Stimulated Raman adiabatic passage in molecular electronic states

Chuan-Cun Shu, Jie Yu, Kai-Jun Yuan, Wen-Hui Hu, Jing Yang, and Shu-Lin Cong*Department of Physics, Dalian University of Technology, Dalian 116024, China

�Received 27 November 2008; published 17 February 2009�

We explore theoretically the rovibrational dynamics of stimulated Raman adiabatic passage �STIRAP� inmolecular electronic states including both the rotational and the vibrational degrees of freedom, with the LiHmolecule as an example. By using a pure rovibrational state as initial state, we find that, under the condition oftwo-photon resonance, an efficient rovibrational population transfer can be achieved via STIRAP in molecularelectronic states and the desired target state can be selected by adjusting laser-pulse parameters. Besides theinterested rovibrational states, however, some unwanted rovibrational states may affect population transferprocess, especially while the two-photon detuning is taken into account. By inspecting the evolution process ofwave packet, we can easily search these unwanted rovibrational states and study their influences. Moreover, theeffect of rotational temperature on the population transfer process is discussed using thermal mixed state asinitial state.

DOI: 10.1103/PhysRevA.79.023418 PACS number�s�: 33.80.Be, 42.65.Dr, 33.20.Vq, 33.20.Fb

I. INTRODUCTION

Preparation of well-defined quantum states for atoms ormolecules is an important topic in quantum optics, atomicand molecular physics, quantum information processing, andcontemporary laser physics �1–3�. Controlling populationtransfers of atoms or molecules by strong laser pulses hasalso attracted attention of researchers �4–7�. Several ap-proaches are theoretically proposed to realize this goal bypulse timing control, multipath interference, optimal control,and adiabatic passage �8–12�. The stimulated Raman adia-batic passage �STIRAP� technique, which was proposed andexperimentally carried out by Bergmann and co-workers�13,14�, is one of efficient ways used for controlling popula-tion transfer with two counterintuitive and partially overlap-ping laser beams. The STIRAP method has been widelystudied, both in atomic and molecular systems with a three-level or multilevel structure �15–18�.

Initially, the STIRAP was developed to steer efficientlythe rovibrational excitation of molecules with continuouswave �cw� laser beams �14�. Since the transition dipole mo-ments of many molecules are too small, currently availablecw lasers are not powerful enough to induce sufficiently alarger Rabi frequency for successful application of STIRAP.Therefore, it is of significant interest to explore the feasibil-ity of STIRAP with laser pulses. Recently, manipulation ofrovibrational wave-packet composition has been studied boththeoretically and experimentally with laser pulses �19–22�.Chang and Solá �23� studied the Raman excitation of rovi-brational coherent and incoherent states using long pulseswith bandwidth �frequency width� much less than the rota-tional constant. However, molecular process is very compli-cated due to the presence of the nuclear motion. When laserfield is strong but still insufficiently intense to cause multipleionization of molecules, the bound electronic states will besignificantly modified by the laser field. These modifiedpotential-energy surfaces will in turn affect the motion of the

nuclei �12�. Therefore, it has to be kept in mind that, forlonger laser-molecule interaction time, the motion of nucleiwill play an important role in excitation process because theelectric dipole moment function depends on the vibrationalcoordinate R. On the other hand, since molecule has a lot ofvibrational and rotational energy levels in each electronicstate, and these rovibrational levels are very congested, someunwanted transitions among different rovibrational statesmay occur. In addition, if the bandwidth of laser pulse islarge enough, the situation will become more complex: thefield may induce the population transfer from an initial stateto more rovibrational states. Therefore, it is not appropriateto explore these complicated processes by using a simplelevel model. Based on this consideration and the recent in-vestigation of vibrational population transfer between mo-lecular electronic states �24�, we think that a more detailedtheoretical description of STIRAP including both the rota-tional and the vibrational degrees of freedom is necessary.

In the framework of the Born-Openheimer approximation�BOA�, the time-dependent quantum wave-packet method isvery efficient and successful for dynamics calculations ofdiatomic and triatomic molecules. An attractive aspect is thatmuch insight can be obtained by inspecting snapshots of thewave packet at regular time intervals. From the snapshots, itis easy to visualize the nuclear motion and the underlyingphysics. In the present work, we explore the rovibrationaldynamics of STIRAP in molecular electronic states, takingthe LiH molecule for example. The molecular vibration androtation are taken into account in the calculation, and theeffect of unwanted rovibrational states on STIRAP is exam-ined in detail. We also discuss the effect of temperature onthe rovibrational wave-packet manipulation using thermalmixed state as initial state.

