Fast-Forward Assisted STIRAP - American Chemical Society

Article pubs.acs.org/JPCA

Fast-Forward Assisted STIRAP Shumpei Masuda*,†,‡ and Stuart A. Rice*,† †

James Franck Institute, The University of Chicago, Chicago, Illinois 60637, United States Department of Physics, Tohoku University, Sendai 980, Japan

‡

ABSTRACT: We consider combined stimulated Raman adiabatic passage (STIRAP) and fast-forward field (FFF) control of selective vibrational population transfer in a polyatomic molecule. The motivation for using this combination control scheme is 2-fold: (i) to overcome transfer inefficiency that occurs when the STIRAP fields and pulse durations must be restricted to avoid excitation of population transfers that compete with the targeted transfer and (ii) to overcome transfer inefficiency resulting from embedding of the actively driven subset of states in a large manifold of states. We show that, in a subset of states that is coupled to background states, a combination of STIRAP and FFFs that do not individually generate processes that are competitive with the desired population transfer can generate greater population transfer efficiency than can ordinary STIRAP with similar field strength and/ or pulse duration.

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INTRODUCTION It is now well established that it is possible to actively control the quantum dynamics of a system by manipulating the frequency, phase, and temporal character of an applied optical field.1,2 The underlying mechanisms of all the proposed and experimentally demonstrated active control methods rely on coherence and interference effects embedded in the quantum dynamics. Although the various control protocols provide prescriptions for the calculation of the control field, in general, the manifold of states of the driven system is too complicated to permit exact calculation of that field. That difficulty has led to the consideration of control of the quantum dynamics with a simplified Hamiltonian, e.g., within a subset of states of the full manifold of states without regard for the influence of the remaining background states. One example of this class of control methods is the use of stimulated Raman adiabatic passage (STIRAP)3−7 to transfer population within a three state subset of a larger manifold of states. Various extended STIRAP methods, involving more than three states, also have been proposed,8−13 but we do not concern ourselves with the latter in this article. However, the simplification achieved by restricting the dynamics to a submanifold of states is not always acceptable. In particular, when the transition dipole moments between a selected subset of states and the other states of the manifold are not negligible, it is necessary to account for the influence of transitions involving the background states on the efficiency of the population transfer. Furthermore, STIRAP relies on adiabatic driving in which the populations in the instantaneous eigenstates of the Hamiltonian are constant. Because an adiabatic process must be carried out very slowly, at a rate much smaller than the frequencies of transitions between states of the Hamiltonian, the field strength and/or pulse duration imposed must be restricted to avert unwanted © 2015 American Chemical Society

processes. Recognition of this restriction has led to the development of control protocols, which we call assisted adiabatic transformations; these transformations typically use an auxiliary field to produce, with overall weaker driving fields and/or in a shorter time and without excitation of competing processes, the desired target state population. In an early study of a version of assisted adiabatic population transfer, Kurkal and Rice11 used the extended STIRAP process devised by Kobrak and Rice10 to study vibrational energy transfer between an initial state and two nearly degenerate states in nonrotating SCCl2. The Kobrak−Rice extended STIRAP process, which is designed to control the ratio of the populations transferred to the target states, uses three pulsed fields: a pump field, a Stokes field, and a field that couples the target state to a so-called branch state. The ratio of populations of the target states that can be achieved depends on, and is limited by, the ratio of the dipole transition moments between the branch state and the target states and is discretely controllable by suitable choice of the branch state from the full manifold of states. Because this extended STIRAP process exploits adiabatic population transfer, the field strengths and pulse durations used must satisfy the same constraints as for a simple adiabatic population transfer. Other assisted adiabatic transformation control methods that have been proposed include the counter-diabatic protocol,14−17 the invariant-based inverse engineering protocol,18,19 and the fast-forward protocol.20−24 Complete population transfer induced by modified pump and Stoke pulses has also been studied by using the invariant-based engineering method for Received: January 18, 2015 Revised: March 13, 2015 Published: March 16, 2015 3479

