The dynamic Stark effect due to a strong nonresonant but nonionizing laser field provides a route to quantum control via the creation of novel superposition states. We consider the creation of a field-free "switched" wave packet through adiabatic turn-on and sudden turn-off of a strong dynamic Stark interaction. There are two limiting cases for such wave packets. The first is a Raman-type coupling, illustrated by the creation of field-free molecular axis alignment. An experimental demonstration is given. The second case is that of dipole-type coupling, illustrated by the creation of charge localization in an array of quantum wells

Ivanov, MY

Mottershead, J

Spanner, M

Stolow, A

Sussman, BJ

Underwood, JG

English

UCL Discovery

Switched Wave Packets: A Route to Nonperturbative Quantum ControlJonathan G. Underwood,1 Michael Spanner,1,2 Misha Yu. Ivanov,1 Jeff Mottershead,1Benjamin J. Sussman,1,3 and Albert Stolow1,3,*1Steacie Institute for Molecular Sciences, National Research Council of Canada,100 Sussex Drive, Ottawa, Ontario, Canada K1A 0R62Department of Physics, University of Waterloo,Waterloo, Ontario, Canada N2L 3G13Department of Physics, Queen’s University, Kingston, Ontario, Canada K7L 3N6(Received 19 February 2003; published 4 June 2003)The dynamic Stark effect due to a strong nonresonant but nonionizing laser ﬁeld provides a route toquantum control via the creation of novel superposition states. We consider the creation of a ﬁeld-free‘‘switched’’ wave packet through adiabatic turn-on and sudden turn-off of a strong dynamic Starkinteraction. There are two limiting cases for such wave packets. The ﬁrst is a Raman-type coupling,illustrated by the creation of ﬁeld-free molecular axis alignment. An experimental demonstration isgiven. The second case is that of dipole-type coupling, illustrated by the creation of charge localizationin an array of quantum wells.DOI: 10.1103/PhysRevLett.90.223001 PACS numbers: 32.80.Qk, 42.50.HzThe development of coherent laser sources and thecontrol over optical phase implied has emerged as an im-portant tool for the control ofquantum processes. Brumerand Shapiro as well asTannor and Rice developed coher-ent optical phase control schemes based on the quantuminterference between two or more pathways to a degener-ate target state [1]. Although these coherent controlschemes are often based on a perturbative description inthat the eigenstates of relevance are those of the ﬁeld-free Hamiltonian, schemes using strong resonant or near-resonant laser ﬁelds have also been developed [2].An alternative approach based on the nonlinear interac-tion of an adaptively shaped laser pulse using feedback-learning algorithms to optimize a chosen product wasproposed by Rabitz [3]. This was most successfully dem-onstrated in the strong ﬁeld fragmentation ionization ofpolyatomic molecules [4] and atomic ionization [5].These schemes are nonperturbative in that the ﬁeld-freeHamiltonian may not be pertinent, and they may takeadvantage of many processes including multiphoton reso-nances, dynamic Stark effects, propagation on multiplepotential energy surfaces, and strong ﬁeld ionization-fragmentation processes [6]. In nonionizing laser ﬁelds,strong shaped pulses have been used to control processesin condensed phases [7] and may take advantage of multi-photon resonances and intrapulse interferences [8] anddynamic Stark shifts. Although the underlying mecha-nisms are complex and difﬁcult to elucidate, strong ﬁeldshaped pulse feedback control has been as successful ascoherent quantum interference control. Here we considernonperturbative control schemes which are restrictedonly to the dynamic Stark effect induced by nonresonant,nonionizing laser ﬁelds. By choice of laser frequencyand intensity, complex competing processes might beeliminated.As a ﬁrst step to this nonperturbative control, we studyexperimentally and theoretically the creation of ﬁeld-freeground state superpositions via the adiabatic applicationof a strong nonresonant dynamic Stark coupling which issuddenly truncated, thus forming a wave packet. This‘‘switched wave packet’’ will be very different in natureand composition than that created via resonant excitationor strong ﬁeld ionization. We consider the interactionof a strong nonresonant laser ﬁeld of the form E t  ^ e eE t cos !t  with a quantum system, where ^ e e is thedirection of polarization of the electric ﬁeld of frequency! with a slowly varying envelope E t . The Hamiltonianof the system in the