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Journal of Physics Conference Series

Higher-order amplitude squeezing

Evgeny V Shchukin, W. Vogel

Published Apr 1, 2006• 3 citations
Publisher

Abstract

A brief review of nonclassicality conditions in terms of moments of the creation and annihilation operators is given. By introducing k -th power amplitude squeezing, the notions of ordinary quadrature squeezing and amplitude-squared squeezing are generalized. Minimum uncertainty states are considered as a special class of k -th power amplitude squeezed states. These states can be characterized by a special, rather simple nonclassicality condition.

DOI
10.1088/1742-6596/36/1/027

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A brief review of nonclassicality conditions in terms of moments of the creation and annihilation operators is given. By introducing k -th power amplitude squeezing, the notions of ordinary quadrature squeezing and amplitude-squared squeezing are generalized. Minimum uncertainty states are considered as a special class of k -th power amplitude squeezed states. These states can be characterized by a special, rather simple nonclassicality condition.
Practically applicable criteria for the nonclassicality of quantum states are formulated in terms of different types of moments. For this purpose the moments of the creation and annihilation operators, of two quadratures, and of a quadrature and the photon number operator turn out to be useful. It is shown that all the required moments can be determined by homodyne correlation measurements. An example of a nonclassical effect that is easily characterized by our methods is amplitude-squared squeezing.
Normal squeezing is described in terms of variables which are linear in the field creation and annihilation operators. Amplitude-squared squeezing, on the other hand, is defined in terms of variables which are quadratic in these operators. As a result processes, such as second-harmonic generation, which couple one mode amplitude to the square of another convert this kind of squeezing into normal squeezing. In particular, the variables one considers are the hermitian part and one over i times the antihermitian part of the square of the annihilation operator. The variances of these observables obey an uncertainty relation. In order to better understand the properties of amplitude-squared squeezing it is useful to find the minimum uncertainty states corresponding to this uncertainty relation. These states are described by two parameters; the first,, gives the expectation of the square of the annihilation operator and the other λ, describes the amount of amplitude-squared squeezing. AH of these states have a mean amplitude of zero. These states for which =0 have unusual noise properties in that their Q functions have a four-fold rotational symmetry. We also find that a squeezed vacuum state is an amplitude-squared minimum uncertainty state.
The real and imaginary parts of the square of the field amplitude obey an uncertainty relation. If the uncertainty in either of these variables is less than it would be in a minimum uncertainty state with equal uncertainties in both variables, the state is amplitude-squared squeezed. This is a nonclassical effect which can be produced by second harmonic generation, two-photon absorption, and degenerate parametric amplification, as well as by higher-order processes. Amplitude-squared squeezing can be converted into normal squeezing by interactions in which the square of the amplitude of one field is coupled to the amplitude of another. Second harmonic generation and some forms of four-wave mixing provide examples of such processes.
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.
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 real and imaginary parts of the square of the field amplitude are the variables that describe amplitude-squared squeezing. These quantities obey an uncertainty relation. Here we find a particularly simple subset of the states that satisfy the uncertainty relation as an equality. These states are constructed by applying a squeeze operator to a state that consists of a Hermite polynomial, whose argument is the mode creation operator multiplied by a constant, acting on the vacuum. The squeezed vacuum is such a state. These states may or may not be squeezed in the normal sense, and may or may not have sub-Poissonian photon statistics.