The rest of this paper is organized as follows. In Sec. II,we summarize a standard model of STIRAP in a three-levelsystem and introduce the theoretical treatment of STIRAP inmolecular electronic states. The results and discussions aredemonstrated in Sec. III. Finally, some conclusions and fu-ture perspective are given in Sec. IV*shlcong@dlut.edu.cn

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1050-2947/2009/79�2�/023418�10� ©2009 The American Physical Society023418-1

II. THEORETICAL TREATMENTS

Figure 1 shows the Raman excitation process for one-photon-allowed rovibrational transitions in the lowest twoelectronic states of the LiH molecule. The pump lasercouples the ground rovibrational state �g�=X 1�+�� ,J ,M� tothe intermediate rovibrational state �e�=A 1�+��� ,J� ,M�,and the Stokes pulse couples the state �e�=A 1�+��� ,J� ,M�to final rovibrational state �f�=X 1�+��� ,J� ,M�. For conve-nience, the ground electronic state X 1�+ and the excitedelectronic state A 1�+ are abbreviated to X and A, respec-tively. The total laser fields ��t� is expressed as

��t� = �k=P,S

Ekfk�t�cos��k�t − tk�� , �1�

where Ek, fk�t�, tk, and �k are the field amplitude, normalizedpulse envelope, center time, and carrier frequency of the kthpulse, respectively. The subscripts P and S represent thepump and the Stokes laser pulses, respectively.

When linearly polarized laser pulses are used to excite themolecule, the allowed coupling between two rovibrationallevels is determined by ��� ,J� ,M���� ,J ,M�=��,���J,M

J�,M�J,J��1�M,M, where � is the Delta Kronecker and��,�� is the transition dipole moment between two vibrationalstates. The angular overlaps are calculated by

�J,MJ−1,M = �J−1,M

J,M = J2 − M2

�2J + 1��2J − 1��2�

for P��J,MJ−1,M� and R��J−1,M

J,M � branches of the rotational spec-trum �23�, as shown in Fig. 1. The electric dipole selectionrules indicate that the rotational quantum number J changesby �1 for one-photon absorption, and J=J�−J=0,2 and−2 correspond to Q, O, and S branches of the Raman spec-trum, respectively.

A. STIRAP in a three-level system

The goal of STIRAP is to control the population transferfrom the initial rovibrational state �g�= �� ,J ,M� to the final

rovibrational state �f�= ��� ,J� ,M� while keeping the popula-tion of the intermediate rovibrational state �e�= ��� ,J� ,M� assmall as possible. We summarize the standard three-levelmodel of STIRAP �17�. The molecular wave function can beexpanded as

��t�� = Cg�t��g� + Ce�t��e� + Cf�t��f� , �3�

where, Cg�t�, Ce�t�, and Cf�t� are the probability amplitudesof the three levels �g�, �e�, and �f�, respectively. Using thetime-dependent Schrödinger equation one can derive thecoupled equation:

i��

�tC�t� = H�t�C�t� , �4�

where C�t� is a three-component column matrix, with theelements Cg�t� ,Ce�t� ,Cf�t��. The Hamiltonian under rota-tion wave approximation �RWA� reads �25�

H�t� = ��0

1

2�P�t� 0

1

2�P�t� P

1

2�S�t�

01

2�S�t� P − S

. �5�

Here �P and �S are the Rabi frequencies of the pump andStokes pulses, respectively. �P=Ee−Eg−��P and �S=Ee−Ef −��S are the one-photon detunings of the pump andStokes pulses from their respective transitions, respectively.An essential condition for STIRAP is the two-photon reso-nance, i.e., P=S=.

The theoretical treatment of STIRAP is greatly facilitatedby introducing an adiabatic basis. It is straightforward toverify that the following linear combinations of the barestates �g�, �e�, and �f� are eigenstates of H�t�:

�+�t�� = sin �t�sin ��t��g� + cos ��t��e�

+ cos �t�sin ��t��f� , �6a�

�0�t�� = cos �t��g� − sin �t��f� , �6b�

�−�t�� = sin �t�cos ��t��g� − sin ��t��e�

+ cos �t�cos ��t��f� , �6c�

where the mixing angles �t� and ��t� are defined�modulo �� as �t�=arctan��P�t� /�S�t�� and ��t�= 1