DOI: 10.1021/acs.jpca.5b00525 J. Phys. Chem. A 2015, 119, 3479−3487

Article

The Journal of Physical Chemistry A three level systems,25 as has optimal control of laser-induced population transfer (see e.g., refs 26 and 27). In related work relevant to modifying a STIRAP population transfer, Unanyan et al.28 showed that an additional radiative coupling between the initial state and the final state reduces nonadiabatic losses and that the pulse area of the additional field need not be precisely π to improve the population transfer. We have shown elsewhere29 that, in a subset of states that is coupled to background states, a combination of STIRAP fields and a counter-diabatic field (CDF) can generate greater population transfer efficiency than can ordinary STIRAP with necessarily restricted field strength and/or pulse duration, and that the exact CDF for an isolated three level system is a useful approximation to the CDF for three and five state submanifolds embedded in a large manifold of states. In this article we propose a combined phase-controlled STIRAP and fast-forward field (FFF) to control selective vibrational population transfer in a polyatomic molecule under conditions that require restriction of the STIRAP field strength and/or pulse duration. The phase-controlled STIRAP + FFF control is more general than the STIRAP + CDF control in the sense that the former includes the latter as a particular case. The pulse area of the FFF can be chosen to be larger than π, while that of the CDF must be π. We find, using the model system described below, that STIRAP + FFF control of population transfer depends on the area of the FFF when the driven subset of states is coupled to background states. Specifically, since the purpose of our investigation is to ascertain the effectiveness of STIRAP + FFF control of population transfer between a subset of states embedded in a larger manifold of states, we have used as a vehicle the same model system studied in ref 11. This model spectrum represents the vibrational energy levels of a nonrotating SCCl2 molecule and we show that the STIRAP + FFF that affects complete transfer of population in an isolated three-state system is a useful approximation to the control field that affects efficient transfer of population for a three-state system embedded in background states. Although the FFF is designed for the isolated three-state system it suppresses the influence of the background states strongly coupled to the STIRAP pumped subset of states.

systems with time-dependent off-diagonal elements and with general unitary transformations.31 Fast-Forward Protocol for Discrete Systems. We consider a manifold of discrete states {|i⟩} and time-dependent transition (hopping) amplitudes ωl,m between states |l⟩, |m⟩ ∈ {| i⟩}. Note that these transition amplitudes depend on the applied field and are the analogues of the Rabi frequencies of the pump and Stokes pulses in a STIRAP process; they are not the conventional transition probabilities. The derivation of the fast-forward driving fields proceeds in the same manner as that described in ref 24. The equation of motion of the system wave function takes the form i

dΨ(m , t ) = dt

∑ ωm,l(R(t ))Ψ(l , t ) + l

V0(m , R(t )) Ψ(m , t ) ℏ

(1)

where Ψ(m,t) is the coefficient of |m⟩ and R is a timedependent parameter characterizing the temporal dependence of ωm,l. Equation 1 describes the population transfer among molecular states with ωm,l corresponding to the Rabi frequency of the laser field coupling the states |l⟩ and |m⟩ and V0(m) the energy of the field-free state |m⟩. Hereafter we refer to V0(m) as a potential. We specify the form of the wave function of the intermediate state by using an energy eigenstate of the Hamiltonian corresponding to eq 1 in order to derive the form of the FFF. Now let ϕn(m,R) and En(R) be the wave function (coefficient of |m⟩) and energy of the nth eigenstate of the instantaneous Hamiltonian; they satisfy the timeindependent discrete Schrödinger equation

∑ ℏωm,l(R)ϕn(l , R) + V0(m , R)ϕn(m , R) l

= En(R )ϕn(m , R )

(2)

We seek the transition amplitudes and potential that generates T ϕn(m,Rf) exp[−(i/ℏ)∫ 0 FEn(R(t′))dt′] from ϕn(m,Ri), where R(TF) = Rf. Although such dynamics is realized as a solution of eq 1 if dR(t)/dt is sufficiently small, corresponding to an adiabatic process, if dR(t)/dt is not very small unwanted excitations occur. We consider a time-dependent intermediate state wave function ΨFF that evolves from ϕn(m,Ri) to ϕn(m,Rf) exp[−(i/ℏ)∫ 0TFEn(R(t′))dt′] in time TF. The Schrödinger equation for ΨFF is

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FAST-FORWARD ASSISTED STIRAP The fast-forward protocol is constructed to control the rate of evolution of particles between selected initial and target states in a continuous system. It can be regarded as defining a trajectory in the state space connecting the initial and final states for which the control field that accelerates the initial-tofinal state transition is realizable. The time-dependent intermediate states acquire, relative to the states along the original trajectory of the initial-to-final state transition without the FFF acceleration20 or the adiabatic transition,21 an additional time-dependent phase. A variant of the fast-forward protocol that uses a more general unitary transformation than that leading to the additional phase has been applied to population transfer in a two-state system.30 Additionally, the fast-forward protocol has been extended to treat spatially discrete systems, e.g., accelerated manipulation of a Bose− Einstein condensate (BEC) in an optical lattice by Masuda and Rice.24 The analysis reported in ref 24 assumed that the offdiagonal elements of Hamiltonian are constants. Takahashi extended the fast-forward protocol for more general discrete

i

dΨFF(m , t ) = dt

∑ ωmFF,l(t )ΨFF(l , t ) + l

VFF(m , t ) ΨFF(m , t ) ℏ (3)

