Table Of ContentManipulating Quantum States of Molecules Created via Photoassociation of
Bose-Einstein Condensates
Xiao-Ting Zhoua, Xiong-Jun Liub∗, Hui Jingc,d, C. H. Laib and C. H. Ohb†
a. Theoretical Physics Division, S. S. Chern Institute of Mathematics, Nankai University, Tianjin 300071, P.R.China
b. Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542
c. Department of Physics, The University of Arizona, 1118 East 4th Street, Tucson, AZ 85721
d. State Key Lab of Magnetic Resonance and Atomic and Molecular Physics,
Wuhan Institute of Physics and Mathematics, CAS, Wuhan 430071, P.R. China
(Dated: February 1, 2008)
We show the quantum state transfer technique in two-color photoassociation (PA) of a Bose-
Einstein condensate, where a quantized field is used to couple thefree-bound transition from atom
statetoexcitedmolecular state. Undertheweak excitation condition,wefindthatquantumstates
7 ofthequantizedfieldcanbetransferredtothecreatedmolecularcondensate. Thefeasibility ofthis
0
technique is confirmed by considering the atomic and molecular decays discovered in the current
0
PA experiments. The present results allow us to manipulate quantum states of molecules in the
2
photoassociation of a Bose-Einstein condensate.
n
a PACSnumbers: 34.50.Rk,03.75.Nt,32.80.Pj,33.80.Ps
J
4
1 Overthelastfewyears,aparticularlyimportantdevel- sate. Themulti-modeassociatinglightisappliedandthe
opment in ultracoldquantum systems has been the abil- effectofatomic/moleculardecaysisdiscussedindetailin
1 ity to form and manipulate quantum degenerate molec- thismodel. Thepresentresultconfirmsthe possibilityof
v
ular gases via photoassociation (PA) in Bose-Einstein quantum state transfer in the current PA experiments,
0
Condensates (BECs). Besides the creation of molecu- and will have wide applications to, e.g. quantum infor-
8
0 lar quantum gases with Feshbach resonances [1], a more mation science with cold molecules [9].
1 general method, say, a stimulated optical Raman tran-
0 sition (STIRAP) has been employed to directly produce
7
deeply bound molecules in recent years [2, 3]. STIRAP
0
was proposedas a promising way for a fast, efficient and
/
h robust process to convert a BEC of atoms into a molec-
p
ular condensate [4, 5, 6]. The central idea for this kind
-
t of STIRAP is the realization of the dark superposition
n
state of a BEC of atoms and a BEC of molecules, which
a
u has been observed in recent experiments [7].
q Although PA process has been widely studied in both
:
v theoretical and experimental aspects, the possibility of
i manipulating quantum states of the created cold molec-
X FIG. 1: Level scheme of two-color PA process induced via a
ulargasisnotwelldiscoveredinsuchprocess. Controlof quantizedandaclassicalfield. δand∆denotethedetunings.
r
a quantum states of molecular gas may be achieved by re-
alizingquantumstatetransferfromphotonstomolecules Turning to the situation of Fig. 1, we con-
formed in the PA process. Most recently, one of us sider the quasi one-dimensional model of N identical
studied the possibility of quantum conversion between atoms that have condensed into the same one-particle
lightandmoleculesviacoherenttwo-colorPAbyusinga state with wave vector k = 0. Through a quasi-
single-modeassociatinglight[8]. Hereweshallstudyfur- one dimensional quantized associating field Eˆ(z,t) =
ther the validity of quantum state transfer from a quan- ~ν Eˆ(z,t)exp(iν(z−ct)),PAthenremovestwoatoms
tizedassociatinglighttomoleculargasinthePAprocess. 2ǫ0L c
