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Effects of permanent dipole moments on stimulated Raman adiabatic passage
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Chemical Physics 342 (2007) 16–24 www.elsevier.com/locate/chemphys
Effects of permanent dipole moments on stimulated Raman adiabatic passage Alex Brown
*
Department of Chemistry, University of Alberta, Edmonton, AB, Canada T6G 2G2 Received 27 June 2007; accepted 5 September 2007 Available online 12 September 2007
Abstract The effects of permanent dipole moments on stimulated Raman adiabatic passage (STIRAP) are considered. Analytic expressions for the Hamiltonian including the effects of permanent dipole moments are developed for the STIRAP process. The potential detrimental effect of permanent dipoles on population transfer using standard STIRAP techniques is demonstrated using model three-level systems. However, the presence of permanent dipole moments can allow the utilization of alternative multi-photon mechanisms for STIRAP. Here two-photon plus two-photon STIRAP is highlighted as a potential new mechanism. 2007 Elsevier B.V. All rights reserved. Keywords: STIRAP; Permanent dipole moments
1. Introduction The stimulated Raman adiabatic passage (STIRAP) technique has been widely utilized to control population transfer in both atomic and molecular systems, see the recent reviews [1,2]. STIRAP was originally designed for use in a three-level system, where no transition dipole moment connects the initial and final states directly but the intermediate state is dipole coupled to both initial and final states [3–6]. In its simplest form for the three-level system, the STIRAP method uses two suitably timed coherent laser pulses: the ‘‘pump’’, coupling the initial state j1i to the intermediate state j2i, and the ‘‘Stokes’’ coupling states j2i and j3i. The STIRAP process transfers population adiabatically from the initial state j1i to final state j3i without significantly populating intermediate state j2i. In the STIRAP process, the Stokes pulse precedes but overlaps the pump pulse. Criteria containing constraints on both the strengths and the rates of change of the Stokes and pump fields that satisfy the condition of adiabatic *
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transfer of population can be found in the literature [2]. If these conditions are satisfied, population can be transferred completely between the initial and final states. Importantly, the transfer is relatively insensitive to fluctuations in the pulse energies, durations, and time delay. A variety of extensions of the three-level STIRAP technique to multiple laser excitation in N-level systems have been proposed [7–9]. Also, multi-photon STIRAP techniques involving m pump photons and n Stokes photons, so-called (m + n)-STIRAP have been demonstrated [10–15]. While STIRAP is often viewed from a population transfer perspective, a variety of studies on the utility of STIRAP-like techniques for controlling molecular dynamics have also been carried out, see [16] and references therein. Recently, STIRAP has been examined in extended K-systems [17] and it has been shown that the presence of permanent dipole moments can lead to laser phase dependence of target state populations. The authors used a previously developed [18] rotating wave approximation (RWA) for the interaction of a single frequency laser with a system having permanent dipole moments to help qualitatively interpret the results of their calculations. Motivated by the ad hoc use of the one-colour (one frequency)
A. Brown / Chemical Physics 342 (2007) 16–24
RWA to describe the STIRAP process (an inherently twocolour one), an RWA incorporating the effects of permanent dipole moments to predict STIRAP for a three-level system has been developed. Here, the formal development of this RWA is presented. The RWA is used to highlight the potential effects of permanent dipole moments on STIRAP including modification of the laser–molecule couplings (Rabi frequencies) through both n-photon absorptions and zero-photon ones and the introduction of new multi-photon absorption mechanisms that depend upon the presence of permanent dipole moments. The RWA is tested versus exact calculations on a model three-level system for both (1 + 1)-STIRAP and (2 + 2)-STIRAP. This paper is organized in the following manner. The underlying theory of STIRAP when permanent dipoles are neglected, i.e., lii = 0, is briefly reviewed in Section 2.1. In Section 2.2, new expressions for the Hamiltonian describing the STIRAP process including the effects of permanent dipole moments are developed. Results illustrating the effects of permanent dipoles on STIRAP and demonstrating the utility of the new RWA expressions in interpreting them are presented in Section 3. Both final state populations as a function of pump and Stokes field strengths and time-dependent state populations for fixed field parameters are considered. The three-level models used for the example calculations are introduced in Section 3.1. In Section 3.2, the regular (1 + 1)-STIRAP is discussed, while Section 3.3, illustrates (2 + 2)-STIRAP, where the pump and Stokes processes both involve the absorption of two photons. For (2 + 2)-STIRAP, the presence of permanent dipole moments, lii, is critical as the two-photon absorption is forbidden (for the three-level system considered) if lii = 0. Finally, some brief conclusions are presented in Section 4. Atomic units are utilized throughout this paper. The unit of energy E is the Hartree, for the transition and permanent dipole moments ljk is ea0, the field frequencies x is EH h1, time t is hE1 H , and for the field strength ej is EH(ea0). EH is the Hartree of energy, e is the absolute value of the charge of an electron, a0 is the Bohr radius, and h is the reduced Planck constant. The following conversion factors to SI units (and commonly occurring units) will be useful in what follows: ea0 = 8.478 · 1030 C m (=2.541 D), EH h1 = 4.134 · 1016 s1 (=2.19474 · 106 cm1), hE1 H ¼ 2:4189 1017 s (1015 fs = 1012 ps = 1 s), and the relationship between field intensity I and the electric field is I = 3.509 · 1016[e(a.u.)]2 W/cm2.
2. Theory
daðtÞ ¼ ½E ~ l ~ eðtÞaðtÞ: dt
In Eq. (1), E is a diagonal N · N matrix containing the energies, Ei, of the stationary states and ~ l is an N · N matrix with the dipole moment matrix elements ~ ljk ¼ h/j j~ lj/k i. These are the permanent (i = j) and transition (i 5 j) dipole moments. Often the permanent dipole moments are neglected, i.e., lii = 0. In this paper, the interaction of two pulsed laser fields with a three-level system (N = 3) is considered. 2.1. Three-level model: lii = 0 The STIRAP process when lii = 0 has been discussed extensively in the literature [2] and, hence, will only be reviewed briefly here. The STIRAP method uses two time-delayed laser pulses (pump and Stokes) to transfer population from an initial state j1i to a final state j3i via an intermediate state j2i. In the three-level system, there is no transition dipole moment connecting the initial and final states directly but the intermediate state is dipole coupled to both initial and final states. In this paper, we consider the K-configuration where E2 > E3 > E1, see Fig. 1. The STIRAP process is designed to transfer adiabatically all of the population from the initial state to the final state without significant population in the intermediate state at any time during the interaction. The two pulses are designated as ‘‘pump’’, ep(t), that couples the initial state j1i to the intermediate state j2i, and ‘‘Stokes’’, eS(t), that couples j2i to the final state j3i. In its simplest form, the pump pulse has a frequency equal to the energy difference between states 1 and 2, xp = E2 E1, and the frequency of the Stokes pulse is equal to that between states 2 and 3, xS = E2 E3. In the STIRAP process, the pulses are counterintuitively ordered with the Stokes pulse preceding but overlapping the pump pulse. Therefore, the total laser field is ~ eðtÞ ¼ e~S ðtÞ þ e~p ðtÞ ¼ ^eS eS fS ðtÞ cosðxS t þ dS Þ þ ^ep ep fp ðt td Þ cos½xp ðt td Þ þ dp :
ð2Þ
In Eq. (2), ^ei is the linear polarization vector, ei is the peak field strength, fi(t) is the pulse envelope, xi is the circular carrier frequency, and di is the carrier-envelope phase of laser i, which in the following derivation is set equal to zero for both the pump and Stokes pulses. The time delay between the laser pulses is given by td and the time delay is set to be equal to the pulse width [5]. We consider Gaussian
Table 1 Dipole moment matrix used in the three-models
The time-dependent wave equation for an N-level system interacting with a laser field, e(t), is given within the semi-classical dipole approximation by i
17
ð1Þ
j1i j2i j3i
j1i
j2i
j3i
1.00 0.01 (0.1) 0.00
0.01 (0.1) 0.00 or 0.99 0.01 (0.1)
0.00 0.01 (0.1) 1.00
Transition dipole moments given in brackets are for the model used in two-photon transitions. All dipole moments expressed in a.u.
