Cross phase modulation in a five--level atomic medium: Semiclassical theory

0.79 MB·English

by  Carlo Ottaviani

Cross phase modulation in a five–level atomic medium: Semiclassical theory
Carlo Ottaviani,∗ Stojan Rebi´c, David Vitali, and Paolo Tombesi
Dipartimento di Fisica Universit`a di Camerino, I-62032 Camerino, Italy
(Dated: February 2, 2008)
The interaction of a five-level atomic system involving electromagnetically induced transparency
with four light fields is investigated. Two different light-atom configurations are considered, and
theirefficiencyingeneratinglargenonlinearcross-phaseshiftscompared. Thedispersiveproperties
of those schemes are analyzed in detail, and the conditions leading to group velocity matching for
6
two of the light fieldsare identified. An analytical treatment based on amplitudeequations is used
0
in order to obtain approximate solutions for the susceptibilities, which are shown to fit well with
0
thenumerical solution of thefull Bloch equations in a large parameter region.
2

n PACSnumbers: 42.50.Gy, 42.65.-k,03.67.Hk
a
J
PSfragrepla
ements

5 I. INTRODUCTION
(a) (b)
2 j2i
j3i Æ2
An efficient cross-phase modulation (XPM) in quan-
s] tum and semiclassical regimes is both interesting and Æ1 (cid:10)2
c
useful in many possible applications, such as those in
i
t optical communications [1], optical Kerr shutters [2], (cid:10)2 j2i
p (cid:10)1 Æ1
o quantumnon-demolitionmeasurements[3]andquantum
s. phasegates[4]. Inallofthese,butthelasttwoespecially, (cid:10)1
c a large XPM is desirable for low pump powers and high
j3i j1i
i sensitivities.
s j1i
y In a standard three-level cascade scheme, shown in
FIG. 1: (a) Three-level Λ–scheme for EIT. Transitions are
h Fig. 1b, nonlinear effects are obtained alongside absorp-
drivenby theprobe and couplingfields, with Rabifrequency
p tion, which increase as the fields are tuned closer to the
Ω andΩ respectively. Whentheprobedetuningδ matches
[ 1 2 1
atomic transition[5]. To reduce the absorptionto anac-
thetwophotonRaman-resonanceconditionwiththecoupling

2 ceptable level, light fields need to be strongly detuned field, the atomic medium becomes transparent for the probe
v from the intermediate atomic level 2 , simultaneously field. (b)Three-levelcascade configuration. Toobtain signif-
| i
0 reducing however the size of the nonlinearity, since both icantnonlineareffectsfortheprobefieldwithRabifrequency
0 are inversely proportional to the square of the detuning. ω ,alarge detuningδ from theintermediatelevel|2i isnec-
1 1
2
essary.
0
Extensivestudiesaimedatavoidingthis problemhave
1
5 been performed in recent years. A promising candidate
0 emerged with the use of quantum coherence effects in
/ the interaction of light with multilevel atoms. Coherent ular system in the presence of EIT, usually arise by one
s
c population trapping (CPT) [6] and in particular the re- of the two following related mechanisms. One is to vi-
i lated effect of electromagnetically induced transparency olate the strict two-photon resonance condition, with
s
y (EIT) [7, 8] have been studied theoretically [5, 9, 10] a frequency mismatch smaller than the width of the
h and experimentally [3, 11, 12, 13] in various energy-level transparency window [3, 14, 15]. Alternatively, one can
p schemes based on a generic Λ-scheme (see Fig. 1a). At add additional energy level(s) in order to induce an ac-
:
v resonance (δ1 = 0 in Fig. 1a), the presence of the cou- Stark shift and effectively tune the signal out of reso-
i pling field (with Rabi frequency Ω ) cancels, by destruc- nance[5,10,16]. Bothmechanismsresultinlargenonlin-
X 2
tive interference, the absorption on the probe transition earities,accompaniedbyveryweakabsorption. Recently,
r (withRabifrequencyΩ ),andrendersthemediumtrans- theso-calledM-scheme,showninFig.2,hasbeenstudied
a 1
parentforthe probebeam. Amoregeneralconditionfor and proposed as a promising source of giant nonlineari-
EITistwo-photonresonance,aconditionthatissatisfied ties that canbe utilized for XPM[14, 17]. The double Λ
whenthefrequencydifferencebetweenthefieldsmatches nature of this M configuration offers the opportunity of
the energy gap between levels 1 and 3 . However, on a simultaneous group velocity reduction for pulses prop-
| i | i
the exact EIT resonance, probe field decouples from the agatinginsidetheatomicsample. Groupvelocitymatch-
atoms, making the dynamics purely linear. ing, originally pointed out by Lukin and Imamog˘lu [18],
Optical nonlinearities in a multilevel atomic or molec- is important to obtain a large XPM. In fact, it has been
shownbyHarrisandHau[10]thatifequalgroupvelocity
reduction is not achieved for both fields, the nonlinear
phase accumulation will saturate at a certain constant
∗Electronicaddress: [email protected] value. The consequence is that increasing the length of
2
thesampleinwhichthenonlinearinteractiontakesplace the physics of the symmetric M-scheme is described by
is not useful. On the other hand, if group velocities are following the same order as in Sec. II. Conclusions are
equal, the nonlinear phase accumulation becomes linear drawn in Sec. IV.
in the interaction length [18] and it may become very
large.
A large cross Kerr phase shift is very useful for II. THE ASYMMETRIC M SCHEME
photonic-basedimplementationsofquantuminformation
(QI) processing systems [4, 19]. In fact, a fundamental A. The System
buildingblockforquantuminformationprocessingisthe
quantum phase gate (QPG). In a QPG, one qubit gets The M-systemunder considerationhas a double adja-
a phase conditionalto the other qubit state accordingto
centΛstructureasshowninFig.2,whereatomswithfive
the transformation [4, 20] i j exp iφ i j
1 2 ij 1 2 levels (three ground states 1 , 3 , 5 , and two excited
| i | i → { }| i | i
where i,j = 0,1 denote the logical qubit bases. This | i | i | i
states 2 , 4 ) interact with four electromagnetic fields.