qfrom the state |bi, creating a molecule in the excited
In this letter, we identify the quantum state transfer
state |ei. Here Eˆ(z,t) is slowly varying amplitude, L
processintwo-colorPAofaBose-Einsteincondensateby
is the quantization length in the z direction and ν
quantizing associating light. Under the weak excitation
the frequency of associating field. Including a second
condition,weshowquantumstatesofthequantizedfield
coherent classical laser characterized by the coupling
canbe fullytransferredtothecreatedmolecularconden-
strength(Rabifrequency)Ω(t),bound-boundtransitions
remove excited molecules from state |ei and create sta-
ble molecules in state |ci. In second-quantized notation,
∗Electronicaddress: [email protected] we denote the boson field operators for atoms, primar-
†Electronicaddress: [email protected] ily photoassociated molecules, and stable molecules, re-
2
spectively, by φˆb,φˆe and φˆc. Atomic and quantized field φˆ†2φˆ = − i ∂ (φˆ†2φˆ)− i (γ +i∆+i2δ)φˆ†2φˆ
b c b e be b e
operators satisfy the following commutation relations: Ω∂t Ω
[φˆ (z,t),φˆ†(z′,t)]=δ δ(z−z′), [Eˆ(z,t), Eˆ†(z′,t)]= +gEˆ†φˆ†φˆφˆ†φˆ − gEˆφˆ†2φˆ2. (7)
(~µν)δ(z −νz′). With µthνese definitions, the interaction Ω b b e c Ω b b
ǫ0
Hamiltonian of our system can be given by Here γ = 2γ +γ and γ = 2γ +γ are transversal
bc b c be b e
decay rates. From Eqs. (6) and (7), we further obtain
Hˆ = − dz ~gEˆ(z,t)φˆ†(z,t)φˆ(z,t)φˆ(z,t)
e b b
+ ~ΩZφˆe†(z(cid:0),t)φˆc(z,t)+h.c. +~∆ dzφˆe†(z,t)φˆe(z,t) φˆb†2φˆc = Ω2+(γbe+i2δΩ+2 i∆)(γbc+i2δ)(cid:18)−gΩEˆφˆb†2φˆb2
+ ~δ dzφˆ†(z,t)φˆ(z,t), (cid:1) Z (1) − Ωi ∂∂t −Ω1(1+γbe+γbc+4iδ+i∆)φˆb†2φˆc
b b
Z + 1(∆(cid:2)−iγ )φˆ†2φˆ + gEˆ†φˆ†φˆφˆ†φˆ
where g is the coupling coefficient between the atoms Ω be b e Ω b b e c
apnhdotqounadnetitzuendinfigesl,d,re∆spaenctdivδelrye.prWeseenntottheehoenree-ganhdastwthoe- + gEˆ†φˆb†φˆbφˆe†φˆc . (cid:3) (8)
Ω
unitsofs−1m1/2,sincehereweconsiderquasi-onedimen- (cid:19)
sional system [6, 7]. The above equation is hard to tackle. However, ac-
The evolution of the quantum field can be described cording to the present experiments [7], we can simplify
in the slowly varying amplitude approximation by the itwithseveralapproximations. Firstly,weshallconsider
propagationequation: the weak-excitation condition, i.e. only a small ratio of
atoms are converted into molecules, thus the last term
(∂ +c ∂ )Eˆ(z,t)=igLφˆ†(z,t)φˆ†(z,t)φˆ(z,t). (2) in the right-hand side (r.h.s.) of above formula is very
b b e
∂t ∂z
small and can be safely neglected. Secondly, we assume
Ontheotherhand,theevolutionofatomicfieldoperators Ω changes sufficiently slowly with time so that adiabatic
is governed by a set of Heisenberg equations condition is fulfilled. For this only the first term in the
r.h.s. of (8) plays important role in the dynamical evo-
φˆ˙ = −iδφˆ −γ φˆ +igEˆ†φˆ†φˆ, lution of atomic fields. On the other hand, in this paper
b b b b b e
φˆ˙ = i∆φˆ −γ φˆ ++igEˆφˆφˆ +iΩφˆ, (3) we shall first neglect the effect of two-photon detuning
e e e e b b c (δ) and atomic-molecular decay (γ ) that will be dis-
bc
φˆ˙ = −γ φˆ +iΩφˆ, cussed later in detail. By introducing a normalized time
c c c e
τ = t/T, where T is a characteristic time scale, and ex-
whereγb,γc andγe denotethedecayratesofcorrespond- panding the r.h.s. of (8) in powers of 1/T, we find in
ing atomic or molecular states. With these formulas we lowest nonvanishing order,
find the atom-molecule evolutions can be described by
