Quantum Trajectories

Edited by Keith H. Hughes School of Chemistry University of Wales Bangor Bangor LL57 2UW United Kingdom and Gérard Parlant Institut Charles Gerhardt UniversitéMontpellier 2, CNRS Equipe CTMM, Case Courrier 1501 Place Eugène Bataillon 34095 Montpellier France Suggested Dewey Classification: 541.2

ISBN 978-0-9545289-9-7

Published by

Collaborative Computational Project on Molecular Quantum Dynamics (CCP6) Daresbury Laboratory Daresbury Warrington WA4 4AD United Kingdom

c CCP6 2011

ii Preface v Mixed Quantum/Classical Dynamics: Bohmian and DVR Stochastic Trajectories C. Meier, J. A. Beswick, T. Yefsah 1 Trajectory-Based Derivation of Classical and Quantum Mechanics Bill Poirier 6 The Bipolar Reaction Path Hamiltonian (BRPH) Approach for Multi- Dimensional Reactive Scattering Systems Jeremy B. Maddox and Bill Poirier 9 An Iterative Finite Diﬀerence Method for Solving the Quantum Hydro- dynamic Equations of Motion Brian K. Kendrick 13 Kinematic Quantum Trajectories Timothy M. Coﬀey 20 Complex Trajectories and Dynamical Origin of Quantum Probability Moncy V. John 25 Energy Rays for Electromagnetic Pulses Scattering from Metal- Dielectric Structures Robert E. Wyatt 28 Quantum Dynamics through Quantum Potentials S. Duley, S. Giri, S. Sengupta and P. K. Chattaraj 35 Conceptual Issues, Practicalities and Applications of Bohmian and other Quantum Trajectories in Nanoelectronics John R. Barker 43 Principles of Time Dependent Quantum Monte Carlo Ivan P. Christov 49 Types of Trajectory Guided Grids of Coherent States for Quantum Prop- agation Dmitrii V. Shalashilin and Miklos Ronto 54

iii Accurate Deep Tunneling Description by the Classical Schrödinger Equa- tion Xavier Giménez and Josep Maria Bofill 62 The Bohmian Model, Semiclassical Systems and the Emergence of Clas- sical Trajectories Alex Matzkin 66 Quantum Trajectories for Ultrashort Laser Pulse Excitation Dynamics Gérard Parlant 69 Quantum Dynamics and Super-Symmetric Quantum Mechanics Eric R. Bittner and Donald J. Kouri 72 Bohmian Trajectories of Semiclassical Wave Packets Sarah Römer 79 Quantum Trajectories in Phase Space Craig C. Martens 83 The Semiclassical Limit of Time Correlation Functions by Path Integrals G. Ciccotti 86 Path Integral Calculation of (Symmetrized) Time Correlations Functions S. Bonella 91 On-the-fly Nonadiabatic Bohmian DYnamics (NABDY) Ivano Tavernelli, Basile F. E. Curchod and Ursula Rothlisberger 96 Quantum Many-Particle Computations with Bohmian Trajectories: Ap- plication to Electron Transport in Nanoelectronic Devices A. Alarcón, G.Albareda, F.L.Traversa and X.Oriols 103 An Account on Quantum Interference from a Hydrodynamical Perspec- tive A. S. Sanz 111 Quantum, Classical, and Mixed Quantum-Classical Hydrodynamics I. Burghardt and K. H. Hughes 116

iv K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury Preface

This booklet was produced as a result of the CCP6 Workshop on “Quantum Trajectories” held at the Bangor University, UK, between July 12-14, 2010. The workshop was sponsored by the UK Collaborative Computation Project 6 (CCP6) on molecular quantum dynamics. Details of CCP6 and its activities can be found at http://www.ccp6.ac.uk. The main focus of the workshop was the hydrodynamic formulation of quantum mechanics. Over the last 10-13 years the hydrodynamic formulation of quantum mechanics has grown into a quantum trajectory methodology with widespread applications in physics and chemistry. Furthermore, theoretical investigations into the hydrodynamic form of quantum mechanics have spawned, or inspired, a number of other trajectory based approaches to quantum mechanics. The workshop brought together researchers involved in the computational and numerical development of quantum trajectories and researchers involved in use of quantum trajectories as an insightful and intuitive approach for studying and interpreting a wide range of quantum phenomena in a diverse range of fields. Each speaker was asked to provide a brief article which could be collected into a workshop booklet that reviews their work and the topics covered in their talk. This booklet should be of interest to both the specialist and non-specialist in this field. The booklet covers computational approaches to the topic of quantum trajectories and should serve as a guide to many of the recent developments in this field. The editors would like to thank all those who participated and contributed to the workshop.

K. H. Hughes G. Parlant August 2011

v K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

Mixed Quantum/Classical Dynamics: Bohmian and DVR Stochastic Trajectories

C. Meier, J. A. Beswick and T. Yefsah∗ Laboratoire Collisions, Agrégats, Réactivité, IRSAMC UniversitéPaul Sabatier, Toulouse, France

I. INTRODUCTION

In many systems comprising a large number of particles, even though a detailed quantum treatment of all degrees of freedom is not necessary, there may exist subsets that have to be treated quantum mechanically under the influence of the rest of the system treated classically. In these cases, mixed quantum/classical approaches have to be used to describe its dynamics [1]. The most popular of these are the mean-field approximation [2], the surface hopping trajectories [4, 5] or the methods based on quantum/classical Liouville space representations [6–12]. In the mean-field treatment the force for the classical motion is calculated by averaging over the quantum wavefunction. In the surface hopping scheme the classical trajectories move according to a force derived from a single quantum state with the possibility of transitions to other states. Based on these ideas, Bastida et al [13] have proposed a scheme (MQCS for mixed quantum classical steps) in which the quantum wave function is expanded in a discrete variable representation (DVR) rather than in the usual finite (spectral) basis representation (FBR). The interpretation of the dynamics of the quantum degree of freedom is provided in terms of stochastic hops from one grid point to another, governed by the time evolution of the wave packet. Another way to mix quantum mechanics with classical mechanics, proposed in refs. [14–20], is based on Bohmian quantum trajectories for the quantum/classical connection. In this approach, which was called MQCB (Mixed Quantum/Classical Bohmian) trajectories, the wave packet is used to define de Broglie-Bohm quantum trajectories [21–24] which in turn are used to calculate the force acting on the classical variables. The main

∗ present address: Laboratoire Kastler Brossel, D´epartement de Physique de l’Ecole Normale Sup´erieure, Paris, France

1 diﬀerence between MQCB and MQCS lies in the way the force on the classical variables is calculated. In the MQCS method, the force is evaluated using diﬀerent discrete points of the quantum degree of freedom, while in the MQCB method, it is calculated at the continuous points of the quantum trajectory. Hence, the underlying question is to determine what is the connection between the sequence of grid points obtained by stochastic hopping methods and the deterministic, continuous quantum trajectory. In his paper Quantum mechanics in terms of discret beables [25], Jeroen C. Vink has shown that indeed a connection exists for a three-point approximation of the kinetic energy operator. In this contribution, we generalize Vink’s formulation to an arbitrary N-point expression. The paper is organized as follows. In Sec. (II) we give a brief review of MQCS method; in Sec. (III) we address the problem of representing a quantum wave packet by stochastic hops between discrete points in space, and under which conditions the sequence of hopping trajectories converge towards a deterministic quantum trajectory. Finally Sec. (IV) gives a general discussion of the relationship between all these treatments.

II. STOCHASTIC DVR AND MQCS

Considering a Hamiltonian with x being the quantum degree of frredom and X(t) the classical trajectory. Choosing a one-dimensional (diabatic) DVR basis |xni with eigenvalue xn of the position operator, for the quantum degree of X freedom, the wavefunction can be expressed by ψ(x, (t),t) = n cn(t) |xni, where the wave packet Ψ(x,t) propagates with time along the x coordinate, and P is a DVR basis function with eigenvalue xn of the position operator, such that cn(t)= hxn|ψ(t)i = ψ(xn, X(t),t). The values of the time-dependent coefficients can be obtained by solving the time-dependent Schrödinger equation using any conventional method (split operator, etc...). Initially the wave packet is a coherent superposition of DVR states. The time evolution of the coefficients of the wave packet in these DVR states is given by

dcn i i X = −~ cm(t)hxn|H|xmi = −~ cm(t)hxn|T |xmi + V (xn, (t)) (1) dt m " m # X X where we have used the fact that the non diagonal matrix elements of the Hamil- tonian reduce to the matrix elements of the kinetic energy operator T in the DVR basis and the potential energy matrix is diagonal. The classical degrees of freedom are calculated using the force evaluated at one single DVR point,

1 ∂ (V (x,X)) P˙ = − (2) M ∂X x=xn(t),X=X(t)

2 the dynamics in the quantum subspace is accounted for by the possiblity of quantum jumps between these points. Hence, xn is a sequence of DVR points, forming a stochastic quantum trajectory, deﬁned as follows: Eq.(1) determines the transition probabilities between DVR states. Let us introduce the diagonal 2 2 density matrix elements Pn = |cn| = |ψ(xn, X(t),t)| representing the probability for the system to be in state |xni at time t. The time evolution of Pn(t) is deduced from Eq.(1) can be written as [26]

dPn = (Tnm Pm − Tmn Pn) (3) dt m X with ~ ∗ (2) cm cn Tnm = D ℑ if Tnm > 0 (4) M nm |c |2 m (2) 2 2 where we have denoted by Dnm = hxn|∂ /∂x |xmi the matrix elements of the second derivative. In this form, it can be viewed as a rate equation, which can be solved by a stochastic approach using Pn→m = Tnmδt as probability to hop from DVR point |xni to |xmi during a timestep δt. This approach is similar to the way the quantum potential energy surface for the classical particle dynamics is selected in Tully’s fewest switches method [3] for mixed quantum-classical dynamics. MQCS method can thus be thought as the implementation of the surface hopping method in the discrete variable representation, rather than in the usual ﬁnite basis representation.

III. DISCRET BEABLES

In this section, we discuss the connection between a continuous quantum trajectory and stochastic hopping between discrete grid points ﬁrst proposed by Vink in 1993 [25]. To this end, it is enough to consider only one quantum degree of freedom. We consider a particle of mass M in a one dimensional space extending from −∞

~ 2 ∂Pn ∗ ∂ ψ = − ℑ ψn 2 (5) ∂t M ( ∂x ) xn

Using the three-point centered expansion of the second derivative and a Taylor ′ expansion of the wavefunction according to ψn±1 = ψn ± ǫψn, this equation can again be written in the form (3) with [25, 27]

3 ′ ′ Sm ′ Sm ′ Tnm = δm,n if S ≥ 0 ; Tnm = − δm,n if S ≤ 0 (6) Mǫ −1 m Mǫ +1 m with ψn = An exp(iSn/~) and the prime denoting a ﬁrst derivative with respect to x. Finally Tnn is determined by the normalization Tnn dt = 1− n6=m Tnm dt Mǫ with dt ≤ ′ ′ in order to have Tnn dt ≥ 0. Vink’s has shown [25], that if |Sn−1−Sn+1| P one uses of (6) it is possible to recover the causal Bohmian trajectories through a stochastic algorithm with minimal choice [27] for the transition density. The proof is straightforward [26]. For a trajectory starting in point n at t = 0 the ′ probability to jump to the position n + 1 is, if we assume that Sn 6= 0,

′ Sn Pn = dt (7) +1 Mǫ

The mean average displacement after dt will then be given by dx = ǫ Pn+1 which using (7) gives in the limit ǫ → 0

∂x 1 ∂S = (8) ∂t M ∂x which is precisely the Bohmian equation for the quantum trajectories. Further- more, one can also show that the second moment of the distribution, i.e., the dispersion in the average displacement, vanishes in the limit ǫ → 0 [26]. These results can be generalized to an arbitrary number of points to express the second derivative by a discrete N-point formula. Using the expressions given in Fornberg [28], one can show that eq. (5) can always be written in the form of eq. (3), however, the Tnm now couple sites beyond nearest neighbors [26]. Again, in the limit of ǫ → 0, the mean-squared dispersion vanishes, and the sequence of hopping points converges towards the Bohmian trajectory.

IV. DISCUSSION

Comparing MQCB and MQCS, we see that both methods calculate the back- reaction of the quantum system onto the classical system not by averaging the force over the quantum wave function (as does the mean ﬁeld method), but by choosing to evaluate this force only at one single point. The time evolution of this point, however, is diﬀerent in both methods: the MQCB method uses the deterministic Bohm trajectory associated with the quantum wave function, the MQCS method generated these points by stochastic jumps between discrete DVR points of an underlying DVR basis. On the other hand, generalizing the Vink’s formulation, we have shown that Bohmian trajectories can be obtained by stochastic jumps between discrete points in the limit of a large number of

4 points. Clearly, the MQCS method is based on the same idea except that the discret points are chosen according to the DVR representation. In the limit of a large number of DVR points the MQCS trajectories should converge towards the Bohmian MQCB trajectories.

[1] Classical and Quantum Dynamics in Condensed Phase Simulations (B. Berne and G. Cicotti and D. Coker Eds.; World Scientific, Singapour,1998) [2] G. Billing, Int. Rev. Phys. Chem. 13,309 (1994). [3] J. C. Tully, J. Chem. Phys. 93, 1061 (1990) [4] J.C. Tully, Int. J. Quantum Chem. 25, 299 (1991). [5] J. C. Tully. Nonadiabatic dynamics. In Modern methods for multidimensional dynamics computations in chemistry, edited by D. L. Thompson, page 34. (World Scientific, Singapore, 1998) [6] D. A. Micha and B. Thorndyke Adv. Quantum Chem. 47, 292-312 (2004). [7] B. Thorndyke and D. A. Micha, Chem. Phys. Lett. 403 (2005) 280-286. [8] A. Donoso, C. C. Martens, J. Chem. Phys. 102, 4291 (1998) [9] R, Kapral, G. Cicotti, J. Chem. Phys. 110, 8919 (1999) [10] S. Nielsen, R, Kapral, G. Cicotti, J. Chem. Phys. 115, 5805 (2001) [11] I. Horenko, C. Salzmann, B. Schmidt, C. Schütte, J. Chem. Phys. 117, 11075 (2002) [12] I. Horenko, M. Weiser, B. Schmidt, C. Schütte, J. Chem. Phys. 120, 8913 (2004) [13] A. Bastida, J. Zuniga, A. Requena, N. Halberstadt, and J.A. Beswick, PhysChem- Comm 7, 29 (2000) [14] E. Gindensperger and C. Meier and J.A.Beswick, J. Chem. Phys. 113, 9369 (2000). [15] E. Gindensperger, C. Meier and J. A. Beswick, Adv. Quantum Chem. 47, 331-346 (2004). [16] E. Gindensperger and C. Meier and J.A.Beswick,J. Chem. Phys. 116,8 (2002). [17] E. Gindensperger and C. Meier and J.A. Beswick and M.-C. Heitz, J. Chem. Phys. 116, 10051 (2002). [18] O.V. Prezhdo and C. Brooksby Phys. Rev. Lett. 86, 3215 (2001). [19] L.L. Salcedo Phys. Rev. Lett. 90, 118901 (2003). [20] O.V. Prezhdo and C. Brooksby Phys. Rev. Lett. 90, 118902 (2003). [21] The quantum theory of motion P. R. Holland (Cambridge University Press, 1993). [22] L. de Broglie, C. R. Acad. Sci. Paris 183, 447 (1926); 184, 273 (1927). [23] D. Bohm, Phys. Rev. 85, 166 (1952); 85, 180 (1952). [24] R. E. Wyatt, Quantum Dynamics with trajectories, Springer, New York (2005) [25] J.C. Vink, Phys. Rev. A 48, 1808 (1993) [26] C. Meier, J. A. Beswick, T. Yefsah, in Quantum Trajectories, edited by P. K. Chatteraj, Taylor & Francis, (2010) [27] J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge Univer- sity Press, (1987) [28] B. Fornberg, A practical guide to pseudospectral methods, Cambridge University Press (1998)

5 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

Trajectory-Based Derivation of Classical and Quantum Mechanics

Bill Poirier Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409-1061, United States

Why are the laws of nature what they are? It is well known that a relatively small change in any of the fundamental physical constants would lead to a radically different and far less complex universe, incapable of sustaining intelligent life that could make such queries [1]. Interpretations of this anthropic principle vary. But what of the fundamental laws themselves, i.e. the mathematical equations that govern dynamics? Much less attention has been paid here. Fundamentally, there are only two dynamical equations of profound importance, i.e. Newton’s (classical) laws, and the Schrödinger (quantum) equation. On the surface, these appear nothing like one another—which is interesting of itself (e.g., why should there be a classical limit?) Yet connections have been known since the earliest days of the quantum theory. D. Bohm [2] developed these into a full-fledged quantum formulation, a hybrid approach in which the state of a system is represented as a wavefunction plus a quantum trajectory. The latter obeys Newton-like equations with a modified potential, V (x)+Q(x,t) [one-dimensional (1D) systems are presumed here], with the quantum correction Q obtained from the wavefunction. More recently, it has been shown that the wavefunction may be dispensed with entirely [3, 4]—i.e., one may construct a complete formulation of (spin-free) quantum mechanics based solely on trajectories, thus enabling true comparisons between classical and quantum mechanics. In a recent article [4], the author derived a 4th-order but otherwise Newton- like ordinary differential equation (ODE), describing the quantum trajectories for 1D stationary scattering states. What is it that makes this particular ODE— together with the Newton ODE—so special? Are there certain properties that only these satisfy, or could a viable dynamical law be constructed from essentially any ODE? Some clues are provided in Ref. [4], and extended there also to the non-stationary case. In this paper, we show that the classical and quantum ODEs are the simplest that can be constructed so as to satisfy the two bedrock physical principles of energy conservation and action extremization. Some other ODE’s satisfying these conditions—representing “alternative dynamical laws”— are also presented, although they are very rare, and only a handful have been discovered thus far. The dynamical laws of nature are evidently extremely spe-

6 cial. We begin with a derivation of the classical dynamical law. Consider a candidate trajectory, x(t), together with potential V [x] and kinetic T [˙x] “functionals” (term used somewhat loosely). Neither functional form is yet specified; they are for the moment regarded as unknowns. For essentially any choice of V [x] and T [˙x], an allowed dynamical trajectory x(t) is defined to be one which extrem- izes the action, i.e. the time integral of the Lagrangian, L[x, x˙] = T [˙x] − V [x]. This condition leads (via Euler-Lagrange) to a 2nd-order ODE for x(t), whose precise form depends on the specification of the V and T functionals. If in addition to action extremization, one also imposes the condition that energy, E[x, x˙]= T [˙x]+V [x] be conserved, this results in a constraint on the functionals themselves. Specifically, though V [x] is still found to be unconstrained, T [˙x] must take the form of a constant timesx ˙ 2. Identification of that constant with one half the mass, m/2, leads at once to Newton’s ODE. Thus, only the standard forms of classical physics emerge as permissible, i.e., no other candidate kinetic energy (e.g., T [˙x] = Ax˙ 4) would lead to a consistent dynamical law satisfying both conditions above. One might argue that the above analysis is overly simplistic, in that a fairly specific decomposition of the Lagrangian functional into a pure x component plus a purex ˙ component is presumed. In particular, this implies a coordinate x in terms of which space is homogeneous (albeit another standard principle of physics), as a result of which the kinematic kinetic energy quantity T [˙x] must be x-independent. In any case, a generalized coordinate treatment changes nothing fundamental about the above conclusions. With q denoting the generalized coordinate, one has the more general form T [q, q˙]= T [f[q]˙q], where both T and f are essentially arbitrary. The two conditions above then imply that E[q, q˙]= H[q, q˙], (1) where the Hamiltonian H is the usual (minus) Legendre transform of L[q, q˙] in the variable p = ∂L/∂q˙. Equation (1) is not satisfied in general; but the allowed forms that do satisfy this equation correspond precisely to those of the standard generalized Lagrangian formulation of mechanics (i.e., encompassing arbitrary point transformations). A further generalization—the broadest possible—would consider completely arbitrary L[q, q˙] functionals; but in this case, T and V contributions (and therefore E) can no longer be defined (though H still can be). The functionals above involve only 1st-order time derivatives of x(t); deriving additional dynamical laws therefore clearly requires higher-order time deriva- ... tives. A kinematic quantum correction functional, Q[˙x, x,¨ x ,...], is posited, in terms of which the Lagrangian and energy functionals become: ... L[x, x,˙ x,...¨ ] = T [˙x] − V [x] − Q[˙x, x,¨ x ,...] (2) ... E[x, x,˙ x,...¨ ] = T [˙x]+ V [x]+ Q[˙x, x,¨ x ,...] (3) Application of the action extremization and energy conservation conditions then

7 does two important things. First, it imposes an extremely severe constraint on the allowed functional form for Q. Second, it leads to a 2nd-or-higher-order ODE for x(t)—i.e., the new dynamical law—whose solutions are the corresponding quantum trajectories. A systematic determination of all permissible dynamical laws has thus been ... performed, for Q[˙x, x,¨ x ,...] functionals of successively higher orders, up to 3rd order. The complete set of meromorphic solutions is presented below:

V [x] = completely unconstrained m T [˙x] = x˙ 2 2 A order 0 (classical mechanics)  no solutions order 1 Q[˙x, x,...¨ ] =  (4) no solutions order 2 n n−2... x¨ − 2n x¨ x B x2n n x2n−1 order 3  ˙ 4 +2 ˙  To date, no solutions at higher than 3rd order have been found, although their existence has not been disproven. Any linear combination of the above solutions is also a solution, as the dynamical law problem itself is linear (as opposed to the corresponding quantum trajectory ODEs). With regard to the value of the parameter n, meromorphicity requires that this must be an integer. For n< 2, undesirable singular trajectories result. The simplest solution with well-behaved trajectories is therefore n = 2. This choice leads exactly to the trajectories of standard quantum mechanics [4], with the identiﬁcation B = −(5/4)(~2/2m). A 1st-order non-meromorphic solution has also been discovered, which may be relevant for bipolar quantum trajectory methods [5, 6]. Other future work will include generalization for spin, non-stationary states, and multiple dimensions. Acknowledgments. B. Poirier gratefully acknowledges a grant from The Robert A. Welch Foundation (Grant No. D-1523) and a Small Grant for Exploratory Research from The National Science Foundation (Grant No. CHE-0741321).

[1] N. Bostrum, Anthropic Bias: Observation selection effects in science and philosophy (Routledge New York, 2002). [2] D. Bohm, Phys. Rev. 85 (1952) 166; Ibid. 180. [3] P. Holland, Ann. Phys. 315 (2005) 505. [4] B. Poirier, Chem. Phys. 370 (2010) 4. [5] N. Fröman and P. O. Fröman, JWKB Approximation (North-Holland, 1965). [6] C. Trahan and B. Poirier, J. Chem. Phys. 124 (2006) 034115; Ibid. 034116.

8 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

The Bipolar Reaction Path Hamiltonian (BRPH) Approach for Multi-Dimensional Reactive Scattering Systems

Jeremy B. Maddox(a) and Bill Poirier(b) (a) Department of Chemistry, Western Kentucky University, Bowling Green, Kentucky 42101-1079, United States and (b) Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409-1061, United States

One generic goal of chemical physics is to predict the reaction probabilities of elementary chemical reactions using quantum theory. Fundamentally, these reactive collisions can be viewed as quantum scattering processes, and the reaction probabilities are related to the elements of the scattering matrix, or S-matrix. The S-matrix may be calculated from the solutions of either the time-dependent or time-independent Schrödinger equations (TDSE and TISE, respectively) [1]. In the present work we follow the latter approach and describe a new method for calculating stationary wave functions of a multi-dimensional (multi-D) reactive scattering system. We build upon the bipolar counter-propagating wave methodology (CPWM) developed by Poirier and co-workers for various applications in chemical physics [2–5]. The term “bipolar” refers to a specific representation of a stationary wave function in terms of two traveling waves that propagate in opposite directions. We invoke an adiabatic representation of the scattering system’s Hamiltonian that effectively reduces the multi-D problem to a 1D problem involving multiple scattering amplitudes. This approach also provides a convenient framework to connect CPWM algorithms [6] with the reaction path Hamiltonian formalism developed by Miller and co-workers for applications in polyatomic reactive scattering [7]. Hence, we refer to the method as the bipolar reaction path Hamiltonian (BRPH) approach to reactive scattering. To illustrate the BRPH we consider a 2D scattering problem with a linear reaction coordinate x and a vibrational coordinate y. The Hamiltonian of the system is given by

ˆ − ~2 2 2 H = ( /2m)(∂x + ∂y )+ V (x,y) (1) where m is a reduced mass and V is a potential energy function. For simplicity we assume that V is asymptotically (x → ±∞) symmetric, such that the re- actants and products have the same eigenfunctions with energies Ei, which are

9 identified with the different scattering channels of the system. Furthermore, we define a set of adiabatic eigenfunctions φi(x,y) and energies ǫi(x) that depend parametrically on the reaction coordinate via the x-dependence of V . It is im- plied that V (x,y) → V (y), ǫi(x) → Ei, and φi(x,y) → φi(y), asymptotically. A given stationary state with energy E is expanded as a sum over scattering chan- n nels φE(x,y)= i ai(x,t)φi(x,y), where n is the number of open channels (i.e., φi) satisfying E>EP i. The scattering amplitudes ai will satisfy different boundary conditions, thus distinguishing between members of the n-fold degenerate set of total scattering states, i.e., φE. We invoke the so-called “constant velocity” bipolar decomposition of the amplitudes ai = ai+ + ai−, where ai± are referred to as the BRPH components. These are associated with a pair of counter-propagating traveling waves

ai±(x,t)= αi±(x) exp[(i/~)(±pix − Et)] (2) where pi = 2m(E − Ei) is the momentum of a free particle with kinetic energy E − Eip. We adopt a scattering convention, such that the ai+’s represent incident/transmitted waves that move with constant positive momenta, and the ai−’s represent reflected waves with negative momenta. The αi±’s depend on the scattering potential and are determined numerically using CPWM algorithms. The fact that the superposition of the BRPH components must be a solution of the TISE constrains the relationship between the various ± amplitudes. Some time ago, Fröman and Fröman (FF) derived this constraint in the context of a generalized semiclassical theory [8]. In the present work, the FF condition is given by ∂xai = (i/~)pi(ai+ − ai−) and may be combined with the TISE to construct a set of Lagrangian-like hydrodynamic equations of motion for the BRPH components

dtai± = ∂tai± ± vi∂xai± = Fiai± + Gi(ai+ + ai−)+ Hi (3) where

i − Fi = ~(E 2Ei) (4) − i − Gi = ~(ǫi Ei) (5) ~ i ∗ ∗ 2 Hi = 2 φ ∂xφjdy ∂xaj + φ ∂ φjdy aj (6) 2m Z i Z i x Xj

The BRPH equations and their solutions can be qualitatively understood in terms of competing fluxes. The scattering boundary conditions impose a flux of incident probability amplitude. The Fi term is associated with the flux carried by a plane-wave, such that both incident/transmitted and reflected amplitude propagate through the scattering region. The Gi term couples the ± components

10 and leads to intra-channel flux that essentially converts incident/transmitted amplitude into reflected amplitude, and vice versa, within the same scattering channel. The Hi term depends on all of the BRPH components and leads to inter-channel flux. Both Gi and Hi will vanish asymptotically. The competitive action of the scattering boundary conditions and various fluxes eventually leads to a steady state at long times, such that both the FF condition and the TISE are satisfied. In Figure 1 we present benchmark results for a 2D scattering problem defined by an Eckart barrier along x and a Morse oscillator along y

2 2 V (x,y) = V0 sech (αx)+ D(x)(1 − exp[−β(y − Y (x))]) (7) 2 D(x) = D0 + V0 sech (αx) (8) 2 Y (x) = y0 +∆y sech (αx) (9) where V0=0.01, D0=0.05, α=3, β=1.2, ∆y=1.25, and m=2000; all quantities have atomic units. The left panel shows slices of the potential at x → −∞ and x = 0. The potential is not separable in x and y. The right panel shows state- to-state reaction probabilities for the lowest three scattering channels. For each energy value we have used independent calculations using ﬁnite diﬀerences to test whether the calculated total stationary states satisfy both the FF condition and the TISE. The numerical test for the percent error in the energy is evaluated as ((Hφˆ E)/φE − E)/E over the spatial grid. The median percent error, with respect the grid, in the calculated total energies were less than 0.002 for all energy points shown. We are presently extending the BRPH approach to more realistic scattering problems with curvilinear reaction coordinates. Acknowledgments. J. B. Maddox gratefully acknowledges start-up support from the Department of Chemistry and the Applied Research and Technology Program (ARTP) at Western Kentucky University. B. Poirier gratefully acknowledges a grant from The Welch Foundation (Grant No. D-1523) and a Small Grant for Exploratory Research from The National Science Foundation (Grant No. CHE-0741321).

