Extensions to quantum optimal control algorithms and applications to special problems in state selective molecular dynamics Karsten Sundermann and Regina de Vivie-Riedle Citation: J. Chem. Phys. 110, 1896 (1999); doi: 10.1063/1.477856 View online: http://dx.doi.org/10.1063/1.477856 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v110/i4 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 21 Sep 2013 to 128.153.5.49. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions
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Extensions to quantum optimal control algorithms and applications tospecial problems in state selective molecular dynamicsKarsten Sundermann and Regina de Vivie-Riedle Citation: J. Chem. Phys. 110, 1896 (1999); doi: 10.1063/1.477856 View online: http://dx.doi.org/10.1063/1.477856 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v110/i4 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors
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Achievement of molecular control by manipulating costructive and destructive interferences of optical fields issubject to current theoretical and experimental efforts. Fconcepts for controlling chemical reactions using sequenof broad band laser pulses have been developed theoretiby Tannor, Rice and Kosloff1,2 followed by some examplefirst realized experimentally in the group of Zewail anGerber.3,4 Another approach to active control is based onprinciple of feedback control.5 Experimentally progress habeen made in the development of ultrafast pulse shaptechniques6,7 providing a flexible tool for feedback controstrategies. On the theoretical side highly sophisticated omal control schemes and algorithms have been develoand refined.8 A new family of iteration methods incorporaing feedback from the control field was presented recentlythe group of Rabitz.9,10 They proposed new algorithmswhich exhibit a very fast convergence behavior in combition with high accuracy and selectivity. This new algorithwas tested and extended in our group for different typescontrol problems. There was special emphasis on the deof optimized electric fields which can be experimentallyalized. For this purpose the iteration algorithm was extento allow for smooth switch on and off behavior of the dsigned fields. Strategies taken from signal processing areplied to analyze the frequency sequences of the optimlaser field with the intention to construct a new, experimetally more feasible laser pulse. Robustness of the resulfields with respect to variations in the phase of the la
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field is tested. The purpose of this paper is to outline anddiscuss the implementation of these experimentally mvated constraints in the control scheme. The effect ofextended algorithm on the optimized laser field is demstrated for several examples, including selective vibratiopopulation transfer with infrared~IR! laser pulses, vibra-tional population transfer via stimulated emission pump~SEP!, and photoisomerization processes induced by pumdump laser pulses.
II. THEORY
A. Field optimization
In quantum optimal control theory, one tries to find lasfields e(t) which drive a system from an initial stateC i(0)5F i at time t50 to a final target stateF f at a fixed timet5T. For the determination of such optimal fields many aproaches have been published; in this paper we will conctrate on algorithmic schemes. These schemes allow uformulate the problem in terms of the maximization offunctional, which includes as the most important part toverlapu^C i(T)uF f&u2 of the laser-driven initial wave function C i with the target stateF f at time t5T. Additionalterms that allow us to minimize the laser energy and to sisfy the Schro¨dinger equation are added. The most recfunctional which allows an algorithmic treatment was pulished by Zhuet al.9 together with an iterative optimizationscheme. For further considerations we first presentfunctional9
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1897J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 K. Sundermann and R. de Vivie-Riedle
~1!
HereC i(t) denotes the wave function which shall be drivby the electric fielde(t) from the initial stateC i(0)5F i tothe final stateF f at time t5T. To limit the resulting fieldintensities the parametera serves as a penalty factor to thelectric field energy. The energy-like term*0
Tue(t)u2dt of theresulting optimized fielde(t) decreases with increasinga.The additional termT2 secures the boundary condition foC i(t) to fulfill the Schrodinger equation. This is formulatein integral form and inserted into the functional together wa Lagrangian multiplierC f(t)u. The additional termT1 to-gether with the real part Re is only needed in order toeasier differential equations describing an extreme pointhe functional. The importance of termT1 will be seen in thederivation of the differential equations, where it will bneeded in Eqs.~9! and ~10!. The field optimization is nowreduced to find the maximum of the functional, Eq.~1!,which is solved by a variation of the variablesC i(t), C f(t),ande(t) for
dJ50. ~2!
