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American Journal of Physics

Expansion coefficients of a squeezed coherent state in the number state basis

Jiaxin Gong, P. K. Aravind

Published Oct 1, 1990• 30 citations
Publisher

Abstract

An analytic expression is obtained for the expansion coefficients of a single-mode squeezed coherent state in the number state (or Fock) basis. This is done by deriving a set of coupled recursion relations for the coefficients that are solved in closed form. The result coincides with that of Stoler and Yuen but the present method has the advantage (for neophytes) that it does not involve any sophisticated operator ordering techniques. Thus the present approach is suitable for inclusion in a course on quantum mechanics dealing with squeezed states.

DOI
10.1119/1.16337

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An analytic expression is obtained for the expansion coefficients of a single-mode squeezed coherent state in the number state (or Fock) basis. This is done by deriving a set of coupled recursion relations for the coefficients that are solved in closed form. The result coincides with that of Stoler and Yuen but the present method has the advantage (for neophytes) that it does not involve any sophisticated operator ordering techniques. Thus the present approach is suitable for inclusion in a course on quantum mechanics dealing with squeezed states.
k:th power (amplitude-)squeezed states are defined as the normalized states giving equality in the Schroedinger-Robertson uncertainty relation for the real and imaginary parts of the k:th power of the one-mode annihilation operator. Equivalently they are the set of normalized eigenstates (for all possible complex eigenvalues) of the Bogolubov transformed "k:th power annihilation operators". Expressed in the number representation the eigenvalue equation leads to a three term recursion relation for the expansion coefficients, which can be explicitly solved in the cases k = 1, 2. The solutions are essentially Hermite and Pollaczek polynomials, respectively. k = 1 gives the ordinary squeezed states, i.e. displaced squeezed vacua. For k equal to or larger than three, where no explicit solution has been found, the recursion relation for the symmetric operator given by the real part of the k:th power of the annihilation operator defines a Jacobi matrix corresponding to a classical Hamburger moment problem, which is undetermined. This implies that the operator has an infinity of self-adjoint extensions, all with disjoint discrete spectra. The corresponding squeezed states are well-defined, however.
Two-mode squeezing is central to entangled-photon generation and nonlinear interferometry, yet standard perturbative low-gain treatments and Gaussian formalisms can obscure the interference of photon-number amplitudes, especially in nonlinear interferometers and at high gain. Here we derive a closed-form Fock-basis expression for the action of the two-mode squeezing operator on arbitrary number states, enabling the direct analysis of nonlinear interferometers in the photon-number basis at arbitrary squeezing strength. Within this framework, we provide intuitive physical interpretations of several known quantum-interference effects and identify a new multi-photon interference phenomenon in a four-crystal geometry that could readily be observed in laboratories. Our work provides a compact analytic toolkit and explicit design rules for engineering multi-photon interference, with applications in quantum sensing, precision metrology, and quantum state generation.
We examine the generalized squeezed states defined as eigenstates of a linear combination of the lowering and raising operators a2{\mathrm{a}}^{2}a2 and (${\mathrm{a}}^{\mathrm{\ifmmode\dagger\else\textdagger\fi{}}}$)2{)}^{2})2, respectively. This approach is entirely equivalent to the minimum-uncertainty method applied to the amplitude-squared operators. We solve the eigenvalue equation in Glauber's coherent-state representation and find two independent solutions. Their Fock-state expansions, one containing only even and the other only odd number states, reveal a strong nonclassical character. We show that the calculation of the mean photon number is sufficient to obtain the expectation values of interest. Consequently, photon statistics is investigated in both cases by using the generating function of the photon-number distribution. We find the conditions under which the second-order squeezed states display photon antibunching and quadrature squeezing. Also discussed is the preservation of their amplitude-squared squeezing by linear amplification at gains exceeding 2. Analytically, our results are simple formulas in terms of Kummer and Gauss hypergeometric functions that allow straightforward numerical calculations.
The Fock expansion of multimode pure Gaussian states is derived starting from their representation as displaced and squeezed multimode vacuum states. The approach is new and appears to be simpler and more general than previous ones starting from the phase-space representation given by the characteristic or Wigner function. Fock expansion is performed in terms of easily evaluable two-variable Hermite–Kampé de Fériet polynomials. A relatively simple and compact expression for the joint statistical distribution of the photon numbers in the different modes is obtained. In particular, this result enables one to give a simple characterization of separable and entangled states, as shown for two-mode and three-mode Gaussian states.
Squeezed state is theoretically obtained by performing the squeezing operator on an initial state.Due to non-commutable operators each other,it is not easy to directly obtain the explicit expression of squeezed state.Here,a general approach is provided for deriving the squeezed state′s explicit form,namely,by virtue of the Weyl-Wigner correspondence scheme the explicit forms of single-mode and two-mode squeezed state is obtained,respectively.Without disentangling the squeezing operator,this adopted method is applicable to explicitly solve the general multimode case.