2arctan��P2 �t�+�S

2�t� /�. Instead of applying the plusesin the intuitive sequence, the Stokes pulse precedes the pumppulse. If the two pulses overlap sufficiently and are strongenough, �0�t�� is an adiabatic dark state, and the populationof the initial state �g� can be transferred completely to thetarget state �f�. Thus, the time-dependent populations of �g�and �f� can be written as

Pg�t� = cos2 �t�, Pf�t� = sin2 �t� . �7�

vJ=1J=0

J=2

J=1J=0

J=2

v J=1J=0

J=2

v

Li

H

ε(t=0)µ θ

PR

R

1 3 5 7 9 11

0

1

2

3

4

5

6

''''

" """

FIG. 1. �Color online� Potential-energy curves and rovibrationallevels of the ground X 1�+ and excited A 1�+ states of the LiHmolecule. The solid and dashed arrows represent the R and Pbranches the Raman transition, respectively.

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B. STIRAP in molecular electronic states

In the present work, we describe STIRAP in molecularelectronic states and do not invoke the RWA. Since the lin-early polarized light is employed to excite the molecule, thequantum number M is conserved, i.e., M =0. The time-dependent wave function can be expressed as

�M�R,�,�,t� = eiM��M�R,�,t�/R , �8�

where �M�R ,� , t� is calculated by

i��

�t�M�R,�,t� = H�R,�,t��M�R,�,t� , �9�

with

�M�R,�,t� = ��X�R,�,t��A�R,�,t�

� , �10�

where �X�R ,� , t� and �A�R ,� , t� are the nuclear wave func-tion in the electronic states X and A, respectively. The mo-lecular Hamiltonian including the laser-molecule interactionis given by

H�R,�,t� = �TR + T��I + W�R,�,t� , �11�

where I is a unit matrix, R is the internuclear separation, and� is the angle between the molecular axis and the polariza-tion direction of laser field, as shown in Fig. 1. The kinetic-energy terms are given by

TR = −�2

2m

�2

�R2 �12�

and

T� = −�2

2mR2

1

sin �

�

���sin �

�

��� +

�2

2mR2

M2

sin2 �, �13�

where m represents the reduced mass of the molecule. If thelaser pulse contains a resonant frequency of rovibrationaltransition, the first-order dipole interaction is dominant com-pared with the polarizability term �26�, and therefore thelaser-molecule interaction W�R ,� , t� can be written as

W�R,�,t� = �VX�R� 0

0 VA�R�� − ��t�cos �

� ��XX�R� �XA�R��XA�R� �AA�R�

� , �14�

where VX�R� and VA�R� are the bare molecular potentials.�ij�R� �i , j=X ,A� are the transition �i� j� and permanent �i= j� dipole moments at R.

Before propagating the time-dependent Schrödinger equa-tion in Eq. �9�, its initial form has to be defined. The initialwave packet is given by

�M�R,�,t = t0� = ���J�R�PJM�cos ��0

� , �15�

where PJM�cos �� is the associated Legendre polynomial and

��J�R� is J-dependent radial vibrational wave function.PJ

M�cos �� satisfy the following equation �27,28�:

�−1

sin �

�

���sin �

�

��� +

M2

sin2 �− J�J + 1��PJ

M�cos �� = 0.

�16�

Using Fourier grid Hamiltonian method �29�, one can com-pute ��J�R� by solving numerically the radial Schrödingerequation,

�−�2�2

2m � R2 + VX�R� +J�J + 1��2

2mR2 − ��J���J�R� = 0,

�17�

where ��J is the rovibrational energy of the ground electronicstate X.

The time propagation is accomplished by using the splitoperator method �30,31�:

��R,�,t + t� = U�t���R,�,t�

= e−�i/��t�TR+T�+W�R,�,t����R,�,t�

� e−�i/2��tTRe−�i/2��tT�

� e−�i/��tW�R,�,t�e−�i/2��tTR

� e−�i/2��tT���R,�,t� . �18�

It is worth noting that the kinetic-energy operators, TR andT�, are nonlocal in coordinate space. A basis-set expansion inthe terms of the normalized associated Legendre polynomialsare used to treat the angular part, and an evenly spaced gridis employed to calculate vibrational degree of freedom. Thus,in the calculation, TR is transformed between momentum andcoordinate spaces by using the fast Fourier transform methodand T� is switched forward and backward between the poly-nomial representation and the coordinate space by utilizingdiscrete variable representation �DVR� technique �32�. TheDVR is a unitary transformation of an finite basis represen-tation �FBR� �33�. It is defined in terms of points � andweights �� of the N-point Gaussian quadrature associatedwith the orthogonal polynomial �34�:

T��i,J� = ��iPJ

M�cos �i� , �19�

where �i is a discrete angular basis. W�R ,� , t� acts on thewave function directly by multiplication in coordinate space.