Even though a solution for the electric fielde(t) of dJ50will provide a good driving laser field for the populatiotransfer from the initial stateC i(0) to the target stateF f , anexperimental realization of such a field will hardly be fesible. Typically it does not have a smooth switch on andbehavior, i.e., the field starts and ends spontaneouslyhigh laser intensities, see Ref. 9. This problem is overcoby introducing a time dependence of the penalty factora,which means that for different times different intensities wbe allowed. Therefore we rewrite
a5a~ t !5a0
s~ t !, ~3!
with s(t) serving as a shape function for the laser pulse.a first approach we consider
s~ t !5sin2S p•
t
TD ~4!
with T the overall time for the population transfer; pulsfollowing this shape were first treated in Ref. 11. The effof such a shape function is that the value ofa(t) goes toinfinity for t50 and t5T, which means that the penaltfactor in those regions will be very high and therefore fieintensities very low, in particulare(0)5e(T)50. The maxi-mum of the allowed field intensities occurs at the minimu
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tf
fthe
s
t
of a(t) which is reached for the shape function of Eq.~4! atmid time t5T/2. To derive the equations used for the fieoptimization we insert Eq.~3! into Eq.~1!, thus obtaining ournew functionalK:
K@C i~ t !,C f~ t !,e~ t !#
5u^C i~T!uF f&u22a0•E0
T ue~ t !u2
s~ t !dt
22 ReH ^C i~ t !uF f&E0
T
^C f~ t !ui
\@H2me~ t !#
1]
]tuC i~ t !&dtJ ~5!
and apply the variation of the variablesC f(t), C i(t), ande(t) to the extreme value problemdK50. In the followingvariation x(t)1dx(t) of the variables, we will omit termsdepending quadratically ondx(t) and will use' to indicatethese omissions. Starting with the variation ofC f(t) we get
dC fK5K@C i~ t !,C f~ t !1dC f~ t !,e~ t !#
2K@C i~ t !,C f~ t !,e~ t !#
522 ReH ^C i~T!uF f&E0
T
^dC f~ t !ui
\@H2me~ t !#
1]
]tuC i~ t !&dtJ 50. ~6!
SincedC f has been chosen arbitrarily, Eq.~6! can only besatisfied if
S i
\@H2me~ t !#1
]
]t DC i~ t !50, ~7!
⇔ i\]
]tC i~ t !5@H2me~ t !#C i~ t !. ~8!
This means, as expected, that the functionC i(t) has to fulfillthe time-dependent Schro¨dinger equation, with the boundarconditionC i(0)5F i . Applying the variation ofC i(t) to thefunctionalK yields
dC i ~ t !K5K@C i~ t !1dC i~ t !,C f~ t !,e~ t !#
2K@C i~ t !,C f~ t !,e~ t !#
'2 Re$^C i~T!uF f&^F f udC i~T!&%
22 ReH ^C i~T!uF f&
3E0
T
^C f~ t !ui
\@H02me~ t !#1
]
]tudC i~ t !&dtJ
22 ReH ^dC i~T!uF f&
3E0
T
^C f~ t !ui
\@H02me~ t !#1
]
]tuC i~ t !&dtJ . ~9!
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1898 J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 K. Sundermann and R. de Vivie-Riedle
According to Eq.~7! the last term in Eq.~9! vanishes andtherefore we obtain
dC i ~ t !K52 ReS ^C i~T!uF f&
3H ^F f udC i~T!&2E0
T
^C f~ t !ui
\@H2me~ t !#
1]
]tudC i~ t !&dtJ D .
With the choice of a time-independentdC i(t)5dC i thetime derivative]/]t vanishes and by moving the adjoint othe operatori /\@H2me(t)# to the left, in order to indicate itshall work onC f(t), we obtain:
dC i ~ t !K52 ReS ^C i~T!uF f&3H ^F f udC i&
2E0
TK 2i
\@H2me~ t !#C f~ t !UdC i L dtJ D 50.
~10!
Equation~10! holds for aC f(t) satisfying the Schro¨dingerequation with boundary conditionC f(T)5F f , therefore wedemand
i\]
]tC f~ t !5@H2me~ t !#C f~ t !, C f~T!5F f . ~11!
