Geometry and Dynamics of Gaussian Wave Packets and their Wigner Transforms
We find a relationship between the dynamics of the Gaussian wave packet and the dynamics of the corresponding Gaussian Wigner function from the Hamiltonian/symplectic point of view. The main result states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space yields the covariance matrix of the corresponding Gaussian Wigner function. This fact, combined with Kostant’s coadjoint orbit covering theorem, establishes a symplectic/Poisson-geometric connection between the two dynamics. The Hamiltonian formulation naturally gives rise to corrections to the potential terms in the dynamics of both the wave packet and the Wigner function, thereby resulting in slightly different sets of equations from the conventional classical ones. We numerically investigate the effect of the correction term and demonstrate that it improves the accuracy of the dynamics as an approximation to the dynamics of expectation values of observables.
Key words and phrases:Gaussian wave packet, Wigner function, momentum maps, Hamiltonian dynamics, Lie–Poisson equation, coadjoint orbit, semiclassical mechanics
Coherent states play a crucial role in quantum dynamics, and their mathematical properties have been exploited over the decades in many different fields, especially quantum optics and chemical physics; see, e.g., Berceanu , Bialynicki-Birula and Morrison , Bonet-Luz and Tronci , Combescure and Robert . This is due to the fact that coherent states behave like classical states, in the sense that the expectation values of the quantum canonical operators undergo classical Hamiltonian dynamics; see, e.g., de Gosson , Combescure and Robert . Indeed, it is well known that, for quadratic Hamiltonians defined on , the time evolution equation of the Wigner function becomes identical to the Liouville equation
for the corresponding classical system, where is the canonical Poisson bracket on , i.e., for any ,
using Einstein’s summation convention. Besides their interesting properties relating classical and quantum systems, coherent states have always attracted much attention due to their intriguing geometric properties. Specifically, coherent states are defined (up to phase factors) as orbits of the representation of the Heisenberg group on the space of wave functions . In particular, it is customary to select the particular orbit corresponding to the Gaussian wave function arising as the vacuum (or ground) state solution of the harmonic oscillator. This interpretation of coherent states in terms of group orbits led Perelomov  to define generalized coherent states in terms of orbits corresponding to other group representations. For example, spin coherent states are orbits of for its natural representation on the space of Pauli spinors. Also, squeezed coherent states or Gaussian wave packets are orbits of the Lie group—sometimes called the Schrödinger group—given by the semidirect product of the metaplectic group and the Heisenberg group [28, 7]: Applying the representation of the Schrödinger group on the vacuum state of the harmonic oscillator yields the squeezed coherent state or the Gaussian wave packet, which is among the most studied quantum states in the literature; see e.g., Heller [21, 22], Littlejohn , Hagedorn [17, 18, 19, 20], Combescure and Robert .
The emergence of the metaplectic group in the structure of the Gaussian wave packet makes their mathematical study particularly interesting and also somewhat intricate, due to the form of the metaplectic representation [7, 6]. However, in the phase space picture of quantum mechanics, the subtlety of the metaplectic representation disappears and one may work with the corresponding symplectic matrices instead: Indeed, the symplectic group possesses a natural action on functions on the phase space. The Wigner transform of a Gaussian wave packet is a Gaussian function in the phase space that is entirely characterized by its mean (phase space center) and symplectic covariance matrix ; see (1.2) below. It is common in the literature to describe the dynamics of the mean by the classical Hamiltonian system and that of the covariance matrix by the congruence transformation given by the symplectic matrix , which in turn evolves according to the linearization of the classical Hamiltonian system. Upon extending to a more general positive-definite covariance matrix , this also applies to any Gaussian Wigner function on phase space .
The main focus of this paper is the geometry and dynamics of the Gaussian wave packet
and its Wigner transform. We are particularly interested in establishing a connection between the dynamics of the two in a symplectic/Poisson-geometric manner.