{ }
gate is universalwhen the conditional phase shift (CPS) | i | i
This configurationcanbe realizedinZeeman-splittedal-
kali atoms, such as 87Rb atoms. The Rabi frequencies
associated with the lasers driving the atomic transitions
φ=φ +φ φ φ , (1)
11 00− 10− 01 are defined as
is nonzero, and it is equivalent to a CNOT gate up to µ
ij k
local unitary transformations when φ=π [4, 20]. Ωk = E , (2)
− ¯h
To obtain a CPS of φ = π, one looks for a strong in-
teraction between qubits, ideally accompanied by weak
where is the electric field amplitude, µ is the rel-
decoherence. Photonsareaparticularlyattractivechoice k ij
E
ative dipole matrix elements induced on the transition
for qubits due to their robustness against decoherence
i j . On transitions 3 2 and 5 4 we
during the processing and transmission of information.
| i ↔ | i | i ↔ | i | i ↔ | i
apply twostrongfields,the coupler Ω andthe tuner Ω
Thisfeatureshouldideallypermitthetransmissionofthe 2 4
respectively. On the transition 1 2 a probe field
quantuminformationstoredinveryweakquantumpulses
| i ↔ | i
is applied (with Ω ), while on the transition 3 4
oververylongdistanceswithanegligiblysmallreduction 1
| i ↔ | i
a trigger field (with Ω ) is applied. In this paper, we
of the initial signal. There is however an important dif- 3
will analyze the XPM and the group velocity matching
ficulty in the implementation of an all–optical–QPG: to
between the probe and the trigger fields. We call the
processtheinformationoneneedsstrongphoton–photon
scheme of Fig. 2 the asymmetric M scheme due to the
interaction. In fact, to implement QI with photons, a
asymmetricdistributionoftheinitialatomicpopulation.
nonlinear interaction is needed either to build a two-
All the atoms are in fact assumed to be initially in state
photon gate operation [14, 15, 21, 22] or at the detec-
1 so that they directly feel the effect of the probe field
tion stage in linear optics quantum computation [23]. It
| i
only, while the effect of the trigger field is only indirect.
should also be mentioned that the generation of single-
Due to this inherent asymmetry, the dynamics experi-
photons(whichisalsonecessaryinlinearopticsquantum
enced by probe and trigger fields are always different,
computation) also relies on nonlinear interactions.
even when the corresponding parameters (Rabi frequen-
In this paper we perform a semiclassical analysis of
the interaction of light with atoms in the M configura- cies, decay rates, detunings) are equal. The symmetric
version of this scheme will be analyzed in Sec. III.
tion, inwhichthe amplitude ofthe four fields involvedis
The detunings δ (see Fig. 2) are defined as follows
described in terms of the corresponding Rabi frequency. i
The aim is to estimate the effects of noise sources, such
E E = h¯ω +h¯δ (3a)
asdephasingandspontaneousemission,bothonthenon- 2− 1 1 1
linear interaction and on group velocity matching. The E2 E3 = h¯ω2+h¯δ2 (3b)
−
semiclassical regime offers a clear picture of the physical E E = h¯ω +h¯δ (3c)
4 3 3 3
−
aspects involvedin EIT-based nonlinear optics, and well
E E = h¯ω +h¯δ , (3d)
4 5 4 4
describes a number of recent experiments [13, 24]. To −
thisendweconsidertwodifferentconfigurationsofatom- where E , (i=1,...,5) is the energy of level i , and ω
i i
field interactions, which we will call the asymmetric (see is the frequency of the field with Rabi frequen|cyi Ω .
i
Fig. 2) and the symmetric (see Fig. 10) M scheme. The The Hamiltonian of the system is
paper is thus composed of two main parts. In Sec. II
we describe the physics of the asymmetric M-scheme. 5
We start by defining the system and calculating the sus- HA = Ei i i +h¯ Ω1e−iω1t 2 1 +Ω2e−iω2t 2 3
| ih | | ih | | ih |
ceptibilities using an approximate treatment employing i
X (cid:0)
amplitude equations. These analytical calculations are +Ω e−iω3t 4 3 +Ω e−iω4t 4 5 +h.c. , (4)
3 4
thencomparedwiththe resultsofthenumericalsolution | ih | | ih |
of the full system of Blochequations. Finally, the condi- (cid:1)
tionsforgroupvelocitymatchingareanalyzed. InSec.III where h.c. denotes the hermitian conjugate. Moving to
3
become
j4i
PSfragrepla
ements
j2i ΓAV
Æ3 Æ4 b˙1 = − 12 b1−iΩ⋆1b2, (9a)
Æ1 Æ2 b˙ = ΓA2V +iδ b iΩ b iΩ b , (9b)
2 1 2 1 1 2 3
− 2 − −
(cid:10)4 (cid:18) (cid:19)
ΓAV
(cid:10)3 b˙ = 3 +iδ b iΩ⋆b iΩ⋆b , (9c)
(cid:10)2 3 − 2 12 3− 2 2− 3 4
(cid:10)1 (cid:18) (cid:19)
ΓAV
b˙ = 4 +iδ b iΩ b iΩ b , (9d)
4 13 4 3 3 4 5
