Quantum Interferences and Their Classical Limit in Laser Driven Coherent Control Scenarios

Chemical Physics 370 (2010) 143–150

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Chemical Physics

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Quantum interferences and their classical limit in laser driven coherent control scenarios

Ignacio Franco *,1, Michael Spanner 2, Paul Brumer

Chemical Physics Theory Group, Department of Chemistry, Center for Quantum Information and Quantum Control, University of Toronto, Toronto, ON, Canada M5S 3H6 article info abstract

Article history: The analogy between Young’s double-slit experiment with matter and laser driven coherent control Received 24 September 2009 schemes is investigated, and shown to be limited. To do so, a general decomposition of observables in In ﬁnal form 21 February 2010 the Heisenberg picture into direct terms and interference contributions is introduced, and formal quan- Available online 25 February 2010 tum-classical correspondence arguments in the Heisenberg picture are employed to deﬁne classical analogs of quantum interference terms. While the classical interference contributions in the double-slit Keywords: experiment are shown to be zero, they can be nonzero in laser driven coherent control schemes and lead Coherent control to laser control in the classical limit. This classical limit is interpreted in terms of nonlinear response Quantum-classical correspondence theory arguments. Quantum interference Heisenberg dynamics Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction tum phase on the system. Hence, by varying the laser phases one can vary the magnitude and sign of quantum interference Identifying physical phenomena as quantum, as distinct from contributions. classical, has become of increasing interest due to modern develop- In this paper we examine the accuracy and generality of this ments in ‘‘quantum technologies”, such as quantum information analogy, with the goal of understanding whether laser driven and computation, and quantum control [1,2]. Significantly, a coherent control is an explicitly quantum phenomenon or whether variety of new results are emerging which enlighten as to what the same physical process is manifest classically as well. We do so constitutes an inherently quantum feature. For example, several using a framework, developed below, based upon the Heisenberg attributes often regarded as quantum in nature, such as the no- representation in quantum mechanics, an approach particularly cloning theorem, the Born rule for probabilities and a Hilbert space well suited to examine the analogy between quantum mechanical structure for the dynamics, have been found to be features of clas- and classical processes because it deals with dynamical variables sical mechanics as well [3–6]. directly instead of superposition states and probability amplitudes Here we focus upon the role of quantum effects in the laser dri- [10–12]. ven coherent control of atomic and molecular processes. Many of The question posed herein is somewhat subtle and requires these processes have long been understood in terms of quantum clarification. Specifically, our focus is on whether various coherent interference effects exactly analogous to the ones in the double-slit control scenarios qualitatively require a quantum description, or experiment with matter. That is, the current view on coherent con- whether they can qualitatively be described by classical mechanics. trol phenomena is that the property that makes possible the active If the former is the case, then we will call the phenomenon inher- control of molecular dynamics is quantum interference, which ently quantal in nature. This is in contrast to the circumstance arises from the fact that the behavior of material particles at where classical mechanics provides a qualitative description but molecular scales can be described in terms of waves [2,7–9]. The fails to quantitatively describe observed results. For example, with- tool that permits practical exploitation of quantum interference in this definition, tunnelling is a quantum process, but light for control of molecular dynamics is the laser, as it imparts a quan- absorption by molecules is not [13,14]. In addition to shedding light on the nature of laser control, the question posed is also of relevance practically. Specifically, if a phe- * Corresponding author. nomenon is indeed describable classically, then it is sensible to E-mail address: [email protected] (I. Franco). consider modelling the process in the classical limit using classical 1 Present address: Department of Chemistry, Northwestern University, Evanston, IL dynamics. This is not the case if it is a pure quantum phenomenon. 60208-3113, United States. 2 Present address: Steacie Institute for Molecular Sciences, National Research Nonetheless, there is no implication that if a process can indeed be Council of Canada, Ottawa, ON, Canada K1A 0R6. defined classically that a classical computation will provide an ade-

