Fast-Forward Assisted STIRAP
FFast-Forward Assisted STIRAP
Shumpei Masuda a) and Stuart A. Rice b) James Franck Institute, The University of Chicago, Chicago,IL 60637 (Dated: 26 October 2018)
We consider combined stimulated Raman adiabatic passage (STIRAP) and fast-forward field (FFF) control of selective vibrational population transfer in a polyatomicmolecule. The motivation for using this combination control scheme is twofold: (i)to overcome transfer inefficiency that occurs when the STIRAP fields and pulse du-rations must be restricted to avoid excitation of population transfers that competewith the targeted transfer and (ii) to overcome transfer inefficiency resulting fromembedding of the actively driven subset of states in a large manifold of states. Weshow that, in a subset of states that is coupled to background states, a combinationof STIRAP and FFFs that do not individually generate processes that are compet-itive with the desired population transfer can generate greater population transferefficiency than can ordinary STIRAP with similar field strength and/or pulse dura-tion. The vehicle for our considerations is enhancing the yield of HNC in the drivenground state-to-ground state nonrotating HCN → HNC isomerization reaction andselective population of one of a pair of near degenerate states in nonrotating SCCl . I. INTRODUCTION
It is now well established that it is pos-sible to actively control the quantum dy-namics of a system by manipulating the fre-quency, phase and temporal character of anapplied optical field . The underlying mech-anisms of all the proposed and experimen-tally demonstrated active control methods a) Department of Physics, Tohoku University, Sendai980, Japan; Electronic mail: [email protected] b) Electronic mail: [email protected] rely on coherence and interference effects em-bedded in the quantum dynamics. Althoughthe various control protocols provide pre-scriptions for the calculation of the controlfield, in general, the manifold of states ofthe driven system is too complicated to per-mit exact calculation of that field. That dif-ficulty has led to the consideration of con-trol of the quantum dynamics with a sim-plified Hamiltonian, e.g. within a subset ofstates without regard for the influence of theremaining background states. One exam-1 a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t le of this class of control methods is theuse of stimulated Raman adiabatic passage(STIRAP) to transfer population withina three state subset of a larger manifold ofstates. Various extended STIRAP meth-ods, involving more than three states, alsohave been proposed . This simplificationis not always acceptable: when the transitiondipole moments between a selected subset ofstates and the other (background) states ofthe manifold are not negligible it is neces-sary to account for the influence of transi-tions involving the background states on theefficiency of the population transfer. Further-more, STIRAP relies on adiabatic driving inwhich the populations in the instantaneouseigenstates of the Hamiltonian are constant.Because an adiabatic process must be car-ried out very slowly, at a rate much smallerthan the frequencies of transitions betweenstates of the Hamiltonian, the field strengthand/or pulse duration imposed must be re-stricted to avert unwanted processes. Recog-nition of this restriction has led to the devel-opment of control protocols which we call as-sisted adiabatic transformations; these trans-formations typically use an auxiliary field toproduce, with overall weaker driving fieldsand/or in a shorter time, and without ex-citation of competing processes, the desiredtarget state population.In an early study of a version of assisted adiabatic population transfer, Kurkal andRice used the extended STIRAP processdevised by Kobrak and Rice to study vibra-tional energy transfer between an initial stateand two nearly degenerate states in nonro-tating SCCl . The extended STIRAP pro-cess, which is designed to control the ratioof the populations transferred to the targetstates, uses three pulsed fields: a pump field,a Stokes field, and a field that couples thetarget state to a so-called branch state. Theratio of populations of the target states thatcan be achieved depends on, and is limitedby, the ratio of the dipole transition mo-ments between the branch state and the tar-get states, and is discretely controllable bysuitable choice of the branch state from themanifold of states. Because the extendedSTIRAP process exploits adiabatic popula-tion transfer, the field strengths and pulsedurations used must satisfy the same con-straints as for a simple adiabatic populationtransfer.Other assisted adiabatic