Coherent states in mathematical physics explained

Coherent states have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see also^[1]). However, they have generated a huge variety of generalizations, which have led to a tremendous amount of literature in mathematical physics.In this article, we sketch the main directions of research on this line. For further details, we refer to several existing surveys.^{[2] [3] [4]}

A general definition

Let

ak H be a complex, separable Hilbert space, X a locally compact space and d\nu a measure on X . For each x in X , denote |x\rangle a vector in ak H . Assume that this set of vectors possesses the following properties:

1. The mapping

x \mapsto | x \rangle is weakly continuous, i.e., for each vector |\psi\rangle in ak H , the function \Psi (x) = \langle x| \psi\rangle is continuous (in the topology of X ).

1. The resolution of the identity $$\backslash int\_X\; |\; x\backslash rangle\backslash langle\; x|\backslash ,\; d\backslash nu\; (x)\; =\; I\_$$ holds in the weak sense on the Hilbert space

ak H , i.e., for any two vectors | \phi\rangle , | \psi \rangle in ak H , the following equality holds: $$\backslash int\_X\; \backslash langle\backslash phi|\; x\backslash rangle\backslash langle\; x|\backslash psi\backslash rangle\backslash ,\; d\backslash nu\; (x)\; =\; \backslash langle\backslash phi|\backslash psi\backslash rangle\backslash ,.$$A set of vectors | x\rangle satisfying the two properties above is called a family of generalized coherent states. In order to recover the previous definition (given in the article Coherent state) of canonical or standard coherent states (CCS), it suffices to take X\equiv C , the complex plane and $d\backslash nu\; (x)\; \backslash equiv\; \backslash frac\; d^2x.$

Sometimes the resolution of the identity condition is replaced by a weaker condition, with the vectors

| x \rangle simply forming a total set in { ak H} and the functions \Psi (x) = \langle x | \psi\rangle , as| \psi \rangle runs through { ak H} , forming a reproducing kernel Hilbert space.The objective in both cases is to ensure that an arbitrary vector | \psi \rangle be expressible as a linear (integral) combination of these vectors. Indeed, the resolution of the identity immediately implies that$$|\; \backslash psi\; \backslash rangle\; =\; \backslash int\_X\; \backslash Psi\; (x)\; \backslash left|\; x\; \backslash right\backslash rangle\backslash ,\; d\backslash nu\; (x)\backslash ,,$$where \Psi (x) = \langle x | \psi\rangle .

These vectors

\Psi are square integrable, continuous functions on X and satisfy the reproducing property$$\backslash int\_X\; K\; (x,y)\backslash Psi\; (y)\backslash ,\; d\backslash nu\; (y)\; =\; \backslash Psi\; (x)\backslash ,,$$where K (x, y) = \langle x | y \rangle is the reproducing kernel, which satisfies the following properties

Some examples

We present in this section some of the more commonly used types of coherent states, as illustrations of the general structure given above.

Nonlinear coherent states

A large class of generalizations of the CCS is obtained by a simple modification of their analytic structure. Let

\varepsilon _{1} \leq \varepsilon _{2} \leq ... \leq \varepsilon _{n} \leq … be an infinite sequence of positive numbers (\varepsilon _{1} ≠ 0 ). Define \varepsilon _{n} ! = \varepsilon _{1} \varepsilon _{2} \ldots \varepsilon _{n} and by convention set \varepsilon _{0} ! = 1 . In the same Fock space in which the CCS were described, we now define the related deformed or nonlinear coherent states by the expansion

$$\backslash vert\; \backslash alpha\backslash rangle\; =\; (\backslash vert\; \backslash alpha\backslash vert^2)^\; \backslash ,\; \backslash sum\_^\backslash infty\; \backslash frac\; |\; n\backslash rangle\backslash ,\; .$$The normalization factor