− 2 − −
(cid:18) (cid:19)
j5i ΓAV
b˙ = 5 +iδ b iΩ⋆b . (9e)
j3i 5 − 2 14 5− 4 4
(cid:18) (cid:19)
j1i
The system’s initial state is assumed to be the ground
FIG. 2: Asymmetric M scheme. The probe and the trig-
state 1 . SinceanefficientXPMrequiresadispersivein-
ger fields, with Rabi frequencies Ω1 and Ω3 respectively, to- | i
teraction,we tailorthe dynamics in sucha waythat this
gether with the stronger pump fields, the coupler and the
initial condition on the populations remains essentially
tuner (with Rabi frequencies Ω and Ω , respectively) drive
2 4
the corresponding transitions. All the atoms are assumed to unaltered, even when the system reaches the steady-
be in state |1i and thedetuningsare definedin Eqs. (3). state, i.e.,
bss 1. (10)
1 ≃
the interaction picture with respect to the following free TothisendweassumethatthecontrolfieldΩ isstronger
2
Hamiltonian then the probe field Ω , with the system being approxi-
1
mately on Raman resonance for the first and the second
H = E 1 1 +(E ¯hδ )2 2 +(E ¯hδ )3 3
0 1 2 1 3 12
| ih | − | ih | − | ih | Λ subsystems (δ δ and δ δ ). Equations (9) are
1 2 3 4
+(E ¯hδ )4 4 +(E ¯hδ )5 5, (5) ∼ ∼
4 13 5 14 then solved in the steady-state. In order to get a consis-
− | ih | − | ih |
tent expression for the nonlinear susceptibilities one has
where to considerhigherordercontributionsto Eq.(10), which
is obtained by imposing the normalization of the atomic
δ12 = δ1−δ2, (6a) wave-functionofEq.(8)atsecondorderin|Ω1/Ω2|. One
δ = δ δ +δ , (6b) gets the following expression for the steady state ampli-
13 1 2 3
−
tudes
δ = δ δ +δ δ , (6c)
14 1 2 3 4
− −
we get the following effective Hamiltonian Ω 2 d 2+ Ω 2
1 3 2
bss = 1 | | | | | | , (11a)
HeAfSf =h¯δ1|2ih2|+h¯δ12|3ih3|+h¯δ13|4ih4|+h¯δ14|5ih5| 1 − 2|d2hd3−|Ω2|2|2 i
++hh¯¯ΩΩ⋆11||21iihh12||++h¯h¯ΩΩ2⋆2||23iihh32||++h¯h¯ΩΩ3⋆3||43iihh34||++h¯h¯ΩΩ4⋆4||45iihh54||. (7) bs2s = Ω1d3(cid:2)|Ω4|2−dD4da5(cid:3)+|Ω3|2d5bs1s (11b)
Ω 2 d d
bss = Ω Ω⋆| 4| − 4 5bss (11c)
3 − 1 2 D 1
B. Amplitude variables approach a
Ω Ω⋆Ω d
bss = 1 2 3 5bss (11d)
We now study the dynamics driven by Eq. (7). How- 4 − Da 1
ever, we have to include the effects of spontaneous emis- bss = Ω1Ω⋆2Ω3Ω⋆4bss, (11e)
sion and dephasing, and we first treat them in a phe- 5 D 1
a
nomenological manner by including decay rates ΓAV for
i
where we have defined
each atomic level i in the equations for the amplitude
| i
variables (AV) of the atomic wave-function. From anin-
d =δ ıΓAV/2, (12a)
tuitive point of view, for the excited levels 2 and 4 2 1− 2
these rates describe the total spontaneous d|eciay rat|esi, d3 =δ12−ıΓA3V/2, (12b)
whileforthegroundstatestheassociateddecayratesde- d =δ ıΓAV/2, (12c)
scribe dephasing processes [8]. Therefore, the evolution d4 =δ13−ıΓA4V/2, (12d)
equations for the amplitudes b (t) of the atomic state 5 14− 5
i
5
ψ(t) = b (t)i (8)
i
| i | i D = d d Ω 2 d d Ω 2 d d Ω 2. (13)
Xi=1 a 2 3−| 2| 4 5−| 4| − 2 5| 3|
(cid:2) (cid:3)(cid:2) (cid:3)
4
These results can be used to determine the probe and third-order cross-Kerr susceptibilities χ(3,ck). Eqs. (15)
P,T
trigger susceptibilities, which are defined as clearly show the asymmetry of the scheme between the
probe and trigger fields, with the latter possessing a
χ = Nµ12 bssbss,⋆ = N|µ12|2bssbss,⋆, (14a) nonzero cross-Kerr susceptibility only. This is a conse-
P Vε 2 1 −V¯hε Ω 2 1 quence of the asymmetry of the population distribution,
0 1 0 1
E
Nµ N µ 2 which essentially remains in the ground state 1 all the
χT = Vε 34 bs4sbs3s,⋆ =−V¯h|ε34Ω| bs4sbs3s,⋆, (14b) time. This means that the trigger field drives a| viirtually
0 3 0 3
E empty transition,hence the contributiontothe suscepti-
whereN isthenumberofatomsinteractingwiththeelec- bility comes only from higher order (see [15] for discus-
tromagnetic field, V is the volume occupied by the gas, siononthe link betweenthe populationdistributionand
andε isthevacuumdielectricconstant. Dopplerbroad- a linear contribution to susceptibility). It will be shown
0
ening is neglected here. It is well known that first order in Sec. III that the symmetric M-scheme brings about
Doppler effect can be cancelled by using co-propagating both a linear and a self-Kerr contribution to the trigger
laser fields [6]. In particular we emphasize that this is susceptibility.
valid for cold atomic media in a magneto-opticaltrap as
ByusingEqs.(11)andthedefinitionsofEqs.(12)into
well as for a standard gas cell.
