Input-Output Theory
In the early 1980’s calculations were performed on the degree of squeezing that could be achieved within cavity parametric amplifiers. These calculations indicated that the field inside the cavity could be squeezed by no more than a factor of two under st
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Input-Output Theory
B. Yurke
In the early 1980's calculations were performed on the degree of squeezing that could be achieved within cavity parametric amplifiers. These calculations indicated that the field inside the cavity could be squeezed by no more than a factor of two under steady state conditions [1]. As a result lore spread that cavities were bad for squeezing. It turns out, however, that because of a remarkable interference effect between the field that is reflected off of the input port mirror, and the field that - after entering the cavity - is squeezed and then re-emitted, the field external to the cavity can exhibit arbitrarily large amounts of squeezing: even though the field inside the cavity only exhibits a factor of two squeezing [2]. How to calculate the quantum statistical properties of the field exiting a cavity given the input field and the internal field had become an important issue that needed to be solved for the advancement of the squeezed state field. Works by Yurke and Denker [3], Yurke [4], and Collett and Gardner [5-7] were instrumental in showing the way. This is, in part, what input-output theory is about. But input-output theory has much more to offer. The equations of motion of the external fields and the canonical commutation relations that the field operators must satisfy place powerful constraints, in fact, often dictating the quantum behaver of the systems with which they interact. This theme will be developed through much of this chapter. Recent developments in inputoutput theory include the formulation of better mathematical machinery for dealing with nonlinearities [8], the development of machinery for dealing with random media and cavities of complex shape [9], and the application of inputoutput theory to fermion systems [10]. For a book whose subject matter currently finds its primary application in optics this chapter, with its emphases on mechanical systems and exactly solvable systems, may seem a bit quirky. However, it is not out of line with how quantum mechanics is usually taught. Quantization of the electromagnetic field is often taught almost as a footnote or afterthought once one has mastered the quantum mechanics of mechanical systems. In most cases it will be obvious how to apply the techniques presented here to the corresponding optics problem. Besides, once the equations are cast in terms of creation and annihilation operators and characteristic time scales, quantities P. D. Drummond et al. (eds.), Quantum Squeezing © Springer-Verlag Berlin Heidelberg 2004
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B. Yurke
having units such as mass or length can be stripped away and the equations become identical with those used in quantum optics. Also, an emphasis on mechanical systems is, perhaps, not out of line from a historical perspective. Much early development of squeezed state theory concerned its application to gravitational wave detection using mechanical oscillators [11,12]. Quantum mechanical calculations were carried out with one ton mechanical resonators in mind. To give the reader advance war
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