This equation also states that the Lagrangian multipC f(t) can be regarded as a wave function obeying the sSchrodinger equation asC i(t) simply with a differentboundary condition. So far we got from the variation of tfunctional the time-dependent Schro¨dinger equations forC i(t) and C f(t) with corresponding boundary conditionC i(0)5F i andC f(T)5F f . In order to derive an equatiofor the optimal fielde(t), we apply the variation ofe(t)
de~ t !K5K@C i~ t !,C f~ t !,e~ t !1de~ t !#2K@C i~T!,F f ,e~ t !#
5a0•E0
T ue~ t !u2
s~ t !dt2a0•E
0
T ue~ t !1de~ t !u2
s~ t !dt
22 ReH ^C i~T!uF f&
3E0
T
^C f~ t !u2i
\mde~ t !uC i~ t !&dtJ , ~12!
52a0•E0
T 2e~ t !de~ t !1de2~ t !
s~ t !
22
\•Im E
0
T
$^C i~T!uF f&
3^C f~ t !umuC i~ t !&de~ t !dt%, ~13!
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re
'2E0
TH 2a0
e~ t !
s~ t !1
2
\•Im$^C i~T!uF f&
3^C f~ t !umuC i~ t !&%J de~ t !dt50. ~14!
Since we did not impose any restrictions onde(t), Eq. ~14!can be satisfied for allde(t) only by a vanishing integrandi.e.,
a0
e~ t !
s~ t !1
1
\•Im$^C i~T!uF f&^C f~ t !umuC i~ t !&%50,
~15!
or expressed as an equation for the optimized fielde(t):
e~ t !52s~ t !
\a0•Im$^C i~T!uF f&^C f~ t !umuC i~ t !&%. ~16!
Making use of the Schro¨dinger equations~7! and ~11! therelation
^C i~T!uF f&5^C i~ t !uC f~ t !& ~17!
holds and together with the results from the variations ofthree variablesC f(t), C i(t), and e(t), we obtain the fol-lowing set of differential equations, which describe the opmal field e(t) and the evolving wave functionsC i(t) andC f(t):
e~ t !52s~ t !
\a0•Im$^C i~T!uF f&^C f~ t !umuC i~ t !&%
52s~ t !
\a0•Im$^C i~ t !uC f~ t !&^C f~ t !umuC i~ t !&%,
~18!
i\]
]tC i~ t !5@H2me~ t !#C i~ t !, C i~0!5F i , ~19!
i\]
]tC f~ t !5@H2me~ t !#C f~ t !, C f~T!5F f . ~20!
As these equations are similar to those of Ref. 9 we sothem by applying a unidirectional version of the iteratioptimization scheme proposed in Ref. 9, which we derivethe following and formulate it in terms of a fixed point problem ~Ref. 12, p. 67, Ref. 13, p. 151!. This means we try tofind an easy computable functionF, which maps an electricfield e(t) to another oneF@e(t)# and has the property thatfield e(t), which remains unchanged under this mappini.e., e(t)5F@ e(t)#, is a solution to the set of the coupledifferential equations~18!, ~19!, and ~20!. The followingfunction F
F@e~ t !#ª2s~ t !
\a0•Im$^C i~ t !uC f~ t !&^C f~ t !umuC i~ t !&%,
~21!
andC i(t), C f(t) obeying
i\]
]tC f~ t !5@H2me~ t !#C f~ t !, C f~T!5F f , ~22!
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1899J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 K. Sundermann and R. de Vivie-Riedle
i\]
]tC i~ t !5S H1m
s~ t !
\a0•Im$^C i~ t !uC f~ t !&
3^C f~ t !umuC i~ t !&% DC i~ t !, C i~0!5F i
~23!
has the property that a fixed pointe(t) of F, together withthe wave functionsC i(t) andC f(t), satisfies Eq.~18!, ~19!,and ~20! as can be easily seen. A fixed pointe(t) to F canthen be obtained by iteratively computingF on an initialguesse(t), or in other words, the sequence
limn→`
Fn@e~ t !#→ e~ t ! ~24!
converges towards a fixed pointe(t) of F. @To justify thisapproach one might consider the theorem of Banach on fipoints~Ref. 12, p. 67, Ref. 13, p. 151! which proves that fora contractive mapping F, i.e., uF(x)2F(y)u,aux2yu;(x,y) anda,1, the sequence limn→`Fn@e(t)#→ e(t)converges towards a fixed point ofF. A detailed proof forthe convergence of this iterative scheme is given in Ref.#The last question is how to compute the functionF. This caneasily be done by first propagatingC f backwards in timefrom t5T to t50 and then propagatingC i andC f forwardwith simultaneous evaluation of expression~18! for the elec-tric field e(t). For more detailed information see Ref. 9.comprehension the optimal field can be determined by itetively computing the mappingF on an initial guess for theelectric field. With this scheme, which is a unidirectionversion of the scheme of Zhuet al.,9 we observed convergence towards the optimal fielde(t).
