Wigner Phase-Space Distribution As a Wave Function

PHYSICAL REVIEW A 88, 052108 (2013)

Wigner phase-space distribution as a wave function

Denys I. Bondar,* Renan Cabrera,† Dmitry V. Zhdanov,‡ and Herschel A. Rabitz§ Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA (Received 17 December 2012; revised manuscript received 5 August 2013; published 11 November 2013) We demonstrate that the Wigner function of a pure quantum state is a wave function in a specially tuned Dirac bra-ket formalism and argue that the Wigner function is in fact a probability amplitude for the quantum particle to be at a certain point of the classical phase space. Additionally, we establish that in the classical limit, the Wigner function transforms into a classical Koopman–von Neumann wave function rather than into a classical probability distribution. Since probability amplitude need not be positive, our ﬁndings provide an alternative outlook on the Wigner function’s negativity.

DOI: 10.1103/PhysRevA.88.052108 PACS number(s): 03.65.Ca, 03.65.Sq, 03.67.Ac

I. INTRODUCTION II. BACKGROUND In his seminal work [1], Wigner deﬁned the combined Around the time the Wigner distribution was conceived, distribution of the quantum particle’s coordinate and momen- Koopman and von Neumann [27–30] (for modern develop- tum in terms of the wave function. Since then, the Wigner ments and applications, see Refs. [30–44]) recast classical function has played a paramount role in the phase space mechanics in a form similar to quantum mechanics by formulation of quantum mechanics [2–4], is a standard tool introducing classical complex valued wave functions and for establishing the quantum-to-classical interface [2,5–7], representing associated physical observables by means of com- and has a broad range of applications in optics and signal muting self-adjoint operators. In particular, it was postulated processing [8,9] as well as quantum computing [10–16]. that the wave function |(t) of a classical particle obeys the Techniques for the experimental measurement of the Wigner following equation of motion: function are also developed [7,17–19]. Despite its ubiquity, the d pˆ Wigner function is haunted by the obscure feature of possibly i |(t)=Lˆ |(t), Lˆ = λˆ x − U (xˆ)λˆ p, (1) being negative. Wigner [20] demonstrated that his function is dt m the only one satisfying a reasonable set of axioms for a joint [x,ˆ λˆ x ] = [p,ˆ λˆ p] = i, probability distribution. This feature of the Wigner function (2) x,ˆ pˆ = x,ˆ λˆ = p,ˆ λˆ = λˆ ,λˆ = . has been a subject of numerous interpretations [13,21–23], [ ] [ p] [ x ] [ x p] 0 including the development of a mathematical framework for Without loss of generality one-dimensional systems are con- handling negative probabilities [24,25]. The Wigner function’s sidered throughout. Since the self-adjoint operators represent- negativity was also associated with the exponential speedup in ing the classical observables of coordinate xˆ and momentum pˆ quantum computation [14,16]. commute, they share a common set of orthogonal eigenvectors In this paper we provide insight into the negativity by |px such that 1 = dpdx |pxpx|. In the KvN classical advocating the following interpretation: The Wigner func- mechanics, all observables are functions of xˆ and pˆ.The tion is a probability amplitude for a quantum particle to expectation value of an observable Fˆ = F (x,ˆ pˆ) at time t is be at a certain point of the classical phase space, i.e., (t)|Fˆ|(t). The probability amplitude px|(t) for a the Wigner function is a wave function analogous to the classical particle to be at point x with momentum p at time t Koopman–von Neumann (KvN) wave function of a classical is found to satisfy particle. The remainder of the paper is organized as follows: The ∂ p ∂ ∂ + − U (x) px|(t)=0. (3) necessary background on the KvN classical mechanics and ∂t m ∂x ∂p operational dynamical modeling [26] is reviewed in Sec. II. The basic equations, on which our interpretation rests, are This is the evolution equation for the classical wave function in = ˆ =− = derived in Secs. III to V. A connection between these equations the xp representation, where xˆ x, λx i∂/∂x, pˆ p, and ˆ =− and quantum mechanics in phase space is established in λp i∂/∂p in order to satisfy the commutation relations (2). Sec. VI. Conclusions are drawn in Sec. VII. Utilizing the chain rule and Eq. (3), we obtain the well known classical Liouville equation for the phase space probability distribution ρ(x,p; t) =|px|(t)|2, ∂ p ∂ ∂ + − U (x) ρ(x,p; t) = 0. (4) ∂t m ∂x ∂p *[email protected] †[email protected] Newtonian trajectories emerge as characteristics of either ‡[email protected]; Present address: Department Eq. (3) or (4). Hence, the essential difference between KvN and of Chemistry, Northwestern University, Evanston, Illinois 60208, Liouville mechanics lies in weighting individual trajectories USA (see Fig. 1): Arbitrary complex weights underlying the classi- §[email protected] cal wave function | can be utilized in KvN mechanics (3),