0301-0104/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2010.02.016 144 I. Franco et al. / Chemical Physics 370 (2010) 143–150 quate quantitative result. Further, the resulting classical analogues, rier has no slits and that at time t ¼ 0 two slits are opened. The if they exist, offer alternative perspectives on well established system is described by the Hamiltonian coherent control schemes. Lastly, decoherence effects on laser con- b b b trol could be analyzed from a completely classical perspective [15] HðtÞ¼H0 þ V ðtÞ; ð3Þ and may well offer insight into novel ways to combat it. b where H0 is the Hamiltonian of the system plus barrier and This problem may be regarded as ill-posed since, after all, nat- b b b ure is quantum and even the stability of matter requires a quantum V ðtÞ¼V 1ðtÞþV 2ðtÞð4Þ description. Further, coherent control schemes often employ man- b describes the opening of the two slits, where V n describes opening ifestly quantum phenomenon such as electronic transitions. Again, b slit n. Under the influence of HðtÞ the system evolves into a super- we do not question whether a quantitative account of laser control position state scenarios requires a quantum treatment. However, from a qualita- b tive perspective the question still remains: is the wave character of jWðtÞi ¼ UðtÞj/0i; ð5Þ matter necessary for the appearance of laser-induced interferences b determined by the evolution operator UðtÞ. Any interference effects that can be used to control molecular dynamics? that may arise during the experiment are completely characterized This manuscript is organized as follows: In Section 2 we discuss b by UðtÞ and by the initial state. the formal analogy between the double-slit experiment and laser At this point it is convenient to express the evolution operator driven coherent control schemes. For this we present a general as decomposition of observables in the Heisenberg picture into direct terms and interference contributions, as well as a procedure to b b b UðtÞ¼U 0ðtÞU IðtÞ; ð6Þ isolate the contributing terms. Then, in Section 3 we study such b b a decomposition in the classical limit and use it to define classical where U 0ðtÞ¼expðiH0t=hÞ characterizes the slit-free evolution of b analogues of interference terms. With it, we extract interference the system while U IðtÞ quantifies the influence of the slitsh on the b d b contributions in a classical double-slit experiment and a classical dynamics. Since UðtÞ satisfies the Schrödinger equation ih dt UðtÞ b b b laser control scenario. It is shown that the classical interference ¼ HðtÞUðtÞ, the dynamics of U I is governed by contributions in laser control can be nonzero and give origin to d b b b ih U ðtÞ¼V IðtÞU ðtÞ; ð7Þ classical laser control. Our main results as well as a qualitative dt I I discussion of the nature of the ‘‘interference in the classical limit” are provided in Section 4. where b I b y b b b y b b y b I b I V ðtÞ¼U 0 V ðtÞU 0 ¼ U 0 V 1ðtÞU0 þ U 0V 2ðtÞU0 V 1ðtÞþV 2ðtÞð8Þ 2. Quantum interference is the slit-opening component of the Hamiltonian in Interaction picture. Eq. (7) can be solved iteratively, yielding the well-known 2.1. The double-slit experiment series [20]

X1 The usual example used to explore quantum interference phe- b b ðmÞ U IðtÞ¼ U I ðtÞ; ð9Þ nomena is Young’s double-slit experiment with matter. In it, parti- m¼0 cles drawn, e.g., from a thermal distribution, are directed, at low b ð0Þ ^ where U I ðtÞ¼1 is a zeroth-order term and where intensity, at a barrier with two slits. A detector is placed at some Z t distance behind the slits and the position of those particles that b ðmÞ i b b ðm1Þ U t dt0 V I t0 U t0 m > 0 10 pass through the slits and arrive at the detection screen are re- I ð Þ¼ ð Þ I ð Þð ÞðÞ h 0 corded. In the classical case each particle can pass either through b b the upper or lower slit and there is no correlation between the describes the mth order contribution to U I, induced by V ðtÞ. two possibilities. The resulting probability distribution at the The above equations suggest an exact partition of the evolution operator that exposes the interference phenomenon. Speciﬁcally, detection screen PðyÞ is therefore the sum of the probability distri- the evolution operator can be written as butions PnðyÞ that would have been obtained if the experiment had been carried out with only slit n ¼ 1 or slit n ¼ 2, open, b b b b b UðtÞ¼U 0ðtÞþU 1ðtÞþU 2ðtÞþU 12ðtÞ; ð11Þ b PðyÞ¼P1ðyÞþP2ðyÞ: ð1Þ where U 0ðtÞ is the slit-free evolution operator, the term X1 By contrast, in quantum mechanics there is an additional contribut- b b b ðmÞ U nðtÞ¼U 0ðtÞ U I;n ðtÞðn ¼ 1; 2Þ ing term to PðyÞ; P ðyÞ, to give 12 m¼1 Z ð12Þ t b ðmÞ i 0 b I 0 b ðm1Þ 0 PðyÞ¼P1ðyÞþP2ðyÞþP12ðyÞ: ð2Þ with U I;n ¼ dt V nðt ÞU I;n ðt Þðm > 0Þ; h 0

The term P12 arises from quantum interference effects, may be po- is the additional contribution that would have been obtained if only b b sitive or negative, and is responsible for the appearance of fringes in slit n was open (i.e. if V ðtÞ¼V nðtÞ), while the detection screen that cannot be accounted for through classical b b b b b considerations. U 12ðtÞ¼UðtÞðU 0ðtÞþU 1ðtÞþU 2ðtÞÞ ð13Þ Several compelling accounts of the double-slit experiment exist represents all contributions to the evolution operator that explicitly (see, for example, Refs. [16–19]). Below, we choose to describe it by b b b depend on the existence of both V 1 and V 2. When UðtÞ is applied to considering the actual dynamics of the underlying scattering pro- the initial state it produces a superposition state of the form cess in the Heisenberg picture. As will become evident, this per- b b b b spective is particularly suitable to draw a parallel with coherent jWðtÞi ¼ ðU 0ðtÞþU 1ðtÞþU 2ðtÞþU 12ðtÞÞj/0i control phenomena and to study quantum interferences in the ¼jw ðtÞiþjw ðtÞiþjw ðtÞiþjw ðtÞi: ð14Þ classical limit. 0 1 2 12