transformationcontrol methods include the counter-diabaticprotocol , the invariant-based inverse en-gineering protocol and the fast-forwardprotocol .We have shown elsewhere that, in a sub-set of states that is coupled to backgroundstates, a combination of STIRAP fields anda counter-diabatic field (CDF) can generate2reater population transfer efficiency thancan ordinary STIRAP with necessarily re-stricted field strength and/or pulse duration.And it has been shown that the exact CDFfor an isolated three level system is a use-ful approximation to the CDF for three andfive state sub-manifolds embedded in a largemanifold of states.In this paper we complement our previ-ous study with an examination of the use ofcombined phase-controlled STIRAP and fast-forward fields (FFFs) to control selective vi-brational population transfer in a polyatomicmolecule under conditions that require re-striction of the STIRAP field strength and/orpulse duration. Again using selective vibra-tional energy transfer to drive the rotation-less HCN → HNC isomerization reaction andstate-to-state vibrational energy transfer inan isolated nonrotating SCCl molecule asvehicles for our study it is shown that thephase-controlled STIRAP + FFF that affectscomplete transfer of population in an isolatedthree-level system is a useful approximationto the control field that affects efficient trans-fer of population for a three-state system em-bedded in background states. The FFF sup-presses the influence of background statesstrongly coupled to the STIRAP pumpedsubset of states. II. FAST-FORWARD ASSISTEDSTIRAP
The fast-forward protocol is constructedto control the rate of evolution of particlesbetween selected initial and target states ina continuous system. It can be regardedas defining a trajectory in the state spaceconnecting the initial and final states forwhich the control field that accelerates theinitial-to-final state transition is realizable.The time-dependent intermediate states ac-quire, relative to the states along the originaltrajectory of the initial-to-final state transi-tion without the FFF acceleration or theadiabatic transition , an additional time-dependent phase. The fast-forward proto-col has been extended to treat spatially dis-crete systems, e.g. accelerated manipulationof a Bose-Einstein condensate (BEC) in anoptical lattice by Masuda and Rice , andspin systems , and a variant of this methodcan be used to accelerate selective populationtransfer between states in a discrete spectrumof states of a molecule. A. Fast-Forward Protocol forDiscrete Systems
We consider a manifold of discrete states {| i (cid:105)} and time dependent transition (hop-ping) rates ω l,m between states | l (cid:105) , | m (cid:105) ∈ | i (cid:105)} . Note that these transition rates dependon the applied field and are the analogues ofthe Stokes and Raman frequencies in a STI-RAP process; they are not the conventionaltransition probabilities. The derivation ofthe fast-forward driving fields proceeds in thesame manner as described in Ref. 23. Theequation of motion of the system wave func-tion takes the form i d Ψ( m, t ) dt = (cid:88) l ω m,l ( R ( t ))Ψ( l, t )+ V ( m, R ( t )) (cid:126) Ψ( m, t ) , (1)where Ψ( m, t ) is the coefficient of | m (cid:105) and R is a time-dependent parameter character-izing the temporal dependence of ω m,l . Eq.(1) describes the population transfer amongmolecular states with ω m,l corresponding tothe Rabi frequency of the laser field couplingthe states | l (cid:105) and | m (cid:105) and V ( m ) the energy ofthe field-free state | m (cid:105) . Hereafter we refer to V ( m ) as a potential. Now let φ n ( m, R ) and E n ( R ) be the wave function (coefficient of | m (cid:105) ) and energy of the n th eigenstate of theinstantaneous Hamiltonian; they satisfy thetime-independent discrete Schr¨odinger equa-tion (cid:88) l (cid:126) ω m,l ( R ) φ n ( l, R )+ V ( m, R ) φ n ( m, R ) = E n ( R ) φ n ( m, R ) . (2)We seek the transition ratesand potential that generates φ n ( m, R f ) exp[ − ( i/ (cid:126) ) (cid:82) T F E n ( R ( t (cid:48) )) dt (cid:48) ]from φ n ( m, R i ), where R ( T F ) = R f . Al-though such dynamics is realized as asolution of Eq. (1) if dR ( t ) /dt is suffi-ciently small, corresponding to an adiabaticprocess, if dR ( t ) /dt is not very small un-wanted excitations occur. We consider atime-dependent intermediate state wavefunction Ψ FF that evolves from φ n ( m, R i )to φ n ( m, R f ) exp[ − ( i/ (cid:126) ) (cid:82) T F E n ( R ( t (cid:48) )) dt (cid:48) ] intime T F . The Schr¨odinger equation for Ψ FF is i d Ψ FF ( m, t ) dt = (cid:88) l ω FF m,l ( t )Ψ FF ( l, t )+ V FF ( m, t ) (cid:126) Ψ FF ( m, t ) , (3)and the transition rates ω FF m,l between | l (cid:105) and | m (cid:105) are time-dependent and/or tunable. Thewave function Ψ FF ( m, t ) is assumed to be rep-resented, with the additional phase f ( m, t ),in the formΨ FF ( m, t ) = φ n ( m, R ( t )) exp[ if ( m, t )] × exp (cid:104) − i (cid:126) (cid:90) t E n ( R ( t (cid:48) )) dt (cid:48) (cid:105) . (4)We require that f ( m,
