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arXiv:1109.2171 · Journal of Physics A Mathematical and Theoretical

Almost complete coherent state subsystems and partial reconstruction of wavefunctions in the Fock–Bargmann phase-number representation

M Calixto, J Guerrero, J C Sánchez-Monreal

Published May 30, 2012• 3 citations
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Abstract

We provide (partial) reconstruction formulas and discrete Fourier transforms for wave functions in standard Fock-Bargmann (holomorphic) phase-number representation from a finite number $N$ of phase samples $\{θ_k=2πk/N\}_{k=0}^{N-1}$ for a given mean number $p$ of particles. The resulting Coherent State (CS) subsystem ${\cal S}=\{|z_k=p^{1/2}e^{iθ_k}>\}$ is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact. For an unbounded number of particles, ${\cal S}$ is "almost complete" (a \textit{pseudo-frame}) and partial reconstruction formulas are provided along with an study of the accuracy of the approximation, which tends to be exact when $p

DOI
10.1088/1751-8113/45/24/244029

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We provide (partial) reconstruction formulas and discrete Fourier transforms for wave functions in standard Fock-Bargmann (holomorphic) phase-number representation from a finite number NNN of phase samples {θk=2πk/N}k=0N−1\{θ_k=2πk/N\}_{k=0}^{N-1}{θk​=2πk/N}k=0N−1​ for a given mean number ppp of particles. The resulting Coherent State (CS) subsystem S={∣zk=p1/2eiθk>}{\cal S}=\{|z_k=p^{1/2}e^{iθ_k}>\}S={∣zk​=p1/2eiθk​>} is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact. For an unbounded number of particles, S{\cal S}S is "almost complete" (a \textit{pseudo-frame}) and partial reconstruction formulas are provided along with an study of the accuracy of the approximation, which tends to be exact when $p
We provide (partial) reconstruction formulas and discrete Fourier transforms for wave functions in standard Fock-Bargmann (holomorphic) phase-number representation from a finite number N of phase samples {θ k = 2πk/N } N -1 k=0 for a given mean number p of particles.The resulting Coherent State (CS) subsystem S = {|z k = p 1 2 e iθ k } is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact.For an unbounded number of particles, S is "almost complete" (a pseudo-frame) and partial reconstruction formulas are provided along with an study of the accuracy of the approximation, which tends to be exact when p < N and/or N → ∞.
We investigate the completeness of coherent states subsystems with respect to several classes of states on rational lattices. Our main results show that the phase-space shifts of every finite linear combination of Hermite functions with respect to a rational lattice are complete in L^2(R), thus generalizing a remark of von Neumann (and proved by Bargmann, Perelomov et al.). An analogous result is proven for windows that factor into certain rational functions and the Gaussian. The results are also interesting from a conceptual point of view since they show a vast difference between the completeness and the frame property of a discrete set of coherent states.
In a first part we give a brief presentation of general Fock space setting to describe quantum field theory. Bosons are quantum particles with integer spin and have symmetric wave functions; Fermions are quantum particles with half-integer spin and are represented by anti-symmetric wave functions. The functional setting is given by a symmetric or anti-symmetric tensor product of Hilbert spaces. We describe these spaces and transformations between these spaces. We shall follow the references [23, 42]. Coherent states are defined by translating the vacuum states with the Weyl operators. This is easily done here for bosons. We shall see in the next chapter how to deal with fermions. In a second part we give an interesting application of bosonic coherent states to the study of the classical limit