Geometry of quantum active subspaces and of effective ... - CNRS

JOURNAL OF MATHEMATICAL PHYSICS 48, 052102 共2007兲

Geometry of quantum active subspaces and of effective Hamiltonians David Viennota兲 Observatoire de Besançon, Institut UTINAM (CNRS UMR 6213), 41bis Avenue de l’Observatoire, BP1615, 25010 Besançon Cedex, France 共Received 8 January 2007; accepted 16 March 2007; published online 9 May 2007兲

We propose a geometric formulation of the theory of effective Hamiltonians associated with active spaces. We analyze particularly the case of the time-dependent wave operator theory. This formulation is related to the geometry of the manifold of the active spaces, particularly to its Kählerian structure. We introduce the concept of quantum distance between active spaces. We show that the time-dependent wave operator theory is, in fact, a gauge theory, and we analyze its relationship with the geometric phase concept. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2723552兴

I. INTRODUCTION

The numerical study of the interaction of a molecule with a strong laser field leads to a need for long computational times and large computer memory capacity when use is made of a wave packet approach, which involves a direct integration of the time-dependent Schrödinger equation. This computational problem is even greater for the study of control processes, since the repetition of many propagations is needed to find the optimum values of several adjustable parameters which describe the nonlinear effects due to ultrashort laser pulses. Fortunately, it turns out that a reasonable description of the matter-field interaction can often be made by using an active space of small dimension, provided that the basis sets used consist of instantaneous Floquet eigenvectors1 or generalized Floquet eigenvectors.2 This feature makes an effective Hamiltonian approach attractive, and Guérin and Jauslin3 have proposed such an approach, based on the quantum analog of the Kolmogorov-Arnold-Moser 共KAM兲 transformation,4 with resonant effects being treated by a rotating wave approximation. The superadiabatic Floquet approach,5 which uses a sequence of unitary transformations to produce bases which follow the nonadiabatic evolution, is also very similar to an effective Hamiltonian approach. Finally, the constrained adiabatic trajectory method6 is a method consisting to define an effective Hamiltonian by adding a time-dependent complex potential. The time-dependent wave operator theory is another example of effective Hamiltonian theory for time-dependent systems. It has long been used to describe photoreactive processes7,8 and has several features which make it useful in the search for an efficient description of nonadiabatic effects. The theory of time-dependent wave operators, and their stationary equivalents, the Bloch wave operators, has been the subject of several works.9–11 We consider a separable Hilbert space H which is used to describe the states of a quantum system. The dynamical system is described by a time-dependent Hamiltonian H共t兲, and its associated time-dependent Schrödinger equation and has the time evolution operator U共t , 0兲共∀t 苸关0 , T兴兲. The idea of the wave operator theory is to consider a fixed active subspace S0 of H, such that the dynamics projected into this subspace can be integrated by using an effective Hamiltonian. S0 should be chosen to describe the strong and fast part of the dynamics issuing from the initial state; this choice has been made previously by using artificial intelligence

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approaches12,13 and a wave operator sorting algorithm.8,14 After the solution of the Schrödinger equation in the active space S0, the wave operator ⍀ which generates Heff is used to transform the solution in S0 into the true solution in the Hilbert space H. The above description can be summarized as follows: ∀t,

⍀共t兲:

S共t兲 → H

共1兲

␺0共t兲哫 ␺共t兲,

where ␺共t兲 is a solution of the Schrödinger equation ıប⳵t␺共t兲 = H共t兲␺共t兲

共2兲

and where ␺0共t兲, defined as the projection of ␺ into S0, is a solution of the equation ıប⳵t␺0共t兲 = Heff共t兲␺0共t兲.

共3兲

The effective Hamiltonian 共which describes the approximate dynamics in S0兲 is defined by Heff共t兲 = P0H共t兲⍀共t兲, and the target space is S共t兲 = P0U共t , 0兲S0 傺 S0, with P0 being the projector on S0. The time-dependent wave operator can be written as ⍀共t兲 = U共t兲共P0U共t,0兲P0兲−1 , where 共P0U共t , 0兲P0兲−1 = P0共P0U共t , 0兲P0兲−1 P0 is the 关dom共P0U共t , 0兲P0兲−1 = S0兴. Finally, the wave function can be written as follows:

inverse

共4兲 of

U共t兲

␺共t兲 = ⍀共t兲Ueff共t,0兲␺0 ,

within

S0

共5兲

where ıប⳵tU 共t , 0兲 = H 共t兲U 共t , 0兲. It is possible to define a stationary analog to the time-dependent wave operator. We will now consider 共in the same separable Hilbert space H兲 the operator H and the eigenvalue equation eff

eff

eff

H␺ = ␭␺ .

共6兲

We consider two subspaces S0 and S of H. We call them the active and target subspaces, and we denote the projectors of these subspaces by P0 and P. We are interested in eigenvectors included in S such that P␺ = ␺. As for the previous time-dependent problem, we reduce this problem to one within the active subspace. We then introduce the Bloch wave operator: ⍀:

S0 → S

␺0 哫 ␺

,

共7兲

where ␺ is a solution of the eigenvalue equation 关Eq. 共6兲兴 and ␺0 is a solution of Heff␺0 = ␭␺0 .

共8兲

The effective operator is defined by Heff = P0H⍀. The Bloch wave operator is formally given by the expression ⍀ = P共P0 PP0兲−1 ,

共9兲

where 共P0 PP0兲−1 is the inverse of P within S0. The time-dependent wave operator is a generalization of the Møller wave operators ⍀± −1 −1 = limt→⫿⬁e−ıប Hteıប H0t, which compare the dynamics induced by a time-independent Hamiltonian H with the dynamics induced by a simplest Hamiltonian H0. The time-dependent wave operator has the same function; it compares the true dynamics with the effective dynamics condensed in a fixed active space. The time-dependent wave operator techniques consist then to separate the subdynamics governed by Heff共t兲 from the dynamics outside S0 which is induced by


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⍀共t兲. The aim of this paper is to show that this method has a geometric formulation very close to usual geometric theories of dynamical systems. We show that the separation in two parts of the dynamics is associated with a fiber bundle formalism; the effective dynamics is associated with the base manifold of the bundle, whereas the true dynamics is associated with the total space of the bundle. The wave operator action, consisting to built the true wave function from the effective one, is, in fact, a horizontal lift of the effective dynamics in the bundle with respect to a particular connection explicated in this paper. The effective dynamics, which is condensed in the base manifold, can be geometrically formulated by a complexification of the Poincaré form formulation of classical dynamics. With this geometric formulation of the dynamics in the base manifold, we have a consistent geometric point of view of the time-dependent wave operator method. Section III is dedicated to this analysis. Since the time-dependent wave operator geometric formulation is associated with a principal bundle, it is very close to the non-Abelian geometric phase theory. We show in this paper that the time-dependent wave operator can be assimilated with an operator which multiplies the effective wave function by a non-Abelian geometric phase. In 1984, Berry15 proved, in the context of the standard adiabatic approximation, that the wave function of a quantum dynamical system takes the form

␺共t兲 = e−ıប

−1兰t E 共R共t 兲兲dt −兰t 具a,R共t 兲兩⳵ 兩a,R共t 兲典dt ⬘ ⬘ 0 ⬘ t⬘ ⬘ ⬘ 0 a

兩a,R共t兲典,

共10兲

where Ea is a nondegenerate instantaneous eigenvalue isolated from the rest of the Hamiltonian spectrum and having instantaneous eigenvector 兩a , R共t兲典. R is a set of classical control parameters used to model the time-dependent environment of the system. The set of all configurations of R is supposed to form a C⬁-manifold M. The important result is the presence of the extra phase term, t called the Berry phase e−兰0具a,R共t⬘兲兩⳵t⬘兩a,R共t⬘兲典dt⬘. Simon16 found the mathematical structure which models the Berry phase phenomenon, namely, a principal bundle with base space M and with structure group U共1兲. If we eliminate the dynamical phase by a gauge transformation which involves redefining the eigenvector at each time, then expression 共10兲 is the horizontal lift of the curve C described by t 哫 R共t兲 with the gauge potential A = 具a , R兩dM兩a , R典. If C is closed, then the Berry phase e−养CA 苸 U共1兲 is the holonomy of the horizontal lift. In 1987, Aharonov and Anandan17 proved that the appearance of geometric phases such as the Berry phase is not restricted to the use of adiabatic approximation but arises in a more general context. Let t 哫 ␺共t兲 be a wave function such that ␺共T兲 = eı␾␺共0兲 and H共t兲 be the Hamiltonian of the system. Suppose that the Hilbert space is n dimensional 共the case n = + ⬁ is not excluded兲; then the wave function defines a closed curve C in the complex projective space CPn−1. If one redefines the wave function such that ˜␺共T兲 = ˜␺共0兲, then

␺共t兲 = e−ıប

t ˜ ˜ ˜ −1兰t 具˜ 0 ␺共t⬘兲兩H共t⬘兲兩␺共t⬘兲典dt⬘−兰0具␺共t⬘兲兩⳵t⬘兩␺共t⬘兲典dt⬘

˜␺共t兲.