FF and the transition amplitudes ωm,l between |l⟩ and |m⟩ are timeFF dependent and/or tunable. ωm,l will be defined later. The wave function ΨFF(m,t) is assumed to be represented, with the additional phase f(m,t), in the form

⎡ i ΨFF(m , t ) = ϕn(m , R(t )) exp[if (m , t )] exp⎢ − ⎣ ℏ

∫0

t

⎤ En(R(t ′))dt ′⎥ ⎦ (4)

We require that f(m,0) = f(m,TF) = 0. Assuming ΨFF(m,t) ≠ 0 (ϕn(m,R(t)) ≠ 0) we divide eq 3 by ΨFF(m,t), substitute into eq 4, and then decompose the equation into real and imaginary parts. The imaginary part of the equation leads to 3480


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The Journal of Physical Chemistry A ∂ϕ (m , R ) ⎤ dR ⎡ * ⎥ Re⎢ϕn (m , R ) n dt ⎣ ∂R ⎦ =

Ω p(t ) = μ12 Ep(e)(t )/ℏ ΩS(t ) = μ23 ES(e)(t )/ℏ

∑ Im[ϕn*(m , R )ϕn(l , R )(ωmFF,l(t )

where is the envelope of the amplitude of the pump (Stokes) field and μij the transition dipole moment between states |i⟩ and |j⟩. Note that, by assumption, μ13 = 0. The timedependent field-dressed eigenstates of this system are linear combinations of the field-free states with coefficients that depend on the Stokes and pump field magnitudes and the transition dipole moments. The field-dressed state of interest to us is

l

× exp[i(f (l , t ) − f (m , t ))] − ωm , l(R(t )))]

(5)

and the real part leads to the driving potential VFF(m , t ) = V0(m , R(t )) ⎡ ϕ (l , R(t )) + ∑ Re⎢ℏ n (ω (R(t )) ⎢⎣ ϕn(m , R(t )) m , l l

|ϕ2(t )⟩ = cos Θ(t )|1⟩ − sin Θ(t )|3⟩

⎤ df (m , t ) − ωmFF, l (t ) exp[i(f (l , t ) − f (m , t ))])⎥ − ℏ ⎥⎦ dt −ℏ

∂ϕn(m , R(t )) ⎤ dR ⎡ 1 ⎥ Im⎢ ⎥⎦ dt ⎢⎣ ϕn(m , R(t)) ∂R

tan Θ(t ) = (6)

(7)

and eq 2 becomes

∑ ωm,l(R(t ))ϕn(l , R(t )) = 0 l

Ω̇ pΩS − Ω pΩ̇S

(8)

ΩS 2 + Ω p 2

If ϕn(m,R(t)) = 0 for any t, the driving potential is arbitrary because it has no influence in the Schrödinger equation. Application to a STIRAP Process. In its simplest form STIRAP is used to transfer population between states |1⟩ and |3⟩ in a three state manifold in which transitions |1⟩ → |2⟩ and |2⟩ → |3⟩ are allowed but |1⟩ → |3⟩ is forbidden. The driving optical field consists of two suitably timed and overlapping laser pulses with the (Stokes) pulse driving the |2⟩ → |3⟩ transition preceding the (pump) pulse driving the |1⟩ → |2⟩ transition. The field dressed states of this system are combinations of the bare states |1⟩ and |3⟩ with coefficients that depend on the Rabi frequencies of the pump (Ωp) and Stokes (ΩS) fields. Consequently, as those fields vary in time there is an adiabatic transfer of population from |1⟩ to |3⟩. In the three-state system the efficiency of STIRAP is relatively insensitive to the details of the pulse profile and the pulse separation when the adiabatic condition32

Ω̇ pΩS − Ω pΩ̇S ΩS 2 + Ω p 2

ΩS(t )

(12)

