 2ic2ε reflectedbeam. Wechoseaprobelengthofa=1cmand
(cid:0) (cid:1) 0 adetuningof∆ =3 106s−1. FortheJ =1 J =1
s e g
this results in a coupling between E and E . If E had transition of the D ×Line of 87Rb, the recoil f↔requency,
1 2 2 1
notbeenintroduced,theterm Ω ǫ(−) wouldnothavea the rate of spontaneousemission andthe dipole moment
1
corresponding term ǫ(−) on∼the left-hand side, so that takethevalues[17]E =2.3 104s−1,γ =3.61 107s−1
r
∼ × ×
the equation would be inconsistent. Sorting the terms and d=2.99ea /√12, where e is the elementary charge
0
accordingtotheirpolarizationandphasefactorsleadsto anda isBohr’sradius. Inaddition,weassumedamean
0
a coupled equation for the amplitudes, atomic density ρ¯ of 2 1010cm−3 which is close to the
experimental condition×s (ρ¯ 1010cm−3) described in
∂z(cid:18)EE12 (cid:19)=(cid:18)−iinn00 i(n0in02∆k)(cid:19)(cid:18)EE12 (cid:19) , (24) Rfeerrf.ed[2t].o tAhbeoruetfle3c0t%edobfeathme,i≈anlclolomsisnegsainretednusietytoissptoranntas--
− −
neous emission.
where Simplified explanation of thebackscattered beam: Since
the above derivation is somewhat involved we will give
1 ω2ρ¯ d2
n (ω ):= s | | χ (ω ) . (25) here another, more physical explanation of the effect.
0 s 2k 2c2ε ~ 0 s
s 0
I(cid:144)I
0
The boundary conditions on the physical solution are 1
determinedbythebehaviorofE andE outsidethegas
1 2
andMaxwell’sequation E = B˙ ,whichimpliesthat
∇× −
the transverse electric field is continous at the bound-
ary of the gas (see Fig. 2). Since E1 corresponds to the 0.5
incoming signal beam that travels along the positive z-
axis, its amplitude is given by a fixed value E (z) = E 0.3
1 0
for z < 0 (region I). On the other hand, the counter-
propagatingfield E2 is initially empty so that E2(z)=0 z@cmD
0.5 1
for z > a (region III). The boundary conditions for the
amplitudes E in region II are thus given by [16]
1,2
FIG.3: Intensityoftheincoming(solidline)andthereflected
(dashedline)beaminaVSCPT-gasoflengtha=1cm. Both
! !
E1(z =0)=E0 and E2(z =a)=0. (26) intensitiesarerelatedtotheintensityI0oftheincomingbeam
in region I (Fig. 2).
Introducing the notation δ := (2n ∆k)∆k the full
0
−
p
5
The incoming signal beam transfers the initial atomic proceed as in section III to derive the macroscopic po-
state VSCPT tothe superposition ψ VSCPT + larization (we only write down the part induced by E )
1 1
| i | i∼| i
H VSCPT VSCPT + e,p = 0 (for simplicity
s
| i ∼ | i | i
we here set ∆k = k k to zero). Hence, the absorp-
tion of an incomingss−ignpal photon leads to a (partial) P(+)(z,t)= ρ¯d∗Ω I eikszǫ(+) I e−ikszǫ(−) e−iωst
1 α β
transfer of the initial coherence between , ~k and − 2 (cid:16) − (cid:17)
p (31)
+,~k to one between e,0 and +,~k|−. −The ilatter
p p with
| i | i | i
corresponds to an induced dipole moment which, due
∞
to angular momentum conservation, can only lead to
emission of photons with polarisation σ−. In Maxwell’s Iα := χ(ωs,p)f(p)dp,
Z
equations such photons are coupled to the coherence
−∞
ρ (z,z) = e,z ψ ψ +,z at the position z of the
e+ 1 1
h | ih | i ∞
atom. An elementary calculation leads to ρ (z,z)
e+
he,z|e,0ih+,~kp|+,zi∼ exp(−ikpz) so that the emitte∼d Iβ := Z χ(ωs,p)f(p)eiωr(p)tdp
photons are propagating in the opposite direction of the
−∞
incoming beam. The associated change in the photon’s
momentum is providedby the different atomic momenta γ 1 −1
χ(ω ,p) := E +i +∆ ω (p) .
inthetwogroundstates. Theσ−polarizedbackscattered s (cid:20) r 2 s− 2 r (cid:21)
lightbeamE is thus notonlyasignalforthe coherence
2
of the VSCPT state, but also a direct signal of the en- In the derivation we have used the approximationsEr
tanglement between atomic internal and center-of-mass ~∆k2/(2m) Er andkp ∆k kp as wellas k2 k−s.