共11兲

The extra phase in addition to the dynamical phase is called the Aharonov-Anandan phase 共or nonadiabatic Berry phase兲. We can eliminate the dynamical phase by a gauge transformation; the Aharonov-Anandan phase then appears as the horizontal lift of C in the principal bundle with base space CPn−1, the structure group U共1兲, and with the 共2n − 1兲-dimensional sphere S2n−1 as total space. The Berry-Simon model and the Aharonov-Anandan model are related by the universal classifying theorem of principal bundles;18,19 more precisely, the Aharonov-Anandan principal bundle is a universal bundle for the Berry-Simon principal bundle. The base space CPn−1 is endowed with a natural metric called the Fubini-Study metric. Anandan and Aharonov20 have proved that this metric measures the quantum distance between the quantum states. The connection bewteen geometric phases and universal classifying spaces is explained in Refs. 21–23 A non-Abelian geometric phase 共based on the gauge group U共M兲兲 was introduced by Wilczek and Zee in 1984 Ref. 24 and its universal structure was studied by Bohm and Mostafazadeh in 1994.22 In a previous article,25 we have proposed a geometric structure based on the concept of


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principal composite bundle to describe in the context of the adiabatic approximation, the nonAbelian geometric phase when it does not commute with the dynamical phase 共see also the works of Sardanashvily26,27兲. The wave operator formalism is associated with a fixed active space. In the present work, we are also interested by the possibility to use an evolutive active space, i.e., a time-dependent active space which adapts to the time evolution rather than the use of a time-dependent wave operator to lift an effective dynamics within a fixed subspace. Although these two concepts seems to have radically different philosophies, we show that they have very close geometric formulations and that they differ only from the choice of the connection in the principal bundle of the active spaces. Section II is devoted to the geometric formalism of the evolutive active space theory. The methods of effective Hamiltonians 关Eqs. 共1兲–共6兲兴 based on a Floquet eigenbasis can be viewed as methods using an evolutive active space, the space spanned by the Floquet eigenvectors depending on the time through the Floquet variable. The results of this paper could permit to clarify the connections between some techniques used in quantum dynamics 共effective Hamiltonians, geometric phases, and time-dependent basis representation兲 by showing that the geometric formulation unifies them. The theory of active subspaces could be a fruitful framework for the study of problems of quantum computing in the context of realistic dynamical systems 共atoms or molecules interacting with a field兲. Indeed, an elementary quantum computation unit, a qubit, is a two-dimensional space which can be identified with a two-dimensional active space of a quantum dynamical system. More generally, a system of n qubits can be assimilated to a 2n-dimensional active space. Moreover, the logical operations are represented by unitary operators. In a geometric method of quantum computing called holonomic quantum computation 共see Refs. 28–34兲, the logical operations are described by a gauge theory in the bundle of the active spaces. The present work, analyzing the geometric representation of dynamical systems in the context of evolutive active spaces, could be a first step to the application of the efficient methods of the quantum dynamics 共wave operator theory, adiabatic assumption, evolutive active space method, effective Hamiltonian methods, etc.兲 to quantum computation. At this point we give a brief summary of our notation. We denote a principal bundle with base space M, total space P, projection ␲, and structure group G by 共P , M , G , ␲兲. The set of the sections of a bundle is denoted by ⌫共M , P兲. The set of differential n forms of M is denoted by ⍀nM. The tangent space of a manifold M at the point p 苸 M is denoted by T pM. The set of complex matrices with n rows and p columns is denoted by Mn⫻p共C兲. The ¯Z denotes the complex conjugation, whereas Z* or Z# denote other involutions specified by the context. II. GEOMETRY OF EVOLUTIVE QUANTUM ACTIVE SPACES

We consider a quantum dynamical system described by the Hamiltonian t 哫 H共t兲 in the Hilbert space H. Let U共t , 0兲 be the evolution operator of the dynamical system, i.e., the unitary operator which is the solution of the Schrödinger equation ıប⳵tU共t,0兲 = H共t兲U共t,0兲,

U共0,0兲 = 1.

共12兲

We denote by 兵兩i典其 a reference orthonormal basis of H. In practice, if the quantum system is a molecule interacting with a field, then 兵兩i典其i is the eigenbasis of the unperturbed molecule. We want to describe the quantum dynamical system by using evolutive active subspaces. This section presents the mathematical theory of this concept. We first need to define the evolutive active subspaces. Definition 1: An M dimensional evolutive active subspace of a quantum dynamical system 共H共t兲 , H兲 is defined by specifying two kinds of data: • a map t 哫 S共t兲 from the time line to the M-dimensional subspaces of H such that ∀␺0 苸 S共0兲, U共t , 0兲␺0 苸 S共t兲 ∀ t ⬎ 0 and • a map t 哫兵兩␣共t兲典其␣ from the time line to a basis of S共t兲.


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We note that if S共t兲 is an evolutive active subspace associated with a self-adjoint Hamiltonian, then the orthogonal projector associated with S共t兲, P共t兲, satisfies the Schrödinger–von Neumann equation ıប

dP共t兲 = 关H共t兲, P共t兲兴. dt

共13兲

共The equation for a non-self-adjoint Hamiltonian is studied in Sec. II C兲. The concept of an evolutive active space is particularly interesting if it is possible to find a basis 兵兩␣共t兲典其␣ without integrating the Schrödinger equation, as, for example, happens in the adiabatic theory where 兵兩␣共t兲典其␣ is a set of instantaneous eigenvectors. Regarding Eq. 共13兲, the time-dependent projection operator can be identified with a specific example of a dynamical invariant 共Ref. 35兲. Indeed, the definition of an active subspace ␺0 苸 S0 ⇒ U共t , 0兲␺0 苸 S共t兲 is similar to the definition of an invariant subspace S under the action of the semigroup t 哫 U共t , 0兲, i.e., ␺0 苸 S ⇒ U共t , 0兲␺ 苸 S. The evolutive active space is an example of “dynamical invariant subspaces” which are generally spanned by eigenvectors of a Lewis invariant. The relation between periodic or cyclic dynamical invariants and non-Abelian geometric phases is discussed in Refs. 36–38. The present work focuses on the connection between geometric phases with nonperiodic and noncyclic dynamical invariants which satisfy the projection prescription P共t兲2 = P共t兲. Since we want to relate this analysis to the wave operator theory, we prefer to refer to the range of P共t兲共the active space兲 rather than to the dynamical invariant theory. Sections II A and II B explore the connection between the geometric phase concept and the evolutive active space theory. They show that the complete dynamics of the wave function can be computed by a horizontal lift from the dynamics of the evolutive active space governed by the Schrödinger–von Neumann equation. This dynamics appears then as a subdynamics condensed in the base space of the bundle where the horizontal lift takes place. Section II C shows that this subdynamics, analytically defined by the Eq. 共13兲, can be geometrically formulated. We have then a consistent geometric formulation of the quantum dynamics in the active space formalism. A. Generalized geometric phases

The following proposition proves that the geometric phase phenomenon is associated with the evolutive active space description. Proposition 1: Let 共H共t兲 , H兲 be a quantum dynamical system and 共S共t兲 , 兵兩␣共t兲典其␣=1,. . .,M 兲 be an evolutive active space. Let 兵兩␣共t兲#典其␣=1,. . .,M be a set biorthogonal to 兵兩␣共t兲典其␣=1,. . .,M , i.e., ∀t, ∀ ␣, ␤ = 1, . . . ,M,

具␣共t兲兩␤共t兲典 = ␦␣␤ .

共14兲

(We do not suppose that 兵兩␣共t兲典其 is orthonormal). Let ␺␣共t兲 be the solution of the Schrödinger equation ıប⳵t␺␣共t兲 = H共t兲␺␣共t兲,

␺␣共0兲 = 兩␣共0兲典.

共15兲

By the definition of the evolutive active space, we have ␺␣共t兲苸 S共t兲. Then, the representation of ␺␣ on the active space is M

␺␣共t兲 = 兺关Te−ıប

−1兰t E共t 兲dt −兰t A共t 兲dt ⬘ ⬘ 0 ⬘ ⬘ 0

␤=1

兴␤␣兩␤共t兲典,

共16兲

where the matrices E and A of MM⫻M 共C兲 are defined by E共t兲␣␤ = 具␣共t兲兩H共t兲兩␤共t兲典,

A共t兲␣␤ = 具␣共t兲兩⳵t兩␤共t兲典.

共17兲

t 兰0X共t兲

Te is the time-ordering exponential (the Dyson series), i.e., D共t兲 = Te is a solution of ⳵tD共t兲 = X共t兲D共t兲 for all X 苸 MM⫻M 共C兲. The matrices E and A generate, respectively, the non-Abelian dynamical and geometric phases.


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Proof: We know that ␺␣共t兲苸 S共t兲 ∀ t, since 兵兩␣共t兲典其 is a basis of S共t兲, ∀␣ and ∀t, ∃兵U␤␣共t兲苸 C其␤=1,. . .,M such that M

␺␣共t兲 = 兺 U␤␣共t兲兩␤共t兲典.

共18兲

␤=1

We group the numbers 兵U␤␣其␣␤ into a matrix U 苸 MM⫻M 共C兲. ␺␣ is a solution of the Schrödinger equation. By using expression 共18兲, we find the equation ˙ 共t兲兩␤共t兲典 + ıប 兺 U 共t兲⳵ 兩␤共t兲典 = 兺 U 共t兲H共t兲兩␤共t兲典. ıប 兺 U ␤␣ ␤␣ t ␤␣ ␤

␤

共19兲

␤

We project this equation on 具␥共t兲#兩 to give ˙ = − ıប−1 兺 U 具␥共t兲兩H共t兲兩␤共t兲典 − 兺 U 具␥共t兲兩⳵ 兩␣共t兲典. U ␥␣ ␤␣ ␤␣ t ␤

共20兲

␤

This equation can be written in a matrix form,

⳵tU = 共− ıប−1E共t兲 − A共t兲兲U.

共21兲

This last equation has for its solution the Dyson series U共t兲 = Te−ıប

−1兰t E共t 兲dt −兰t A共t 兲dt ⬘ ⬘ 0 ⬘ ⬘ 0

共22兲

.

䊏

By virtue of the intermediate representation theorem 共Ref. 40兲, we can write U共t兲 = Te−兰0A共t⬘兲dt⬘Te−ıប t

−1兰t Te兰t0⬘A共t⬙兲dt⬙E共t 兲Te−兰t0⬘A共t⬙兲dt⬙dt ⬘ ⬘ 0

共23兲

.

We can then isolate for study the pure geometric term Te−兰0A共t⬘兲dt⬘. The precise relation between the pure non-Abelian geometric phase and the non-Abelian geometric and dynamical phase is explained in Refs. 25–27. This proposition defines a more general context in which a geometric phase appears in quantum dynamics. If S共t兲 is an instantaneous spectral subspace, with 兵兩␣共t兲典其 being an orthonormalized set of instantaneous eigenvectors of a self-adjoint Hamiltonian H共t兲, then the proposition reproduces the non-Abelian Berry phase of the adiabatic approximation; if 兵兩␣共t兲典其 is an orthonormalized cyclic basis, then the proposition reproduces the non-Abelian Aharonov-Anandan phase 共in these cases, 兩␣ 典 = 兩␣典兲. If H共t兲 is a non-self-adjoint Hamiltonian, with 兵兩␣共t兲典其 a set of generalized eigenvectors of H共t兲 and 兵兩␣共t兲典其 the associated generalized eigenvectors of H共t兲†, then the proposition gives rise to the Berry phase of a non-self-adjoint Hamiltonian in the adiabatic approximation. If H is finite dimensional with dim H = N, then one can then introduce the matrices Z 苸 MN⫻M 共C兲 and Z 苸 MN⫻M 共C兲 defined by t

Z共t兲 =

冢

具1兩1共t兲典 . . . 具1兩M共t兲典 ]

]

具N兩1共t兲典 . . . 具N兩M共t兲典

冣

,

Z 共t兲 =

冢

具1兩1共t兲典 . . . 具1兩M共t兲典 ]

]

具N兩1共t兲典 . . . 具N兩M共t兲典

冣

,

共24兲

where 兵兩i典其 is a reference Hermitian basis on which the Hamiltonian has a finite N ⫻ N matrix representation, E共t兲 = Z共t兲†H共t兲Z共t兲,

A共t兲 = Z共t兲†

d Z共t兲. dt

共25兲

We now introduce an important property.