The time-dependent population of the molecule in theelectronic state X�A� is computed by �35�

PX�A� =� d� sin �� dR��X�A��R,�,t��2. �20�

The time-dependent rovibrational population P�0J0M in therovibrational eigenstate ��0J0M =��0J0

�R�PJ0

M�cos �� is calcu-lated by �36�

P�0J0M�t� = ����0J0M��M�R,�,t���2. �21�

III. RESULTS AND DISCUSSION

In the calculation, we let R to vary in a range of 1.75–11.0in atomic units with a 1024-point Fourier grid. The angular

STIMULATED RAMAN ADIABATIC PASSAGE IN… PHYSICAL REVIEW A 79, 023418 �2009�

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grid points are 60 Gauss-Legendre quadrature points. Thelength of time step t=0.05 fs. The data of the twoR-dependent potential-energy curves and the dipole momentsare taken from Ref. �37�. A Gaussian pulse profile,

fk�t� = e−4 ln 2�t − tk�2/�k2, �22�

is employed throughout this work, where �k are full-widths athalf-maximum �FWHMs� with �P=�S=5.0 ps. The pumppulse is centered at time tP=0 ps, while the Stokes pulsewith the center time tS=−2.5 ps is applied before the pumppulse.

A. Rovibrational wave-packet manipulation via STIRAP using��=0,J=0,M=0‹ as initial state

We consider a case in which the initial rovibrational stateis �g�= ��=0,J=0,M =0�, the intermediate state �e�= ���=9,J�=1,M =0�, and the target state �f�= ���=6,J�=0,M=0�. The center frequency ��P is 28 954.37 cm−1 with thedetuning �P=−7 cm−1, and ��S is 21 482.29 cm−1 withthe detuning �S=−7 cm−1. The peak intensities of pumpand Stokes pulses are 3.1�1010 W /cm2 and 2.32�1010 W /cm2, respectively. Figure 2 shows the time-

dependent populations in the electronic states X and A andthe rovibrational states ��=0,J=0,M =0�, ���=9,J�=1,M=0�, and ���=6,J�=0,M =0�. As can be seen in Fig. 2, thepopulation almost completely stays in ground state X duringthe laser-molecule interaction, and the population on the ex-cited state A is nearly the same as the rovibrational popula-tion on the state ���=9,J�=1,M =0�. This indicates that thepopulation transfer process mainly takes place in the threerovibrational states �g�= ��=0,J=0,M =0�, �e�= ���=9,J�=1,M =0�, and �f�= ���=6,J�=0,M =0� without involvingother rovibrational states. A STIRAP relies on the existenceof a dark state �0�t�� in Eq. �6b�. From Fig. 2�b�, we canfind that the time-dependent rovibrational population P000�t�on ��=0,J=0,M =0� decreases smoothly from 1 to 0, whileP600�t� on ���=6,J�=0,M =0� increases smoothly from 0 to1. The population evolutions with time are well consistentwith Eq. �7�. Therefore, a STIRAP is successfully con-structed in molecular electronic states.

Figure 3 displays the real parts of the wave packet�M�R ,� , t� at the initial time t=−9.0 ps and the final timet=7.0 ps. In Fig. 3�a�, the initial wave packet is prepared onthe rovibrational state ��=0,J=0,M =0�. At the final time t=7.0 ps, the rovibrational wave packet in angular directionis very similar to the initial wave packet whose angular func-tion is P0�cos ��, whereas in radial direction there are sevenpeaks which means ��=6. Therefore a rovibrational wavepacket on the eigenstate ���=6,J�=0,M =0� is obtained. As

|ν=0,J=0,M=0>

|ν=9,J=1,M=0>' '

|ν=6,J=0,M=0>'' ''

X

A

0.0

0.2

0.4

0.6

0.8

1.0

Popula

tion

0.0

0.2

0.4

0.6

0.8

1.0

Popula

tion

FIG. 2. �Color online� �a�The time-dependent population in theelectronic states X and A. �b�The time-dependent rovibrationalpopulation in ��=0,J=0,M =0�, ���=9,J�=1,M =0�, and ���=6,J�=0,M =0�. The laser parameters are chosen to be IP=3.1�1010 W /cm2, IS=2.32�1010 W /cm2, �P=�S=5 ps, tP=0,tS=−2.5 ps, ��P=28 954.37 cm−1, and ��S=21 482.29 cm−1.

t=-9.0 ps

0.0

0.05

ReΨ

Μ(R,θ,t) (a)

3060

90120

150180 o

Angle θ

oo

oo

o 0 o 23

45

6

R (a.u.)