B. Analysis of the optimized laser field
Electric laser fields obtained by optimal control theoryfirst sight often appear to be hardly realizable. Thereforeperform a more detailed analysis in order to provide expmentalists with information needed, such as temporal ording of the main carrier frequencies, time resolution, timelay between pulse sequences, and phase relation to rebuoptimized pulse e(t) using fast pulse shapintechniques,14–16 which have been developed only recentMore techniques for time-frequency analysis are givenRef. 17. An additional problem which may occur for mulcolor experiments is that their frequency spectrum isbroad to be realized by a single short pulse. In this casepulse might be realized by applying pulse shaping techniqon a sequence of different single-colored subpulses. Thesented pulse analysis is generally applicable, but in thelowing discussion we will focus on the two-color problem.possible approach is to divide the optimal pulse into aquence of subpulses with distinct but localized spectrafrequenciesV i . For this approach the temporal orderingthe active frequencies must be determined. Subsequentlyoriginal pulsee(t) is reconstructed as a superposition ofnsubpulsesek(t) with spectra localized around the frequencV i . The active frequencies at a timet cannot be determinedby a conventional Fourier transform over the whole pulsethe time window@0,T# since it contains no time informatio
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at all, even though it provides the highest possible frequeresolution. For the time–frequency analysis we employwindowed Fourier transform, which basically narrows toverall time window@0,T# to smaller intervals@t,t1Dt#with width Dt at different timestP@0,T#. A Fourier trans-form of the electric fielde(t) is now performed for all timestP@0,T# in the corresponding windows@t,t1Dt#. Theoriginal pulse can therefore be written as
e~ t !5 (k>0
ek~ t ! ~25!
5Re(k>0
ak~t!exp@ ivk~ t2t!#, ~26!
tP@0,T# and tP@t,t1Dt#
with ak(t) being the Fourier coefficients of the pulse in thtime domain tP@t,t1Dt# and vk5k•2p/Dt the corre-sponding Fourier frequencies. Decreasing the overall twindow @0,T# to @t,t1Dt# has the effect that more timelocalized information on the frequencies is supplied atexpense of lowering the frequency resolution. In the wind@t,t1Dt#, there are fewer~discrete! frequencies than in theoverall window, and therefore they provide a tool to appromate the original pulse by a sequence of a few pulses wdifferent carrier frequenciesV i corresponding to some of thFourier frequenciesvk . Taking into account the experimental fact that it is difficult to control the phases between indpendent pulsesek(t) with different frequencies, we allowthem to have phase shiftsfk , which are inserted into Eq~26! by additional phase factors exp(ifk)
e~ t !5Re(k>0
ak~t!exp@ ivk~ t2t!1fk#, ~27!
tP@0,T# and tP@t,t1Dt#.The evaluation of Eq.~27! at the beginningt of the time
windows @t,t1Dt# gives
e~t!5 (k>0
Re@ak~t!exp~ ifk!#, tP@0,T#, ~28!
with ai(t) denoting again the Fourier coefficients of thpulsee(t) in the time window@t,t1Dt#. The effect of thephase shiftsfk on the efficiency of the pulse will be discussed in Sec. III C. The determination of the widthDt of thetime window @t,t1Dt# will be demonstrated for a pulsfield e(t) containing two carrier frequenciesV0 and V1 ,which shall be approximated by two subpulsesE0(t) andE1(t) ~here we useEi since they correspond to the frequeciesV i , unlike thee i to v i!. Pulses with such spectra typcally occur for pump–dump-like processes~see Sec. III C!.The choice ofDt determines the maximum frequency reslution Dv52p/Dt which can be achieved by the windoweFourier transform. Since we intend to separate two pulwith frequencies V0 and V1 we have to chooseDt>2p/(V12V0). Subsequently, we determine those Fourfrequenciesvm5m•2p/Dt and vn5n•2p/Dt, m,n,PN,which are closest to the desired frequenciesV0 and V1
~sinceDv does not necessarily divideV0 andV1! and thenextract the two subpulses
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1900 J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 K. Sundermann and R. de Vivie-Riedle
E0~t!5em~t!5Re@am~t!exp~ ifm!#, ~29!