presence of the laser ﬁeld is thenH t  H0   d   E t , where H0 is the ﬁeld-free Hamil-tonian and d is the dipole moment of the system. Theﬁeld-free eigenstatescanoftenbeseparated intobands,asdepicted in Fig. 1, representing the energy bands of amolecule, a set of quantum wells, a Stark or Zeemanmanifold, or some other structure. If the optical fre-quency is greater than   and smaller than  ,t h edynamics may be represented by an effectiveHamiltonian where all other bands have been adiabati-cally eliminated. The equations of motion for the wavefunction   t  Pnan t e iEntjni expressed as a super-position of the initially populated band (N   0) eigen-states are [9]δ N,nN+1,nυ h∆FIG. 1. A quantum system described by quantum numbers Nandn. The laser frequency is chosen to be much larger than theintraband spacing   (described by n) and much smaller than theinterband spacing   (described by the N).PHYSICAL REVIEW LETTERS week ending6 JUNE 2003 VOLUME 90, NUMBER 22223001-1 0031-9007=03=90(22)=223001(4)$20.00  2003 The American Physical Society 223001-1i _ a an t  Xmam t ei!nmtHeffnm t ; (1)where the leading terms of the effective Hamiltonian areHeff t  Vdipole t  VRaman t ; (2)Vdipolenm  t    E t cos !t ^ e e   dnm; (3)VRamannm  t    14jE t j2^ e e    nm !  ^ e e; (4)where dnm   d0n0m is the transition dipole moment and nm !  XN 0Xid0nNidNi0m 1!Ni   ! 1!Ni   ! :(5)In the situation where the intraband dipole moments arelarge, Vdipolenm  t  will dominate and the response followsthe instantaneous electric ﬁeld. Conversely, when theinterband dipole moments are large, VRamannm  t  dominatesandtheresponsefollowstheelectricﬁeld envelopejE t j2.These situations represent the two limiting cases forswitched wave packets, and an example of each will bepresented below.When the interaction is turned on adiabatically [10],each eigenstate of the ﬁeld-free Hamiltonian evolves intoa single instantaneous eigenstate of the laser dressedsystem. The accumulated phase is given by the timeintegral of the instantaneous eigenenergy. Adiabatic be-havior is achieved through sufﬁciently slow turn-on ofthe laser ﬁeld and limiting the intensity to avoid crossingof the dressed states [11]. For the limiting case of aRaman dominated interaction, the application of theadiabatic approximation is clear and the relevant statesand quasienergies are easily found from Eq. (4) [11]. Forthe case of a dipole dominated interaction, the Floquetstates which explicitly include the fast electric ﬁeld os-cillations provide a useful description [12]. The adiabaticcondition here refers to varying the envelope and carrierfrequency slowly enough to avoid transferring populationbetween the Floquet states. In either limiting case, anonadiabatic change in the interaction leads to a super-position of dressed states and beats at frequencies corre-sponding to the differences between quasienergies of thedressed states [10]. This results in Rabi oscillations inthe ﬁeld-free state populations. A sudden truncation ofthe ﬁeld will project the dressed state onto the ﬁeld-freestates forming a wave packet which subsequently evolvesunder the ﬁeld-free Hamiltonian. The content of thiswave packet will depend upon the nature of the stronginteraction, the turn-on time  on, and the switchingtime  sw.We ﬁrst consider the limiting case of a switched wavepacket in which the interaction is dominated by theRaman term, Eq. (4), an example being molecular rotors.An anisotropic linear molecule placed in a linearly po-larized laser ﬁeld experiences a potential of the form [13]V  ;t      14  k    ? jE t j2 cos2   ; (6)where   is the angle between the laser polarization andthe molecular axis,  k and  ? are the polarizabilitycomponents parallel and perpendicular to the molecularaxis, and the term in square brackets corresponds tothe well depth U t . The Hamiltonian is then H   BJ2  V  ;t , where J is the angular momentum operator and Bis the molecular rotational constant. The aligned (pendu-lar) molecule is described by a superposition of ﬁeld-freerotational eigenstates   t  PaJMjJMi. Since V  ;t  isindependent of  , the Hamiltonian is separable in   and , implying conservation of M. We consider here a pulseshape of the formjE t j2  8> <> :E20sin2  t2 on ;t    on;E20cos2   t  on 2 sw ;  on <t   on    sw:(7)For each initial J,a single pendular statewill be created if on is long enough for adiabatic behavior. If  on is tooshort, the state created will be a superposition of pendulareigenstates and the wave packet will