[ˆx , pˆ ]=i κ [ˆx, pˆ]=0 q q [ˆx, pˆ]=i 0 κ 1

FIG. 1. (Color online) The conceptual difference between (a) Li- ouville and (b) KvN classical mechanics. In the Liouville mechanics [Eq. (4)], classical particles, moving along Newtonian trajectories, are FIG. 2. (Color online) Schematic flow showing the derivation of tagged by positive weights according to the probability distribution quantum mechanics, classical mechanics, and quantum mechanical ρ. In the KvN mechanics [Eq. (3)], classical particles, following the phase space representation within the operational dynamical model- same trajectories, are tagged by arbitrary real (or complex) weights ing proposed in [26]. representing the KvN classical wave function |. whereas only positive weights having probabilistic meaning and limκ→0 Hˆ qc = h¯ Lˆ . See Fig. 2 for a pictorial summary of are permitted in Liouville mechanics (4). these derivations presented in [26]. Note that the classical wave function and the classical A crucial point for our current analysis is that this unified probability distribution satisfy the same dynamical equation, wave function |κ (t is dropped henceforth) in the xp which reflects the physical irrelevance of the phase of a representation is proportional to the Wigner function W (see classical wave function. Ref. [26]), In recent work [26], we put forth operational dynamical √ modeling as a systematic theoretical framework for deducing px|κ = 2πhκW¯ (x,p), equations of motion from the evolution of average values. (7) First, starting from the Ehrenfest theorems [45], we obtained dy y y W(x,p) = ρ x − ,x + eipy/(¯hκ); the Schrodinger¨ equation if the momentum and coordinate 2πhκ¯ κ 2 2 operators obeyed the canonical commutation relation, and the moreover, KvN equation (1) if the momentum and coordinate operators commuted. Then, applying the same technique to the Ehrenfest √ xλ | = hκρ¯ (u,v), theorems, p κ κ ∂ (¯hκ)2 ∂2 ∂2 d ihκ¯ − − − U(u) + U(v) ρ (u,v) = 0, 2 2 κ m κ (t)|xˆq |κ (t)=κ (t)|pˆq |κ (t), ∂t 2m ∂v ∂u dt (8) (5) = − = + d u x hκλ¯ p/2,v x hκλ¯ p/2. (t)|pˆ | (t)= (t)|−U (xˆ )| (t), dt κ q κ κ q κ with a generalization [xˆq ,pˆq ] = ihκ¯ (0 κ 1) and demanding a smooth classical limit κ → 0, we established III. DERIVATION OF MAIN RESULTS the existence of the uniquely defined operator Hˆ qc such that Since xˆ and λˆ p are commuting self-adjoint operators, the following resolution of the identity is valid in the Hilbert phase d | =Hˆ | ih¯ κ (t) qc κ (t) , space H (i.e., a vector space with the scalar product | dt xp h¯ 1 hκ¯ of functions of two variables): Hˆ = pˆλˆ + U xˆ − λˆ qc x p ˆ = | | | ∈H m κ 2 1Hxp dxdλp xλp xλp , xλp xp , (9) 1 hκ¯ | = | ˆ | = | − U xˆ + λˆ , xˆ xλp x xλp , λp xλp λp xλp ; κ 2 p likewise, xˆq = xˆ − hκ¯ λˆ p/2, pˆq = pˆ + hκ¯ λˆ x/2, (6) ˆ = | | | ∈H 1Hxp dλx dp λx p λx p , λxp xp , where xˆq and pˆq represent the quantum coordinate and (10) pˆ|λx p=p|λxp, λˆ x |λxp=λx|λx p. momentum, respectively, xˆ, pˆ, λˆ x , and λˆ p are the same classical operators as in Eq. (2), and κ denotes the degree of A consequence of the commutators [x,ˆ λˆ x ] = [p,ˆ λˆ p] = i is quantumness or commutativity: κ → 1 corresponds to quan- ipλp −ixλx tum mechanics whereas κ → 0 recovers classical mechanics, λx p|xλp=e /(2π).