and
E1~t!5en~t!5Re@an~t!exp~ ifn!# ~30!
whereak(t) is thekth Fourier coefficient fore(t) in the timewindow @0,T#. The algorithm for the decomposition ofpulse having two distinct frequenciesV0,V1 can be sum-marized as follows:
~i! select two frequenciesV0 andV1 close toV0 andV1
with DV5V12V0 such that
V05nDV, V15~n11!•DV, nPN, ~31!
~ii ! determineDt52p/DV and select the time window@t,t1Dt#;
~iii ! for each timetP@0,T# perform a Fourier transform othe pulsee(t) in the windowtP@t,t1Dt# giving theFourier coefficientsak(t),
~iv! use theak(t) and the phasesfk to compute fortP@0,T# the subpulses
E0~t!5Re@an~t!exp~ifn!# ~32!
andE1~t!5Re@an11~t!exp~i•fn11!#, ~33!
~v! construct an efficient driving fieldE~t!5E0~t!1E1~t!, tP@0,T#. ~34!
This decomposition yields two subpulses with spectra loc
ized aroundV0 and V1 and a fixed phase relation. The aplication of this scheme will be shown in Sec. III C for thphotoisomerization of Li2Na via a controlled pump–dumcycle.
III. APPLICATIONS
Out of a variety of possible applications for optimal cotrol theory, we have chosen three which are of current inest in modern spectroscopy and femtosecond chemistry18,19
~i! selective IR vibrational transitions,~ii ! selective vibra-tional excitation using SEP,20 and ~iii ! laser controlled pho-toisomerization. All examples have been treated numericby use of the split operator technique.21,22
A. IR excitation of OH vibrations
In order to demonstrate the improvement in the pushape by including the correct asymptotic behavior oflaser field in the iteration scheme, we selected the vibratioexcitation of the OH molecule for direct comparison wiprevious work.9 The potential energy surfaceV(x) and di-pole moment functionm(x):
V~x!5D0~e2b~r 2r 0!21!2, ~35!
m~x!5m0re2t/t* , ~36!
are taken from Ref. 9 together with the parameters givenatomic unitsD050.1994,b51.189,r 051.821,m053.088,and r * 50.6 ~as in Ref. 9!. The optimal control scheme oZhu et al. and the modified one using a shape functions(t)was applied to the vibrational transitionn50→n51. Theconverged optimized electric fields are shown in Fig. 1
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gether with their corresponding frequency distribution. Tlaser pulse on the top results from the original approachthe laser pulse below results from the optimization schewith the shape functions(t)5sin2(pt/T). The populationtransfer is comparable 0.99 for the pulse following the aproach of Zhuet al.9 and 0.98 when using the shape functios(t)5sin2(pt/T). The main difference is the smooth switcon and off behavior of the pulse, which beautifully followthes(t)5sin2(pt/T) shape for the whole pulse duration. Bopulses do not contain higher frequencies in the spectrum.additional analysis showed no chirp or time-dependent phprofile on them. This shows that the new constraint ontime-dependent penalty factor yields laser pulses whichmuch more likely to be realized experimentally.
B. Vibrational excitation of K 2 via stimulatedemission pumping
Beside IR excitation, SEP is a well known and ofteapplied technique to populate excited vibrational levelsthe electronic ground state.20 A first approach in this direc-tion using ultrashort laser pulses was made by Tannor, Rand Kosloff.2 A first laser pulse, the pump pulse, transfethe vibrational ground state wave packet from the electrocally lower lying surface to the electronically excited onwhere the wave packet propagates for a timeDT and is thendumped back to the electronic ground state by a secpulse. However, this scheme only allows the control aexcitation of wave packets and not the selective populaof one vibrational level. To achieve state selectivity we eployed optimal control in order to determine whether itpossible, in theory, to achieve population inversion fromvibrational ground state to a selected excited vibratiostate. This approach fundamentally relies on interferenceseveral pump–dump cycles of the wave packets evolvingtime under the driving laser fielde(t). In Fig. 2 this~SEP!scheme is illustrated for our selected example, the K2 mol-ecule. The arrows indicate that the wave packet has to
FIG. 1. Optimized laser pulses for the vibrational transitionv50→v51 ofthe OH molecule, together with their corresponding spectra. On topoptimized pulse for the shape functions(t)51 below the one for the shapefunction s(t)5sin2(pt/T).