18
A. Brown / Chemical Physics 342 (2007) 16–24
where T is a diagonal transformation matrix given by Z t eðt0 Þdt0 T jk ¼ djk exp½iEk ðt t0 Þ exp iljk
E2=0.02a.u., μ22 μ23,ΩS
t0
¼ djk exp½iEk ðt t0 Þ exp½icjk
Ωp,μ12
and t0 is the time that the pulse-system interaction begins. Upon transforming to the interaction representation, we have an expression for the time-dependent coefficients
E3=0.006a.u., μ33 E1 = 0.0 a.u., μ11 Fig. 1. Schematic diagram of the three-level system including corresponding energies. Dipole moments are given in Table 1.
pulse envelopes, where fi ðtÞ ¼ expðt2 =s2i Þ. The pulse duration is si, with i = Stokes or pump, and the temporal full width at half maximum (FWHM) is related to the pulse duration by Dt = 2si(ln 2)1/2. In the interaction representation and after invoking the rotating wave approximation (RWA), the Hamiltonian for two pulse excitation of the three-level system depicted in Fig. 1 with lii = 0.0 can be written as 0 1 0 Xp ðtÞ 0 1B C H ¼ @ Xp ðtÞ 2D XS ðtÞ A: ð3Þ 2 0 XS ðtÞ 0 In Eq. (3), Xi(t) = l Æ ei(t) are the Rabi frequencies, where l is the relevant transition dipole moment and e(t) the timedependent amplitude of the corresponding electric field. Here D is the one-photon detuning of the laser frequencies from the respective transitions, i.e., E21 xp and E23 xS, where Eij = Ei Ej. For generality, the detuning of the lasers from the pump and Stokes transitions has been included but, in this paper, we only consider situations exactly on resonance, i.e., D = 0. Criteria for variation of the Rabi frequencies that satisfy the condition of adiabatic transfer of population can be found in the literature [2]. These criteria contain constraints on both the strengths and the rates of change of the Stokes and pump fields. However, if they are satisfied, 100% population transfer between the initial and final states with negligible population of the intermediate state can be achieved and this transfer is relatively insensitive to fluctuations in the pulse energies, durations, and time delay. 2.2. Three-level model: lii 5 0 We now consider the effects of permanent dipole moments (diagonal dipole matrix elements) by examining the Hamiltonian when lii 5 0. As has been done in the literature previously when examining the effects of permanent dipole moments [19], Eq. (1) is transformed to an interaction representation defined by aðtÞ ¼ T bðtÞ;
ð5Þ
ð4Þ
b_ j ¼ i
3 X
ljk ~ eðtÞexp½iðEk Ej Þðt t0 Þ exp½iðckk cjj Þbk :
k¼1;k6¼j
ð6Þ The crucial part of this equation is the quantity Z t 0 0 I ¼ exp½iðckk cjj Þ ¼ exp iðlkk ljj Þ eðt Þdt :
ð7Þ
t0
Following the procedure outlined previously for an RWA for the interaction of a pulsed laser with a two-level system possessing permanent dipole moments [20], we can substitute an expression for the electric field, Eq. (2), and integrate by parts to obtain Z d kj ^eS eS d kj ^eS eS t dfS ðt0 Þ ckk cjj ¼ fS ðtÞ sinðxS tÞ dt0 xS xS t0 d kj ^ep ep sinðxS t0 Þdt0 þ fp ðt td Þ sin½xp ðt td Þ xp Z d kj ^ep ep t dfp ðt0 td Þ sin½xp ðt0 td Þdt0 ; 0 dt xp t0 ð8Þ where dkj = lkk ljj is the difference in permanent dipole moments between states k and j. Generally, the time derivative of the pulse envelope is inversely proportional to the pulse duration s. Therefore, if (xisi) 1, the second and fourth terms in the above equation can be neglected relative to the first and third terms. With this assumption, one obtains I ¼ expfizpkj fp ðt td Þ sin½xp ðt td Þg exp½izSkj fS ðtÞ sinðxS tÞ; ð9Þ zakj
where ¼ ½d kj ^ea ea =xa is a parameter shown to be important in other laser–polar molecule interactions [19– 25]. Using a known identity [26], Eq. (9) can be written in terms of Bessel functions of integer order n, Jn(x), as 1 1 X X J m ðzSkj fS ðtÞÞJ l ðzpkj fp ðt td ÞÞ I¼ m¼1 l¼1
expðimxS tÞ exp½ilxp ðt td Þ:
ð10Þ
In order to obtain a compact approximation to Eq. (10), we follow analogously the methods introduced for other problems considering the effects of permanent dipole moments [18–20]. These steps involve the following: expand the cosine functions of e(t) in complex exponential form,
A. Brown / Chemical Physics 342 (2007) 16–24
change the summation indices to m 0 = m + 1, m 0 = m 1, l 0 = l + 1, and l 0 = l 1, and utilize the identity [26] J kþ1 ðxÞ þ J k1 ðxÞ ¼
2k J k ðxÞ: x
ð11Þ
Following this procedure, the expression for the timedependent coefficients, Eq. (6), can be written as " # 3 1 1 X X X ^ ^ ml e ll e S eS p ep jk jk b_ j ¼ i þ zpkj zSkj k¼1;k6¼j m¼1 l¼1 J m ðzSkj fS ðtÞÞJ l ðzpkj fp ðt td ÞÞ exp½iðEkj mxS lxp Þt exp½ilxp td bk :
ð12Þ
At this point, the RWA can be made where off-resonant terms, i.e., Ekj mxS lxp 5 0, are neglected. We assume that the pump transition involves the absorption of l pump photons, and zero Stokes photons, while the Stokes transition involves the absorption of m Stokes photons and zero pump photons. After making the RWA, the Hamiltonian has exactly the same form as Eq. (3) with D = 0. However, the usual Rabi frequencies XS(t) and Xp(t) are replaced by l d 32 eS fS ðtÞ d 32 ep fp ðt td Þ ðm;0Þ C S ðtÞ ¼ 2m 23 xS J m J0 xS xp d 32 ð13Þ and C ð0;lÞ p ðtÞ ¼ 2l
l12 d 21 eS fS ðtÞ d 21 ep fp ðt td Þ xp J 0 Jl ; xS xp d 21 ð14Þ
respectively. We shall refer to the above approximation as the RWA(t) since the arguments of the Bessel functions are explicitly time-dependent. Generally, the determination of the time-dependent populations using the above expressions for the Rabi frequencies will require numerical evaluation. One can proceed analytically by using the expansion, 1 X ½zð1 f 2 ðtÞÞ=2n J mþn ðzÞ: J m ðzf ðtÞÞ ¼ f m ðtÞ ð15Þ n! n¼0 In this case, the modified Rabi frequencies are l d 32 eS d 32 ep ðm;0Þ C S ðtÞ ¼ 2m 23 xS J m J0 fS ðtÞ d 32 xS xp
ð16Þ
and
l12 d 21 eS d 21 ep C ð0;lÞ ðtÞ ¼ 2l J x J fp ðt td Þ: p 0 l p d 21 xS xp
19
photon transitions are allowed. For example, two-photon transitions, which are forbidden in two-level systems, are allowed. Note that in many-level systems, the ‘‘direct’’ permanent dipole mechanism for two-photon transitions can be more effective than that involving virtual states [27]. Secondly, although the pump (Stokes) transition involves the absorption of zero Stokes (pump) photons, the Stokes (pump) field can affect the pump (Stokes) Rabi frequency through the term J 0 ðzakj Þ. If the argument of the Bessel function is small, e.g., d is small, the effect of the J 0 ðzakj Þ Bessel function can be neglected since limzakj !0 J 0 ðzakj Þ ¼ 1. Therefore, the Rabi frequencies for the Stokes and pump pulses are independent and can be written as l d 32 eS ðmÞ C S ðtÞ ¼ 2m 23 xS J m ð18Þ fS ðtÞ d 32 xS and C ðlÞ p ðtÞ ¼ 2l
l12 d 21 ep xp J l fp ðt td Þ: d 21 xp
ð19Þ
In the limit that d21 and d32 equal zero, Eqs. (18) and (19) give the correct lii = 0 expressions for the Rabi frequencies, see Eq. (3). Expressions similar to Eqs. (18) and (19) were used previously to help analyze phase-sensitive STIRAP in dipolar extended systems [17]. The above approximations to the Rabi frequencies, Eqs. (18) and (19) will be referred to as the RWA(no 0). It is also possible to consider a similar neglect of the J 0 ½zakj ðtÞ Bessel functions but retaining the explicit time-dependence of the Bessel function arguments, i.e., Eqs. (14) and (15) without the J 0 ½zakj ðtÞ terms. This approximation will be referred to as the RWA(t, no 0). In this paper, two examples using three-level systems are considered to illustrate the utility of the new RWA expressions: the standard STIRAP model involving one-photon transitions, i.e., m = 1, l = 1, and a new two-photon STIRAP mechanism for which the presence of permanent dipoles is critical, i.e., m = 2, l = 2. As discussed previously [28], RWA expressions incorporating the effects of d 5 0 for two-level systems must be modified to quantitatively consider situations where the number of photons absorbed is 3, 5, 7. . .. 3. Results and discussion 3.1. Model system
ð17Þ
The above approximations to the Rabi frequencies, Eqs. (16) and (17) will be referred to as the RWA. The advantage of this approximation, where the arguments of the Bessel functions are time-independent, is that an analytic solution analogous to the d = 0 one exists for the state populations [1,2]. From these expressions, two effects of permanent dipole moments are evident. First, alternate mechanisms for N-
The model K-system including the stationary state energies is illustrated in Fig. 1. The transition and permanent dipole moments are taken to be aligned with each other and with the directions of the applied laser fields. The model we consider has transitions in the IR region and, thus, the energies and dipoles roughly correspond to rovibrational transitions in an asymmetric double well system, e.g., HCN ! HNC isomerization [29]. However, the results should scale to different energy regimes. The transition dipole moments are set to be equal, l12 = l23. Two dif-
