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Physical Review Letters

Higher-Order Squeezing of a Quantum Field

C. K. Hong, L. Mandeļ

Published Jan 28, 1985• 359 citations
Publisher

Abstract

The concept of $2n\mathrm{th}$-order squeezing of a quantum field is introduced as a natural generalization of the usual second-order squeezing. It is shown that the processes of degenerate parametric down conversion, harmonic generation, and resonance fluorescence all exhibit higher-order squeezing.

DOI
10.1103/physrevlett.54.323

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Physical Review A•Oct 1, 1986•30 citations
The concept of 2nth2n\mathrm{th}2nth-order squeezing of a quantum field is introduced as a natural generalization of the usual second-order squeezing. It is shown that the processes of degenerate parametric down conversion, harmonic generation, and resonance fluorescence all exhibit higher-order squeezing.
The concept of squeezing, which generally implies that one quadrature component of an optical field has measured fluctuations below the vacuum limit, is generalized to higher moments of even order. Higher-order squeezing is encountered in many of the nonlinear processes that generate second-order squeezing. It is shown that the field produced in degenerate parametric downconversion is squeezed to all even orders and is intrinsically squeezed to orders 2,6,10…. Higher-order squeezing also shows up in second harmonic generation and in resonance fluorescence from an atom. However, in those cases there is no intrinsic squeezing beyond the second order, although a strong second-order squeezing effect is manifest in the higher-order moments of the field.
The concept of Nth-order squeezing and intrinsic Nth-order squeezing is introduced and applied to several physical situations that are known to exhibit second-order squeezing. It is shown that the field produced in degenerate parametric down-conversion is squeezed to all orders, as well as being intrinsically squeezed to orders 2,6,10,. . . . The same is true for the fundamental mode in second-harmonic generation within the short-time approximation, but in this case there is only intrinsic second-order squeezing. In resonance fluorescence from a two-level atom squeezing is again found to be intrinsically a second-order phenomenon, although an intrinsic second-order effect can give rise to a weak form of higher-order squeezing.
It is found that two-mode output quantum electromagnetic field in two-mode squeezed states exhibits higher-order squeezing to all even orders. And the generalized uncertainty relations are also presented for the first time. The concept of higher-order squeezing of the single-mode quantum electromagnetic field was first introduced and applied to several processes by Hong and Mandel in 1985. Lately Li Xizeng and Shan Ying have calculated the higher-order squeezing in the process of degenerate four-wave mixing and presented the higher-order uncertainty relations of the fields in single-mode squeezed states. In this paper we generalize the above work to the higher-order squeezing in two-mode squeezed states. The generalized uncertainty relations are also presented for the first time.
It is found that the field of the combined mode of the probe wave and the phase-conjugate wave in the process of degenerate four-wave mixing exhibits higher-order squeezing to all even orders. The degree of squeezing increases with the order N, and the higher-order squeeze parameter qN{q}_{N}qN​ may approach -1.