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Property 1: Let 兵t 哫 ␺␣共t兲其␣ be a set of wave functions within the evolutive active S共t兲 such that ␺␣共0兲 = 兩␣共0兲典. 兵␺␣其 are the wave functions of the evolutive active space without dynamical phase, i.e., M

␺␣共t兲 = 兺关Te−兰0A共t⬘兲dt⬘兴␤␣兩␤共t兲典,

共26兲

∀ ␣, ␤,

共27兲

t

␤=1

if and only if 具C共t兲␺␣共t兲兩⳵t兩␺␤共t兲典 = 0,

where C 共t兲 is an operator of S共t兲 defined by

∀ ␣,

C共t兲兩␣共t兲典 = 兩␣共t兲典.

共28兲

Proof: First, we suppose that ␺␣ is the evolutive active space wave function, i.e., ␺␣共t兲 t = 兺␤U␤␣兩␤共t兲典, with U共t兲 = Te−兰0A共t⬘兲dt⬘. We then have

⳵t␺␣ = − 兺 A␤␥U␥␣兩␤共t兲典 + 兺 U␤␣⳵t兩␤共t兲典. ␤,␥

␤

共29兲

¯ 具␦共t兲兩, we obtain By projecting this equation on to 具C␺⑀兩 = 兺␦U ⑀␦ 具C␺⑀兩⳵t兩␺␣典 = − 关U†AU兴⑀␣ + 关U†AU兴⑀␣ = 0.

共30兲

Second, we suppose that 具C ␺␥兩⳵t兩␺␣典 = 0. ␺a共t兲苸 S共t兲 and 兵兩␣共t兲典其 is a basis of S共t兲, then ∃U␤␣共t兲苸 C such that ␺␣共t兲 = 兺␤U␤␣共t兲兩␤共t兲典. Then

¯ U ¯ ˙ 具C␺␥兩⳵t兩␺␣典 = 0 ⇒ 兺 U ␦␥ ␤␣具␦共t兲兩␤共t兲典 + 兺 U␦␥U␤␣具␦共t兲兩⳵t兩␤共t兲典

共31兲

˙ 兴 = − 关U†AU兴 , ⇒关U†U ␥␣ ␥␣

共32兲

␦,␤

␤,␦

˙ = −AU. and finally U

䊏

B. The principal bundle of the active spaces

To simplify the discussion, we suppose that H is self-adjoint and that we can choose an orthonormal basis of the active space 共i.e., Z = Z兲. In this section we explore the geometric structure describing the evolutive active space of the dynamical system. Initially, we suppose that the Hilbert space is finite dimensional, H = CN, and we select evolutive active spaces with dimension equal to M. By using a matrix representation associated with a basis of CN, we can consider the evolution operator U共t , 0兲 associated with the dynamical system as an application from the time line to U共N兲共the Lie group of unitary matrices of order N兲. We do not choose the fixed basis 兵兩i典其, but the following time-dependent basis B共t兲 = 共兩1共t兲典 , . . . , 兩M共t兲典 , 兩1共t兲 ⬜ 典 , . . . , 兩N − M共t兲 ⬜ 典兲, where B0共t兲 = 共兩1共t兲典 , . . . , 兩M共t兲典兲 is the orthonormal basis of S共t兲, the evolutive active space, and where B⬜共t兲 = 共兩1共t兲 ⬜ 典 , . . . , 兩N − M共t兲 ⬜ 典兲 is a basis of S⬜共t兲, the orthogonal supplement of S共t兲. The time-dependent representation DB共t兲共U共t , 0兲兲 of U共t , 0兲 in the basis B共t兲 is a unitary matrix. Since S共t兲 is an evolutive active space, and since we choose the initial conditions of the dynamics inside S共0兲, then a transformation which only affects the region outside the active space has no influence on the wave function. In the Lie group U共N兲, we have then the equivalence relation defined by ∀U, V 苸 U共N兲, U ⬃ V ⇔ U = VG, with G 苸 U共N − M兲. It is then clear that the space of evolution operators associated with the inner dynamics of the active space is U共N兲 / U共N − M兲. This manifold is known in the literature18 as the complex Stiefel manifold VM 共CN兲 = U共N兲 / U共N − M兲. As we have seen in previous paragraph, the evolution inside the active space gives rise to a non-


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J. Math. Phys. 48, 052102 共2007兲

David Viennot −1

Abelian phase Te−ıប 兰t0E共t⬘兲dt⬘ − 兰t0A共t⬘兲dt⬘. The manifold describing the evolution without this non-Abelian phase is G M 共CN兲 = U共N兲 / 共U共N − M兲 ⫻ U共M兲兲 called the complex Grassmannian manifold 共Ref. 18兲. We can show Ref. 18 that V M 共CN兲 = 兵Z 苸 MN⫻M 共C兲兩Z†Z = I M 其.

共33兲

A matrix Z 苸 V M 共CN兲 can be interpreted as the matrix of the basis vectors of an active space, as expressed in the fixed basis of CN, 兵兩i典其. The Stiefel manifold V M 共CN兲 is then the space of all M-dimensional active spaces of CN endowed with an orthonormal basis. The Grassmannian manifold G M 共CN兲 = V M 共CN兲 / U共M兲 is the space of all M-dimensional active spaces without a particular basis. Moreover, we can show that G M 共CN兲 = 兵P 苸 MN⫻N共C兲兩P2 = P, P† = P,tr P = M其.

共34兲

P 苸 G M 共CN兲 can be identified with the orthogonal projector of the active space. To summarize, we now have produced the structure of the principal bundle on the right U = 共VM 共CN兲 , G M 共CN兲 , U共M兲 , ␲U兲 with ∀Z 苸 V M 共CN兲, ␲U共Z兲 = ZZ†. Let E = 共E , G M 共CN兲 , C M , ␲E兲 be the associated vector bundle of U. The active space 共with a vector space structure兲, defined on P 苸 G M 共CN兲, is ␲E−1共P兲. In other words, if P共t兲苸 GM 共CN兲 is the solution of a Schrödinger–von Neumann equation, then the evolutive active space is S共t兲 = ␲E−1共P共t兲兲 and it is endowed with a basis chosen in ⌫共GM 共Cn兲 , V M 共Cn兲兲共the set of the sections of U兲. By virtue of the Narasimhan-Ramaman theorem,41 we can endow U with a natural connection, called universal, defined by the gauge potential A共ZZ†兲 = Z†dZ 苸 ⍀1共G M 共CN兲,u共M兲兲,

共35兲

where u共M兲 is the Lie algebra of U共M兲共the set of anti-self-adjoint matrices of order M兲. Let t 哫共兩1共t兲典 , . . . , 兩M共t兲典兲 = Z共t兲 be a section of U over the path ␲U共Z共t兲兲 = P共t兲苸 GM 共CN兲关P共t兲 being a solution of the Schrödinger–von Neumann equation兴. The horizontal lift of this path which passes by 兩␣共0兲典苸 E is M

t † ␺␣共t兲 = 兺关Te−兰0Z 共t⬘兲⳵t⬘Z共t⬘兲dt⬘兴␤␣兩␤共t兲典.

共36兲

␤=1

This is precisely the non-Abelian geometric phase. It is possible to explain the connection in more detail. ∀X 苸 u共M兲, its associated fundamental vector field is Xˆ defined by ∀F 苸 VM 共CN兲,

Xˆ共F兲 =

冋

d R共e−tX兲F dt

册冋册 =

t=0

d tX Fe dt

= FX,

共37兲

t=0

where R is the canonical right action of U共M兲 on V M 共CN兲. The tangent space TFV M 共CN兲 can be identified with M vectors of CN and so to a matrix of MN⫻M 共C兲. Let ␻ 苸 ⍀1V M 共CN兲 be the 1-form defined by ∀F 苸 V M 共CN兲,

∀ ⌽ 苸 TFV M 共CN兲,

␻共⌽兲 = F†⌽.

共38兲

It is clear that ␻共FX兲 = X 苸 u共M兲, and the equivariance of ␻ follows immediately from its definition; ␻ is then a connection 1-form. Let 关−1 , 1兴苹 t 哫 F共t兲苸 VM 共CN兲, ␻共关dF / dt兴t=0兲 = 关F†共d / dt兲F兴t=0 苸 u共M兲. Let Z共t兲苸 V M 共CN兲 be such that ␲U共Z共t兲兲 is a solution of the Schrödinger– von Neumann equation. Letting ⌿共t兲 be a matrix of M wave functions such that ␲U共⌿共t兲兲 = ␲U共Z共t兲兲, then ⌿共t兲 = Z共t兲Te−ıប

−1兰t Z†共t 兲⳵ Z共t 兲dt ⬘ t⬘ ⬘ ⬘ 0

⇔ ⌿†⳵t⌿ = 0 ⇔ ␻共⳵t⌿兲 = 0.