Eqs. (14), and comparing with Eqs. (15) at the corre-
Inserting Eqs. (11) into Eqs. (14) and expanding in
sponding orderin the electric fields, one getsthe explicit
seriesatthelowestordersintheprobeandtriggerelectric
dependenceofthelinearandnonlinearsusceptibilitiesas
fields, and respectively, one gets
E1 E3 a function of the system parameters, i.e.,
χ χ(1)+χ(3,sk) 2+χ(3,ck) 2 (15a)
P ≃ P P |E1| P |E3|
χT ≃ χT(3,ck)|E1|2, (15b) χ(1) = N|µ12|2 δ12−iΓA3V/2
P V¯hε δ iΓAV/2 δ iΓAV/2 Ω 2
where we have introduced the linear susceptibility χ(P1), 0 1− 2 12− 3 −| 2(|16)
(3,sk) (cid:0) (cid:1)(cid:0) (cid:1)
the third-order self-Kerr susceptibility χ and the for the probe linear susceptibility, and
P
χ(3,sk) = N|µ12|4 − δ12−iΓA3V/2 δ12−iΓA3V/2 2+|Ω2|2 , (17a)
P V¯h3ε0 δ1−iΓA2V/2 δ12−(cid:0) iΓA3V/2 −|Ω(cid:1)2h|(cid:12)(cid:12)2 δ1−iΓA2V(cid:12)(cid:12)/2 δ12−iiΓA3V/2 −|Ω2|2 2
χ(3,ck) = N|µ12|2|(cid:2)µ(cid:0)34|2 (cid:1)(cid:0) (cid:1) |Ω2(cid:3)|(cid:12)(cid:12)2(cid:0) δ14−iΓA5V/(cid:1)2(cid:0) (cid:1) (cid:12)(cid:12) (,17b)
P V¯h3ε0 δ1−iΓA2V/2 δ12−iΓA3V/2 −|Ω(cid:0)2|2 2 δ13−iΓ(cid:1)A4V/2 δ14−iΓA5V/2 −|Ω4|2
N µ 2 µ 2(cid:2)(cid:0) (cid:1)(cid:0) (cid:1)Ω 2 δ (cid:3) h(cid:0)iΓAV/2 (cid:1)(cid:0) (cid:1) i
χ(3,ck) = | 12| | 34| | 2| 14− 5 (,17c)
T V¯h3ε0 δ1−iΓA2V/2 δ12−iΓA3V/2 −|Ω(cid:0)2|2 2 δ13−iΓ(cid:1)A4V/2 δ14−iΓA5V/2 −|Ω4|2
(cid:12)(cid:0) (cid:1)(cid:0) (cid:1) (cid:12) h(cid:0) (cid:1)(cid:0) (cid:1) i
(cid:12) (cid:12)
for the third-order nonlinear susceptibilities. The typical EIT situation we are considering in which
two cross-Kerr susceptibilities are identical whenever Ω is large enough. In fact, when Ω 2
2 2
the quantity δ iΓAV/2 δ iΓAV/2 Ω 2 is | δ| iΓAV/2 δ iΓAV/2 , one has [14] | | ≫
1− 2 12− 3 − | 2| 1− 2 12− 3
(at least approximately) real. This happens in the
(cid:0) (cid:1)(cid:0) (cid:1) (cid:12)(cid:0) (cid:1)(cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
N µ 2 µ 2 δ iΓAV/2
χ(3,ck) =χ(3,ck) = | 12| | 34| 14− 5 . (18)
P T V¯h3ε0 |Ω2|2 δ13−iΓA4V/2 δ14−iΓA5V/2 −|Ω4|2
h(cid:0) (cid:1)(cid:0) (cid:1) i
We shall see in the next subsection that these approxi- of the system.
mateexpressionsforthenonlinearsusceptibilitiesfitvery TheasymmetricM-schemecanbeseenasanextension
well with the numerical solution of the exact dynamics ofthefourlevelN-schemeintroducedinRef.[5],withthe
5
addition of the coupling to an additional level 5 pro- tions (OBE), which allow to describe spontaneous emis-
| i
videdbythe tunerfield withRabifrequencyΩ . Infact, sion and dephasing rigorously and no more phenomeno-
4
it easy to check that upon setting Ω = 0 in Eq. (17b), logicallyasinthe AVtreatmentpresentedinthe preced-
4
one recovers the third-order nonlinear susceptibility of ingsubsection. Weconsidersixspontaneousdecaychan-
the four-level N-scheme derived in Refs. [5, 13]). As we nels,i.e.,thedecayoftheexcitedstate 2 ontothethree
| i
willseebelow,theroleofthetunerfieldistoenableafine ground state sublevels 1 , 3 , and 5 with rates Γ ,
21
| i | i | i
tuning of the group velocities, in order to achieve group Γ andΓ respectively,andthe correspondingdecayof
23 25
velocitymatchingbetweenprobeandtrigger[14,16,18]. the excitedstate 4 ontothe three sublevels 1 , 3 ,and
| i | i | i
5 with rates Γ , Γ and Γ respectively. Moreover
41 43 45
| i
we consider dephasing of the each level i with dephas-
| i
C. Comparison with the Optical Bloch Equations ing rate γ , so that the master equation for the atomic
ii
density operator ρ is given by
We now study the dynamics of the asymmetric M
scheme of Fig. 2 by means of the optical Bloch equa-
5
i Γ γ
ρ˙ = HAS,ρ + lk 2σˆ ρσˆ† σˆ† σˆ ρ ρσˆ† σˆ + kk (2σˆ ρσˆ σˆ ρ ρσˆ ), (19)
−¯h eff 2 kl kl− kl kl − kl kl 2 kk kk − kk − kk
(cid:2) (cid:3) lX=2,4k=X1,3,5 (cid:16) (cid:17) Xk=1
whereHAS isgivenbyEq.(7)andσˆ = k l . Thecor- this reasonwe will analytically derive from the OBE the
eff kl | ih |
respondingsystemofOBE’sforthemeanvaluesσ (t) probe linear susceptibility only, and we will then use the
ij
≡
σˆ (t) ρ (t)is displayedinAppendix AasEqs.(A1) OBEonly for the numericaldeterminationof the atomic
ij ji
h i≡
and(A2),wherewehavedefinedforconveniencethetotal steady state. Additionally, deriving this result will en-
decay rates able us to draw a formal analogy between the AV and
OBE treatments (see Eqs. (26) below).
Γ = Γ +Γ +Γ , (20) The probe susceptibility is defined in terms of the
2 21 23 25
Γ4 = Γ41+Γ43+Γ45, (21) atomic coherence σ12 as (see also Eq. (14a))
and the composite dephasing rates Nµ12 N µ12 2
χ = σ = | | σ . (24)
P 12 12
Vε −V¯hε Ω
0 1 0 1
E
γ =γ +γ , i=1...5. (22)
ij ii jj
UsingEqs.(23)andperformingaseriesexpansionatthe
The OBE for the M scheme are quite involved and less lowest order in the probe and trigger fields, we arrive
suited for an approximate analytical treatment with re- at an approximate solution for σ , which, inserted into
12
spect to the AV equations of the preceding subsection. Eq.(24), givesthe followingexpressionforthe probelin-
In fact, if we consider again the condition Ω /Ω 1 ear susceptibility
1 2
| | ≪
and, consistently with Eq. (10), we assume that
N µ 2 δ iγ /2
(1) 12 12 13
χ = | | − .