{ l N}(\vert \alpha\vert ^{2)} is chosen so that \langle \alpha \vert \alpha \rangle = 1 . These generalized coherent states are overcomplete in the Fock space and satisfy a resolution of the identity$$\backslash int\_\; \backslash vert\; \backslash alpha\; \backslash rangle\backslash langle\; \backslash alpha\; \backslash vert\backslash ;\; (\backslash vert\; \backslash alpha\backslash vert^2)\backslash ;\; d\backslash nu\; (\backslash alpha,\; \backslash overline)\; =\; I\backslash ,,$$l D being an open disc in the complex plane of radius L , the radius of convergence of the series$\backslash sum\_^\backslash infty\; \backslash frac$ (in the case of the CCS, L = inf ty .) The measure d\nu is generically of the form d\thetad λ (r) (for \alpha = re ^{i\theta} ), where d λ is related to the \varepsilon _{n} ! through the moment condition.

Once again, we see that for an arbitrary vector

| \phi\rangle in the Fock space, the function \Phi(\alpha) = \langle \phi | \alpha\rangle is of the form \Phi(\alpha) = { l N}(\vert \alpha\vert ^{2)} ^{- (12)/()} f(\alpha) , where f is an analytic function on the domain l D . The reproducing kernel associated to these coherent states is$$K(\backslash overline,\; \backslash alpha\text{'})\; =\; \backslash langle\; \backslash alpha|\; \backslash alpha\text{'}\backslash rangle\; =\; \backslash left[\{\backslash mathcal\; N\}(\backslash vert\; \backslash alpha\backslash vert^2)\; \{\backslash mathcal\; N\}(\backslash vert\; \backslash alpha\text{'}\backslash vert^2)\backslash right]^\; \backslash sum\_^\backslash infty\; \backslash frac\; \backslash ,\; .$$

Barut–Girardello coherent states

A by its action on the vectors | \alpha\rangle ,$$A|\; \backslash alpha\; \backslash rangle\; =\; \backslash alpha|\; \backslash alpha\; \backslash rangle\backslash ,,$$and its adjoint operator A ^{\dagger} . These act on the Fock states | n\rangle as$$A\; |\; n\backslash rangle\; =\; \backslash sqrt|\; n\; -1\; \backslash rangle\backslash ,,\; \backslash qquad\; A^\backslash dagger|\; n\; \backslash rangle\; =\; \backslash sqrt|\; n+1\; \backslash rangle\backslash ,\; .$$Depending on the exact values of the quantities \varepsilon _{n} , these two operators, together with the identity I and all their commutators, could generate a wide range of algebras including various types of deformed quantum algebras. The term 'nonlinear', as often applied to these generalized coherent states, comes again from quantum optics where many such families of states are used in studying the interaction between the radiation field and atoms, where the strength of the interaction itself depends on the frequency of radiation. Of course, these coherent states will not in general have either the group theoretical or the minimal uncertainty properties of the CCS (they might have more general ones).

Operators

A and A ^{\dagger} of the general type defined above are also known as ladder operators . When such operators appear as generators of representations of Lie algebras, the eigenvectors of A are usually called Barut–Girardello coherent states.^[5] A typical example is obtained from the representations of the Lie algebra of SU(1,1) on the Fock space.

Gazeau–Klauder coherent states

A non-analytic extension of the above expression of the non-linear coherent states is often used to define generalized coherent states associated to physical Hamiltonians having pure point spectra. These coherent states, known as Gazeau–Klauder coherent states, are labelled by action-angle variables.^[6] Suppose that we are given the physical Hamiltonian $H\; =\; \backslash sum\_^\backslash infty\; E\_n\; \backslash left|\; n\; \backslash right\backslash rangle\; \backslash left\backslash langle\; n\backslash right|$, with