052108-2 WIGNER PHASE-SPACE DISTRIBUTION AS A WAVE ... PHYSICAL REVIEW A 88, 052108 (2013)

Let Hq denote the Hilbert space of single particle wave Comparing Eqs. (8) and (13), we conclude that functions (i.e., a vector space with the scalar product || √ | = = || || = of functions of one variable), which differs from the Hilbert xλp κ hκ¯ xq u ρˆκ xq v . (14) phase space Hxp deﬁned above. We denote by xˆq and pˆq ([xˆq ,pˆq ] = ihκ¯ ) two pairs of operators: one acting on Hxp [as H Utilizing Eqs. (9), (11), and (14), we obtain deﬁned in Eq. (6)], the other acting on q , | = dxdλ |xλ xλ | xˆ ||x = x ||x , pˆ ||p = p ||p , κ κ p κ p p κ q q q q q q q q ˆ = || || = || || = ∗ 1Hq dxq xq xq dpq pq pq , hκ¯ dxdλp ρκ (u,v)ρκ (u,v) ix p /(¯hκ) e q q = || || || || xq ||pq = √ , ||xq ∈ Hq , ||pq ∈ Hq . (11) dxq dxq xq ρˆκ xq xq ρˆκ xq 2πhκ¯ = || 2|| = 2 dxq xq ρˆκ xq Tr ρˆκ ; (15) The density matrixρ ˆκ , a self-adjoint operator acting in the space H , obeys the Liouville–von Neuman equation q hence, 2 | = ⇐⇒ 2 = d pˆq κ κ 1 ρˆκ ρˆκ . (16) ihκ¯ ρˆκ = + U(xˆq ), ρˆκ =⇒ (12) dt 2m 2 2 2 ∂ (¯hκ) ∂ ∂ Settingρ ˆκ =||φκ φκ ||, φκ ||φκ = 1, where ||φκ ∈ ihκ¯ − − − U(u) + U(v) H = ∂t 2m ∂v2 ∂u2 q is a one-particle wave function, and denoting φκ (u) = || = = || = + × = || || = = xq u φκ , φκ (v) xq v φκ , η p hκλ¯ x/2, xq u ρˆκ xq v 0. (13) and ξ = p − hκλ¯ x/2, we derive

| | = | | | | | κ G(xˆq )F (pˆq ) κ dxdλpdx dλpdλx dp κ xλp xλp G(xˆq )F (pˆq ) λxp λx p x λp x λp κ = ∗ | | ∗ hκ¯ dxdλpdx dλpdλx dp φκ (u)φκ (v)G(u) xλp λx p F (η) λxp x λp φκ (u )φκ (v ) dudvdu dv dξdη = φ∗(u)φ (v)G(u)F (η)φ (u)φ∗(v)eiη(u−u )/(¯hκ)e−iξ(v−v )/(¯hκ) (2πhκ¯ )2 κ κ κ κ iηu/(¯hκ) −iηu/(¯hκ) ∗ ∗ e e = φκ (v)φ (v)dv dη φ (u)G(u)√ du F (η) φκ (u ) √ du κ κ 2πhκ¯ 2πhκ¯ − ipq xq /(¯hκ) ipq xq /(¯hκ) = ∗ e√ e√ dpq φκ (xq )G(xq ) dxq F (pq ) φκ (xq ) dxq 2πhκ¯ 2πhκ¯ = || || || || || || dxq dxq dpq φκ G(xˆq ) xq xq pq pq F (pˆq ) xq xq φκ . (17)