0) = f ( m, T F ) = 0. As-suming Ψ FF ( m, t ) (cid:54) = 0 ( φ n ( m, R ( t )) (cid:54) = 0) wedivide Eq. (3) by Ψ FF ( m, t ), substitute intoEq. (4), and then decompose the equationinto real and imaginary parts. The imagi-4ary part of the equation leads to dRdt Re (cid:104) φ ∗ n ( m, R ) ∂φ n ( m, R ) ∂R (cid:105) = (cid:88) l Im (cid:104) φ ∗ n ( m, R ) φ n ( l, R ) × (cid:16) ω FF m,l ( t ) exp (cid:2) i (cid:0) f ( l, t ) − f ( m, t ) (cid:1)(cid:3) − ω m,l (cid:0) R ( t ) (cid:1)(cid:17)(cid:105) (5)and the real part leads to the driving poten-tial V FF ( m, t ) = V ( m, R ( t ))+ (cid:88) l Re (cid:104) (cid:126) φ n ( l, R ( t )) φ n ( m, R ( t )) (cid:16) ω m,l ( R ( t )) − ω FF m,l ( t ) × exp (cid:2) i (cid:0) f ( l, t ) − f ( m, t ) (cid:1)(cid:3)(cid:17)(cid:105) − (cid:126) df ( m, t ) dt − (cid:126) dRdt Im (cid:104) φ n ( m, R ( t )) ∂φ n ( m, R ( t )) ∂R (cid:105) . (6)When φ n ( m, R ) = 0 for any R , theSchr¨odinger equation (3) takes the form (cid:88) l ω FF m,l ( t ) e if l φ n ( l, R ( t )) = 0 , (7)and Eq. (2) becomes (cid:88) l ω m,l ( R ( t )) φ n ( l, R ( t )) = 0 . (8)If φ n ( m, R ( t )) = 0 for any t the driving po-tential is arbitrary because it has no influencein the Schr¨odinger equation. B. Application to a STIRAP Process
In its simplest form STIRAP is used totransfer population between states | (cid:105) and | (cid:105) in a three state manifold in which transi-tions | (cid:105) → | (cid:105) and | (cid:105) → | (cid:105) are allowed but | (cid:105) → | (cid:105) is forbidden. The driving opticalfield consists of two suitably timed and over-lapping laser pulses with the (Stokes) pulsedriving the | (cid:105) → | (cid:105) transition preceding the(pump) pulse driving the | (cid:105) → | (cid:105) transi-tion. The field dressed states of this systemare combinations of the bare states | (cid:105) and | (cid:105) with coefficients that depend on the Rabifrequencies of the pump (Ω p ) and Stokes (Ω S )fields. Consequently, as those fields vary intime there is an adiabatic transfer of pop-ulation from | (cid:105) to | (cid:105) . In the three-statesystem the efficiency of STIRAP is relativelyinsensitive to the details of the pulse profileand the pulse separation when the adiabaticcondition ∆ T (Ω S + Ω p ) / >
10 can be met,where ∆ T is the pulse overlap. Using theinteraction representation and the rotatingwave approximation (RWA), the Hamiltonianof the three-state system with resonant pump | (cid:105) → | (cid:105) and Stokes | (cid:105) → | (cid:105) fields can berepresented in the form H RWA ( t ) = − (cid:126) p ( t ) 0Ω p ( t ) 0 Ω S ( t )0 Ω S ( t ) 0 , (9)with Ω p and Ω S the Rabi frequencies definedby Ω p ( t ) = µ E ( e ) p ( t ) / (2 (cid:126) ) , Ω S ( t ) = µ E ( e ) S ( t ) / (2 (cid:126) ) , (10)where E ( e ) p ( S ) is the envelope of the amplitudeof the pump (Stokes) field and µ ij the transi-5ion dipole moment between states | i (cid:105) and | j (cid:105) . Note that, by assumption, µ = 0.The time-dependent field-dressed eigenstatesof this system are linear combinations of thefield-free states with coefficients that dependon the Stokes and pump field magnitudes andthe transition dipole moments. The field-dressed state of interest to us is | φ ( t ) (cid:105) = cos Θ( t ) | (cid:105) − sin Θ( t ) | (cid:105) , (11)where tan Θ( t ) = Ω p ( t )Ω S ( t ) . (12)Because the Stokes pulse is applied before butoverlaps the pump pulse, initially Ω p (cid:28) Ω S and all of the population is initially in field-free state | (cid:105) . At the final time Ω p (cid:29) Ω S soall of the population in | φ ( t ) (cid:105) projects ontothe target state | (cid:105) . Note that | φ ( t ) (cid:105) hasno projection on the intermediate field-freestate | (cid:105) . Suppose now that either the pulsedfield duration or the field strength must berestricted to avoid exciting unwanted pro-cesses that compete with the desired popula-tion transfer, with the consequence that thecondition ∆ T (Ω S + Ω p ) / >