σ11 ≈ 1, (23a) P V¯hε0 [δ12−iγ13/2][δ1−i(Γ2+γ12)/2]−|Ω2|2
σ 0, j =2,...,5, (23b) (25)
jj
≈ By comparing Eq. (25) with Eq. (16), one can imme-
at the steady state, it is possible to see that by insert- diately see that the AV and OBE predictions for the
ing Eqs. (23) into Eqs. (A2) for the coherences, one gets probe linear susceptibility coincide provided that the
a satisfactory expression for the probe linear suscepti- phenomenological decay rates ΓAV are appropriately in-
i
bility only. To be more specific, only the approximate terpreted, i.e.,
linear susceptibility fits well with the numerical solution
of the OBE, while it turns out to be extremely difficult ΓA2V ↔ Γ2+γ12, (26a)
to deriveanalyticalexpressionsfromEqs.(A1)and(A2) ΓAV γ . (26b)
3 ↔ 13
for the nonlinear susceptibilities, as simple as those of
Eqs. (17), and which reproduce in the same way the ex- This comparison shows therefore that the AV approach
actnumericalsolutionintheEITregimewearestudying. provides a treatment of the atomic dynamics simpler
Obviously, one can exactly solve analytically the OBE, than the OBE’s approach, but roughly equivalent, and
but the resulting expressions are very cumbersome and that the intuitive interpretation of its phenomenologi-
notphysicallytransparentsuchasthoseofEqs.(17). For caldecay ratesΓAV as spontaneous emissiontotaldecay
i
6
ratesfortheexcitedstates,andasdephasingratesinthe s)0.04
case of ground state sublevels, is essentially correct, es- unit0.03
y 0.02
pecially in the typical case in which dephasing rates are ar0.01
muchsmallerthanspontaneousemissiondecayrates(see bitr 0
EanqWds.ew(2et6hc)eo)n.mcpoanrseidietrwtihthe nthuemaerniaclayltiscoalulttiroenatomfetnhtebOasBeEd (1)χ} (arRe{P−−−−0000....00004321 NAnuamlyetriiccaall
−5 −4 −3 −2 −1 δ0/Γ 1 2 3 4 5
on the AV approach presented above. The numerical 1 4
calculations are performed in the range of parameters s)0.07
correspondingto EIT, i.e., |Ω1|,|Ω2|≪|Ω3|,|Ω4| and we y unit00..0056 NAnuamlyetriiccaall
stay near two-photon resonance for both the probe and ar
the trigger field. In Figs. 3-6 we compare the analytical bitr0.04
ar0.03
ssoolluuttiioonnsooffthEeq.co(m16p)leatnedseEtqosf.B(1lo7c)hweiqthuatthioennsugmiveernicianl (1)χ} (P00..0012
the Appendix A. From these plots it is evident that our m{ 0
I −5 −4 −3 −2 −1 0 1 2 3 4 5
δ/Γ
analytical treatment works satisfactorily well, except for 1 4
a small interval of values of the detuning, corresponding
to the maximum probe (or trigger) absorption. In such FIG.3: Comparisonofthenumericalsolution(dottedline)of
a case, the detunings match the Rabi frequencies of the theOBE with theanalytical prediction of Eq.(16) (full line)
two pumps, and the probe (or trigger) field is in reso- fortherealpart(above)andimaginarypart(below)ofthelin-
earprobesusceptibilityversusthenormalizedprobedetuning
nance with a single atomic transition. The atoms are
δ /Γ . Theparameterusedarethefollowing: ΓAV =Γ =36
significantly pumped to the excited levels and the pop- 1 4 2 2
MHz, ΓAV = Γ = 38 MHz, δ = δ = δ = 0, ∀ i,j
uthlaetidoinscraespsuanmcpytiboentwoefeEnqt.he(1e0x)aicstnnoutmfeurlificlalelds.oluIntiofnacotf, γij = Γ4A3V = Γ4A5V = 10−4Γ4, Ω21 = 30.08Γ44, Ω2 = 2Γ4,
Ω =0.04Γ , Ω =Γ .
3 4 4 4
the OBE and the AV approach is strictly related to the
atomic population out of level 1 which, in the case of
| i
Fig. 3, is about 14% of the total population.
Figs.3-6refertoasituationwithsmalldephasingrates so that, using Eqs. (15), the nonlinear cross-phase shift
(∀i,j, γij =ΓA3V =ΓA5V =10−4Γ4 ∼few kHz)whichare for the two fields of interest is given by
typical for not too dense gases. For larger values of the
dephasingrates(sometensofkHz),wehaveseenthatthe ω l
φck = 1 dzRe[χ3,ck] ε (z,t)2, (30a)
analyticalpredictionoftheAVapproachofthepreceding P 2c P | T |
subsectionstartstodepartfromtheexactsolutionofthe Z0
ω l
OBE. φck = 3 dzRe[χ3,ck] ε (z,t)2, (30b)
T 2c T | P |
Z0
where l is the length of the atomic medium. These non-
D. Group velocity matching
linear cross-phase shifts are of fundamental importance
alsoforquantuminformationprocessingapplications. In
The propagation equation for the slowly varying elec-
fact,theCPSofEq.(1)isdeterminedonlybythesecross-
tric field amplitudes ε (z,t), i=P,T, defined as
i Kerr contributions to the total phase shift, because the
linearandself-Kerrcontributionscancelout,asshownin
(z,t)=ε (z,t)exp ik z iω t +c.c. i=P,T,
i i i i
E { − } Refs. [14, 15].