E _{0} = 0 , i.e., it has the energy eigenvalues E _{n} and eigenvectors | n\rangle , which we assume to form an orthonormal basis for the Hilbert space of states { ak H} . Let us write the eigenvalues as E _{n} = \omega\varepsilon _{n} by introducing a sequence of dimensionless quantities \{\varepsilon _{n\}} ordered as: 0 = \varepsilon _{0} < \varepsilon _{1} < \varepsilon _{2} < … . Then, for all J \geq 0 and \gamma \in R , the Gazeau–Klauder coherent states are defined as

$$|J,\backslash gamma\backslash rangle\; =\backslash mathcal\; N(J)^\backslash sum\_^\backslash infty\backslash ,\backslash frac|n\backslash rangle\backslash ,,$$where again

l N is a normalization factor, which turns out to be dependent on J only. These coherent states satisfy the temporal stability condition,

$$e^\backslash vert\; J,\; \backslash gamma\; \backslash rangle\; =\; \backslash vert\; J,\; \backslash gamma\; +\; \backslash omega\; t\; \backslash rangle\backslash ,,$$and the action identity,$$\backslash langle\; J,\; \backslash gamma\; |\; H|\; J,\; \backslash gamma\; \backslash rangle\_\; =\; \backslash omega\; J\backslash ,\; .$$While these generalized coherent states do form an overcomplete set in

{ ak H} , the resolution of the identity is generally not given by an integral relation as above, but instead by an integral in Bohr's sense, like it is in use in the theory of almost periodic functions.

Actually the construction of Gazeau–Klauder CS can be extended to vector CS and to Hamiltonians with degenerate spectra, as shown by Ali and Bagarello.^[7]

Heat kernel coherent states

Another type of coherent state arises when considering a particle whose configuration space is the group manifold of a compact Lie group K. Hall introduced coherent states in which the usual Gaussian on Euclidean space is replaced by the heat kernel on K.^[8] The parameter space for the coherent states is the "complexification" of ; e.g., if is then the complexification is . These coherent states have a resolution of the identity that leads to a Segal-Bargmann space over the complexification. Hall's results were extended to compact symmetric spaces, including spheres, by Stenzel.^{[9] [10]} The heat kernel coherent states, in the case

K = SU ( 2 ) , have been applied in the theory of quantum gravity by Thiemann and his collaborators.^[11] Although there are two different Lie groups involved in the construction, the heat kernel coherent states are not of Perelomov type.

The group-theoretical approach

Gilmore and Perelomov, independently, realized that the construction of coherent states may sometimes be viewed as a group theoretical problem.^{[12] [13] [14] [15] [16]}

In order to see this, let us go back for a while to the case of CCS. There, indeed, the displacement operator

D(\alpha) is nothing but the representative in Fock space of an element of the Heisenberg group (also called the Weyl–Heisenberg group), whose Lie algebra is generated by X, P and I . However, before going on with the CCS, take first the general case.

Let

G be a locally compact group and suppose that it has a continuous, irreducible representation U on a Hilbert space ak H by unitary operators U(g), g \in G . This representation is called square integrable if there exists a non-zero vector | \psi \rangle in ak H for which the integral$$c(\backslash psi)\; =\; \backslash int\_G\; \backslash vert\backslash langle\backslash psi\; |\; U(g)\backslash psi\backslash rangle\backslash vert^2\backslash ,\; d\backslash mu\; (g)$$converges. Here d\mu is the left invariant Haar measure on G . A vector | \psi\rangle for which c(\psi) < inf ty is said to be admissible, and it can be shown that the existence of one such vector guarantees the existence of an entire dense set of such vectors in ak H . Moreover, if the group G is unimodular, i.e., if the left and the right invariant measures coincide, then the existence of one admissible vector implies that every vector in ak H is admissible. Given a square integrable representation U and an admissible vector |\psi\rangle , let us define the vectors

$$|\; g\backslash rangle\; =\; \backslash frac\; 1\backslash ,\; U(g)|\; \backslash psi\; \backslash rangle,\; \backslash text\; g\backslash in\; G.$$These vectors are the analogues of the canonical coherent states, written there in terms of the representation of the Heisenberg group (however, see the section on Gilmore-Perelomov CS, below). Next, it can be shown that the resolution of the identity$$\backslash int\_G\; \backslash left|\; g\; \backslash right\backslash rangle\; \backslash left\backslash langle\; g\backslash right|\; \backslash ,\; d\backslash mu\; (g)\; =\; I\_$$holds on