have additional timedependence to its alignment [13]. A switched rotationalwave packet calculation is depicted in Fig. 2. A strongﬁeld pulse with  on   15 ps was used to create a switchedwave packet in N2 at 50 K. Alignment is measured by theexpectation value of cos2   .A tt i m etsw, the pendularstate is suddenly projectedwith sw   0:1p sontothefreerotor states, forming a ﬁeld-free rotational wave packet.We present an experimental demonstration of a switchedrotational wave packet. Brieﬂy, a 1:064  m laser pulsewith on of 125 psand  sw of  100 fs wasgeneratedusinga plasma shutter method. A femtosecond Ti:Sa oscillator(80 fs, 800 nm) and a picosecond Nd:YAG oscillator(125 ps, 1:064  m) were electronically synchronizedwith a timing jitter of  1p s . Both oscillator pulseswere ampliﬁed at a repetition rate of 1 kHz. A two stageTi:Sa ampliﬁer system produced 2.5 mJ, 80 fs pulses. The0 5 10 15 20 250102030Well Depth (meV)0 5 10 15 20 250.20.40.6<<cos2θ>>Time (ps)(a) (b) FIG. 2. Switched rotational wave packet in N2 at 50 K.(a) The well depth, U t , created by a laser ﬁeld with  on  15 ps and  sw   0:1p s . (b) The time-dependent alignment ofthe ensemble showing wave packet revivals and ﬁeld-freealignment.PHYSICAL REVIEW LETTERS week ending6 JUNE 2003 VOLUME 90, NUMBER 22223001-2 223001-2picosecond ampliﬁer was a diode pumped Nd:YVO4 laser[14] producing 1 mJ, 125 ps pulses. The 1:064  m beamwas focused into an ethylene glycol jet with an intensityof  5   1011 Wcm  2, below the threshold for break-down. A fraction (600  J) of the 800 nm beam wasfocused to the same spot position and size in the jetwith a peak intensity of  5   1014 Wcm  2, well abovethe threshold for breakdown. The 800 nm triggeredplasma acted as a shutter for the 1:064  m pulse, truncat-ing itstransmission. In our case, the plasmawassustainedby the intense 1:064  m pulse. The transmitted1:064  m light had  on   125 ps and  sw   100 fs,a sseen in Fig. 3(a). Residual scattered 800 nm light copro-pagating with the 1:064  m pulse was removed by di-chroic beam splitters.The 1:064  m switching pulse was focused into a gascell containing CO2 (300 torr) with peak intensity of 1011 Wcm  2 corresponding to a well depth of 0:7cm  1. The alignment induced by the switchingpulse was probed via optical Kerr effect (OKE) rotationof a copropagating weak 80 fs probe pulse, derived fromthe remainder of the 800 nm 80 fs pulse. Prior to the gascell, this probe passed through a Glan-Taylor polarizeroriented at  45  clockwise with respect to the switchingpulse. The 800 nm probe energy transmitted through asecond crossed polarizer (   45 ) was recorded as afunction of time delay between the switching and probepulses ( t   tprobe    sw). The extinction ratio of thecrossed polarizer pair was <10 6.The transmitted probepulse was dispersed through a 0.4 m monochromator anddetected with a photomultiplier tube.The OKE signal intensity is given by I  t   hcos2 i 13   C 2 [15]. The constant C describes theheterodyned signal contribution. Here the polarizationrotation by the cell windows provided the local oscillatorﬁeld for the heterodyne contribution. In Fig. 3(b), weplot the observed signal as a function of time delaybetween the switching and probe pulses. During theiroverlap, there is both an electronic and a nuclear compo-nent to the signal. After the switching of the 1:064  mpulse, there is clear rotational revival structure and asso-ciated molecular axis alignment indicative of wavepacket formation (we note, however, that, due to theroom temperature of the gas sample, the macroscopicalignment is small). In this case, the revival structuredecays in amplitude due to collisional decoherence. Wenote that the extension of the switched rotational wavepacket method using elliptically polarized light providesa convenient route to ﬁeld-free three-dimensional mo-lecular alignment [16].We next consider the limiting case of a switched wavepacket in which the interaction with the laser ﬁeld isdominated by the dipole interaction term, Eq. (3), anexample being a coupled quantum well system. Thesesystems respond to the instantaneous electric ﬁeld andshow dynamics dependent upon the carrier phase of theelectric ﬁeld relative to the envelope. Quantum well sys-tems typically exhibit miniband widths much smallerthan the interband separation. We therefore neglect theeffects of interband transitions, and consider only theintraband dynamics. Within the tight binding (nearestneighbor) approximation, the Hamiltonian