is given by For Gaussian probe and trigger pulses of time dura-
tions τ and τ , and with peak Rabi frequencies Ωpeak
P T P
∂ + 1 ∂ ε (z,t)=ikiχ (z,t)ε (z,t), i=P,T, and Ωpeak respectively, the nonlinear cross-phase shifts
(cid:18)∂z vgi ∂t(cid:19) i 2 i i can beTwritten as (see also Refs. [14, 15])
(27)
wc/h(e1r+e vngiig)is, wthitehgcrotuhpevsepleoecdityo,fgliegnhetrainllyvadceufiunmedaansdvgi = φcPk = ω41cl√π¯hµ2|3Ω4pT2eak|2 erfζ[PζP]Re[χ3P,ck], (31a)
| |
nig = 21Re[χi]+ ω2i ∂R∂eω[χi] (28) φcTk = ω43cl√π¯hµ2|1Ω2pP2eak|2 erfζ[TζT]Re[χ3T,ck], (31b)
(cid:18) (cid:19)ωi | |
the group index, ωi being the frequency of field i. The where ζP = (1−vgP/vgT)√2l/vgPτT and ζT is obtained
solution of Eq. (27) is from ζP upon interchanging the indices P T. Large
↔
nonlinear cross-phase shifts take place for appreciably
ε (z,t)=ε (0,t z )exp iki zdz′χ (z′,t) , (29) large values of the two cross-Kerr susceptibilities real
i i − vi 2 i parts, and especially when probe and trigger velocities
g (cid:26) Z0 (cid:27)
7
(3)(3)χχRe{} (arbitrary units)m{} (arbitrary units)PP−−−−−−−0240124243215xx 1100−−77−4 −3 −2 −1 δ10/Γ4 1 2 3 NANAnunuamamllye4yetrtiriicicccaaaallll 5 (3)(3)χχRe{} (arbitrary units)m{} (arbitrary units)S−KS−K−−−−−−111−012345024685432120245xx 1100−−77−4 −3 −2 −1 δ10/Γ4 1 2 3 NANAnunuamamlly4eyettririicicccaaaallll 5
I −5 −4 −3 −2 −1 δ0/Γ 1 2 3 4 5 I −5 −4 −3 −2 −1 δ0/Γ 1 2 3 4 5
1 4 1 4
FIG. 4: Comparison of the numerical solution (dotted line) FIG. 5: Comparison of the numerical solution (dotted line)
of the OBE with the analytical prediction of Eq. (17b) (full of the OBE with the analytical prediction of Eq. (17a) (full
line) for the real part (above) and imaginary part (below) line) for the real part (above) and imaginary part (below) of
of the probe cross-Kerr susceptibility versus the normalized theprobeself-Kerrsusceptibilityversusthenormalizedprobe
probe detuning δ1/Γ4. To reduce as much as possible the detuning δ1/Γ4. To reduce as much as possible the influence
influence of the self-Kerr susceptibility we have considered a of the cross-Kerr susceptibility we have considered a trigger
probeRabifrequencyΩ1muchsmallerthanthatofthetrigger RabifrequencyΩ3 muchsmallerthanthatoftheprobefield.
fiMeHldz.,Pδ2ar=amδe3te=rsδa4re=: Γ0,A2V∀ =i,jΓγ2ij==36ΓA3MVH=z,ΓΓA5A4VV ==Γ104−=4Γ348, PMaHrazm, δe2te=rsδa3re=: δΓ4A2=V 0=, ∀Γ2i,j=γij36=MΓA3HVz,=ΓA4ΓVA5V==Γ140−=4Γ348,
Ω1 =0.004Γ4, Ω2 =2Γ4, Ω3 =0.04Γ4, Ω4 =Γ4. Ω1 =0.5Γ4, Ω2 =2Γ4, Ω3 =0.005Γ4, Ω4 =Γ4.
where [14]
become equal, i.e., when ζ 0, in which case the
P,T → δ2 + Ω 2 δ δ Ω 2 2 δ2 ΓAV/2 2
erf[ζ]/ζ reaches the maximum value 2/√π. In this limit 14 | 4| 13 14−| 4| − 14 4
β = .
the cross-phase phase shifts linearly increase with the (cid:0) (δ δ(cid:1)h(cid:0) Ω 2)2+δ2(cid:1) ΓAV/2(cid:0)2 2 (cid:1) i
lengthoftheatomicmediuml. Thisexplainswhyachiev- 13 14−| 4| 14 4
ing groupvelocity matching, vgP =vgT, is of fundamental h (cid:0) (cid:1) i (33)
importance. Moreover group velocities become small for Inthe EIT situationwe areconsideringitis nP,nT 1,
g g ≫
large group indices and this condition can be achieved so that, using Eqs. (32),
within the EIT transparency window, where Re[χ] van-
ishes, and the group velocity is strongly reduced due to vP c 2h¯ǫ0c|Ω2|2 , (34a)
a large dispersion gradient ∂Re[χ]/∂ω. g ≃ nP ≃ (N/V)µ 2ω (1+ Ω 2β)
g | 12| 1 | 3|
c 2h¯ǫ cΩ 2
Letusseehowsmallandequalprobeandtriggergroup vT 0 | 2| . (34b)
velocitiescanbe obtained. We considerthe approximate g ≃ nTg ≃ (N/V)|µ34|2ω3|Ω1|2β
analyticalexpressionsforthesusceptibilitiesofEqs.(15)-
(17) derived above within the AV approach, and which As expected, the asymmetric M-scheme does not yield
we have seen to work very well in the EIT regime. As- equalslowdownofbothtriggerandprobepulseautomat-
suming to stayat the center ofthe transparencywindow ically as,for example, the scheme ofPetrosyanandKur-
for the probe (δ = 0) where the dispersion gradient izki [16] does. In fact, the two expressions of the group
12
is maximum, and neglecting dephasing rates ΓAV and velocities are generally different. Nonetheless, Eqs. (34)
3
ΓAV,whicharetypicallymuchsmallerthanalltheother show that group velocity matching is always achievable
5
parameters, one gets by properlyadjusting the parameterβ, whichmeans ad-
justing the tuner intensity Ω 2 and the composite de-
4
| |
tuning δ . This shows that the present asymmetric M-
14
schemecanbeseenasamodifiedversionoftheN-scheme
N µ 2ω
nP | 12| 1 (1+ Ω 2β), (32a) of Ref. [5], in which the tuner pump field is added just
g ≃ V 2h¯ǫ Ω 2 | 3|
0 2 in orderto “tune” the groupvelocity ofthe triggerpulse
| |
nT N |µ34|2ω3 Ω 2β, (32b) so to make it equal to that of the probe. The possibility
g ≃ V 2h¯ǫ Ω 2| 1| to achieve group velocity matching is shown in Fig. 7,
0 2
| |
8
s) 4x 10−8 4000
nit 3 Numerical
bitrary u 012 Analytical 3000
3)} (ar −−21 2000
(χRe{t−−43 m/s)
−5x 10−8−4 −3 −2 −1 δ30/Γ4 1 2 3 4 5 Vg(p,t1000
s) 8
unit Numerical 0
y 6 Analytical
ar
bitr 4 −1000 trigger numerical
ar trigger analytical
3)} ( 2 pprroobbee naunmalyetriiccaall
(χm{t0 −200200.03 20.034 20.038 20.042 20.046 20.05
I −5 −4 −3 −2 −1 δ0/Γ 1 2 3 4 5 δ3/Γ4
3 4
FIG.7: Groupvelocityoftheprobeandtriggerpulsesversus
FIG. 6: Comparison of the numerical solution (dotted line)
the normalized trigger detuning δ /Γ . Full lines denote the
of the OBE with the analytical prediction of Eq. (17c) (full 3 4
analytical predictions of Eqs. (34) (the thick line refers to
line) for the real part (above) and imaginary part (below)
the probe and the thin line to the trigger). Circles and dots
of the trigger cross-Kerr susceptibility as a function of the
refer tothenumerical solution of theOBEfor theprobeand
normalized trigger’s detuning δ /Γ . Parameters are similar
3 4
to those of Fig. 3, ΓAV = Γ = 36 MHz, ΓAV = Γ = 38 triggergroupvelocity,respectively. Thisfigureshowshowitis
2 2 4 4
MHz, δ1 = δ2 = δ4 = 0, ∀ i,j γij = ΓA3V = ΓA5V = 10−4Γ4, pMo-ssscibhleemteo:obtwtaoindigffreoruepntvevlaolcuiteys mofatδcheinxgisitnftohrewahsyicmhmvePtr=ic
Ω1 =0.08Γ4,Ω2 =2Γ4, Ω3 =0.04Γ4, Ω4 =Γ4. 3 g
vT ≃1000 m/s. The parameters are those of the D and D
g 1 2
line in the 87Rb spectrum: ΓAV = Γ ≃ 36 MHz, ΓAV =
2 2 4
Γ ≃ 38 MHz, δ = δ = 0, δ ≃ δ ≃ 20Γ , Ω = 0.08Γ ,
4 1 2 4 3 4 1 4
where both the numerical result derived form the OBE Ω2 =2Γ4, Ω3 =0.04Γ4, Ω4 =Γ4, ∀ i,j, γij = ΓA3V = ΓA5V =
10−4Γ , N/V =3.0·1013 cm−3.
