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

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Quantum Interferences and Their Classical Limit in Laser Driven Coherent Control Scenarios Chemical Physics 370 (2010) 143–150 Contents lists available at ScienceDirect Chemical Physics journal homepage: www.elsevier.com/locate/chemphys 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 ana- logs 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 ðmÀ1Þ 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 Þ¼À 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 ðmÀ1Þ 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.
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