[1] D. J. Tannor, Introduction to quantum mechanics: A time-dependent perspective (University Science Books, 2007). [2] B. Poirier, J. Chem. Phys. 121 (2004) 4501. [3] C. Trahan and B. Poirier, J. Chem. Phys. 124 (2006) 034115; Ibid. 034116. [4] B. Poirier and G. Parlant, J. Phys. Chem. A 111 (2007) 10400. [5] B. Poirier, J. Chem. Phys. 128 (2008) 164115; Ibid. 129 (2008), 084103. [6] B. Poirier, J. Theor. Comput. Chem. 6 (2007) 99. [7] W. H. Miller, N. C. Handy and J. E. Adams, J. Chem. Phys. 72 (1980) 99. [8] N. Fr¨oman and P. O. Fr¨oman, JWKB Approximation (North-Holland, 1965).

11 12 1.0

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E æ

6 æ

V æ æ ç ç + ç ç 0.4 +

probability ç 4 + ç æ + ç + + ç + + + ç æ 0.2 + + 2 ç + + + + + æ + ç æ + ç æ ç æ ç 0 0.0 çæ+ ç+ ç+ ç+ ç+ ç+ ç+ ç+ ç+ ç ç ç ç 0 1 2 3 4 5 1 2 3 4 5 6

y HbohrL EE0

FIG. 1: (Left) Potential energy slices for the Eckart+Morse oscillator problem. (Right) State-to-state reaction probabilities: (ﬁlled) 0-0, (empty) 1-1, and (cross) 0-1. Horizon- tal/vertical lines indicate the lowest eigenenergies of the asymptotic Morse oscillator.

12 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

An Iterative Finite Diﬀerence Method for Solving the Quantum Hydrodynamic Equations of Motion

Brian K. Kendrick Theoretical Division (T-1, MS-B268) Los Alamos National Laboratory Los Alamos, NM 87545 USA

I. INTRODUCTION

The quantum hydrodynamic equations of motion associated with the de Broglie-Bohm[1–5] description of quantum mechanics describe a time evolving probability density whose “fluid” elements evolve as a correlated ensemble of particle trajectories. These equations are intuitively appealing due to their similarities with classical fluid dynamics and the appearance of a generalized Newton’s equation of motion in which the total force contains both a classical and quantum contribution. However, the direct numerical solution of the quantum hydrodynamic equations (QHE) is fraught with challenges: the probability “fluid” is highly-compressible, it has zero viscosity, the quantum potential (“pressure”) is non-linear, and if that weren’t enough the quantum potential can also become singular during the course of the calculations. Collectively these properties are responsible for the notorious numerical instabilities associated with a direct numerical solution of the QHE. The most successful and stable numerical approach that has been used to date is based on the Moving Least Squares (MLS) algorithm.[6–8] The improved stability of this approach is due to the repeated local least squares fitting which effectively filters or reduces the numerical noise which tends to accumulate with time. However, this method is also subject to instabilities if it is pushed too hard. In addition, the stability of the MLS approach often comes at the expense of reduced resolution or fidelity of the calculation (i.e., the MLS filtering eliminates the higher-frequency components of the solution which may be of interest). Recently, a promising new solution method has been developed which is based on an iterative solution of the QHE using finite differences.[9] This method (referred to as the Iterative Finite Difference Method or IFDM) is straightforward to implement, computationally efficient, stable, and its accuracy and convergence properties are well understood. A brief overview of the IFDM will be presented followed by a couple of benchmark applications on one- and two-dimensional

13 Eckart barrier scattering problems.

II. METHODOLOGY

In the de Broglie-Bohm description of quantum mechanics, the complex wave function is written in polar form ψ(x,t) = exp[C(x,t)] exp[iS(x,t)/~] and substituted into the time dependent Schrödinger equation. The velocity field is defined as v(x,t)= ∇S/m where m is the mass. The resulting quantum hydrodynamic equations (QHE) are given by (in a fixed Eulerian frame) dC 1 = − ∇· v − v ·∇C, (1) dt 2

dv m = −∇ (V + Q) − m v ·∇v . (2) dt The classical potential is denoted by V and the quantum potential Q is given by ~2 Q = − (∇2C + |∇C|2) . (3) 2m

The classical and quantum forces are given by fc = −∇ V and fq = −∇Q, respectively. The probability density is ρ = e2C and the quantum trajectories are obtained by solving the equation x˙ = v. The IFDM is based on an iterative solution (Newton’s method) of the nonlinearly coupled QHE using an implicit second-order central differencing in both space and time. A “control volume” approach is used for which the velocity grid is “staggered” relative to the spatial grid. The use of a staggered velocity grid is important in order to prevent the appearance of spurious oscillations in the numerical solutions.[10] In one-dimension, the x coordinate is discretized using a set of N computational cells with centers located at xj where j = 0,...N +1 and width ∆x. The velocities vj+1/2 are defined on the “cell walls” which lie half-way between the cell centers (i.e., xj+1/2 = (xj + xj+1)/2). The time coordinate is discretized using a set of evenly spaced time intervals tn (n = 0, 1,...) separated by ∆t. The 2nd-order finite difference expressions for the QHE (Eqs. 1 and 2) are given by[9]

n+1 n n+1 n+1 n+1 n+1 (Cj − Cj )/∆t = − h(vj+3/2 − vj−1/2)+(vj+1/2 − vj−3/2) n n n n +(vj+3/2 − vj−1/2)+(vj+1/2 − vj−3/2)i/(16 ∆x) n+1 n+1 n+1 n+1 − h(vj+1/2 + vj−1/2)(Cj+1 − Cj−1 ) n n n n +(vj+1/2 + vj−1/2)(Cj+1 − Cj−1)i/(8∆x) , (4)

14 and

n+1 n n+1 n n+1 n+1 n+1 m (vj+1/2 − vj+1/2)/∆t = (fj+1/2 + fj+1/2)/2 − m h vj+1/2(vj+3/2 − vj−1/2) n n n + vj+1/2(vj+3/2 − vj−1/2)i/(4∆x) , (5) where f = fc +fq denotes the total force. Equations 4 and 5 are solved iteratively by rewriting these two equations as

n+1 n f1(C,v)= Cj − Cj − ∆t RHSCeq , (6) n+1 n f2(C,v)= vj+1/2 − vj+1/2 − ∆t RHSveq/m , (7) where RHSCeq and RHSveq denote the right-hand-side of Eqs. 4 and 5, re- n+1 n+1 spectively. Using an abbreviated notation where C = Cj and v = vj+1/2, Newton’s method in two variables consists of expanding f1 and f2 in a Taylor series expansion with respect to C = Co +∆C and v = vo +∆v where Co and vo represent an initial guess (typically the solution from the previous time step: n n Co = C and vo = v )

∂f1 ∂f1 f1(C,v)= f1(C ,v )+ ∆C + ∆v, (8) o o ∂C ∂v ∂f2 ∂f2 f2(C,v)= f2(C ,v )+ ∆C + ∆v. (9) o o ∂C ∂v

By setting f1(C,v) = 0 and f2(C,v) = 0, the ∆C and ∆v can be expressed in terms of the known quantities f1(Co,vo) and f2(Co,vo) and their derivatives with respect to C and v. Analytic expressions for the derivatives of f1 and f2 with respect to C and v can be derived by taking the appropriate derivatives of Eqs. 6 and 7.[9] Once the ∆C and ∆v are computed, the values of C and v are updated and these updated values become the new guess (i.e., the Co and vo). Equations 8 and 9 are then solved for new values of ∆C and ∆v and the process is repeated until convergence is achieved. In practice, the absolute values of the ∆C,∆v, f1, and f2 are monitored until all four are simultaneously less than some user specified convergence threshold at all of the grid points. The converged C and v are now the correct values at the future time step tn+1. The time index is incremented n → n + 1 and the process is repeated to find the values of C and v at the next time step and so on. The iterative solution method converges exponentially with respect to the iteration count which typically varies between 10 - 20 iterations depending upon the size of the time step (∆t) and the desired level of convergence. At each time step, the values of the C0, CN+1, v−1/2, and vN+3/2 at the edges of the grid are determined by applying the appropriate finite- difference expressions for the Gaussian-like boundary conditions: ∂2C/∂x2 = constant and ∂2v/∂x2 = 0.[9]

15 A stability analysis of the IFDM was performed using the method of truncation error analysis.[9] In this approach, Taylor series expansions of the C and v with respect to ∆x and ∆t are substituted into the finite difference Eqs. 4 and 5 and the truncation error terms (i.e., the terms multiplying ∆x2 and ∆t2) are explicitly derived. Some of these error terms represent numerical diffusion 2 2 2 2 and take the form γv ∂ v/∂x and γC ∂ C/∂x (the explicit functional forms for γv and γC are given in Ref. 9). The effective diffusion coefficients are given 2 2 by Γv = ∆t γv and ΓC = ∆t γC which can be positive or negative and can also change sign during the course of the calculation. Even a temporary occurrence of a negative diffusion coefficient can lead to instabilities and a complete breakdown of the calculations. To ensure a stable propagation, the overall diffusion coefficient must remain positive. This can be accomplished using the well-known hydrodynamic technique called “artificial viscosity”.[11–13] In this approach, a positive diffusion term with the same functional form as that derived in the truncation error analysis is introduced into the finite difference equations. The positive diffusion coefficient multiplying this new diffusive term is chosen large enough to ensure that the overall diffusion coefficient is always positive so that the propagation will remain stable. In other words, the overall diffusion (truncation error + artificial viscosity) is always dissipative or stabilizing. It is important to note that the diffusive artificial viscosity term is of the same order in ∆t2 as the truncation error. Thus, the stability of the calculation can be maintained even as the grid spacing and time step are decreased (i.e., the resolution of the calculation is increased).

III. APPLICATIONS

The IFDM has been applied to three different test cases: (1) a one-dimensional free Gaussian wave packet, (2) a one-dimensional Gaussian wave packet scattering off an Eckart barrier, and (3) a two-dimensional Gaussian wave packet scattering off an Eckart barrier coupled with a harmonic oscillator potential in the other degree of freedom.[9, 14, 15] The IFDM results for these three different applications were compared to the results of two other numerical methods: the Crank-Nicholson algorithm[11] (which is based on a finite difference solution of the time-dependent Schrödinger equation) and the MLS/ALE approach[8] (which is another quantum hydrodynamic method). These comparisons provide an initial assessment of the convergence properties, numerical accuracy, computational efficiency, and stability of the IFDM. The accuracy of the IFDM solution for the one-dimensional free Gaussian wave packet was found to be in excellent agreement with the analytic result with an average percent difference of less than 5 × 10−5 % using a grid spacing of ∆x = 0.05 au. In contrast the Crank-Nicholson method required a much smaller grid spacing of ∆x = 0.0005 au to obtain an average percent difference

16 of 6 × 10−3 %. This is due to the fact that the Crank-Nicholson algorithm is directly solving for the highly oscillatory real and imaginary parts of the wave function. The accuracy of the 4-th order MLS/ALE solution for the free Gaussian wave packet was found to be similar to the IFDM. The stability of the IFDM was verified out to the edges of the fixed (Eulerian) grid where the probability density is extremely small ρ< exp(−40, 000). For the one- and two-dimensional Eckart barrier tunneling problems, the IFDM transmission probabilities were found to be in excellent agreement with both the Crank-Nicholson and MLS/ALE methods. In all three applications, the IFDM proved to be more computationally efficient than either the Crank- Nicholson or MLS/ALE approach. For the one-dimensional Eckart barrier tunneling problem, the IFDM transmission probabilities were computed 1.8 and 6.9 times faster than those computed using the Crank-Nicholson and MLS/ALE methods, respectively.[9] For the two-dimensional Eckart barrier tunneling problem, the IFDM transmission probabilities were computed in 15 minutes compared to the 1 hour required by the ALE/MLS approach.[15] The latest version of the IFDM requires no filtering of the second derivative of C in the evaluation of the quantum force.[15] The original version used a moving window average of ∂2C/∂x2 to help reduce the numerical noise and improve stability.[9] We have found that this window averaging can be eliminated 1 all together by simply increasing the artificial viscosity coefficient to νo = 1 × 105 and truncating the fixed (Eulerian) grid to some reasonable value of the density (i.e., ρ> 1 × 10−8). A “smart” Eulerian grid is then implemented which “activates” or adds new edge points to the grid when the density at the edge point increases above a user specified threshold (i.e., ρ> 1 × 10−8). The values of C and v at the edge point are initialized in a stable way by using a finite difference representation of the appropriate Gaussian like boundary conditions at the edge (i.e., ∂2C/∂x2 = constant and ∂2v/∂x2 = 0). These modifications not only improve the accuracy and resolution of the method by eliminating the window averaging of the second derivative of C, they also significantly improve the computational efficiency of the method by reducing the size of the grid. The calculations are performed only at those “active” Eulerian grid points where the density is significant. In this way, the fixed Eulerian grid is optimized so that it exhibits similar computational advantages as a Lagrangian or ALE grid.

IV. CONCLUSIONS AND FUTURE WORK

The Iterative Finite Difference Method (IFDM) is a promising new approach for the direct numerical solution of the quantum hydrodynamic equations of motion.[9] The set of non-linearly coupled finite differenced quantum hydrodynamic equations are solved iteratively at each time step using Newton’s method. The IFDM converges exponentially with respect to the iteration count and is

17 second order accurate in both space and time. The stability of the method was investigated using truncation error analysis and the functional form of the potentially negative (unstable) numerical diffusion terms were identified. The stability of the IFDM is ensured by introducing a positive (stabilizing) numerical diffusion term (“artificial viscosity”) into the finite difference equations. This stabilizing term is of the same order as the errors which are already present in the calculation. Thus, the stability of the method can be maintained even as the resolution of the calculation is increased. The methodology was briefly reviewed and the results of several initial benchmark calculations discussed. These initial applications show that the IFDM is computationally efficient, stable, and numerically accurate. Recent improvements to the IFDM include the elimination of all filtering or averaging of the second derivative of C and a truncation of the fixed (Eulerian) grid. A “smart” Eulerian grid is used for which the calculations are performed only at active grid points which contain significant probability density. In its present form, the IFDM does not solve the “node problem” (i.e., the singularities in the quantum potential and force which can occur in the regions where the density approaches zero due to the interference effects in the reflected component of a wave packet scattering off a barrier).[16] However, an important feature of the IFDM is that it remains stable even in the presence of nodes which allows the propagation to continue and accurate transmission probabilities can be computed. Also, the artificial viscosity technique can be used to mitigate the “node problem” by preventing the formation of nodes (at the expense of smoothing out the solution in the region near the node).[8, 9] Future work will include investigation of a possible analytic treatment of the “node problem” within the IFDM, its extension to higher dimensions, and most importantly its application to real molecular systems.

V. ACKNOWLEDGMENTS

This work was done under the auspices of the US Department of Energy at Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Los Alamos National Security, LLC, for the National Nuclear Security Adminis- tration of the US Department of Energy under contract DE-AC52-06NA25396.

[1] L. de Broglie, C. R. Acad. Sci. Paris. 183, 447 (1926); 184, 273 (1927). [2] D. Bohm, Phys. Rev. 85, 166 (1952); 85, 180 (1952). [3] P. R. Holland, “The Quantum Theory of Motion”, Cambridge University Press, New York, 1993.

18 [4] R. E. Wyatt, “Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics”, Springer, New York, 2005. [5] “Quantum Trajectories”, edited by P. K. Chattaraj, CRC Press/Taylor & Francis Group, 2010. [6] C. Lopreore and R. E. Wyatt, Phys. Rev. Lett. 82, 5190 (1999). [7] R. E. Wyatt, Chem. Phys. Lett. 313, 189 (1999). [8] B. K. Kendrick, J. Chem. Phys. 119, 5805 (2003). [9] B. K. Kendrick, J. of Molec. Struct: THEOCHEM 943, 158 (2010). [10] S. V. Patankar, “Numerical Heat Transfer and Fluid Flow”, Hemisphere Publish- ing Co., New York, 1980. [11] W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, “Numerical Recipes; The Art of Scientiﬁc Computing”, Cambridge University Press, New York, 1986. [12] J. VonNeumann and R. D. Richtmyer, J. Appl. Phys. 21 (1950) 232. [13] E. Scannapeico and F. H. Harlow, Los Alamos National Laboratory report LA- 12984, (1995). [14] D. K. Pauler and B. K. Kendrick, J. Chem. Phys. 120 (2004) 603. [15] B. K. Kendrick, unpublished. [16] R. E. Wyatt and E. R. Bittner, J. Chem. Phys. 113 (2000) 8898.

19 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Bangor

Kinematic Quantum Trajectories

Timothy M. Coﬀey∗ Department of Physics and Center for Complex Quantum Systems, 1 University Station C1600, University of Texas, Austin, TX 78712, USA

The typical route to quantum trajectories by Bohm [1, 2] uses Schrödinger’s wave equation. The complex wave function is written in a polar form ψ(~x, t)= R(~x, t)exp[−iS(~x, t)] for real and single-valued functions R and S. The wave equation then implies a continuity equation and a quantum Hamilton-Jacobi’s equation. The particles evolve according to a guidance law, ~p = ∇S, and in addition to the classical potential, they are also influenced by a quantum potential Q = −(~2/2mR)∇2R. Bohm’s construction is dynamical in nature since it provides causes for the particle motions: masses, forces, potentials. In addition, for Bohm these trajectories are ontological, that is, an electron, for example, actually is a particle with a specific position and momentum that evolves along a trajectory [2]. The kinematic route to quantum trajectories, on the other hand, doesn’t utilize or solve any equations of motion, but rather develops a geometrical or informational description of only the evolution of the probability density ρ = ψ∗ψ. By design kinematic particle trajectories are conserved, and for all times are distributed according to the quantum probability density. There- fore, by their respective continuity equations the velocity fields are related, ~vkinematic = ~vBohm + ~va/ρ with ∇ · ~va = 0. In one-dimension the gauge velocity va is necessarily zero, and thus in one-dimension all kinematic trajectories will be identical to Bohm’s trajectories. One possible way to achieve kinematic trajectories is by simply sampling the density, sorting the sampled points, and then connecting the points in order [3]. This sampling method, however, is limited to one-dimensional problems (and in higher dimensions if the wave function is separable in each coordinate) due to the lack of a natural sorting order in higher dimensions. A recent method [4] that employs the geometrical construction of centroidal Voronoi tessellations(CVT) [5, 6], though, overcomes this ordering problem by maintaining particle identity throughout the calculation. Using the CVT method for separable and non-

∗Email: [email protected]

20 separable two-dimensional examples suggest that the velocity gauge ~va, might typically be zero, which means that the CVT and Bohm trajectories typically might be identical. The only disparity known at this time between the CVT method and Bohm trajectories is around a persistent stationary node. Around the node the Bohm trajectories might have circular orbits or be at rest, while the CVT trajectories are always at rest. Neither case is experimentally veriﬁable, however, but the CVT method seems to provide a more consistent description of the situation. The CVT approach realizes that the particles themselves are some sort of representation of the probability density. For a ﬁnite number of particles, N, the measure of the quality of the representation is done by introducing a novel distortion functional,

N D = (~x − ~x )2ρ(~x)γ d~x, (1) Z i Xi=1 Ci where γ = (k + 2)/k, k is the number of dimensions (the length of each position vector ~xi), and each integration is taken over the Voronoi volume [7], Ci = {~x | k ~x − ~xi k≤k ~x − ~xj k for all j 6= i}.. This distortion functional is similar to the distortion measure in the field of vector quantization or signal compression [5], though the word ‘quantization’ in this field has nothing to do with quantum mechanics. In vector quantization the distortion functional has the same form as (1) except with γ = 1. The best representation of the probability density, the set of particle ~xi’s, is defined to be the one which minimizes the distortion. The characteristic of the minimum particle configuration is that they form a CVT. The global minimization of the distortion function introduces a non-locality behavior of the CVT trajectories, which is also present for Bohm trajectories. Frequently, the CVT’s are computed using the Lloyd-Max iterative algorithm (also known in the literature simply as the Lloyd algorithm) [8, 9]. The Lloyd- Max algorithm typically begins with a random sampling of the density as shown by the dots in Figure 1(a); the density has two high dense regions in the lower- left and upper-right corners of the figure. The lines in the figure are the Voronoi tessellation of this particle configuration. During each iteration the algorithm computes the Voronoi tessellation of the particle positions, then each particle ~xi ′ is moved to the center of mass or centroid of its particular Voronoi cell Ci by, ~xi =

Ci ρ(~x)~xdV/ Ci ρ(~x) dV. The algorithm continues until some stopping criterion isR satisfied; typicallyR either a fixed number of iterations, or the maximum distance any particle moves during the iteration is less than some small predetermined value. Shown in Figure 1(b) is the resulting CVT after 200 such iterations. Notice the uniformity of the structure in part (b) as opposed to the tessellation in part (a). The Lloyd-Max algorithm is beneficial since it is easy to implement, and has several non-degeneracy and global minimum or fixed-point convergence proofs in one and many dimensions [10–12]. More importantly, the algorithm

21 (a) (b)

FIG. 1: (a) A Monte-Carlo sampling of a probability density with high dense regions in the lower-left and upper-right. The single dots represent the possible particle positions. The straight lines are the Voronoi tessellation of these particle positions. (b) The centroidal Voronoi tessellation after 200 iterations of the Lloyd-Max algorithm that began from the initial sampling.

keeps track of each particle’s position during the entire computation, which is necessary for identifying each particle to build its trajectory. The CVT trajectory method begins by sampling the probability distribution at t = 0 to get N particle positions. The initial sampling, then, is used to construct an initial CVT at t = 0. Time is then advanced a small amount δt, and rather than resampling the probability density, the CVT at t = 0 is recycled and used as the initial particle conﬁguration for the CVT computation at the new time. The calculation of the center of mass of each Voronoi cell Ci during the Lloyd- Max algorithm is done not with the usual ρ(~x; t), but instead with ρ(~x; t)(k+2)/k in keeping with the distortion functional in (1). Recall that the Lloyd-Max algorithm keeps track of each particle’s position during the CVT computation, therefore, one can chain the i-th particle’s positions at the various times to form a trajectory. In Figure 2 is shown an example of a two-dimensional CVT at three times, and the construction of a particular trajectory. Notice that the CVT method will never have trajectory intersections, which again is a familiar behavior of the Bohm trajectories as well. In Figure 3 is a random subset of 20 trajectories from the total ensemble of N = 400 trajectories computed for a two-dimensional free gaussian packet. The packet begins concentrated at the center of the box, and then spreads out in time. For each CVT trajectory (+) the corresponding Bohm trajectory (solid line) is calculated. In the ﬁgure, we can see that the CVT trajectories match the Bohm trajectories quite well. In fact, for the whole ensemble the Pearson correlation

22 FIG. 2: At each time a centroidal Voronoi tessellation(CVT) is computed using the particle positions from the previous time as input. The Lloyd-Max algorithm begins with the old positions, but uses the probability distribution at the new time. Each particle’s position is tracked during each iteration of the Lloyd-Max algorithm. After the algorithm stops the trajectories are constructed by mapping a particle’s old position to the new position as shown in the ﬁgure.

coeﬃcients between the components of the CVT and Bohm trajectories were rx = 0.996 and ry = 0.997. Recall that the Bohm trajectories are derived from dynamical equations of motion, while the CVT method utilizes a geometrical depiction of the density. By their nature, the CVT trajectories are simply kinematically portraying the evolution of the probability density. In the cases in which the CVT method results in Bohm trajectories, it would seem it is unreasonable to interpret the Bohm trajectories as ontological. Rather, the Bohm trajectories, like the CVT trajectories, should be thought to simply be kinematically depicting the evolution of the probability density as well.

23 yHtL 80

70 + + + + + + + + + + 60 + + + + + + + + + +++ ++ ++ + +++ + + + +++ ++ +++ + ++++ ++ ++ +++++++ ++ ++++ +++++++ ++ ++++++++++++++++++++++++++ 50 +++++++++++ ++++++++++ + + ++++ ++++ + + ++ + +++ + ++ ++ + + + +++ + + + + ++ + + + ++ + + + 40 + + ++ + + + + + + + + + + + + + + ++ + + + ++ + + + 30 + + +

20 xHtL 20 30 40 50 60 70 80

FIG. 3: A comparison of the Bohm trajectories (solid line) and the CVT trajectories (+) for a two-dimensional gaussian wave packet. The gaussian begins concentrated at the middle of the ﬁgure, and as time progresses the gaussian spreads. The ﬁgure shows a random subset of 20 trajectories from the total of 400 particles used in the calculation.

[1] D. Bohm, Phys. Rev. 85 (1952) 166, 180. [2] D. Bohm and B. J. Hiley, The Undivided Universe (Routledge, New York, 1993). [3] T. M. Coﬀey, R. E. Wyatt and W. C. Schieve, J. Phys. A: Math. Theor. 41 (2008) 335304. [4] T. M. Coﬀey, R. E. Wyatt and W. C. Schieve, J. Phys. A: Math. Theor. 43 (2010) 335201. [5] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression (Kluwer Academic, Boston, 1992). [6] Q. Du, V. Faber and M. Gunzburger, SIAM Rev. 41 (1999) 4 637. [7] A. Okabe, B. Boots and K. Sugihara, Spatial Tessellations (John Wiley & Sons, New York, 1992). [8] S. P. Lloyd, IEEE Trans. Inf. Theo. IT-28 (1982) 129, Reprinted in Quantization, edited by P.F. Swaszek (Van Nostrand Reinhold, New York, 1985). [9] J. Max, IEEE Trans. Inf. Theo. IT-6 (1960) 7, Reprinted in Quantization, edited by P.F. Swaszek (Van Nostrand Reinhold, New York, 1985). [10] Q. Du, M. Emelianenko and L. Ju, SIAM J. Num. Anal. 44 (2006) 1 102. [11] Q. Du and M. Emelianenko, SIAM J. Num. Anal. 46 (2008) 3 1483. [12] M. Emelianenko, L. Ju and A. Rand, SIAM J. Num. Anal. 46 (2008) 3 1423.