and the approximate analytical expressions of Eqs. (34) 4
are plotted versus the trigger detuning δ . Two different
3
values of δ exist for which vP = vT 1000 m/s (see
3 g g ≃
Fig.7). Parametervaluesherecorrespondto typicalval-
ity.
ues for a cell of 87Rb atoms, i.e., ΓAV = Γ 36 MHz,
ΓAV =Γ 38MHz,N/V 3 10123cm−3,2δ≃=δ =0, It will be shown in this Section that the above con-
4 4 ≃ ≃ × 1 2 clusion is an artifact of approximations made to obtain
δ δ 20Γ , Ω = 0.08Γ , Ω = 2Γ , Ω = 0.04Γ ,
4 3 4 1 4 2 4 3 4
Ω ≃= Γ≃, i,j, γ = ΓAV = ΓAV = 10−4Γ . More- a closed and compact expression for group velocities. In
4 4 ∀ ij 3 5 4 particular, the pulse propagation in this approximation
over Fig. 7 clearly shows that the simple expressions of
is described by Eqs. (27), with group velocities vi given
Eqs. (34) well reproduce the exact numerical solution of g
byEqs.(34)andnonlinearsusceptibilitiesχ beingthose
the OBE. i
of Eqs. (17). This is equivalent to the adiabatic elimi-
nation of the atomic degrees of freedom. Such adiabatic
elimination, strictly speaking, is not valid in the param-
E. Pulse propagation eter regime explored in this paper: strong nonlinear in-
teractionbetweenprobe andtriggerpulses suggeststhat
In previous Section, we have addressed the problem the contribution of the atomic medium is far from being
of group velocity matching between probe and trigger adiabatic. Also, it should be noted that the dephasing
fields in the asymmetric M-scheme. It should be empha- processeshavebeenneglectedinthederivationofvi. For
g
sized that the analysis and the results presented there the adiabatic case, the above conclusion is correct: the
are strictly valid for the continuous-wave (cw) fields. trigger pulse suffers anomalous dispersion and its group
We would now address that same problem but with the velocity becomes singular towards the edges of a probe
pulsed probe and trigger fields in mind. At the first pulse. However, pulse propagation through the asym-
look,Eqs.(34)appearto suggestthatthe groupvelocity metricM-systemdonotfollowsuchasimpleapproximate
matchingwouldnotbepossibleinthepulsedregime. As evolution. Fullpropagationproblemmustthenbesolved
the group velocity of the trigger pulse is inversely pro- which includes adding the time-dependent equations for
portionaltothe squareofthe probepulse, triggersuffers the pulses
anomalous dispersion, i.e. in the presence of a pulsed
probe, the trigger pulse will get distorted, splitting into ∂ 1 ∂ kiNµi
+ ε (z,t)=i σ (z,t), i=P,T,(35)
severalcomponents, each having a different groupveloc- ∂z c∂t i 2 Vǫ i
(cid:18) (cid:19) 0
9
FIG. 8: (Color online) Propagation of probe and trigger pulses through the asymmetric M medium. Pulses are taken to be
Gaussian at time t=0 and are sufficiently long (τi >1/∆ωtir),i=P,T. Units are arbitrary, with c=1.
FIG. 9: (Color online) Propagation of probe and trigger pulses through the asymmetric M medium. Pulses are taken to be
Gaussian at time t=0 and short (τi<1/∆ωtir), i=P,T. Unitsare arbitrary,with c=1.
totheOBEs[Eqs.(A1,A2)],andnumericallysolvingthe pulse distortion comes from the absorption, as the pulse
resultingsystemofequations. Intheaboveequation,itis spreads outside of the transparency window. Trigger
understood that σ =σ and σ =σ in the notation pulse shows the same absorption effect, but moreover it
P 12 T 34
of Eqs. (A2). also splits into several components which then continue
to propagatewith adifferentgroupvelocities each. Note
Results are shown in Figs. 8 and 9 for the same set
that the singularity present in the adiabatic approach is
of parameters that yields group velocities matching in
not present here. This is due to the fact that the de-
Fig. 7. Vertical axes have been scaled appropriately to
phasing,neglectedinthe adiabatictreatment,effectively
obtainRabifrequenciesΩ . Twooperatingregimescould
i regularizes the equations, removing the singularity.