共39兲

We see that the relation ⌿ ⳵t⌿ = 0 proved in the previous section is, in fact, the horizontality condition. †


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J. Math. Phys. 48, 052102 共2007兲


If H is infinite dimensional, then we can define the manifolds of the active spaces by an inductive limit, GM 共C⬁兲 = 兵P 苸 B共H兲兩P2 = P, P† = P,tr P = M其,

共40兲

V M 共C⬁兲 = 兵共␺1, . . . , ␺ M 兲苸 HM 兩 ∀ i, j具␺i兩␺ j典 = ␦ij其,

共41兲

with B共H兲 being the Banach space of bounded operators of H. If H共t兲 is not self-adjoint, then the structure is the same with the replacement of U共M兲 by GL共M , C兲, of V M 共CN兲 by the noncompact Stiefel manifold 共see Ref. 18兲 V*M 共CN兲 = GL共N , C兲 / GL共N − M , C兲, and of Z† by Z*†, where Z* is the matrix of the eigenvectors of H†. Note that V*M 共CN兲 = V M 共CN兲 ⫻ T共M,C兲,

共42兲

with T共M , C兲 being the manifold of upper triangular matrices of order M with positive diagonal elements. To simplify the analysis, in the sequel we will suppose that H is self-adjoint. Let 关0 , T兴苹 t 哫 Z共t兲Z†共t兲苸 G M 共CN兲 be a map which satisfies the Schrödinger–von Neumann equation, and let C be the path in G M 共CN兲 associated with this map. The non-Abelian geometric phase associated with this map is then the horizontal lift of C, i.e., T † 共t兲⳵tZ共t兲dt

Te−兰0 Z

共43兲

= Pe−兰CA ,

where Pe is the exponential order with respect to the path 共see Ref. 42兲. In general, the nonAbelian geometric phase does not commute with the dynamical phase. It is then necessary to have a structure describing simultaneously the two phases. This structure is a principal composite bundle as explained in Refs. 25–27. Following Ref. 25, let U be the principal bundle describing the geometric phase endowed with the connection with gauge potential A = Z†dZ, and let Q = 共Q , R , U共N兲 , ␲Q兲 be the principal bundle with the time line as base space, endowed with the connection describing the evolution operators. Consider the space G M 共CN兲 ⫻ R as the trivial bundle S with base space the time line R and with the typical fiber G M 共CN兲. If we consider U as a structure bundle, Q as a transversal bundle, and S as a base bundle 共see Ref. 25 for the definitions of these notions兲, we construct a composite principal bundle 共this construction is based on the local trivializations of the different bundles, see Ref. 25兲. The connections of U and of Q define a natural connection in the composite principal bundle 共see Ref. 25兲 associated with the gauge potential, A+,H共ZZ†,t兲 = ıប−1Z†H共t兲Zdt + Z†dZ 苸 ⍀1共G M 共CN兲 ⫻ R兲.

共44兲

Let t 哫 Z共t兲Z†共t兲 be the solution of ıប⳵t共Z共t兲Z†共t兲兲 = 关H共t兲 , Z共t兲Z†共t兲兴, 关0 , T兴苹 t 哫共Z共t兲Z†共t兲 , t兲 that appears as a path in G M 共CN兲 ⫻ R, and its horizontal lift in the principal composite bundle is U共T兲 = Te−ıប

−1兰TE共t兲dt−兰TZ共t兲†⳵ Z共t兲dt t 0 0

.

共45兲

The dynamics is then described as follows. Let ıប H共t兲dt be a connection in Q; then the horizontal lift of 关0 , T兴 in Q induces a path in GM 共CN兲 parametrized by Z共t兲Z†共t兲 satisfying the Schödinger–von Neumann equation. t 哫共Z共t兲Z†共t兲 , t兲 is a path in GM 共CN兲 ⫻ R for which the horizontal lift in the composite bundle, associated with the connection A+,H, describes the non-Abelian total phase. The integration of the dynamics is then decomposed in two steps. The first one consists to find the path t 哫 Z共t兲Z共t兲† 苸 G M 共CN兲 and the second one consists to lift the result of this first step. Section II C presents the geometric formulation of the subdynamics in G M 共CN兲 constituting the first step. −1


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David Viennot

C. Geometric formulation of the subdynamics of the evolutive active space in the base manifold

This section shows that the subdynamics of the evolutive active space can be geometrically formulated. 1. The self-adjoint case

Definition 2: Let C 苸 L共H兲 be a self-adjoint operator of the Hilbert H = CN 共with possibly N = + ⬁兲 considered as an observable of the quantum system. Let f: G M 共CN兲 → C be the function defined by f共Z,Z†兲 = Cij¯Z␣j Zi␣ = tr共Z†CZ兲.

共46兲

We call f the weak version of the observable C. Let f and g be the weak versions of two observables C and B, and the Poisson bracket of f and g is the weak version of the commutator of C and B; indeed, 兵f,g其 =

⳵ f ⳵g ⳵ f ⳵g − ⳵Zi␣ ⳵¯Zi ⳵¯Zi ⳵Zi␣ ␣ ␣

共47兲

¯ j 共Ci Bk − Bi Ck兲Z␣ =Z ␣ j i j i k

共48兲

=tr共Z†关C,B兴Z兲.

共49兲

Proposition 2: Let H be the self-adjoint Hamiltonian of the quantum system supposed to be ¯ i ∧ dZ␣ be the Kähler time independent. Let H共Z , Z†兲 = Hij¯Z␣j Zi␣ be its weak version. Let F = tr F = dZ ␣ i N † N form of G M 共C 兲共see the Appendix兲. A curve t 哫 ⌿共t兲⌿ 共t兲苸 G M 共C 兲 satisfies the Schrödinger–von Neumann equation if and only if iXF = ıប−1dH,

共50兲

¯ i / dt兲共⳵ / ⳵¯Zi 兲 is the tangent vector field to the curve and where i where X = 共d⌿i␣ / dt兲共⳵ / ⳵Zi␣兲 + 共d⌿ ␣ ␣ N is the inner product of GM 共C 兲. Proof: ¯i d⌿ d⌿i␣ ¯ i ␣ dZi␣ − dZ␣ . dt dt

共51兲

¯ j + Hi ¯Z j dZ␣ . dH = HijZi␣dZ i ␣ j ␣

共52兲

iXF = ıប−1dH共⌿,⌿†兲

共53兲

i XF = Moreover,

Then,

⇔ıប

¯i d⌿i␣ ¯ i d⌿ ␣ ¯j ␣ ¯ j − Hi ⌿ dZi␣ − ıប dZ␣ = − Hij⌿i␣dZ ␣ j ␣dZi , dt dt

共54兲

and then,


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J. Math. Phys. 48, 052102 共2007兲


冉

ıប

冊冉

冊

␣ ¯i d⌿ ␣ ¯ j dZ␣ + − ıប d⌿ j + Hi ⌿␣ dZ ¯ j = 0. + Hij⌿ ␣ i j i ␣ dt dt

共55兲

Since 共Zi␣ , ¯Z␣i 兲 are degenerate coordinates for GM 共CN兲, these variables are not independent. We have the relation Z †Z = I M

共56兲

⇒dZ†Z + Z†dZ = 0

共57兲

⇔ ∀ ␣, ␤,

¯ i + ¯Zi dZ␣ = 0. Zi␣dZ ␤ ␤ i

共58兲

¯ i + ¯Zi dZ␣兲 = 0. 兵⌳␤共t兲其 are Lagrange factors assoFor all functions 兵⌳␣␤共t兲其, we have ⌳␣␤共Zi␣dZ ␤ ␤ i ␣ ciated with the constraint on the variables. Equation 共55兲 is then equivalent to the two following equations: ıប

− ıប

d⌿i␣ d −1 t −1 t + ⌳␤␣⌿i␤ = Hij⌿i␣ ⇔ ıប 共⌿Te−ıប 兰0⌳共t⬘兲dt⬘兲 = H⌿Te−ıប 兰0⌳共t⬘兲dt⬘ , dt dt

冉

¯i d⌿ d ␣ −ıប−1兰t0⌳共t⬘兲dt⬘ ¯ i = Hi ⌿ ¯j + ⌳␣␤⌿ 兲 ␤ j ␣ ⇔ ıប 共⌿Te dt dt

冊

†

= 共H⌿Te−ıប

共59兲

−1兰t ⌳共t 兲dt † ⬘ ⬘ 0

兲 .

共60兲

Finally, we have ıប

⳵⌿⌿† ⳵⌿⌼⌼†⌿† = ıប ⳵t ⳵t

共61兲

⳵⌿⌼ † † ⳵共⌿⌼兲† ⌼ ⌿ + ıប⌿⌼ ⳵t ⳵t

共62兲

=H⌿⌼⌼†⌿† + ⌼⌼†⌿†H

共63兲

=关H,⌿⌿†兴,

共64兲

=ıប

−ıប−1 t

兰0⌳共t⬘兲dt⬘ with ⌼†⌼ = ⌼⌼† = 1. 䊏 where ⌼ = Te We remark that we recover the invariance of the Schrödinger equation with respect to the local −1 gauge choice Te−ıប 兰t0⌳共t⬘兲dt⬘. We suppose now that the system Hamiltonian is time dependent. Let A+,H = ıប−1Z†H共t兲Zdt † + Z dZ be the connection of the composite bundle. Let F+,H = dA+,H + A+,H ∧ A+,H be the curvature of the composite bundle. Moreover, we consider the following forms: A+ = tr共A+,H兲 = ¯Z␣i dZi␣ + ıប−1Hdt

共65兲

¯ i ∧ dZ␣ + ıប−1dH ∧ dt, F+ = tr共F+,H兲 = dZ ␣ i

共66兲

and

¯ j Z ␣. where H共Z , Z† , t兲 = Hij共t兲Z ␣ i Proposition 3: Let H共Z , Z† , t兲 be the weak version of the time-dependent self-adjoint Hamiltonian of the quantum system. A curve t 哫共⌿共t兲 , ⌿†共t兲兲 satisfies the Schrödinger equation if and only if


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David Viennot

TABLE I. Comparison between classical dynamics and quantum dynamics with active spaces. Classical dynamical systems

Quantum dynamical systems

Phase space: R2M Variables: qi Conjugate variables: pi

Grassmannian: G M 共CN兲

2

Symplectic metric: dᐉ = dpidq

Variables: Zi␣ Conjugate variables: ¯Zi␣ ¯ i dZ␣ − dZ ¯ i Z␤¯Z j dZ␣ Kählerian metric: dᐉ2 = dZ ␣ i ␣ i ␤ j Weak observables algebra: 共兵Ci ¯Z j Z␣其 , + , 兵. , . 其兲

i

Observables algebra: 共C⬁共R2M , C兲 , + , 兵. , . 其兲

j ␣ i

Poincaré 1-form: ␭ = pidqi

Trace of the gauge potential: A = ¯Zi␣dZi␣ A = ¯Zi dZ␣ + ıប−1Hdt

⌳ = pidqi − Hdt

␣

+

Poincaré 2-form: ␻ = dpi ∧ dqi

i

⍀ = dpi ∧ dqi − dH ∧ dt

¯ i ∧ dZ␣ Kähler form: F = dZ ␣ i i ␣ −1 ¯ ∧ dZ + ıប dH ∧ dt F = dZ

Fundamental equation: iX␻ = −dH i X⍀ = 0

Fundamental equation: ıបiXF = −dH ıបiXF+ = 0

+

␣

i

iXF+ = 0,

共67兲

¯ i / dt兲共⳵ / ⳵¯Zi 兲 − ⳵ / ⳵t is the tangent vector field to the curve and where X = 共d⌿i␣ / dt兲共⳵ / ⳵Zi␣兲 + 共d⌿ ␣ ␣ where i is the inner product of G M 共CN兲 ⫻ R. Proof: iXF+共Z,Z†,t兲 = 0