24 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

Complex Trajectories and Dynamical Origin of Quantum Probability

Moncy V. John Department of Physics, St. Thomas College, Kozhencherry, Kerala 689641, INDIA

Complex quantum trajectories were ﬁrst obtained and drawn by adopting a modiﬁed de Broglie-Bohm approach to quantum mechanics [1]. First we notice ˆ that substituting eiS/~ for Ψ in the Schrodinger equation gives the quantum Hamilton-Jacobi equation [2]. Inspired by its similarity with the corresponding classical equation, we postulate an equation of motion similar to that in the pilot-wave theory of de Broglie [3],

∂Sˆ ~ 1 ∂Ψ mx˙ ≡ = , (1) ∂x i Ψ ∂x for the particle. To use this, one needs to find Sˆ from the standard solution Ψ of the Schrodinger equation. The trajectories x(t) are obtained by integrating the above equation with respect to time. In general they will lie in a complex x-plane. The eigentrajectories x(t) ≡ xr(t)+ixi(t) in the free particle, harmonic oscillator and potential step problems and trajectories for a wave packet solution were obtained in [1]. As an example, some of the complex trajectories in the n = 1 harmonic oscillator are shown in figure 1. These appear as the famous Cassinian ovals. The Jacobi lemniscate that passes through xr = 0, xi = 0 is a special case of these ovals. One of the challenges before this complex quantum trajectory representation is to explain the quantum probability axiom. In a recent work which explores the connection between probability and complex quantum trajectories [4], the Born’s probability density to find the particle around some point on the real axis x = xr0 was found to be given by

xr0 ⋆ 2m Ψ Ψ(xr0, 0) ≡ P (xr0)= N exp − x˙ idxr , (2) ~ Z where the integral is taken along the real line. Since it is deﬁned and used only along the real axis, the continuity equation for probability in the standard

25 FIG. 1: quantum mechanics is valid here also, without any modifications. This possibility of regaining the quantum probability distribution from the velocity field is a unique feature of the complex trajectory formulation. For instance, in the de Broglie approach, the velocity fields for all bound eigenstates are zero everywhere and it is not possible to obtain a relation between velocity and probability. In addition, we consider it desirable to extend the probability axiom to the xrxi-plane and hence look for the probability of a particle to be in an area dxrdxi around some point (xr,xi) in the complex plane. Let this quantity be denoted as ρ(xr,xi)dxrdxi. Here it is natural to impose a boundary condition that ρ agrees with Born’s rule along the real line. Then one must see whether a continuity equation for probability holds everywhere in the xrxi-plane. An extended probability density that satisfies these two conditions in most regions of the complex plane was proposed in [4] as

t −4 1 2 ′ ρ(xr,xi)= ρ0 exp ~ Im mx˙ + V (x) dt , (3) Zt0 2 ′ ′ where the integral is taken along the trajectory [xr(t ),xi(t )]. One needs to know ρ0 at some initial point (xr0,xi0) on the trajectory and if we choose this as (xr0, 0), the point of crossing of the trajectory on the real line, then ρ0 may take the value P (xr0) and may be found using (2). It was shown in [4] that ρ in (3) satisﬁes the desired continuity equation for the particle, as it moves along. However, it is seen that a solution of the continuity equation cannot satisfy the boundary condition for some regions of the complex plane, such as the region inside the lemniscate where trajectories do not enclose the poles ofx ˙. In the context of solving the continuity equation, it is easy to see that this is due to the boundary condition overdetermining the problem. In the trajectory integral approach, one can explain it as a disagreement of the values of ρ at two consecu- tive points of crossing of the trajectory on the real axis, with that prescribed by Born’s rule. Given this situation, we look for a trajectory integral deﬁnition for ρ that can agree with the Born’s rule (on the real line) in such regions, even if it

26 does not obey the continuity equation in the extended region. Such a deﬁnition, similar to that in equation (3), was found in [5] as

− t ′ ∝ 4 1 2 ′ ρ (xr,xi) P (xr0)exp ~ Im mx˙ dt . (4) Zt0 2 Comparing with (3), we note the absence of the potential term V (x) in the integrand. This trajectory integral definition for such regions can be seen to give Ψ⋆(x)Ψ(x) with x complex. This is independently considered in [6, 7], but for the entire complex plane. Since here it is defined only for the subnests which do not enclose the poles ofx ˙, there is no difficulty in normalising the combined probability for the entire plane, which is shown in figure 2.

FIG. 2:

Another important observation we make in this connection is regarding the classical limit of quantum mechanics. It may be noted that the probability axiom in this modiﬁed de Broglie formulation helps to distinguish the classical limit of quantum harmonic oscillator as one in which the oscillator is probable to be found only very close to the real axis [5]. This result is very signiﬁcant for complex quantum trajectories, for it explains why the complex extension is not observable even indirectly in the classical limit. We anticipate that this property is generally true.

[1] M.V. John, Found. Phys. Lett. 15 (2002) 329; quant-ph/0102087 (2001). [2] H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, MA, 1980. [3] G. Bacciagaluppi, A. Valentini, Quantum Theory at the Crossroads, Cambridge University Press, Cambridge, (2009); quant-ph/0609184v1 (2006). [4] M.V. John, Ann. Phys. 324 (2009) 220; arXiv:0809.5101 (2008). [5] M.V. John, Ann. Phys. DOI. 10.1016/j.aop.2010.06.008 (2010) [6] C.-C. Chou, R.E. Wyatt, Phys. Rev. A 78 (2009) 044101. [7] C.-D. Yang, Chaos, Solitons Fractals 42 (2009) 453.

Energy Rays for Electromagnetic Pulses Scattering from Metal-Dielectric Structures

Robert E. Wyatt

Department of Chemistry and Biochemistry University of Texas Austin, Texas 78712

In this study, time-dependent electromagnetic (EM) energy distributions are described for femtosecond pulses scattering from a micron-scale metal- dielectric structure. Associated with these distributions are ensembles of energy rays, the integral curves of the Poynting vector field. The energy rays demonstrate how- and when- EM energy is redistributed in space and time. Typically, energy rays show bending and undulations, even in ‘free space’ and in regions where the refractive index is invariant. Energy rays for the EM field are analogous to Bohm trajectories in quantum mechanics [1], which in the Hydrodynamic Formulation of Quantum Mechanics [2, 3] are integral curves for the probability flux vector field. By analogy, energy rays in EM theory form a central element in the Hydrodynamic Formulation of Electrodynamics [3, 4]. Several earlier investigations have reported computational results on energy rays [5-10]. Unlike the results reported here, these studies dealt with monochromatic scattering processes. Energy rays for EM scattering generally have very different properties from the rays in geometrical optics (GO). In the traditional approach to GO [11], geometrical rays are solutions to the ray equation, d / ds(ndr / ds) n, where 28 s is the arc length along the ray. It is important to note that these rays bend only when the refractive index changes. The ray equation is equivalent to a set of 1-st order equations [12]: dr/ dt ( c / 2 k0 )(2 k ), dk/, dt  G where

2 kS and the geometrical potential is given by G( r ) ( c / 2 k00 )( nk ) . Not very well known is the approach to optical rays that Luneburg developed in the mid-1940s [13]. By applying the method of characteristics to the two Maxwell curl equations, he showed that the ray equation can be obtained without approximation. As a result, it can be stated that the short wavelength approximation involved in the traditional GO derivation is a sufficient but not necessary condition. More recently, an alternate derivation of these results has been presented [14] which does not rely on the method of characteristics. Also, Orefice et al [12] recently derived exact ray tracing equations from the Helmholtz equation. These equations are similar to the 1-st order equations mentioned previously, except that the total potential in the second equation is the

2 sum of G and the wave potential, defined by W ( c / 2 k0 )(  R )/ R , where R is the wave amplitude In this study, energy rays and EM energy densities will be presented for a time-dependent scattering environment involving a micron-scale composite structure having both dielectric and metallic regions. The fields used to generate these results were obtained by numerically integrating the Maxwell curl equations using the FDTD (finite-difference time-domain) algorithm [15-17]. The current study, along with the recent work of Chu and Wyatt [18], are the first to blend the accurate computation of time-dependent EM with synchronous generation of energy rays. For the model studied here, the grating problem, there are three dielectric segments, ‘optical holes’, in the grating and they are separated by

29 reflective metallic segments. Circular electromagnetic wave fronts emanating from the source point first impinge on the central dielectric segment and then encounter the two outer segments at slightly later times. Maxwell’s two curl equations are given by:

E   B/  t, H  J   D/  t. The following constitutive relations for linear isotropic non-magnetic media are assumed in these studies, DE, 0r

B 0 H, where 0 and 0 are the vacuum permittivity and permeability, and

 r is the relative permittivity (dielectric constant). For these studies of scattering ˆ in two dimensions, E Ez k is perpendicular to the x-y plane and the components of H in this plane are given by (Hxy ,H ). At one grid point, a modulated Gaussian source was used to create the electric field. The perfectly matched layer method [19] was used to prevent echoes from the edges of the grid (also see refs. [16, 17]). Equations for the energy rays may be introduced by first considering

Poynting’s theorem [20], which is given by U /  t  S   J  E, in which

J is the current density. In this equation, the Poynting vector and energy density are given by where S is the SEB/, 0 U (1/2)00 EE   BB/   , energy flux and U is the energy density. From S and U, the equation of motion for an energy ray, given the starting position r0 , is dr / dt S/ U. In the current study, an ensemble involving several hundred trajectories were launched at a delayed starting time, chosen so that the field at the source point had decayed to a negligible value. The equation for energy rays is analogous to 30 one of the equations that can be used to compute quantum trajectories [2] in terms of the probability flux and the probability density.

FIGURE 1. Energy rays for the grating problem superimposed upon a contour map of the electric field at the final time step. An ensemble of rays is launched from the grid (green dots) near the left-center region of the figure and some of these rays (red) transmit through the dielectric segments in the grating. The reflected rays (yellow) on the left side of the grating experience large deflections.

FIGURE 2. Enlargement of the lower-left region of Fig. 1. One transmitted ray (red) and three reflected rays (pink and green) are highlighted. The pink reflected ray makes several bounces near the metal surface, and the two green rays make abrupt turns before departing from the reflection zone.

Computational results for grating scattering problem are illustrated in Figs. 1 and 2. The grating is shown as the vertical strip in Fig. 1, with the metal segments (gray) separating three dielectric windows. The Gaussian pulse was created at a source point on the left side of the figure and an ensemble of rays was launched from a set of grid points (green dots) located between the source point and the grating. Rays that transmit the dielectric segments are plotted in red at the final time (800 time steps, or 136 fs) and rays that reflect from the grating are shown in yellow. These rays are superimposed upon a contour map of the electric field. Although it appears that rays may intersect, they never pass through the

32 same spatial point at the same time. Rays that transmit the outer dielectric segments are focused toward larger values of the y coordinate when they emerge from the grating. In addition, there is extensive bending of the reflected rays and the turning points for many of these rays occur to the left of the metal segments. These rays are reflecting in ‘free space’, relatively far from the reflective metal segments. An enlargement of the lower-left region of this figure is shown in Fig. 2. One transmitted ray (red) and three reflected rays (pink and green) are highlighted. The transmitted ray first bounces from the metal surface, then makes it into the dielectric segment. The pink reflected ray makes several bounces near the metal surface, and the two green rays make sharp turns before departing from the reflection zone. Many of the reflected rays bounce several times from the metal surface before finally escaping from this region. For monochromatic scattering, we have already mentioned that a system of coupled 1-st order equations has been derived for the energy rays [12]. In an extension of earlier work dealing with spin 1/2 particles, Holland replaced the Maxwell curl equations by a matrix equation for the Riemann-Silberstein vector [29], which specifies the state of the fields. This state function was then represented in polar form and Hamilton-Jacobi (HJ) and continuity equations were derived for the amplitude and action functions. The HJ equation contains an effective potential having a structure similar to the Bohm quantum potential. Further investigation of the computation and properties of wave potentials would be very informative. A novel approach to the computation of energy rays has recently been described by Coffey and Wyatt [30]. These rays were generated without employing the Poynting vector and no equations of motion were assumed or used. A similar method was used previously to generate quantum trajectories directly from the time evolving probability density [31]. 33

Acknowledgment

This work was supported in part by a research grant from the Robert Welch Foundation (grant number F-0362). We thank Tim Coffey and Chia-Chun Chou for many informative discussions.

[1] D. Bohm, Phys. Rev. 85, 166 (1952). [2] R. E. Wyatt, Quantum Dynamics with Trajectories (Springer, New York, 2005). [3] A. S. Sanz and S. Miret-Artes, to be published. [4] I. Bialynicki-Birula, Nonlinear Dynamics, Chaotic and Complex Systems, E. Infeld, R. Aelazny, and A. Galkowski (Cambridge University Press, Cambridge, 1997), p. 64. [5] R. D. Prosser, Int. J. Theor. Phys. 15, 169 (1976); H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, Phys. Rev. E 62, 7330 (2000); T. Wunscher, H. Hauptmann, and F. Herrmann, Am. J. Phys. 70, 600 (2002); E. Hesse, J. Quant. Spect. & Rad. Trans. 109, 1374 (2008); A. S. Sanz, M. Davidovic, M. Bozic, S. Miret-Artes, Ann. Phys. 325, 763 (2010); M. Gondran and A. Gondran, Am. J. Phys. 78, 598 (2010). [6] M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 2003). [7] A. Orefice, R. Giovanelli, and D. Ditto, Found. Phys. 39, 256 (2009). [8] R. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964). [9] C. Belevier-Cebrerus and M. Rodriguez-Danta, Am. J. Phys. 69, 360 (2001). [10] K. S. Yee, IEEE Transactions on Antennas and Propagation, 14, 302 (1966). [11] A. Taflove and S. C. Hagness, Computational Electrodynamics, The Finite-Difference Time- Domain Method (Artech House, Boston, 2005). [12] C. C. Chu and R. E. Wyatt, to be published [13] J. P. Berenger, J. Comp. Phys. 114, 185 (1994). [14] F. Richter, M. Floian, and K. Henneberger, Europhys. Lett. 81, 67005 (2008) [15] P. R. Holland, Proc. Roy. Soc. A 461, 3659 (2005). [16] T. M. Coffey and R. E. Wyatt, to be published. [17] T. M. Coffey, R. E. Wyatt, and W. C. Schieve, J. Phys. A 43, 335301 (2010).

Quantum Dynamics through Quantum Potentials

S. Duley, S. Giri, S. Sengupta and P. K. Chattaraj*

Department of Chemistry and Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur 721302, India * Author for correspondence: Email: [email protected]

The hydrodynamic interpretation of quantum mechanics was first formulated by Madelung [1] soon after the introduction of Schrodinger’s wave equation. In this formalism the Schrödinger time dependent equation for a single particle is transformed into two fluid dynamical equations: an equation of continuity, and an Euler type equation of motion wherein the probability density 2 2 defined by  is interpreted as the charge density,    and the velocity v of this charge fluid is obtained from the phase of the complex-valued wavefunction. The introduction of the concept of a fluid with associated density and velocity to study time evolution of a quantum system provides a “classical” approach to describe quantum phenomena. It was through the works of Bohm [2,3], de Broglie [4,5] and Takabayashi [6,7] the hydrodynamic interpretation has gained acceptance and has seen wide use in different areas of physics and chemistry. To obtain the quantum fluid dynamical equations, the time dependent single-particle wavefunction is expressed as an ansatz (polar form) 1 i S (r,t)  (r,t)   2 (r,t)exp (1)  Where   2  R 2 ; R and S being real functions of position and time. Substituting this ansatz in the single particle time dependent Schrödinger equation  2   2   V  i  (2) 2m t and separating the real and imaginary parts, the two quantum fluid dynamical equations are obtained as follows, the continuity equation

   .( v)  0 (3) t and the equation of motion; dv m    (V  V ) (4) dt qu where 1 v   S (5) m dv v   (v.)v (6) dt t and  2 1 1 V    2  2 /  2 (7) qu 2m Like the time dependent Schrödinger equation from which they are derived, the fluid dynamical equations essentially constitute an initial value problem. Since the wavefunction  is a single valued function of position, the associated fluid 1 dynamical variables R or equivalently  2 and S need to satisfy certain conditions. The single valuedness of requires that R be single-valued as well, while the phase function S is not so constrained. In fact due to the polar form of the ansatz (equation 1), S can be multivalued so that any two S functions differing by an integral multiple of 2  may give rise to the same . In a nodal region the phase function is not well defined and may undergo a discontinuous jump. It has been pointed out by Wallstrom [8,9] that this condition on the multivalued S function has to be introduced properly in the solution of the Schrödinger equation. It has been shown [8] that unless this condition on S is imposed for regions separated by nodes where   0 an infinite number of different solutions to the fluid dynamical equations may arise. The fluid dynamical equation of motion (equation 4) contains two potential terms, viz. the classical potential V and a quantum potential or the so called Bohm potential [2,10] Vqu . So unlike classical hydrodynamics the quantum fluid is subjected to an additional potential which is of purely quantum origin, so that the dynamics of local fluid dynamical quantities are dependent on this quantum potential [11]. However the global behaviour of physically observable quantities of the quantum systems are found to be unaffected by this quantum potential [11], as the expectation values of position and momentum coordinates for the particle are dependent only on the externally applied potential V. In this regard the basic 36 character of the fundamental fluid dynamical variables, viz., charge density  and the fluid velocity v requires some attention. While the charge density is a physical observable the local velocity field v cannot be so regarded. It has been shown explicitly by Kan and Graffin [12] that the current density j   v being a physical observable the fluid velocity v cannot be represented by a linear Hermitian operator so that it cannot be a physical observable. Although the quantum fluid velocity field v is not a physical observable it offers a significant advantage in carrying over the common classical pictures into the quantum domain and to “classically” explain certain quantum processes. The quantum trajectories [13] introduced in the de Broglie-Bohm causal interpretation of quantum mechanics play a key role. The quantum trajectories are obtained from the equation 5 by solving dr 1 v   S (9) dt m S to obtain the function r(t) . It should be noted that the fluid velocity v  is m related to the complex momentum p as  1 p  mv  ln  2 . (10) i and the individual trajectories computed from the equation 9 for different initial positions do not represent the actual single particle quantum system. The quantum trajectories have been computed by Hirschfelder et al [14] for a Debye-Picht [15,16] wavepacket incident on a two dimentional squre potential barrier and other [17,18] dynamical systems. The single-valuedness of the wavefunction  , implies in the Madelung transcription a similar single valuedness on the probability amplitude R and a quantization condition on the gradient of the phase S function S. Since the fluid velocity is expressed as v  , then this condition m can be expressed as nh  v.dr (11) C m where n  0, 1, 2...... Using Stoke’s theorem this can be related to a surface integral over the area bounded by C as nh  (v).dA  (12) A m

37 where n  0, 1, 2...... But the quantity (v) is the well known hydrodynamical vorticity  allowing the equation 12 to be recast as nh  .dA  (13) A m with the quantization condition, . For an area containing a node of the wavefunction , n  0 implies a nonzero value of the vorticity . Since satisfies the quantization condition as given by equation 12, the vorticity around wavefunction nodes is quantized, as first discovered by Dirac [19, 20]. Furthermore it has been pointed out that [21] since the wavefunction can be generally divided into a real and an imaginary component, a wavefunction node requires that both components are simultaneously equal to zero. In their study of a Hydrogen atom colliding with a collinear Hydrogen molecule McCullough and Wyatt [22] have reported occurrence of a vortex in the probability current plot, which have been termed by them as “the quantum whirlpool effect”. This has been further confirmed by studies of Kuppermann and coworkers [23]. A detailed study of nodal topology of wavefunctions giving rise to such vortices has been carried out by Reiss [24-27] and Heller et al [28] The quantum fluid dynamical equations for a single particle have been extended to situations where there is a magnetic field along with the scalar field. If E and B are the electric and magnetic fields, respectively, the fluid dynamical equations read [29] as   .( v)  0 (14) t v e m    (eE  v B)   (V  V ) (15) t c qu where the fluid velocity is 1 e v  ( S  A) . (16) m c The spin-magnetic interaction term has been included [28] and the modified fluid dynamical equations have been obtained. The electron spin has been incorporated more rigorously using the Pauli spin and the relevant equations [7,30,31] have been derived, while the quantum fluid dynamical equations for the relativistic case have been obtained [32]. As in the case of single particle systems, the Schrödinger equation can be transformed into a continuity equation and an Euler-type equation of motion for many particle systems as well. For an N- particle system these equations have the general form

 N (17)  .( vi )  0 t i 1 N vi (18)  (vi . i )vi  i Vcl (r1,....., rN )  Vqu  t i 1 where i denotes the ith particle of the N- particle system. However, the fluid- dynamical quantities like the density and velocity field correspond to a fluid in the 3N dimensional configuration space, and do not have any direct physical significance for N>1. In this case the interpretation of the essentially fluid dynamical features like vortices and streamlines are no longer germane and the power of visualization so unique to the single particle quantum fluid dynamics appear to be lost. Attempts to directly project the fluid dynamical equations for such many-particle systems onto the 3D Euclidean space have not seen much success [33], except within some approximate single body theories like the Hartree and the Hartee-Fock theories. While Takabayashi [6] formulated the quantum fluid dynamical equations for a many-particle system using Hartree theory, Wong and coworkers [34, 35] have applied the time dependent Hartree- Fock theory for the N- body quantum fluid system. Using the concept of natural orbital theory [36], a many-electron system subject to electric and magnetic fields has been treated within a 3D quantum fluid dynamics [37]. The formulation of density functional theory (DFT) [38, 39] for many particle systems has been very successful in explaining the electronic structure, bonding and properties of atoms and molecules for time independent situations and for ground states in terms of single-particle density  (r,t) as the fundamental variable. The essence of the density functional theory is that the electron density contains all information about the ground state of a many- electron system and the true charge density distribution minimizes the related energy functional [38]. The density functional theory has been further extended to a time dependent situation [40, 41]. To treat the dynamical problem for many electron systems the quantum fluid dynamics (QFD) [42,43] and the time dependent density functional theory (TDDFT) have been combined to obtain the quantum fluid density functional theory (QFDFT) [44-56]. In the quantum fluid density functional theory an N- electron system is mapped onto a system of N noninteracting particles moving under the influence of an effective potential veff (r,t) obeying a generalized nonlinear Schrödinger equation as (in au)

 1 2  (r, t)    veff (r, t) (r, t)  i (19)  2  t

39 which is solved to yield the time dependent single “orbital”  (r,t)for the many- electron system considered. The quantum fluid dynamical quantities of this system viz.  (r,t) and the current density j(r,t) or the velocity is obtained as 2    (20) j     (21) where 1  (r,t)   2 exp(i ) (22) Recently QFDFT has been successfully applied in solving the ion-atom collision problems [46-50, 55,56] and atom-field interaction problems [51,52, 54]. The quantum fluid dynamics of nonlinear oscillators have been investigated [57-59] to study the signatures of nonintegrability through the hydrodynamic formalism. Certain distinctive features in the processes involving many particle systems such as an ion colliding with a many – electron atom [60] have been studied using the quantum fluid density functional theory in order to understand the associated charge transfer process and related electronic structure principles. Counter to the conventional quantum mechanics, de Broglie [61-65] introduced the idea that the wavefunction should instead of replacing the concept of material point be in coexistence with the point particle thus extending the classical concepts into quantum domain. It was suggested that in the nonrelativistic situation an ensemble of identical particles located in different places may be attached to the Schrödinger wavefunction, so that their probability 2 of finding is governed by  as proposed by Born. Essentially de Broglie assigned a dual role to be played by the Schrödinger wavefunction  so that along with its conventional interpretation mentioned above, it also causally directs the trajectory of an ensemble element. This guidance principle was used by de Broglie [4,10] to construct the stationary state orbits of hydrogen atom. In summary, various quantum potential based approaches like quantum fluid dynamics, quantum theory of motion and quantum fluid density functional theory compliment the conventional quantum dynamics through its classical interpretation.

Acknowledgment

We want to thank CSIR, New Delhi for financial support and Professor A. S. Sanz for going through the ms. 40

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Conceptual Issues, Practicalities and Applications of Bohmian and other Quantum Trajectories in Nanoelectronics

John R. Barker Department of Electronics and Electrical Engineering University of Glasgow, Glasgow G12 8LT, UK

1. INTRODUCTION

The study of quantum transport in semiconductor devices, particularly in nano-structured systems, has benefited from the interpretive power of the concept of quantum trajectories either in direct space or in phase-space. Here we briefly review progress since the 1980s.

2. WAVE-PACKET TRAJECTORIES AND TRAVERSAL TIMES

The simple classical notion of a particle arriving at a particular place at a particular time is problematic in quantum mechanics and led to considerable practical interest in the concept of a tunnelling time [1]. The rapid development of novel quantum devices that exploited semiconductor heterostructures, such as resonant tunnelling diodes, led to several detailed studies of the traversal time problem based on generating trajectories derived from monitoring features of scattered wave-packets obtained from direct solution of the time-dependent Schrödinger equation [2-5]. Further interest followed the advent of single- electron devices [6-9]. The concept of traversal/tunnelling time was reviewed under the PHANTOMS programme [10] and has been resurrected again [11] in studies of SiGe devices and in the context of experimental data in [12].

3. BOHMIAN TRAJECTORIES

For many years the theory of quantum transport in nanostructures has benefited from the interpretive power of Bohmian trajectories [13] by the post- processing [11, 14-29] of direct quantum calculations for either wave-functions or non-equilibrium Green’s Functions . The resulting trajectories are deterministic [20] with interesting topological properties. Non-deterministic trajectories have also been advocated in an extension of Bohmian mechanics to a stochastic form

43 but have had limited application. The possibilities for using ab initio Bohmian mechanics as a direct self-contained simulation tool has been discussed in the light of modern semiconductor device simulation but runs into difficulties when vortex motion occurs [20, 23]. Time-dependent studies of transport in semiconductor quantum waveguide structures have revealed a rich source of vortex flows corresponding to highly complex quantum potentials [24]. The Bohm picture has proved useful in interpreting novel logic devices that rely on non-invasive measurement [25].