20
A. Brown / Chemical Physics 342 (2007) 16–24
ferent values of the transition dipole moments are considered in what follows: 0.01 a.u. for the one-photon transitions considered in Section 3.2 and 0.1 a.u. for the twophoton transitions considered in Section 3.3. Smaller transition dipole moments are considered for one-photon transitions as the effects of permanent dipole moments only manifest themselves for large field strengths in this case, see Section 3.2. The permanent dipole moments of the initial and final states are taken to be equal in magnitude but of opposite sign: l11 = l33 = 1.00 a.u. The dipole moment of the intermediate state is taken to be either 0 a.u. or 0.99 a.u. The FWHM of the pump and Stokes pulses are chosen to be 10 ps and the time delay is chosen to be equal to the FWHM. Since l12 = l23, the pump and Stokes fields strengths are chosen to be equal in all of the following calculations. The time-dependent Schro¨dinger equation has been solved using a standard fourth-order Runge–Kutta algorithm [30] with a time step of 0.5 fs, when the RWA is invoked, and 0.01 fs, for exact calculations which do not use the RWA. Initially all the population is in the ground state j1i. 3.2. One-photon STIRAP In order to achieve STIRAP, the field strengths must be optimized for the chosen pulse durations and time delay. The population of the final state j3i at the end of the interaction of the pulses, P3(t = 1), is determined as a function of the field strength, where ep = eS, see Fig. 2. The frequencies of the pump and Stokes fields are set to xp = E21 = 0.02 a.u. and xS = E23 = 0.014 a.u., i.e., they are set equal to the one-photon transition frequencies. For the case where lii = 0 and lij = 0.01 a.u., 100% population transfer is readily achieved and maintained as the field
strength is increased. For dkj = 0, the results determined using the exact Hamiltonian (RWA not invoked) and the RWA Hamiltonian, Eq. (1), are indistinguishable. If l11 = l33 = 1.00 a.u., the exact dkj 5 0 calculations demonstrate that small effects of the permanent dipoles are seen at large fields strengths, e > 0.045 a.u. In particular, the population of the final target state j3i begins to decrease from one as the fields strength is increased. The decrease in final state population is a direct consequence of the presence of permanent dipole moments (discussed in more detail for Fig. 3). Four different calculations have been carried out for the case when dkj 5 0 in order to assess successive approximations in the RWA: exact, RWA(t), RWA, and RWA(t, no 0). The RWA(t) and the RWA(t, no 0) provide near quantitative agreement with the exact calculations. On the other hand, the RWA overestimates the effects of the permanent dipoles. This is primarily due to the time-independent ‘‘zero-photon’’ terms J 0 ðzakj Þ in the laser–molecule coupling that do not reflect the pulse shapes of the pump and Stokes fields. Therefore, while this approximation provides a (very) qualitative prediction of the effects of permanent dipoles and allows for an analytic solution to the problem, it cannot be used to quantitatively predict the final state populations for STIRAP. In order to demonstrate a more significant effect of permanent dipole moments involving one-photon transitions, a model where l11 = l33 = 1.00 a.u. and l22 = 0.99 a.u. is considered, i.e., the intermediate state has a dipole moment similar to the initial state but of opposite sign to the final state. Since d21 = 0.01 a.u. and d23 = 1.99 a.u., the difference in permanent dipole moments will primarily affect the Stokes transition, see for example Eq. (13). Fig. 3 illustrates the final population of the target state, P3(t = 1), as function of the field strengths of the pump
1.0
1.0 0.8 0.6
P3(t=∞)
P3(t=∞)
0.8
0.4
0.6 0.4
0.2
0.2 0.0
0
0.01
0.02
0.03
0.04
0.05
Field strength, ε(a.u.) Fig. 2. Final population of the target state, P3(t = 1), as a function of the field strength of the pump and Stokes lasers, e = ep = eS, for one-photon STIRAP in the three-level model system with l22 = 0 a.u. Results are shown for the following calculations (see text for details): exact/RWA d = 0 (solid line), exact d 5 0 (bold dot-dashed line) and RWA(t) d 5 0 (dot-dashed line), RWA d 5 0 (dotted line), and RWA(t, no 0) (dashed line).
0.0
0
0.01 0.02 0.03 0.04 Field Strength, ε (a.u.)
0.05
Fig. 3. Final population of the target state, P3(t = 1), as a function of the field strength of the pump and Stokes lasers, e = ep = eS, for one-photon STIRAP in the three-level model system with l22 = 0.99. Results are shown for the following calculations (see text for details): exact/RWA d = 0 (solid line), exact d 5 0 (dashed line), RWA d 5 0 (dotted line), RWA(t) d 5 0 (dot-dash line), and RWA(t, no 0) (dot-dot-dashed line).
A. Brown / Chemical Physics 342 (2007) 16–24
1 0.8 0.6 0.4 0.2 0
-5
0
5
10
15
20
25
Time (ps)
ð20Þ
Xeff Ds > 10;
where Ds is the period during which the pulses overlap, is no longer fulfilled. For the case dkj 5 0 and for one-photon transitions, the rms Rabi frequency is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b
1 0.8
ð1;0Þ
Xeff ¼ X2S þ X2p and the rms Rabi frequency will always increase as the field strengths are increased. The question then arises as to whether the decrease in ð1;0Þ the Rabi frequencies, C S and C pð0;1Þ , is due to the one-phoa ton, J 1 ðzkj Þ, or the zero-photon, J 0 ðzakj Þ, part of the coupling. Clearly from the good agreement between the RWA(t, no 0) and exact results, see Figs. 2 and 3, the zero-photon part of the coupling plays a minor role for the molecular and laser parameters considered here. One can also examine this numerically for specific field strengths of interest. For instance in the second example considered, there is no population transfer to the target state for the field strengths e = eS = ep = 0.0275 a.u., see Fig. 3. For these field strengths, the J 0 ðzakj Þ term is 0.99998 and ð1;0Þ
0.76228 for the C pð0;1Þ ðtd Þ and C S ð0Þ couplings, respecð1;0Þ tively. On the other hand, the J 1 ðzSkj Þ term for C S ð0Þ is approximately zero (magnitude = 0.03076). The presence of the permanent dipole difference, d32 = 1.99 a.u., has ‘‘turned off’’ the Stokes transition and, thus, there is no population transfer to the target state. In addition to predicting the final target state population, the RWA(t) and RWA(t, no 0) can be utilized to determine the time-dependent behaviour of the state populations. For a field strength in the plateau region of Fig. 3, e.g., e = 0.01 a.u., the time-dependent state populations determined using the RWA(t) and RWA(t, no 0) are in near quantitative agreement with the results of exact calculations (results not illustrated). For the interesting case where the Stokes transition is ‘‘turned off’’, i.e., e = 0.0275 a.u., the time-dependent state populations as determined using the RWA(t) and exact calculations are illustrated in Fig. 4. The Stokes transition has not been
0.6
P2(t)
Xeff ¼ C S þ C pð0;1Þ , where C S and C pð0;1Þ , see Eqs. (13) and (14), depend on Bessel functions that are oscillatory in nature, i.e., the Rabi frequencies can decrease with increasing field strength. On the other hand, for dkj = 0, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0.4 0.2 0
c
-5
0
5
15 10 Time (ps)
20
25
-5
0
5
10 15 Time (ps)
20
25
0.3
0.2
P3(t)
ð1;0Þ
a
P1(t)
and Stokes lasers, ep = eS. The effects of the permanent dipole moments are significantly increased relative to those shown in Fig. 2. For example, the target state population can be decreased to zero as the field strengths are increased. Also, the effects of permanent dipoles manifest themselves for smaller fields (e > 0.02 a.u.) than for the case considered in Fig. 2. As has been seen in Fig. 2, both the RWA(t) and RWA(t, no 0) provide near quantitative agreement with the exact results while the RWA predicts the qualitative behaviour of the target state population but does not provide a quantitative prediction. The population of the target state begins to decrease as the field strength is increased because the ‘‘global’’ adiabatic criterion [2]
21
0.1
0
Fig. 4. Time-dependent population of the states, Pj(t), for STIRAP in the three-level model system with l22 = 0.99 a.u.: (a) state j1i, initial state, (b) state j2i, intermediate state, and (c) state j3i, target state. The field strength of the pump and Stokes lasers, e = ep = eS = 0.0275 a.u. Results are shown for the following calculations (see text for details): exact d 5 0 (solid line) and RWA(t) d 5 0 (dashed line).