be identified, long-pulse regime and short-pulse regime,
where ‘long’ and ‘short’ denotes the pulses’ length in
time. This length is compared to the inverse width of
the transparency window. Long pulses fit well into the
transparency window, while short pulses do not. Fig. 8
showstheresultsofoursimulationfortheinitiallyidenti-
It is also noted that in the long-pulse regime, both of
callongGaussianpulses. Itisclearthatthepulsesprop-
the pulses propagate virtually undistorted, with a group
agate undistorted with the equal group velocities. Tiny
velocity uniform across each of the pulses. Our simula-
amplitude decay is present due to the small imaginary
tions suggest that the approximate Eqs. (34) are valid,
part of the nonlinear susceptibility.
as long as the Rabi frequencies there are considered to
Short pulses (Fig. 9) however, show distortion. Probe be taken at the peak of the pulse, i.e. Ω Ωpeak.
i → i
10
III. THE SYMMETRIC M–SCHEME
j4i
PSfragrepla
ements
j2i
In this section we analyze the symmetric M–scheme, Æ4
schematically shown in Fig. 10. The initial conditions Æ3
Æ1
and the configuration of the fields are slightly different Æ2
from those of the asymmetric case of Sec. II. The same
(cid:10)4
five levels could be used, but all the atoms are now ini- (cid:10)3
tially prepared in level 3 (see Fig. 10). Moreover, the (cid:10)1
| i
roleofthe probeandofthe couplerfields areexchanged, (cid:10)2
i.e., now the probe field (still with Rabi frequency Ω
1
andcentralfrequencyω )coupleslevels 2 and 3 ,while
1
thecoupler(stillwithRabifrequencyΩ| aindce|ntiralfre- j5i
2
quencyω2)inducestransitionsbetweenlevels 1 and 2 . j3i
| i | i
The role of trigger and tuner fields remains unchanged.
j1i
In such a way, the scheme becomes symmetric for probe
FIG. 10: Symmetric M scheme. The probe and the trig-
and trigger, and the two fields experience exactly the
ger fields, with Rabi frequencies Ω and Ω respectively, to-
same dynamics whenever the corresponding parameters 1 3
gether with the stronger pump fields, the coupler and the
are made equal, i.e., when the Rabi frequencies are cor-
tuner (with Rabi frequencies Ω and Ω , respectively) drive
2 4
respondingly equal (Ω = Ω , Ω = Ω ), as well as the
1 3 2 4 the corresponding transitions. All the atoms are assumed to
detunings,(δ =δ ,δ =δ ),whicharenowdefinedsim-
1 3 2 4 bein state |3i and thedetuningsare defined in Eqs. (36).
ilarlytothoseoftheasymmetricM scheme(seeEqs.(3))
except for probe-coupler exchange, i.e.,
E E = h¯ω +h¯δ , (36a)
2 1 2 2 we get the following effective Hamiltonian
−
E E = h¯ω +h¯δ , (36b)
2− 3 1 1 HS =h¯δ 2 2 +h¯δ 1 1 +h¯δ 4 4 +h¯δ 5 5
E E = h¯ω +h¯δ , (36c) eff 1| ih | 12| ih | 3| ih | 34| ih |
4 3 3 3
− +h¯Ω 2 3 +h¯Ω 2 1 +h¯Ω 4 3 +h¯Ω 4 5
E E = h¯ω +h¯δ . (36d) 1| ih | 2| ih | 3| ih | 4| ih |
4− 5 4 4 +h¯Ω⋆ 3 2 +h¯Ω⋆ 1 2 +h¯Ω⋆ 3 4 +h¯Ω⋆ 5 4. (40)
1| ih | 2| ih | 3| ih | 4| ih |
In this way, the scheme can be seen again as formed
by two adjacent Λ, one for the probe and one for the
trigger, now however symmetrically placed with respect A. Amplitude Variables Approach
to state 3 . As we have done for the asymmetric M
| i
scheme, we assume to stay close to the two-photon res- We first study the system dynamics by means of the
onance conditions, δ1 δ2 and δ3 δ4, and more- AVapproach,inwhichthe stateoftheatomisdescribed
≃ ≃
over that Ω1 Ω2 , and Ω3 Ω4 , so that both by the wave-function of Eq. (8), whose time evolution
| | ≪ | | | | ≪ | |
probe and trigger will experience EIT. As we have seen is determined by the Hamiltonian of Eq. (40), supple-
above,alargeXPMisobtainedwhenthegroupvelocities mentedwith phenomenologicaldecayratesΓAV for each
i
are equal [10, 14, 15, 16, 18, 22], and the advantage of atomic level i . The corresponding evolution equations
the present symmetric M–scheme is that group velocity for the ampli|tuides b (t) are
i
matchingisautomaticallyachievedoncethatthescheme
is exactly symmetric between probe and trigger. b˙ (t) = ıd b (t) ıΩ⋆b (t), (41a)
1 − 1 1 − 2 2
The Hamiltonian of the system is b˙ (t) = ıd b (t) ıΩ b (t) ıΩ b (t), (41b)
2 2 2 2 1 1 3
− − −
5 b˙ (t) = ıd b ıΩ⋆b (t) ıΩ⋆b (t), (41c)
H = E i i +h¯ Ω e−iω1t 2 3 +Ω e−iω2t 2 1 3 − 3 3− 1 2 − 3 4
S i| ih | 1 | ih | 2 | ih | b˙ (t) = ıd b (t) ıΩ b (t) ıΩ b (t), (41d)
+XiΩ3e−iω3t|4ih3(cid:0)|+Ω4e−iω4t|4ih5|+h.c. . (37) b˙45(t) = −−ıd45b45(t)−−ıΩ3⋆4b34(t),− 4 5 (41e)
(cid:1) where,similarlytowhatwehavedonefortheasymmetric
Moving to the interaction picture with respect to the case, we have defined
following free Hamiltonian
d = δ ıΓAV/2, (42a)
H0′ = E3|3ih3|+(E2−¯hδ1)|2ih2|+(E1−¯hδ12)|1ih1| d21 = δ112−−ıΓA21V/2, (42b)
+(E4−¯hδ3)|4ih4|+(E5−¯hδ34)|5ih5|, (38) d3 = −ıΓA3V/2, (42c)
d = δ ıΓAV/2, (42d)
4 3− 4
where d = δ ıΓAV/2. (42e)
5 34− 5
δ = δ δ , (39a)
12 1 2 Since we choose again Ω /Ω 1 and Ω /Ω 1,
− 1 2 3 4
| | ≪ | | ≪
δ = δ δ , (39b) it is reasonable to assume that the atomic population
34 3 4
−