≪ |ω0 − ω+(−)| cannot be met. Then the STIRAP

process generates incomplete population transfer, and we propose to assist the population transfer with a fast-forward driving field. So far we assumed that μ1,3 = 0. However, in general the initial and target vibrational states are connected by a nonzero transition dipole moment, although it might be much smaller than the transition dipole moments μ1,2 and μ2,3. Here we FF derive the phase-controlled STIRAP and FFFs, ωFF 1,2, ω2,3, and ωFF , that avoid unwanted excitation for the case with nonzero 1,3 μ1,3. We note that our purpose goes beyond generating complete population transfer in three-state systems. In the next section it is shown that the FFF accompanied by phasecontrolled STIRAP fields decreases the influence of the background states when |1⟩, |2⟩, and |3⟩ are embedded in background states. The analysis of the preceding subsection can be applied to a three-state STIRAP process with V0 = 0 and the identifications ω1,3 = 0, ω1,2(R(t)) = −Ωp(R(t))/2, and ω2,3(R(t)) = −ΩS(R(t))/2. We choose R(t) = t, in which case ω1,2 and ω2,3 correspond to the Rabi frequencies of the pump and Stokes pulses. The Hamiltonian corresponding to the time-independent Schrödinger eq 2 is represented as eq 9. We now consider a field-dressed state

≪ |ω0 − ω+(−)| is satisfied, where ω0 is

the energy of an instantaneous eigenstate to be followed in the adiabatic dynamics and ω+(−) are the energies of the other instantaneous eigenstates. Using the rotating-wave representation and the rotating wave approximation (RWA), the Hamiltonian of the three-state system with resonant pump |1⟩ → |2⟩ and Stokes |2⟩ → |3⟩ fields can be represented in the form ⎛ 0 Ω p(t ) 0 ⎞ ⎟ ⎜ ℏ 0 ΩS(t )⎟ HRWA(t ) = − ⎜ Ω p(t ) ⎟ 2⎜ ⎟ ⎜ ΩS(t ) 0 ⎠ ⎝ 0

Ω p(t )

Because the Stokes pulse is applied before but overlaps the pump pulse, initially Ωp ≪ ΩS and all of the population is initially in field-free state |1⟩. At the final time, Ωp ≫ ΩS, so all of the population in |ϕ2(t)⟩ projects onto the target state |3⟩. Note that |ϕ2(t)⟩ has no projection on the intermediate fieldfree state |2⟩. Suppose now that either the pulsed field duration or the field strength must be restricted to avoid exciting unwanted processes that compete with the desired population transfer, with the consequence that the condition

l

l

(11)

where

When ϕn(m,R) = 0 for any R, the Schrödinger eq 3 takes the form

∑ ωmFF,l(t )eif ϕn(l , R(t )) = 0

(10)

E(e) p(S)

|ϕ2(R )⟩ =

∑ ϕ2(m , R)|m⟩

(13)

m

with ϕ2(1,R) = cos Θ(R), ϕ2(2,R) = 0, ϕ2(3,R) = −sin Θ(R), and ϕ2(3,R)/ϕ2(1,R) = −Ωp(R)/ΩS(R). As mentioned earlier, m = 2 is treated separately because ϕ2(2,R) = 0. For m = 2 eqs 7 and 8 take the form FF FF ω2,1 (t ) e if1ϕ2(1, R ) + ω2,3 (t ) e if3ϕ2(3, R ) = 0

(9)

with Ωp and ΩS the Rabi frequencies defined by

(14)

and 3481


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The Journal of Physical Chemistry A ω2,1(R )ϕ2(1, R ) + ω2,3(R )ϕ2(3, R ) = 0

(15)

respectively. Combining eqs 14 and 15 we obtain FF ω2,1 (t ) FF ω2,3 (t )

= eiΔf

ω2,1(R(t )) ω2,3(R(t ))

(16)

with

Δf (t ) ≡ f3 (t ) − f1 (t )

(17)

Figure 1. Schematic diagrams of three different trajectories of A: (a) Re[A(t)] = 0; (b) Re[A(t)] = A1 exp[−t2/ τ2] (solid curve) and Re[A(t)] = A2(t/τ) exp[−t2/τ2] (dotted curve) with A1(2) and τ constants.

Noting that ϕ2(2,R) = 0 and ϕ2(1,R), ϕ2(3,R) ∈ R, eq 5 can be rewritten as dR(t ) ∂ϕ2(1, R ) FF (t ) e iΔf ] = ϕ2(3, R )Im[ω1,3 dt ∂R dR(t ) ∂ϕ2(3, R ) FF (t ) e−iΔf ] = ϕ2(1, R )Im[ω3,1 dt ∂R

FF ω2,1 (t ) FF ω2,3 (t )

(18)

ω2,3 ω2,3 ω1,2

Re[A(t )] −

Re[A(t )] −

df1 dt df3 dt

FF ω1,3 =i

(20)

(21)