‘‘turned off’’ completely as there is transient population in the target state j3i, see Fig. 4c. The transient population ð1;0Þ arises because the Stokes coupling C S ð0Þ is only near zero at the maximum in the Stokes field. Examining the time-dependent populations between 2 ps < t < +3 ps,
A. Brown / Chemical Physics 342 (2007) 16–24
the mechanism leading to the population of state j3i becomes clear. Between 2 ps and +2 ps, population is transferred via a one-photon transition from state j1i to state j2i by the leading edge of the pump pulse; recall that the pump pulse follows the Stokes pulse and is resonant with the E21 energy difference. Once population is in state j2i, it can be transferred via a one-photon transition directly to state j3i by the trailing edge of the Stokes pulse. For t > +8 ps, the Stokes pulse does not induce further population transfer between states j2i and 3i, i.e., the Stokes pulse is ‘‘over’’. The overall behaviour of the state populations as a function of time are well described by the RWA(t). However, while the populations of states j1i and j2i oscillate between 0 and 1 for the RWA(t) calculation, they only oscillate between 0.07 and 0.93 for the exact ones, see Fig. 4a and b. The oscillations of the exact (RWA) populations between 0.07 and 0.93 (0 and 1) reflects the distribution of state populations when the Stokes pulse is ‘‘over’’. Note that different choices of field strength, or slight changes in the time delay between the pulses, leads to different maximal and minimal populations in the oscillations. At these relatively large field strengths (e = eS = ep = 0.0275 a.u.), the minor difference between the RWA(t) and exact results can be attributed to the approximations made within the RWA, i.e., neglect of counter-rotating terms and neglect of Bloch–Siegert shifts [31] in the resonance frequency. For one-photon transitions using the current model system, the effects of permanent dipole moments are of minor importance except for large fields strengths. In general, their importance in one-photon transitions can be estimated by considering the magnitudes of the Bessel function arguments, zakj ¼ d kj ea =xa . Therefore, the effects would be larger for systems where there is a large difference in permanent dipole moments between the states of interest (dkj is large) or for transitions between states with much smaller energy separations (xa is small). Our focus in this model treatment is on the effect of permanent dipole moments on STIRAP, and, therefore, we are not overly concerned with the large field strengths used in the calculations. The field strengths could be reduced inversely proportionally to the length of the pulses – this would also diminish the effects of the permanent dipole moments for the one-photon excitations.
multi-photon STIRAP scenarios. For (2 + 2)-STIRAP, a similar model to that used for STIRAP, as discussed in Section 3.2, has been employed except the transition dipole moments have been increased to 0.1 a.u. The final population of the target state, P3(t = 1), as function of the pump and Stokes field strengths is shown in Fig. 5. As with the one-photon STIRAP case, both the RWA(t) and RWA(t, no 0) provide near quantitative agreement with the exact results. On the other hand, the RWA simply provides qualitative agreement. Since weaker field strengths are employed here as compared to the onephoton STIRAP model, the effects of the zero-photon terms are small because J 0 ðzakj Þ is approximately one across the range of field strengths considered in Fig. 5 for both the pump and Stokes transitions. For example, for the field strengths e = 0.01 a.u., J 0 ðzS21 Þ ¼ 0:93213 and J 0 ðzp21 Þ ¼ 0:96645. The RWA(t) can also be utilized to determine the timedependent behaviour of the state populations. The timedependent state populations as determined using the RWA(t) and exact calculations are illustrated in Fig. 6. The overall behaviour of the state populations as a function of time are well described by the RWA(t). However, the exact calculations exhibit small amplitude high-frequency oscillations; the RWA(t) results are smooth. These oscillations have been observed previously in both one-colour pulsed [32,33] and one-colour continuous wave calculations [34–36]. Their presence is due to the counter-rotating terms that have been neglected in the RWA and their positions correspond to the zeros in the electric fields. Although the RWA(t) does not reproduce these high-frequency details, it provides an excellent description of the overall time-dependent behaviour of the state populations.
1.0 0.8 P3(t=∞)
22
0.6 0.4
3.3. Two-photon STIRAP 0.2
We now consider the case of (2 + 2)-STIRAP where the pump and Stokes transitions both involve the absorption of two photons. The frequencies of the pump and Stokes fields are set to xp = E21/2 = 0.01 a.u. and xS = E23/2 = 0.007 a.u., i.e., they are set equal to the two-photon transition frequencies. In the three-level model considered, the two-photon transitions would be forbidden if lii = 0. Hence, these results do not have to be computed as there would be no population transfer – a situation that emphasizes the critical role permanent dipole moments can play in
0.0
0
0.005 0.01 Field Strength, ε (a.u.)
0.015
Fig. 5. Final population of the target state, P3(t = 1), as a function of the field strength of the pump and Stokes lasers, e = ep = eS for (2 + 2)STIRAP in the three-level model system with l22 = 0 a.u. Results are shown for the following calculations (see text for details): exact d 5 0 (solid line), RWA d 5 0 and RWA(no 0) (dotted line, results are indistinguishable), RWA(t) d 5 0 (dashed line), and RWA(t, no 0) d 5 0 (dot-dash line).
A. Brown / Chemical Physics 342 (2007) 16–24
1 〈
〈
⏐1
⏐3
0.8
Pj(t)
0.6 0.4 0.2 0 2.5
〈
⏐2
5
7.5
10
Time (ps) Fig. 6. Time-dependent population of the states, Pj(t), for (2 + 2)STIRAP in the three-level model system with l22 = 0 a.u. The field strength of the pump and Stokes lasers, e = ep = eS = 0.01 a.u. Results are shown for the following calculations (see text for details): exact d 5 0 (solid line) and RWA(t) d 5 0 (dashed line).
23
STIRAP processes to occur. For example, in Section 3.3, we have considered (2 + 2)-STIRAP where both the pump and Stokes processes involve the absorption of two photons. For the three-level model considered, two-photon transition would be formally forbidden if d = 0. From the examples presented, it clear that the effects of permanent dipole moments must be carefully considered when examining STIRAP in molecular systems. Also, their effects may manifest themselves in more complex scenarios such as the phase-dependent populations that have been explored for STIRAP in extended K-systems [17]. The newly derived RWAs should help in the interpretation of STIRAP processes where permanent dipoles play an important role. For example, permanent dipoles may play a role in STIRAP population transfer in asymmetric double well systems, e.g., control of isomerization reactions [15,17,37], or in selective population transfer in molecules [38]. Acknowledgements
4. Conclusions The effects of permanent dipole moments on STIRAP have been carefully considered using simple three-level model systems. In order to understand the role of permanent dipole moments, RWAs including the effects of djk 5 0 have been developed for the STIRAP process. The RWAs have been tested by comparing with exact results for both the target state j3i population as a function of field strengths of the pump and Stokes lasers and for the time-dependent population of the states for fixed laser field strengths. These tests have demonstrated that the RWAs can qualitatively (simplest RWA, see Eqs. (16) and (17)) and also quantitatively [RWA(t), see Eqs. (13) and (14)] predict the exact results. Three potential effects of the permanent dipole moments on the STIRAP process can be predicted from the RWA expressions that have been derived. First, the laser–molecule coupling for the N-photon pump or Stokes transition depends upon the Bessel function, J N ðzajk Þ where a = ‘‘pump’’ or ‘‘Stokes’’, see for example Eqs. (13) and (14). Since the Bessel functions are oscillatory, the STIRAP condition, see Eq. (20), may not be fulfilled as the field strengths are increased. This has been demonstrated for the regular STIRAP process in both Figs. 2 and 3. For the molecular and laser parameters considered in the model, this has lead to a decrease in the field strength region for which STIRAP occurs. Secondly, the N-photon ð1;0Þ Stokes and pump Rabi frequencies, C S and C pð0;1Þ , respectively, depend on zero-photon absorption in the corresponding pump and Stokes transitions through the term J 0 ðzakj Þ, see for example Eqs. (13) and (14). For the examples considered, the zero-photon absorption was shown to play a very minor role. However, in general, the zerophoton effect should be considered. Thirdly, the presence of permanent dipole moments can allow new multi-photon
The author thanks the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canadian Foundation for Innovation and the University of Alberta for financial support. The author thanks W.J. Meath (University of Western Ontario) for useful discussions. References [1] N.V. Vitanov, T. Halfmann, B.W. Shore, K. Bergmann, Annu. Rev. Phys. Chem. 52 (2001) 763. [2] K. Bergmann, H. Theuer, B. Shore, Rev. Mod. Phys. 70 (1998) 1003. [3] J. Oreg, F.T. Hioe, J.H. Eberly, Phys. Rev. A 29 (1984) 690. [4] U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Ku¨lz, K. Bergmann, Chem. Phys. Lett. 149 (1988) 463. [5] U. Gaubatz, P. Rudecki, S. Schiemann, K. Bergmann, J. Chem. Phys. 92 (1990) 5363. [6] J.R. Kuklinski, U. Gaubatz, F.T. Hioe, K. Bergmann, Phys. Rev. A 40 (1989) 6741. [7] B.W. Shore, K. Bergmann, J. Oreg, S. Rosenwaks, Phys. Rev. A 44 (1990) 7442. [8] B.W. Shore, J. Martin, M.P. Fewell, K. Bergmann, Phys. Rev. A 52 (1995) 566. [9] V.S. Malinovsky, D.J. Tannor, Phys. Rev. A 56 (1997) 4929. [10] L.P. Yatsenko, S. Gue´rin, T. Halfmann, K. Bo¨hmer, B.W. Shore, K. Bergmann, Phys. Rev. A 58 (1998) 4683. [11] S. Gue´rin, L.P. Yatsenko, T. Halfmann, B.W. Shore, K. Bergmann, Phys. Rev. A 58 (1998) 4691. [12] K. Bo¨hmer, T. Halfmann, L.P. Yatsenko, B.W. Shore, K. Bergmann, Phys. Rev. A 64 (2001) 023404. [13] S. Gue´rin, H.R. Jauslin, Eur. Phys. J. D 2 (1999) 99. [14] J. Oreg, K. Bergmann, B.W. Shore, S. Rosenwaks, Phys. Rev. A 45 (1992) 4888. [15] I. Vra´bel, W. Jakubetz, J. Chem. Phys. 118 (2003) 7366. [16] J.B. Gong, S.A. Rice, J. Chem. Phys. 121 (2004) 1364. [17] C.A. Marx, W. Jakubetz, J. Chem. Phys. 125 (2006) 234103. [18] S. Nakai, W.J. Meath, J. Chem. Phys. 96 (1992) 4991. [19] M.A. Kmetic, W.J. Meath, Phys. Lett. A 108 (1985) 340. [20] A. Brown, W.J. Meath, P. Tran, Phys. Rev. A 63 (2000) 013403.