The above Gaussian wave packet (2) is parametrized by , , , and , where is the set of symmetric complex matrices (symmetric in the real sense) with positive-definite imaginary parts, i.e.,
and is called the Siegel upper half space ; and stand for the set of complex matrices and the set of symmetric real matrices, respectively. A practical significance of the Gaussian wave packet (2) is that it is an exact solution of the time-dependent Schrödinger equation with quadratic Hamiltonians if the parameters , as functions of the time, satisfy a certain set of ODEs. It also possesses other nice properties as approximations to the exact solution; see Heller [21, 22] and Hagedorn [17, 18, 19, 20] and also Section 2.1 below.
Recently, inspired by the work of Lubich  and Faou and Lubich , Ohsawa and Leok  described the (reduced) dynamics of the Gaussian wave packet (2) as a Hamiltonian system on (as opposed to just ): One has a symplectic structure on that is naturally induced from the full Schrödinger dynamics as well as a Hamiltonian function on given as the expectation value of the Hamiltonian operator with respect to the Gaussian wave packet.
Upon normalization, (2) becomes
and its Wigner transform—called the Gaussian state Wigner function throughout the paper—is also a Gaussian defined on the phase space or the cotangent bundle :
where and are both in and is the covariance matrix defined as
Recently, Bonet-Luz and Tronci  discovered a non-canonical Poisson bracket that describes the dynamics of the Gaussian Wigner function
as a Hamiltonian system with the Hamiltonian function given by the expectation value
where is the Weyl symbol of the Hamiltonian operator .
These recent works  and  shed a new light on the dynamics of the Gaussian wave packet and the Gaussian Wigner function. In fact, these Hamiltonian formulations yield a slightly different form of equations for the phase space variable from those conventional results in the earlier literature mentioned above: The symplectic Gaussian wave packet dynamics in  yields a correction force term in the evolution equation for the momentum (see (12a) below), and the Hamiltonian dynamics of the Gaussian Wigner function in  also possesses a similar property. To put it differently, in the conventional work, the phase space variables evolves according to a classical Hamiltonian system and is decoupled from the dynamics of or ; as a result, the entire system is not Hamiltonian. In contrast,  and  recast the systems for and , respectively, as Hamiltonian systems along with the natural symplectic and Poisson structures and Hamiltonians. These formulations naturally give rise to correction terms as a result of the coupling.
Our main motivation is to unfold the geometry behind the relationship between the Hamiltonian dynamics systems for the variables and . Given that both systems are Hamiltonian and require modifications of the conventional picture, it is natural to expect a symplectic/Poisson-geometric connection between them.
1.3. Main Results and Outline
This paper exploits ideas from symplectic geometry to build a bridge between the above-mentioned recent works  and  on the Gaussian wave packet (4) and its Wigner transform (1.2). The main result, Theorem 3.2, states that the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space gives the covariance matrix (6) of the Gaussian state Wigner function. Its consequence, summarized in Corollary 4.1 in Section 4, is that the dynamics of the covariance matrix—under quadratic potentials—is a collective dynamics, and is hence given by the Lie–Poisson equation on the coadjoint orbits in . Finally, Section 5 generalizes this result to non-quadratic potentials by relating the geometry and dynamics of the Gaussian wave packets with those of the Gaussian state Wigner functions. Particularly, Proposition 5.1 relates the symplectic structure (and the Poisson bracket) found in  with the Poisson bracket found in , thereby establishing a geometric link between the two formulations. We also numerically demonstrate that our dynamics gives a better approximation to the dynamics of expectation values than the classical solutions do.
2. Hamiltonian Dynamics of Gaussian Wave Packet and Gaussian Wigner Function
2.1. Symplectic Structure and Gaussian Wave Packets
It is well known that the Gaussian wave packet (2) gives an exact solution to the time-dependent Schrödinger equation
coupled with additional equations for the rest of the variables .