Suppose then that the particles to be diffracted are initially pre- Here jw0ðtÞi represents that part of the wavefunction that gets pared in some state j/0i. For times t < 0 we imagine that the bar- reﬂected by the barrier, jwnðtÞi describes the part that passes I. Franco et al. / Chemical Physics 370 (2010) 143–150 145

through slit n, and jw12ðtÞi contains any additional contributions 2.2.1. N vs. M control b that involve the two slits. Consider then a molecular system with Hamiltonian H0 inter- In light of Eq. (11) it follows that the evolution of any observable acting with a radiation ﬁeld EðtÞ in the dipole approximation. b H b y b b b b b O ðtÞ¼U ðtÞOUðtÞ in the Heisenberg picture admits the The total Hamiltonian of the system is HðtÞ¼H0 þ V ðtÞ, where decomposition b V ðtÞ¼l^ EðtÞð20Þ b H b H b H b H b H O ðtÞ¼O0 ðtÞþO1 ðtÞþO2 ðtÞþO12 ðtÞ: ð15Þ is the radiation-matter interaction term and l^ is the system’s dipole operator. For simplicity we focus on the case where each competing Here dynamical process is determined by a different frequency compo-

b b y b b nent of the radiation ﬁeld. This physical situation encompasses, OHðtÞ¼U OU ð16Þ 0 0 0 for example, the 1 vs. 2 and the 1 vs. 3 coherent control schemes in which x þ 2x and x þ 3x ﬁelds are used to photoexcite the sys- describes the dynamics of the observable in the absence of slits, the tem, respectively. These control scenarios are two important exam- term ples of the general class of N vs. M control schemes [2].

b H b y b b b y b b b y b b When a two-color laser EðtÞ¼1E1ðtÞþ2E2ðtÞ is incident on O ðtÞ¼U OU n þ U OU 0 þ U OU n ð17Þ n 0 n n the system the radiation-matter interaction term can be written b b b quantiﬁes the additional contribution to the observable if only slit n as V ðtÞ¼V 1ðtÞþV 2ðtÞ, where is open, while b V nðtÞ¼l^ nEnðtÞðn ¼ 1; 2Þð21Þ

b H b y b b b y b b b y b y b y b b O12ðtÞ¼U 1 OU 2 þ U 2 OU 1 þðU 0 þ U 1 þ U 2ÞOU 12 describes the interaction of the system with the nth component of b y b b b b b y b b the field. As before, the evolution operator admits the decomposition þ U 12 OðU 0 þ U 1 þ U 2ÞþU 12 OU 12 ð18Þ b b b b b UðtÞ¼U 0ðtÞþU 1ðtÞþU 2ðtÞþU 12ðtÞ; ð22Þ describes the interference contributions. b b b The structure of Eq. (15) is characteristic of situations that where U 0 ¼ expðiH0t=hÞ is the field-free evolution operator, U n is b involve interference. It contains two direct terms OH and an inter- the correction term that would have been obtained if the system b n b b H evolved solely under V nðtÞ, while U12 represents contributions that ference contribution O12. Note that Eq. (2) is a particular case of Eq. (15) obtained by demanding that the relevant operator measures appear when both colors are present. In analogy with Eq. (14), when b the position y of the particles on a detector located at x ¼ L > 0, UðtÞ acts on the initial state it produces a superposition state of the b b i.e. O ¼ P ¼jx ¼ Lijyihyjhx ¼ Lj. If the particles are initially pre- form b pared to the left of the barrier x < 0 then hPHi¼0 as particles can- b b b b 0 jWðtÞi ¼ ðU ðtÞþU ðtÞþU ðtÞþU ðtÞÞj/ i not cross the barrier when the two slits are closed. Hence, from Eq. 0 1 2 12 0 (15) it follows that ¼jw0ðtÞiþjw1ðtÞiþjw2ðtÞiþjw12ðtÞi: ð23Þ Here w t represents the field-free evolution of the system, w t Pðy; tÞ¼P1ðy; tÞþP2ðy; tÞþP12ðy; tÞ; ð19Þ j 0ð Þi j nð Þi the excitation produced by the nth component of the field, while b where P ðy; tÞ¼hPHi is the probability that the particle crossed the jw ðtÞi contains any additional corrections that explicitly involve n n 12 b b barrier through slit n and arrived at the screen at position y and both V 1 and V 2. time t, while P ðy; tÞ quantifies the interference between the two As in the double-slit experiment, the Heisenberg dynamics of 12 b possibilities. The angle brackets here and below denote an average any observable O admits the decomposition over the initial wavefunction. In the actual experiment only an inte- b H b H b H b H b H grated signal is recorded and Eq. (2) is recovered by integrating Eq. O ðtÞ¼O0 ðtÞþO1 ðtÞþO2 ðtÞþO12ðtÞ: ð24Þ (19) over time. b b Here OH describes the field-free dynamics, the direct terms OH An important aspect of the partitioning in Eq. (15) is that by using 0 n quantify contributions that arise when only the nth color of the field it one can design an experimental procedure to isolate interference b is turned on, while OH represents the interference contributions contributions. In a first experiment, the two slits are closed and the 12 b H that are at the heart of laser control phenomena. Further, using observable of interest is measured, giving hO0 ðtÞi. Subsequently, two additional experiments are performed in which only slit n is Eq. (24) one can design a set of experiments to isolate the interfer- b H b H b H b H ence contributions. In a first experiment the system is allowed to opened and hO ðtÞi measured, giving hOn ðtÞi ¼ hO ðtÞi hO0 ðtÞi. b evolve in the absence of lasers, giving OH t . Then two additional In a final experiment the dynamics of the observable is followed h 0 ð Þi with both slits open. The contributions due to interference are then experiments with either laser, n = 1 or 2, on are performed from b H b H b H b H b H b H which hOn ðtÞi can be extracted. Last, by subjecting the system to given by hO12ðtÞi ¼ hO ðtÞiðhO0 ðtÞiþhO1 ðtÞi þ hO2 ðtÞiÞ. b both laser fields and measuring hOHðtÞi one has sufficient information to isolate the interference contributions. 2.2. Laser control