{ ak H} . Thus, the vectors | g\rangle constitute a family of generalized coherent states. The functions F(g) = \langle g| \phi\rangle for all vectors | \phi \rangle in ak H are square integrable with respect to the measure d\mu and the set of such functions, which in fact are continuous in the topology of G , forms a closed subspace of L ^{2} (G, d\mu ) . Furthermore, the mapping \phi \mapsto F is a linear isometry between ak H and L ^{2} (G, d\mu ) and under this isometry the representation U gets mapped to a subrepresentation of the left regular representation of G on L ^{2} (G, d\mu) .

An example: wavelets

A typical example of the above construction is provided by the affine group of the line,

G _{Aff} . This is the group of all 2×2 matrices of the type,a and b being real numbers with a ≠ 0 . We shall also write g = (b,a) , with the action on R given by (b,a) ⋅ x = b + ax . This group is non-unimodular, with the left invariant measure being given by d\mu (b,a ) = a ^{-2} db da (the right invariant measure being a ^{-1} db da ). The affine group has a unitary irreducible representation on the Hilbert space L ^{2} ( R, dx ) . Vectors in L ^{2} ( R, dx ) are measurable functions \varphi(x) of the real variable x and the (unitary) operators U(b,a) of this representation act on them as$$(U(b,a)\backslash varphi)(x)\; =\; \backslash frac\; 1\backslash ,\backslash varphi\; \backslash left(\backslash frac\; a\backslash right)\; =\; \backslash frac\; 1\backslash ,\backslash varphi\; \backslash left((b,a)^\backslash cdot\; x\backslash right)\backslash ,\; .$$If \psi is a function in L ^{2} ( R, dx ) such that its Fourier transform\widehat\psi satisfies the (admissibility) conditionthen it can be shown to be an admissible vector, i.e.,Thus, following the general construction outlined above, the vectors$$|\; b,\; a\backslash rangle\; =\; \backslash frac\; 1\; \backslash ,\; U(b,a)\backslash psi\backslash ,,\; \backslash qquad\; (b,a)\; \backslash in\; G\_$$define a family of generalized coherent states and one has the resolution of the identity$$\backslash int\_\; \backslash left|\; b,\; a\backslash right\backslash rangle\; \backslash left\backslash langle\; b,a\backslash right|\; \backslash frac\; =\; I$$on L ^{2} ( R, dx ) .In the signal analysis literature, a vector satisfying the admissibility condition above is called a mother wavelet and the generalized coherent states | b, a\rangle are called wavelets. Signals are then identified with vectors | \varphi \rangle in L ^{2} ( R, dx ) and the function$$F(b,a)\; =\; \backslash langle\; b,a|\; \backslash varphi\backslash rangle\backslash ,,$$is called the continuous wavelet transform of the signal \varphi .^{[17] [18]}

This concept can be extended to two dimensions, the group

G _{Aff} being replaced by the so-called similitude group of the plane, which consists of plane translations, rotations and global dilations. The resulting 2D wavelets, and some generalizations of them, are widely used in image processing.^[19]

Gilmore–Perelomov coherent states

The construction of coherent states using group representations described above is not sufficient. Already it cannot yield the CCS, since these are not indexed by the elements of the Heisenberg group, but rather by points of the quotient of the latter by its center, that quotient being precisely