⇔

␣ ¯j ¯i d⌿ d⌿i␣ ¯ i d⌿ ␣ ␣ ¯ j d⌿i dt = 0 dt − ıប−1Hij共t兲⌿ dZi␣ − dZ␣ + ıប−1Hij共t兲⌿i␣ ␣ dt dt dt dt

⇔ıប

¯i d⌿ d⌿i␣ ¯ i ␣ ¯ j − Hi 共t兲⌿ ¯ j dZ␣ . dZi␣ − ıប dZ␣ = Hij共t兲⌿i␣dZ ␣ j ␣ i dt dt

共68兲

共69兲

共70兲

The demonstration is then totally equivalent to the previous one. 䊏 We see that the quantum dynamics with active spaces has a geometric formalism very close to that of classical dynamics, with A or A+ playing the role of the Poincaré 1-form and F or F+ playing the role of the Poincaré 2-form. A comparison between classical and quantum systems is given Table I. To summarize, we obtain the active space dynamics as being a curve on GM 共CN兲 satisfying iXF+ = 0, with the state dynamics being obtained by the horizontal lift of this curve with respect to the composite connection of gauge potential A+,H. 2. The non-self-adjoint case

If H共t兲 is not self-adjoint, the dynamics is quite different. Let 兵兩␣共t兲典其␣ be the generalized eigenvectors of H共t兲, i.e., the states satisfying for some n 苸 N the equation 共H共t兲 − Ei共t兲兲n兩␣共t兲典 = 0, where Ei共t兲 is an eigenvalue of H共t兲. The number of generalized eigenvectors satisfying this equation for a particular Ei is equal to the algebraic multiplicity of Ei 共see Ref. 39, Remark 6.5兲. 兵兩␣共t兲典其␣ are not orthonormalized but are biorthogonal to the generalized eigenvectors of H共t兲† : 兵兩␣ ⴱ 共t兲典其␣. The operator C such that C兩␣共t兲典 = 兩␣ ⴱ 共t兲典 is antilinear with CH = H†C 共see Refs. 44 and 45兲. The general case can be difficult since, in general, C is time dependent. However, if we suppose that H is symmetric 共i.e., Ht = H, where the symbol t denotes the transposition兲, it is easy to show that C is then the complex conjugation 共without the transposition兲, and then 兩␣ ⴱ 共t兲典 = 兩␣共t兲典. In the sequel, we assume that H is symmetric or more generally that the antilinear


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operator C associated with the transformation of the eigenvectors of H共t兲 to the eigenvectors of H†共t兲 is independent of time and so defines an operation extensible to any vectors of H independent of t. Let t 哫 ⌿␣共t兲 be the solution of the Schödinger equation with ⌿␣共0兲 = 兩␣共0兲典共now 兵兩␣共t兲典其␣ is the basis of the active space and it is not necessarily generated by eigenvectors兲. The equation of the dual solution ⌿*␣ = C⌿␣ is obtained by setting

冉

d⌿␣ = C共H⌿␣兲, dt

共71兲

− ıប

dC⌿␣ = H†C⌿␣ , dt

共72兲

d⌿*␣ = H †⌿ *␣ . dt

共73兲

C ıប

− ıប

冊

The operator P共t兲 = ⌿共t兲⌿共t兲*† does not satisfy the Schrödinger–von Neumann equation but obeys the equation ıបP˙ = HP + PH = 关H , P兴+. We refer to this equation as being the anti– Schrödinger–von Neumann equation. Let Z , Z* 苸 MN⫻M 共C兲 be such that Z*†Z = I M , where Z* is the conjugate matrix by C of Z. Z now represents a point of the noncompact Stiefel manifold VM 共CN兲. Zi␣ and ¯Z␣*i can be considered as another degenerate coordinate system for G M 共CN兲. Let J be the almost complex structure of G M 共CN兲共see Refs. 42 and 43兲 defined by ∀␣ , i:

⳵ ⳵ =ı , ⳵Zi␣ ⳵Zi␣

共74兲

⳵ ⳵ =−ı , *i ¯ ¯ ⳵Z␣ ⳵Z␣*i

共75兲

J

J

where J2 = −1. Moreover, let J‡ be the adjoint of J for the duality bracket of the tangent space, i.e., ∀X 苸 TZZ*†G M 共CN兲 and ␾ 苸 ⍀1G M 共CN兲, 具␾ , JX典 = 具J‡␾ , X典. It is easy to show that J‡dZi␣ = − ıdZi␣ ,

共76兲

¯ *i = ıdZ ¯ *i . J‡dZ ␣ ␣

共77兲

Proposition 4: Let H be the non-self-adjoint Hamiltonian of the quantum system supposed to ¯ *i ∧ dZ␣ be be time independent. Let H共Z , Z*†兲 = Hij¯Z␣j*Zi␣ = tr共Z*†HZ兲 be its weak version. Let F = dZ i ␣ * N the pseudo-Kähler form. A curve t 哫 ⌿共t兲⌿ 共t兲苸 G M 共C 兲 satisfies the anti-Schrödinger–von Neumann equation if and only if J‡បiXF = − dH

共78兲

⇔បiJXF = − dH,

共79兲

¯ *i / dt兲共⳵ / ⳵¯Z*i兲 is the tangent vector field to the curve and where where X = 共d⌿i␣ / dt兲共⳵ / ⳵Zi␣兲 + 共d⌿ ␣ ␣ N i is the inner product of G M 共C 兲.


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David Viennot

Proof: i XF =

¯ *i d⌿ d⌿i␣ ¯ *i ␣ dZi␣ − dZ␣ , dt dt

J‡បiXF = − ıប

共80兲

¯ *i d⌿ d⌿i␣ ¯ *i ␣ dZi␣ − ıប dZ␣ . dt dt

共81兲

By a demonstration similar of the previous one, we have J‡បiXF = − dH

冉

⇔ − ıប

冦冉

ıប

⇔

共82兲

冊冉

冊

␣ ¯ *i d⌿ ␣ ¯ *j dZ␣ + − ıប d⌿ j + Hi ⌿␣ dZ ¯ *j = 0 + Hij⌿ i j i ␣ ␣ dt dt

d −1 t −1 t 共⌿Te−ıប 兰0⌳共t⬘兲dt⬘兲 = H⌿Te−ıប 兰0⌳共t⬘兲dt⬘ dt

− ıប

d * ıប−1兰t ⌳共t⬘兲dt⬘ 0 共⌿ Te 兲 dt

冊

†

= 共H†⌿*Teıប

−1兰t ⌳共t 兲dt † ⬘ ⬘ 0

兲 .

冧

共83兲

共84兲

䊏 By comparison of the equation J‡បiXF = −dH with the equation of the self-adjoint case ıបiXF = −dH, we see that the non-self-adjoint case arises mainly by substituting ı by the almost complex structure J‡. In the same way, if H is a time-dependent non-self-adjoint Hamiltonian, the dynamical equation is

.

J‡បiXF+ = 0,

共85兲

¯ *i ∧ dZ␣ + ıប−1dH ∧ dt F+ = dZ i ␣

共86兲

where

共and with, by convention, J‡dt = ıdt兲. III. GEOMETRIC FORMULATION OF THE TIME-DEPENDENT WAVE OPERATORS

The previous section has analyzed the geometry of a dynamics in the evolutive active space formalism. Now we consider the status of the time-dependent wave operator in the bundle of the active spaces. We assume in this section that the Hamiltonian is self-adjoint, and the non-selfadjoint case can be easily obtained by substitutions as Z† by Z*† for example. We use in this section the distances in G M 共CN兲 defined in the Appendix. A. Bloch wave operators and time-dependent wave operators −1 −1 Let P0 , P 苸 G M 共CN兲, Z 苸 ␲U 共P兲, and Z0 苸 ␲U 共P0兲. If distFS共Z , Z0兲 ⬍ ␲ / 2 then 共P0 PP0兲−1 exists and we can define a type of Bloch wave operator P共P0 PP0兲−1. Let 兵兩␣0典其 and 兵兩␣典其 be the bases defined by Z0 and Z. We have

P0 PP0 =

兺兩␣0典具␣0兩␤典具␤兩␥0典具␥0兩

␣␤␥

共87兲


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= 兺关Z†0ZZ†Z0兴␣␥兩␣0典具␥0兩

共88兲

共P0 PP0兲−1 = 兺关共Z†Z0兲−1共Z†0Z兲−1兴␣␥兩␣0典具␥0兩,

共89兲

␣␥

and then

␣␥

from which we find P共P0 PP0兲−1 =

兺关共Z†Z0兲−1共Z†0Z兲−1兴␣␥兩␤典具␤兩␣0典具␥0兩

␣␤␥

共90兲

= 兺关Z†Z0共Z†Z0兲−1共Z†0Z兲−1兴␤␥兩␤典具␥0兩

共91兲

= 兺关共Z†0Z兲−1兴␤␥兩␤典具␥0兩.

共92兲

␤␥

␤␥

The Bloch wave operator P共P0 PP0兲−1 is then a linear map from ␲E−1共P0兲 to ␲E−1共P兲 represented by the matrix 共Z†0Z兲−1. Now we focus on the time-dependent wave operator ⍀共t兲 = U共t , 0兲共P共0兲U共t , 0兲P共0兲兲−1. Let t −1 t V共t兲 = Te−ıប 兰0E共t⬘兲dt⬘−兰0A共t⬘兲dt, where E共t兲 = Z†共t兲H共t兲Z共t兲, A共t兲 = Z†共t兲⳵tZ共t兲, and ␲U共Z共t兲兲 = P共t兲 is a solution of the Schrödinger–von Neumann equation. We have

␺␣共t兲 = U共t,0兲兩␣共0兲典 = 兺 V共t兲␤␣兩␤共t兲典, ␤

共93兲

and then 具␤共t兲兩U共t , 0兲兩␣共0兲典 = V共t兲␤␣ and U共t,0兲 = 兺 V共t兲␤␣兩␤共t兲典具␣共0兲兩. ␣␤

共94兲

We can then write P共0兲U共t,0兲P共0兲 =

兺

␣␤␥␦

V共t兲␤␣兩␥共0兲典具␥共0兲兩␤共t兲典具␣共0兲兩␦共0兲典具␦共0兲兩

共95兲

= 兺关Z†共0兲Z共t兲V共t兲兴␥␣兩␥共0兲典具␣共0兲兩,

共96兲

共P共0兲U共t,0兲P共0兲兲−1 = 兺关V共t兲−1共Z†共0兲Z共t兲兲−1兴␥␣兩␥共0兲典具␣共0兲兩

共97兲

␥␣

giving

␥␣

and thus U共t,0兲共P共0兲U共t,0兲P共0兲兲−1 =

V共t兲␤␦关V共t兲−1共Z†共0兲Z共t兲兲−1兴␥␣兩␤共t兲典具␦共0兲兩␥共0兲典具␣共0兲兩兺 ␣␤␥␦ 共98兲

= 兺关V共t兲V共t兲−1共Z†共0兲Z共t兲兲−1兴␤␣兩␤共t兲典具␣共0兲兩 ␤␣

共99兲


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= 兺关共Z†共0兲Z共t兲兲−1兴␤␣兩␤共t兲典具␣共0兲兩.