The above analysis is general and can be used to study interfer- 2.2.2. Bichromatic control b ences induced by any perturbing potential V ðtÞ with two or more The bichromatic control scenario [2,7] represents yet another components. Through the set of experiments described above, or class of control schemes, with formal equivalence to pump–dump slight variations of them, correlations between competing dynam- schemes and to the 2 vs. 2 control scenario [2]. In this scenario the b ical processes embedded within V ðtÞ can be quantified. Consider system is first prepared in a superposition of two Hamiltonian b then these ideas applied to laser control scenarios in which V ðtÞ, eigenstates. This initial superposition is subsequently photodisso- instead of opening slits on a barrier, describes the interaction of ciated to a given energy in the continuum using a two-color laser the system with components of a laser field. The procedure is com- field with frequency components x2 and x3. The frequencies of pletely parallel to the one presented above. Nevertheless, we re- the dissociating pulse are selected such that jx3 x2j matches peat aspects of it in order to stress the analogy between the two exactly the Bohr transition frequency between the states involved physical situations. in the initial superposition. 146 I. Franco et al. / Chemical Physics 370 (2010) 143–150

Consider this scenario in the Heisenberg picture. For this we each symbol OWðx; pÞ defines an operator in Hilbert space. Using suppose that the preparation and dissociation stages occur at Groenewold’s identity [23] for the Wigner transform of a product two different times, so that of operators: 2 0 13 b b b ! ! ; ; ; > > : Uðt t0Þ¼U dissðt t1ÞU prepðt1 t0Þðt t1 t0Þ ð25Þ b b ih @ @ @ @ ðABÞ ¼ A ðx; pÞ exp 4 @ A5B ðx; pÞ; ð31Þ b b W W 2 @x @p @p @x W Here U prepðt1; t0Þ and U dissðt; t1Þ are the evolution operators during preparation and dissociation, respectively. The initial superposition state is achieved by exciting the system between times t0 and t1 where the arrows indicate the direction in which the derivatives with a laser (field 1) resonantly coupling the two desired levels. act, Eq. (29) becomes 8 0 19 The dynamics during the process is characterized by < ! ! = d b b 2 h @ @ @ @ b b b b ðOHÞ ¼ðOHÞ sin @ A ðHHÞ ; ð32Þ U prepðt1; t0Þ¼U 0ðt1; t0ÞþU 1ðt1; t0Þ; ð26Þ dt W W h :2 @x @p @p @x ; W b b where U 0ðt1; t0Þ is the field-free evolution operator, while U 1ðt1; t0Þ describes the excitation process induced by field 1. During dissoci- which provides a phase space description of the Heisenberg dynamics. ation (for times t > t1) the evolution operator is given by Eq. (32) has a well-defined classical ðh ¼ 0Þ limit provided that b b b b b b H b H U dissðt; t1Þ¼U 0ðt; t1ÞþU 2ðt; t1ÞþU 3ðt; t1ÞþU 23ðt; t1Þ: ð27Þ ðO ÞW and ðH ÞW are semiclassically admissible [10]. That is, that b one can express these functions in a series Here U nðt; t1Þ with n = 2 or 3 represents the influence of the nth dis- b X1 n sociating field on the dynamics, while U23ðt; t1Þ quantifies any addi- h ðOHÞ ðx; p; tÞ¼Ocðx; p; tÞþ o ðx; p; tÞð33Þ tional corrections to the evolution that depend on the presence of W n! n both fields. Using this partitioning, one can decompose the Heisen- n¼1 berg evolution of any observable as: that is asymptotically regular at h ¼ 0 for finite times. If this is the b H b H b H b H b H b H b H case, by letting h ¼ 0 in Eq. (32) one recovers Hamilton’s equations O ðtÞ¼O0 ðtÞþO1 ðtÞþO2 ðtÞþO3 ðtÞþO12ðtÞþO13ðtÞ of motion b H b H þ O23ðtÞþO123ðtÞ: ð28Þ d c c c b H b H O ðx; p; tÞ¼fO ðx; p; tÞ; H ðx; p; tÞg; ð34Þ Here O0 ðtÞ is the field-free evolution, On ðtÞ with n ¼ 1; 2; 3 is the dt term that would have been obtained if only field n was turned on, c @f @Hc @f @Hc c b H where ff ; H g¼@x @p @p @x is the Poisson bracket and O ðx; p; tÞ O ðtÞ represents the interferences between pairs of competing c b nm and H ðx; p; tÞ are regarded