10 cannot bemet. Then the STIRAP process generatesincomplete population transfer and we pro-pose to assist the population transfer with afast-forward driving field.The analysis of the preceding subsectioncan be applied to a three-state STIRAP pro-cess with V = 0 and the identifications ω , = 0, ω , ( R ( t )) = Ω p ( R ( t )), ω , ( R ( t )) =Ω S ( R ( t )). We choose R ( t ) = t , in whichcase ω , and ω , correspond to the Rabifrequencies of the pump and Stokes pulses.The Hamiltonian corresponding to the time-independent Schr¨odinger equation (2) is rep-resented as Eq. (9). We now consider a field-dressed state | φ ( R ) (cid:105) = (cid:88) m φ ( m, R ) | m (cid:105) (13)with φ (1 , R ) = cos Θ( R ), φ (2 , R ) =0, φ (3 , R ) = − sin Θ( R ), and φ (3 , R ) /φ (1 , R ) = − Ω p ( R ) / Ω S ( R ). Asmentioned earlier, m = 2 is treated sepa-rately because φ (2 , R ) = 0. For m = 2 Eqs.(7) and (8) take the form ω FF2 , ( t ) e if φ (1 , R )+ ω FF2 , ( t ) e if φ (3 , R ) = 0 , (14)and ω , ( R ) φ (1 , R ) + ω , ( R ) φ (3 , R ) = 0 , (15)respectively. Combining Eqs. (14) and (15)we obtain ω FF2 , ( t ) ω FF2 , ( t ) = e i ∆ f ω , ( R ( t )) ω , ( R ( t )) , (16)with ∆ f ( t ) ≡ f ( t ) − f ( t ) . (17)Noting that φ (2 , R ) = 0 and φ (1 , R ) , φ (3 , R ) ∈ R , Eq. (5) can be6ewritten as dR ( t ) dt ∂φ (1 , R ) ∂R = φ (3 , R )Im (cid:104) ω FF1 , ( t ) e i ∆ f (cid:105) ,dR ( t ) dt ∂φ (3 , R ) ∂R = φ (1 , R )Im (cid:104) ω FF3 , ( t ) e − i ∆ f (cid:105) . (18)It can be shown that the two equations(18) are identical by using the relations( ω FF3 , ) ∗ = ω FF1 , and ∂ R φ (1 , R ) /φ (3 , R ) = − ∂ R φ (3 , R ) /φ (1 , R ), which are directly de-rived from ∂ R (cid:0) | φ (1 , R ) | + | φ (3 , R ) | (cid:1) = 0.Equations (16) and (18) determine the Rabifrequencies.We consider a fast-forwarded STIRAPprocess with finite f m and vanishing diag-onal elements of the driving Hamiltonian, V FF = 0. Equation (6) and V FF = 0 leadto ω , ω , Re (cid:2) A ( t ) (cid:3) − df dt = 0 ,ω , ω , Re (cid:2) A ( t ) (cid:3) − df dt = 0 , (19)with A ( t ) = ω FF1 , ( t ) e i ∆ f ( t ) . (20) d (∆ f ) /dt is determined when we chooseRe[ A ( t )] to be d ∆ f ( t ) dt = (cid:16) t ) − tan Θ( t ) (cid:17) Re (cid:2) A ( t ) (cid:3) . (21)Equation (18) determines the imaginary partof A to be Im (cid:2) A ( t ) (cid:3) = d Θ dt . (22) Then ω FF1 , is represented as ω FF1 , = e − i ∆ f (cid:16) Re[ A ] + i d Θ dt (cid:17) = e − i ∆ f (cid:104) sin 2Θ2 cos 2Θ d ∆ fdt + i d Θ dt (cid:105) . (23)The Rabi frequency ω FF1 , ( t ) is complex, andmust be realized by controlling the time de-pendences of the phases of the laser fields aswell as the relative phase between and ω FF1 , and ω FF2 , . There is an arbitrariness in thechoice of Re[ A ( t )] or ∆ f ( t ). Three differ-ent trajectories of A are depicted schemat-ically in Fig. 1, for Re[ A ( t )] = 0,Re[ A ( t )] = A exp[ − t /τ ] and Re[ A ( t )] = A ( t/τ ) exp[ − t /τ ], with A and τ con-stant. Re [A]Im [A] Re [A]Im [A](a) (b)
FIG. 1: Schematic diagrams of threedifferent trajectories of A : (a) Re[ A ( t )] = 0;(b) Re[ A ( t )] = A exp[ − t /τ ] (solid curve)and Re[ A ( t )] = A ( t/τ ) exp[ − t /τ ] (dottedcurve) with A and τ constants.It can be shown that the fast-forward as-sisted STIRAP protocol gives the same Rabi7requencies as does the counter-diabatic fieldassisted STIRAP protocol with f m = 0 . (24)Equations (16) and (23) lead to ω FF2 , ( t ) ω FF2 , ( t ) = ω , ( R ( t )) ω , ( R ( t )) , (25)and ω FF1 , = i d Θ dt . (26) ω FF1 , in Eq. (26) is the same as the Rabifrequency of the CDF. The trajectory of A with Re[ A ( t )] = 0 depicted in Fig. 1(a)corresponds to the CDF. Equation (25) de-termines the ratio of ω FF2 , ( t ) and ω FF2 , ( t ) buttheir intensities are arbitrary and can evenbe zero, consistent with the observation thatthe CDF alone can generate complete popu-lation transfer in a two-level system. WhenRe[ A ( t )] (cid:54) = 0 the pulse area of the FFF pulseis larger than π , in contrast to the pulse areaof the CDF, which is π . The restric-tion of the pulse area that is characteristic ofthe CDF protocol is eased in the fast-forwardprotocol. III. HCN → HNC ISOMERIZATIONREACTION
Previous studies of STIRAP generatedpopulation transfer in laser-assisted HCN → HNC isomerization have revealed that background states coupled to the subset ofstates used by the driving STIRAP processdegrade the population transfer efficiency.Mitigation of this inefficiency is sought in anassisted STIRAP process.The three-dimensional potential energysurface for non-rotating HCN/HNC has beenwell studied . The key degrees of freedomthat characterize this surface are the CH,NH and CN stretching motions and the CNHbending motion. These are combined in thesymmetric stretching, bending and asymmet-ric stretching normal modes, with quantumnumbers ( ν , ν , ν ), respectively. The vibra-tional energy levels of HCN and HNC havebeen calculated by Bowman et al . Driv-ing the ground state-to-ground state HCN → CNH isomerization with a conventional STI-RAP process that uses two monochromaticlaser fields is difficult because the Franck-Condon factors between the ground vibra-tional states (0 , ,