≈ ± ≈ ≈−
degrees of freedom. The evaluation of the integrals Iα and Iβ can be simpli-
This method of probing entanglement in atomic gases fied by using that χ(ωs,p) as a function of p is almost
shouldnotonlybeapplicabletoaVSCPTgasbuttoany constant over the momentum range of a VSCPT gas.
state in which an entanglement between internal states We assume here that the momentum width is given by
and momentum states is achieved. The VSCPT gas is σp = ~kp/2 (see Sec. II) and are thus allowed to replace
onlyspecialinthattheentangledstateisalsoadarkstate χ(ωs,p) by χ0(ωs), which yields
with respect to the pump laser field. We will see below
I χ (ω ) , I χ (ω )g , (32)
that this will lead to EIT for the pair of signal fields. α ≈ 0 s β ≈ 0 s
We also emphasize that a mere mixture of states with with g(t):=e−2Er2t2.
different momentum and internal degrees of freedom, as
These results enable us to describe the long-term be-
it is created by other cooling methods such as Raman
haviorofthesystem. Fort 1/E ,theIntegralI isap-
cooling [18], would not display the backscattered beam. ≫ r β
proximatelyzeroandthetermproportionaltoǫ(−) ineq.
(31) can be neglected. Consequently, the backscattered
wave is equal to zero. The reason is that the dephasing
IV. FINITE MOMENTUM WIDTH:
DEPHASING OF THE REFLECTED BEAM ofthe coherencesindifferentdarkstates D(p,t) of(30)
| i
leads to a destructive interference between the backscat-
tered signal beams for different momenta.
Inrealitytheatomsarenotpreparedintheidealstate
Although the backscattered signal is suppressed for
VSCPT VSCPT but have a finite momentum width,
| ih | continuously operating signal beams, it is reasonable to
described by ̺ of Eq. (8). Since this is not a stationary
f expect that it will be strong enough if one employs sig-
state, the free time evolutionof the unperturbed density
nal pulses instead. Incoming and backscattered pulses
operator̺ ofEq.(15)afterthepumplasersareswitched
0 can formally be described by assuming that the ampli-
off reads
tudes E of Eqs. (12) and (14) are slowly varying both
1,2
∞
in space and in time,
̺ (t) = f(p) D(p,t) D(p,t) dp, (29)
0
Z | ih |
−∞ ∂2E (z,t) k ∂ E (z,t) ,
z 1,2 ≪ s | z 1,2 |
where (cid:12) (cid:12)
(cid:12)∂2E (z,t)(cid:12) ω ∂ E (z,t) ω2 E (z,t) .
D(p,t) = |−,p−~kpi e−iωr(p)t|+,p+~kpi (30) (cid:12) t 1,2 (cid:12) ≪ s | t 1,2 |≪ s | 1,2 |
| i √2 − √2 Ma(cid:12)xwell’s equa(cid:12)tion (10) becomes now a coupled set of
first-order partial differential equations [19] for the en-
and ω (p):=2k p/m. The energy ~ω (p) is just the dif-
r p r velopes E and E ,
1 2
ference between the kinetic energiesof the groundstates
,p ~k and +,p+~k . 1
p p
|− − i | i ∂ + ∂ E = in E g E (33)
̺0(t) describes a mixture ofinitially darkstates which (cid:18)c t z(cid:19) 1 0 2 − 1
(cid:0) (cid:1)
evolve into a corresponding bright state at rates which
depend linearly on p. To demonstrate that this behav- 1
∂ ∂ E = in E g E ,
ior results in a dephasing of the backscattered wave we (cid:18)c t − z(cid:19) 2 0 1 − 2
(cid:0) (cid:1)
6
where n is defined in Eq. (25). V. ELECTROMAGNETICALLY INDUCED
0
TRANSPARENCY
WehavenumericallysolvedtheseequationsforaGaus-
sian amplitude E incoming from the left, with the
1
We will now demonstrate that a VSCPT-gas creates
boundary condition that E is zero on the right-hand
2
electromagneticallyinducedtransparency[6,7,8,20,21,
side ofthe medium, andthatthe fields arecontinuousat
22] if the signalandthe pump field interactwith the gas
the boundary of the medium. At t = 0, the E pulse is
1
at the same time. In presence of the pump field, the
completely outside of the medium and E is zero every-
2
Hamiltonian in Eq. (6) has to be replaced by H +H .