共100兲

␤␣

We see that the time-dependent wave operator is, in fact, a type of Bloch wave operator ⍀共t兲 = P共t兲共P共0兲P共t兲P共0兲兲−1, where P共t兲 is solution of the Schrödinger–von Neumann equation. B. The time-dependent wave operator as a horizontal lift of the bundle of active spaces

Let Z 苸 V M 共CN兲, ∀W 苸 V M 共CN兲, such that distFS共W , Z兲 ⬍ ␲ / 2, and ZW = W共Z†W兲−1 is such that † indeed, ZW Z = 共W†Z兲−1W†Z = I M . The set of vectors defined by ZW is biorthogonal to the basis defined by Z. Now consider Z共t兲 such that ␲U共Z共t兲兲 = P共t兲 is a solution of the Schrödinger– von Neumann equation and such that ∀t, distFS共Z共t兲 , Z共0兲兲 ⬍ ␲ / 2. We know by Proposition 1 that the wave function ⌿, a solution of the Schrödinger equation, is

† Z = IM; ZW

⌿共t兲 = Z共t兲Te−ıប

−1兰t E共t 兲dt −兰t A共t 兲dt 0 0 ⬘ ⬘ 0 0 ⬘ ⬘

,

共101兲

E0 共t兲 = 共Z†共0兲Z共t兲兲−1Z†共0兲H共t兲Z共t兲.

共102兲

where A0 共t兲 = 共Z†共0兲Z共t兲兲−1Z†共0兲⳵tZ共t兲,

† † ˜ 共t兲 = 共Z†共0兲Z共t兲兲−1, then 共t兲⳵tZ and E0 = ZZ共0兲共t兲H共t兲Z共t兲. Letting ⍀ In other words, A0 = ZZ共0兲

dZ共t兲 † d † 共Z 共0兲Z共t兲兲−1 = − 共Z†共0兲Z共t兲兲−1Z†共0兲共Z 共0兲Z共t兲兲−1 . dt dt

共103兲

d˜ ˜ 共t兲. ⍀共t兲 = − A0 共t兲⍀ dt

共104兲

⍀共t兲 = 兺关Te−兰0A0 共t⬘兲dt⬘兴␤␣兩␤共t兲典具␣共0兲兩.

共105兲

In other words,

We conclude that t

␣␤

The time-dependent wave operator is a horizontal lift for the connection defined by the gauge † † potential AZ共0兲共ZZZ共0兲兲 = 共Z共0兲†Z兲−1Z†共0兲dZ 苸 ⍀1共KZ共0兲 , gl共M , C兲兲, where KW = 兵ZZW 兩Z N N 苸 V M 共C 兲 , distFS共Z , W兲 ⬍ ␲ / 2其关KW can be injected in its universal classifying space G M 共C 兲兴. In fact, the time-dependent wave operator theory is associated with a family of connections defined by V M 共CN兲 → ⍀1共KW,gl共M,C兲兲, † † W 哫 AW 共ZZW 兲 = 共W†Z兲−1W†dZ = ZW dZ.

共106兲

This family of connections is also defined by a family of connection 1-form ␻W: ∀F 苸 V*M 共CN兲, ∀ ⌽ 苸 TFV*M 共CN兲,

␻W共⌽兲 = 共W†F兲−1W†⌽,

共107兲

for which the horizontality conditions are 共W†⌿共t兲兲−1W†⳵t⌿共t兲 = 0.

共108兲

We thus recover the general structure of generalized geometric phases. In the usual timedependent wave operator theory, we choose the connection associated with the initial condition Z共0兲.


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Note that to define the wave operator connection, we have considered the group extension 共V M 共CN兲 , KW , U共M兲 , ␲K兲共VM 共CN兲 , KW , GL共M , C兲 , ␲K兲, because the wave operator theory does not preserve the self-adjointness of the Hamiltonian representation. In the expression for ␺␣, we wish to separate the geometric phase from the dynamical phase. By virtue of the intermediate representation theorem 关Eq. 共40兲兴, we have

␺␣共t兲 = 兺关Te−ıប

−1兰t E共t 兲dt −兰t A共t 兲dt 0 0 ⬘ ⬘ 0 0 ⬘ ⬘

␤

兴␤␣兩␤共t兲典

共109兲

共110兲 However, we also have ˜ −1共t兲E共t兲⍀ ˜ 共t兲 = Z†共0兲Z共t兲共Z†共0兲Z共t兲兲−1Z†共0兲H共t兲Z共t兲共Z†共0兲Z共t兲兲−1 ⍀ 0

共111兲

=Z†共0兲H共t兲Z共t兲共Z†共0兲Z共t兲兲−1 ,

共112兲

˜ −1共t兲E共t兲⍀ ˜ 共t兲兴 = 兺具␣共0兲兩H兩␥共t兲典具␥共t兲兩⍀共t兲兩␤共0兲典. 关⍀ ␣␤ 0

共113兲

˜ −1共t兲E共t兲⍀ ˜ 共t兲兴兩␣共0兲典具␤共0兲兩 = P H共t兲⍀共t兲 = Heff共t兲, 关⍀ 兺 ␣␤ 0 0 ␣␤

共114兲

and thus

␥

Finally, we obtain

and thus

␺␣共t兲 = ⍀共t兲Te−ıប

−1兰t Heff共t 兲dt ⬘ ⬘ 0

兩␣共0兲典.

共115兲

We see that the time-dependent wave operator theory is just a geometric phase phenomenon where we have separated the geometric and the dynamical phases. The effective Hamiltonian of the wave operator theory emerges spontaneously as being the generator of the dynamical phase conjugated by the geometric phase 共the wave operator兲. We wish to add an important comment. In contrast with the universal connection, the wave operator connections are not defined for all the manifold VM 共CN兲. Letting W 苸 V M 共CN兲, the wave = 共W†Z兲−1W†dZ is only defined for ␲U共Z兲 in the open ball of center operator gauge potential AW ␲U共W兲 and of radius ␲ / 2 共in the Fubini-Study distance兲; in other words, the domain of the connection 1-form is

再

= Z 苸 V*M 共CN兲兩distFS共W,Z兲 ⬍ dom ␻W

冎

␲ . 2

共116兲

The wave operator connection is flat; indeed, if we compute the wave operator curvature, we find = dAW + AW ∧ AW = − 共W†Z兲−1W†dZ共W†Z兲−1 ∧ W†dZ + 共W†Z兲−1W†dZ ∧ 共W†Z兲−1W†dZ = 0. FW

共117兲 Let ⌿ 苸 MN⫻M 共C兲 be a solution of the Schrödinger equation ıប⳵t⌿共t兲 = H共t兲⌿共t兲 and let ⌿W共t兲 = W共⌿共t兲†W兲−1, ıប

d⌿W d⌿† = − Wıប共⌿†W兲−1 W共⌿†W兲−1 dt dt

共118兲


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David Viennot

=W共⌿†W兲−1⌿†H†W共⌿†W兲−1

共119兲

=W共⌿†W兲−1共共W†⌿兲−1W†H⌿兲†

共120兲

† =⌿W共⌿W H⌿兲†

共121兲

,† . =⌿WEW

共122兲

,† † † ⇔ −ıប⳵t⌿W = EW ⌿W, is the dual equation of the This last equation, ıប⳵t⌿W = ⌿WEW i Schrödinger equation with the new local conjugate variables ¯ZW ␣. In a geometric framework, the + ıប−1EW dt of the wave operator theory consists then to the lift with respect to the connection AW † † † † † curve ⌿共t兲⌿共t兲W obtained as a solution of the equation ıប共d⌿⌿W / dt兲 = H⌿⌿W − ⌿⌿WH⌿⌿W .

IV. CONCLUSION

We have seen that the theory of active spaces, effective Hamiltonians, and wave operators is, in fact, a gauge theory. The active space theory is defined by the principal bundle U = 共V M 共CN兲 , G M 共CN兲 , U共M兲 , ␲U兲, where the Grassmannian manifold models the space of the active space projectors, and where the Stiefel manifold models the space of the active space basis. The fibers ␲E−1共P兲 of the associated vector bundle E = 共E , GM 共CN兲 , C M , ␲E兲 model the active spaces with the vector space structure. The Grassmannian manifold is endowed with a metric associated with its Kählerian structure. This metric measures the quantum distance between the active spaces, i.e., the probabilistic compatibility between the active spaces 共see Appendix兲. U is endowed with the connection A = Z†dZ, which is universal in the sense of the NarasimhanRamaman theorem 共see Ref. 41兲. The universal connection is, in fact, the single connection compatible with the Kählerian metric of the Grassmannian, like the Levi-Civita connection in the Riemannian case. The horizontal lifts in this connection define the non-Abelian AharonovAnandan geometric phases. Moreover, U is also endowed with a family of connections AW = 共W†Z兲−1W†dZ for which the horizontal lifts define the time-dependent wave operators. The wave operator appears then as a geometric phase, and the effective Hamiltonian of the wave operator theory emerges spontaneously in the geometric framework as being the generator of the dynamical phase conjugated by the geometric phase. In a more general framework, if Z 苸 MN⫻M 共C兲 represents a basis of an evolutive active space with Z 苸 MN⫻M 共C兲, a biorthogonal basis 共Z†Z = I M 兲 such that the passage from Z to Z is a time-independent procedure. The evolution path of the active space on the Grassmannian is then a solution of ıបiXF+ = 0 J‡បiXF+ = 0

if H is self-adjoint, if H is non-self-adjoint,

共123兲

where X is the tangent vector field of the curve, and with ¯ i ∧ dZ␣ + ıប−1d tr共Z†HZ兲 ∧ dt, F+ = dZ i ␣

共124兲

with J‡ being the dual of the almost complex structure J

⳵ ⳵ , ␣ =ı ⳵Zi ⳵Zi␣

J

⳵ ⳵ =−ı . ⳵¯Z*i ⳵¯Z*i ␣

共125兲

␣

The wave functions are obtained by the horizontal lift of this curve with respect to the composite connection A+ = Z†dZ + ıប−1Z†HZdt. Z†HZ 共or its conjugate by the geometric phase兲 is the effective Hamiltonian associated with the evolutive active space.