0) of HCN and CNH andthe vibrational levels close to the top of theisomerization barrier (e.g. (5 , , eleven vibra-tional states, shown schematically in Fig. 2,are considered; rotation of the molecule isneglected. Kurkal and Rice proposed over-coming the Franck-Condon barrier with se-quential STIRAP, consisting of two succes-sive STIRAP processes. The use of this se-8uence is intended to avert the unwantedcompetition with other processes that canbe generated by the very strong fields thatwould be needed to overcome the Franck-Condon barriers encountered in a single STI-RAP process. In the first step of the se-quential STIRAP, Kurkal and Rice chose the(0 , , , ,
1) and (5 , ,
1) states of HCNas the initial, intermediate and final states,respectively; in the second STIRAP process,the (5 , , , ,
1) and (0 , ,
0) states ofHNC are taken as the initial, intermediateand final states, respectively. Other states,shown with dashed lines in Fig. 2, are re-garded as background states. The pump 1 ( 6, 0, 2 ) |9>( 5, 0, 1 ) |3>( 2, 0, 1 ) |4>( 2, 0, 0 ) |10>( 1, 0, 1 ) |11>( 0, 0, 0 ) |5>( 0, 0, 0 ) |1>( 1, 0, 1 ) |6>( 2, 0, 1 ) |2>( 3, 0, 1 ) |7>( 3, 0, 2 ) |8>( 0.0 /cm )( 8585.87 /cm )( 5393.70 /cm )( 11674.45 /cm )( 13702.24 /cm )( 17574.40 /cm )( 19528.57 /cm ) ( 14154.3 /cm )( 12139.9 /cm )( 5023.2 /cm )( 10651.9 /cm )
HCN HNC
FIG. 2: Schematic diagram of thevibrational spectrum of states used for thenumerical simulations. The states selectedfor use in the successive STIRAP processesare represented with thick lines, and thebackground states are represented with thindashed lines. field is resonant with the transition from the(0,0,0) state of HCN to the (2 , ,
1) state ofHCN; the Stokes 1 field is resonant with thetransition from the (2 , ,
1) state of HCN tothe (5 , ,
1) state; the pump 2 field is resonantwith the transition from the (5 , ,
1) state tothe (2 , ,
1) state of HNC; and the Stokes 2field is resonant with the transition from the(2 , ,
1) state to the (0 , ,
0) state of HNC.The transition dipole moments and the ener-gies of the vibrational states denoted by | i (cid:105) are listed in Ref. 12.Kurkal and Rice showed that the first STI-RAP process is not sensitive to coupling withthe background states caused by the Stokes1 pulse . However the second STIRAP pro-cess is influenced by interference with thebackground states because the intermediatestate of the second STIRAP process has largetransition dipole moments with the back-ground states . The time-dependences ofthe populations of states | (cid:105) − | (cid:105) in the se-quential STIRAP process are displayed inFig. 3. We take the strengths of the pumpand the Stokes fields to be E ( e ) j,p ( S ) ( t ) = ˜ E j,p ( S ) exp (cid:104) − ( t − T j,p ( S ) ) (∆ τ ) (cid:105) (27)where ∆ τ = FWHM / (2 √ ln 2), and FWHMis the full width at half maximum of theGaussian pulse with maximum intensity˜ E j,p ( S ) that is centered at T j,p ( S ) , and j =91 ,
2) denotes the first ( j = 1) and the sec-ond STIRAP ( j = 2) process. We solved thetime-dependent Schr¨odinger equation numer-ically with a fourth order Runge-Kutta inte-grator in a basis of bare matter eigenstateswith T j,p − T j,S = FWHM / (2 √ ln 2). The pa-rameters of the laser fields used in our calcu-lations are shown in Table I. It is seen clearlyin Fig. 3 that the fidelity of the first STIRAPin the sequential STIRAP process is robustwith respect to interference from the back-ground states . For that reason we assumethat the population is transferred from | (cid:105) to | (cid:105) completely, and we focus attention on thesecond STIRAP process, choosing | (cid:105) as theinitial state of the assisted STIRAP controlprocess. States | (cid:105) and | (cid:105) are now and here-after regarded as background states.TABLE I: Strengths and widths of thepump 1, 2 and Stokes 1, 2 laser pulses. ˜ E j,p ( S ) (a.u.) T j,p ( S ) FWHM (ps)Stokes 1 0.00692 133 85pump 1 0.00728 194 85Stokes 2 0.00575 423 85pump 2 0.00220 484 85
We now take the three vibrational states | (cid:105) , | (cid:105) , | (cid:105) in Fig. 2 as the initial, interme-diate and target states of both a STIRAP +FFF and a STIRAP + CDF process. We usethe amplitudes and FWHMs for the pump time (ps) P opu l a ti on FIG. 3: Time-dependence of the severalstate populations for | (cid:105) → | (cid:105) in thesequential STIRAP driven HCN → HNCisomerization .and Stokes laser pulses listed in Table II, andchoose the time-dependence of Re[ A ( t )] to beRe[ A ( t )] = A exp (cid:104) − t ∆ τ (cid:105) (28)with A = 0 .