as the classical counterparts of OH and dynamical processes onset by pulses n and m ðn – mÞ, while b b HH, respectively. OH ðtÞ is the interference contributions that depend on the three b b 123 Now, for ðOHÞ and ðHHÞ to satisfy Eq. (33) only requires that fields. As this scenario involves three fields, eight experiments are W W b ð^xHÞ and ðp^HÞ are semiclassically admissible. This property is an required to isolate all contributing terms to hOHðtÞi. W W important consequence of Eq. (31) which shows that the product of two semiclassically admissible operators is also semiclassically b b 3. Interferences in the classical limit admissible, and holds provided that O and H are expressible in terms of products of the position ^x and momentum p^ operators. A crucial property of the partitions introduced in Section 2 is At the initial time common observables such as position, b H that each of the contributing terms entering into O ðtÞ has a momentum, angular momentum and the Hamiltonian are semi- well-defined classical ðh ¼ 0Þ limit. In this section we exploit this classically admissible. Further, the leading term in the h-expansion property to define classical analogues of quantum interference coincides with the familiar quantities of classical mechanics. terms. Central to this analysis is the quantum-classical correspon- Hence, the main assumption in this quantum-classical correspondence principle in the Heisenberg picture. We thus begin by study- dence principle is that semiclassical admissibility is stable under ing the classical limit of the Heisenberg dynamics [10,11],asit time evolution for the position and momentum operators. If this permits identifying the h ¼ 0 limit of quantum observables in the is the case, then one can identify the h ¼ 0 limit of an observable Heisenberg picture with the phase-space flow of the corresponding in the Heisenberg picture with the phase-space flow of the corre- classical observable. sponding classical observable. Recent work by Osborn [24] has established conditions for the stability of Eq. (33) under time 3.1. Heisenberg dynamics in the classical limit evolution. b Note that this formal correspondence between quantum and Consider an observable O in the Schrodinger picture with no b classical observables does not apply to the density matrix by itself. explicit time dependence. The dynamics of O in the Heisenberg The Wigner representation for pure state density matrices is singu- picture is governed by the Heisenberg equations of motion lar [10] at h ¼ 0 and hence it is not semiclassically admissible even d b 1 b b at the initial time. OHðtÞ¼ ½OHðtÞ; HHðtÞ: ð29Þ dt ih b The interest here is to study OHðtÞ, and the above differential equa- 3.2. Classical partitions of observables tion for operators, in the classical limit. For this it is convenient to b represent OHðtÞ and Eq. (29) in the phase space of c-number posi- Insofar as the above correspondence rule holds, one can apply tion x and momentum p variables, and then take the classical the analysis in Section 2 to study correlations between competing b (h ¼ 0) limit. A phase space picture of OHðtÞ can be obtained using dynamical processes in classical systems. Consider, for example, the Wigner transform [21,22], defined as the general partitioning of observables in Eq. (15): Z b H b H b H b H b H ipv=h 1 b 1 O ðtÞ¼O0 ðtÞþO1 ðtÞþO2 ðtÞþO12ðtÞ: ð15Þ OWðx; pÞ¼ dv e x þ vjOjx v : ð30Þ 2 2 b If O is semiclassically admissible and the four Hamiltonians deter- b b b Through this transformation each quantum operator O is repre- mining the different contributions to the dynamics (H0; H0þ b b b b b b sented by a unique phase space function OWðx; pÞ and, conversely, V 1; H0 þ V 2 and H0 þ V 1 þ V 2) are also semiclassically admissible I. Franco et al. / Chemical Physics 370 (2010) 143–150 147 with nonzero classical counterpart then each term in the above 0.08 equation has a well defined classical limit of the form