R ^{2} . The key observation is that the center of the Heisenberg group leaves the vacuum vector | 0 \rangle invariant, up to a phase. Generalizing this idea, Gilmore and Perelomov^{[12] [13] [14] [15]} consider a locally compact group G and a unitary irreducible representation U of G on the Hilbert space ak H , not necessarily square integrable. Fix a vector |\psi\rangle in { ak H} , of unit norm, and denote by H the subgroup of G consisting of all elements h that leave it invariant up to a phase, that is,$$U(h)\; \backslash left|\; \backslash psi\; \backslash right|\backslash rangle\; =\; e^\; \backslash left|\backslash psi\backslash right\backslash rangle\backslash ,,$$where \omega is a real-valued function of h . Let X = G/H be the left coset space andx an arbitrary element in X . Choosing a coset representative g(x) \in G , for each coset x , we define the vectors$$|\; x\backslash rangle\; =\; U(g(x))\; \backslash left|\backslash psi\backslash right\backslash rangle\; \backslash in\; .$$The dependence of these vectors on the specific choice of the coset representative g(x) is only through a phase. Indeed, if instead of g(x) , we took a different representative g(x)' \in G for the same coset x , then since g(x)' = g(x)h for some h \in H , we would have U(g(x)')| \psi\rangle = e ^{i\omega} | x\rangle . Hence, quantum mechanically, both | x\rangle and U(g(x)') | \psi\rangle represent the same physical state and in particular, the projection operator \left| x\right\rangle \left\langle x \right| depends only on the coset. Vectors | x\rangle defined in this way are called Gilmore–Perelomov coherent states. Since U is assumed to be irreducible, the set of all these vectors as x runs through G/H is dense in ak H . In this definition of generalized coherent states, no resolution of the identity is postulated. However, if X carries an invariant measure, under the natural action of G , and if the formal operator B defined as$$B\; =\; \backslash int\_X\; \backslash left|\; x\backslash right\backslash rangle\; \backslash left\backslash langle\; x\backslash right|\; d\backslash mu\; (x)\backslash ,,$$is bounded, then it is necessarily a multiple of the identity and a resolution of the identity is again retrieved.

Gilmore–Perelomov coherent states have been generalized to quantum groups, but for this we refer to the literature.^{[20] [21] [22] [23] [24] [25]}

Further generalization: Coherent states on coset spaces

The Perelomov construction can be used to define coherent states for any locally compact group. On the other hand, particularly in case of failure of the Gilmore–Perelomov construction, there exist other constructions of generalized coherent states, using group representations, which generalize the notion of square integrability to homogeneous spaces of the group.^{[2] [3]}

Briefly, in this approach one starts with a unitary irreducible representation

U and attempts to find a vector | \psi\rangle , a subgroup H and a section \sigma : G/H \to G such that$$\backslash int\_\; \backslash left|\; x\backslash right\backslash rangle\; \backslash left\backslash langle\; x\backslash right|\; d\backslash mu\; (x)\; =\; T\backslash ,,$$where | x\rangle = U(\sigma (x)) \left| \psi \right\rangle , T is a bounded, positive operator with bounded inverse and d\mu is a quasi-invariant measure on X = G/H . It is not assumed that |\psi\rangle be invariant up to a phase under the action of H and clearly, the best situation is when T is a multiple of the identity. Although somewhat technical, this general construction is of enormous versatility for semi-direct product groups of the type R ^{n} \rtimes K , where K is a closed subgroup of GL(n, R ) . Thus, it is useful for many physically important groups, such as the Poincaré group or the Euclidean group, which do not have square integrable representations in the sense of the earlier definition. In particular, the integral condition defining the operator T ensures that any vector | \phi \rangle in ak H can be written in terms of the generalized coherent states | x\rangle namely,$$|\; \backslash phi\; \backslash rangle\; =\; \backslash int\_X\; \backslash Psi\; (x)\; |\; x\backslash rangle\; \backslash ,\; d\backslash mu\; (x)\backslash ,,\; \backslash qquad\; \backslash Psi\; (x)\; =\; \backslash langle\; x\; |\; T^\backslash phi\; \backslash rangle\; \backslash ,,$$which is the primary aim of any kind of coherent states.