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ACKNOWLEDGMENT

The author thanks Professor John P. Killingbeck for his help. APPENDIX: GEOMETRY OF THE BASE MANIFOLD

G M 共CN兲 is the base manifold of the bundle of active spaces. In this present appendix, we explore some geometric properties of G M 共CN兲 where the subdynamics of the evolutive active space or where the effective dynamics of the wave operator theory takes place. Let A be the gauge potential of U, so that 共A1兲

A = Z†dZ,

where Z 苸 MN⫻M 共C兲 such that Z†Z = I M . Z represents a point in the Stiefel manifold VM 共CN兲, and the equivalence class of Z for the relation Z ⬃ ZU with U 苸 U共M兲 represents a point in the Grassmann manifold G M 共CN兲共␲U共Z兲兲. The curvature associated with the connection is obtained by the Cartan structure equation F = dA + A ∧ A = dZ† ∧ dZ + 共Z†dZ兲 ∧ 共Z†dZ兲,

共A2兲

and it satisfies the Bianchi identity dF + 关A,F兴 = 0.

共A3兲

Z†Z = 1 ⇒ dZ†Z = − Z†dZ ⇔ A† = − A.

共A4兲

We note, moreover, that

We know that the complex Grassmannian G M 共CN兲 is endowed with the structure of a Kählerian manifold,42,43 with its Kähler potential being K=

1 ln det共W†Z兲, 2

共A5兲

with ln being the principal value of the complex natural logarithm. The Kähler form F can be deduced from K by the relation F = 共2ı¯⳵⳵K兲W=Z ,

共A6兲

where ⳵ and ¯⳵ are the Dolbeault differentials.42,43 Introducing Zi␣ = 具i 兩 ␣典 and ¯Z␣i = 具␣ 兩 i典 with ␣ = 1 , . . . , M and i = 1 , . . . , N, as a degenerate coordinate system, we find that ¯ j ∧ dZ␤ − 共Z␥dZ ¯ k 兲 ∧ 共Z ¯ j dZ␤兲. F = tr F = dZ j k ␤ ␥ j ␤

共A7兲

Proof: In this section, we adopt the Einstein convention for the Latin indices but not for the Greek indices. We denote by S M the group of permutations of 共1 , . . . , M兲 and by 共−1兲␴ the signature of the permutation ␴. The detailed derivative associated with Eq. 共A7兲 is as follows. det W†Z =

兺 ␴苸S

M

M

¯ i Z␣ , 共− 1兲␴ 兿 W ␴共␣兲 i ␣=1

共A8兲

M

1 ¯ i Z ␣W ¯ j dZ␤ , ⳵ ln det W Z = W 兺共− 1兲␴␤兺=1 ␣兿 ␴共␣兲 i ␴共␤兲 j det W†Z ␴苸SM ⫽␤ †

共A9兲


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David Viennot M

¯⳵⳵ ln det W†Z =

1 兺共− 1兲␴␤兺=1 ␥兺兿 W¯␴i 共␣兲Zi␣W¯␴j 共␤兲Zk␥dW¯␴k 共␥兲 ∧ dZ␤j det W†Z ␴苸SM ⫽␤ ␣⫽␤,␥ M

+

1 1 ¯ i Z␣dW ¯ j ∧ dZ␤ − 共− 1兲␴ 兺兿 W 兺 ␴共␤兲 j ␴共␣兲 i † det W Z ␴苸SM 共det W†Z兲2 ␤=1 ␣⫽␤

冊冉兺

M

共− 1兲␴ 兺

兿 ␤=1 ␣⫽␤

¯j ¯ i Z␣Z␤dW W ␴共␣兲 i j ␴共␤兲 ∧

␴苸S M

冉兺

␴苸S M

M

共− 1兲␴ 兺

¯ i Z ␣W ¯ j dZ␤ W 兿 ␴共␣兲 i ␴共␤兲 j ␤=1 ␣⫽␤

冊

,

共A10兲

共¯⳵⳵ ln det W†Z兲W=Z =

兺 ␴苸S

M

共− 1兲␴ 兺

¯ k ∧ dZ␤ 共Z†Z兲␴␣共␣兲¯Z␴j 共␤兲Zk␥dZ 兺兿 ␴共␥兲 j ␤=1 ␥⫽␤ ␣⫽␤,␥

M

+

兺

␴苸S M

−

M

共− 1兲

冉兺冉兺

␴苸S M

∧

M

兺 ␤=1 ␥⫽␤

=兺

␴苸S M

␴

兺兿共Z†Z兲␴␣共␣兲dZ¯␴j 共␤兲 ∧ dZ␤j

␤=1 ␣⫽␤ M

共− 1兲

␴

兺兿共Z†Z兲␴␣共␣兲Z␤j dZ¯␴j 共␤兲

␤=1 ␣⫽␤ M

共− 1兲␴ 兺

共Z†Z兲␴␣共␣兲¯Z␴j 共␤兲dZ␤j 兿 ␤=1 ␣⫽␤

M

¯Z j Z␥dZ ¯ k ∧ dZ␤ − ␤ k ␥ j

兺兺 ␤=1 ␥⫽␤

冊冊

M

¯Z j Z␥dZ ¯ k ∧ dZ␤ + ␥ k ␤ j

兺 dZ¯␤j ∧ dZ␤j −

␤=1

冉

M

共A11兲

冊冉兺冊

兺 Z␤j dZ¯␤j ∧

␤=1

M

␤=1

¯Z j dZ␤ ␤ j

共A12兲

=

冉

M

冊冉

兺 Zk␥dZ¯␥k ∧

␥=1

兺 ¯Z␤j dZ␤j −

␤=1

M

+

兺

␤=1

冊

M

M

兺 Zk␤¯Z␤j dZ¯␤k ∧ dZ␤j −

␤=1

M

¯k ∧ Zk␤¯Z␤j dZ ␤

dZ␤j +

M

兺 dZ¯␤j ∧ dZ␤j −

␤=1

¯ j ∧ dZ␤ − = 兺 dZ ␤ j ␤=1

M

M

¯ k 兲 ∧ 共Z ¯ j dZ␤兲共Zk␥dZ 兺兺 ␤ ␥ j ␤=1 ␥=1

冉兺冊冉兺冊 M

␤=1

M

¯j Z␤j dZ ␤

∧

␤=1

¯Z j dZ␤ ␤ j

共A13兲

M

兺

␤,␥=1

¯ k 兲 ∧ 共Z ¯ j dZ␤兲. 共Zk␥dZ ␤ ␥ j

共A14兲

䊏 The natural metric of the Grassmannian is then the Kähler form where we “replace” the i ¯j ¯ j ⳵Zi exterior product ∧ by the tensor product, i.e., dl2 = 2ı共⳵2K / ⳵W ␤ ␣W=Z兲dZ␣dZ␤. By the same approach as that given in this appendix, we then have dl2 = tr共dZ†dZ兲 + tr共共Z†dZ兲2兲

共A15兲

=tr共dZ†dZ兲 − tr共dZ†ZZ†dZ兲

共A16兲

¯ i dZ␣ − dZ ¯ i Z␤¯Z j dZ␣ . =dZ ␣ i ␤ j ␣ i

共A17兲

¯ j ∧ dZ␤ since tr共关Z†dZ , Z†dZ兴兲 = 0. Note that F = tr F = dZ ␤ j


052102-21

J. Math. Phys. 48, 052102 共2007兲


The equivalence class ␲U共Z兲 of Z represents an M-dimensional active space. The particular choice of a particular element of the equivalence class is associated with a particular choice of basis 共兩␣典兲␣ for the active space. We wish to find a distance in GM 共CN兲 which characterizes the quantum distance between active spaces. We have two choices. The first is the chordal distance distch共W,Z兲 = 冑2冑M − tr共W†ZZ†W兲 = 冑2冑M − 储W†Z储F2 , where 储 · 储F is the Frobenius norm of the matrices: possible choice is the Fubini-Study distance

N 储A储F2 = tr共A†A兲 = 兺i=1 兺␤M=1兩Ai␤兩2.

distFS共W,Z兲 = arccos共det共W†Z兲det共Z†W兲兲 = arccos兩det W†Z兩2 .

共A18兲 The second 共A19兲

Note that these are distances for G M 共CN兲共the set of equivalence classes兲, and not for VM 共CN兲. We see that 0 艋 distch共W,Z兲艋冑2M ,

0 艋 distFS共W,Z兲艋

共A20兲

␲ . 2

共A21兲

The fact that the distances are bounded just reflects the fact that G M 共CN兲 is a compact manifold. The interpretation as quantum distances follows from the fact that if W and Z represent the same active space W = ZU, then the distance between W and Z is zero, and also from the following property. Property 2: Let E1 be the active space represented by W and E2 be the active space represented by Z. Then distch共W , Z兲 = 冑2M if and only if E1 ⬜ E2 and distFS共W , Z兲 = ␲ / 2 if and only if ⬜ E⬜ 1 艚 E2 ⫽ 兵0其 or E2 艚 E1 ⫽ 兵0其. Proof: W = 共兩␾1典 , . . . , 兩␾ M 典兲 and Z = 共兩␺1 , . . . , 兩␺ M 典兲, tr共W†ZZ†W兲 = 0 ⇔ 兺兩具␺i兩␾ j典兩2 = 0 ⇔ ∀ i, j,

具␺i兩␾ j典 = 0.