4. QUANTUM POTENTIAL/DENSITY GRADIENT

In modern semiconductor device modelling many groups now routinely use the Quantum Potential or Density Gradient as a calibratable function for incorporating effects of quantum confinement, quantum transport and tunnelling within ab initio drift-diffusion modelling [29-36] where the band structure may have a profound impact [23, 36, 65] on the standard form [13] of the quantum potential. This approach generally compares quite well with full non-equilibrium Green function methodology except where macro-vortex formation occurs. As with any attempt to forward integrate the Bohmian equations of motion there are difficulties when strong vortex flows occur because the quantisation of vorticity stems from an integrability condition on the phase [20]. However, there are severe limitations to using the quantum potential alone, particularly if vortex motion is present in the current flow. It is then not possible to derive the velocity field from the carrier density alone (through the continuity equation) because the flow is not irrotational. Generalisations, based on gauge invariance, to include a quantum vector potential[37] have been suggested and explicit examples constructed. These simple pictures have been corroborated by full self-consistent Non-Equilibrium Green’s function studies of nano-structured devices where the quantum hydrodynamic velocity field is obtained directly(reviewed[29]). Various important generalisations of the quantum potential are reviewed in [38].

5. QUANTUM HYDRODYNAMIC FLOWS

The Bohm picture may also be viewed as a quantum hydrodynamic picture as foreseen by Madelung [39]. In the quantum transport modelling of devices whether by solving the time dependent Schrödinger equation or by the full power of non-equilibrium Green’s function (NEGF) methodology, a well- defined charge density and current density may be computed with the aid of Poisson’s equation from which device modellers extract the current-voltage characteristics. A velocity field may be extracted from the ratio of current density 44 to the charge density. The resulting density field, velocity field pair may be energy resolved in steady state problems or may be computed for the macroscopic density and velocity fields. This construction is essentially a post- processing method that reveals trajectories (in Bohmian view) or streamlines (quantum hydrodynamic view). We have performed extensive studies of quantum transport in realistic nano-devices (where one assumes the presence of non-self- averaging) atomistic micro-variability using this approach under-pinned by NEGF methods [28-29, 40-59]. An approach to a full many body field theoretic quantum hydrodynamics is outlined in [66]

6. HEURISTIC AND TOPOLOGICAL METHODS

Topologically-based methods have been used to give post-processing interpretations of velocity fields and density fields in inhomogeneous devices [21]. For example, the presence of strong nodes in the density field gives rise to quantised vortex flow in the velocity field [21, 24, 27, 29, 60-63]. From these studies we have proposed heuristic topologically-based methods to derive ab initio velocity fields (trajectory families) which allow the incorporation of dissipative processes such as inelastic scattering, charge capture and de-trapping [22-24] using non-Hermitian effective potentials [64]. A quasi-string formalism has been devised as a computational model to show that coherent quantum states are reconstructible from a variational principle for quantum trajectories [23]. These tools have been particularly useful in qualitative predictions of the effects of atomistic micro-variability in semiconductor devices [24, 29, 65].

7. TRAJECTORIES BASED ON QUANTUM DISTRIBUTIONS

Applications of the Wigner function for quantum electronic transport appeared in the 1980s [67-74], mostly of a formal nature. The first computation for a realistic time-dependent physical semiconductor system was made in [75]. Quantum Monte Carlo for device simulation has been extensively investigated since then [76-79]. In many cases the non-positive definiteness of the Wigner distribution was overcome by using damping theoretic methods and by averaging procedures. The equations of motion for carriers have been obtained by moment expansions of the non-local Wigner integral equation of motion and solved using Ensemble Monte Carlo simulation by trajectory tracking [78]. Recently a powerful method [79] has emerged for handling the non-positive definite Wigner function by particle tracking but it is highly compute-intensive. Unfortunately even this method suffers from second problem that follows because the Wigner function does not have compact support in phase space [17]. This problem 45 derives from the geometric centre-of-mass construction. Thus a wave-packet incident on a simple 1D tunnel barrier may split into two well-defined reflected and transmitted packets; but in the Wigner representation this situation leads to well-defined exit packets in phase space plus a wildly oscillating structure midway between the exiting packets in a region where the position distribution and momentum distributions are essentially zero. This problem has been overcome by our introduction of a new unique quantum distribution [80, 81] which we have called a C-distribution that has manifest compact support in phase space (C for compact support and complex valued). The C-distributions may be derived from the density matrix using a mixed real space-momentum representation and the approach generalises to double-time double-space non- equilibrium Green’s functions. The equations of motion and possible Monte Carlo trajectory computational schemes are discussed within exactly soluble models [81] that illustrate the formalism and its interpretation. The near-classical limit is easily obtained and lends itself to path-variable iterative methods including Monte Carlo trajectory schemes. The formalism has well-defined phase space trajectories for stationary states, time-dependent states and open systems.

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Sokolovsky, Report on the first European Workshop on Tunnelling Times, Phantoms Newsletter 7, 5 (1994). [11] J R Barker and J.R. Watling, Microelectronic Engineering 63, 97 (2002). [12] G. Nimtz, Foundations of Physics 39, 1346 (2009). [13] D. Bohm, Phys. Rev. 85, 166–193, (1952). [14] J.R. Barker, “Granular Nanoelectronics”, NATO ASI Series B 251 (Plenum Press, New York), 19 (1991). [15] J.R. Barker, “Granular Nanoelectronics”, NATO ASI Series B 251 (Plenum Press, New York), 327 (1991). [16] J.R. Barker, S. Roy and S. Babiker,”Science and Technology of Mesoscopic Structures”, (London:Springer Verlag), Ch 22, 213 (1992). [17] J.R. Barker in “Handbook on Semiconductors, volume 1” (Elsevier-North Holland) Ch 19, 1079 (1992). [18] J R Barker, Semiconductor Sci.Tech. 9, 911 (1994). 46 [19] J R Barker, “Quantum transport in ultra-small devices”, (Plenum Press, New York) 171 (1995). [20] J R Barker, Semiconductor Science and Technology 13A, 93 (1998). [21] J. R. Barker , D. K. Ferry, and R. Akis, Superlattices and Microstructures 27, 319 (2000). [22] J R Barker and J.R. Watling, Superlattices and Microstructures 27, 347 (2000). [23] J R Barker, VLSI Design 13, 237 (2001) [24] J.R. Barker, Microelectronic Engineering 63, 223 (2002). [25] J. R. Barker, Semiconductor Science and Technology 13 A, 93 (1998). [26] J R Barker , "Progress in Non-equilibrium Green's Functions II", (World Scientific Publ., Singapore), 198 (2003). [27] J.R. Barker, Physica E 19, 62 (2003). [28] J. R. Barker, Semiconductor Science and Technology 19S, 56, (2004). [29] J.R. Barker, A. Martinez, A. Svizhenko, M.P. Anantram and A. Asenov, J. Phys. Conf. Ser. 35, 233 (2006). [30] M. G. Ancona, Phys. Rev. B 42, 1222 (1990). [31] C. S. Rafferty, B. Biegel, Z. Yu, M. G. Ancona, J. Bude, and R. W. Dutton, in “Simulation of Semiconductor Processes and Devices”, (K. De Meyer and S. Biese- mans, Eds. Berlin, Germany: Springer), 137 (1998). [32] A. G. Ancona, Z. Yu, R. W. Dutton, P. J. Vande Vorde, M. Cao, and D. Vook, in Proc. SISPAD ’99, 235 (1999). [33] A. Asenov, G. Slavcheva, A. Brown, J. H. Davies and S. Saini, IEEE Transactions on Electron Devices 48, 722 (2001). [34] M.G. Ancona et al, J. App. Phys. 104, 073726 (2008). [35] A. Asenov, A. R. Brown, G. Roy, B. Cheng, C. L. Alexander, C. Riddet, U. Kovac, A. Martinez, N. Seoane and S. Roy, Journal of Computational Electronics 8, 349 (2009). [36] J.R. Watling, J.R. Barker, S. Roy, J. Computational Electronics 1, 279 (2002). [37] J.R. Barker, J. Computational Electronics 1, 17 (2002). [38] D. Vasileska, H.R. Khan, A. S. Ahmed, C. Ringhofer, C. Hetzinger, Int. J. Nanoscience 4, 305 (2005). [39] E. Madelung, Z. Phys. 40, 322 (1926). [40] A. Martinez, N. Seone, A. R. Brown, J. R. Barker and A. Asenov, IEEE Transactions on Electron Devices 57, 1626 (2010). [41] A. Martinez, N. Seoane, A. Brown, J. Barker, and A. Asenov, J. Phys. Conf. Ser. 220, (2010). [42] A. Martinez, N. Seone, A. R. Brown, J. R. Barker and A. Asenov, IEEE Transactions on Nanotechnology 8, 603 (2009). [43] N. Seoane, A. Martinez, A.R. Brown, J.R. Barker, A. Asenov, IEEE Transactions on Electron Devices 56, 1388 (2009). [44] A. Martinez, K.Kalna, P.V.Sushko, A.L. Schluger, J.R. Barker and A. Asenov, IEEE Transactions on Nanotechnology 8, 159 (2009). [45] A. Martinez, M. Bescond, A. R. Brown, J. R. Barker and A. Asenov, J. Computational Electronics 7, 359 (2008). [46] A Martinez, J R Barker, M Bescond, A R Brown and A Asenov, J. Phys. Conf. Ser. 109, 012026 (2008). [47] K.Kalna , A. Martinez, A. Svizhenko, M. P. Anantram, J. R. Barker and A. Asenov, J. Computational Electronics 7, 288 (2008). [48] A. Martinez, K. Kalna, J. R. Barker, and A. Asenov, Physica Status Sol.C-Current topics in Solid State Physics 5, 47 (2008). [49] A. Martinez, M. Bescond, J.R. Barker, A. Svizhenko, M. P. Anantram, C. Millar and A. Asenov, IEEE Transactions Electron Devices 54, 2213 (2007). [50] A. Martinez, J. R. Barker, A. Svizhenko, M. P. Anantram, and A. Asenov, 47 IEEE Transactions on Nanotechnology 6, 438 (2007). [51] A. Martinez, K. Kalna, J.R. Barker and A. Asenov, Physica E 37, 168, (2007) [52] A. Martinez, J. R. Barker, A. Asenov, A. Svizhenko and M.P. Anantram, J. Computational Electronics 6, 215 (2007). [53] A. Martinez, J.R. Barker, A. Svizhenko, M.P. Anantram and A. Asenov, Springer Proc. in Physics, 110 (August 15th ) (2006). [54] A. Martinez, A. Svizhenko, M.P. Anantram, J.R. Barker, and A. Asenov, A., J. Phys. Conf. Ser. 35, 269 (2006). [55] A. Martinez, J.R. Barker , A. Svizhenko, M.P. Anantram, A. Brown, B. Biegel, and A. Asenov, J. Phys. Conf. Ser. 38, 192 (2006). [56] A. Martinez, A. Svizhenko, M.P. Anantram, J.R. Barker, A.R. Brown and A. Asenov, IEDM 2005, IEDM Technical Digest, San Francisco, December, 613 (2005). [57] J.R. Barker, J. Computational Electronics 2, 153 (2003). [58] J. R. Barker, Superlattices and Microstructures 34, 361 (2004). [59] J. R. Barker, Semiconductor Science and Technology 19S, 56, (2004). [60] J.R. Barker, Physics of Semiconductors: Proceedings of the 26th International Conference on the Physics of Semiconductors, Edinburgh, 2002, Institute of Physics Conference Series 171, ed A R Long and J H Davies, IoP Publishing, Bristol (UK), P231 (2003). [61] J. R. Barker and A. Martinez, J. Computational Electronics 3, 401, (2004). [62] J.R. Barker, Physics of Semiconductor, ed J. Menedez and C.G. Van de Walle, AIP Press 27 1493 (2005). [63] J.R. Barker, AIP Conference Proceedings 995, Nuclei and Mesoscopic Physics, 104 (2008). [64] D.K. Ferry and J. R. Barker, Applied Physics Letters 74, 582 (1999). [65] J. R. Barker and J. R. Watling, VLSI Design 13, 453 (2001). [66] J.R. Barker , J. Computational Electronics 1, 23 (2002). [67] J.R. Barker, “Physics of Non-linear Transport in Semiconductors”, NATO ASI Series B 52 (Plenum Press, New York), Ch. 5, 126 (1980). [68] J. R. Barker, J. Physique 42 245 (1981). [69] J. R. Barker, J. Physique 42 293 (1981). [70] J. R. Barker, "Handbook of Semiconductors" 1, (North Holland: Oxford) Ch. 13, 617 (1982). [71] J. R. Barker and S. Murray, Phys. Letters A 93, 271 (1983). [72] J. R. Barker, D. Lowe and S. Murray, in "Physics of Sub-Micron Structures" ed H.L. Grubin, D K Ferry and K Hess, (Plenum Press:New York), 277 (1984). [73] J. Lin and L.C. Chiu, J. Applied Physics 57, 1373 (1984) [74] W.R. Frensley, Phys. Rev. B 36, 1570 (1987). [75] J. R. Barker, Physica B 134, 22 (1985). [76] J. R. Barker, "Physics and Fabrication of Microstructures and micro-devices",Springer- Verlag, Proc. in Physics 13, 210 (1986). [77] J.R. Barker, “Granular Nanoelectronics”, NATO ASI Series B 251 (Plenum Press, New York), 19 (1991). [78] H. Kosina, International Journal of Computational Science and Engineering 2, 100 (2006). [79] L. Shifren, C. Ringhofer, and D. K. Ferry, IEEE Transactions on Electron Devices 50, 769 (2003). [80] J.R. Barker, Physica E 42, 491 (2010). [81] J.R. Barker, J. Computational Electronics,A new approach to modelling quantum distributions and quantum trajectories for density matrix and Green function simulation of nano-devicesaccepted for publication, on-line from October 9th, (2010).

Principles of Time Dependent Quantum Monte Carlo

Ivan P. Christov Physics Department, Sofia University 1164 Sofia, Bulgaria

1. INTRODUCTION

The recent advent of lasers that produce pulses with duration below one femtosecond (attosecond pulses) [1] has allowed to probe events on the charateristic time scale of correlated electronic motion in atoms, molecules and solid state. Since the computational cost for directly solving many-body problems in quantum mechanics scales exponentially with the system dimensionality the realistic description of correlated electrons in attosecond time scale requires the development of new efficient methods for solving time-dependent Schrödinger equation (TDSE). It is generally believed that the exponential-time scaling inherent to the many-body quantum systems is related to the non-local quantum correlation effects. Some of the existing methods to treat the electron correlation effects approximately include time-dependent density functional theory (TDDFT) [2] where the many-body problem is reduced to single-body problems of non- interacting electrons moving in an effective exchange-correlation potential, which is however generally unknown. More reliable, but computationally very expensive is the multiconfiguration time dependent Hartree-Fock method [3]. Recently, we have introduced a new approach to solve many-body quantum problems which uses both particles and waves and reduces the many- body TDSE to a set of coupled single-body TDSE and equations of motion for Monte Carlo particles where each particle (walker) is attached to separate guiding wave (de Broglie-Bohm pilot wave). In this way the new method named time dependent quantum Monte Carlo (TDQMC) [4-7] recovers the symmetry that is due to the particle-wave dualism in quantum mechanics. It is important to stress that in TDQMC all calculations are performed in physical space for both the particles and the associated guiding waves, unlike in traditional Quantum Monte Carlo techniques where the evolution occurs in configuration space. It is assumed in TDQMC that the walker distribution in space reproduces the electron density where each individual particle samples its own distribution given by the modulus square of the corresponding guide wave. Thus the many-body probability distribution in space is an intersection of the single-body distributions sampled 49 by the individual Monte Carlo particles, which is consistent with the standard interpretation of quantum mechanics. As a matter of fact, the TDQMC method can be considered to be a reconciliation of some aspects of the Copenhagen and the de Broglie-Bohm theories, with direct application to quantum calculations. Some advantages of TDQMC as sompared to other QMC and particle methods are that the density function is positive everywhere and therefore the “fermion sign” problem is avoided. Also, TDQMC does not use Bohmian quantum potential which usually causes numerical difficulties. Since TDQMC uses particles and waves in a symmetric way it allows to treat self cositently complex quantum systems of different kinds of particles (e.g. electrons and nuclei) without invoking the Born-Oppenheimer approximation.

2. TDQMC DYNAMICS

For a non-relativistic system consisting of nuclei and electrons TDQMC reduces the many-body TDSE to a set of coupled single-body TSDE for the different replicas of the quantum sub-system. For example, for the electronic degree these equations read [4-7]:

2 K  k 2 eff k ii(r i , t )    i  V e n [ r i  R J ( t )] tm 2 e J 1 N eff k k Ve e[r i  r j ( t )]  V ext ( r i , t )  i ( r i , t ) , (1) ji  and similarly for the nuclear degree. The effective potentials are introduced in Eq.(1) in order to incorporate various local and non-local quantum correlation effects. For the electron-electron interaction we have:

lk 1 M rrjj()()tt Veff[r r k ( t )]  V [ r  r l ( t )]Κ , (2) , e e i jk  e e i j kk z l1  r ,t j jj  k where K is a smoothing kernel, Z j are weighting factors, and M is the total number of Monte Carlo walkers. To account for the nonlocality, the effective potentials in Eq.(2) involve weighted nonlocal Coulomb interactions experienced by a given trajectory from the i-th electron ensemble from the trajectories that

50 belong to the j-th electron ensemble. The width of the kernel kk plays the  jjr ,t role of a characteristic length of the nonlocal quantum correlations that depends on the density of the walkers in the quantum system. It is calculated at each instant kk of time using kernel density estimation procedure. The limit jjr ,0t  we call the ultra-correlated case where each walker from a given electron ensemble interacts with only one walker from each of the rest of the ensembles. The motion of the Mote Carlo walkers can be calculated in the simplest case using the de Broglie-Bohm guidance equation:

1 v( rkk ) Im   ( r ,t ) (3) ik i i me i (,)r t

The TDQMC algorithm involves no free parameters.

3. RESULTS

As a simple illustration of TDQMC method we show here the results for the ground state walker distribution and ionization probability of one-dimensional helium atom exposed to a strong femtosecond laser pulse. After propagation over 300 complex time steps, the initial random walker ensemble evolves towards steady state, with distribution in (2D) configuration space shown in Figure 1.

FIGURE 1. Walker distribution in configuration space for 1D Helium for symmetric ground state.

It is seen from Fig.1 that the walker’s density exhibits a butterfly shape where the particles are pushed away from the region of equal coordinates (X1=X2 in Fig.1) due to the repulsive Coulomb potential. Another exmple of the correlated electron dynamics is shown in Fig.2 where the helium atom is ionized in the field of ultashort laser pulse. It is seen that the TDQMC prediction is very close to the exact result for the ionization probability while the time dependent Hartree-Fock (TDHF) and the ultra-correplated calculations significatly underestimete and overestimate the ionization yield, respectively.

FIGURE 2. Time dependent ionization probability for 1D helium.

4. CONCLUSION

The TDQMC method presented describes the evolution of quantum systems by using ensembles of classical particles and quantum waves. The guide waves obey a set of coupled linear Schrödinger equations where the use of effective potentials accounts for the local and nonlocal correlation effects between the particles. Unlike other many-body quantum methods TDQMC does not involve calculation of overlap, exchange and correlation integrals, which significantly improves its scaling properties. Since particles and waves are used in a symmetric manner in TDQMC approximations such as Born-Oppenheimer are obsolescent.

Acknowledgment

The author gratefully acknowledges support from the National Science Fund of Bulgaria under contracts DO-02-115/2008 and DO-02-167/2008.

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[2] M. Petersilka, U. J. Gossmann, and E. K. U. Gross, Phys. Rev. Lett. 76, 1212 (1996). [3] J. Zanghellini, M. Kitzler, C. Fabian, T. Brabec, and A. Scrinzi, Laser Phys. 13, 1064 (2003). [4] I. P. Christov, Opt. Express 14, 6906 (2006). [5] I. P. Christov, J. Chem. Phys. 127, 134110 (2007). [6] I. P. Christov, J. Chem. Phys. 128, 244106 (2008). [7] I. P. Christov, J. Chem. Phys. 129, 214107 (2008).

Types of Trajectory Guided Grids of Coherent States for Quantum Propagation

Dmitrii V. Shalashilin and Miklos Ronto School of Chemistry University of Leeds, LS2 9JT, UK * Author for correspondence: Email: [email protected]

1. INTRODUCTION

Several methods have been suggested in the literature, which employ various trajectory guided grids of Frozen Gaussian wave packets as a basis set for quantum propagation. They exploit the same idea that a grid can follow the wave function. Therefore a basis does not have to cover the whole Hilbert space of the system. Among the methods considered in this article are the method of variational Multiconfigurational Gaussians (vMCG) technique and related Gaussian Multiconfigurational Hartree method (G-MCTDH) [1], the method of Coupled Coherent States (CCS) [2] and the recently developed Multiconfigurational Ehrenfest approach (MCE) [3,4]. The methods differ in exact way their grids are guided. The goal of this article is to present all four techniques from the same perspective and compare their mathematical structure.

2. THEORY

It is well known that the time dependence of a wave function 1 , 2,..., n  is simply that of its parameters, and equations for the “trajectories”  j t can be worked out from the variational principle S  0 (1)

54 by minimizing the action S   Ldt of the Lagrangian

L *1 ,..., *n ,  *1 ,..., *n , 1 ,..., n , 1 ,..., n  ˆ (2)  * ,* ,..., *  i  Hˆ  , ,...,  1 2 n t 1 2 n The variational principle (1) straightforwardly leads to the Lagrange equations of motion L d L , (3)    0 α dt α which can also can be written in Hamilton’s form.   Hˆ  D α  (4) α which is a system of linear equations for the time derivatives of the parameters of the wave function and D is a matrix p i   D jl   i  *1 ,*2 ,..., *n  1 , 2,..., n  (5)  * j  * j  l

The importance of Eq.(4) obtained by Kramer and Saraceno [5] is in showing that mathematical structures of quantum and classical mechanics are identical and therefore all methods and achievements of classical dynamics can be directly applied to quantum mechanics. For example the introduction of quantum ergodoicity becomes very straightforward. The vMCG method applies the above approach to the wave function which is a superposition of several Gaussian Coherent States.

   al t z l t (6) l1,N

In some versions of vMCG the Coherent State can also be “distributed” among two (or more) electronic potential energy surfaces 1 and 2 .

(t)  a1k (t) 1  a2k (t) 2  zk (t) (7) k1,N The wave function (7) is similar in spirit with G-MCTDH approach which describes the „system“ a1k (t) 1  a2k (t) 2 ... on a regular basis and the „bath“ by Gaussian wave packets z k (t) . Here we use Klauder’s z- notations for the Coherent states which labels a 1D Gaussian Coherent State

1    4   2 i ipq  x z    exp x  q  px  q  (8)    2  2  characterized by its position q and momentum p with a single complex number z

1 1   2 q  i1 2 p z  (9) 2 The equations of vMCG, which are simply the Lagrange Eq.(3) or Hamilton’s Eq.(4) written for the vector of parameters is

α  (a11 ,..., a1N ,a21 ,..., a2N , z11 ,..., z1M , z N1 ,..., z NM ) of the wave function (7). The equations of vMCG are sometimes numerically unstable. Figures 1 and 2 illustrate such instability on the example of the wave function propagation in the simple 1D Morse potential on the basis of only 3 basis Coherent States. Figure 1 shows a jump of the norm due to numerical instability of the equations, which occurs when two trajectories z1 and z3 overlap in phase space as shown at the Fig.2.

Figure 1 Discontinuity in the norm due to numerical instability and stiffness of the equations. The jump can be eliminated by decreasing the time step, which makes propagation rather inefficient

Figure 2 Two trajectories of states in phase-space intersect and numerical instability occurs at the moment of their intersection. 57

In this very simple calculation the problem can be eliminated by decreasing the time step. More generally the vMCG approach achieves numerical stability by regularization techniques so that good and well converged results can be obtained [1]. The vMCG approach simplifies if only one configuration is used for the wave function (7)

(t)  a1 (t) 1  a2 (t) 2  ... z(t) (10) which yields the well known Ehrenfest approach. For the wave function (10) the Kramer-Saraceno equations are simply those of the standard Ehrenfest method where the trajectory for z is the one by average Hamiltonian  H Ehr i z   (11) z * where

H a * a  H a * a  H a * a  H a * a H Ehr   Hˆ   11 1 1 22 2 2 12 1 2 21 2 1 a1 * a1  a2 * a2 (12)

and trajectory for the amplitudes a are given by a system of coupled equations     z z *  z *z a  ii  H a  iH a 1  2 11  1 12 2   (13)     z z *  z *z a  ii  H a  iH a 2  2 22  2 21 1  

58 which can be rewritten in a more compact form if the amplitudes are presented as a product of oscillating exponent of the classical action and relatively smooth

preexponential factor a1,2  d1,2 expiS1,2 , for which the equations (13) become

 d1  iH12d2 expiS2  S1  (14)  d2  iH21d1 expiS1  S2 

Ehrenfest dynamics is very stable and robust but the wave function (10) is not flexible enough to converge to exact quantum result. Recently a new method of Multi-Configuration Ehrenfest dynamics was suggested, which combines the robustness and stability of Ehrenfest method with the flexibility and accuracy of vMCG. The idea of MCE is to use the wave function (7) but assuming predetermined Ehrenfest trajectories (11) for z to guide the basis and apply variational principle to the amplitudes a only. Then a system of coupled equations for a is obtained [3]. The most recent version of MCE replaces Eq.(7) with the ansatz

(t)   Dk tk t   Dk ta1k (t) 1  a2k (t)  2  z k (t) k 1,N k 1,N (15) representing the wave function as a superposition of Ehrenfest configurations [4]. Assuming Ehrenfest trajectories (11-14) for a and z the coupled equations for the amplitudes D of each Ehrenfest configuration are obtained and can be found in.

For the case when only single state 1 is present in each configuration in the wave function (15) the MCE is equivalent to the Coupled Coherent States method. Both MCE and CCS are very robust and numerically stable and have 59 been used in a number of simulations. In addition they reduce the number of expensive variational equations thus economizing computational cost.

2. CONCLUSIONS This article shows that (1) The equations of vMCG technique can be obtained as those of Kramer and Saraceno theory for multiconfigurational wave function. (2) The Ehrenfest method can also be obtained variationally for the single configurational wave function. (3) The MCE technique uses Ehrenfest trajectories to guide trajectories of Gaussians and fully variational equations for their amplitudes. On a single potential energy surface MCE becomes equivalent to to CCS. (4) The advantage of CCS and MCE is that they reduce the number of expensive variational equations and improve numerical stability without any regularization (5) Using “approximate” trajectories instead of those yielded by fully variational approach is not an approximation but simply a choice of a basis set.

[1] I.Burghardt, H.-D.Meyer, and L.S.Cederbaum, J.Chem.Phys., 111, 2927 (1999);. G.A.Worth and

I.Burghardt, Chem.Phys.Lett. 368, 502 (2003); I. Burghardt, M. Nest, and G.A. Worth, J.Chem.Phys.

119, 5364 (2003); I. Burghardt, K. Giri, and G. A. Worth,

J.Chem.Phys.. 129 174101 (2008).