Finally, we have demonstrated that all expectation values for the evolution equation for a classical KvN wave function (1). the state ||φκ coincide with those for |κ , Hence, in the classical limit, the Wigner function maps a |G(xˆ )F (pˆ )| =φ ||G(xˆ )F (pˆ )||φ , (18) quantum wave function into a corresponding KvN classical κ q q κ κ q q κ wave function rather than a classical phase space distribution. | where G(xˆq ) and F (pˆq ) are arbitrary functions of the quantum Since the vectors px are identical in both KvN and Wigner position and momentum, respectively. representations, W(x,p) is proportional to the probability Furthermore, in the context of the Maslov noncommutative amplitude that a quantum particle is located at a point (x,p) calculus [46–48] based on the most general operator ordering, of the classical phase space. Note it is important to distinguish identity (18) can be generalized to the classical (x,ˆ pˆ) and quantum (xˆq ,pˆq ) phase spaces because the notion of a phase space point arises naturally only |F (xˆ ,pˆ )| =φ ||F (xˆ ,pˆ )||φ , (19) κ q q κ κ q q κ in the commutative classical variables (x,ˆ pˆ). One may take valid for an arbitrary function F of two variables. this distinction further and interpret the Wigner function as the Equations (16) and (18) reveal that the Wigner distribution KvN wave function of a classical counterpart of the analogous of a pure state behaves like a wave function. As shown in Fig. 2, quantum system. Like any wave function, the Wigner function the Wigner function’s dynamical equation (6) transforms into need not be positive.

052108-3 BONDAR, CABRERA, ZHDANOV, AND RABITZ PHYSICAL REVIEW A 88, 052108 (2013)

IV. A GENERALIZATION TO MIXED STATES that A,ˆ B,ˆ Cˆ ∈ L(Hq ). In particular, the set L(Hq ) includes Equation (17) offers a method to extend the developed all density matrices and observables of the form F (xˆq ,pˆq ). ˆ formalism to an arbitrary state. In particular, we find a fixed Define linear operators (i.e., superoperators) ω(B) and (Cˆ ) acting in L(H )as[49] vector |1∈Hxp such that the following equation is valid for q all density matricesρ ˆκ , ω(Bˆ )Aˆ = (1/2){B,ˆ Aˆ},(Cˆ )Aˆ = [C,ˆ Aˆ], (24)