where. The numerical solution is easily obtained using C s
Thesameperturbativemethodsthathavebeenemployed
standard mathematical software packagessuch as Math-
insectionIIIcanbeappliedhere,providedthesignalfield
ematica. The results can be summarized as follows.
is much weaker than the pump field (Ω /Ω 1).
s p
| | | | ≪
The time dependence of ̺0 manifests itself solely in We consider the simplified situation where all atoms are
the presence of the function g. For a signal pulse whose initially in the unperturbed stationary state VSCPT ;
| i
length is short enough so that g 1, the efficiency of the previous section indicates that this is justified for
≈
the backscattering-effect is as high as for the idealized sufficientlyshortsignalpulses. Thefirstordercorrection
VSCPT-gas(Sec.III). Fig.4showsaplotofg asafunc- ̺ is now determined by
1
tion of time for the Helium and Rubidium parameters.
i
∂ ̺ = [H ,̺ ] , (34)
The small mass of 4He results in a large recoil fre- (cid:2) t−LC(cid:3) 1 −~ s 0
quency (E =2.7 105s−1) and causes a rapid decay of
r where is defined as
× C
g within 5µs. Consequently, the length of the incoming L
pulsemustbeasshortas2 10−7sinordertomaximize i
the peak intensity of the b×ackscattered pulse. However, LC(·):=−~[HC, ·] + Lγ(·)
oneshouldbearinmindthatthesolutiontothehomoge-
nous part of eq. (16) cannot be neglected for t 1/γ. with H =H +H . The unitary transformation
≤ C 0 p
Therefore, the predictions of our theory are only correct
for pulses of a few µs or longer. In the case of Helium, U := e e eiωpt+ eikpzˆ+ + + e−ikpzˆ
the efficiency of the backscattering-effectis less than 1% | ih | |−ih−| | ih |
if a Gaussian pulse of 2µs is shone into the probe.
removes the operators exp[ ik zˆ] as well as the time
s
±
Acompletelydifferentbehaviorshouldbeobservedfor dependence from the interaction Hamiltonian Hp. The
a VSCPT gas of Rubidium atoms. The recoil frequency transformed operator ̺˜1 := U̺1U† then obeys a more
of Rubidium is aboutten times smaller than for Helium, convenient equation than Eq. (34). The remaining cal-
andg is almostconstantwithinthe first5µs. Evenfora culation follows exactly the procedure of Sec. III. In
Gaussianpulseof4µs,thepeakintensityofthebackscat- particular, the matrix element e ̺1 e vanishes even
h | | i
tered signal beam is almost as high as in the case of in the presence of the pump field. This is a consequence
the idealized VSCPT gas (section III). Thus, we expect of the initial atomic state VSCPT being an eigenstate
| i
that for signal pulses traveling through a Rubidium gas of the complete Hamiltonian HC. If the finite width of
the backscattered beam should be a good signal for the theatomicmomentumdistributionistakenintoaccount,
entanglement between the electronic and center-of-mass excitations will occur and, consequently, side bands of
degrees-of-freedomin a VSCPT state. frequency ωsb :=ωs 2∆ω will be present.
−
The result for the field amplitudes is again of the
form (27) and (28) if one replaces δ, n and χ by
0 0
δ := (2n ∆k)∆k,
p p
−
gHtL p
1 1 ω2ρ¯d2
n (ω ):= s | | χ (ω ) (35)
p s 2k 2c2ε ~ p s
s 0
and
0.5
4He 87Rb
∆ω
χ (ω ):= . (36)
p s ∆ω E +iγ +∆ 2Ω 2
r 2 s − | p|
0 t@ΜsD (cid:0) (cid:1)
5 10 15 20
We exploited that ~∆k2/(2m) and ~k ∆k/m are much
p
FIG. 4: Plot of g(t) = exp[−2Er2t2] as a function of time. smaller than the frequency difference ∆ω := ωs −ωp =
Therecoilfrequencyof87Rb(Er =2.3×104s−1)isabout10 c∆k between the signal and the pump field.