Equation 18 determines the imaginary part of A to be

Then

ωFF 1,3

(22)

is represented as

⎡ sin 2Θ dΔf ⎛ dΘ ⎤ dΘ ⎞⎟ FF +i ω1,3 = e−iΔf ⎜Re[A] + i = e−iΔf ⎢ ⎥ ⎝ ⎠ ⎣ 2 cos 2Θ dt dt ⎦ dt (23)

The half Rabi frequency ωFF 1,3(t) is complex and must be realized by controlling the time dependences of the phases of the laser FF fields as well as the relative phase between ωFF 1,2 and ω2,3. There is an arbitrariness in the choice of Re[A(t)] or Δf(t). Three different trajectories of A are depicted schematically in Figure 1, for Re[A(t)] = 0, Re[A(t)] = A1 exp[−t2/τ2], and Re[A(t)] = A2(t/τ) exp[−t2/τ2], with A1(2) and τ constant. It can be shown that the fast-forward assisted STIRAP protocol gives the same Rabi frequencies as does the counterdiabatic field assisted STIRAP protocol with

fm = 0

(26)

VIBRATIONAL ENERGY TRANSFER IN THIOPHOSGENE In this section we examine the efficiency with which population transfer can be selectively directed by STIRAP + FFF control to one of a pair of nearly degenerate states in the presence of background states. As stated earlier, the vehicle for our studies is a model of the vibrational spectrum of the nonrotating SCCl2 molecule. The SCCl2 molecule has three stretching (ν1, ν2, ν3) and three bending (ν4, ν5, ν6) vibrational degrees of freedom; it suffices, for our purposes, to use the same set of energies and transition dipole moments as used by Kurkal and Rice,11 covering the range 0−21 000 cm−1, determined by Bigwood, Milam, and Gruebele.33 These energy levels are displayed in Figure 2 and tabulated in ref 11. This model manifold of states is sufficiently complex to permit qualitative investigation of the influence of background states on the efficiency of energy transfer within an embedded subset of states. We consider a STIRAP process within the subset of three states (|200 000⟩, |300 000⟩, |200 020⟩) embedded in the full manifold of states. Hereafter we refer to these three states as |1⟩, |5a⟩, and |6⟩, respectively. The STIRAP + FFF control process is intended to generate higher population transfer from |200 000⟩ to |200 020⟩ than does the pure STIRAP process. We note that |210 011⟩, hereafter called |9⟩, with energy 5658.1828 cm−1, is nearly degenerate with |6⟩, with energy 5651.5617 cm−1, and that the transition moment coupling states |1⟩ and |6⟩ are one order of magnitude smaller than those coupling states |1⟩ and |9⟩ and |5a⟩ and |9⟩. States |1⟩, |5a⟩ and |6⟩ are taken as the initial, intermediate, and final states of both a STIRAP + FFF and a STIRAP + CDF process. Other states, shown in Figure 2, are regarded as

d(Δf)/dt is determined when we choose Re[A(t)] to be

dΘ dt

dΘ dt

■

(19)

with

Im[A(t )] =

(25)

eq 26 is the same as the half Rabi frequency of the CDF. The trajectory of A with Re[A(t)] = 0 depicted in Figure 1a corresponds to the CDF. Equation 25 determines the ratio of FF ωFF 2,1(t) and ω2,3(t), but their intensities are arbitrary and can even be zero, consistent with the observation that the CDF alone can generate complete population transfer in a two-level system. When Re[A(t)] ≠ 0, the pulse area of the FFF pulse is larger than π, in contrast to the pulse area of the CDF, which is π.14,17 The restriction of the pulse area that is characteristic of the CDF protocol is eased in the fast-forward protocol.

=0

⎞ dΔf (t ) ⎛ 1 =⎜ − tan Θ(t )⎟Re[A(t )] dt ⎝ tan Θ(t ) ⎠

ω2,3(R(t ))

ωFF 1,3in

=0

FF A(t ) = ω1,3 (t ) e iΔf (t )

ω2,1(R(t ))

and

It can be shown that the two eqs of 18 are identical by using the FF relations (ωFF 3,1)* = ω1,3 and ∂Rϕ2(1,R)/ϕ2(3,R) = −∂Rϕ2(3,R)/ ϕ2(1,R), which are directly derived from ∂R(|ϕ2(1,R)|2 + |ϕ2(3,R)|)2 = 0. Equations 16 and 18 determine the Rabi frequencies. We consider a fast-forwarded STIRAP process with finite f m and vanishing diagonal elements of the driving Hamiltonian, VFF = 0. Equation 6 and VFF = 0 lead to ω1,2

=

(24)

Equations 16 and 23 lead to 3482


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The Journal of Physical Chemistry A