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S. Child, J. Chem. Phys., 114, 9296 (2001).; D. V. Shalashilin and M. S. Child, J. Chem. Phys., 115,

5367 (2001); D. V. Shalashilin and M. S. Child, J. Chem. Phys., 121, 3563 (2004).; P.A.J. Sherratt, 60

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[5] P. Kramer and M. Saraceno, Geometry of the Time-Dependent Variational Principle in Quantum

Mechanics (Springer, NewYork, 1981).

Accurate Deep Tunneling Description by the Classical Schrödinger Equation

Xavier Giménez1,3 and Josep Maria Bofill2,3 1)Departament de Química Física, 2)Departament de Química Orgànica, 3)Institut de Química Teòrica i Computacional. Universitat de Barcelona. Martí i Franquès, 1. 08028 Barcelona, Spain.

The description of classical processes in quantum terms, is actually known for almost five decades, the Classical Schrödinger Equation (CSE).1 It is a non–linear differential equation arising from real quantities, as a limiting case of the Time– Dependent Schrödinger Equation (TDSE), i.e. a linear differential equation for complex functions. The CSE provides an equation of motion for classical particles, in the closest possible language to quantum mechanics. It was introduced, in perhaps the most extensive work on the subject, by Schiller, 2-4 who develops a complete class of classical analogs of quantum algebra objects, in an analysis of van Vleck’s classical limit of quantum mechanics.5 The author explores as well a number of specific cases analytically solvable, such as the hydrogen atom or asymptotic Rutherford scattering. Shortly thereafter, N. Rosen rederived CSE in a rather heuristic form1,6, emphasizing its non–linear character, as well as discussing the use of mixtures, rather than combinations, for correct classical–limit densities.6 Holland7 did an extensive analysis of the implications of the non–linearity of CSE, pointing out the interest in having available generic solutions, based on expansion schemes for the wavefunction. Ghose, 8 and Ghose and Samal9 used CSE to study a continuous transition from quantum to classical

62 mechanics. They added to the later term in CSE a multiplicative factor accounting for decoherence, such that for short times it displays quantum behaviour, whereas for later times the system naturally relaxes with a phenomenological bath and undergoes classical. These authors develop a numerical algorithm to solve CSE and compute individual trajectories, but they assume that the later term in CSE equation arises from TDSE (thus avoiding an iteration process, see next chapter). Later on, Wyatt10 stresses the connection between CSE and classical trajectories, whereas Nikolic,11 on the contrary, analyses CSE in probabilistic terms. To the best of our knowledge, no practical, sufficiently general algorithm appears to be available for solving CSE. One may actually consider, in advance, that the non–linearity of CSE prevents any gaining in computational capability from such calculations. However, there have been some remarkable advances in solving a generic class of non–linear Schrödinger equations,12 a class that includes CSE. Therefore, a new impetus might be ready to understanding classical mechanics in quantum terms, by solving CSE, as applied to problems going beyond those having solutions in closed form. It is our hope to provide, in addition, different standpoints for looking at the classical limit of quantum mechanics, as well as compelling evidence on quantitative indicators for this classical limit, at least for some representative systems and selected physical quantities. In the present work, a numerical test considers a scattering process across an Eckart potential barrier. This is a well–known system, used to test the ability of semiclassical methods to reproduce the so–called deep tunneling regime.13-16 Figure 1 shows several time snapshots of a coherent state wavepacket colliding against an Eckart barrier. This makes the central wavepacket energy to be one half of the barrier height, so that any transmission is due solely to tunneling.

63 Figure 1 Time snapshots for tunneling transmission of a wavepacket traversing an Eckart barrier. Continuous red line: Accurate quantum mechanical results; continuous blue: present Classical Schrödinger Equation calculations.

Figure 1 shows that accurate and CSE values for the wavepacket density remarkably agree for all times. In particular, it is outstanding the ability of CSE to reproduce the oscillatory fringes of the reflected packet, at intermediate times, and the tunneling transmitted packet. In addition, Classical Schrödinger Equation results essentially capture the whole deep tunneling behaviour, at the conditions of the present study. Other physical conditions, corresponding to smaller masses 64 and smaller widths (not shown here), have also been explored. The observed trend is that quantum tunneling is again fairly well reproduced, but convergence rates are much slower and, in some cases, the number of iterations dramatically increases for large collision times. Nevertheless, these results for tunneling Eckart transmission come to a bit of surprise, mainly when one recalls the well– known difficulties that this physical system caused to several other semiclassical techniques.17,18 Even though CSE is formally equivalent to Hamilton–Jacobi, it is clear that its specific implementation in the present work does allow for the occurrence of quantum effects. More specifically, the present algorithm is based on a complex canonical transformation, a matrix equation cast, an allowance for complex momentum, as well as the application of quantum mechanical continuity conditions. All these features contribute to a remarkable change from the equivalent conditions imposed to the classical equations of motion, so that dwelling inside the classically forbidden region is allowed and quantum wave– like behaviour is certainly taken into account.

[1] N. Rosen, Am. J. Phys. 32, 597 (1964). [2] R. Schiller, Phys. Rev. 125, 1100 (1962). [3] R. Schiller, Phys. Rev. 125, 1109 (1962). [4] R. Schiller, Phys. Rev. 128, 1402 (1962). [5] J.H. van Vleck, Proc. Nat. Acad. Sci. 14, 178 (1928). [6] N. Rosen, Am. J. Phys. 33, 146 (1965). [7] P.R. Holland, The Quantum Theory of Motion, Cambridge Univ. Press., Cambridge, 1993. [8] P. Ghose, Found. Phys. 32, 871 (2002). [9] P. Ghose, K. Samal, Found. Phys. 32, 893 (2002). [10] R.E. Wyatt, Quantum dynamics with trajectories, Springer, New York, 2005. [11] H. Nikolic, Found. Phys. Lett. 19, 553 (2006). [12] F.W. Strauch, Phys. Rev. E 76, 046701 (2007). [13 S. Keshavamurthy, W.H. Miller, Chem. Phys. Lett. 218, 189 (1994). [14] N.T. Maitra, E.J. Heller, Phys. Rev. Lett. 78, 3035 (1997). [15] F. Grossmann, Phys. Rev. Lett. 85, 903 (2000). [16] M. Saltzer, J. Ankerhold, Phys. Rev. A 68, 042108 (2003). [17] F. Grossmann, Phys. Rev. Lett. 85, 903 (2000). [18] D.J. Tannor, S. Garaschuck, Annu. Rev. Phys. Chem. 51, 553 (2000).

65 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

The Bohmian Model, Semiclassical Systems and the Emergence of Classical Trajectories

Alex Matzkin Lab. de Physique Théorique et Modélisation (CNRS Unité8089), Universitéde Cergy-Pontoise, 95302 Cergy-Pontoise cedex, France

I. INTRODUCTION

The de Broglie–Bohm interpretative framework [1] (abbreviated as BM, for ’Bohmian model’) is the main alternative to standard quantum mechanics (QM). While BM and QM give equivalent results, the trajectory-based representation and flow-based numerical methods specific to BM have given rise to an increasing number of works employing BM as a tool to compute and interpret the physics of various systems [2]. Notwithstanding, BM was not introduced as a computational scheme, but rather as manner of providing QM with an ontological framework [3]: the aim of BM is to give a realist [4] description of quantum phenomena in terms of the motion of point-like particles following well-defined trajectories, allowing to unify the classical and quantum descriptions of nature. Although the programme is attractive, we argue it cannot work. Our arguments are based on the investigation of semiclassical systems. These systems, while being purely quantum, display a striking correspondence between their properties and those of the equivalent classical system. The properties of the Bohmian particle are radically different and spoil this quantum-classical correspondence. An important consequence is that Bohmian trajectories can never become classical in these systems, raising the problem of whether BM can account for the emergence of classical dynamics.

II. SEMICLASSICAL SYSTEMS AND THE QUANTUM-CLASSICAL CORRESPONDENCE

The investigations of the quantum-classical correspondence, which has its ori- gins in the early days of quantum mechanics were revived in the 1990’s in the context of quantum chaos [5, 6]. It is today well-established that several types of quantum systems – known generically as semiclassical systems – display the manifestations of properties belonging to the classical analog of these systems.

66 This is due to the fact that the wavefunction propagates essentially along the trajectories of the corresponding classical system; indeed in these cases the semiclassical approximation to the path integral propagator, given by [7]

2S 1/2 x x 1 ∂ k S x x ℏ K( 0, ,t)= X det exp(i k( 0, ,t)/ + iφk) , (1) 2iπℏ ∂x∂x0 k becomes excellent. Here the sum runs on all the classical trajectories k connecting x0 to x in the time t. Sk is the classical action for the kth trajectory and the determinant is the inverse of the Jacobi field familiar from the classical calculus of variations, reflecting the local density of the paths; φk is a phase accounting for reflections and conjugate points encountered along the kth trajectory. As a result individual classical trajectories are ”visible” in these quantum systems (in the wavefunction or through observable quantities), and their spectral properties such as the distribution of the energy levels depend on the average properties of families of classical orbits. This is why the quantum counterpart of classically chaotic systems have specific universal features quite different from the quantum analogue of a classically regular system.

III. BOHMIAN TRAJECTORIES AND THE DYNAMICAL MISMATCH

Bohmian trajectories are defined from the probability density current, that depends most crucially on the choice of the initial distribution. Hence Bohmian trajectories in a given system can be chaotic or regular depending on the chosen initial distribution. This is also the case in semiclassical systems: Bohmian trajectories bear no relation with the underlying classical dynamics, spoiling the quantum-classical correspondence characteristic of semiclassical systems. The fact that by way of the propagator (1) the wavefunction evolves along classical trajectories does not mean that the current density will necessarily do so. The reason is that an initial localized wavefunction spreads and interferes yielding Bohmian trajectories quite different from the classical ones, even though each part of the wavepacket moves along a classical trajectory. This ”dynamical mismatch” was discussed at length in Ref. [8], and specific examples have been given for Rydberg atoms [9], a hydrogen atom in a magnetic field [8], and a square billiard [10]. The dynamical mismatch between Bohmian trajectories and classical motion has serious implications [8] concerning the empirical acceptability of the de Broglie-Bohm theory as describing the real behaviour of the quantum world. The usual hand-waving argument – quantum and classical particle motions belong to different domain, the former approaching the latter in some limit as the quantum potential vanishes – is clearly not applicable: it seems indeed untenable

67 to explain the appearance of classical structure in the wave and in the statistical distribution of the particles while upholding that the real dynamics of the particle remains highly non-classical, without any type of correspondence with the classical dynamics that should emerge at some point. From within BM, the only option to account for the emergence of classical trajectories from the quantum Bohmian dynamics is to rely on decoherence and non-spreading wavepackets. These two ingredients, however, are not speciﬁc to BM (and therefore do not need any of its ontological propositions), but form part of the practical recipe standard QM employs in order to account for the quantum-classical transition. It is well-known that this practical recipe is inconsistent [11, 12], since it gives incompatible meanings to the reduced density matrices. The study of the BM properties in semiclassical systems leads us to conclude that despite its attractive appeal and high interest including in practical computations, BM is unable to account for the emergence of classical trajectories.

[1] P. R. Holland, ”The Quantum Theory of Motion” (Cambridge University Press, Cambridge, 1993). [2] R. E. Wyatt, ”Quantum Dynamics with Trajectories” (Springer, Berlin, 2005). [3] D. Bohm. and B. J. Hiley, ”The Undivided Universe: an ontological interpretation of quantum theory” (Routledge, London, 1993). [4] A. Matzkin Eur J Phys 23 285 (2002). [5] M. C. Gutzwiller, ”Chaos in Classical and Quantum Mechanics” (Springer, Berlin, 1990). [6] M. Brack M and R. Badhuri, ”Semiclassical Physics” (Westview Press, Boulder, USA, 2003). [7] W. Dittrich and M. Reuter ”Classical and Quantum Dynamics” (Springer, Berlin, 2001). [8] A. Matzkin and V. Nurock, Studies In History and Philosophy of Science B 39, 17 (2008). [9] A. Matzkin, Phys. Lett. A 361, 294 (2007). [10] A. Matzkin, Found. Phys. 39, 903 (2009). [11] S. L. Adler, Studies In History and Philosophy of Science B 34, 135 (2003). [12] A. Leggett, J. Phys.: Condens. Matter 14, R415-R451 (2002).

68 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

Quantum trajectories for ultrashort laser pulse excitation dynamics

Gérard Parlant Institut Charles Gerhardt, UniversitéMontpellier 2, CNRS, Equipe CTMM, Case Courrier 1501, Place Eugène Bataillon, 34095 Montpellier, France

The Quantum Trajectory Method [1] (QTM) is a computational implementation of Madelung’s hydrodynamical approach to quantum mechanics based on the ansatz ψ = A exp(iS/~). The probability density A2 is partitioned into a finite number of “particles” which evolve along quantum trajectories guided by the gradient of the action ∇S. Quantum trajectories obey Newton-like equations with a modified potential V (x)+ Q(x,t), where the quantum potential Q couples trajectories to one another. In QTM, coupled equations of motion for the densities and action functions of all particles are propagated in time, and from these the wave function can be synthesized at each instant. A computational drawback of QTM is that the quantum potential Q may become very large, or even singular, in some situations, thus giving rise to numerically unstable trajectories. This is the case, in particular, of quantum interferences that may induce very small amplitude in the wave function. This so-called node problem [1] is currently the major obstacle that prevents the development of QTM. QTM has been extended to the dynamics of electronic nonadiabiatic collisions [1–3]. In the multisurface QTM, a constant number of quantum trajectories are propagated on each electronic state (there is no “trajectory hop” between surfaces) and probability density and phase information are transferred from one state to another through communications between trajectories. By substituting the polar form ψj(x,t) = Aj (x,t)exp[iSj(x,t)/~] into the time-dependent Schrödinger equation (TDSE), one can derive [1, 2] the hydrodynamic equations for the nuclear motion of a two-state system, where for each state j, Aj is the (positive) amplitude, and Sj is the action function, both real quantities, and Pj(x,t) = ∇Sj(x,t) is the momentum associated with the flow velocity vj(x,t) = Pj(x,t)/m of the probability fluid. Like their single-state 2 counterparts, the continuity equations for the densities ρj = Aj ,

dρ1/dt = −ρ1∇v1 − λ12; dρ2/dt = −ρ2∇v2 − λ21, (1) express the conservation of the probability ﬂuid on each individual surface; in

69 1/2 addition, the extra source/sink terms, λ12 = −λ21 = (2V12/~)(ρ1ρ2) sin (∆), take into account the transfer of probability density from one state to the other one, with ∆ = (S1 − S2)/~. (n.b. we assume an observer “going with the ﬂow of probability” [1], so that Eqs. (1), (2), and (4) involve the total time derivative d/dt.) In the Newtonian equations,

dp1/dt = −∇ (V11 + Q11 + Q12); dp2/dt = −∇ (V22 + Q22 + Q21) , (2) one can see that any given particle evolving on state j is subject to three force components: (i) the classical force, (ii) the quantum force, that reflects the influence of all other particles on the same surface, and (iii) an extra coupling force, that derives from the off-diagonal quantum potential Qjk,

1/2 Q12 = V12 (ρ2/ρ1) cos (∆) , with ρ1Q12 = ρ2Q21. (3) Finally, equations for the rate of change of the action functions read: 2 2 dS1/dt = mv1/2 − V11 − Q11 − Q12; dS2/dt = mv2/2 − V22 − Q22Q21. (4) In the hydrodynamical equations of motion for nonadiabatic dynamics, Eqs (1)–(4), it appears that particle motions and interstate transitions are closely interconnected. In particular, interstate transfer forces, deriving from Qij [Eq. (3)] can noticeably modify the course of trajectories, and eventually change the result of the calculation. Unfortunately, the multistate QTM inherits the numerical drawbacks of its single-state counterpart mentioned above. In addition, extra propagation difficulties, related to the interstate electronic coupling per se, may be anticipated, especially in case of large values of V12. In this work, we use a form of the multisurface QTM that formally separates interstate transitions from single-state nuclear motions [6, 7] (see also [4]). On each state, quantum trajectories are propagated in a moving frame attached to a decoupled Gaussian wave packet, while transition probabilities are obtained from coupled continuity equations similar to Eqs (1). In practice, this is done by substituting into the TDSE, for each state j = 1, 2, a split polar form of the nuclear wave function, ψj (x,t) = φj (x,t) × χj (x,t), with φj = aj exp(isj/~) and χj = αj exp(iσj/~), where the φj’s refer to single-state motion while the χj’s correspond to inter-state exchange of density and phase. For each state j, the “decoupled” wave function φj(x,t) is solution to the single-state TDSE ~2 ∂ − ∇2φ + V φ = i~ φ , (5) 2m j jj j ∂t j while the “coupled” wave function χj (x,t) satisfies a modified TDSE 2 2 ~ 2 ~ ∇φj ∂ φk − ∇ χj − ∇χj = i~ χj − Vjk χk, (6) 2m m φj ∂t φj

70 where one can notice an interstate coupling term on the right-hand side of Eq. (6). Bohmian equations of motion for φj(x,t) and χj(x,t), solutions to Eqs (5) and (6), are easily derived [6, 7]. It must be mentioned that while this decoupling procedure is formally exact, in the laser excitation example given below, the radiative coupling is so large that the source/sink terms λjk and the oﬀ-diagonal quantum potentials Qjk dominate the behavior of the “coupled” part of the wave function, so that the Bohmian equations of motion for χj(x,t) can be simpliﬁed [7] to give:

dωj/dt ≈−λjk/fj; dσj/dt ≈−Qjk, (7)

2 2 where fj = aj and ωj = αj are the probability densities relative to φj and χj, respectively. The decoupled-representation QTM approach described above has been applied to a model problem [5] of two adiabatic potential energy curves coupled through a laser pulse. The system is initially in its ground vibrational state and is promoted to a repulsive state by the laser field. Due to large values of the laser field, electronic state populations exhibit several oscillations (Rabi flops) during the short laser pulse. A small number of trajectories (typically 30 to 100) is sufficient to obtain an excellent agreement with exact quantum calculations, a number that can be contrasted with the 105 semiclassical trajectories used in Ref. 5. Moreover, an interesting “hole” structure in the initially Gaussian wave function is observed. It has been previously interpreted as a momentum kick from the laser pulse to the wave packet [8]. We suggest that it might be of interest to investigate this dynamical effect in terms of the interstate transfer forces deriving from the off-diagonal quantum potentials Qjk.

[1] R. E. Wyatt, Quantum Dynamics with Trajectories: Introduction to Quantum Hy- drodynamics, Springer, New York, 2005. [2] R. E. Wyatt, C. L. Lopreore, and G. Parlant, J. Chem. Phys., 114 (2001) 5113. [3] I. Burghardt and L. S. Cederbaum, J. Chem. Phys., 115 (2001) 10312. [4] S. Garashchuk, V. A. Rassolov, and G. C. Schatz, J. Chem. Phys., 123 (2005) 174108; V. A. Rassolov and S. Garashchuk, Phys. Rev. A, 71 (2005) 032511. [5] F. Grossmann, Phys. Rev. A, 60 (1999) 1791. [6] G. Parlant, Quantum Trajectories, Chap. 17, CRC Press/Taylor & Francis, Boca Raton, 2010. [7] J. Julien and G. Parlant, unpublished. [8] U. Banin, A. Bartana, S. Ruhman, and R. Kosloﬀ, J. Chem. Phys., 101 (1994) 8461.

71 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

Quantum dynamics and super-symmetric quantum mechanics.

Eric R. Bittner∗ and Donald J. Kouri Department of Chemistry, University of Houston, Houston, TX 77204

In my talk I will present an overview of our recent work involving the use of supersymmetric quantum mechanics (SUSY-QM). I begin by discussing the mathematical underpinnings of SUSY-QM and then discuss how we have used this for developing novel theoretical and numerical approaches suitable for studying molecular systems. I will conclude by discussing our attempt to extend SUSY- QM to multiple dimensions.

I. INTRODUCTION

We advance a new approach which casts the problem of excited states of a physical Hamiltonian in terms of nodeless ground states associated with a hierarchy of “auxiliary Hamiltonians”. This hierarchy of Hamiltonians results from the implementation of the “super-symmetric quantum mechanics” (SUSY- QM) formalism introduced some years ago in particle physics and quantum ﬁeld theory[1, 2]. In high-energy physics (HEP), SUSY is a fundamental symmetry which relates elementary particles of one spin to another particle with identical mass/energy but whose spin diﬀers by ±~/2 . In essence, SUSY predicts that for every boson there exists a corresponding fermion with the same mass/energy. As of date, superpartners of the particles of the Standard Model have not been observed; supersymmetry, if it exists, must be a broken symmetry allowing the ’sparticles’ to be heavy. Supersymmetric quantum mechanics, on the other hand, borrows the basic formal ideas from HEP to solve quantum mechanics problems. In short, SUSY-QM introduces pairs of Hamiltonians that share a particular mathematical relationship (termed partner Hamiltonians) such that for every eigenstate of one Hamiltonian, its partner has a corresponding eigenstate with the same energy but with a lower quantum number. In brief, one introduces a “superpotential”, W , such that the original Hamiltonian can be written in the

∗corresponding author

72 form

+ H =(−∂x + W )(+∂x + W )= Qˆ Qˆ (1) where Qˆ+ and Qˆ are “charge operators” analogous to the creation/annihilation operators in the quantum treatment of the 1D harmonic oscillator. From a historical context, it is interesting to note that Schrödinger used a SUSY-like technique in his original solution of the hydrogen atom. In fact the final result for the H-atom spectrum and wavefunctions can be derived with greater easy using SUSY than with the more traditional approach involving the recursion relation for Laguerre polynomials. While SUSY-QM has also been explored for one dimensional, non-relativistic quantum mechanical problems[3–8], thus far these studies have focused on the formal aspects and on obtaining exact, analytical solutions for the ground state for specific classes of problems. In several recent papers[9–12] we have begun exploring the SUSY-QM approach as the basis of a general computational scheme for bound state problems. Of course, all this begs the question: Can this approach be generalized to higher numbers of dimensions and to more than a single particle? There has been substantial effort in the past to do just this.[4–7, 13– 24] However, until now, no such generalization has been found that is able to generate all the excited states and energies even for so simple a system as a pair of separable, 1-D harmonic oscillators (HO) or equivalently, for a separable 2-D single HO. Here we briefly report on our generalization of SUSY-QM to higher dimensions and showed that it does, in fact, yield the correct analytical results for separable and non-separable problems. [12] We present a succinct summary of our approach as applied to simple problems and discuss extensions to more complex systems such as atomic clusters.

II. BASIC FORMALISM

We begin with a brief summary of our new generalization of SUSY-QM to treat higher dimensionality and more than one particle. Previous attempts generally involved introducing additional, “spin-like” degrees of freedom.[5, 7, 16–19, 21, 22, 25–27] However, no practical extension to higher dimensions or reduction to numerical method has been produced to date following these lines. The extension to more general systems involves introducing separate coordinate systems for each particle in the system. Consequences of this include: 1) a simple expression for the Laplacian for the system 2) vector generalizations of the SUSY superpotential and of the two charge operators that generate the Hamiltonians for the ﬁrst and second sectors 3) the original Hamiltonian remains exactly the same as before except that it is now in the SUSY factored form 4) then the ﬁrst sector problem can be addressed using any of the standard approaches 5) the

73 (a) (b)

FIG. 1: (a) Convergence of first excitation energy for a model anharmonic potential using a n-point DVR. Gray squares: standard variational approach. Here we plot the error between the numerical and exact values ǫ = log10 |E(n) − E(exact)|. Black squares: SUSY. Dashed lines are linear fits used to guide the eye.(From Ref. [11]) (b) Location of excited state node using the Monte Carlo/SUSY approach on a model double well potential.(From Ref. [10]). second sector Hamiltonian is, however, a second rank tensor (or matrix) and its wave function is a vector. This is analogous to the situation in relativistic quantum mechanics where tensor Hamiltonians and vector wave functions result from the embedding of coordinate systems separately in each particle (with the difference being that time continues to be a parameter here as opposed to each particle having its own proper time) 5) despite this complication, we derived the Rayleigh-Ritz variational principle for this sector and were able to solve for the vector energy eigenstates of sector 2. To illustrate the general form of the theory we write the quantum Hamiltionian (using ~2/2m = 1)

~ † ~ 1 ~ ~ ~ 1 H1 = Q1 · Q1 + E0 ; Q1 = ∇ + W1; W1 = −∇ ln ψ0 (2)

1 ~ where ψ0 is the usual ground-state of H1 and ∇ is the 3n dimensional gradient operator. It is straightforward to verify that Eq. 1 yields a factorization of the original Schr¨odinger equation. The superpotential W~ 1 is a real-valued column vector. Within our approach, we write the sector 2 Hamiltonian as tensor prod- ←→ ~ ~ † 1←→ uct: H 2 = Q1Q1 + E0 I . The degeneracies between sector 1 and sector 2 are the consequence of the “inter-twining” relations:

~ ←→ ~ ~ † ←→ ~ † Q1H1 = H 2 · Q1 & Q1 · H 2 = H1Q1

74 The full SUSY algebraic structure is obtain by writing this in matrix form by deﬁning the super-charge operators

0 0 0 Q~ † Q = & Q† = 1 (3) Q~ 0 0 0 1 and writing

H 0 H = 1 ←→ = {Q†, Q} (4) 0 H 2 This formalism applies to any number of distinguishable particles and has been proved to yield correct numerical results for the case of 2 particles (or equivalently, for a 2-D 1 particle system) with non-separable interaction. Then we deﬁne the sector 3 scalar Hamiltonian by

~ ~ + (2) H3 = Q2 · Q2 + E0 (5) with the ground state wave equation

(3) (3) (3) H3ψ0 = E0 ψ0 . (6)

(3) (2) 2 It is easily seen that E0 = E1 − E0 . This procedure continues until all bound states of the original Hamiltonian are exhausted. It should also be clear that the sector 2 excited state wave function is obtained from the nodeless sector 3 ~ + ground state by applying Q2 to it. Then the second excited state for sector ~ + ~(2) 1 results from taking the scalar product of Q1 with ψ1 . The approach thus leads to an alternating sequence of scalar and tensor Hamiltonians, with the lowest energy state of the nth member of the hierarchy being isoenergetic with the ﬁrst-excited state of the n − 1 sector (for n> 0). The critical observation is that in all cases we need only determine nodeless ground states and we can use this fact to develop novel numerical approaches for determining the excitation spectrum for complex multi-dimensional systems.

III. SUMMARY AND OUTLOOK

The idea of using “out-side the box” ideas to advance quantum computational techniques certainly is within the spirit of this CCP6 conference. While we have not emphasized the dynamical aspects of our approach and have focused solely upon ﬁnding stationary states, we have begun to formulate a “quantum trajectory” approach for determining the eigenstates of the tensor-sector Hamil- tonians in our approach using the Dirac-Frenkel-McLachlan variational principle extended to imaginary time. We next sketch brieﬂy how this may be used.