times smaller than for 4He(Er =2.7×105s−1). This result allows us to investigate the behavior of a
σ+ polarized signal pulse. The electrical fields E and
1
7
Re[χp] With the help of the residue theorem we arrive at the
2´10-8s following expressions for the field amplitudes
-2´10-8s -107 0 107 DΩ@s-1D E1(z,t) = E0 exp(cid:20)−2κσ2+ω2(cid:21) (cid:16)1− az −h+(η+)az2(cid:17) (40)
κ2 z z a
E (z,t) = E exp − 1 h (η ) − ,
-5´10-8s Im[χp] 2 0 (cid:20)−2σω2(cid:21)(cid:18) − a − − − a2 (cid:19)
where
FIG. 5: Real and imaginary part of χp(ωs) for thefollowing
set of parameters: Ωp = 107s−1, Er = 2.3 ×104s−1 and π eη±2
γ =3.61×107s−1. Wealsoassumedthatthedetuningofthe h±(η±) = r2 σ n′(ω )(Erf(η±)+1) (41)
pump field vanishes, i.e. ωp =ω0. ω p p
1 1
η = κ +
± (cid:18) ± an′(ωp)(cid:19) √2σω
E2 within the medium are then given by z 4
κ = t σ2
± (cid:18) ∓ c − σ (cid:19) ω
∞ ω
E(1+)(z,t) := Z E1ei(ωc z−ωt)dωǫ(+) (37) and Erf(η±) denotes the error function.
−∞ Fig. 6 (a) shows a contour plot of the intensity of
the E pulse just before and inside the medium, which
1
1 n (ω)z
(z,ω) := (ω) 1 p stretches from z = 0 to z = 10cm. This unrealistic
1 0
E √2π E (cid:18) − np(ω)a i(cid:19) probe length has been chosen to better visualize the re-
−
ducedgroupvelocityoftheE pulsewithinthemedium,
1
and
which also appears in other EIT-media [6, 7, 8]. Start-
ing from z = 0, the trajectory of the incoming pulse is
∞
E(2+)(z,t) := Z E2e−i(ωcz+ωt)dωǫ(−) (38) toiflttehdetogrrwouaprdvsetlhoeciptyosriteidvuectt-iaoxnisinwshidicehtihseaVcoSnCsePqTu-egnacse.
−∞ Fig. 6 (b) shows the intensity of the backscatteredbeam
E . It can be seen that the incoming pulse gives rise
1 n (ω)(a z) 2
p
(z,ω) := (ω) − . to two backscatteredbeams which can be understood as
2 0
E √2π E (cid:18) np(ω)a−i (cid:19) follows. Since np(ω) vanishes for ω = ωp, the Fourier
At z = 0, just outside the medium, the electric field E components close to ∆ω =0 do not contribute in the in-
1 tegralinEq.(38). Consequently,the Gaussian issplit
reduces to E0
into two parts which build up the two reflected beams.
E (z =0,t)=E (t)ǫ(+) + c.c. , (39) We finally note that an increased atomic density ρ¯, a
1 0
broadened frequency width σ or a less intense pump
ω
where E (t) := 1/√2π ∞ (ω)e−iωtdω is the Fourier field amplifies the backscattering effect and reduces the
0 −∞E0 transparencyfor the incoming pulse. For ρ¯=1011cm−3,
transform of (ω). R
Fig. 5 showE0s the real and imaginary part of χp(ω) σthωe=pe1a0k6,inΩtpen=sit5y×of10th6eanredflaecpterdobbeealemng(trheloaftiave=to1ctmhe,
which is related to n (ω) through Eq. (35). Since χ (ω)
p p peakintensityoftheincomingpulse)isabout0.3,forthe
vanishes for ∆ω = 0, the VSCPT-gas will be transpar-
parameters of Fig. 6 it is given by 0.08.