The idea of reformulating this whole set of ODEs for as a Hamiltonian system has been around for quite a while; see, e.g., Pattanayak and Schieve , Faou and Lubich , and Lubich [29, Section II.4]. Ohsawa and Leok  built on these works from the symplectic-geometric point of view and came up with a Hamiltonian system on with an phase symmetry, and by applying the Marsden–Weinstein reduction  (see also Marsden et al. [32, Sections 1.1 and 1.2]), obtained the Hamiltonian system
on the reduced symplectic manifold that is equipped with the symplectic form
Note that we use Einstein’s summation convention throughout the paper unless otherwise stated. Given a Hamiltonian , (9) determines the vector field on ; in coordinates it is written as
where stands for the matrix whose -entry is . In our setting, it is natural to select the Hamiltonian as
where denotes the Hessian matrix of the potential . In fact, it is an approximation to the expectation value of the Hamiltonian operator . Then we have
This equation differs from those of Heller [21, 22] and Hagedorn [17, 18, 19, 20] by the correction term to the potential in the second equation; see Ohsawa and Leok  and  for the effects of this correction term.
The corresponding Poisson structure on is defined as follows: For any , let and be the corresponding Hamiltonian vector fields, i.e., and similarly for , then
where each of the brackets is calculated by holding the remaining variables (that are not involved in the bracket) fixed; we employ this convention throughout the paper to simplify the notation.
2.2. Lie–Poisson Structure for Gaussian Moments
The center and the covariance matrix (assumed to be positive definite) of the Gaussian Wigner function from (7) are given in terms of the first two moments of the Gaussian Wigner function (1.2), that is
where we have used the following expectation value notation with respect to as well as more general Wigner function : For an observable ,
In , the first two moments of the Wigner quasiprobability density were characterized as the momentum map corresponding to the action
of the Jacobi group , i.e., the semidirect product of the symplectic group and the Heisenberg group . This group structure has attracted some attention over the years mainly because of its relation to squeezed coherent states  and, more recently, because of its connections to certain integrable geodesic flows on the symplectic group [4, 23]. Here, the space of quasiprobability densities is equipped with a Poisson structure given by the following Lie–Poisson bracket on the space of the set of Wigner functions  (see Appendix A for more details):
whereas the Heisenberg group
is equipped with the multiplication rule
where is the standard symplectic form on , i.e., setting and ,
The semidirect product is defined in terms of the natural -action on , i.e., with ; as a result, the group multiplication for the Jacobi group is given by
In the context of Gaussian quantum states, the Jacobi group plays exactly the same role as in classical Liouville (Vlasov) dynamics , so that the momentum map structure of the first two Wigner moments has an identical correspondent in the classical case (one simply replaces the Wigner function by a Liouville distribution). More specifically, the momentum map corresponding to the action (18) is (see Appendix A for a verification)
Here, is the dual of the Lie algebra of , with and being the Lie algebras of and , respectively. The momentum map is equivariant and hence is Poisson (see, e.g., Marsden and Ratiu [30, Theorem 12.4.1]) with respect to the above and the -Lie–Poisson bracket on defined as follows: For any ,
where is the standard commutator on ; we also identified via the usual dot product. The differential is defined in terms of the natural dual pairing as follows: For any
where we identified with via the inner product on defined in (33) below. The other differentials denoted with are defined similarly.
Notice however that the image of the momentum map in (19) is not quite identified with those moments of interest from (16); in other words, is not a natural space in which those moments live. However, by exploiting the “untangling map” of Krishnaprasad and Marsden [26, Proposition 2.2] and the identification of with outlined in Section 3.2, the Lie–Poisson bracket (20) gives rise to the Poisson bracket
on ; this space is naturally identified as the space of the Gaussian moments . See Appendix B for the details of the derivation of the above Poisson bracket. With a Hamiltonian , we have
which are equivalent to (5.2) in Bonet-Luz and Tronci  (up to a sign misprint therein).
If the classical Hamiltonian is quadratic, then (22) with describes the time evolution of the Gaussian state Wigner function (1.2) corresponding to the dynamics of the Gaussian wave packet (4) as an exact solution to the corresponding Schrödinger equation. See Section 5 for the dynamics with non-quadratic Hamiltonians.