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[21] W.J. Meath, R.A. Thuraisingham, M.A. Kmetic, Adv. Chem. Phys. 73 (1989) 307. [22] A.E. Kondo, W.J. Meath, Mol. Phys. 74 (1991) 113. [23] R.A. Thuraisingham, W.J. Meath, Mol. Phys. 56 (1985) 193. [24] T. Hattori, T. Kobayashi, Phys. Rev. A 35 (1987) 2733. [25] M.A. Kmetic, W.J. Meath, Phys. Rev. A 41 (1990) 1556. [26] V. Mangulis, Handbook of Series for Scientists and Engineers, Academic Press, New York, 1965. [27] A.E. Kondo, W.J. Meath, S.H. Nilar, A.J. Thakkar, Chem. Phys. 186 (1994) 375. [28] A. Brown, W.J. Meath, Phys. Rev. A 53 (1996) 2571. [29] J.M. Bowman, S. Irle, K. Morukuma, A. Wodtke, J. Chem. Phys. 114 (2001) 7923.
[30] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing, second ed., Cambridge University Press, Cambridge, 1992. [31] F. Bloch, A. Siegert, Phys. Rev. 57 (1940) 522. [32] G.F. Thomas, Phys. Rev. A 32 (1985) 1515. [33] R.W. Ziolkowski, J.M. Arnold, D.M. Gogny, Phys. Rev. A 52 (1995) 3082. [34] W.R. Salzman, Phys. Rev. A 16 (1977) 1552. [35] G.F. Thomas, W.J. Meath, J. Phys. B 16 (1983) 951. [36] J.V. Moloney, W.J. Meath, Mol. Phys. 31 (1976) 1537. [37] V. Kurkal, S.A. Rice, Chem. Phys. Lett. 344 (2001) 125. [38] V. Kurkal, S.A. Rice, J. Phys. Chem. B 105 (2001) 6488.
Chemical Physics 342 (2007) 16–24 www.elsevier.com/locate/chemphys
Effects of permanent dipole moments on stimulated Raman adiabatic passage Alex Brown
*
Department of Chemistry, University of Alberta, Edmonton, AB, Canada T6G 2G2 Received 27 June 2007; accepted 5 September 2007 Available online 12 September 2007
Abstract The effects of permanent dipole moments on stimulated Raman adiabatic passage (STIRAP) are considered. Analytic expressions for the Hamiltonian including the effects of permanent dipole moments are developed for the STIRAP process. The potential detrimental effect of permanent dipoles on population transfer using standard STIRAP techniques is demonstrated using model three-level systems. However, the presence of permanent dipole moments can allow the utilization of alternative multi-photon mechanisms for STIRAP. Here two-photon plus two-photon STIRAP is highlighted as a potential new mechanism. 2007 Elsevier B.V. All rights reserved. Keywords: STIRAP; Permanent dipole moments
1. Introduction The stimulated Raman adiabatic passage (STIRAP) technique has been widely utilized to control population transfer in both atomic and molecular systems, see the recent reviews [1,2]. STIRAP was originally designed for use in a three-level system, where no transition dipole moment connects the initial and final states directly but the intermediate state is dipole coupled to both initial and final states [3–6]. In its simplest form for the three-level system, the STIRAP method uses two suitably timed coherent laser pulses: the ‘‘pump’’, coupling the initial state j1i to the intermediate state j2i, and the ‘‘Stokes’’ coupling states j2i and j3i. The STIRAP process transfers population adiabatically from the initial state j1i to final state j3i without significantly populating intermediate state j2i. In the STIRAP process, the Stokes pulse precedes but overlaps the pump pulse. Criteria containing constraints on both the strengths and the rates of change of the Stokes and pump fields that satisfy the condition of adiabatic *
Tel.: +1 780 492 1854; fax: +1 780 492 8231. E-mail address: [email protected]
0301-0104/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2007.09.007
transfer of population can be found in the literature [2]. If these conditions are satisfied, population can be transferred completely between the initial and final states. Importantly, the transfer is relatively insensitive to fluctuations in the pulse energies, durations, and time delay. A variety of extensions of the three-level STIRAP technique to multiple laser excitation in N-level systems have been proposed [7–9]. Also, multi-photon STIRAP techniques involving m pump photons and n Stokes photons, so-called (m + n)-STIRAP have been demonstrated [10–15]. While STIRAP is often viewed from a population transfer perspective, a variety of studies on the utility of STIRAP-like techniques for controlling molecular dynamics have also been carried out, see [16] and references therein. Recently, STIRAP has been examined in extended K-systems [17] and it has been shown that the presence of permanent dipole moments can lead to laser phase dependence of target state populations. The authors used a previously developed [18] rotating wave approximation (RWA) for the interaction of a single frequency laser with a system having permanent dipole moments to help qualitatively interpret the results of their calculations. Motivated by the ad hoc use of the one-colour (one frequency)
A. Brown / Chemical Physics 342 (2007) 16–24
RWA to describe the STIRAP process (an inherently twocolour one), an RWA incorporating the effects of permanent dipole moments to predict STIRAP for a three-level system has been developed. Here, the formal development of this RWA is presented. The RWA is used to highlight the potential effects of permanent dipole moments on STIRAP including modification of the laser–molecule couplings (Rabi frequencies) through both n-photon absorptions and zero-photon ones and the introduction of new multi-photon absorption mechanisms that depend upon the presence of permanent dipole moments. The RWA is tested versus exact calculations on a model three-level system for both (1 + 1)-STIRAP and (2 + 2)-STIRAP. This paper is organized in the following manner. The underlying theory of STIRAP when permanent dipoles are neglected, i.e., lii = 0, is briefly reviewed in Section 2.1. In Section 2.2, new expressions for the Hamiltonian describing the STIRAP process including the effects of permanent dipole moments are developed. Results illustrating the effects of permanent dipoles on STIRAP and demonstrating the utility of the new RWA expressions in interpreting them are presented in Section 3. Both final state populations as a function of pump and Stokes field strengths and time-dependent state populations for fixed field parameters are considered. The three-level models used for the example calculations are introduced in Section 3.1. In Section 3.2, the regular (1 + 1)-STIRAP is discussed, while Section 3.3, illustrates (2 + 2)-STIRAP, where the pump and Stokes processes both involve the absorption of two photons. For (2 + 2)-STIRAP, the presence of permanent dipole moments, lii, is critical as the two-photon absorption is forbidden (for the three-level system considered) if lii = 0. Finally, some brief conclusions are presented in Section 4. Atomic units are utilized throughout this paper. The unit of energy E is the Hartree, for the transition and permanent dipole moments ljk is ea0, the field frequencies x is EH h1, time t is hE1 H , and for the field strength ej is EH(ea0). EH is the Hartree of energy, e is the absolute value of the charge of an electron, a0 is the Bohr radius, and h is the reduced Planck constant. The following conversion factors to SI units (and commonly occurring units) will be useful in what follows: ea0 = 8.478 · 1030 C m (=2.541 D), EH h1 = 4.134 · 1016 s1 (=2.19474 · 106 cm1), hE1 H ¼ 2:4189 1017 s (1015 fs = 1012 ps = 1 s), and the relationship between field intensity I and the electric field is I = 3.509 · 1016[e(a.u.)]2 W/cm2.
2. Theory
daðtÞ ¼ ½E ~ l ~ eðtÞaðtÞ: dt
In Eq. (1), E is a diagonal N · N matrix containing the energies, Ei, of the stationary states and ~ l is an N · N matrix with the dipole moment matrix elements ~ ljk ¼ h/j j~ lj/k i. These are the permanent (i = j) and transition (i 5 j) dipole moments. Often the permanent dipole moments are neglected, i.e., lii = 0. In this paper, the interaction of two pulsed laser fields with a three-level system (N = 3) is considered. 2.1. Three-level model: lii = 0 The STIRAP process when lii = 0 has been discussed extensively in the literature [2] and, hence, will only be reviewed briefly here. The STIRAP method uses two time-delayed laser pulses (pump and Stokes) to transfer population from an initial state j1i to a final state j3i via an intermediate state j2i. In the three-level system, there is no transition dipole moment connecting the initial and final states directly but the intermediate state is dipole coupled to both initial and final states. In this paper, we consider the K-configuration where E2 > E3 > E1, see Fig. 1. The STIRAP process is designed to transfer adiabatically all of the population from the initial state to the final state without significant population in the intermediate state at any time during the interaction. The two pulses are designated as ‘‘pump’’, ep(t), that couples the initial state j1i to the intermediate state j2i, and ‘‘Stokes’’, eS(t), that couples j2i to the final state j3i. In its simplest form, the pump pulse has a frequency equal to the energy difference between states 1 and 2, xp = E2 E1, and the frequency of the Stokes pulse is equal to that between states 2 and 3, xS = E2 E3. In the STIRAP process, the pulses are counterintuitively ordered with the Stokes pulse preceding but overlapping the pump pulse. Therefore, the total laser field is ~ eðtÞ ¼ e~S ðtÞ þ e~p ðtÞ ¼ ^eS eS fS ðtÞ cosðxS t þ dS Þ þ ^ep ep fp ðt td Þ cos½xp ðt td Þ þ dp :
ð2Þ
In Eq. (2), ^ei is the linear polarization vector, ei is the peak field strength, fi(t) is the pulse envelope, xi is the circular carrier frequency, and di is the carrier-envelope phase of laser i, which in the following derivation is set equal to zero for both the pump and Stokes pulses. The time delay between the laser pulses is given by td and the time delay is set to be equal to the pulse width [5]. We consider Gaussian
Table 1 Dipole moment matrix used in the three-models
The time-dependent wave equation for an N-level system interacting with a laser field, e(t), is given within the semi-classical dipole approximation by i
17
ð1Þ
j1i j2i j3i
j1i
j2i
j3i
1.00 0.01 (0.1) 0.00
0.01 (0.1) 0.00 or 0.99 0.01 (0.1)
0.00 0.01 (0.1) 1.00
Transition dipole moments given in brackets are for the model used in two-photon transitions. All dipole moments expressed in a.u.