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1901J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 K. Sundermann and R. de Vivie-Riedle
pumped and dumped at different internuclear distances fthe potential energy surfaces in order to get a selective tsition to the target state.
The system is treated for the one-dimensional moHamiltonian including two surfaces:
H~ t !5S T1Vg
00
T1VeD2e~ t !S 0
meg
mge
0 D5H02e~ t !•m, ~37!
with
T52\2
2m
]2
]r 2 , ~38!
where m is the reduced mass of K2, Vg is the electronicground stateX 1Sg
1 , Ve is the electronically excited statA 1Su
1 , andr is the internuclear distance. The potential eergy surfaces and the dipole transition function are tafrom Ref. 23. The laser pulse optimization has been pformed using the shape functions(t)5sin2(pt/T), where theoverall time T for this process has been limited toT536 000\/Eh'870 fs. The optimized pulse transfers 89%
FIG. 3. Optimized laser pulse for the vibrational transitionn0→n5 of the K2
molecule using the SEP mechanism.
FIG. 2. The vibrational excitation of the K2 molecule from the vibrationalground to the fifth excited state using the electronicA 1Su
1 surface as anintermediate state. The dashed lines indicate that the pump–dump chave to occur over the whole region of the target state.
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mn-
l
-nr-
f
the initial wave packet to the fifth vibrationally excited stain the electronic ground state. Taking into account thatremain on the electronically excited stateA 1Su
1 the targetstate is hit up to 96%. At first sight the optimized electrfield ~see Fig. 3! seems to be highly counter intuitive, however two interesting features can be extracted leadingmore physical insight. First, the optimized field is built up ba sequence of short subpulses with a delay ofDT'3100Eh /\'75 fs, indicated by the dashed lines at tpulse maxima, and a pulse width of approximately the sasize. The periodDT52p/DE'3040Eh /\573.5 fs derivedby the energy spacingDE52.06731023Eh'452 cm21 be-tween n50 and n55 corresponds very well to the delatime DT'75 fs of the subpulses and also their bandwidjust covers this energy spacing between the initial and tastate.
The second interesting aspect is that the spectral w~see Fig. 4! of the laser pulse covers the energy differencof the electronic ground and excited state in the internucranges spanned by the vibrational target staten55, wherethe energy difference ranges fromDE56.331022Eh
'13 825 cm21 at r 53.44 Å to DE54.631022Eh'10 095cm21 at r 54.6 Å. Both features emphasize that in contrato the conventional pump–dump scheme transitions ocover a very broad spectral region defined by the vibratioenergy spacing of initial and target states and by the difence potential of the electronic states involved in the spaspreading of the vibrational target state. This observatstrongly supports the hypothesis that interferences of partthe pumped and dumped wave packet are needed whening to selectively populate a vibrational target state by Swith more nodes than the initial state. The optimized pumay be interpreted as a sequence of pulses with the prtemporal resolution, time delay, and phase relation.
C. Laser controlled pump–dump transitions for Li 2Na
As a more complex system for optimal control we hachosen the mixed alkali cluster Li2Na for which quantumchemical calculations predict two isomers in the shape ofacute and an obtuse triangle24 ~see Fig. 5!. The obtuse tri-angle is energetically higher by 0.07 eV and is separatedan energy barrier of 0.03 eV from the energetically preferacute triangular form.
FIG. 4. Spectrum of the optimal pulse.
les
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1902 J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 K. Sundermann and R. de Vivie-Riedle
The objective is to control the isomerization from thacute to the obtuse conformation by exciting the system frits acute ground state to an electronically excited surfawhere it propagates and is dumped back to the electrground state in the region for the obtuse target state~see Fig.6!.
Taking into account the full dimensionality of the sytem, the PESs have been calculated inCs symmetry, usingfirst a restricted Hartree–Fock calculation for open shell stems, which subsequently has been refined by a multicfiguration self-consistent field~MCSCF! and a configurationinteraction~CI! calculation for the three valence electronThese calculations have been performed using the progMOLPRO25 with a valence double zeta basis. For the electrocally excited state, the 42 A8 state has been chosen, sincepotential shape allows the excited wave packet to evotowards the configuration of the obtuse isomer.