75 The essential step in implementing ND-SUSY will be in how we choose to represent the many-body wave function. Let us assume that for an arbitrary N dimensional system we can write a sector 1 state as

ψo = g(~x, {λn}) (7) n Y where g(~x; {λn}) is a multidimensional gaussian parameterized by {λn}. For example we can write

1 g(~x; {λ }) = exp − (~x − ~µ )T S−1(~x − ~µ )+ a (8) n 2 n n n n S−1 Re where {λn} = {an, ~µn, n } ∈ . Here, an is the weight ascribed to a given S−1 gaussian, µn is the center, and n is the covariance matrix. Note that we are writing this in the most general form since ~x represents the cartesian coordinates of all the particles in the system. In practice, however, we ascribe a factorized gaussian to each physical particle so that the total number of coeﬃcients 10×np where np = N/3 is the number of particles in the system. We can construct the ground state by requiring that δE = 0 and determining the coeﬃcients via integrating the McLachlan/Dirac/Frenkel variational principle in imaginary time (i.e. we make the analytic continuation by taking it/~ → τ/~ ) viz

∂ψ ∗ ∂λ ∂ψ n (Hψ + i )d~x = 0. (9) ∂λ ∂τ ∂λ Z j i Now, let’s write our ground state in the product basis

ψo = g(~xk, {λk}) (10) k Y where k denotes a given physical particle and {λk} are the variational coeﬃcients associated with that particle. Now, construct the superpotential W~ 1 (eq. 13 in our JPC paper),

∂ W1kn = − ln ψo (11) ∂xkn

Since ψo is a product of gaussians and taking Sk to be symmetric, one ﬁnds a simple expression for the superpotential.

S−1 W1kn = ( k )nm(xkm − µkm) (12) m X where m = 1, 2, 3 labels the cartesian coordinates for particle k.

76 This notion can be extended to the tensor sectors by writing Eq. 13 as

† ∂ψ~ ←→ ∂λ ∂ψ~ · ( H · ψ~ + i )d~x = 0. (13) ∂λ 2 ∂τ ∂λ Z j ! i where ψ~ is a trial state in sector-2. This results in a series of coupled non-linear “trajectory” equations for the coeﬃcients. Integrating this forward in imaginary time will result in a robust variational estimate for the sector-2 ground state which then can be used to determine the excitation energy or sector-1excited state wave function. We are currently exploring this idea.

Acknowledgments

This work was supported in part by the National Science Foundation (ERB: CHE-0712981) and the Robert A. Welch foundation (ERB: E-1337, DJK: E- 0608). The authors also acknowledge Prof. M. Ioﬀe for comments regarding the extension to higher dimensions.

[1] E. Witten, Nuclear Physics B (Proc. Supp.) 188, 513 (1981). [2] E. Witten, J. Differential Geometry 17, 661 (1982). [3] H. Baer, A. Belyaev, T. Krupovnickas, and X. Tata, Physical Review D (Particles and Fields) 65, 075024 (pages 8) (2002), URL http://link.aps.org/abstract/ PRD/v65/e075024. [4] A. A. Andrianov, N. V. Borisov, M. V. Ioffe, and M. I. Eides, Theoretical and Mathematical Physics 61, 965 (1984). [5] A. A. Andrianov, N. V. Borisov, M. I. Eides, and M.V.Ioffe, Phys. Lett. A 109, 143 (1985). [6] F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995). [7] A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Phys. Lett. A 105, 19 (1984). [8] A. Gangopadhyaya, P. K. Panigrahi, and U. P. Sukhatme, Phys. Rev. A 47, 2720 (1993). [9] D. J. Kouri, T. Markovich, N. Maxwell, and B. G. Bodman, J. Phys. Chem. A 113, 7698 (2009). [10] E. R. Bittner, J. B. Maddox, and D. J. Kouri, The Journal of Physical Chemistry A 113, 15276 (2009), URL http://dx.doi.org/10.1021/jp9058017. [11] D. J. Kouri, T. Markovich, N. Maxwell, and E. R. Bittner, The Journal of Physical Chemistry A 113, 15257 (2009), URL http://dx.doi.org/10.1021/jp905798m. [12] D. J. Kouri, K. Maji, T. Markovich, and E. R. Bittner, The Journal of Physical Chemistry A 114, 8202 (2010), http://pubs.acs.org/doi/pdf/10.1021/jp103309p, URL http://pubs.acs.org/doi/abs/10.1021/jp103309p.

77 [13] P. T. Leung, A. M. van den Brink, W. M. Suen, C. W. Wong, and K. Young, Journal of Mathematical Physics 42, 4802 (2001), URL http://link.aip.org/ link/?JMP/42/4802/1. [14] R. de Lima Rodrigues, P. B. da Silva Filho, and A. N. Vaidya, Physical Review D (Particles, Fields, Gravitation, and Cosmology) 58, 125023 (pages 6) (1998), URL http://link.aps.org/abstract/PRD/v58/e125023. [15] A. Contreras-Astorga and D. J. F. C., AIP Conference Proceedings 960, 55 (2007), URL http://link.aip.org/link/?APC/960/55/1. [16] A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Theoretical and Mathematical Physics 72, 748 (1987). [17] A. A. Andrianov and M. V. Ioffe, Phys. Lett. B 205, 507 (1988). [18] A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Theoretical and Mathematical Physics 61, 1078 (1984). [19] A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, JETP Lett. 39, 93 (1984). [20] A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Phys. Lett. B 181, 141 (1986). [21] F. Cannata, M. V. Ioffe, and D. N. Nishnianidze, Journal of Physics A: Mathe- matical and General 35, 1389 (2002), URL http://stacks.iop.org/0305-4470/ 35/1389. [22] A. Andrianov, M. Ioffe, and D. Nishnianidze, Phys. Lett. A 201, 103 (2002). [23] A. Das and S. A. Pernice, arXiv:hep-th/9612125v1 (1996). [24] M. A. Gonzalez-Leon, J. M. Gullarte, and M. de la Torre Mayado, SIGMA 3, 124 (2007). [25] A.A.Andrianov, M.V.Ioffe, and V.P.Spiridonov, Phys. Lett. A 174, 273 (1993). [26] R. I. Dzhioev and V. L. Korenev, Phys Rev Lett 99, 037401 (2007), ISSN 0031- 9007 (Print). [27] A. A. Andrianov, M. V. Ioffe, and D. N. Nishnianidze, Theoretical and Mathe- matical Physics 104, 1129 (1995).

78 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

Bohmian Trajectories of Semiclassical Wave Packets

S. Römer joint work with D. Dürr Mathematisches Institut, LMU München, Theresienstr. 39, 80333 München, Germany∗

I. INTRODUCTION

There are many ways to formulate the classical limit of quantum mechanics. The strongest assertion would be about “quantum particle trajectories” becom- ing Newtonian. Particle trajectories, however, are not ontological elements of orthodox quantum theory and thus the “classical limit” must be defined in some operational way. In contrast, Bohmian mechanics, which for all practical purposes is equivalent to quantum mechanics, is a quantum theory of point particles moving, so the study of the classical limit becomes a straightforward task [1, 2]: Under which circumstances are the Bohmian trajectories of particles approximately Newtonian trajectories? Here “approximately” can be understood in various manners. The technically simplest but also weakest is that at every time t the Bohmian particle’s position is close to the centre of a “classically moving” very narrow wave packet ψ. This essentially amounts to showing that ψ(t) 2 is more or less transported along a Newtonian flow (see [3] for a recent| work| on this). The strongest and clearly most direct assertion would be that almost every Bohmian trajectory converges to a Newtonian trajectory in the uniform topology. We shall prove here a slightly weaker statement, namely that the uniform closeness holds in probability. We shall establish this result for a particular class of wave packets which were defined by Hagedorn in [4] and which move along classical paths.

∗Electronic address: [email protected]

79 II. BOHMIAN TRAJECTORIES OF HAGEDORN WAVE PACKETS

In Bohmian mechanics the state of a particle is described by a wave function ψ(y,s) (y R3, s R) and by its position Y R3. The wave function evolves according to∈ Schr¨odinger’s∈ equation (~ = m =∈ 1)

∂ 1 i ψ(y,s)= Hψ(y,s)= y + V (y) ψ(y,s) (1) ∂s −2△ with the potential V . The wave function governs the motion of the particle by

d ψ yψ(Y (y0,s),s) Y (y0,s)= v (Y (y0,s),s)=Im ∇ , Y (y0, 0) = y0 . (2) ds ψ(Y (y0,s),s) For a wave function ψ the position Y is a random variable the distribution of which is given by the equivariant probability measure Pψ with density ψ(y) 2 (Born’s statistical rule; see [2, 5] for a precise assertion). This means that| at any| time t the particle will typically be somewhere in the “main” support of ψ(y,t) . Thus for a narrow wave packet which, according to Ehrenfest’s theorem,| moves| – at least for some time – along a classical trajectory, at every instance of time t the position of the particle will typically be close to a classical position. To be sure: this does not imply that a typical Bohmian trajectory stays close to the classical trajectory for the whole duration of a given time interval, since it may every now and then make a large excursion. We consider a sufficiently smooth potential and a special class of initial wave functions where the potential V varies on a much larger scale than the wave functions, see e.g. [1] for a physical discussions of the scales. More precisely, we choose V ε(y) := V (εy) for some small parameter ε, thus defining a microscopic (y,s) and a macroscopic scale (x,t) := (εy,εs). As initial wave functions we ε cl cl take the semiclassical wave packets Φk(X (0), P (0), ) defined by Hagedorn in [4, 6]. They are non-isotropic three dimensional generalised· Hermite polynomials of order k := k multiplied by a Gaussian wave packet centred around the | | classical phase space point (Xcl(0), P cl(0)). On the macroscopic scale, i.e. on the scale of variation of the potential, their standard deviation is of order √ε both in position and momentum, that is they vary on an intermediate scale. This is the best order of ε allowed, since by Heisenberg’s uncertainty relation σyσp 1 on the microscopic scale, so on the macroscopic scale σxσp = εσyσp must be∼ of order ε. In the following, we change to macroscopic coordinates (x,t)=(εy,εs). With 3 ε 2 x t := x, := x and ψ (x,t) := ε− ψ( ε , ε ) Schrödinger’s equation then △reads △ ∇ ∇ ∂ ε2 iε ψε(x,t)= Hεψε(x,t)= + V (x) ψε(x,t) . (3) ∂t − 2 △

80 In this setting Hagedorn [4, 6] proved: with an error of order √ε in L2-norm ε ε ε cl cl the solution ψk(x,t) of (3) with initial data ψk(x, 0) = Φk(X (0), P (0), x) ε cl cl cl cl is given by Φk(X (t), P (t), x), where (X (t), P (t)) is the corresponding classical phase space trajectory, that is the solution of the Newtonian law of motion with initial data (Xcl(0), P cl(0)). Now consider the Bohmian trajectories on the macroscopic scale, i.e. solutions of the diﬀerential equation

d ε Xε(x ,t)= vψk (Xε(x ,t),t)= dt 0 0 ε ε (4) ψk(X (x0,t),t) ε = εIm ∇ ε ε , X (x0, 0) = x0 . ψk(X (x0,t),t) Our main result [9] is their convergence in probability: For all T > 0 and γ > 0 there exists some R< such that ∞ ε ψk( ,0) 3 ε cl P · ( x0 R max X (x0,t) X (t) R√ε ) > 1 γ (5) { ∈ | t [0,T ] | − |≤ } − ∈ for all ε small enough.

III. IDEA OF THE PROOF

ε Since the Bohmian trajectories X (x0,t) (as solutions of (4)) are continuous in t, none of those starting close to Xcl(0) (with x B (Xcl(0)) = 0 ∈ R√ε R3 cl cl x X (0) x < R√ε ) can leave the vicinity BR√ε(X (t)) of the { ∈ | | −cl | } cl classical trajectory X (t) without crossing the moving sphere ∂BR√ε(X (t)) = x R3 Xcl(t) x = R√ε . According to Hagedorn’s results [4, 6] { ∈ | | − | } cl nearly all Bohmian trajectories start in BR√ε(X (0)), so one is left to con- cl troll the probability that a Bohmian trajectory crosses ∂BR√ε(X (t)) in the time interval [0,T ]. The latter is equivalent to controlling the propability that the random conﬁguration-space-time trajectory (Xε( ,t) , t) crosses the surface ε cl · ΣT = (x, t) t [0,T ], x ∂BR√ε(X (t)) . Invoking the probabilistic { | ∈ ∈ } ε ε meaning of the quantum probability current density J ψk := (jψk , ψε 2) with ε k ψk ε ε | | j := εIm (ψ )∗ ψ one can show [7, 8] that an upper bound for this crossing k ∇ k propability is given by the modulus of the ﬂux across this surface

ε J ψk (x,t) U dσ Z · ε ΣT where U denotes the local unit normal vector at (x,t). Then the main technical ε challenge is to establish the pointwise estimates for the wave function ψk and

81 ε its gradient ψk that are needed to calculate this integral. For these pointwise estimates and∇ the precise proof of (5) see [9].

[1] V. Allori, D. Dürr, S. Goldstein and N. Zangh`ı, Journal of Optics B 4 (2002) 482, arXiv: quant-ph/0112005. [2] D. Dürr and S. Teufel, Bohmian Mechanics (Springer, Berlin, 2009), revised translation of Dürr, D.: Bohmsche Mechanik als Grundlage der Quantenmechanik, Springer, Berlin, 2001. [3] P. Markowich, T. Paul and C. Sparber, Journal of Functional Analysis 259 (2010) 6 1542 . [4] G. A. Hagedorn, Ann. Physics 269 (1998) 77. [5] D. Dürr, S. Goldstein and N. Zangh`ı, Journal of Statistical Physics 67 (1992) 843. [6] G. A. Hagedorn, Ann. Inst. H. PoincaréPhys. Théor. 42 (1985) 4 363. [7] K. Berndl, Zur Existenz der Dynamik in Bohmschen Systemen, Ph.D. thesis, Ludwig-Maximilians-Universität München, (1994). [8] K. Berndl, D. Dürr, S. Goldstein, G. Peruzzi and N. Zangh`ı, Comm. Math. Phys. 173 (1995) 3 647. [9] D. Dürr and S. Römer, Journal of Functional Analysis 259 (2010) 9 2404 .

82 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

Quantum Trajectories in Phase Space

Craig C. Martens Department of Chemistry University of California, Irvine Irvine, CA 92697-2025, USA

In this paper we review a method for simulating quantum processes using classical-like molecular dynamics in phase space. Our approach is based on solving the quantum Liouville equation using ensembles of classical trajectories. The nonlocality of quantum mechanics is incorporated in the trajectory representation as nonclassical interactions between the members of the ensemble, leading to an entanglement of their evolution. The statistical independence of the individual trajectories making up an ensemble in the classical limit is lost when quantum effects are included, and the entire state of the system must be propagated as a unified whole. We describe implementations of this approach in the Wigner and Husimi representations of quantum mechanics in phase space and apply the method to several model problems. A gauge-like freedom in the representation of phase space distribution function evolution with quantum trajectory ensembles is described. Finally, we briefly consider the general problem of solving evolution equations using trajectories in contexts beyond quantum dynamics. Quantum mechanics is the proper theoretical framework for describing the behavior of atoms and molecules [1, 2]. For simple systems, a direct numerical solution of the time-dependent Schrödinger equation is quite feasable, thanks to advances in both theoretical methodology and computer performance. This approach ceases to be practical for complex many-body problems, and approximate methods must be employed. A broad range of such approaches have been developed, including mean-field methods, semiclassical and mixed classical-quantum methods, phenomenological reduced descriptions, and others. One surprisingly effective approach in many cases is to simply ignore quantum effects altogether and use classical mechanics to describe the motion of atoms in molecular systems. The result is the method called classical molecular dynamics (MD) [3], a commonly used tool for studying many particle systems where high temperatures, large masses, or other factors allow quantum effects in the atomic motion to be neglected. An MD simulation is performed by solving the appropriate Hamilton’s or Newton’s equations of motion for the particles making up the system given their mutual forces of interaction and appropriate initial

83 conditions. An individual classical trajectory for a multidimensional problem is much easier to integrate numerically than the time-dependent wave packet of the corresponding quantum system. Unless the anecdotal information revealed by a single trajectory is sufficient, however, collections of trajectories—ensembles— must in general be considered. A distribution of trajectories evolving in phase space is the most direct classical analogue of an evolving quantum wave packet, and statistical averages of dynamical variables over the classical ensemble par- allel the corresponding quantum expectation values of operators. In order to discuss quantum systems from an analogous ensemble perspective, we adopt a phase space representation of quantum mechanics—the Wigner representation [4–7]. The nonlocality of quantum mechanics forbids arbitrarily fine subdivision of the quantum distribution into individual independent elements as is possible in classical mechanics, and insists that the entire state be propagated as a unified whole. If a trajectory ensemble representation of nonlocal quantum motion is to be achieved, the statistical independence of the trajectories must be given up and the individual members of the ensemble must interact with each other. We represent the time-dependent state of the system ρ(q,p,t) as an ensemble of trajectories. In classical mechanics, the ensemble members evolve independently of each other. A quantum state, however, is a unified whole, and the uncertainty principle prohibits an arbitrarily fine subdivision and independent treatment of its constituent parts. We incorporate the non-classical aspects of quantum mechanics explicitly as a breakdown of the statistical independence of the members of the trajectory ensemble. We derive non-classical forces acting between the ensemble members that model the quantum effects governing the evolution of the corresponding nonstationary wave packet. The realization of our formalism in the context of a classical molecular dynamics simulation is accomplished by generating an ensemble of initial conditions representing ρW (q,p, 0) and then propagating the trajectory ensemble. The entangled trajectory formalism gives a unique and appealing physical picture of the quantum tunneling process. Rather than “burrowing” through the obstacle, trajectories that successfully escape the metastable well do so by “borrowing” enough energy from their fellow ensemble members to surmount the barrier. This loan is then paid back, always keeping the mean energy of the ensemble a constant. We have outlined an approach to the simulation of quantum processes using trajectory integration and ensemble averaging [8–12]. The general method has been applied in the context of quantum tunneling through a potential barrier. The basis of the method is the Liouville representation of quantum mechanics and its realization in phase space via the Wigner function formalism. The evolution of the phase space functions is approximated by the motion of the corresponding trajectory ensembles. In the classical limit, the members of the ensemble evolve independently under Hamilton’s equations of motion. When

84 quantum eﬀects are included, however, the corresponding “quantum trajectories” are no longer separable from each other. Rather, their statistical independence is destroyed by nonclassical interactions that reﬂect the nonlocality of quantum mechanics. Their time histories become interdependent and the evolution of the quantum ensemble must be accomplished by taking this entanglement into account.

[1] C. Cohen-Tannoudji and B. Diu and F. Laloe, Quantum Mechanics, (Wiley, New York, 1977). [2] G. C. Schatz and M. A. Ratner, Quantum Mechanics in Chemistry, (Prentice Hall, Englewood Cliﬀs, 1993). [3] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, (Clarendon Press, Oxford, 1987). [4] E. P. Wigner, Phys. Rev., 40, (1932), 749. [5] K. Takahashi, Prog. Theor. Phys. Suppl., 98, (1989), 109. [6] H. -W. Lee, Phys. Rep., 259, (1995), 147. [7] S. Mukamel, Principles of Nonlinear Optical Spectroscopy, (Oxford University Press, Oxford, 1995). [8] A. Donoso and C. C. Martens, Phys. Rev. Lett., 87, (2001), 223202. [9] A. Donoso and C. C. Martens, Int. J. Quantum Chem., 87, 2002, 1348. [10] A. Donoso and C. C. Martens, J. Chem. Phys., 116, (2002), 10598. [11] A. Donoso and C. C. Martens, J. Chem. Phys., 119, (2003), 5010. [12] H. L´opez and C. C. Martens and A. Donoso, J. Chem. Phys., 125, (2006), 15411.

85 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

The Semiclassical Limit of Time Correlation Functions by Path Integrals

G. Ciccotti School of Physics, Room 302b UCD-EMSC, University College Dublin, Belﬁeld, Dublin 4, Ireland∗

I. INTRODUCTION

The exponential scaling of the computational cost of quantum time evolution with the number of degrees of freedom motivates current attempts to approximate and interpret quantum dynamics via classical trajectories. These can in fact be computed with essentially linear eﬀort and provide a more intuitive representation of the dynamics. In spite of these tempting properties of the trajectories, the accuracy and generality of such attempts requires careful analysis since it is unclear whether they can be successful for condensed phase systems. To illustrate this point, we comment on how and when quantum evolution can be approximated in terms of (generalized) classical dynamics in the calculation of the symmetrized time correlation function [1]

1 i ∗ i ~ Htˆ − ~ Htˆ c GAB(t,β)= Tr Aeˆ c Beˆ (1) Z { } i~β in semiclassical conditions. In the expression above, tc = t , β = 1/kBT − 2 (T is the temperature and kB Boltzmann’s constant), Hˆ is the Hamiltonian of the system and Z = Tr e−βHˆ is the canonical partition function. Eq. (1) is equivalent via a relationship{ in} Fourier space to the standard time correlation − ˆ i ˆ − i ˆ 1 βH ˆ ~ Ht ˆ ~ Ht function CAB(t,β)= Z Tr e Ae Be , but it also shares some formal properties with classical correlation{ functions, for} example it is by construction a real function, and this suggests that it might be a convenient starting point for describing semiclassical systems (see for example[2–8]). The analysis presented in the following is described in [9] and we refer to that paper for a detailed derivation of the results summarized here.

∗ On leave from Dipartimento di Fisica and CNISM Unit`a1, Universit`a“La Sapienza”, Piazzale Aldo Moro 5, 00185 Rome, Italy

86 II. THEORY

The starting point of our considerations is the Feynman path integral expression (in a mixed coordinate and momenta representation) of the forward and backward propagators in complex time that appear in eq. (1). To examine the semiclassical limit of that expression, mean and diﬀerence paths in the coordinates and momenta are introduced and the exponent of the overall path integrals written as a Taylor series expansion in the diﬀerence paths.

A. First order result

Retaining only terms up to linear order in the Taylor series expansion, the symmetrized function can be written as

p¯2 −β 1 +V (¯r ) (1) 1 2m 0 G (t,β)= dr¯ dp¯ e Aw(¯r , p¯ )Bw(¯rt, p¯t) (2) AB Z 0 1 0 1 Z where Ow(¯r, p¯) stands for the Wigner transform [10] of operator Oˆ, and (¯rt, p¯t) are the end points of the classical trajectory evolved from (¯r0, p¯1) for a time t. Both the dynamics and the statistical weight in the correlation function above thus reduce to their fully classical counterparts. The Fourier relationship with the standard time correlation function mentioned in the Introduction can, however, be used to restore some non-classical properties (such as detailed balance) of this quantity, and it is in fact formally identical to the so-called quantum correction procedure that was introduced by Schoﬁeld in ref. [1]. However, it is well known that this correction can fail at low temperature even when the system is non-interacting (see [9] for an explanation of this fact) and, more in general, that the temperature and mass range in which it is valid are quite limited.

87 B. Second order result

The result of a second order truncation of the series expansion of the exponent, instead, can be expressed as

N−1 (2) Zcl(2ǫβ/~) dp¯N G (t,β)= dr¯ dr¯N dr¯kdp¯k AB Z(β) 0 2π~ ( k ) Z Y=1 Z ǫ 2 2 2 β p¯ N−1 [p¯k+1−p¯k+ǫt∇V (¯rk)] − 1 V r N p¯ 2 − ~ 2m + (¯0) − 1 − − k k 2 k 2 [r¯k r¯k−1 ǫt ] =1 2~ǫ |∇ V (¯r )| e e =1 2σ m e β k P P −  ~ 2 N N 1 2  × Zcl(2ǫβ/ ) (√2πσ ) 2π~ǫβ V (¯rk)  k=1 |∇ |  p ǫ 2 Q  2 β N p¯k  − +V (¯rk−1)  ~ k=2 2m e Aw(¯r , p¯ )Bw(¯rN , p¯N )F ( p¯k , r¯k ; ǫβ,ǫt) × P 0 1 Ω { } { }

(3) where r¯k− , p¯k (k = 1,...,N) are the positions and momenta along the path, { 1 } Zcl(2ǫβ/~) is the classical partition function at inverse temperature 2ǫβ/~, and FΩ is discussed below. The factors in the curly bracket are a probability density and the approximate symmetrized correlation function can then be computed as ǫ 2 2 β N p¯k − +V (¯rk ) ~ k=2 2m −1 the expectation of e Aw(¯r , p¯ )Bw(¯rN , p¯N )F ( p¯k , r¯k ). P 0 1 Ω { } { } The variables r¯k− , p¯k can be sampled as follows: The zero time val- { 1 } uesr ¯0, p¯1 are obtained from the high temperature classical Boltzmann factor 2ǫ p¯2 − β 1 +V (¯r ) ~ 2m 0 e /Zcl(2ǫβ/~), while the other variables are generated recursively from

r¯k = rk− + ǫtp¯k/m + σξk k [1,N] 1 ∈ (4)  2 p¯k =p ¯k ǫt V (¯rk)+ ~ǫβ V (¯rk) ηk k [1,N 1]  +1 − ∇ |∇ | +1 ∈ − p whereξk and ηk+1 are Gaussian white noises. This scheme illustrates how quantum mechanical delocalization sets in this semiclassical representation of the correlation function. Within this approximation in fact, both the new coordinates and momenta are sampled at each complex time step from Gaussian distributions centered around classically evolved phase space points. The dis- ~ 2 ǫβ persion around the classical path is determined by the variances σ = m and 2 ~ǫβ V (¯rk). The classical limit is restored for ~ 0 and/or β 0 when these variances∇ tend to zero. For ﬁnite values of Planck’s→ constant or→ of the inverse temperature, the non classical nature of the time evolution of the system appears at each time step in the form of Gaussian random displacements from the

88 ”driving” classical propagation. While this interpretation is intriguing, the actual interest of the driving classical trajectory depends crucially on the system. If the potential is everywhere convex, the function FΩ in the integrand reduces to a constant and the estimate of the average as a mean over paths generated as outlined above is viable. In the more general case of potentials with regions of negative curvature, on the other hand, this function does not have an explicit form, and there is no reason to expect that it will be localized around the complex paths generated via the sampling scheme of eq. (4). Furthermore, it can be shown that small variations in its argument result in ”explosively” different values for FΩ. Attempts to interpret or estimate the average above via a scheme based on localized paths are therefore doomed to failure for two reasons: first, the integrand is not peaked around the sampling function, second it is a numerically unstable function. These characteristics are a direct mani- festation of delocalization, an intrinsic property of quantum mechanics that it is very difficult, if not impossible, to represent within this semiclassical scheme. A possible scheme to build upon the considerations presented here and to go beyond the semiclassical approximation systematically and still employing the path integral representation of the symmetrized correlation function has been introduced in [11] and is summarized in the contribution by S. Bonella.

III. CONCLUSIONS

The path integral expression of the symmetrized correlation function is a useful tool to examine how and when quantum evolution can be approximated via (generalized) classical trajectories. In particular, the second order result presented in the previous section shows how, in the semiclassical limit, the most relevant contributions to the path integral localize or, pathologically, de-localize around guiding or poorly guiding classical trajectories for general systems. While we employed the path integral formalism to illustrate how a picture based on classical dynamics is usually not enough to compute quantum properties, the diﬃculty to account for delocalization appears also in other approximations of quantum mechanics (e.g. Wigner-Liouville, semiclassical IVR) pointing to the inherent diﬃculty of using a trajectory based picture to represent this phenomenon.