01 /ps. The time-dependence ofTABLE II: Strengths and widths of thepump 2 and Stokes 2 laser pulses. ˜ E ,p ( S ) (a.u.) FWHM (ps)pump 2 0.0009295 212.5Stokes 2 0.002875 212.5 ∆ f is shown in Fig. 4(a). The phase of theStokes field is changed by ∆ f (see Eq. (16))and the trajectory of A ( t ) = ω FF1 , ( t ) e i ∆ f ( t ) isshown in Fig. 4(b). The amplitudes of theRabi frequencies coupling the three states are10hown in Fig. 5(a) and the time-dependenceof the population of each state in the three-state system decoupled from the backgroundstates is shown in Fig. 5(b). The data dis-played clearly show that 100% populationtransfer is generated. As seen from Eq. (23)and Fig. 5(a) the amplitude of ω FF1 , is largerthan that of the CDF. The restriction of thepulse area that is characteristic of the CDFprotocol is eased in the fast-forward protocol.We now examine the efficiency of the STI-RAP + FFF control when the subset of states | (cid:105) , | (cid:105) , | (cid:105) is embedded in the manifold ofstates depicted in Fig. 2. We consider aFFF corresponding to Rabi frequency ω FF1 , ac-companied with a pump pulse and a phase-controlled Stokes field with all the back-ground states in Fig. 2; the time-evolutionof the system is calculated exactly, withoutuse of the rotating wave approximation. InFig. 6 the time-dependences of the popula-tions of the initial, intermediate and the tar-get states for (a) STIRAP and (b) STIRAP+ FFF control for A = 0 . T ,p − T ,S = FWHM / (2 √ ln 2).The efficiencies of both STIRAP and STI-RAP + FFF controls are degraded due tointerference with background states stronglycoupled to the intermediate state. However -6 0 -500 500 f time (ps) (a) Re [ A ]Im [ A ](b) FIG. 4: The time-dependence of (a) ∆ f and(b) A for FWHM = 212.5 ps and T ,p − T ,S = FWHM / (2 √ ln 2).the influence of the background states is sup-pressed in the STIRAP + FFF control com-pared to that in the STIRAP control becauseof the direct coupling of the initial and tar-get states by the FFF. Suppose now that thepeak amplitude of the laser field coupling theinitial and final states is larger than that ofthe CDF in Eq. (26) and is fixed, whilst itsphase and that of the Stokes field are con-trollable with respect to time. As shown in11 time [ps] popu l a ti on (b) time [ps] H opp i ng r a t e FF CD FF FF (a) -500 500 FIG. 5: (a) The time-dependences of theamplitudes of the Rabi frequencies forFWHM = 212.5 ps and T ,p − T ,S = FWHM / (2 √ ln 2). (b) Thetime-dependences of the populations.Fig. 7 the fidelity of the STIRAP + CDFcontrol decreases when the amplitude of thelaser field coupling the initial and final statesis larger than that of the CDF in Eq. (26).In such cases the decrease of the fidelity ispartially avoidable via phase control of the time [ps] popu l a ti on time [ps] popu l a ti on (a)(b) -500 500 -500 500 FIG. 6: Time-dependences of thepopulations of the initial, intermediate andthe target states for (a) STIRAP and (b)STIRAP + FFF control for peak field ratio= 1.2, FWHM = 212.5 ps and T ,p − T ,S = FWHM / (2 √ ln 2).laser fields. Five values of A are used; theratio of the peak amplitude of the FFF to theCDF amplitude ranges from 1 to 1.5. In Fig.7 we compare the calculated fidelities to thatof the STIRAP + CDF control for the case12hat the CDF strength is greater than thatof the CDF in Eq. (26). The decrease of thefidelity due to variance of the amplitude ofthe CDF is reduced by phase control of thelaser pulses. peak field ratioCD+STIRAPFF+STIRAP F i d e lit y FIG. 7: The dependence of the fidelity onthe peak amplitude of the laser fieldcoupling the initial and target states. Thehorizontal axis is the ratio of the peak laserfield amplitude to the amplitude of the CDFin Eq. (26) for FWHM = 212.5 ps and T ,p − T ,S = FWHM / (2 √ ln 2).As seen from Eq. (16), the FFF alone cangenerate complete population transfer to thetarget state in a two-level system. Howeverthe efficiency of the single pulse control isdegraded when there is interaction with thebackground states, and is not stable to vari-ation of the area of the pulse. So as to studythe stability of the efficiency of the popula-tion transfer driven by a variable FFF we rep- resent the total driving field in the form E ( t ) = E p ( t ) + E S ( t ) + λE FF ( t ) , (29)where λ = 0 corresponds to driving the sys-tem with only the STIRAP fields and λ = 1to driving the system with the STIRAP andthe FFF; E p ( S ) is the pump (Stokes) field; E FF is the FFF corresponding to the Rabifrequency in Eq. (23). In Fig. 8 the stabilityof the STIRAP + FFF control and the FFFalone control to the variation is monitoredby the fidelity as a function of λ . Clearly,the sensitivity to the variation of amplitudeof STIRAP + FFF control is decreased com-pared to that of FFF control. The STIRAP+ FFF control generates higher fidelity thando STIRAP or FFF individually for a widerange of the field ratio λ . The value of λ corresponding to the peak of the fidelity ofSTIRAP + FFF control is smaller than onein Fig. 8, because of the interference with thebackground states generated by the strongfields.The FFF, whose peak amplitude is pro-portional to 1/FWHM, can degrade the effi-ciency of the control when the FWHM of thelaser pulses is too short. And an increaseof the field strengths in a simple STIRAPprocess does not generate greater popula-tion transfer efficiency because those strongerfields also generate greater interference be-tween the active subset of states and the13 F i d e lit y λ FF+STIRAPFF field only
FIG. 8: Comparison of the fidelity ofSTIRAP+FFF control and FFF controlwith FWHM = 212.5 ps and T ,p − T ,S = FWHM / (2 √ ln 2) for various λ .background states . It is seen in Fig. 9that the FFF with λ = 0 . λ = 1 for 8.5 ps ≤ FWHM ≤
34 ps. The drop of fidelity when FWHMdecreases is due to the large intensity of theFFF. The fields associated with STIRAP andFFF generated population transfer are com-plementary if the amplitude of the FFF is nottoo large.