Due to theC2v symmetry of the two isomer states, thsystem is treated in a two-dimensional model using Jaccoordinates. WithR denoting the Li2–Na distance,m1
5mLi2mNa/(mLi2
1mNa) denoting the reduced massLi2–Na, r denoting the Li–Li distance, andm25 1
2mLi denot-ing the reduced mass of Li–Li, the kinetic energy operatogiven by:
FIG. 5. Electronic ground state PES for Li2Na. In the upper left at a Li–Lidistance of 2.75 Å and Li2–Na distance of 3.1 Å, the acute triangular forof the isomer is located, whereas the obtuse form is located at a Ldistance of 3.35 Å and Li2–Na distance 2.57 Å in the lower right.
FIG. 6. One-dimensional section through the PES across the two minprojected onto the Li–Li distance. An intuitive pump–dump cycle, transring the system from the initial acute triangular form to the obtuse onindicated by the arrows.
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me,ic
-n-
.m
i-
e
bi
s
T52\2
2m1
]2
]R22\2
2m2
]2
]r 2 . ~39!
For the pump–dump scheme with two PESs coupled bytransition dipole momentmeg , the system Hamiltonian is:
H~ t !5S T1Vg
00
T1VeD2e~ t !S 0
meg
mge
0 D5H02e~ t !•m. ~40!
The pulse optimization has been performed with the totime limited to T536 000Eh /\'870 fs and employing theshape function of Eq.~4!. The penalty factor has been seta58.0. The complex structure of the resulting laser puwhich transfers 64% of the initial wave packet to the desitarget state together with its spectrum is shown in Fig. 7
The power spectrum of this pulse shows two peaksabout V054.4331022Eh and V154.731022Eh , whichmatches the resonant frequencies between the PES inregion of the initial and target states. As the frequency dtribution of this pulse is very broad and shows two peaks,apply the algorithm of Sec. II B to decompose the pulse itwo subpulses with smaller bandwidths. Following the algrithm we choose a time window@t,t1Dt# with Dt
52200Eh /\;53 fs to get a frequency resolutionDV52p/2200Eh'2.85631023Eh'628 cm21, which corre-sponds approximately to the energy difference of the tfrequency peaks. Within this frequency resolution we det
mine the two frequenciesV0516•DV54.5731022Eh
'10 028 cm21 and V1517•DV54.8631022Eh'10 655cm21 close to the peak frequenciesV054.4331022Eh
'9721 cm21 andV154.731022Eh'10 315 cm21. Follow-ing the algorithm of Sec. II B we perform for each timetP@0,T# a Fourier analysis in the windowtP@t,t1Dt# giv-ing the Fourier coefficientsa16(t) anda17(t) which are used
to extract the pulses corresponding to the frequenciesV0 and
V1 . The decomposition achieved in this way is shownFig. 8, with the original pulse on the left, followed by thtwo subpulses with spectra near the pump and dump freqcies and in the last place the superposition of the two spulses as an approximation of the original pulse. Therefotime-dependent frequency profile for the pump and dufrequencies can be extracted from the envelopes of thepump and dump pulses shown in the middle. The transitprobability of 43% of the reconstructed pulse is below tyield of 64% resulting from the optimal pulse, but signifi
Li
a,-is
FIG. 7. The optimized pulse which transfers 65% from the initial state totarget state, together with its spectrum.
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ese
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1903J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 K. Sundermann and R. de Vivie-Riedle
cantly higher than the yield of 26% which could be achievby intuitive time delayed Gaussian type pump–dump pulas discussed in Ref. 26.
The critical question concerning the effect of the phaf i @Eq. ~27!#, which are difficult to control for such types otwo-color experiments, is investigated by variation of tphase relation [email protected]#. Considering that the time foelectronic dephasing in the gas phase is long compared topulse duration, control of the phase in a two-color expement may be the key problem in an experimental realizaof optimal pump–dump isomerization. Figure 9 showstransition probability as a function of the phase differenDf5f12f0 . The solid line represents the transition proability to the vibrational ground state, whereas the dasline shows the transition probability to any of the obtuisomer vibrational states. The transfer yield stays at a hlevel over three quarters of the interval, but it decreasesmatically for [email protected],2.5#. This shows that phase shiftmight have a big effect on pump–dump processes and tcolor experiments with interfering pulses, even thoughthe example of Li2Na the transfer is stable for most of thpossible phase shifts.