∂ (φˆb†2φˆe) = −(i∆+i2δ+2γb+γe)φˆb†2φˆe+igEˆφˆb†2φˆb2 φˆb†2φˆc ≈ −gΩEˆφˆ†b φˆ†b φˆb φˆb. (9)
∂t
−igEˆ†φˆ†φˆφˆ†φˆ +iΩφˆ†2φˆ, (4) Thus Eq. (6) can be recast into
b b e e b c
∂∂t(φˆb†2φˆc) = −(i2δ+2γb+γc)φˆb†2φˆc−igEˆ†φˆb†φˆbφˆe†φˆc φˆb†2φˆe ≈igΩn22∂∂tEˆ(z,t)−igΩn32∂Ω∂(tt)Eˆ(z,t). (10)
+ iΩφˆ†2φˆ. (5)
b e
Here n2 = φˆ†2φˆ2 is the atomic BEC density that is
b b
Itshouldbenotedthatwehavedisregardedthemotionof
assumed to be constant in weak excitation case. Substi-
the trappedatoms and molecules. This is valid since the
tuting this result into Eq. (2) yields
time scale for coherentSTIRAP is shorterthan the time
scale for the motion of the atoms and/or molecules in ∂ c ∂
( + )Eˆ(z,t)=
thetrap[4,5,6]. Also,aslongaslaserintensitiespermit ∂t 1+ g2nN ∂z
STIRAP during a time interval much shorter than the Ω2
1 g2nN ∂Ω
time scales for collisions between atoms and molecules, = Eˆ(z,t), (11)
collisions are negligible as well. The above two Eqs. can 1+ g2Ωn2N Ω3 ∂t
be recast into
where N =hφˆ†φˆi is the total atomic number in the in-
φˆb†2φˆe = − i ∂ (φˆb†2φˆc)− i (γbc+i2δ)φˆb†2φˆc teractingregiobn. bNotingthatΩ(t)istimedependent,the
Ω∂t Ω
+gEˆ†φˆb†φˆbφˆe†φˆc. (6) rm.ha.ns.aobfsaobrpotvieoneqourateinohna(n1c1e)mreenptreosfentthseaqnuaadnitaibzeadticfiRelda-.
Ω
3
The group velocity of the associating field is given by strong classical field is applied so that Ω2 ≫ g˜2N2 or
vg = ccos2θ = c/ 1+ g˜2ΩN22 . Here g˜ = gn1/2N−1/2 is θ(0) = 0, one has φˆc(z,0) = 0. Then, after Ω is adi-
relatedtoatomicdensityn1/2 [7]andcanbe regardedas abatic turned off, i.e. θ(t) = π/2, from (13) we have
constant in the we(cid:0)ak-excitat(cid:1)ion condition. The mixing Lφˆc(z,t) = −Eˆ(z − 0tvgdt′,0), say, quantum states of
angle is defined according to tan2θ = g˜2N2/Ω2(t) that the associating light are fully transferred into the cre-
R
is governed by the atomic number and the strength of ated molecular BEC. The similar result is well-known
external classical field. Different from the usual electro- in quantum memory technique with EIT. Nevertheless,
magneticallyinducedtransparency(EIT)casewherethe identification of quantum state transfer in the present
groupvelocityofassociatinglightisproportionaltoN−1 model suggests a novel technique to create molecules in
[10], here the groupvelocity of quantized field is propor- non-classicalstates via a PA process.
tionaltoN−2 duetothenonlinearityintheHamiltonian In above discussions, we have ignored the decay γ =
bc
(1). When the classical field is adiabatically turned off 2γ +γ of ground atomic and molecular states. How-
b c
sothatthemixingangleθ →π/2,onehasv →0,which ever, recent experiments [7] indicate γ is not very small
g c
means the initial photon wave packet is decelerated to a in a PA process. As a result, it is important to further
fullstop. Equation(11)hasthefollowingsimplesolution verify the validity of quantum state transfer technique
when decay of atomic and molecular states is included.
Eˆ(z,t)= cosθ(t) Eˆ(z− tv dt′,0). (12) Noting that one- and two-photon detunings can be ad-
g
cosθ(0) justed in experiment [7], we would like to consider here
Z0
small detuning case so that δ2 ≪ γ2 and δ2,∆2 ≪ γ2 .
bc be
On the other hand, from Eq. (9), one can easily find
The lowest order in Eq. (8) then reads
gn Ω2(0)+g˜2N2 t
φˆc(z,t)≈−Ω(0)sΩ2(t)+g˜2N2Eˆ(z−Z0 vgdt′,0).(13) φˆb†2φˆc = Ω−+gφˆ1b†γ2φˆγb2 Eˆ(z,t). (14)
Ω be bc
With the expression (13) one can readily reach a
full quantum state transfer from photons to created Substituting this result into Eq. (6) and then into Eq.