OcðtÞ¼Oc ðtÞþOc ðtÞþOc ðtÞþOc ðtÞ; ð35Þ 0 1 2 12 0.06 where the different contributions to OcðtÞ are the classical counterparts of the corresponding terms in Eq. (15). Using the example of c the double-slit experiment, O0 describes the classical flow of the 0.04 c observable in the absence of slits, On quantifies the additional con- c tributions to the observable if only slit n is open, while O12 quantifies any classical correlations between the two possibilities. The 0.02 latter term is the classical limit of the quantum interference contributions to the observable. By analogy with the quantum language we will refer to it as the classical interference term. 0 It is important to note that this connection between quantum and classical mechanics makes no reference to the initial state. In a quantitative comparison, quantum contributions arising from −0.02 the dynamics of observables and from the quantum nature of the −4 −3 −2 −1 0 1 2 3 4 initial state need to be taken into account. However, from a qualitative perspective this aspect is immaterial; for if matter interfer- Fig. 2. Probability density distribution at the detection screen of the double-slit ence effects are in principle the origin of laser control then under experiment schematically shown in Fig. 1 after launching 300,000 particles. The ’s completely classical conditions no laser control should be possible, correspond to the case in which the two slits are open. The ’s and ’s to the one in c which only the lower or upper slit is open, respectively. The broken line represents i.e. O12ðtÞ should be zero. the classical interferences in the observable Pc ðyÞ. Hence, in this analysis classical averages are obtained by 12 weighting OcðtÞ by an ensemble of classical initial conditions q ðx; pÞ: c Fig. 2 shows the results for the remaining three experiments and c c the resulting classical interference terms in the observable Pc ðyÞ hO ðtÞic ¼ TrfO ðtÞqcðx; pÞg: ð36Þ 12 (broken line). We observe that within statistical errors the classical c The classical interference contributions to the observable hO12ðtÞic particles do not exhibit any kind of correlations between the two can be isolated and quantified by repeating the set of experiments classical possibilities. That is, as expected, the probability density described above for the quantum case but by using a classical initial observed at the screen is merely the sum of the probabilities of distribution that evolves according to the classical equations of particles going through each of the slits. The observed classical motion. We now consider two examples: the double-slit experi- behavior is a consequence of the fact that classical particles do ment and the 1 vs. 2 laser control scenario. If the analogy between not have a wave character. these two physical situations is faithful, one would expect the interference contributions in both cases to be zero. 3.2.2. Example: 1 vs. 2 control In this scenario a spatially symmetric system is driven with an 3.2.1. Example: the double-slit experiment x þ 2x field. In quantum mechanics the interference between the Fig. 1 shows the setup of the classical double-slit numerical one-photon absorption induced by the 2x-component and the experiment. The observable of interest is the distribution of parti- two-photon absorption process induced by the x-component c c cles at the detection screen P ðyÞ. In this example, P0ðyÞ¼0 since results in phase-controllable transport. We now study this scenario the particles cannot cross the barrier when the two slits are closed. from a completely classical perspective. As a system we choose an ensemble of noninteracting particles of mass m and charge q initially confined in the one-dimensional

−0.2

−0.4

−0.6

−0.8

−1

Fig. 1. Scheme of the double-slit experiment. Particles are directed at the barrier −8 −6 −4 −2 0 2 4 6 8 starting from a Gaussian distribution centered at the black dot in the ﬁgure

ðx0; y0Þ¼ð0; 0Þ with spatial width Dx ¼ Dy ¼ 20 lm. The mean velocities of the 0 1 0 1 0 initial state are vx ¼ 200 m s and vy ¼ 0ms with a spread of Dvx ¼ Fig. 3. Bounding potential VðxÞ¼E0= coshðxÞ for the one-dimensional model 0 1 Dvy ¼ 7:5ms . The barrier is 0.2 m in width. The slits are 0.15 m long and system. The initial state is an ensemble of initial conditions with constant energy 0.35 m apart. E ¼0:95E0. Here x0 is some characteristic length. 148 I. Franco et al. / Chemical Physics 370 (2010) 143–150

potential schematically shown in Fig. 3. As an initial distribution the relative phase 2/x /2x between the two laser components, we choose a microcanonical ensemble with energy E ¼0:95E0, i.e., one of the control variables. Hence the calculations presented where E0 is the minimum of the potential. At time t ¼ 0 the sys- here are averaged over the carrier envelope phase a 2½0; 2pÞ of tem is allowed to interact with an x þ 2x Gaussian laser pulse, so the laser field, done by randomly selecting an a for each member that the components ðn ¼ 1; 2Þ of the radiation-matter interaction of the ensemble. term are given by As an observable we choose the velocity distribution PcðvÞ of "# the photoejected particles after the laser field is turned off. Any 2 c t Tc rectification effects should manifest as anisotropy in P ðvÞ. Using V nðtÞ¼qxnx cosðnxt þ /nx þ naÞ exp : ð37Þ Tw the decomposition in Eq. (35),