Figure 2. Schematic diagram of the vibrational spectrum of SCCl2.

background states. The pump field is resonant with the transition from the |1⟩ to |5a⟩; the Stokes field is resonant with the transition from |5a⟩ to |6⟩. We take the strengths of the pump and the Stokes fields to be ⎡ (t − T )2 ⎤ p(S) (e) ⎥ ̃ exp⎢ − Ep(S) (t ) = Ep(S) ⎢⎣ (Δτ )2 ⎥⎦

Figure 3. (a) Time-dependence of Δf and (b) the trajectory of A for fwhm = 21.5 ps, A0 ≃ 0.087 ps−1, and Tp − TS = fwhm/(2(ln 2)1/2). The solid lines correspond to Ẽ p = 3.11 × 10−5 a.u., Ẽ S = 3.44 × 10−5 a.u., and the broken lines to Ẽ p = 3.11 × 10−5 a.u., Ẽ S = 2.09 × 10−5a.u.

(27)

where Δτ = fwhm/(2(ln 2)1/2), and fwhm is the full width at half-maximum of the Gaussian pulse with maximum intensity Ẽ p(S) that is centered at Tp(S). We solved the time-dependent Schrödinger equation numerically with a fourth order Runge− Kutta integrator in a basis of bare matter eigenstates with Tp − TS = fwhm/(2(ln 2)1/2) without use of the rotating wave approximation. We choose the time-dependence of Re[A(t)] to be ⎡ t2 ⎤ Re[A(t )] = A 0 exp⎢ − 2 ⎥ ⎣ Δτ ⎦

each state in the three-state system decoupled from the background states is shown for the STIRAP + FFF control in Figure 4b and for the ordinary STIRAP process in Figure 4c. The data displayed clearly show that 100% population transfer is generated in the STIRAP + FFF control. As seen from eq 23 and Figure 4a, the amplitude of ωFF 1,3 is larger than that of the CDF. The restriction of the pulse area that is characteristic of the CDF protocol is eased in the fast-forward process. We now examine the efficiency of the STIRAP + FFF control when the subset of states |1⟩, |5a⟩, and |6⟩ is embedded in the manifold of states depicted in Figure 2. We consider a FFF corresponding to ωFF 1,3 accompanied by a pump pulse and a phase-controlled Stokes field with all the background states in Figure 2; the time-evolution of the system is calculated exactly, without use of the rotating wave approximation. In Figure 5 the time-dependences of the populations of the initial, intermediate, and the target states for (a) STIRAP and (b) STIRAP + FFF control are shown for the same parameter set as that of Figure 4. The efficiencies of both STIRAP and STIRAP + FFF controls are degraded due to interference with background states strongly coupled to the subset of states |1⟩, |5a⟩, and |6⟩. However, the influence of the background states is suppressed in the STIRAP + FFF control compared to that in the STIRAP control because of the direct coupling of the initial and target states by the FFF. To compare the efficiency of the STIRAP + FFF control to that of the STIRAP + CDF control we examine the dependence of the STIRAP + FFF generated population transfer on the peak ratio of the FFF to the CDF. The peak ratio is tuned by using A0. The range of the phase tuned in the STIRAP + FFF control increases when the peak field ratio becomes large, while the population transfer when the peak field ratio is one is

(28)

with the constant A0. The time-dependence of Δf is shown in Figure 3a for the parameter set: fwhm = 21.5 ps, Ẽ p = 3.11 × 10−5 a.u., Ẽ S = 3.44 × 10−5 a.u., and A0 ≃ 0.087 ps−1. The pulse area of the pump and Stokes pulses are chosen to be the same as the ones used in ref 29. As mentioned above, the pulse area of the FFF pulse is larger than that of the CDF. The amplitude of A0 was chosen so that the peak amplitude of the FFF is 1.5 times larger than that of the CDF. The phase of the Stokes field is changed by Δf (see eq 16) and the trajectory of A(t) = iΔf(t) ωFF is shown in Figure 3b. The time-dependence of Δf 1,3(t) e depends on Ẽ p and Ẽ S. When Ẽ p and Ẽ S are chosen to be Ẽ p = 3.11 × 10−5 a.u., Ẽ S = 2.09 × 10−5a.u. with which the peak amplitude of the Rabi frequencies of the pump and Stokes pulse are the same, the Δf becomes zero at the end of the application of the pulsed field (see the broken lines in Figure 3a). The STIRAP + FFF control generates complete population transfer if the subset of states |1⟩, |5a⟩, and |6⟩ is decoupled from the other background states. We first examine the effect of the FFF with the subset of states |1⟩, |5a⟩, and |6⟩ decoupled from the other background states for reference. The amplitudes of the half Rabi frequencies coupling the three states are shown in Figure 4a, and the time-dependence of the population of 3483


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Figure 5. (a) Time-dependences of the populations for the STIRAP + FFF control for A0 ≃ 0.087 ps−1, fwhm = 21.5 ps, Ẽ p = 3.11 × 10−5 a.u., Ẽ S = 3.44 × 10−5 a.u., and Tp − TS = fwhm/(2(ln 2)1/2). (b) Time-dependences of the populations for the STIRAP control.