1|G(xˆq )F (pˆq )|κ =Tr [G(xˆq )F (pˆq )ˆρκ ], ∀ρˆκ , (20) where {B,ˆ Aˆ}=Bˆ Aˆ + AˆBˆ and ω(Bˆ )Aˆ denotes the result of the action of the linear map ω(Bˆ )onAˆ as an element of L(Hq ); a where the trace on the right hand side is calculated over the similar interpretation holds for the notation (Cˆ )Aˆ. One can H {|| } H = space q .Let φn be a basis in q such that φn(u) readily demonstrate that x = u||φ . Substitutingρ ˆ = ρ ||φ φ || and q √n κ n,m n,m n m ˆ ˆ ˆ = ˆ ˆ ˆ = ˆ ˆ ˆ | = [(B),(C)]A 4[ω(B),ω(C)]A [[B,C],A], 1 xλp hκχ¯ (u,v) into Eq. (20) and following the steps (25) in Eq. (17), we obtain the equation for the unknown χ [ω(Bˆ ),(Cˆ )]Aˆ = (1/2){A,ˆ [B,ˆ Cˆ ]}. H L H χ(u,v)φ∗(v)dv = φ∗(u), (21) Hence, the Hilbert phase space xp can be equated to ( q ) n n with the classical operators (2) realized as which has a unique solution χ(u,v) = δ(u − v); therefore, xˆ = ω(xˆq ), λˆ x = (pˆq )/(¯hκ), δ(λp) dx (26) xλp|1= √ , |1= √ |xλp = 0. (22) pˆ = ω(pˆq ), λˆ p =−(xˆq )/(¯hκ), hκ¯ hκ¯ where xˆ ,pˆ ∈ L(H ). Other realizations of H can be Note that according to Eq. (7), the vector |1 corresponds to q q q xp similarly constructed. an identity density matrix. Equations (20), (23), and (26) bridge the developed Hilbert Equation (20) is as a generalization of Eq. (18) to the case phase space formalism with the Mukunda approach to the when | represents the Wigner function of arbitrary mixed κ Wigner function [49]. states [see Eq. (7)]. Additionally, the ket vector |1 maps an observable VI. IMPLICATIONS FOR PHASE SPACE FORMULATION F (xˆq ,pˆq ) ∈ L(Hxp ), which is an element of the Hilbert space OF QUANTUM MECHANICS L(Hxp ) of linear operators acting on Hxp , into a vector from H xp as The Hilbert phase space is an alternative formulation of quantum mechanics obtained as the merger of the wave F (xˆq ,pˆq )|1=|F (xq ,pq )∈Hxp , (23) function and phase space representations via the operational F (xˆq ,pˆq ) ∈ L(Hxp ). dynamical modeling (Fig. 2). While the connection with the wave-function formulation has already been elucidated in In other words, each observable corresponds to a vector in previous sections [in particular, see Eqs. (16), (18), and (20)], the Hilbert phase space, such that the observable’s average is there is more to be considered [in addition to Eq. (7)] regarding calculated as the scalar product F (xq ,pq )|κ (20). the connection with quantum mechanics in phase space [2–4]. The Moyal bracket {{ , }}—a cornerstone of the quantum mechanical phase space formalism—is deﬁned V. A REALIZATION OF THE HILBERT PHASE SPACE Hxp as [3,4,50,51] Both Hxp and Hq have been considered so far as abstract ←−−→ ←−−→ 2 hκ¯ ∂ ∂ hκ¯ ∂ ∂ inﬁnite-dimensional vector spaces with no direct connection {{f,g}} = f (x,p)sin − g(x,p), between them. Following Ref. [49], we shall construct a hκ¯ 2 ∂x ∂p 2 ∂p ∂x H realization of the Hilbert phase space xp in terms of the (27) space of quantum-mechanical wave functions Hq . for any smooth functions f (x,p) and g(x,p). Using the identi- A set of linear operators acting on Hq , endowed with the −−→ ←−− Hilbert-Schmidt inner product (A,ˆ Bˆ ) = Tr (Aˆ†B), forms the ties exp(a∂/∂y)f (y) = f (y)exp(a∂/∂y) = f (y + a), we ex- Hilbert space L(Hq ). Throughout this section, we assume pand the Moyal bracket

←−−→ ←−−→ ←−−→ ←−−→ 1 ihκ¯ ∂ ∂ ihκ¯ ∂ ∂ ihκ¯ ∂ ∂ ihκ¯ ∂ ∂ {{f,g}} = f (x,p) exp − − exp − + g(x,p) ihκ¯ 2 ∂x ∂p 2 ∂p ∂x 2 ∂x ∂p 2 ∂p ∂x 1 ihκ¯ ∂ ihκ¯ ∂ ihκ¯ ∂ ihκ¯ ∂ = f x + ,p − − f x − ,p + g(x,p) ihκ¯ 2 ∂p 2 ∂x 2 ∂p 2 ∂x 1 hκ¯ hκ¯ hκ¯ hκ¯ = px| f xˆ − λˆ ,pˆ + λˆ − f xˆ + λˆ ,pˆ − λˆ |g, (28) ihκ¯ 2 p 2 x 2 p 2 x