FIG. 8. Pulse decomposition starting in the upper left with the original pufollowed first by the pump pulse and second by the dump pulse. In the uright the reconstruction for a phase differenceDf5f02f150 is shown.The second row shows the spectra corresponding to the electric fields a
FIG. 9. Influence of the phase difference between the pump and the dpulse on the transition rate. The solid line shows the transition to the vitional ground state of the obtuse isomer, whereas the dashed line givetransfer rate into any vibrational state of the obtuse isomer.
Downloaded 21 Sep 2013 to 128.153.5.49. This article is copyrighted as indicated in the abstract.
ds
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IV. CONCLUSIONS
In this paper we propose an approach to quantumnamical control problems, which should have a better chato be experimentally realizable. Our extensions are basethe new iteration method of Zhu, Botina and Rabitz. Talgorithm was modified to take into account constraintsthe form of the driving field. In accordance with experimetal feasibility smooth switch on and off behavior was dmanded. This was obtained by introducing a time-dependpenalty factora into the iterative optimization scheme. Thmethod was tested for different types of control problemsdemonstrate its general applicability.
An example of the vibrational excitation of an OH moecule was chosen for direct comparison of the method wand without additional constraints on the laser field. Thesults showed that for the IR excitation the incorporated cstraints yield nearly Gaussian-type laser pulses with onlyloss of transfer efficiency compared to the unbiasedproach. The envelope and the frequency spectrum of thesulting pulse suggest that it can be realized experimenta
The treatment of the vibrational overtone excitationn50→n55 in K2 shows that it is theoretically possible tselectively populate highly excited vibrational levels viaSEP mechanism. In contrast to a conventional pump–duapproach, an optimal laser field, obtained by algorithmictimal control, is able to hit a preselected target state witprobability up to 96%. At first sight this pulse seems tohighly nonintuitive, but a more detailed investigation revestructures that can be assigned to energy spacings of vtional states and potential energy surfaces involved.
The results for the photoisomerization of Li2Na showedthat optimal control schemes are also applicable to mcomplex two-dimensional systems with two PESs involveThe optimized laser field which controls the isomerizatisuggests a realization of this pulse by a two-color expment. Therefore this pulse was decomposed into a supesition of two single-colored subpulses. This decompositwas achieved by a windowed Fourier transform. Our resshowed that the reconstruction of a two-color pulse keephigh transfer rate, but that experimental efforts also havebe made on the control of phase relations between lapulses, since it can be a critical parameter for the whprocess.
In conclusion, the theoretical results indicate that opmal control schemes, incorporating constraints on the pushape together with a scheme to decompose multicolopulses into a sequence of single-colored subpulses, leaelectric fields closer to experimental feasibility. For mocomplex transition mechanisms, where the target statenot be accessed directly via a Franck–Condon window,optimized laser fields become more complex and seemfirst sight to be highly nonintuitive. However, ana posteriorianalysis of the optimized pulses can on one hand give minsight into the physical processes taking place, as anample of the SEP mediated overtone excitation in K2 dem-onstrates or on the other hand, decomposes a pulse by acation of a windowed Fourier transform into a wecharacterized sequence of subpulses, as was shown fophotoisomerization of Li2Na. This demonstrates that optim
eer
ve.
pa-the
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1904 J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 K. Sundermann and R. de Vivie-Riedle
control algorithms provide a flexible tool not only to detemine driving laser fields to control a system, but alsoanalyze its physical properties.
ACKNOWLEDGMENTS
The authors would like to thank Professor J. Manz~FU-Berlin! for fruitful discussions and financial support of thDFG through Project No. Sfb 337 is gratefully acknowedged. For invaluable help parallelizing the employed pgrams for a Cray T3E the authors thank Professor R. Bising ~University of Utrecht! and Konrad Zuse RechenzentruBerlin ~ZIB! for the computational equipment. K. Sundemann would also like to thank Dr. P. Saalfrank~FU-Berlin!for many discussions and the Max-Planck-Institut fu¨r Quan-tenoptik ~Garching!, for its kind hospitality during his staythere.
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