[1] P. Schoﬁeld. Phys. Rev. Lett, 4:239, 1960. [2] V.S. Filinov. Mol. Phys., 88:1517, 1996. [3] V.S. Filinov. Mol. Phys., 88:1529, 1996. [4] W.H. Miller, S.D. Schwartz, and J.W. Tromp. J. Chem. Phys., 79:4888, 1983. [5] V. Jadhao and N. Makri. J. Chem. Phys., 129:161102, 2009. [6] J.A. Poulsen, H. Li, and G. Nyman. J. Chem. Phys., 131:024117, 2009.

89 [7] G. Krilov, E. Sim, and B.J. Berne. Chem. Phys., 268:21, 2001. [8] N. Chakrabarti, T. Carrington, and B. Roux. Chem. Phys. Letts., 293:209, 1998. [9] S. Bonella, M. Monteferrante, C. Pierleoni, and G. Ciccotti. J. Chem. Phys., 133:164104 2010. [10] E. Wigner. Phys. Rev., 40:749, 1932. [11] S. Bonella, M. Monteferrante, C. Pierleoni, and G. Ciccotti. J. Chem. Phys., 133:164105 2010.

90 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

Path Integral Calculation of (Symmetrized) Time Correlations Functions

S. Bonella Department of Physics, Universit`adi Roma La Sapienza, P.la A.Moro 2, 00185 Rome, Italy

I. THEORY

The path integral expression of the symmetrized time correlation function [1], 1 i ˆ ∗ − i ˆ ˆ ~ Htc ˆ ~ Htc GA,B(t,β) = Z Tr{Ae Be } can be used to derive a computational scheme that, in principle, allows to systematically include quantum dynamical eﬀects starting from a zero order approximation of this function based upon classical trajectories. In the following, we summarize this scheme, illustrated in detail in [2]. Introducing resolutions of the identity in the coordinate representation, we can isolate matrix elements of the operators and use the time compo- sition property to rewrite the forward and backward propagators in GAB(t,β) as products of short complex time propagators. Thus (indicating with tilde the variables referring to the backward propagator)

L−1 1 GAB(t; β) = drLdr˜L < r˜L|Bˆ|rL > drJ dr˜J K(rJ+1, r˜J+1,rJ , r˜J ) Z (J=1 ) Z Y Z × dr0dr˜0K(r1, r˜1,r0, r˜0) (1) Z In the expression above, we introduced the product of the real and imaginary time propagators for the (finite) intervals τβ = β/(2L) and τt = t/L, and defined the short complex time propagator for the leg that evolves the system forward from configuration rJ to configuration rJ+1 and backward fromr ˜J+1 tor ˜J as

ˆ i ˆ ν ν −τβ H ν ν ~ τtH K(rJ+1, r˜J+1,rJ , r˜J ) = drJ dr˜J < r˜J |e |r˜J >< r˜J |e |r˜J+1 >

Z i ˆ ˆ − ~ τtH ν ν −τβ H × < rJ |e |rJ > (2)

The propagators in inverse temperature in the expression above can be computed exactly via relatively straightforward path integral methods for the evaluation

91 of the density matrix [3, 4] in convenient variables (sum and difference). The product of propagators in real time, instead, is approximated by introducing a path integral representation in sum and difference variables [5] (see also contribution by G. Ciccotti), and truncating to first order the Taylor series expansion of the representation’s phase in the difference variables. The effect of this linearization approximation is to reduce the generation of the path in sum variables to a time-stepping propagation that is formally identical to a classical evolution algorithm. At the same time, the path in difference variables can be almost completely integrated analytically and the only remaining contributions from these variables appear in two phase factors to be computed at the beginning and the end of the classical propagation along the sum path. The explicit form of the approximate short time propagators can be found in [2]. Substituting this form in eq. (1), the symmetrized correlation function can be interpreted as the ratio of two expectation values over a probability density, P , defined by the product of the Green’s functions corresponding to the (approximate) real and (exact) imaginary time dynamics in each short time propagator K (see ref. [2] for the definition). Indicating with I the identity operator, if we use L short time propagators, this ratio is

L P GAB(t; β)= (3) P where the observable OAB(t; β) contains the matrix elements of the operators, and a product of L − 1 phase factors originated by the surviving difference variable integrals (see ref. [2] for the explicit definition). Since the linearization approximation for the real time propagation becomes exact for L → ∞, i.e. as we increase the number of propagators K, the expression above for the symmetrized correlation function can, in principle, be made as accurate as necessary. The averages in eq.(3) can be computed via Monte Carlo sampling of the probability density P . This sampling is realized by generating paths in complex time according to the propagations in real and imaginary time determined by the appropriate Green’s functions. The structure of these paths is illustrated in Fig. (1). For L = 1 there is only one real time leg of duration τt = t while the imaginary time propagation corresponds to an inverse temperature β/2 for both the mean and difference variables. The combination of these propagations is illustrated in the top panel. In the figure, the horizontal axis is time, the vertical axis temperature. The vertical plane represents the space of configurations associated to the thermal path integral, the intermediate states of the path are indicated with the red spheres (in the figure, we use, as an example, six ”thermal beads”). The harmonic interactions that represent the kinetic part of Hamiltonian in the thermal paths (see for example [3]) are indicated with zigzagged lines connecting adjacent beads, while the interactions among the two paths due to the potential are drawn as dashed lines. The propagation in real time is sketched in the figure as the curve

92 FIG. 1: Graphic representation of the propagators in real and imaginary times contributing to the approximate Schofield function for the case L = 1 (upper panel) and L = 2 (lower panel). The horizontal axis is real time while the vertical axis is inverse temperature. The thermal paths are represented as red dots on the vertical planes. Segments of classical propagation in phase space are represented as continuos red curves in the horizontal planes. The golden circles indicate the connection between the dynamics in real time (horizontal planes) and the representation of the dynamics in imaginary time (vertical planes). on the horizontal plane. This plane represents the phase space of the system. The red and golden circle at t = 0 indicates the structure of the initial conditions for the evolution: the initial coordinate coincides with the last bead of the thermal path in the mean variables, while the initial momentum is sampled from a Maxwellian at inverse temperature β/2. A phase factor is associated with the initial and final point of the classical propagation. The structure of the product of the Green’s functions for generic values of L can be inferred from the lower part of figure 1 where we show what happens for L = 2. In this case, there are two segments of classical dynamics, represented as in the previous case by the curves on the horizontal planes, each of duration t/2 and two propagations of mean and difference variables in imaginary time, represented in the vertical planes, each taking the system to an inverse temperature equal to one half of the actual inverse temperature. As before, the first segment of dynamics starts, with a Gaussian initial momentum, from the last bead of the mean variables

93 thermal path at t = 0. The end point of this leg of propagation is the initial configuration for the mean variable thermal path at t/2, and the second segment of dynamics has as initial conditions the final coordinate of the mean variable thermal path and a new momentum sampled from a Gaussian. The variances of the Gaussians associated to the momentum sampling are doubled with respect to the case L = 1. Two phase factors at the beginning and end of each classical dynamics segment are associated with this product of Green’s functions (so L = 2 has a total of four phase factors). In general, the product of Green’s functions L in GAB(t; β) involves L segments of classical propagation, each of duration t/L, interspersed with L pairs of thermal paths in the mean and difference variables, each at an inverse temperature β/2L. The rules for connecting the coordinate and momenta at the initial and final time of the dynamics with the final and initial points of the thermal paths, and for constructing the 2L phase factors contributing to the observable are completely analogous to the L = 2 case.

II. CONCLUSIONS

The method just outlined becomes exact as the number of segments of propagation in complex time tends to infinity and, in this respect, it is completely analogous to a thermal path integral that becomes exact in the limit of an infinite number of ”slices” in the imaginary time propagator. Unfortunately, since in this limit the number of phase factors in the observable also becomes infinite, the numerical effort required to compute the Monte Carlo average increases dramatically. This is not surprising as any exact expression for the correlation function will manifest the well known dynamical sign problem characteristic of quantum dynamics. For semiclassical systems, however, the number of segments required to achieve convergence of the approximation can be small enough to make the computational strategy described here interesting. The tests performed so far indicate that the L = 1 result provides a viable alternative to standard fully linearized methods [5–7] with the advantage that the expression for the thermal density is simpler than that in those methods. The tests also show that quantum corrections to the dynamics can indeed be introduced by increasing the number of short time propagators. However, the numerical cost associated to higher order iterations of the scheme is very high and more work is necessary to make the algorithm useful for condensed phase simulations.

[1] P. Schoﬁeld. Phys. Rev. Lett, 4:239, 1960. [2] S. Bonella, M. Monteferrante, C. Pierleoni, and G. Ciccotti. J. Chem. Phys., 133:164105 2010.

94 [3] R. P. Feynman. Statistical Mechanics a Set of Lectures. Addison Wesley, New York, 1990. [4] H. Kleinert. Path Integrals in Quantum Mechanics, Statics, Polymer Physics and Financial Markets. World Scientiﬁc, Singapore, 2004. [5] J.A. Poulsen, G. Nyman, and P.J. Rossky. J. Phys. Chem A, 108:8743, 2004. [6] H. Wang, X. Sun, and W.H. Miller. J. Chem. Phys., 108:9726, 1998. [7] Q. Shi and E. Geva. J. Phys. Chem. A, 107:9070, 2003.

95 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

On-the-ﬂy Nonadiabatic Bohmian DYnamics (NABDY)

Ivano Tavernelli, Basile F. E. Curchod, Ursula Rothlisberger Laboratory of Computational Chemistry and Biochemistry, Ecole Polytechnique F´ed´erale de Lausanne.

I. INTRODUCTION

In the semiclassical description of molecular systems, only the quantum character of the electronic degrees of freedom are considered while the nuclear motion is treated at the classical level [1]. In the adiabatic case, this picture corresponds to the Born-Oppenheimer limit where the nuclei move as point charges on the potential energy surface (PES) associated with a given electronic state. This approximation relies on the assumption that the electrons follow adiabatically the motion of the slow nuclei, and thus that the electronic state of the system is not affected by the nuclear displacement. Despite the wide success of this approximation, many physical and chemical processes do not fall in a regime where nuclei and electrons can be considered to be decoupled [2–4]. In particular, for an adequate description of many photochemical and photophysical processes nonadiabatic effects need to be included, e.g. to precisely describe relaxation processes via nonradiative internal conversion or intersystem crossing. Within the mixed-quantum classical approaches, the classical path, or Ehren- fest approximation is the most straightforward one. Here, the classical subsystem evolves under the mean field generated by the electrons, and the electronic dynamics is evaluated along the classical path of the nuclei. An important lim- itation of the classical path approach is the absence of a ”back-reaction” of the classical Degrees of Freedom (DoF) to the dynamics of the quantum DoF. On the other hand, these methods are well suited for the study of the nuclear dynamics in the full phase space (without the need of introducing constraints or low dimensional reaction coordinates) and can easily be implemented in software packages that allow for the ”on-the-fly” calculation of energies and forces. One way is to employ Ehrenfest’s theorem and calculate the effective force on the classical trajectory through a mean potential that is averaged over the quantum DoF [5–9]. Beyond such quasi-classical methods, the semiclassical Wentzel-Kramers- Brillouin (WKB)-type approach has a long tradition in adding part of the missing nuclear quantum effects to classical simulations. Semiclassical methods [10–12]

96 take into account the phase exp(iS(t)/~) evaluated along a classical trajectory and are therefore capable - at least in principle - of describing quantum nuclear effects including tunneling, interference effects, and zero-point energies. The most common semiclassical methods have been extensively reviewed in recent articles [13–16]. The intuitively appealing picture of trajectories hopping between coupled potential energy surfaces gave rise to a number of quasiclassical implementations [17–24]. The most well-known method is Tully’s ”fewest switches” trajectory surface hopping method [17–21, 25] (TSH in this text), which has evolved into a widely used and successful technique. In this framework, the nuclear wavepacket is represented by a swarm of independent classical trajectories (independent trajectory approximation, ITA) while the nonadiabatic couplings (NACs) induce hops between different electronic states occurring according to a stochastic algorithm, see Fig. 1.

FIG. 1: Left: Trajectory surface hopping in the independent trajectory approximation. An initial nuclear wavepacket on an electronic excited state is represented by a swarm of independent trajectories (dots and lines). According to a stochastic algorithm, trajectories can jump to other states (vertical dashed lines). The final distribution of a sufficiently large number of such trajectories is assumed to reproduce the final nuclear wavepacket splitting. Right: Representation of the same process in a nuclear quantum dynamics scheme.

In the adiabatic representation, coupling between electronic and nuclear degrees of freedom are obtained by calculation of the nonadiabatic coupling terms. The nonadiabatic coupling vectors (NACVs) are deﬁned between state I and J for the nuclei γ by:

dγ (R)= {Φ∗ (r; R)[∇ Φ (r; R)]} dr (1) JI Z J γ I and the second order nonadiabatic elements by

2 Dγ (R)= Φ∗ (r; R) ∇ Φ (r; R) dr. (2) JI Z J γ I

97 The latter are often neglected due to their usually small size. The complete set of electronic wavefunctions {ΦJ (r; R)} are the solutions of the time-independent electronic Schrödinger equation ˆ r R r R el R r R Hel( ; )ΦJ ( ; )= EJ ( )ΦJ ( ; ) (3) In TSH, all trajectories are evolved with forces obtained from an adiabatic surface (in contrast to the mean-field approach). In addition, a new equation of motion for each trajectory α on state J is propagated, which contains all the α nonadiabatic information via a time-dependent amplitude CJ (t). The classical trajectories evolve adiabatically according to Born-Oppenheimer dynamics el el until a hop between two potential energy surfaces (EI and EJ ) occurs with a probability given by a Monte Carlo-type procedure. In practice, a swarm of trajectories is propagated independently starting from different initial conditions, and the final statistical distribution of all these trajectories is assumed to reproduce the correct time evolution of the nuclear wavepacket (see Fig 1). It is important to stress that, at present, no formal justification of Tully’s algorithm has been formulated. Furthermore, the independent trajectory approximation neglects nuclear quantum effects such as tunneling, interferences between the nuclear wavepackets or (de)coherence effects. All trajectories are independent and do not carry any quantum information like phase or amplitude, which could be exchanged with other trajectories. At the end of the 90ies, a new quantum trajectory method (QTM) has been developed by Wyatt et al. [26] for adiabatic quantum dynamics based on a Bohmian (or hydrodynamical) interpretation of quantum mechanics. Within this method, it is possible to derive formally exact equations of motion for quantum trajectories (or fluid elements), that explicitly reproduce the nuclear wavepacket dynamics [27]. Trajectories are correlated with one another by means of a quantum potential. Different schemes for multisurface QTM have been proposed in the diabatic representation [28–30]. In this work, we propose to solve the non-relativistic quantum dynamics of nuclei and electrons within the framework of Bohmian dynamics using the adiabatic representation of the electronic states. An on-the-fly trajectory-based nonadiabatic molecular dynamics algorithm is derived (NABDY, NonAdiabatic Bohmian DYnamics), which is able to capture also the nuclear quantum effects that are missing in the traditional trajectory surface hopping approach based on the independent trajectory approximation. The use of correlated trajectories produces a quantum dynamics, which is in principle exact and computationally very efficient. Instead of representing the molecular nuclear wavepacket in terms of uncorrelated trajectories, the wavepacket is now split in fluid elements (set of initial molecular configurations) that carry phase and amplitude information. This method allows for on-the-fly dynamics in the adiabatic representation. Thanks to this formulation, the method can be coupled to the ab initio code cpmd, which provides electronic energies, classical forces and nonadiabatic coupling elements for each configuration at DFT/TDDFT level of theory.

98 For the calculations of nonadiabatic coupling elements and more precisely the nonadiabatic coupling vectors between electronic states J and I, dJI (R), with LR-TDDDFT, the main challenge is to express these quantities initially deﬁned in terms of wavefunctions (see Eq. (1)) as functionals of the electronic den- dγ R dγ R sity JI [ρ]( ) or, equivalently, of the set of Kohn-Sham orbitals JI [{φ.}]( ). Among diﬀerent techniques, we use auxiliary many electron wavefunctions obtained from the ground state and singly excited Kohn-Sham Slater determinants, for which we have shown that the exact results obtained using Many-Body Per- turbation Theory can be recovered [31–33].

II. THEORY AND FIRST APPLICATIONS

Starting from the time-dependent Schrödinger equation, we can use the Born- r R ∞ r R R Oppenheimer Ansatz Ψ( , ,t) = J ΦJ ( ; )ΩJ ( ,t) to describe the total molecular wavefunction Ψ(r, R,t). Here,P r gives the positions of all the electrons of the system and R those of the nuclei. J denotes a specific electronic state, ΦJ (r; R) is a solution of the time-independent electronic Schrödinger equation and ΩJ (R,t) can be seen as a nuclear wavefunction. After some manipulation and using the polar representation for the nuclear wavefunctions, we can extract equations of motion for the nuclear amplitude AJ (R,t)

∂A (R,t) 1 1 2 J = − ∇ A (R,t)∇ S (R,t) − A (R,t)∇ S (R,t) ∂t M γ J γ J 2M J γ J Xγ γ Xγ γ ~ γ R R iφ + DJI ( )AI ( ,t)ℑ e 2Mγ XγI ~ dγ R R iφ − JI ( )∇γ AI ( ,t)ℑ e Mγ γ,IX6=J 1 dγ R R R iφ − JI ( )AI ( ,t)∇γ SI ( ,t)ℜ e , (4) Mγ γ,IX6=J 1 where φ = ~ (SI (R,t) − SJ (R,t)), and a Newton-like equation for the time propagation of ﬂuid elements associated to a discretization of the nuclear con- ﬁgurational space 2R d β J Mβ 2 = −∇β E (R)+ QJ (R,t)+ DJ (R,t) (5) J el (dt )

In these equations, SJ (R,t)/~ represents the phase of the nuclear wavefunction and γ is a general notation for a speciﬁc nucleus with mass Mγ . Eq 5 contains the well-known quantum potential QJ (R,t) and an additional nonadiabatic quantum potential DJ (R,t) which describes all cross-surfaces terms.

99 Numerically, we use a conventional QTM propagation scheme for the adiabatic part of the dynamics on a specific electronic state (say S1) far from any nonadiabatic regions. When the size of the nonadiabatic couplings and the amount of amplitude transferred reach a certain threshold, the multisurface dynamics governed by the nondiagonal elements dJI (R) and DJI (R) is initiated and a new set of fluid elements is attributed to the new electronic state (S2, see Fig 2). The arbitrary Lagrangian-Eulerian (ALE) frame [34] is used to synchronize the time propagation of the fluid elements on the different surfaces.

FIG. 2: Schematic view of NABDY. State S2 is activated through a threshold based on the incoming amplitude from the initial state S1.

In order to validate our approach, we have first applied it to the study of the Tully model 1 system [25]. For initial momenta between 16 and 22 a.u., the NABDY dynamics compared well with the results obtained from the exact nuclear wavepacket propagation (deviation < 0.6%). On the other hand, the final populations obtained from TSH deviate on average by about 2% from the exact result. Interestingly, NABDY dynamics converges with only one tenth of the trajectories needed to converge TSH. We have applied NABDY to other nonadiabatic processes, like for instance the collision of an H atom with a H2 molecule. For this system, we have computed on-the-fly the PESs and the nonadiabatic couplings using DFT/TDDFT with the LDA exchange-correlation functional as implemented in the plane-wave code CPMD [35]. The quality of this level of theory for the H-H2 collision has been assessed in a previous publication [31]. We present here some initial results (see Fig 3), where the H atom is directed almost perpendicular to the H2 bond axis with two different initial momenta, k = 75 and k = 150 a.u.. In these cases, we observed a population of the first excited state of 27.9% and 31.4%, respectively.

In conclusion, we report here nonadiabatic Bohmian dynamics, where all the electronic properties needed are computed on-the-ﬂy by DFT and LR-TDDFT.

100 FIG. 3: Collision of a H atom with a H2 molecule. PESs (blue: ground state, orange: ﬁrst excited state) and nonadiabatic couplings (black dotted line) are computed via DFT/TDDFT. Inset shows the LUMO orbital of the system close to the avoided crossing.

We are currently working on an eﬃcient adaptation of this scheme in the context of general nonadiabatic molecular dynamics.

[1] D. Marx and J. Hutter, Modern Methods and Algorithms of Quantum Chemistry, vol. 1 of NIC Series (Forschungszentrum Juelich, 2000) p. 301, p. 301. [2] A. W. Jasper, S. Nangia, C. Zhu and D. G. Truhlar, Acc. Chem. Res. 39 (2006) 2 101. [3] T. Yonehara, S. Takahashi and K. Takatsuka, J Chem Phys 130 (Jun 2009) 21 214113. [4] K. Takatsuka, International Journal of Quantum Chemistry 109 (2009) 10 2131. [5] R. B. Gerber, V. Buch and M. A. Ratner, J. Chem. Phys. 77 (1982) 32. [6] R. Kosloﬀ and A. Hammerich, Faraday Discuss. 91 (1991) 239. [7] F. A. Bornrmann, P. Nettesheim and C. Schutte, J. Chem. Phys 105 (1996) 1074. [8] C. Zhu, A. W. Jasper and D. G. Truhlar, J. Chem. Phys. 120 (2004) 5543. [9] I. Tavernelli, U. F. R¨ohrig and U. Rothlisberger, Mol. Phys. 103 (2005) 963. [10] W. Miller, J. Chem. Phys. 53 (1970) 3578. [11] M. F. Herman, J. Chem. Phys. 81 (1984) 754. [12] E. Heller, J. Chem. Phys. 94 (1991) 2723. [13] M. S. Topaler, T. Allison, D. W. Schwenke and D. G. Truhlar, J. Chem. Phys. 109 (1998) 3321. [14] G. A. Worth and M. A. Robb, Adv. Chem. Phys. 124 (2002) 355. [15] H. D. Meyer and G. A. Worth, Theor. Chem. Acc. 109 (2003) 251. [16] H. D. Meyer, U. Manthe and L. S. Cederbaum, Chem. Phys. Lett. 165 (1990) 73. [17] J. C. Tully and R. K. Preston, J. Chem. Phys. 55 (1971) 562.

101 [18] S. Hammes-Schiffer and J. C. Tully, J. Chem. Phys. 101 (1994) 4657. [19] N. C. Blais, D. Truhlar and C. A. Mead, J. Chem. Phys. 89 (1988) 6204. [20] O. V. Prezhdo and P. J. Rossky, J. Chem. Phys. 107 (1997) 825. [21] I. Krylov and R. Gerber, J. Chem. Phys 105 (1996) 4626. [22] N. L. Doltsinis and D. Marx, Phys. Rev. Lett. 88 (2002) 166402. [23] C. F. Craig, W. R. Duncan and O. V. Prezhdo, Phys. Rev. Lett. 95 (2005) 163001. [24] E. Tapavicza, I. Tavernelli and U. Rothlisberger, Phys. Rev. Lett. 98 (2007) 023001. [25] J. C. Tully, J. Chem. Phys. 93 (1990) 1061. [26] C. L. Lopreore and R. E. Wyatt, Phys. Rev. Lett. 82 (1999) 26 5190. [27] R. E. Wyatt, Quantum dynamics with trajectories: Introduction to quantum hydrodynamics (Interdisciplinary applied mathematics, 2005). [28] R. E. Wyatt, C. L. Lopreore and G. Parlant, J. Chem. Phys. 114 (2001) 12 5113. [29] C. L. Lopreore and R. E. Wyatt, J. Chem. Phys. 116 (2002) 4 1228. [30] B. Poirier and G. Parlant, J. Phys. Chem. A 111 (2007) 41 10400. [31] I. Tavernelli, E. Tapavicza and U. Rothlisberger, J. Chem. Phys. 130 (2009) 124107. [32] I. Tavernelli, E. Tapavicza and U. Rothlisberger, J. Mol. Struc. (Theochem) 914 (2009) 22. [33] I. Tavernelli, B. F. E. Curchod and U. Rothlisberger, J. Chem. Phys. 131 (2009) 19 196101. [34] B. K. Kendrick, J. Chem. Phys. 119 (2003) 5805. [35] CPMD (Copyright IBM Corp 1990-2001, Copyright MPI für Festkörperforschung Stuttgart, 1997-2001), http://www.cpmd.org.

102 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

Quantum Many-Particle Computations with Bohmian Trajectories: Application to Electron Transport in Nanoelectronic Devices

A. Alarcón, G.Albareda, F.L.Traversa and X.Oriols Dept. d’Enginyeria Electrònica, Universitat Autònoma de Barcelona (UAB) 08193, Bellaterra, Spain Email: [email protected]

I. INTRODUCTION

We have recently shown [1] that Bohmian trajectories allow a direct treatment of the many-particle interaction among electrons with an accuracy comparable to Density Functional Theory (DFT) techniques. The computational technique developed in [1], which allows a computation of the quantum correlations among particle without explicitly knowing the many-particle wave-function, can be applied to many diﬀerent open and closed quantum systems. In particular, in this article, we present a general, versatile and time-dependent 3D quantum electron transport simulator, named BITLLES (Bohmian Interacting Transport in Electronic Structures), based on the computation of many-particle Bohmian trajectories mentioned in [1]. As a numerical example, we show the ability of BITLLES simulator to predict the electrical characteristics (DC, AC and ﬂuc- tuations) of a Resonant Tunneling Diode, i.e. a many-particle open quantum system far from equilibrium.

II. THE BITLLES SIMULATOR

A. The Monte Carlo nature of the simulator

It is not possible to take into account all degrees of freedom for a close Hamil- tonian of the whole solid-state system (battery, wires, sample,...). Therefore, when we neglect a large part of the degrees of freedom of the circuit, we dis- regard their inﬂuence into the N(t) explicitly simulated electrons inside the 3D active region of the device, i.e.the simulation box whose volume Ω is depicted in

103 Fig. 1. Thus, we cannot completely specify the initial N(t)-particle wavefunction inside the simulation box because we do not know with certainty the number of electrons N(t), their energies, their positions.... In the BITLLES simulator, the adaptation of Bohmian mechanics to electron transport in open systems, leads to a quantum Monte Carlo algorithm, where randomness appears because of these uncertainties. For this purpose, we take into account two statistical ensembles of the initial properties of the electrons on the numerical simulations. First, a g-distribution that represents the inﬁnite ensemble of all possible distributions in the initial positions of Bohmian particles. Second, an h-distribution that takes into account the uncertainty in the number of electrons in the active region N(t), the mean energy associated to the wavepackets of these electrons and the injection times of each electron [2].