052108-4 WIGNER PHASE-SPACE DISTRIBUTION AS A WAVE ... PHYSICAL REVIEW A 88, 052108 (2013) where g(x,p) =px|g. Thus, we established the Moyal in the introduced Hilbert phase space. The latter is an bracket realization in the Hilbert phase space. The function alternative formulation of quantum mechanics obtained as f of noncommutative variables in Eq. (28) should be defined a meld of the wave function (the Dirac bra-ket) and phase according to the Weyl calculus, as explained in Ref. [52]. The space (the Wigner function) representations as well as the operators ±λˆ x and ±λˆ p (known as the Bopp operators [52,53]) Koopman–von Neumann classical mechanics. The dynamical are analogues of the left and right derivatives. equation (6) is valid for any nonpure state, providing a An elementary consequence of Eq. (28) is the equivalence starting point for an efficient numerical algorithm for the between Eq. (6) and Moyal’s equation of motion [3] (see also Wigner function propagation [55]. In the current paper, Ref. [54]), we have studied dynamics in the Cartesian coordinates. In the next work, we will extend our formalism to rotational ∂W p2 ={{H,W}},H(x,p) = + U(x). (29) dynamics. ∂t 2m Here W is the Wigner function of an arbitrary (in general, mixed) state. ACKNOWLEDGMENTS The authors thank anonymous referees for their crucial VII. CONCLUSIONS comments. In particular, we are grateful for drawing our We demonstrate that the Wigner distribution of a pure attention to Ref. [49]. The authors are financially supported quantum state is a wave function [Eqs. (6), (16), and (18)] by NSF and ARO.

[1] E. Wigner, Phys. Rev. 40, 749 (1932). [23] C. Ferrie, Rep. Prog. Phys. 74, 116001 (2011). [2] A. O. Bolivar, Quantum-Classical Correspondence: Dynamical [24] W. Muckenheim,¨ G. Ludwig, C. Dewdney, P. R. Holland, Quantization and the Classical Limit (Springer, Berlin, 2004). A. Kyprianidis, J. P. Vigier, N. Cufaro Petroni, M. S. Bartlett, [3] C. Zachos, D. Fairlie, and T. Curtright, Quantum Mechanics in andE.T.Jaynes,Phys. Rep. 133, 337 (1986). Phase Space (World Scientiﬁc, Singapore, 2005). [25] P. T. Landsberg, Nature (London) 326, 338 (1987). [4]T.L.CurtrightandC.K.Zachos,Asia Pac. Phys. Newsl. 01,37 [26] D. I. Bondar, R. Cabrera, R. R. Lompay, M. Y. Ivanov, and (2012). H. A. Rabitz, Phys. Rev. Lett. 109, 190403 (2012). [5]W.H.Zurek,Nature (London) 412, 712 (2001). [27] B. O. Koopman, Proc. Nat. Acad. Sci. (USA) 17, 315 [6] D. Dragoman and M. Dragoman, Quantum-Classical Analogies (1931). (Springer, Berlin, 2004), Chap. 8. [28] J. von Neumann, Ann. Math. 33, 587 (1932). [7] S. Haroche and J. M. Raimond, Exploring the Quantum: Atoms, [29] J. von Neumann, Ann. Math. 33, 789 (1932). Cavities and Photons (Oxford University Press, Oxford, 2006). [30] D. Mauro, Ph.D. thesis, Universita` degli Studi di Trieste, 2002, [8] L. Cohen, Proc. IEEE 77, 941 (1989). arXiv:quant-ph/0301172. [9] D. Dragoman, EURASIP J Adv. Signal Process. 2005, 1520 [31] E. Gozzi, Phys. Lett. B 201, 525 (1988). (2005). [32] E. Gozzi, M. Reuter, and W. D. Thacker, Phys. Rev. D 40, 3363 [10] C. Miquel, J. P. Paz, and M. Saraceno, Phys. Rev. A 65, 062309 (1989). (2002). [33] J. Wilkie and P. Brumer, Phys.Rev.A55, 27 (1997). [11] E. F. Galvao,˜ Phys.Rev.A71, 042302 (2005). [34] J. Wilkie and P. Brumer, Phys.Rev.A55, 43 (1997). [12] C. Cormick, E. F. Galvao,˜ D. Gottesman, J. P. Paz, and A. O. [35] E. Gozzi and D. Mauro, Ann. Phys. (NY) 296, 152 (2002). Pittenger, Phys.Rev.A73, 012301 (2006). [36] E. Deotto, E. Gozzi, and D. Mauro, J. Math. Phys. 44, 5902 [13] C. Ferrie and J. Emerson, New J. Phys. 11, 063040 (2009). (2003). [14] V. Veitch, C. Ferrie, D. Gross, and J. Emerson, New J. Phys. 14, [37] E. Deotto, E. Gozzi, and D. Mauro, J. Math. Phys. 44, 5937 113011 (2012). (2003). [15] A. Mari and J. Eisert, Phys. Rev. Lett. 109, 230503 (2012). [38] A. A. Abrikosov, E. Gozzi, and D. Mauro, Ann. Phys. (NY) 317, [16] V. Veitch, N. Wiebe, C. Ferrie, and J. Emerson, New J. Phys. 15, 24 (2005). 013037 (2013). [39] M. Blasone, P. Jizba, and H. Kleinert, Phys. Rev. A 71, 052507 [17] C. Kurtsiefer, T. Pfau, and J. Mlynek, Nature (London) 386, 150 (2005). (1997). [40] P. Brumer and J. Gong, Phys. Rev. A 73, 052109 (2006). [18] A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, [41] P. Carta, E. Gozzi, and D. Mauro, Ann. Phys. (Leipzig) 15, 177 Nature (London) 448, 784 (2007). (2006). [19] S. Deleglise,´ I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J.-M. [42] E. Gozzi and C. Pagani, Phys.Rev.Lett.105, 150604 Raimond, and S. Haroche, Nature (London) 455, 510 (2008). (2010). [20] R. F. O’Connell and E. P. Wigner, Phys. Lett. A 83, 145 [43] E. Gozzi and R. Penco, Ann. Phys. (NY) 326, 876 (2011). (1981). [44] E. Cattaruzza, E. Gozzi, and A. F. Neto, Ann. Phys. (NY) 326, [21] A. J. Bracken and J. G. Wood, Europhys. Lett. 68, 1 (2004). 2377 (2011). [22] O. Man’ko and V. I. Man’ko, J. Russ. Laser Res. 25, 477 (2004). [45] P. Ehrenfest, Z. Phys. 45, 455 (1927).