Coherent states: a Bayesian construction for the quantization of a measure set

We now depart from the standard situation and present a general method of construction of coherent states, starting from a few observations on the structure of these objects as superpositions of eigenstates of some self-adjoint operator, as was the harmonic oscillator Hamiltonian for the standard CS. It is the essence of quantum mechanics that this superposition has a probabilistic flavor. As a matter of fact, we notice that the probabilistic structure of the canonical coherent states involves two probability distributions that underlie their construction. There are, in a sort of duality, a Poisson distribution ruling the probability of detecting

n excitations when the quantum system is in a coherent state |z\rangle , and a gamma distribution on the set C of complex parameters, more exactly on the range R ^{+} of the square of the radial variable. The generalization follows that duality scheme. Let X be a set of parameters equipped with a measure \mu and its associated Hilbert space L ^{2(X,} d\mu) of complex-valued functions, square integrable with respect to \mu . Let us choose in L ^{2(X,} d\mu) a finite or countable orthonormal set l{O}=\{\phi _{n} , n = 0 , 1 , ... \} :$$\backslash langle\; \backslash phi\_m\; |\; \backslash phi\_n\; \backslash rangle\; =\; \backslash int\_\backslash overline\backslash ,\; \backslash phi\_n(x)\backslash ,\; d\backslash mu(x)\; =\; \backslash delta\_\backslash ,\; .$$In case of infinite countability, this set must obey the (crucial) finiteness condition:Let ak{H} be a separable complex Hilbert space with orthonormal basis \{|e _{n\rangle} , n = 0 , 1 , ... \} in one-to-one correspondence with the elements of l{O} . The two conditions above imply that the family of normalized coherent states l{F} _{ak{H} }= \ in ak{H} , which are defined by$$|x\backslash rangle\; =\; \backslash frac\backslash sum\_n\; \backslash overline\backslash ,\; |e\_n\backslash rangle\backslash ,,$$resolves the identity in ak{H} :$$\backslash int\_X\; d\backslash mu(x)\; \backslash ,\backslash mathcal(x)\; \backslash left|x\backslash right\backslash rangle\; \backslash left\backslash langle\; x\; \backslash right|\; =\; I\_\backslash ,\; .$$Such a relation allows us to implement a coherent state or frame quantization of the set of parameters X by associating to a function X \ni x \mapsto f(x) that satisfies appropriate conditions the following operator in ak{H} :$$f(x)\; \backslash mapsto\; A\_f\; :=\; \backslash int\_X\; \backslash mu(dx)\; \backslash ,\backslash mathcal(x)\; \backslash ,\; f(x)\; \backslash left|x\backslash right\backslash rangle\; \backslash left\backslash langle\; x\; \backslash right|\; .$$The operator A _{f} is symmetric if f(x) is real-valued, and it is self-adjoint (as a quadratic form) if f(x) is real and semi-bounded. The original f(x) is an upper symbol, usually non-unique, for the operator A _{f} . It will be called a classical observable with respect to the family l{F} _{ak{H} } if the so-called lower symbol of A _{f} , defined as$$\backslash check\; (x)\; :=\; \backslash langle\; x\; |\; A\_f\; |\; x\; \backslash rangle\; =\; \backslash int\_X\backslash mu(dx\text{'})\; \backslash ,\backslash mathcal(x\text{'})\; \backslash ,\; f(x\text{'})\backslash ,\; \backslash vert\backslash langle\; x|\; x\text{'}\backslash rangle\backslash vert^2\; \backslash ,\; .$$has mild functional properties to be made precise according to further topological properties granted to the original set X .A last point of this construction of the space of quantum states concerns its statistical aspects.There is indeed an interplay between two probability distributions:

See also

• Coherent states
• Lieb conjecture
• Quantization

References

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