as ℏ↘0\hbar \searrow 0ℏ↘0 of non-relativistic boson systems with two body interaction in the neighborhood of a solution of the classical system (here the Hartree equation). The classical limit corresponds here to the mean-field limit as the number of particles goes to infinity. As we have done for finite systems, we use here Hepp’s method, which is a linearization procedure of the quantum Hamiltonian around the classical field. The fluctuations around this solution are controlled by a purely quadratic Hamiltonian. In a series of several important papers [114–116] Ginibre and Velo have proven an asymptotic expansion and remainder estimates for these quantum fluctuations. Finally, following the paper [225] one can show that, in the limit ℏ→0\hbar \rightarrow 0ℏ→0, the marginal distribution of the time-evolved coherent states tends in trace-norm to the projector onto the solution of the classical fields equation (Hartree equation) with a uniform remainder estimates in time.
We define a discrete Bargmann transform for discrete and finite functions by means of the coherent states in the su(2) finite harmonic oscillator model. The transform space is over a corresponding finite square mesh in the complex plane. From there, the inverse discrete Bargmann transform reconstitutes the original function with an average error of 10−7 for functions of unit norm. Three defining properties of the Bargmann transform also hold in this discrete version: (1) the Bargmann transform of finite su(2) harmonic oscillator functions are complex power functions; (2) the discrete su(1,1) repulsive oscillator functions self-reproduce in the chosen finite interval with little error; and (3) the oscillator dynamics under the fractional Fourier transform in Bargmann’s complex space correspond, in this discrete transform, to the dynamics under the su(2) fractional Fourier–Kravchuk transform.
Let Γ(H)\Gamma(\mathcal{H})Γ(H) be the boson Fock space over a finite dimensional Hilbert space H\mathcal{H}H. It is shown that every gaussian symmetry admits a Klauder-Bargmann integral representation in terms of coherent states. Furthermore, gaussian symmetries, gaussian states and second quantization contractions, all of these operators belong to a weakly closed, selfadjoint semigroup E2(H)\mathcal{E}_2(\mathcal{H})E2​(H) of bounded operators in Γ(H)\Gamma(\mathcal{H})Γ(H). This yields, a new parametrization of gaussian states, which is a very fruitful alternative to the customary parametrization by position-momentum mean vectors and covariance matrices. This leads to a rich harvest of corollaries: (i) every gaussian state ρ\rhoρ admits a factorization ρ=Z1†Z1\rho = Z_{1}^{\dagger}Z_{1}ρ=Z1†​Z1​, where Z1Z_{1}Z1​ is an element of E2(H)\mathcal{E}_2(\mathcal{H})E2​(H) and has the form Z1=cΓ(Λ)exp⁡∑r=1nλrar+∑r,s=1nαrsarasZ_{1} = \sqrt{c}\Gamma(\sqrt{\Lambda})\exp{\sum_{r=1}^{n} \lambda_ra_r+\sum_{r,s=1}^{n} \alpha_{rs}a_{r}a_{s}}Z1​=c​Γ(Λ​)exp∑r=1n​λr​ar​+∑r,s=1n​αrs​ar​as​ on the dense linear manifold generated by all exponential vectors, Λ\LambdaΛ being a positive operator in H\mathcal{H}H, ar,1≤r≤na_{r}, 1\leq r \leq nar​,1≤r≤n are the annihilation operators corresponding to the nnn different modes in Γ(H)\Gamma(\mathcal{H})Γ(H), λr∈C\lambda_r\in \mathbb{C}λr​∈C and [αrs][\alpha_{rs}][αrs​] is a symmetric matrix in Mn(C)M_n(\mathbb{C})Mn​(C); (ii) an explicit particle basis expansion of an arbitrary mean zero pure gaussian state vector along with a density matrix formula for a general gaussian state in terms of its E2(H)\mathcal{E}_2(\mathcal{H})E2​(H)-parameters; (iii) an easy test for the entanglement of pure gaussian states and a class of examples of pure nnn-mode gaussian states which are completely entangled; (iv) tomography of an unknown gaussian state in Γ(Cn)\Gamma(\mathbb{C}^n)Γ(Cn) by the estimation of its E2(Cn)\mathcal{E}_2(\mathbb{C}^n)E2​(Cn)-parameters using O(n2)O(n^2)O(n2) measurements with a finite number of outcomes.