Figure 4. (a) Time-dependences of the amplitudes of the half Rabi frequencies for A0 ≃ 0.087 ps−1, fwhm = 21.5 ps, Ẽ p = 3.11 × 10−5 a.u., Ẽ S = 3.44 × 10−5 a.u., and Tp − TS = fwhm/(2(ln 2)1/2). (b) Timedependences of the populations for the STIRAP + FFF control. (c) Time-dependences of the populations for the STIRAP process.

Figure 6. Dependence of the efficiency on the peak field ratio of the FFF to the CDF for the STIRAP + FFF control with fwhm = 21.5 ps, Tp − TS = fwhm/(2(ln 2)1/2), Ẽ p = 3.11 × 10−5 a.u., and Ẽ S = 3.44 × 10−5 a.u.

identical to that generated by STIRAP + CDF without phase tuning. Figure 6 displays the dependence of the STIRAP + FFF generated population transfer on the peak ratio of the FFF to the CDF for the parameters fwhm = 21.5 ps, Tp − TS = fwhm/ (2(ln 2)1/2), Ẽ p = 3.11 × 10−5 a.u., and Ẽ S = 3.44 × 10−5 a.u. For a wide range of peak field ratio the population transfer generated by STIRAP + FFF exceeds that generated by STIRAP + CDF control. Figure 7 displays the dependence of population transfer generated by STIRAP, STIRAP + CDF, and STIRAP + FFF (with peak field ratios 1.2 and 1.5) on the fwhm of the pulses.

The values of Ẽ p,S for each value of the fwhm have been adjusted so that the pulse areas of the pump and Stokes fields are the same as those used for the calculations shown in Figure 6. The drop of the efficiency of the STIRAP process when fwhm decreases is due to the large intensity of pump and Stokes fields. The increase of the field strengths in a simple STIRAP process does not generate greater population transfer efficiency because those stronger fields also generate greater interference between the active subset of states and the background states. The STIRAP + FFF generated population 3484


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Figure 7. Dependence of the efficiency on the fwhm for the ordinary STIRAP, the STIRAP + CDF control, and the STIRAP + FFF control with Tp − TS = fwhm/(2(ln 2)1/2) and the peak field ratios = 1.2 and 1.5.

Figure 9. Comparison of the efficiency of STIRAP + FFF control and FFF control for A0 ≃ 0.087 ps−1, fwhm = 21.5 ps, Ẽ p = 3.11 × 10−5 a.u., Ẽ S = 3.44 × 10−5 a.u., and Tp − TS = fwhm/(2(ln 2)1/2) for various λ.

transfer exceeds those generated by ordinary STIRAP and STIRAP + CDF control for 21.5 ps ⩽ fwhm ⩽86 ps. Demirplak and Rice14 have noted that the STIRAP + CDF process is sensitive to the phase difference between the STIRAP and CDFs. We examine the extent of the phase sensitivity of the STIRAP + FFF control using the phase variation φ of the FFF from the phase used in Figure 4 and determine if there is a domain of the phase variation φ for which the combined STIRAP + FFF control remains more efficient than that of the STIRAP process. The φ-dependence of the efficiency is shown in Figure 8 for peak field ratio = 1.5, fwhm = 21.5 ps, and Tp − TS = fwhm/(2(ln 2)1/2). The combined STIRAP + FFF control is more efficient than the STIRAP process in a wide range of φ.

control is decreased compared to that of FFF control. The STIRAP + FFF control generates higher efficiency than STIRAP or FFF individually for a wide range of λ. As seen from eq 16, the FFF alone can generate complete population transfer to the target state in a two-level system. However, the efficiency of the single pulse control is degraded when there is interaction with the background states and is not stable to variation of the area of the pulse.