c c c c c P ðvÞ¼P0ðvÞþP1ðvÞþP2ðvÞþP12ðvÞ; ð38Þ The pulse haspffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi width Tw ¼ 100t0 and is centered at time Tc ¼ 3Tw where t0 ¼ mx0=E0 is a characteristic time and x0 a characteristic c where P0ðvÞ is the field-free ensemble average of the observable, length. We choose x ¼ 0:66E0=ðqx0Þ; 2x ¼ 0:33E0=ðqx0Þ and c PnðvÞ the additional contributions induced by the nx-component xt0 ¼ 0:45. Interest is in isolating effects that depend solely on c of the field, and P12ðvÞ the classical interferences. In this case, c P0ðvÞ¼0 as no photodissociation can occur in the absence of the field. Fig. 4 shows the results of the experiment when only one of the field’s components is used. The resulting probability density c 1.25 PnðvÞ exhibits a right/left symmetry and no net currents are gener- ated. The situation is completely different when both the x and the 1.00 2x components are applied (Fig. 5). The x þ 2x field generates anisotropy in PcðvÞ. Further, the degree and magnitude of the effect 0.75 can be manipulated by varying the relative phase between the two frequency components of the beam. The resulting classical interfer- 0.50 ence terms are shown in gray. In stark contrast to the double-slit experiment, the interference between the two laser-induced 0.25 dynamical processes is nonzero and is the origin of the rectification effect. Moreover, Pc ðvÞ can take positive or negative values and is 0.00 12 −2.00 −1.50 −1.00 −0.50 0.00 0.50 1.00 1.50 2.00 phase-controllable, just as its quantum counterpart. Supplementary analyses and additional work on this control scenario are worth noting. Ref. [11] provides an analytical treat- Fig. 4. Velocity probability density PcðvÞ for particles dissociated by the ment of the 1 vs. 2 photon case for a quartic oscillator, and a -component Pc (black line) and the 2 component Pc (gray line) of the x 1ðvÞ x 2ðvÞ detailed analysis of the classical limit. Ref. [25] shows how the la- laser field. Here v0 ¼ x0=t0 is a characteristic velocity. ser-induced symmetry breaking effect can be accounted for in both

4.00 4.00 φ −2φ =0 φ −2φ =π/2 3.00 2ω ω 3.00 2ω ω

2.00 2.00

1.00 1.00

0.00 0.00

−1.00 −1.00 −2.00 −1.50 −1.00 −0.50 0.00 0.50 1.00 1.50 2.00 −2.00 −1.50 −1.00 −0.50 0.00 0.50 1.00 1.50 2.00

4.00 4.00 φ −2φ =π φ −2φ =3π/2 3.00 2ω ω 3.00 2ω ω

2.00 2.00

1.00 1.00

0.00 0.00

−1.00 −1.00 −2.00 −1.50 −1.00 −0.50 0.00 0.50 1.00 1.50 2.00 −2.00 −1.50 −1.00 −0.50 0.00 0.50 1.00 1.50 2.00

