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Journal of Mathematical Physics

Nth (even)-order minimum uncertainty products

Robert Lynch, Harry A. Mavromatis

Published Aug 1, 1990• 14 citations
Publisher

Abstract

This paper considers the problem of finding the quantum states that minimize the products of the (even) Nth-order fluctuation of two canonically conjugate operators. The problem is first attacked in an abstract form and an equation derived for the desired states. A consideration of the N=2 case then leads to a connection with the new concept of ‘‘squeezed states’’ of the electromagnetic field, and the usual exact solution involving the coherent state. The concept of ‘‘higher-order squeezing’’ is touched on to motivate the further discussion. The cases N=4, 6, and 8 are then taken up, and approximate solutions to them found via a new numerical technique. During these calculations, it is noted that only the first few terms in the expansion in number states figure in the solution; this observation is then exploited to find an approximate closed solution to the problem valid for all N.

DOI
10.1063/1.528642

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This paper considers the problem of finding the quantum states that minimize the products of the (even) Nth-order fluctuation of two canonically conjugate operators. The problem is first attacked in an abstract form and an equation derived for the desired states. A consideration of the N=2 case then leads to a connection with the new concept of ‘‘squeezed states’’ of the electromagnetic field, and the usual exact solution involving the coherent state. The concept of ‘‘higher-order squeezing’’ is touched on to motivate the further discussion. The cases N=4, 6, and 8 are then taken up, and approximate solutions to them found via a new numerical technique. During these calculations, it is noted that only the first few terms in the expansion in number states figure in the solution; this observation is then exploited to find an approximate closed solution to the problem valid for all N.
The authors consider the problem of finding the quantum states that minimize the products of the 'absolute value' Nth-order fluctuation of two canonically conjugate operators. A scheme is applied to the problem, wherein the exact solution state is approximated by working in a subspace consisting of the ground state plus a finite number of higher-order states, using a variational technique. The solution is first obtained for the ground state, and this is subsequently compared to the solution assuming 2*2 and 3*3 spaces. For positive N they obtain a small, smooth decrease in the value of the fluctuation as the space increases and expect the 3*3 space results to be close to the exact Nth-order fluctuation. For negative N, however, they get an abrupt decrease in the fluctuation as the space increases.
This paper considers the problem of whether mixed states can arise in the Nth order minimum uncertainty expression UN=<Ψ‖‖x‖N‖Ψ≳<Ψ‖‖p‖N‖Ψ≳. Using a method due to R. Jackiw [J. Math. Phys. 9, 339 (1968)], and finding the extremum of the appropriate functional, a set of equations is derived which must be satisfied by such a mixed state. It is easily shown that a sufficient condition that the equations have a solution is that the mixed state in fact reduces to a pure state. It does not appear possible to rigorously prove that such a reduction is necessary, although the authors present approximate calculations which make such a conclusion plausible. In the course of the approximate calculations generalized Nth order ‘‘squeezed state’’ values for UN are also obtained which are of some interest in and of themselves.
We consider the problem of minimization of products of mean values of the high powers of operators x and p. From this point of view, we study several two-term superpositions of the Fock states, as well as three popular families of infinite superpositions: squeezed states, even/odd coherent states, and orthogonal even coherent states (or compass states). The new element is the analysis of products of the corresponding (co)variances and the related generalized (Robertson-Schrödinger) intelligent states (RSIS). In particular, we show that both Fock and pure Gaussian homogeneous states are RSIS for the fourth powers (but not for the sixth ones). We show that lower bounds of the high-order uncertainty products can be significantly below the vacuum values. In this connection, the concept of significant and weak high-order squeezing is introduced.
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.
In this paper, we study first the connection of the k orthonormalized eigenstates of ak{\mathit{a}}^{\mathit{k}}ak with coherent states, then, according to the higher-order squeezing defined by Zhang et al. [Phys. Lett. A 150, 27 (1990)], study the properties of the higher-order squeezing of k orthonormalized eigenstates. Our results show that the k orthonormalized eigenstates can be represented as a linear superposition of k coherent states, and that all of them are minimum uncertainty states of the operators Z1{\mathit{Z}}_{1}Z1​(N) and Z2{\mathit{Z}}_{2}Z2​(N) (N=mk, m=1,2,3, . . .) for even and odd k, and all of them have the Nth-order squeezing [N=(m+1/2)k, m=0,1,2, . . .] for even k.