18
A. Brown / Chemical Physics 342 (2007) 16–24
where T is a diagonal transformation matrix given by Z t eðt0 Þdt0 T jk ¼ djk exp½iEk ðt t0 Þ exp iljk
E2=0.02a.u., μ22 μ23,ΩS
t0
¼ djk exp½iEk ðt t0 Þ exp½icjk
Ωp,μ12
and t0 is the time that the pulse-system interaction begins. Upon transforming to the interaction representation, we have an expression for the time-dependent coefficients
E3=0.006a.u., μ33 E1 = 0.0 a.u., μ11 Fig. 1. Schematic diagram of the three-level system including corresponding energies. Dipole moments are given in Table 1.
pulse envelopes, where fi ðtÞ ¼ expðt2 =s2i Þ. The pulse duration is si, with i = Stokes or pump, and the temporal full width at half maximum (FWHM) is related to the pulse duration by Dt = 2si(ln 2)1/2. In the interaction representation and after invoking the rotating wave approximation (RWA), the Hamiltonian for two pulse excitation of the three-level system depicted in Fig. 1 with lii = 0.0 can be written as 0 1 0 Xp ðtÞ 0 1B C H ¼ @ Xp ðtÞ 2D XS ðtÞ A: ð3Þ 2 0 XS ðtÞ 0 In Eq. (3), Xi(t) = l Æ ei(t) are the Rabi frequencies, where l is the relevant transition dipole moment and e(t) the timedependent amplitude of the corresponding electric field. Here D is the one-photon detuning of the laser frequencies from the respective transitions, i.e., E21 xp and E23 xS, where Eij = Ei Ej. For generality, the detuning of the lasers from the pump and Stokes transitions has been included but, in this paper, we only consider situations exactly on resonance, i.e., D = 0. Criteria for variation of the Rabi frequencies that satisfy the condition of adiabatic transfer of population can be found in the literature [2]. These criteria contain constraints on both the strengths and the rates of change of the Stokes and pump fields. However, if they are satisfied, 100% population transfer between the initial and final states with negligible population of the intermediate state can be achieved and this transfer is relatively insensitive to fluctuations in the pulse energies, durations, and time delay. 2.2. Three-level model: lii 5 0 We now consider the effects of permanent dipole moments (diagonal dipole matrix elements) by examining the Hamiltonian when lii 5 0. As has been done in the literature previously when examining the effects of permanent dipole moments [19], Eq. (1) is transformed to an interaction representation defined by aðtÞ ¼ T bðtÞ;
ð5Þ
ð4Þ
b_ j ¼ i
3 X
ljk ~ eðtÞexp½iðEk Ej Þðt t0 Þ exp½iðckk cjj Þbk :
k¼1;k6¼j
ð6Þ The crucial part of this equation is the quantity Z t 0 0 I ¼ exp½iðckk cjj Þ ¼ exp iðlkk ljj Þ eðt Þdt :
ð7Þ
t0
Following the procedure outlined previously for an RWA for the interaction of a pulsed laser with a two-level system possessing permanent dipole moments [20], we can substitute an expression for the electric field, Eq. (2), and integrate by parts to obtain Z d kj ^eS eS d kj ^eS eS t dfS ðt0 Þ ckk cjj ¼ fS ðtÞ sinðxS tÞ dt0 xS xS t0 d kj ^ep ep sinðxS t0 Þdt0 þ fp ðt td Þ sin½xp ðt td Þ xp Z d kj ^ep ep t dfp ðt0 td Þ sin½xp ðt0 td Þdt0 ; 0 dt xp t0 ð8Þ where dkj = lkk ljj is the difference in permanent dipole moments between states k and j. Generally, the time derivative of the pulse envelope is inversely proportional to the pulse duration s. Therefore, if (xisi) 1, the second and fourth terms in the above equation can be neglected relative to the first and third terms. With this assumption, one obtains I ¼ expfizpkj fp ðt td Þ sin½xp ðt td Þg exp½izSkj fS ðtÞ sinðxS tÞ; ð9Þ zakj
where ¼ ½d kj ^ea ea =xa is a parameter shown to be important in other laser–polar molecule interactions [19– 25]. Using a known identity [26], Eq. (9) can be written in terms of Bessel functions of integer order n, Jn(x), as 1 1 X X J m ðzSkj fS ðtÞÞJ l ðzpkj fp ðt td ÞÞ I¼ m¼1 l¼1
expðimxS tÞ exp½ilxp ðt td Þ:
ð10Þ
In order to obtain a compact approximation to Eq. (10), we follow analogously the methods introduced for other problems considering the effects of permanent dipole moments [18–20]. These steps involve the following: expand the cosine functions of e(t) in complex exponential form,
A. Brown / Chemical Physics 342 (2007) 16–24
change the summation indices to m 0 = m + 1, m 0 = m 1, l 0 = l + 1, and l 0 = l 1, and utilize the identity [26] J kþ1 ðxÞ þ J k1 ðxÞ ¼
2k J k ðxÞ: x
ð11Þ
Following this procedure, the expression for the timedependent coefficients, Eq. (6), can be written as " # 3 1 1 X X X ^ ^ ml e ll e S eS p ep jk jk b_ j ¼ i þ zpkj zSkj k¼1;k6¼j m¼1 l¼1 J m ðzSkj fS ðtÞÞJ l ðzpkj fp ðt td ÞÞ exp½iðEkj mxS lxp Þt exp½ilxp td bk :
ð12Þ
At this point, the RWA can be made where off-resonant terms, i.e., Ekj mxS lxp 5 0, are neglected. We assume that the pump transition involves the absorption of l pump photons, and zero Stokes photons, while the Stokes transition involves the absorption of m Stokes photons and zero pump photons. After making the RWA, the Hamiltonian has exactly the same form as Eq. (3) with D = 0. However, the usual Rabi frequencies XS(t) and Xp(t) are replaced by l d 32 eS fS ðtÞ d 32 ep fp ðt td Þ ðm;0Þ C S ðtÞ ¼ 2m 23 xS J m J0 xS xp d 32 ð13Þ and C ð0;lÞ p ðtÞ ¼ 2l
l12 d 21 eS fS ðtÞ d 21 ep fp ðt td Þ xp J 0 Jl ; xS xp d 21 ð14Þ
respectively. We shall refer to the above approximation as the RWA(t) since the arguments of the Bessel functions are explicitly time-dependent. Generally, the determination of the time-dependent populations using the above expressions for the Rabi frequencies will require numerical evaluation. One can proceed analytically by using the expansion, 1 X ½zð1 f 2 ðtÞÞ=2n J mþn ðzÞ: J m ðzf ðtÞÞ ¼ f m ðtÞ ð15Þ n! n¼0 In this case, the modified Rabi frequencies are l d 32 eS d 32 ep ðm;0Þ C S ðtÞ ¼ 2m 23 xS J m J0 fS ðtÞ d 32 xS xp
ð16Þ
and
l12 d 21 eS d 21 ep C ð0;lÞ ðtÞ ¼ 2l J x J fp ðt td Þ: p 0 l p d 21 xS xp
19
photon transitions are allowed. For example, two-photon transitions, which are forbidden in two-level systems, are allowed. Note that in many-level systems, the ‘‘direct’’ permanent dipole mechanism for two-photon transitions can be more effective than that involving virtual states [27]. Secondly, although the pump (Stokes) transition involves the absorption of zero Stokes (pump) photons, the Stokes (pump) field can affect the pump (Stokes) Rabi frequency through the term J 0 ðzakj Þ. If the argument of the Bessel function is small, e.g., d is small, the effect of the J 0 ðzakj Þ Bessel function can be neglected since limzakj !0 J 0 ðzakj Þ ¼ 1. Therefore, the Rabi frequencies for the Stokes and pump pulses are independent and can be written as l d 32 eS ðmÞ C S ðtÞ ¼ 2m 23 xS J m ð18Þ fS ðtÞ d 32 xS and C ðlÞ p ðtÞ ¼ 2l
l12 d 21 ep xp J l fp ðt td Þ: d 21 xp
ð19Þ
In the limit that d21 and d32 equal zero, Eqs. (18) and (19) give the correct lii = 0 expressions for the Rabi frequencies, see Eq. (3). Expressions similar to Eqs. (18) and (19) were used previously to help analyze phase-sensitive STIRAP in dipolar extended systems [17]. The above approximations to the Rabi frequencies, Eqs. (18) and (19) will be referred to as the RWA(no 0). It is also possible to consider a similar neglect of the J 0 ½zakj ðtÞ Bessel functions but retaining the explicit time-dependence of the Bessel function arguments, i.e., Eqs. (14) and (15) without the J 0 ½zakj ðtÞ terms. This approximation will be referred to as the RWA(t, no 0). In this paper, two examples using three-level systems are considered to illustrate the utility of the new RWA expressions: the standard STIRAP model involving one-photon transitions, i.e., m = 1, l = 1, and a new two-photon STIRAP mechanism for which the presence of permanent dipoles is critical, i.e., m = 2, l = 2. As discussed previously [28], RWA expressions incorporating the effects of d 5 0 for two-level systems must be modified to quantitatively consider situations where the number of photons absorbed is 3, 5, 7. . .. 3. Results and discussion 3.1. Model system
ð17Þ
The above approximations to the Rabi frequencies, Eqs. (16) and (17) will be referred to as the RWA. The advantage of this approximation, where the arguments of the Bessel functions are time-independent, is that an analytic solution analogous to the d = 0 one exists for the state populations [1,2]. From these expressions, two effects of permanent dipole moments are evident. First, alternate mechanisms for N-
The model K-system including the stationary state energies is illustrated in Fig. 1. The transition and permanent dipole moments are taken to be aligned with each other and with the directions of the applied laser fields. The model we consider has transitions in the IR region and, thus, the energies and dipoles roughly correspond to rovibrational transitions in an asymmetric double well system, e.g., HCN ! HNC isomerization [29]. However, the results should scale to different energy regimes. The transition dipole moments are set to be equal, l12 = l23. Two dif-