IV. VIBRATIONAL ENERGYTRANSFER IN THIOPHOSGENE
The calculations reported in Section IIIshow that the efficiency of STIRAP+FFFgenerated population transfer when 8.5 ps
STIRAP F i d e lit y
7 35
FWHM (ps) peak field ratio 1.2peak field ratio 1.5
FIG. 9: FWHM-dependence of the fidelityfor the STIRAP+FFF control (FF fieldratio=1.2 and 1.5), λ = 0 . E ,p = 0 . E ,S = 0 . T ,p − T ,S = FWHM / (2 √ ln 2). ≤ FWHM ≤
34 ps is comparable to or lessthan that of STIRAP+CDF generated pop-ulation transfer for the same pulse widthrange despite the extra flexibility of STI-RAP+FFF compared to STIRAP + CDFcontributed by phase tuning in the former.In this Section we show that that extra flex-ibility of STIRAP+FFF indeed can generatemore efficient population transfer than STI-RAP+CDF in the small FWHM regime, us-ing as an example state-to-state vibrationalenergy transfer in nonrotating SCCl .The SCCl molecule has three stretching( ν , ν , ν ) and three bending ( ν , ν , ν ) vi-brational degrees of freedom; it suffices, forour purposes, to use the same set of ener-14ies and transition dipole moments as usedby Kurkal and Rice , covering the range0 − − , determined by Bigwood,Milam and Gruebele . These energy levelsare displayed in Fig. 10 and tabulated inRef. 11. We will focus attention on the effi-ciency with which population transfer can beselectively directed to one of a pair of nearlydegenerate states in the presence of back-ground states. As in Ref. 24, we consider aSTIRAP process within the subset of threestates ( | (cid:105) , | (cid:105) , | (cid:105) ) embed-ded in the full manifold of states. Hereafterwe refer to these three states as | (cid:105) , | a (cid:105) and | (cid:105) , respectively. The STIRAP+FFF controlprocess is intended to generate higher popu-lation transfer from | (cid:105) to | (cid:105) . Wenote that | (cid:105) , hereafter called | (cid:105) , withenergy 5658.1828 cm − , is nearly degeneratewith | (cid:105) , with energy 5651.5617 cm − , andthat the transition moment coupling states | (cid:105) and | (cid:105) is one order of magnitude smallerthan those coupling states | (cid:105) and | (cid:105) and | a (cid:105) and | (cid:105) .To compare the efficiency of the STI-RAP+FFF control to that of the STI-RAP+CDF control we examine the depen-dence of the STIRAP+FFF generated popu-lation transfer on the peak ratio of the FFFto the CDF. The range of the phase tunedin the STIRAP+FFF control increases whenthe peak field ratio becomes large, while the population transfer when the peak field ra-tio is one is identical to that generated bySTIRAP+CDF without phase tuning. Fig-ure 11 displays the dependence of the STI-RAP+FFF generated population transfer onthe peak ratio of the FFF to the CDF forthe parameters FWHM= 21 . T p − T S =FWHM / (2 √ ln 2), λ = 1, ˜ E p = 0 . E S = 0 .
046 a.u. For a wide range ofpeak field ratio the population transfer gen-erated by STIRAP+FFF exceeds that gen-erated by STIRAP+CDF. Figure 12 displaysthe dependence of population transfer gen-erated by STIRAP, STIRAP+CDF and STI-RAP+FFF (with parameters λ = 1 and peakfield ratios 1.2 and 1.5) on the FWHM of thepulses. The values of ˜ E p,S for each value ofthe FWHM have been adjusted so that thepulse areas of the pump and Stokes fields arethe same as used for the calculations shownin Fig. 11. The STIRAP+FFF generatedpopulation transfer exceeds those generatedby ordinary STIRAP and STIRAP+CDF for21.5 ps ≤ FWHM ≤
86 ps.