for a particlesubject to a periodic potential is [17]H0  X Nn 1Enjnihnj X N 1n 1 n jnihn   1j j n   1ihnj ;(8)where N is the number of wells. This Hamiltonian isexpressed in terms of the Wannier states jni [18] withon-site energy En localized on the well n.  n is thecoupling between the n and n   1 wells. The interactionwith a laser ﬁeld polarized along the interwell directionis HI    E t      edE t PNn 1 njnihnj, where d is thewell spacing. If the interaction is turned on adiabatically,each eigenstate of the ﬁeld-free Hamiltonian will evolveinto a single Floquet state. The quasienergy of the kthFloquet state is given by  k t  J0 edE t =! Ek [17].The degeneracy at the zeros of the Bessel function meansthat the Wannier states jni become eigenstates, electrontunneling is coherently suppressed, and dephasing of theFloquet states is inhibited [17]. This fact has previouslybeen explored in relation to creating charge localizationin such systems [19].We show here that a switched wavepacket generates time-dependent ﬁeld-free charge local-ization. We expand the time-dependent wave functionin terms of Wannier states,   t  Pjaj t jji, and propa-gate numerically the equations of motion of the aj t FIG. 3. Experimental demonstration of switched wave pack-ets. (a) Cross correlation (C.C.) of the switching 1:064  m laserpulse with an 80 fs, 800 nm pulse:  on   125 ps,  sw   110 fs.(b) Optical Kerr effect signal generated by the switching laserpulse from (a), focused (f=30) into 300 torr, 300 K CO2 gas.(c) Expanded region from (b) showing rotational wave packetdynamics and ﬁeld-free alignment in CO2.PHYSICAL REVIEW LETTERS week ending6 JUNE 2003 VOLUME 90, NUMBER 22223001-3 223001-3coefﬁcients. The crossing of the Floquet states restrictsthe value of edE t =! to& 2 if adiabatic behavior is to bemaintained. In Fig. 4, we show the creation of a switchedwave packet in an array of ten wells with equal couplingstrengths. The laser pulse, Fig. 4(a), has the formE t  E t cos !t     ;E t   E0e t2= 2on;t < 0;E0e t2= 2sw;t > 0;(9)where   is the carrier phase of the electric ﬁeld relativetothe envelope. Since the Floquet states are 2 =! periodic,we require  sw   2 =! to maintain the ﬁeld inducedsuperposition. In Figs. 4(b) and 4(c), we plot the induceddipole moment (charge localization) for two choices ofthe absolute carrier phase. During  on, we observe oscil-lations at the ﬁeld frequency ! with only moderate lo-calization as the tunneling splitting is reduced by theoscillating ﬁeld. After  sw, the superposition evolvesunder the ﬁeld-free Hamiltonian and strong time-dependent localization is observed, corresponding to in-duced charge oscillation in the wells. This localization issubstantially larger than during the presence of the laserﬁeld and is seen to depend upon the carrier phase. Thisexample is an illustration of a quantum control scenariorequiring absolute carrier-envelope phase control [20].In this Letter, we have explored the use of a strongdynamic Stark interaction with a quantum system for thecreation of novel ﬁeld-free superposition states. The evo-lution of these ﬁeld-free switched wave packets containsinformation about the nature of the strong interaction aswell as providing ﬁeld-free localization. This nonreso-nant, nonperturbative, but nonionizing regime of inter-action could provide new routes to the control ofmolecular dynamics via the time-dependent modiﬁcationof an effective Hamiltonian. Control is exerted by revers-ibly modifying the propagator during propagation, asopposed to using only the ﬁeld-free propagator or themultiple propagators implicit in strong ﬁeld ionization.*Electronic address: albert.stolow@nrc.ca[1] P. Brumer and M. Shapiro, Annu. Rev. Phys. Chem. 43,257 (1992), and references therein; R.J. Gordon and S. A.Rice, Annu. Rev. Phys. Chem. 48, 601 (1997), and refer-ences therein.[2] K. Bergmann et al., Phys. Rev. Lett. 70, 1003 (1998);A. Shnitman et al., Phys. Rev. Lett. 76, 2887 (1996);M. N. Kobrak and S. A. Rice, J. Chem. Phys. 109,1(1998).[3] R.S. Judson and H. Rabitz, Phys. Rev. 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(a) Laser electric ﬁeldedE0=!   1:5 with  on   30 cycles,  sw   0:1 cycles.(b) Induced dipole moment for carrier phase     0:0.(c) Induced dipole moment for carrier phase      =4.PHYSICAL REVIEW LETTERS week ending6 JUNE 2003 VOLUME 90, NUMBER 22223001-4 223001-4

Switched wave packets: A route to nonperturbative quantum control

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