20
A. Brown / Chemical Physics 342 (2007) 16–24
ferent values of the transition dipole moments are considered in what follows: 0.01 a.u. for the one-photon transitions considered in Section 3.2 and 0.1 a.u. for the twophoton transitions considered in Section 3.3. Smaller transition dipole moments are considered for one-photon transitions as the effects of permanent dipole moments only manifest themselves for large field strengths in this case, see Section 3.2. The permanent dipole moments of the initial and final states are taken to be equal in magnitude but of opposite sign: l11 = l33 = 1.00 a.u. The dipole moment of the intermediate state is taken to be either 0 a.u. or 0.99 a.u. The FWHM of the pump and Stokes pulses are chosen to be 10 ps and the time delay is chosen to be equal to the FWHM. Since l12 = l23, the pump and Stokes fields strengths are chosen to be equal in all of the following calculations. The time-dependent Schro¨dinger equation has been solved using a standard fourth-order Runge–Kutta algorithm [30] with a time step of 0.5 fs, when the RWA is invoked, and 0.01 fs, for exact calculations which do not use the RWA. Initially all the population is in the ground state j1i. 3.2. One-photon STIRAP In order to achieve STIRAP, the field strengths must be optimized for the chosen pulse durations and time delay. The population of the final state j3i at the end of the interaction of the pulses, P3(t = 1), is determined as a function of the field strength, where ep = eS, see Fig. 2. The frequencies of the pump and Stokes fields are set to xp = E21 = 0.02 a.u. and xS = E23 = 0.014 a.u., i.e., they are set equal to the one-photon transition frequencies. For the case where lii = 0 and lij = 0.01 a.u., 100% population transfer is readily achieved and maintained as the field
strength is increased. For dkj = 0, the results determined using the exact Hamiltonian (RWA not invoked) and the RWA Hamiltonian, Eq. (1), are indistinguishable. If l11 = l33 = 1.00 a.u., the exact dkj 5 0 calculations demonstrate that small effects of the permanent dipoles are seen at large fields strengths, e > 0.045 a.u. In particular, the population of the final target state j3i begins to decrease from one as the fields strength is increased. The decrease in final state population is a direct consequence of the presence of permanent dipole moments (discussed in more detail for Fig. 3). Four different calculations have been carried out for the case when dkj 5 0 in order to assess successive approximations in the RWA: exact, RWA(t), RWA, and RWA(t, no 0). The RWA(t) and the RWA(t, no 0) provide near quantitative agreement with the exact calculations. On the other hand, the RWA overestimates the effects of the permanent dipoles. This is primarily due to the time-independent ‘‘zero-photon’’ terms J 0 ðzakj Þ in the laser–molecule coupling that do not reflect the pulse shapes of the pump and Stokes fields. Therefore, while this approximation provides a (very) qualitative prediction of the effects of permanent dipoles and allows for an analytic solution to the problem, it cannot be used to quantitatively predict the final state populations for STIRAP. In order to demonstrate a more significant effect of permanent dipole moments involving one-photon transitions, a model where l11 = l33 = 1.00 a.u. and l22 = 0.99 a.u. is considered, i.e., the intermediate state has a dipole moment similar to the initial state but of opposite sign to the final state. Since d21 = 0.01 a.u. and d23 = 1.99 a.u., the difference in permanent dipole moments will primarily affect the Stokes transition, see for example Eq. (13). Fig. 3 illustrates the final population of the target state, P3(t = 1), as function of the field strengths of the pump
1.0
1.0 0.8 0.6
P3(t=∞)
P3(t=∞)
0.8
0.4
0.6 0.4
0.2
0.2 0.0
0
0.01
0.02
0.03
0.04
0.05
Field strength, ε(a.u.) Fig. 2. Final population of the target state, P3(t = 1), as a function of the field strength of the pump and Stokes lasers, e = ep = eS, for one-photon STIRAP in the three-level model system with l22 = 0 a.u. Results are shown for the following calculations (see text for details): exact/RWA d = 0 (solid line), exact d 5 0 (bold dot-dashed line) and RWA(t) d 5 0 (dot-dashed line), RWA d 5 0 (dotted line), and RWA(t, no 0) (dashed line).
0.0
0
0.01 0.02 0.03 0.04 Field Strength, ε (a.u.)
0.05
Fig. 3. Final population of the target state, P3(t = 1), as a function of the field strength of the pump and Stokes lasers, e = ep = eS, for one-photon STIRAP in the three-level model system with l22 = 0.99. Results are shown for the following calculations (see text for details): exact/RWA d = 0 (solid line), exact d 5 0 (dashed line), RWA d 5 0 (dotted line), RWA(t) d 5 0 (dot-dash line), and RWA(t, no 0) (dot-dot-dashed line).
A. Brown / Chemical Physics 342 (2007) 16–24
1 0.8 0.6 0.4 0.2 0
-5
0
5
10
15
20
25
Time (ps)
ð20Þ
Xeff Ds > 10;
where Ds is the period during which the pulses overlap, is no longer fulfilled. For the case dkj 5 0 and for one-photon transitions, the rms Rabi frequency is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b
1 0.8
ð1;0Þ
Xeff ¼ X2S þ X2p and the rms Rabi frequency will always increase as the field strengths are increased. The question then arises as to whether the decrease in ð1;0Þ the Rabi frequencies, C S and C pð0;1Þ , is due to the one-phoa ton, J 1 ðzkj Þ, or the zero-photon, J 0 ðzakj Þ, part of the coupling. Clearly from the good agreement between the RWA(t, no 0) and exact results, see Figs. 2 and 3, the zero-photon part of the coupling plays a minor role for the molecular and laser parameters considered here. One can also examine this numerically for specific field strengths of interest. For instance in the second example considered, there is no population transfer to the target state for the field strengths e = eS = ep = 0.0275 a.u., see Fig. 3. For these field strengths, the J 0 ðzakj Þ term is 0.99998 and ð1;0Þ
0.76228 for the C pð0;1Þ ðtd Þ and C S ð0Þ couplings, respecð1;0Þ tively. On the other hand, the J 1 ðzSkj Þ term for C S ð0Þ is approximately zero (magnitude = 0.03076). The presence of the permanent dipole difference, d32 = 1.99 a.u., has ‘‘turned off’’ the Stokes transition and, thus, there is no population transfer to the target state. In addition to predicting the final target state population, the RWA(t) and RWA(t, no 0) can be utilized to determine the time-dependent behaviour of the state populations. For a field strength in the plateau region of Fig. 3, e.g., e = 0.01 a.u., the time-dependent state populations determined using the RWA(t) and RWA(t, no 0) are in near quantitative agreement with the results of exact calculations (results not illustrated). For the interesting case where the Stokes transition is ‘‘turned off’’, i.e., e = 0.0275 a.u., the time-dependent state populations as determined using the RWA(t) and exact calculations are illustrated in Fig. 4. The Stokes transition has not been
0.6
P2(t)
Xeff ¼ C S þ C pð0;1Þ , where C S and C pð0;1Þ , see Eqs. (13) and (14), depend on Bessel functions that are oscillatory in nature, i.e., the Rabi frequencies can decrease with increasing field strength. On the other hand, for dkj = 0, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0.4 0.2 0
c
-5
0
5
15 10 Time (ps)
20
25
-5
0
5
10 15 Time (ps)
20
25
0.3
0.2
P3(t)
ð1;0Þ
a
P1(t)
and Stokes lasers, ep = eS. The effects of the permanent dipole moments are significantly increased relative to those shown in Fig. 2. For example, the target state population can be decreased to zero as the field strengths are increased. Also, the effects of permanent dipoles manifest themselves for smaller fields (e > 0.02 a.u.) than for the case considered in Fig. 2. As has been seen in Fig. 2, both the RWA(t) and RWA(t, no 0) provide near quantitative agreement with the exact results while the RWA predicts the qualitative behaviour of the target state population but does not provide a quantitative prediction. The population of the target state begins to decrease as the field strength is increased because the ‘‘global’’ adiabatic criterion [2]
21
0.1
0
Fig. 4. Time-dependent population of the states, Pj(t), for STIRAP in the three-level model system with l22 = 0.99 a.u.: (a) state j1i, initial state, (b) state j2i, intermediate state, and (c) state j3i, target state. The field strength of the pump and Stokes lasers, e = ep = eS = 0.0275 a.u. Results are shown for the following calculations (see text for details): exact d 5 0 (solid line) and RWA(t) d 5 0 (dashed line).