0.0

0.03

-0.03

ReΨ

Μ(R,θ,t)

180

03060

90120

150o

Angle θ

oo

oo

o o 23

45

6

R (a.u.)

t=7.0 ps (b)

FIG. 3. �Color online� The real parts of the wave packet�M�R ,� , t� at �a� the initial time t=−9.0 ps and �b� the final timet=7.0 ps.

SHU et al. PHYSICAL REVIEW A 79, 023418 �2009�

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can be seen in Fig. 3, the wave packet moves away from itsinitial region, and hence the molecular vibration and rotationshould be taken into account.

In the above transition process, J=J�−J=0 correspondsto the Q branch of the Raman transition. We can also selectthe rovibrational state �f�= ���=6,J�=2,M =0� as the targetstate by the O branch of the Raman transition. Figure 4shows the time-dependent populations in the electronic statesX and A and the rovibrational states ��=0,J=0,M =0�, ���=9,J�=1,M =0�, and ���=6,J�=2,M =0�. The center fre-quency ��P is fixed at the transition frequency�28 947.37 cm−1� of ���=9,J�=1,M =0�← ��=0,J=0,M=0�, and ��S is equal to the transition frequency of ���=6,J�=2,M =0�← ���=9,J�=1,M =0�. The peak intensitiesof pump and Stokes pulses are 3.1�1010 and 3.5�1010 W /cm2, respectively. From Fig. 4�b�, we find that thepopulation can be transferred completely from ��=0,J=0,M =0� to ���=6,J�=2,M =0�. Figure 5 shows the realpart of �M�R ,� , t� at t=7.0 ps, and the Legendre functionP2�cos ��. The distribution of rovibrational wave packet inangle direction is very similar to the curve of P2�cos ��. Inradial direction there are seven peaks. Therefore, the popu-lation of the initial state is transferred to the rovibrationalstate ���=6,J�=2,M =0�.

Besides the above three rovibrational states, a smallamount population can be transferred to other rovibrationalstates. From Fig. 4, we can see that the population PA on theexcited state A is evidently larger than rovibrational popula-tion P910 on ���=9,J�=1,M =0�. This indicates that the otherrovibrational states take part in the excitation process, andthe nonadiabatic coupling happens. There are two possibili-ties to increase the population of the excited state A. First,the population may be transferred from the final state ���=6,J�=2,M =0� to the excited state A by the pump andStokes pulses. Second, the two-photon or multiphoton tran-sition path between the X and A states can be opened becauseof the mixed parity in at least one of the two levels for apolar molecule �38,39�. As can be seen in Fig. 4�a�, the popu-lation on the excited state A reaches its peak value at t=−1.35 ps. To explore the population transfer mechanism inFig. 4, we depict the real part of wave packet �A�R ,� , t� att=−1.35 ps and the Legendre function P3�cos �� in Fig. 6.The distribution of the rovibrational wave packet in angledirection is in conformity with the curve of P3�cos ��, andthere are ten clear peaks in radial direction. Thus, we caninfer that the rovibrational state ���=9,J�=3,M =0� iscoupled by Stokes pulse, corresponding to the J=J�−J�=1 transition.

In order to explore the detail of the effect of the othersstates on the population dynamics, we present the time-dependent population distributions on the excited state A and

X

A

|ν=0,J=0,M=0>

|ν=9,J=1,M=0>' '

|ν=6,J=2,M=0>'' ''

0.0

0.2

0.4

0.6

0.8

1.0

Popula

tion

0.0

0.2

0.4

0.6

0.8

1.0

Popula

tion

FIG. 4. �Color online� The time-dependent populations �a� in theelectronic states X and A and �b� in the rovibrational states��=0,J=0,M =0�, ���=9,J�=1,M =0�, and ���=6,J�=2,M =0�.The peak intensities IP=3.1�1010 W /cm2 and IS=3.5�1010 W /cm2. The center frequencies ��P=28 947.37 cm−1 and��S=21 438.29 cm−1.