共A22兲

i,j

If det W†Z = 0, then the column vectors of Z†W are not linearly independent, and then ∃␣i such that 兺 j␣ j具␾i 兩 ␺ j典 = 0, ∀i. Letting 兩˜␺典 = 兺 j␣ j兩␺ j典, ∀␾ 苸 E1, we have 具␾ 兩 ˜␺典 = 0, and then ˜␺ 苸 E⬜ 1 . We ˜␺ 苸 E⬜ 艚 E , ˜␺ ⫽ 0. Since ␺ is an conclude that det W†Z = 0 ⇒ E⬜ 艚 E ⫽ 兵0其. We suppose that ∃ 2 2 j 1 1 , we have ∀i, orthonormal basis of E2, we have ˜␺ = 兺 j␣ j兩␺ j典 with ␣ j = 具␺ j 兩 ˜␺典. Since ˜␺ 苸 E⬜ 1 † ˜ 具␾i 兩 ␺典 = 0 and then ∀i兺 j␣ j具␾i 兩 ␺ j典 = 0. The column vectors of W Z are not linearly independent and 䊏 then det W†Z = 0. The two quantum distances are associated with the two notions of quantum incompatibility of active spaces. We say that two active spaces are incompatible in the strong sense if they are orthogonal: the probability of obtaining the same experimental results with a system in a state of E1 and with a system in a state of E2 is zero. We say that two active spaces are incompatible in the weak sense if E⬜ 1 艚 E2 = 兵0其: there exists a state of E1 for which the probability of obtaining the same measures as that with a system in a state of E2 is zero. Property 3: The two quantum distances induce the Kählerian metric of GM 共CN兲. Proof: distch共Z,Z + dZ兲2 = 2共M − tr共Z†共Z + dZ兲共Z† + dZ†兲Z兲兲

共A23兲

=2共M − tr共共1 + Z†dZ兲共1 + dZ†Z兲兲兲

共A24兲

=2 tr共− Z†dZ − dZ†dZ兲 − tr共Z†dZdZ†Z兲

共A25兲


052102-22

J. Math. Phys. 48, 052102 共2007兲

David Viennot

=2 tr共dZ†dZ兲 − 2 tr共Z†dZdZ†Z兲

共A26兲

=2dl2 ,

共A27兲

where we have used the property 共Z† + dZ†兲共Z + dZ兲 = 1 ⇔ dZ†dZ + Z†dZ + dZ†Z = 0. We recall that if 储A储 is in a neighborhood of 0 and if A is diagonalizable, then det共1 + A兲 = 1 + tr A + 21 共共tr A兲2 − tr共A2兲兲 + O共储A储3兲. We can further obtain the result cos distFS共Z,Z + dZ兲 = det共1 + Z†dZ兲det共1 + dZ†Z兲

共A28兲

冉

冊

共A29兲

冉

冊

共A30兲

1 1 = 1 + tr Z†dZ + 共tr Z†dZ兲2 − tr共Z†dZ兲2 + O共储dZ储3兲 2 2

1 1 ⫻ 1 + tr dZ†Z + 共tr dZ†Z兲2 − tr共dZ†Z兲2 + O共储dZ储3兲 2 2 =1 + tr Z†dZ + tr dZ†Z + 共tr Z†dZ兲共tr dZ†Z兲

共A31兲

1 1 1 1 + 共tr Z†dZ兲2 − tr共Z†dZ兲2 + 共tr dZ†Z兲2 − tr共dZ†Z兲2 + O共储dZ储3兲 2 2 2 2

共A32兲

=1 − tr共dZ†dZ兲 − 共tr Z†dZ兲2 + 共tr Z†dZ兲2 − tr共Z†dZ兲2 + O共储dZ储3兲

共A33兲

=1 − tr共dZ†dZ兲 − tr共Z†dZ兲2 + O共储dZ储3兲

共A34兲

⯝1 − dl2 .

共A35兲

Since cos distFS共Z , Z + dZ兲 ⯝ 1 − 关distFS共Z , Z + dZ兲兴 / 2, we have distFS共Z , Z + dZ兲 = 2dl . Letting C be a geodesic for dl2 in G M 共CN兲 linking Z and W, then 2

冕

C

dl =

1

冑2 distFS共Z,W兲.

2

2

䊏

共A36兲

Moreover, we know that G M 共CN兲傺 MN⫻N共C兲. WW† is self-adjoint ∀W 苸 V M 共CN兲, and then 2 2 G M 共CN兲傺 RN 关a vector vW of RN being the list of the real and the imaginary parts of the 2 2 components 共WW†兲ij with j 艋 i兴. GM 共CN兲 is then a submanifold of RN . We endow RN with the Euclidian metric. ∀WW† 苸 G M 共CN兲, since W†W = I M , we have ¯ i W ␤W ¯j 共WW†兲ij共WW†兲ij = W␣j W ␣ i ␤

共A37兲

¯ i W ␤W ¯ j W␣ =W ␣ i ␤ j

共A38兲

= ␦␣␤␦␤␣

共A39兲

=M .

共A40兲

Then


052102-23


J. Math. Phys. 48, 052102 共2007兲

distch共WW†,ZZ†兲 = 冑2冑M − 共W†ZZ†W兲␣␣

共A41兲

冑

¯ i Z␤¯Z j W␣ = 冑2 M − W ␣ i ␤ j

共A42兲

= 冑2M − 2共WW†兲ij共ZZ†兲ij

共A43兲

= 冑共WW†兲ij共WW†兲ij + 共ZZ†兲ij共ZZ†兲ij − 2共WW†兲ij共ZZ†兲ij

共A44兲

= 冑共WW† − ZZ†兲ij共WW† − ZZ†兲ij

共A45兲

= 冑共vW − vZ兲 · 共vW − vZ兲

共A46兲

=储vW − vZ储.

共A47兲 2

The chordal distance is then the distance through the space RN , whereas the Fubini-Stuty distance is the distance on the “surface” of G M 共CN兲. Remark: In practice, we consider a manifold M associated with the classical control parameters characterizing the environment of the quantum system. For example, if the quantum system is an atom driven by a laser field, then M is the manifold generated by the intensity, the polarization, and the frequency of the field. This control manifold is closer to the experimental situation than the active space manifold G M 共CN兲. The relation between the control parameters and the active 2 spaces is characterized by an immersion map from M to G M 共CN兲共or equivalently to RN 兲. The role of this map is 共locally兲 analyzed in the adiabatic case in Ref. 46. The Kählerian structure of G M 共CN兲 is particularly important, since the metric measures the variations of the evolutive active space and then measures the change of the quantum properties induced by the subdynamics of the evolutive active space. I. Chu, Adv. Chem. Phys. 73, 739 共1989兲. G. Jolicard and J. P. Killingbeck, J. Phys. A 36, R473 共2003兲. 3 S. Guérin and H. R. Jauslin, Adv. Chem. Phys. 125, 147 共2003兲. 4 W. Schere, J. Phys. A 27, 8331 共1994兲. 5 K. Drese and M. Holthaus, Eur. Phys. J. D 5, 119 共1999兲. 6 G. Jolicard, D. Viennot, and J. P. Killingbeck, J. Phys. Chem. A 108, 8580 共2004兲. 7 G. Jolicard, Annu. Rev. Phys. Chem. 46, 83 共1995兲. 8 G. Jolicard and J. P. Killingbeck, Domain-based Parallelism and Problem Decomposition Methods in Computational Science and Engineering 共SIAM, Philadelphia, 1995兲, Chap. 16, p. 179. 9 P. Durand and I. Paidarová, Phys. Rev. A 28, 3184 共1983兲. 10 P. Durand and I. Paidarová, Phys. Rev. A 58, 1867 共1998兲. 11 J. P. Killingbeck and G. Jolicard, J. Phys. A 36, R105 共2003兲. 12 J. V. Tietz and S. I. Chu, Chem. Phys. Lett. 101, 446 共1983兲. 13 J. Chang and R. E. Wyatt, J. Chem. Phys. 85, 1826 共1986兲. 14 R. E. Wyatt and C. Iung, Quantum Mechanical Studies of Molecular Spectra and Dynamics: Dynamics of Molecules and Chemical Reactions 共Dekker, New York, 1996兲. 15 M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 共1984兲. 16 B. Simon, Phys. Rev. Lett. 51, 2167 共1983兲. 17 Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58, 1593 共1987兲. 18 V. Rohlin and D. Fuchs, Premiers Cours de Topologie, Chapitres Géométriques 共Mir, Moscow, 1977兲. 19 N. Steenrod, The topology of Fibre Bundles 共Princeton University Press, Princeton, NJ, 1951兲. 20 J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697 共1990兲. 21 A. Bohm, L. J. Boya, A. Mostafazadeh, and G. Rudolph, J. Geom. Phys. 12, 13 共1993兲. 22 A. Bohm and A. Mostafazadeh, J. Math. Phys. 35, 1463 共1994兲. 23 A. Bohm et al., The Geometric Phase in Quantum Systems 共Springer, Berlin, 2003兲. 24 F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 共1984兲. 25 D. Viennot, J. Math. Phys. 46, 072102 共2005兲. 26 G. Sardanashvily, J. Math. Phys. 41, 5245 共2000兲. 27 G. Sardanashvily, e-priint quant-ph/0004005. 1 2


052102-24

David Viennot

J. Math. Phys. 48, 052102 共2007兲

P. Zanardi and M. Rasetti, Phys. Lett. A 264, 94 共1999兲. J. Pachos, P. Zanardi, and M. Rasetti, Phys. Rev. A 61, 010305 共1999兲. 30 K. Fujii, Rep. Math. Phys. 48, 75 共2001兲. 31 L. M. Duan, J. I. Cirac, and P. Zollen, Science 292, 1695 共2001兲. 32 J. Pachos and P. Zanardi, Am. J. Math. 15, 1257 共2001兲. 33 A. E. Margolin, V. I. Strazhev, and A. Ya. Tregubovich, Phys. Lett. A 303, 131 共2002兲. 34 D. Lucarelli, J. Math. Phys. 46, 052103 共2005兲. 35 H. R. Lewis and W. B. Riesenfeld, J. Math. Phys. 10, 1458 共1969兲. 36 D. B. Monteoliva, H. J. Korsh, and J. A. Núñes, J. Phys. A 27, 6897 共1994兲. 37 A. Mostafazadeh, J. Phys. A 31, 9975 共1998兲. 38 A. Mostafazadeh, J. Phys. A 32, 8157 共1999兲. 39 P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory 共Springer, New York 1996兲. 40 A. Messiah, Quantum Mechanics 共Dunod, Paris, 1965兲. 41 M. Narasimhan and S. Ramaman, Am. J. Math. 83, 563 共1961兲. 42 M. Nakahara, Geometry, Topology and Physics 共IOP, Bristol 1990兲. 43 C. Voisin, Théorie de Hodge et Géométrie Algébrique Complexe 共Société Mathématiques de France/Institut Universitaire de France, Paris 2002兲. 44 P. M. Morse and H. Feshbach, Methods of Theoretical Physics 共McGraw-Hill, New York, 1953兲. 45 A. Maasseen van der Brink and K. Young, J. Math. Phys. 34, 2607 共2001兲. 46 D. Viennot, J. Math. Phys. 47, 092105 共2006兲. 28 29


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