‘‘turned off’’ completely as there is transient population in the target state j3i, see Fig. 4c. The transient population ð1;0Þ arises because the Stokes coupling C S ð0Þ is only near zero at the maximum in the Stokes field. Examining the time-dependent populations between 2 ps < t < +3 ps,
A. Brown / Chemical Physics 342 (2007) 16–24
the mechanism leading to the population of state j3i becomes clear. Between 2 ps and +2 ps, population is transferred via a one-photon transition from state j1i to state j2i by the leading edge of the pump pulse; recall that the pump pulse follows the Stokes pulse and is resonant with the E21 energy difference. Once population is in state j2i, it can be transferred via a one-photon transition directly to state j3i by the trailing edge of the Stokes pulse. For t > +8 ps, the Stokes pulse does not induce further population transfer between states j2i and 3i, i.e., the Stokes pulse is ‘‘over’’. The overall behaviour of the state populations as a function of time are well described by the RWA(t). However, while the populations of states j1i and j2i oscillate between 0 and 1 for the RWA(t) calculation, they only oscillate between 0.07 and 0.93 for the exact ones, see Fig. 4a and b. The oscillations of the exact (RWA) populations between 0.07 and 0.93 (0 and 1) reflects the distribution of state populations when the Stokes pulse is ‘‘over’’. Note that different choices of field strength, or slight changes in the time delay between the pulses, leads to different maximal and minimal populations in the oscillations. At these relatively large field strengths (e = eS = ep = 0.0275 a.u.), the minor difference between the RWA(t) and exact results can be attributed to the approximations made within the RWA, i.e., neglect of counter-rotating terms and neglect of Bloch–Siegert shifts [31] in the resonance frequency. For one-photon transitions using the current model system, the effects of permanent dipole moments are of minor importance except for large fields strengths. In general, their importance in one-photon transitions can be estimated by considering the magnitudes of the Bessel function arguments, zakj ¼ d kj ea =xa . Therefore, the effects would be larger for systems where there is a large difference in permanent dipole moments between the states of interest (dkj is large) or for transitions between states with much smaller energy separations (xa is small). Our focus in this model treatment is on the effect of permanent dipole moments on STIRAP, and, therefore, we are not overly concerned with the large field strengths used in the calculations. The field strengths could be reduced inversely proportionally to the length of the pulses – this would also diminish the effects of the permanent dipole moments for the one-photon excitations.
multi-photon STIRAP scenarios. For (2 + 2)-STIRAP, a similar model to that used for STIRAP, as discussed in Section 3.2, has been employed except the transition dipole moments have been increased to 0.1 a.u. The final population of the target state, P3(t = 1), as function of the pump and Stokes field strengths is shown in Fig. 5. As with the one-photon STIRAP case, both the RWA(t) and RWA(t, no 0) provide near quantitative agreement with the exact results. On the other hand, the RWA simply provides qualitative agreement. Since weaker field strengths are employed here as compared to the onephoton STIRAP model, the effects of the zero-photon terms are small because J 0 ðzakj Þ is approximately one across the range of field strengths considered in Fig. 5 for both the pump and Stokes transitions. For example, for the field strengths e = 0.01 a.u., J 0 ðzS21 Þ ¼ 0:93213 and J 0 ðzp21 Þ ¼ 0:96645. The RWA(t) can also be utilized to determine the timedependent behaviour of the state populations. The timedependent state populations as determined using the RWA(t) and exact calculations are illustrated in Fig. 6. The overall behaviour of the state populations as a function of time are well described by the RWA(t). However, the exact calculations exhibit small amplitude high-frequency oscillations; the RWA(t) results are smooth. These oscillations have been observed previously in both one-colour pulsed [32,33] and one-colour continuous wave calculations [34–36]. Their presence is due to the counter-rotating terms that have been neglected in the RWA and their positions correspond to the zeros in the electric fields. Although the RWA(t) does not reproduce these high-frequency details, it provides an excellent description of the overall time-dependent behaviour of the state populations.
1.0 0.8 P3(t=∞)
22
0.6 0.4
3.3. Two-photon STIRAP 0.2
We now consider the case of (2 + 2)-STIRAP where the pump and Stokes transitions both involve the absorption of two photons. The frequencies of the pump and Stokes fields are set to xp = E21/2 = 0.01 a.u. and xS = E23/2 = 0.007 a.u., i.e., they are set equal to the two-photon transition frequencies. In the three-level model considered, the two-photon transitions would be forbidden if lii = 0. Hence, these results do not have to be computed as there would be no population transfer – a situation that emphasizes the critical role permanent dipole moments can play in
0.0
0
0.005 0.01 Field Strength, ε (a.u.)
0.015
Fig. 5. Final population of the target state, P3(t = 1), as a function of the field strength of the pump and Stokes lasers, e = ep = eS for (2 + 2)STIRAP in the three-level model system with l22 = 0 a.u. Results are shown for the following calculations (see text for details): exact d 5 0 (solid line), RWA d 5 0 and RWA(no 0) (dotted line, results are indistinguishable), RWA(t) d 5 0 (dashed line), and RWA(t, no 0) d 5 0 (dot-dash line).
A. Brown / Chemical Physics 342 (2007) 16–24
1 〈
〈
⏐1
⏐3
0.8
Pj(t)
0.6 0.4 0.2 0 2.5
〈
⏐2
5
7.5
10
Time (ps) Fig. 6. Time-dependent population of the states, Pj(t), for (2 + 2)STIRAP in the three-level model system with l22 = 0 a.u. The field strength of the pump and Stokes lasers, e = ep = eS = 0.01 a.u. Results are shown for the following calculations (see text for details): exact d 5 0 (solid line) and RWA(t) d 5 0 (dashed line).
23
STIRAP processes to occur. For example, in Section 3.3, we have considered (2 + 2)-STIRAP where both the pump and Stokes processes involve the absorption of two photons. For the three-level model considered, two-photon transition would be formally forbidden if d = 0. From the examples presented, it clear that the effects of permanent dipole moments must be carefully considered when examining STIRAP in molecular systems. Also, their effects may manifest themselves in more complex scenarios such as the phase-dependent populations that have been explored for STIRAP in extended K-systems [17]. The newly derived RWAs should help in the interpretation of STIRAP processes where permanent dipoles play an important role. For example, permanent dipoles may play a role in STIRAP population transfer in asymmetric double well systems, e.g., control of isomerization reactions [15,17,37], or in selective population transfer in molecules [38]. Acknowledgements
4. Conclusions The effects of permanent dipole moments on STIRAP have been carefully considered using simple three-level model systems. In order to understand the role of permanent dipole moments, RWAs including the effects of djk 5 0 have been developed for the STIRAP process. The RWAs have been tested by comparing with exact results for both the target state j3i population as a function of field strengths of the pump and Stokes lasers and for the time-dependent population of the states for fixed laser field strengths. These tests have demonstrated that the RWAs can qualitatively (simplest RWA, see Eqs. (16) and (17)) and also quantitatively [RWA(t), see Eqs. (13) and (14)] predict the exact results. Three potential effects of the permanent dipole moments on the STIRAP process can be predicted from the RWA expressions that have been derived. First, the laser–molecule coupling for the N-photon pump or Stokes transition depends upon the Bessel function, J N ðzajk Þ where a = ‘‘pump’’ or ‘‘Stokes’’, see for example Eqs. (13) and (14). Since the Bessel functions are oscillatory, the STIRAP condition, see Eq. (20), may not be fulfilled as the field strengths are increased. This has been demonstrated for the regular STIRAP process in both Figs. 2 and 3. For the molecular and laser parameters considered in the model, this has lead to a decrease in the field strength region for which STIRAP occurs. Secondly, the N-photon ð1;0Þ Stokes and pump Rabi frequencies, C S and C pð0;1Þ , respectively, depend on zero-photon absorption in the corresponding pump and Stokes transitions through the term J 0 ðzakj Þ, see for example Eqs. (13) and (14). For the examples considered, the zero-photon absorption was shown to play a very minor role. However, in general, the zerophoton effect should be considered. Thirdly, the presence of permanent dipole moments can allow new multi-photon
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