0 30 60 90 120 150 180ooooooo

Angle θ

-0.20

-0.10

0.0

0.10

0.20

P2(cosθ)

(b)

t=7.0 ps (a)

0.0

0.03

-0.03

ReΨ

Μ(R,θ,t)

030

6090

120150

180 o

Angle θ

oo

oo

oo 2

34

56

R (a.u.)

FIG. 5. �Color online� �a�The real part of the wave packet�M�R ,� , t� at the final time t=7.0 ps and �b� the Legendre functionP2�cos ��.

STIMULATED RAMAN ADIABATIC PASSAGE IN… PHYSICAL REVIEW A 79, 023418 �2009�

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the related rovibrational states in Fig. 7. The sum of thepopulations on ���=9,J�=1,M =0� and ���=9,J�=3,M =0�is nearly equal to the total population on the excited state A.Therefore, the first population transfer path induced by theStokes pulse is dominant, and the second population transferpath is eliminated. However, if the intensities of laser pulsesbecome intense, and the multiphoton resonant condition issatisfied, the second population transfer path will become

important for the increase in the excited-state population�38,39�.

B. Rovibrational wave-packet manipulation via STIRAP using��=0,J=1,M=1‹ as initial state

We have performed the calculations using the groundrovibrational state as initial state, where the quantum numberM =0. We now present the calculated result using ��=0,J=1,M =1� as initial state. According to the selection rulesJ= �1 and M =0, the intermediate state �e� should be���=9,J�=2,M =1� and the target state �f� may be ���=6,J�=1,M =1� or ���=6,J�=3,M =1�. Figure 8 shows thetime-dependent population distribution when ���=6,J�=1,M =1� is taken as the target state. The center frequency��P is 28 951.61 cm−1 with detuning �P=−8 cm−1, and��S is 21 482.07 cm−1 with detuning �S=−8 cm−1. Thepeak intensities of pump and Stokes pulses are 3.5�1010 and3.9�1010 W /cm2, respectively. From Fig. 8�b�, we can seethat the rovibrational population is nearly 100% transferredfrom the initial state ��=0,J=1,M =1� to the final state ���=6,J�=1,M =1�, and the intermediate state ���=9,J�=2,M=1� is almost unpopulated. As can be seen in Fig. 8, the

0 30 60 90 120 150 180ooooooo

Angle θ

(b)

-0.20

-0.10

0.0

0.10

0.20

P3(cosθ)

0.0

0.006

-0.006

ReΨ

Α(R,θ,t)

(a)

R (a.u.)

24

68

10

t=-1.35 ps

03060

90120

150180o

Angle θ

oo

oo

o o

FIG. 6. �Color online� �a� The real part of the wave packet�A�R ,� , t� at t=−1.35 ps and �b� the Legendre polynomialP3�cos ��.

|ν=9,J=1,M=0>' '

0.0

0.06

0.12

0.18

Popula

tion

|ν=9,J=3,M=0>' '

A

-9 -6 -3 0 3 6

Time (ps)

FIG. 7. �Color online� The time-dependent populations on theexcited state A and the related rovibrational states.

X

A

-9 -6 -3 0 3 6

Time (ps)

|ν=0,J=1,M=1>

|ν=9,J=2,M=1>' '

|ν=6,J=1,M=1>'' ''

(a)

0.0

0.2

0.4

0.6

0.8

1.0

Po

pu

latio

n

0.0

0.2

0.4

0.6

0.8

1.0

Po

pu

latio

n

FIG. 8. �Color online� �a�The time-dependent populations in theelectronic states X and A and �b�the time-dependent rovibrationalpopulations in ��=0,J=1,M =1�, ���=9,J�=2,M =1�, and ���=6,J�=1,M =1�. The center frequency ��P is 28 951.61 cm−1

with the detuning �P=8 cm−1, and ��S is 21 482.07 cm−1 withthe detuning �S=−8 cm−1. The peak intensities of pump andStokes pulses are 3.5�1010 and 3.9�1010 W /cm2, respectively.

SHU et al. PHYSICAL REVIEW A 79, 023418 �2009�

023418-6

population on the rovibrational state ���=9,J�=2,M =1� isvery close to the population on the excited state A. Thismeans that the effect of the other states on the STIRAP pro-cess is very small. The population is adiabatically transferredfrom the initial rovibrational state ��=0,J=1,M =1� to a su-perposition of the states ��=0,J=1,M =1� and ���=9,J�=2,M =1� during the laser-molecule interaction and finallyto the target state ���=6,J�=1,M =1�. Figure 9 shows thereal parts of the wave packet �M�R ,� , t� at t=−9.0 and 7.0ps and the function P1

1�cos ��. We can see that the wavepacket moves perfectly from the initial state ��=0,J=1,M=1� to the target state ���=6,J�=1,M =1� after the interac-tion is over.