B. Self-consistent solution of Poisson equation and many-particle Schr¨odinger equation

Many-particle time-dependent quantum electron simulators have to provide reasonable approximations for handling the many-particle (electron - electron interaction) problem. However, it is well-known that the many-particle Schrödinger equation can be solved for very few -two, three,..- degrees of freedom. To surpass this computational problem, BITLLES simulator is based on a novel algorithm for solving the many-particle Schrödinger equation with Bohmian mechanics [1]. It can include explicitly the Coulomb and exchange correlations (at a level comparable to the time dependent density functional theory). Following reference [1] a many-particle Bohmian trajectory ~ra[t] associated to an a-electron can be computed from the following single-particle wavefunction, Ψa(~ra, t), solution of the single-particle Schrödinger equation:

2 ∂Ψa(~ra, t) ~ 2 i~ = {− ∇ + Ua(~ra, R~ a[t], t) + Ga(~ra, R~ a[t], t) ∂t 2m ~ra

+i · Ja(~ra, R~ a[t], t)}Ψa(~ra, t) (1) where we deﬁne: R~ a[t] = {~r1[t], ~ra−1[t], ~ra+1[t], ~rN [t], t} as a vector that contains all Bohmian trajectories except ~ra[t]. The explicit expression of the potentials Ga(~ra, R~ a[t], t) and Ja(~ra, R~ a[t], t) are explained in [1]. However, their numerical values are unknown and need some educated guesses [1]. Conversely, the term Ua(~ra, R~ a[t], t) can take into account Coulomb interaction without any approximation. In BITLLES simulator the term Ua(~ra, R~ a[t], t) in Eq.(1) can be computed from the following 3D Poisson equation:

∇2 ε( ~r )U ( ~r , R~ [t], t) = ρ (~r , R~ [t], t) (2) ~ra a a a a a a a

104 where ρa(~ra, R~ a[t], t) is the the charge density of an a-electron due to the rest of electrons (Bohmian trajectories) except itself [3]. Therefore, there is a Coulomb potential (or electric field) for each a-electron. The Poisson equation, Eq.(2) with the appropriate boundary conditions, provides the electron-electron Coulomb interaction that reproduces accurately the electrostatics of the system. Then, at each simulation time step, dt, we solve N(t) Poisson equations with N(t) different charge densities. These potential energies solution of the N(t) Poisson equation are, then, introduced into the N(t) Schrödinger equations defined in Eq. (1). It is precisely in the time-dependence of the potentials of Eq. (1) where the correlations with other electrons appear [3]. We do also use a novel boundary conditions for solving the Poisson equation ensuring ’overall charge neutrality’ and ’current conservation’, even for small simulating boxes [4]. The overall procedure explained in this section provides a self-consistent solution of the Poisson and the many-particle Schrödinger equations beyond the mean field approximation. More details are explained in references [2, 3, 5].

C. Time-dependent current computation

In fact, there is an additional argument that justiﬁes the importance of properly introducing the many-particle Coulomb interaction. One has to compute time-dependent variations of the electric ﬁeld (i.e. the displacement current) to assure that the total time dependent current computed in a surface of the simulating box is equal to that measured by an ammeter, i.e. ’current conservation’ [2, 5]. Therefore, the computation of the total (conduction plus displacement) current in BITLLES simulator is made by means of an algorithm based on the Ramo-Shockley theorem [6], [7] to compute current in the volume of Fig. 1.

III. NUMERICAL RESULTS WITH THE BITLLES SIMULATOR

In the following subsections, the BITLLES simulator will be used to predict electrical RTD characteristics. The RTD simulated consists on two highly doped drain-source GaAs regions (the leads), two AlGaAs barriers and the quantum well (the active region).

A. Coulomb interaction in DC scenario

As a ﬁrst example, we consider the inﬂuence of the Coulomb interaction in the prediction of the current-voltage characteristic of a typical RTD. In Figure

105 FIG. 1: Volume Ω: this is a schematic representation of the arbitrary 3D geometry considered in this article as simulation box for the computation of quantum transport with local current conservation.

FIG. 2: RTD Current-voltage characteristic. Results taking into account the Coulomb correlations between the leads and the active region are presented in solid circles. Open circles refer to the same results neglecting the lead-active region interaction. Open triangles refer to a wholly non-interacting scenario, i.e. both coulomb interaction between the leads and the active region and coulomb interaction among electrons within the active region are neglected.

2., we present a comparison between three different I-V characteristics: (i) using our many-particle Coulomb interacting algorithm with our boundary conditions (solid circles), (ii) using the standard Dirichlet external bias boundary conditions (open circles), (iii) using the Dirichlet conditions and switching off Coulomb correlations (open triangles). As it can be observed, the differences between

106 these three approaches appear not only in the magnitude of the current but also in the position and shape of the resonant region. More technical details can be found in references [1, 3, 4].

B. Coulomb interaction in high frequency scenarios

Next, we provide an example of the computation of the total (conduction plus displacement) current in time-dependent scenarios with the BITLLES simulator that includes a time-dependent solution of the 3D Poisson equation. Here, we will consider a single electron crossing a RTD to show the accuracy of our quantum electron transport approach in providing local current conservation, i.e. the sum of the conduction plus the displacement currents is zero when integrated over a closed surface (see Fig. 1) [5]. The computation of the trajectory requires the algorithm explained in the Sec. II. In Fig. 3., we show the total time-dependent current crossing each one of

FIG. 3: Time-dependent total current computed on the six surfaces that form the volume Ω of Fig. 1.. The computation of the current within the direct method (dashed lines) has spurious eﬀects that are not present when the Ramo-Shockley (solid line) is used. the six surfaces of volume Ω of Fig. 1.. The numerical evaluation of the total current through each of the six surfaces is computed from the Ramo-Shockley method mentioned in [5]. In general, the results obtained for the total current on each surface are identical to those obtained from the direct computation of the conduction and displacement current on that particular surface. However, we observe in the plots of the surfaces 1 and 4 two ’spurious’ peaks when the total

107 FIG. 4: Itran(t) an its Fourier transform in inset a and b respectively. The BITLLES numerical results are interpreted from RLC circuits.

current is computed from the direct method. The computation of the current using the Ramo-Shockley method is free from these spurious numerical peaks. More details in [5]. Next, we present the current response in the negative differential conductance region of the I-V characteristic of a RTD to an input step voltage (see both inset of Fig. 4.). The results reported in Fig. 4 have been computed including Coulomb correlation among the N(t) electrons and among these electrons and those in the leads. As pointed out in the inset 1a), Itran(t) manifests a delay of about 0.1ps with respect to the step input voltage, due to the dynamical adjustment of the electric field in the leads. After the delay, the current response becomes an RLC-like (inset 1a), solid line RLC response 2) i.e. purely exponential. Performing the Fourier transform of Itran(t) (inset b solid line) and comparing with the single pole spectra (Fourier transform of RLC-like re- sponses, inset 1b), dashed and dashed dotted lines) as depicted in the inset b, we are able to estimate the cut-off frequency (about 1.6 THz for this device) and the frequency offset (about 0.76 THz ) due to the delay. More details are explained in [2].

C. Current-current correlations

As a last example, we show how the BITLLES simulator can compute noise features. In particular, we briefly discuss on how the many-body Coulomb interaction might affect the Fano factor (zero frequency noise in units of the average current) for RTDs. Specifically, we investigate the correlation between an elec-

108 tron trapped in the resonant state during a dwell time τd and the ones remaining in the left reservoir. This correlation occurs essentially because the trapped electron perturbs the potential energy felt by the electrons in the reservoir. In the limit of non-interacting electrons and mean ﬁeld approximation, the Fano factor will be essentially proportional to the partition noise. However if the Coulomb

FIG. 5: Fano Factor evaluated using the current ﬂuctuations directly available from BITTLES correlation is self-consistently included in the simulations (see Sec. II) this result is no longer reached (see Fig. 5.). Roughly speaking, an electron tunneling into the well from the cathode, raises the potential energy of the well by an amount of e/Cequ, where e is the electron charge and Ceq the structure equivalent ca- pacitance. As a consequence, the density of state in the well is shifted upwards by the same amount and we can obtain sub- and super-poissonian noise. Due to our treatment of the many-particle Coulomb interaction in the BITLLES simulator, these and other Coulomb blockade eﬀects are trivially obtained.

IV. CONCLUSION

We have recently shown [1] that Bohmian trajectories allow a direct treatment of the many-particle interaction among electrons with an accuracy comparable to Density Functional Theory techniques. The computational technique developed in [1], which allows a computation of the quantum correlations among particle without explicitly knowing the many-particle wave-function, can be applied to many diﬀerent open and closed quantum systems. In particular, in this article, we present a general, versatile and time-dependent 3D quantum electron transport simulator, named BITLLES as an example of the previous algorithm to many-particle open quantum system far from equilibrium.

109 Acknowledgment

This work was supported through Spanish MEC project MICINN TEC2009- 06986.

[1] X. Oriols, Quantum trajectory approach to time dependent transport in mesoscopic systems with electron-electron interactions, Phys. Rev. Let. 98(6), 066803–066807, (2007). [2] X.Oriols and J.Mompart, Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology, Editorial Pan Stanford. [3] G. Albareda, J. Suñé, and X. Oriols, Many-particle hamiltonian for open systems with full coulomb interaction: Application to classical and quantum time-dependent simulations of nanoscale electron devices, Phys. Rev. B. 79(7), 075315–075331, (2009). [4] G. Albareda, H. Lopez, X. Cartoixa, J. J. Suñé, X. Oriols, Time-dependent boundary conditions with lead-sample Coulomb correlations:Application to classical and quantum nanoscale electron device simulators, Phys. Rev. B 82, 085301, (2010). [5] A.Alarcón and X.Oriols, Computation of quantum electron transport with local current conservation using quantum trajectories, Journal of Statistical Mechanics: Theory and Experiment. Volume: 2009(P01051), (2009). [6] S. Ramo, Currents induced by electron motion, Proc. IRE. 27(548), 584–585, (1939). [7] X.Oriols, A.Alarcón, and E.Fernández-D´ıaz, Time dependent quantum current for independent electrons driven under non-periodic conditions, Phys. Rev. B. 71, 245322–1–245322–14, (2005).

110 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

An Account on Quantum Interference from a Hydrodynamical Perspective

A. S. Sanz Instituto de F´ısica Fundamental, Consejo Superior de Investigaciones Cient´ıﬁcas, Serrano 123, 28006 - Madrid, Spain

In 1952 David Bohm proposed [1, 2] a physical model to account for the already long-lasting problem of measurement in quantum mechanics and the completeness of the wave function [3]. This physical model, nowadays known as Bohmian mechanics, relies on the assumption that a quantum system consists of a wave and a particle; the wave evolves according to Schrödinger’s equation and rules the particle motion through a guidance condition. This is a very appealing feature: it allows us to understand quantum processes and phenomena on similar grounds as classical ones, i.e., in terms of the motion (in configuration space) displayed by a swarm of trajectories representing the evolution of a quantum fluid. Actually, this has given rise to a rebirth of Bohmian mechanics along the last 10 years, which has passed from being a way to formulate a quantum mechanics “without observers” [4] to a well-known and increasingly accepted resource for new quantum interpretations and computational schemes [5–8]. In this report, I briefly summarize part of the most recent research I have carried out in relation to Bohmian dynamics and quantum phase effects (more detailed information can be found in the bibliography at the end). In particular, the discussion spins around the physical meaning attributed and attributable to concepts such as coherence, interference or superposition, all of them closely related and inherent to quantum physics. As will be seen, when attention is primarily paid to the quantum probability density current instead of to the probability density, very interesting and challenging properties arise. Though such properties manifest very strikingly through Bohmian mechanics, they are general, usually appearing “masked” within the conventional version of quantum mechanics, where we rarely look at quantities dynamically depending on the quantum phase, such as the quantum probability density current. In Bohmian mechanics, the wave function Ψ provides dynamical information about the whole available configuration space to quantum particles, which will move accordingly [1, 2]. This information is mainly encoded in the phase of Ψ, as can be readily seen through the transformation Ψ(r,t) = ρ1/2(r,t)exp[iS(r,t)/~], where ρ and S are the probability density and phase of Ψ, respectively, both being real-valued functions. This relation allows us to

111 pass from Schrödinger’s equation to the system of coupled equations: ∂ρ S + ρ ∇ = 0, (1) ∂t ∇· m ∂S ( S)2 + ∇ + V + Q = 0, (2) ∂t 2m S v = r˙ = ∇ , (3) m where Q is the so-called quantum potential [1, 2], which depends nonlinearly on ρ. The first equation is the continuity equation, which rules the ensemble dynamics, while (2) and (3) govern the particle’s motion —equation (2) is the quantum Hamilton-Jacobi equation, which describes the phase field evolution ruling the motion of quantum particles through (3). Depending on whether one is interested in obtaining the quantum trajectories for interpretive purposes or devising numerical algorithms to synthesize Ψ from them, two different strategies can be considered, the so-called analytic and synthetic approaches [5]. To understand what a Bohmian trajectory is, it is useful to recall the connection between Bohm’s formulation and the hydrodynamical picture of quantum mechanics proposed in 1926 by Madelung [9]. In quantum hydrodynamics, the magnitudes of interest are the probability density, ρ =Ψ∗Ψ, the probability density current, J = ρv, and the drift velocity field, v = J/ρ = S/m. Accordingly, after integration of r˙, one does not obtain trajectories, but∇streamlines, which follow the flow described by the quantum fluid. In this sense, although Bohmian trajectories reproduce all features of the quantum process, this could be exactly the same situation one finds when one puts tracer particles on a classical fluid in order to determine the properties of the fluid flow: these particles help us to visualize the flow dynamics by moving along streamlines, indicating how the fluid current goes or the energy is transported. For example, in a gaseous fluid one can use smoke; in liquid fluids, tinny floating particles (e.g., pollen or charcoal dust) or another liquid (e.g., ink); in the hydrodynamical descriptions employed in Cosmology, the tracer particles can be stars, galaxies or clusters. Similarly, Bohmian trajectories can be regarded as the paths described by point-like tracer particles along quantum streamlines, which allow us to determine the evolution (throughout configuration space) of the quantum flow. Now, in order to understand the implications of quantum coherence, consider a 1 2 r r r / iSi/~ wave packet superposition, Ψ( ,t)= ψ1( ,t)+ψ2( ,t), where the ψi = ρi e are given by counter-propagating (i.e., moving with opposite velocities) Gaus- sian wave packets, characterized by a translation velocity (v0) and a spreading velocity (vs). Depending on whether their translational motion is faster than their spreading or vice versa, we might find two situations [10]: collision-like (v0 vs) and interference-like (v0 vs). In the first case, the wave packets remain≫ well localized after their interference,≪ while in the latter they cannot be resolved due to the permanent presence of interference fringes. If we compute

112 the ﬁeld v, we ﬁnd

1 ρ1 S1 + αρ2 S2 + √α√ρ1ρ2 (S1 + S2) cos ϕ v = ∇ ∇ ∇ m ρ1 + αρ2 + 2√α√ρ1ρ2 sin ϕ 1/2 1/2 1/2 1/2 ~ ρ1 ρ2 ρ2 ρ1 sin ϕ +√α ∇ − ∇ . (4) m ρ1 + αρ2 + 2√α√ρ1ρ2 cos ϕ which can also be derived in standard quantum mechanics. This expression contains the essence of this theory —i.e., the direct meaning of the concept quantum coherence— as well as the explanation for the well-known non-crossing property of Bohmian mechanics —a direct consequence of the presence of quantum coherence. However, not much attention is paid to it or, equivalently, to J (except when computing net fluxes through surfaces, as in processes involving tunneling or scattering), since one usually focuses on probability densities. To stress the importance of quantum probability density currents and quantum velocity fields, consider the above superposition describes the head-on collision of two Gaussian wave packets [10]. Though there is no apparent overlapping between the two wave packets initially, the fact that both are present induces very well-defined phase and velocity fields in the region in between which cannot be neglected regarding the trajectory or phase dynamics [11]. This makes, for example, that in two-slit experiments one can indeed discern the slit traversed by a particle without disturbing it. Indeed, a very important implication readily arises: this fact has to be taken into account very seriously and properly implemented in any trajectory-based algorithm where interferences are involved in order to achieve a correct propagation. Quantum coherence and Bohmian non-crossing give rise to other interesting consequences at a fundamental level, such as the quantum Tal- bot effect [12] or the complexified version of Bohmian mechanics [11, 13], as well as in more applied problems, such as atom-surface scattering [14] or chemical reactivity [15], where interference and quantum phase are relevant. The non-crossing property of Bohmian trajectories leads us to think of the presence of an effective potential, different from Q and coming only from the quantum phase, which can reproduce such an effect. A simple model for this potential looks like [10]

0, x

113 respectively. Thus, while the impenetrable wall gives rise to the bouncing of the trajectories at x = 0, the short-range square well makes the width of the peak closer to the wall half the width of the remaining peaks —in a wave-packet interference pattern that half of the trajectories giving rise to the central maximum comes from dynamical regions with opposite quantum phases or, equivalently, opposite velocity fields. Although more refined models than (5) can be considered in order to get a better correspondence, however the important issue here is to note that, dynamically, a problem involving wave packet superposition is equivalent to the scattering of a single wave packet with an effective potential —classically, something similar happens when two-body collisions are replaced by the collision of an “effective” body, namely the reduced mass system, acted by an “effective” central force. This stresses the difference between the mathematics and the physics of the superposition principle, which becomes more apparent when looking at quantum trajectories, though it is incipiently contained in the standard version of quantum mechanics —one only has to seek for the quantum phase or some associated quantity, such as J or v. Furthermore, notice that the effective potential (5) has nothing to do with the quantum potential associated with a wave-packet superposition (see, for example, Ref. [16] for a picture of the quantum potential associated with the two-slit problem); the origin of (5) is the property of quantum coherence (quantum phase), not the probability density. Finally, I would like to mention the fact that the same discussion sustained above can also be applied when instead of matter wave functions we are dealing with electromagnetic fields described by Maxwell’s equations. Putting aside the question of what a photon (or its wave function) is and regarding it as simply a tracer particle that, as before, allows us to “visualize” (trace) the flow of an electromagnetic field, one can proceed as in Bohmian mechanics and try to reproduce interference patterns by counting single arrivals of such particles. Thus, if the electromagnetic field is characterized by the electric field E(r,t) and the magnetic field H(r,t), then the three key elements will be: the electromagnetic energy den- ∗ ∗ sity, U(r,t)=[ǫ0E(r,t) E (r,t)+ µ0H(r,t) H (r,t)] /4, the electromagnetic energy density current or· Poynting vector, S·(r,t) = Re[E(r,t) H∗(r,t)] /2 and a vector velocity field, dr/dt = S(r,t)/U(r,t), which are analogous× to the quantities of interest in quantum hydrodynamics. The velocity vector field is set up from the fact that the electromagnetic energy density is transported through space in the form of the Poynting vector [17],

S(r,t)= U(r,t)v. (7)

This relations (actually, their time-averaged counterparts) have been used recently to carry out a series of studies within the context of polarized light, the Arago-Fresnel laws and quantum erasure [18], ﬁnding a very nice agreement between the well-known interference patterns and the counting of photon arrivals. Acknowledgements. First, I would like to thank the people with whom I have developed the diﬀerent parts of the work here summarized for many interesting

114 and fruitful discussions: Salvador Miret-Artés, Tino Borondo, Bob Wyatt, Chia- Chun Chou, Josep Maria Bofill, Xavier Giménez, Mirjana Boˇzićand Milena Davidović. Support from the Ministerio de Ciencia e Innovación(Spain) under Project FIS2007-62006 to carry out this work and participate at the CCP6 Workshop on Quantum Trajectories is also acknowledged. Moreover, I also want to thank the Consejo Superior de Investigaciones Cient´ıficas for a JAE-Doc Contract.

[1] D. Bohm, Phys. Rev. 85, 166, 180 (1952). [2] P. R. Holland, The Quantum Theory of Motion (Cambridge University Press, Cambridge, 1993). [3] Zurek and Wheeler, Quantum Theory of Measurement (Princeton University Press, Princeton, NJ, 1983). [4] S. Goldstein, Phys. Today 51(3), 42 (1998); Phys. Today 51(4), 38 (1998). [5] R. E. Wyatt, Quantum Dynamics with Trajectories (Springer, Berlin, 2006). [6] P. K. Chattaraj, Quantum Trajectories (Taylor and Francis, New York, ‘to be published’). [7] X. Oriols and J. Mompart, Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology (Pan Standford Publishing, Singapore, ‘to be published’). [8] A. S. Sanz and S. Miret-Artés, A Trajectory Description of Quantum Processes, from the series Lecture Notes on Physics (Springer, Berlin, ‘to be published’). [9] E. Madelung, Z. Phys. 40, 322 (1926). [10] A. S. Sanz and S. Miret-Artés, J. Phys. A 41, 435303 (2008). [11] A. S. Sanz and S. Miret-Artés, Chem. Phys. Lett. 458, 239 (2008). [12] A. S. Sanz and S. Miret-Artés, J. Chem. Phys. 126, 234106 (2007). [13] C.-C. Chou, A. S. Sanz, S. Miret-Artés and R. E. Wyatt, Phys. Rev. Lett. 102, 250401 (2009); Ann. Phys., doi:10.1016/j.aop.2010.05.009 (2010). [14] A. S. Sanz, F. Borondo and S. Miret-Artés, J. Chem. Phys. 120, 8794 (2004); Phys. Rev. B 69, 115413 (2004). [15] A. S. Sanz, X. Giménez, J. M. Bofill and S. Miret-Artés, Chem. Phys. Lett. 478, 89 (2009); Erratum, Chem. Phys. Lett. 488, 235 (2010). [16] A. S. Sanz, F. Borondo and S. Miret-Artés, J. Phys.: Condens. Matter 14, 6109 (2002). [17] M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 2002), 7th Ed. [18] M. Davidović, A. S. Sanz, D. Arsenović, M. Boˇzićand S. Miret-Artés, Phys. Scr. T135, 014009 (2009); A. S. Sanz, M. Davidović, M. Boˇzićand S. Miret-Artés, Ann. Phys. 325, 763 (2010); M. Boˇzić, M. Davidović, T. L. Dimitrova, S. Miret-Artés, A. S. Sanz and A. Weis, J. Russ. Laser Res. 31, 117 (2010).

115 K. H. Hughes and G. Parlant (eds.) Quantum Trajectories c 2011, CCP6, Daresbury

Quantum, Classical, and Mixed Quantum-Classical Hydrodynamics

I. Burghardt∗ D´epartement de Chimie, Ecole Normale Sup´erieure, 24 rue Lhomond, 75231 Paris cedex 05, France

K. H. Hughes† School of Chemistry, University of Bangor, Bangor, Gwynedd LL57 2UW, United Kingdom

I. INTRODUCTION

This contribution reviews the quantum hydrodynamic formulation for mixed states (density matrices), the classical limit of this formulation, and two mixed quantum-classical hybrid schemes which have been developed in this context. Mixed-state hydrodynamics can be derived by generating the momentum moments of the Wigner distribution [1–6],

hPnρi(q)= dppnρ (q,p) (1) Z W The equations of motion for these coordinate-dependent moment quantities represent an inﬁnite hierarchy which necessitates appropriate truncation strategies [7–10]. In the special case of pure states (wavefunctions), mixed-state hydrodynamics reduces to the pure-state formulation of Bohmian mechanics [2, 4, 11]. The ﬁrst of the hybrid schemes mentioned above involves a hydrodynamic representation of the quantum subspace [12–15] while the second scheme combines a discretized representation of the quantum subspace with a hydrodynamic representation of the classical subsystem [16, 17]. In both cases, the classical limit corresponds to a linearization approximation which is identical to the approximation by which the classical Liouville equation is obtained from the quantum

∗Electronic address: [email protected] †Electronic address: [email protected]

116 Liouville equation [13, 15, 18]. (This is to be distinguished from the classical limit associated with the vanishing of the Bohmian quantum force, see the discussion in Refs. [12, 13, 15]). In the following, a brief summary is given these two quantum-classical schemes.

A. Quantum (hydrodynamic)-classical (Liouvillian) dynamics

The ﬁrst hybrid scheme combines a quantum hydrodynamic (Bohmian) representation with a molecular dynamics (MD) like representation for the classical subspace. The construction of coupled equations of motion for the quantum and classical variables proceeds by introducing the following partial hydrodynamic moment quantities [12, 13],

hPnρi(q, Q, P )= dppnρ (q, p, Q, P ) (2) Z W where momentum moments are defined with respect to the quantum (q,p) subspace, with a parametric dependence on the classical (Q, P ) variables. The dynamical equations for the above moments translate to a Lagrangian picture which couples quantum hydrodynamic equations to classical Hamilton’s equations. A specific feature is a quantum hydrodynamic force which depends both on the quantum coordinate(s) q and on the classical Liouvillian variables (Q, P ). First applications of this approach, which defines a mixed quantum-classical MD, are documented in Refs. [12, 14, 15]. More recently, this scheme has been combined with a non-Markovian effective mode representation of the classical subsystem [19], thus yielding a general approach to delayed dissipation.

B. Quantum (Liouvillian)-classical (hydrodynamic) dynamics

While the approach of the preceding section aims at a trajectory (MD-type) representation of the classical subspace, the second hybrid scheme employs a classical hydrodynamic, mesoscopic representation which can be suitable to describe, e.g., solvation dynamics or transport phenomena. Hybrid moments are now constructed as follows [16, 17]

n n hP fi ′ (q)= dpp f ′ (q,p) (3) ξξ Z ξξ

ˆ ′ ′ based upon the quantum-classical distribution f(q,p) = ξξ′ fξξ (q,p)|ξihξ |, with a discretized representation of the quantum subspace.P Contrary to the quantum hydrodynamic moments of Eq. (2), the moments of Eq. (3) are essentially classical hydrodynamic quantities which carry the indices (ξξ′) of the

117 quantum subsystem. The equations of motion for these hybrid moments have ﬁrst been derived in Ref. [16] and have more recently been constructed rigorously from the microscopic N-particle distribution [17]. Various closure schemes can again be envisaged, in particular a Grad-Hermite type closure [9, 10], a Gaus- sian closure at the level of a quantum-classical Maxwellian distribution, and a dynamical density functional theory (DDFT) approximation by which the hydrodynamic pressure term is replaced by a free energy functional derivative, as in Ref. [16]. The present scheme in its general form [17] thus opens various generalizations of classical-statistical time-dependent density functional approximations [20, 21].

C. Conclusions

Hybrid schemes involving the quantum or classical hydrodynamic picture provide promising dynamical representations which benefit from the Lagrangian trajectory dynamics associated with the hydrodynamic description. The connection of mixed-state hydrodynamics to the underlying phase-space picture allows for a unified description of the classical limit and of dissipative effects. Connections to time-dependent density functional methods may open various generalizations of the latter class of methods.

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118 [17] D. Bousquet, K. H. Hughes, D. A. Micha, and I. Burghardt, Extendend hydrodynamic approach to quantum-classical nonequilibrium evolution I. Theory, J. Chem. Phys., submitted. [18] R. Kapral and G. Ciccotti, J. Chem. Phys. 110, 8919 (1999). [19] K. H. Hughes and I. Burghardt, in: Quantum Trajectories, Ed. P. Cattaraj, Taylor and Francis/CRC Press, Chapter 11, p. 163 (2010). [20] A. J. Archer and R. Evans, J. Chem. Phys. 121, 4246 (2004). [21] A. J. Archer, J. Chem. Phys. 130, 014509 (2009).

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