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CONCLUDING REMARKS We have examined the efficiency of STIRAP + FFF generated selective state-to-state population transfer in the vibrational manifold of nonrotating SCCl2. Neglecting the influence of molecular rotation on the efficiency of vibrational population transfer defines useful models that permit qualitative investigation of the influence of background states on the efficiency of energy transfer within an embedded subset of states, but those models are inadequate for the quantitative description of energy transfer in the corresponding real molecules. It is relevant to ask if our calculations provide a qualitatively valid picture applicable to real situations. We have argued elsewhere29 that, neglecting higher order effects such as vibration rotation interaction, we expect the rotation of a molecule to affect the state-to-state process we describe in two ways. First, the transition dipole moment projection along the field axis differs with rotational state, thereby reducing the rate of excitation. Second, the rotational wave packet created may dephase on a time scale that is comparable with the width of the exciting field, thereby changing the dynamics of the population transfer. If the ratio of the driving field duration to the period of molecular rotation is very small we expect molecular rotation to have negligible influence on the population transfer, and when the period of molecular rotation is comparable to the width of the field pulses that drive the population transfer we must expect less efficient transfer than predicted for the nonrotating molecule. Indeed, noting that the combined STIRAP + FFF control process we describe involves both one and two photon transitions and that the wave-packets of rotational states created by these two excitation processes have different dephasing rates, we expect the evolution of the state of the excited molecule to be complicated when the period of rotation and the exciting field duration are comparable.

Figure 8. Phase dependence of the efficiency of the STIRAP + FFF control for peak field ratio = 1.5, fwhm = 21.5 ps, and Tp − TS = fwhm/(2(ln 2)1/2).

So as to study the stability of the efficiency of the population transfer driven by a variable FFF, we represent the total driving field in the form E(t ) = Ep(t ) + ES(t ) + λE FF(t )

(29)

where λ = 0 corresponds to driving the system with only the STIRAP fields and λ = 1 to driving the system with the STIRAP and the FFF; Ep(S) is the pump (Stokes) field; and EFF is the FFF corresponding to ωFF 1,3 in eq 23. In Figure 9 the stability of the STIRAP + FFF control and the FFF alone control is monitored by the efficiency as a function of λ. Clearly, the sensitivity to the variation of amplitude of STIRAP + FFF 3485


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Postdoctoral Fellowships for Research Abroad for its financial support.

The rotational periods of SCCl2 is of the order of 200 ps. Our calculations of the efficiency of state-to-state population transfer in SCCl2 include cases when the fwhm of the pulsed fields is considerably smaller than 200 ps (see Figure 7). The efficiency of the population transfer is smaller when the fwhm of the pulses is 20 ps than when it is 100 ps, but still usefully large. Since these pulse widths are on the order of one tenth of the rotational period, it is plausible that similar efficiency of state-to-state population transfer can be achieved in the real molecule. Returning to the model cases considered, we have shown that STIRAP + FFF generated state-to-state population transfer is more efficient than STIRAP and STIRAP + CDF generated state-to-state population transfer when applied to a subset of states embedded in and coupled to a larger manifold of states. We have shown that the FFF calculated for an isolated subset of three states can be used to approximate the FFF applicable to a three state subset embedded in a large manifold of states. The FFF is designed to avert unwanted nonadiabatic population transfer at the end of the application of the pulsed field, and it directly couples the initial state to the target state thereby decreasing the sensitivity of the population transfer to the influence of background states. STIRAP + CDF generated population transfer exhibits this same decreased sensitivity for the same reason. However, the pulse area of the FFF is larger than π, in contrast to the pulse area of the CDF for a STIRAP + CDF process in the same system, which is always π. The STIRAP + FFF generated population transfer has, relative to STIRAP + CDF population transfer, an extra control parameter, namely, the FFF amplitude or Re[A(t)]. This parameter can be tuned to optimize the yield of population in a target state. In general, our model calculations show that, when the driven system of states is embedded in a large manifold of states, phase controlled STIRAP + FFF generates more efficient state-to-state population transfer than does STIRAP and STIRAP + CDF processes. In contrast with optimal control theory, in which the form of the external fields is defined with parameters and the parameters varied to find the optimum, we specify the form of the wave function of the intermediate state with the additional phase in eq 4 and then derive the form of the FFF, which is combined with STIRAP fields to generate complete population transfer in a three level system. Although in this study we used a particular form for Re[A(t)] as an example (see eq 28), different forms of Re[A(t)] may increase the efficiency of STIRAP + FFF generated population transfer. The application of optimal control theory to N level systems with several parameters to be optimized is difficult, but we expect that it can be used with the STIRAP + FFF protocol to find the optimal time dependence of the FFF amplitude or Re[A].

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS S.M. is thankful for the Grants-in-Aid for Centric Research of Japan Society for Promotion of Science and the JSPS 3486


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