ent for an incoming cw-field of frequency ω = ω . The
s p
backscattered beam is then equal to zero, see Eqs. (37)
and (38). An EIT-situation can thus be realized for an
VI. QUANTIZATION OF THE SIGNAL FIELD
incoming signal pulse whose Fourier components are
0
E
sharply peaked around the pump laser frequency ω . In
p ToseehowaVSCPTgasinfluencesthequantumstate
order to evaluate the integrals in Eqs. (37) and (38) an-
ofthe signalbeams we considerthe evolutionofthe field
alytically, we assume that
operator for the two signal modes [23],
E (ω ω ) (ω ω )2
(ω):= 0 exp i4 − p exp − p 2
E0 σω (cid:20) σω (cid:21) (cid:20)− 2σω2 (cid:21) Dˆs :=ε0iEωs ai(t)ǫieikiz +H.c., (42)
Xi=1
isgivenbyaGaussiancenteredaroundω =ω ,thephase
p
tsfahmceatomllr,endenius(mωur)easctattnh=abte0.tehxIefptaEhne1dewpdiudaltsshenσisω(ωcoo)fmEp0lneist′e(slωuyffi)ocuie∆tnsωitdlye. wanh~deωrǫe1/t(=h2eεǫm(V+o))dweassitahwreVellcbhaeasirnǫag2ctt=ehreiǫzq(e−uda).nbtyWizkae1thi=oanvkevso,slekut2mE=eω.s−W:k=es
p p ≈ p p · s 0
p
8
(a) 14 theRabifrequenciesΩ1,2byidEωsa1,2/~. Introducingthe
slowly varying operators a˜ (t):= a (t)exp(iω t) one can
i i s
12 derivethe coupledsetoffirst-orderdifferentialequations
10
a˜ 1 1 a˜
t@ΜsD 8 ∂t(cid:18)a˜21 (cid:19)=iβ(cid:18)−1 1(cid:19)(cid:18)a˜21 (cid:19) , (44)
−
6 where second derivatives of the operators a˜1,2 have been
neglected against ω a˜˙ . The coefficient β is given by
4 s 1,2
β := cn , where n can be either n of Eq. (25) or n
x x 0 p
2 ofEq.(35). We arriveatthe followingexpressionforthe
reduced annihilation operators
0
-2 0 2 4 6 8 10
z@cmD a1(t) = e−i(β+ωs)t [cos(βt)a1+isin(βt)a2] (45)
a (t) = e−i(β+ωs)t [isin(βt)a +cos(βt)a ] ,
(b) 2 1 2
14
where the operators a on the right hand side are anni-
12 i
hilation operators in the Schr¨odinger picture. The op-
10 erators ai(t) do not obey the canonical commutation re-
lations since β is a complex parameter. This result is
8
t@ΜsD consistent,sincecanonicalcommutationrelationsarenot
required for reduced Heisenberg operators [24].
6
Solution(45)demonstratesthatasinglephotoninthe
4 incoming signalbeam a will evolve into a superposition
1
of the two signal modes. Such a state corresponds to
2
single-particle entanglement, since the polarization and
0 position degrees-of-freedom of the photon are then en-
0 2 4 6 8 10
z@cmD tangled. Hence, the one-particle entanglement which is
present in the atomic VSCPT state can be transferred
FIG. 6: Contour plots of the intensities of the incoming (a) to a corresponding entanglement of a photon in a weak
andthereflected(b)pulse. Themediumstretchesfromz=0 signal pulse. We remark that, apart from the possible
to z = 10cm, the frequency width of the incoming pulse is appearance of EIT, the creation of single-particle entan-
given by σω = 5×105s−1. We consider a gas of Rubidium glement could also be achieved by a beam splitter fol-
atoms with a mean atomic density of ρ¯= 2×1010cm−3, a lowed by a polarization rotator in one of the two output
RabifrequencyofΩp =107s−1andazerodetuning(ωp =ω0) modes. The distinguishing feature of the VSCPT gas is
for the pump lasers. The trajectory of the incoming pulse in
that this effect is a direct signalofatomic entanglement.
(a) shows a bend at z = 0. This is a consequence of the
In conclusion, we have shown that an atomic gas pre-
reduced group velocity inside themedium.
pared in a VSCPT state exhibits unique optical features
which include the phenomenon of EIT and a backscat-
tered light pulse which is a signal for the entanglement
seekasolutiontoHeisenberg’sequationofmotionforthe associated with the VSCPT state. Detection of the
(transverse) displacement Dˆ , which takes on the form backscattered light pulse should be possible for a gas of
s
Rubidium atoms, while the large recoil velocity in He-
1 ∂2 ∆ Dˆ = Pˆ , (43) lium would make such an experiment unfeasible. On the
(cid:20)c2 t − (cid:21) s ∇×∇× s few-photon level, the VSCPT gas would lead to single-
particle entanglement for a signal photon.
where Pˆ := Tr (̺ Pˆ) as well as a (t) are reduced Acknowledgments We would like to thank
s R R i
Heisenbergoperators[24]. Thetracerunsoverallatomic B. C. Sanders for fruitful discussions. This work was
and radiation degrees-of-freedom except for the two sig- supported by the German Academic Exchange Service
nal modes. It turns out that the operator Pˆ has the (DAAD) and Alberta’s informatics Circle of Research
s
sameformastheclassicalpolarization(22)ifonereplaces Excellence (iCORE).
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