V. CONCLUDING REMARKS
We have examined the efficiency of STI-RAP + FFF generated selective state-to-state population transfer in the vibrationalmanifolds of nonrotating SCCl and theHCN → CNH isomerization. Neglecting the15 |100031>)(|200020>)(|300002>) (Target state) |17> (|210020>)|16> (|201020>)|7> (|300002>)|3> (|310000>)|6> (|200020>)|15> (|100031>)|14> (|201011>) |4> (|302000>)|9> (|210011>)|18> (|100040>)|19> (|210200>)|21> (|301000>)|20> (|100220>)|13> (|201200>)|12> (|211000>)|11> (|131000>)|5 a > (|300000>) (Intermediate state) |10> (|122000>)|5 b > (|013000>)|1> (|200000>)|2> (|210000>) (Initial state) |8> (|230000>) FIG. 10: Schematic diagram of the vibrational spectrum of SCCl .influence of molecular rotation on the effi-ciency of vibrational population transfer de-fines useful models that permit qualitativeinvestigation of the influence of backgroundstates on the efficiency of energy transferwithin an embedded subset of states, butthose models are inadequate for the quantita-tive description of energy transfer in the cor-responding real molecules. It is relevant toask if our calculations provide a qualitativelyvalid picture applicable to real situations.We have argued elsewhere that, neglect- ing higher order effects such as vibration ro-tation interaction, we expect the rotation ofa molecule to affect the state-to-state processwe describe in two ways. First, the transitiondipole moment projection along the field axisdiffers with rotational state, thereby reducingthe rate of excitation. Second, the rotationalwave-packet created may dephase on a timescale that is comparable with the width ofthe exciting field, thereby changing the dy-namics of the population transfer. If the ra-tio of the driving field duration to the period16 F i d e lit y p ea k fi e l d r a ti o
1 2
FIG. 11: The dependence of the fidelity onthe peak field ratio of the FFF to the CDFfor the STIRAP+FFF control withFWHM= 21 . T p − T S = FWHM / (2 √ ln 2), λ = 1,˜ E p = 0 . E S = 0 .
046 a.u.of molecular rotation is very small we expectmolecular rotation to have negligible influ-ence on the population transfer, and whenthe period of molecular rotation is compara-ble to the width of the field pulses that drivethe population transfer we must expect lessefficient transfer than predicted for the non-rotating molecule. Indeed, noting that thecombined STIRAP + FFF control processwe describe involves both one and two pho-ton transitions, and that the wave-packets ofrotational states created by these two excita-tion processes have different dephasing rates,we expect the evolution of the state of theexcited molecule to be complicated when the
20 110
FWHM (ps) F i d e lit y STIRAPFF (ratio=1.5)FF (ratio=1.2)STIRAP+CD
FIG. 12: The dependence of the fidelity onthe FWHM for the ordinary STIRAP, theSTIRAP+CDF control and theSTIRAP+FFF with λ = 1, T p − T S = FWHM / (2 √ ln 2) and the peakfield ratio = 1 . and HCNare of the order of 200 ps and 10 ps, respec-tively. Our calculations of the efficiency ofstate-to-state population transfer in SCCl include cases when the FWHM of the pulsedfields is considerably smaller than 200 ps (seeFig. 12). The efficiency of the populationtransfer is smaller when the FWHM of thepulses is 20 ps than when it is 100 ps, but stillusefully large. And since these pulse widthsare of order one tenth of the rotational periodit is plausible that similar efficiency of state-to-state population transfer can be achievedin the real molecule. Our calculations of the17fficiency of state-to-state population trans-fer in the HCN → CNH isomerization do notinclude cases when the FWHM of the pulsedfields is considerably smaller than the rota-tional period. In this case, as shown in Fig.9, the use of very short pulses severely de-grades the population transfer efficiency.Returning to the model cases considered,we have shown that STIRAP + FFF gen-erated state-to-state population transfer ismore efficient than STIRAP generated state-to-state population transfer when applied toa subset of states embedded in and coupledto a larger manifold of states. Moreover, wehave shown that the FFF calculated for anisolated subset of three states can be used toapproximate the FFF applicable to a threestate subset embedded in a large manifoldof states even when some of the backgroundstates are strongly coupled to the intermedi-ate state of the STIRAP process.The FFF is designed to avert unwantednon-adiabatic population transfer at the endof the application of the pulsed field, and itdirectly couples the initial state to the targetstate thereby decreasing the sensitivity of thepopulation transfer to the influence of back-ground states. STIRAP + CDF generatedpopulation transfer exhibits this same de-creased sensitivity for the same reason. How-ever, 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 samesystem, which is always π . And the STI-RAP + FFF generated population transferhas, relative to STIRAP + CDF populationtransfer, an extra control parameter, namelythe FFF amplitude. This parameter can betuned to optimize the yield of population ina target state. In general, our model calcu-lations show that, when the driven systemof states is embedded in a large manifold ofstates, phase controlled STIRAP + FFF gen-erates more efficient state-to-state populationtransfer than does STIRAP. ACKNOWLEDGMENTS
S.M. thanks the Grants-in-Aid for CentricResearch of Japan Society for Promotion ofScience and the JSPS Postdoctoral Fellow-ships for Research Abroad for its financialsupport.
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