The STIRAP is very sensitive to the two-photon detuning.In Figs. 2 and 8, we have depicted the population transferprocesses in the case of one-photon detuning and two-photonresonance. In the following, we discuss the influence of two-photon detuning on the STIRAP in molecular electronicstates. Figure 10 shows the final rovibrational populations inthe rovibrational states ��=0,J=1,M =1�, ���=9,J�=2,M=1�, and ���=6,J�=1,M =1� for different two-photon detun-ings S−P, where the one-photon detuning P=0, and theother laser-pulse parameters are the same as those in Fig. 9.We can see that the final population is very sensitive to thetwo-photon detuning. Especially at two-photon detuning�S−�P=61 cm−1, the three rovibrational states are al-most unpopulated. The population is completely transferredto the other rovibrational states. To search the “lost” popula-tion, in Fig. 11, we present the real part of �M�R ,� , t� at t=7.0 ps and the function P3

1�cos ��. We can see that thewave packet is almost located in an eigenstate. The real partof �M�R ,� , t� in angular direction is very similar to the func-tion P3

1�cos ��, and there are seven peaks in radial direction.Therefore, the lost population is searched on the rovibra-tional state ���=6,J�=3,M =1� which corresponds to the Obranch of Raman transition. In the typical multilevel STI-RAP model, however, one must take enough levels into ac-count to solve this problem. Thus, our treatment possesses anevident advantage for the complicated situation where otherrovibrational states participate in the population transferprocess.

From the above discussions, we can conclude that thedesired target state can be selected by adjusting laser-pulseparameters, but the two-photon resonance must be main-tained. Goto and Ichimura �15� demonstrated that if thevariation in the Rabi frequency is sufficiently slow, and two-photon detunings are small compared with the Rabi frequen-cies, the adiabatic approximation with two dark states isvalid, and consequently the population transfer process canbe regarded as tripod STIRAP via the two dark states even inthe presence of small two-photon detunings. Based on thisapproach, we can extend our method to investigate the tripodSTIRAP in molecular electronic states with small two-photon detunings by adding a laser pulse.

2

0 30 60 90 120 150 180ooooooo

Angle θ

P11(cosθ)

0.0

-0.05

-0.10

-0.15

-0.20

34

56

R (a.u.)

(a)

t=-9.0 ps

-0.05

ReΨ

Μ(R,θ,t)

0.0

03060

90120

150180 o

Angle θ

oo

oo

oo

03060

90120

150180 o

Angle θ

oo o

o o 23

45

6o

R (a.u.)

t=7.0 ps (b)

0.0

0.03

-0.03

ReΨ

Μ(R,θ,t)

FIG. 9. �Color online� �a� The real part of the wave packet�M�R ,� , t� at t=−9.0 ps, �b� the real part of the wave packet�M�R ,� , t� at t=7.0 ps, and �c� the associated Legendre polyno-mial P1

1�cos ��.

0 10 20 30 40 50 61

Detuning ∆S-∆P(cm )

0.0

0.2

0.4

0.6

0.8

1.0

Po

pu

latio

n

|ν=0,J=1,M=1>

|ν=9,J=2,M=1>' '

|ν=6,J=1,M=1>'' ''

-1

FIG. 10. �Color online� The final rovibrational populations inthe rovibrational states ��=0,J=1,M =1�, ���=9,J�=2,M =1�, and���=6,J�=1,M =1� for different two-photon detuning S−P withfixed one-photon detuning P=0.

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C. Rovibrational wave-packet manipulation via STIRAP usingthermal mixed state as initial state

We now present the calculated results using thermalmixed state as initial state. Considering the effect of the tem-perature is equivalent to statistically averaging over the so-lutions of the Schrödinger equation for all possible initiallyrovibrational states weighed by a Boltzmann factor P�� ,J��28,40�. Since we consider only the case of low temperature,the statistically averaging over vibrational quantum numbers� is omitted. The Boltzmann distribution P�� ,J� associatedwith the rotational states is given by

P��,J� =1

Qexp�− BJ�J + 1�

�BT� , �23�

where

Q = �J=0

�

�2J + 1�exp�− BJ�J + 1��BT

� �24�

is the partition function. B, �B, and T are the molecular ro-tation constant, the Boltzmann constant, and the rotationaltemperature, respectively.