c c Fig. 5. Velocity probability density P ðvÞ for particles dissociated by an x þ 2x field (black lines) for different relative laser phases. The interference contributions P12ðvÞ in each experiment are plotted in gray. I. Franco et al. / Chemical Physics 370 (2010) 143–150 149 quantum and classical mechanics through a single symmetry anal- excitation with an x þ 2x pulse. During the interaction of the sys- ysis of the equations of motion. Recent further studies [26] propose tem with a radiation field the photoinduced dipoles hli are and analyze a specific optical lattice realization of this control ð1Þ ð2Þ 2 ð3Þ 3 scenario, where the approach to the classical limit is expected to hli¼v EðtÞþv E ðtÞþv E ðtÞþ ð41Þ be observable experimentally. Here, for simplicity, the response has been expressed in the adia- batic limit [28], where hli depends only on the instantaneous val- 4. Summary and discussion ues of the field amplitudes. Symmetry breaking in the response is characterized by terms in the polarization that survive time-averag- Our discussion has focused on coherent control scenarios ing. The time-average of Eq. (41) is given by involving multiple laser-induced dynamical pathways. In doing hli¼ ð1ÞEðtÞþ ð3ÞE3ðtÞþ; ð42Þ so we have considered the analogy between the double-slit exper- v v iment and laser driven multiple-path control scenarios. In both where vð2Þ ¼ 0 for symmetric systems, and where the overbar cases the system is subject to two or more competing dynamical denotes time-averaging. For x þ 2x fields, or any other AC field, processes and the quantum evolution results in a superposition EðtÞ¼0, and the first term vanishes. However, the third-order term state with exactly analogous analytical structure [recall Eqs. (14) in the response is not necessarily zero. Rather, for the x þ 2x field and (23)]. Further, the interference contributions that arise from in Eq. (39), it is given by such superpositions are the origin of the characteristic fringes in the double-slit experiment and of the laser control scenarios. How- 3 hli¼vð3Þ 2 cosð2/ / Þ: ð43Þ ever, we have also shown that these two examples are fundamen- 4 x 2x x 2x tally different: in the classical limit the interference contributions in the double-slit experiment are zero, while the interferences in That is, the nonlinear response of the system to an x þ 2x field laser control scenarios may survive and are the origins of classical mixes the frequencies and harmonics of the incident radiation laser control. What then is the difference between them? We pro- and leads to the generation of a phase-controllable zero-harmonic vide several perspectives, below. (dc) component in the response, i.e. the controlled symmetry break- First, consider the issue from the viewpoint of the nature of the ing. This term arises because the net response of the system to the interference term. For example, consider the case of the excitation incident radiation is not just the sum of the responses to the indi- of a state of even parity to a given state in the continuum of energy vidual components of the field [11]. The phenomenon requires a E by a weak x þ 2x field of the form nonlinear response of the system to the laser field, and hence it applies to both quantum or classical systems with anharmonic EðtÞ¼x cosðxt þ /xÞþ2x cosð2xt þ /2xÞ: ð39Þ potentials. The 2x component is chosen to be exactly at resonance with the It is worth reemphazing that the matter interference effects and desired transition, and transfers population from the bound to the other entirely quantum contributions can have an important quan- continuum state through one-photon absorption. In turn, the x titative effect on the response [29,30]. However, the qualitative component couples bound and continuum states through a two- nonlinear response to the laser field that gives rise to laser control- photon resonant excitation. Using second order perturbation theory lable interference contributions does not necessarily rely upon we have that the field creates a superposition of the form: them. At this point the close connection between multiple-field in- jwi¼c j ; oddiþc j ; eveni; ð40Þ 1 E 2 E duced based scenarios in Coherent Control and Nonlinear Optics where the first term is due to the one-photon excitation by the 2x is evident. Here, the crucial physics is the nonlinear response of field, and the second term is due to two-photon excitation by the x matter to incident radiation. The difference between them relies field (a similar results holds even in the nonresonant case [27]). As a on the focus. In Coherent Control the interest is on what happens with matter after the nonlinear interaction, whereas in Nonlinear consequence note specifically that the coefficients c1 and c2 are 2 Optics the concern is on what happens to the light after such an proportional to 2x and x, respectively. The interference term, obtained from the cross product of the two terms, is then proportional interaction. 2 One technical caveat is in order. The validity of the analysis in to the amplitudes of the two fields as 2xx. That is, the interference term is driven by the external field, a feature that is distinctly this paper relies on the correctness of the quantum-classical corre- different from the traditional double-slit experiment where the spondence principle as developed in Section 3.1. It is through this interference term is field-free. It is this significant difference that principle that we are able to establish a connection between the manifests in the classical limit where the external driving field pro- quantum and classical subdivision of observables [Eqs. (15) and vides a classical contribution to the interference as well. (35)], and to define classical analogs of quantum interference con- We emphasize that the interference term arising from Eq. (40), tributions. If this principle does not hold then it is possible that both quantum mechanically and in the classical limit, is then the quantum and classical interference terms do not correspond to collective effect of the two excitation routes. This is quite different the same physical phenomenon. One case where this correspon- from the double-slit experiment in the classical limit. Specifically, dence was clearly demonstrated to hold was our study of the as long as the two slits are separated in space the quantum inter- x þ 2x control where the fields were incident on a quartic oscilla- ference contributions will measure spatial coherences of the wave- tor [11]. Specifically, we obtained an expression for the net dipole function that do not have a classical manifestation. This is so induced by x þ 2x fields in the Heisenberg picture and set h ¼ 0to because classically passing through the upper or lower slit and explicitly recover the classical limit of the interference contribu- arriving at the detection screen constitute two independent and tions. In the quantum case, the net dipole induced by the field is mutually exclusive possibilities. By contrast, the response of a clas- of the form sical system to components of a radiation field do not constitute H 2 ^ ^x ðtÞ¼ 2xC½x; h; ^x; p^ cosð2/ / Þ; ð44Þ independent and mutually exclusive events. x x 2x These distinctions between the double-slit and the multiple- where C^½x; h; ^x; p^, the operator defining the quantum response, is a field induced dynamics are best seen in the response theory treat- function of the laser frequency, the oscillator anharmonicity, and ment [28] of the field induced dynamics. Consider again the case of the position and momentum operators in Schrödinger picture. The 150 I. Franco et al. / Chemical Physics 370 (2010) 143–150 classical solution extracted from the quantum solution is of the where it exists, provides a quantitative approximation to the form: quantum control. xcðtÞ¼2 Cc½x; xcð0Þ; pcð0Þ cosð2/ / Þ; ð45Þ x 2x x 2x Acknowledgment where Cc½x; xcð0Þ; pcð0Þ is the classical limit of C^½x; h; ^x; p^, and xcð0Þ and pcð0Þ are the initial conditions for the classical position We thank NSERC for support of this research. and momentum variables. A comparison of the classical behavior extracted in this way from the quantum solution with an indepen- References dent fully classical calculation showed excellent agreement, explicitly demonstrating that the quantum and classical versions of laser [1] M. Nielsen, I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. control do qualitatively correspond to the same physical phenome- [2] M. Shapiro, P. Brumer, Principles of the Quantum Control of Molecular non. 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