052108-5 BONDAR, CABRERA, ZHDANOV, AND RABITZ PHYSICAL REVIEW A 88, 052108 (2013)

[46] V. P. Maslov, Operational Methods (Mir, Moscow, 1976) [49] N. Mukunda, Pramana 11, 1 (1978). [in Russian]. [50] T. Curtright, D. Fairlie, and C. Zachos, Phys. Rev. D 58, 025002 [47] V. E. Nazaikinskii, B. Y. Sternin, and V. E. Shatalov, in Global (1998). Analysis - Studies and Applications V, edited by Y. Borisovich, [51] T. Curtright, T. Uematsu, and C. Zachos, J. Math. Phys. 42, 2396 Y. Gliklikh, and A. Vershik, Lecture Notes in Mathematics, (2001). Vol. 1520 (Springer, Berlin, 1992), pp. 81–91. [52] M. Hillery, R. O’Connell, M. Scully, and E. Wigner, Phys. Rep. [48] V. E. Nazaikinskii, V. E. Shatalov, and B. Y. Sternin, Methods of 106, 121 (1984). Noncommutative Analysis: Theory and Applications, de Gruyter [53] F. Bopp, Ann. Inst. H. Poincare´ 15, 81 (1956). Studies in Mathematics, Vol. 22 (Walter de Gruyter, Berlin, [54] D. Dragoman, Phys. Lett. A 274, 93 (2000). 1996). [55] R. Cabrera, D. I. Bondar, and H. A. Rabitz, arXiv:1212.3406.

052108-6