For simplicity but without loss of generality, we considera case at rotational temperature T=20 K, where only initialrotational states of J=0 and 1 make a significant contribu-tion. The initial state of J=1 contains the rovibrational states

��=0,J=1,M =0� �statistical weight of 1/3� and ��=0,J=1,M = �1� �statistical weight of 2/3�. Each of them has tobe propagated separately and then combined by using therespective weights. Figure 12 shows the thermally rovibra-tional populations at t=−9.0 and 7.0 ps with the same pulseparameters as those in Fig. 2. When the molecule is initiallyin a thermal mixed state, the population of the initial statecan be transferred to the target state through the differentbranches of Raman transition. However, it is difficult totransfer full population from the thermally mixed state to thetarget state via a STIRAP because different rovibrational lev-els of the thermally initial state correspond to different two-photon resonant conditions. As can be seen in Fig. 12, theefficiency of population transfer is lower than that using a

0 30 60 90 120 150 180ooooooo

Angle θ

P31(cosθ)

0.0

-0.10

-0.20

0.10

0.20(b)

03060

90120

150180 o

Angle θ

oo

oo

oo 2

34

56

R (a.u.)

(a)t=7.0 ps

0.0

0.02

-0.02

ReΨ

Μ(R,θ,t)

FIG. 11. �Color online� �a� The real part of the wave packet�M�R ,� , t� at the final time t=7.0 ps and �b� the associated Leg-endre polynomial P3

1�cos ��.

ν=0,J=0

+-M=0

ν=0,J=1

M=0

ν=0,J=1

M= 1

(a)

0.0

0.2

0.4

0.6

0.8

1.0

Population

t=-9.0psT=20 K

ν=0,J=0 ν=0,J=1 ν=0,J=1

+-M= 1

M=0

M=0

T=20 K

t=7.0ps

(b)

0.00

0.01

0.02

Population

ν =6,J =0 ν =6,J =1 ν =6,J =1

M=0

M=0 +-M= 1

" " " " " "0.0

0.2

0.4

0.6

0.8

1.0

Population

(c)T=20 K

t=7.0ps

FIG. 12. �Color online� The thermally rovibrational populations�a� at t=−9.0 ps, ��b� and �c� � at t=7.0 ps. The pulse parametersare the same as those in Fig. 2.

SHU et al. PHYSICAL REVIEW A 79, 023418 �2009�

023418-8

pure adiabatic states as initial state. According to Eqs. �2�and �16�, for a given J, all rotational states �J ,M� are degen-erate in energy, but the M quantum number influences thestrength of coupling. From Fig. 12, we can find that theefficiency of population transfer from ��=0,J=1,M =0� tothe target state is evidently higher than that from ��=0,J=1,M = �1� because the Rabi frequency for M =0 is lagerthan that for M = �1.

IV. CONCLUSION

We have investigated the rovibrational dynamics of STI-RAP in the molecular electronic states including both therotational and the vibrational degrees of freedom, with theLiH molecule as the example. Using a pure rovibrationalstate as the initial state, we have found that, under the con-dition of the two-photon resonance, the population can betransferred to the selective rovibrational state, and the targetstate can be controlled by adjusting laser-pulse parameters.Besides the interested rovibrational states, we have noticedthat some unwanted rovibrational states affect the dynamicsof STIRAP, especially while the condition of the two-photonresonance was broken. Our calculations show that the rota-

tional and the vibrational degrees of freedom should not beignored when we construct a STIRAP in molecular elec-tronic states. By inspecting the evolution process of wavepacket, we can easily visualize the nuclear motion and theunderlying physics and explore the effect of the unwantedrovibrational states on the population transfer. When the ini-tial state of the LiH molecule is a thermal mixed rovibra-tional state, it is difficult to transfer full population from thethermally mixed state to the target state via a STIRAP andthe efficiency of population transfer was lowered in a certaindegree because different rovibrational levels of the thermallyinitial state correspond to different two-photon resonant con-ditions. In the present work, we only consider the one-photoncoupling between the two electronic states. The theoreticaltreatment can be easily extended to the case of multiphotoncoupling and can also be applied to other diatomic mol-ecules. Naturally, a comparison of the theoretical results withexperimental data is very desirable.

ACKNOWLEDGMENT

This work was supported by the National Science Foun-dation of China under Grants No. 10674022 and No.20633070.

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