molecules via PA process. For example, if initially a (2) yields
g˜2N2 ∂ ∂ g˜2N2 ∂ 1 g˜2N2γ
1+ +c Eˆ(z,t)=− Eˆ(z,t)− bc Eˆ(z,t). (15)
Ω2+γ γ ∂t ∂z Ω ∂t Ω+ 1γ γ Ω2+γ γ
(cid:18) be bc (cid:19) (cid:18) Ω be bc(cid:19) bc be
(cid:0) (cid:1)
Thus the quantumfield propagateswiththe groupve- a decay. The solution can be given by
locity
t
g˜2N2 −1 Eˆ(z,t)=Eˆ(z− vgdt′,0) eR0tdt′α(t′)e−R0tdt′Γ(t′), (17)
v =c 1+ , (16)
g Ω2+γ γ Z0
be bc
(cid:0) (cid:1)
which approaches γ γ c/(g˜2N2+γ γ ) when Ω → 0. where
be bc be bc
The presence of atomic decay results in a nonzero group
velocity even when the classical field is turned off. For tα(t′)dt′ = − t g˜2N2∂t′Ω(Ω2+γbeγbc)−2Ω2∂t′Ω ×
weak associating field case, typical parameter values can Z0 Z0 Ω (Ω2+γbeγbc)2
betakenas[7]γ ≈2.0×107s−1,γ ≈5.0×103s−1,N ≈ dt′
be bc × ,
3.0×106andg˜≈50s−1. Onecanthenevaluatethelimit 1+ g˜2N2
of group velocity by vc ≈ 1.33 km·s−1. Furthermore, Ω2+γbeγbc
g
if we consider a larger number of trapped atoms, say
N ∼108[11],thelimitofgroupvelocitywillbedecreased
byfourordersandyieldsvgc ≈0.13m·s−1. Thenumerical tΓ(t′)dt′ = t g˜2N2γbcdt′ .
result indicates the associating light can still reach an Z0 Z0 Ω2+γbeγbc+g˜2N2
approximate “stop” for the large atomic number case.
The firstterminthe r.h.s. ofaboveequation(15)rep- Theaboveintegralcanbecalculatedstraightforwardly.
resents an adiabatic Raman absorption or enhancement Similar to the former discussion, we assume initially the
of the quantized field, while the second term represents classical field is strong so that Ω2(0) ≫ g˜2N2,γ γ ,
be bc
4
while finally Ω2(t)≪g˜2N2. We then find from Eq. (17) Inconclusion,Wehaveshownthequantumstatetrans-
ferprocessintwo-colorPAofaBose-Einsteincondensate,
t Ω(0) f γ γ +Ω2(t) where a quantized field is used to couple the free-bound
Eˆ(z,t) = Eˆ(z− v dt′,0) be bc ×
g Ω(t) Ω2(0) transition from atom state to excited molecular state.
Z0 (cid:18) (cid:19) (cid:18) (cid:19) Under the weak excitation condition, we find quantum
Ω(0) h t
× exp − Γ(t′)dt′ , (18) statesofthequantizedfieldcanbetransferredtothecre-
(cid:18)γbeγbc+g˜2N2(cid:19) Z0 atedmolecularcondensates. Effectsofatomicandmolec-
(cid:0) (cid:1) ular decays in the present model are discussed in detail.
where f = g˜2N2/(γ γ + g˜2N2) and h = 1/2 +
be bc Our results show that quantum state transfer proposed
γ γ /(2γ γ +2g˜2N2). The above equation reduces
be bc be bc hereisachievableinthecurrentPAexperiments,andwill
into the formula (12) if γ γ → 0. Based on the pre-
be bc have wide range of applications to quantum information
vious parameters used in calculating the limit of group
science. For example, if the created molecular state is
velocity,onecanseeg˜2N2 ≫γ γ . Thuswehavef ≈1
be bc untrapped, we may create molecular lasers with control-
andh≈1/2,andthe createdmolecularBEC is givenby
lable quantum states. Besides, if considering a squeezed
associating light in the PA process, one may study the
t t
Lφˆ =−Eˆ(z− v dt′,0)exp − Γ(t′)dt′ . (19) molecule-light entanglement [14], and so on.
c g
Z0 Z0 This work is supported by NUS academic research
(cid:0) (cid:1)
Grant No. WBS: R-144-000-189-305, and by NSF of
This formula shows quantum state transfer technique
China under grants No. 10575053.
from quantized associating field to created molecules
is still valid when |1 − exp − tΓ(t′)dt′ | ≤ |1 −
0
exp − tγ dt′ | ≪ 1. Therefore, to obtain a sufficient
0 bc (cid:0) R (cid:1)
quantum state transfer, the time duration of a PA pro-
cess(cid:0)shRould sat(cid:1)isfy t ≪ t ∼ γ−1. For 87Rb atomic- [1] D. Kleppner,Phys. Today 57, No. 8, 12 (2004).
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