For Hamiltonians that are linear in , these equations recover the dynamics (12) and (13) in  (suitably specialized to Hermitian quantum mechanics). However, certain approximate models in chemical physics  make use of nonlinear terms in , as they are obtained by Gaussian moment closures of the type . One may perform such closures in the expression of the total energy to obtain the Hamiltonian of the form , and then can formulate, along with the Poisson bracket (21), the dynamics of as a Hamiltonian system.
3. Covariance Matrix as a Momentum Map
Our goal is to establish a link between the symplectic structure (10) or the Poisson bracket (13) on and the Poisson bracket (21) on . In this section, we focus on the correspondence between the second parts— and —of these constituents. The main result, Theorem 3.2 below, states that this link is made via the momentum map corresponding to the natural action of the symplectic group on the Siegel upper half space . We start off with a brief review of the geometry of in Section 3.1, and then after giving a brief account of the identification between and alluded above in Section 3.2, we state and prove the main result that the momentum map yields the covariance matrix (6) in Section 3.3.
3.1. Geometry of the Siegel Upper Half Space
It is well known that the Siegel upper half space defined in (3) is a homogeneous space; more specifically, we can show that
where is the unitary group of degree ; see Siegel  and also Folland [10, Section 4.5] and McDuff and Salamon [33, Exercise 2.28 on p. 48]. To see this, let us first rewrite the definition of using block matrices consisting of submatrices, i.e.,
and define the (left) action of on by the generalized linear fractional transformation
This action is transitive: By choosing
which is easily shown to be symplectic, we have
The isotropy subgroup of the element is given by
where is the orthogonal group of degree ; however is identified with as follows:
Hence and thus . We may then construct the corresponding quotient map as follows:
or more explicitly,
where can be shown to be invertible if . Let be the left multiplication by , i.e., for any . Then it is easy to see that
or the diagram below commutes, i.e., defined in (24) is in fact a left action.
3.2. Symplectic Algebra and Lie Algebra of Symmetric Matrices
The following bracket renders the space of symmetric real matrices a Lie algebra:
We then identify the symplectic algebra with via the following “tilde map”:
where , i.e., it is a real matrix, and . So writing , the identification (31) is written explicitly in terms of the block components as follows:
In fact, it is easy to see that this is a Lie algebra isomorphism: Let be the standard Lie bracket of ; then, for any ,
One may also define inner products on both spaces as follows:
and so we may identify their dual spaces with themselves. As a result, we have
The above inner products are compatible with the identification via the tilde map (31) in the sense that . Therefore, in what follows, we exploit the tilde map identification (31) to write elements in , , and as the corresponding ones in to simplify calculations.
Recall that the symplectic group acts on its Lie algebra via the adjoint action, i.e., for any and , the adjoint action
With an abuse of notation, one may define the corresponding action
Hence the corresponding action on the dual is given by
One then sees easily that, for any , the corresponding is compatible with the Lie bracket (30), i.e.,
and then its adjoint is given by
The coadjoint action (34) defines the coadjoint orbit
for each ; it is well known that is equipped with the following -Kirillov–Kostant–Souriau (KKS) symplectic structures: For any and any ,
The -Lie–Poisson structure on that is compatible with the above KKS symplectic form is given by
3.3. Momentum Map on the Siegel Upper Half Space
Recall that the symplectic group acts on the Siegel upper half space transitively by the action shown in (24). The main ingredient of the paper is the momentum map
corresponding to this action: Let and be its infinitesimal generator (recall that we identify with ), i.e.,
Then is characterized by
for any .
Now our main result is the following:
The image of is the covariance matrix in the Gaussian state Wigner function (1.2), i.e.,
is an equivariant momentum map, i.e., for any ,
Let us first find an expression for the momentum map . Set ; then, using the expression (24) for and writing , we have