IOQuire suddenly means in a time much shorter than the period of oscillation. in .NET

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exposes the true multimode nature of the cavity (i.e., there is a major rearrangement of modes during the size change that breaks down the formal analogy between a singlemode cavity and an oscillator). Can we generate squeezed light from the vacuum by suddenly changing the size of a cavity What happens to the field when the cavity changes size suddenly To answer these questions, we consider a simple one-dimensional model where a cavity is formed by a pair of perfect mirrors placed at x = 0 and x = Li. The mirror at x = Li is suddenly I I moved from its original position to a new position at x = L f (see Figure 4.2).
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Figure 4.2 Can we squeeze the vacuum by physically squeezing a cavity very rapidly This diagram represents our one-dimensional cavity model. A cavity is formed by a pair of perfect mirrors in the vacuum initially located at x :::: 0 and x = Li as in (a). Then suddenly the mirror on the right is pushed from x = Li to x = L f, as in (b). What happens to the quantum field inside the cavity
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The electric and magnetic fields can be written in terms of the normal modes for the perfect cavity J2/ L sin(ml'x/ L) and the external continuum modes J2/7l' sin([xL]k), where L is Li before the sudden change in cavity size and becomes L f after. The discrete cavity modes are normalized to 1, and the external continuum modes are delta-function normalized. Before the sudden change, the fields are given by forO
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(4.88)
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II Suddenly here means this: Any mirror becomes transparent for high-enough frequencies. Let We be the frequency around which the mirror (moving) starts becoming transparent. The sudden change of cavity size we are considering here is such that it happens in a time Te much shorter than l/we Relativity, of course, requires that during the sudden change the mirror never exceed the speed of light. This results in arestrlction on how much Lf can differ from Li: ILf - Ld C/We.
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where k n == mr / L i . After the change they are given by
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(4.89) where k~ == n7r/Lf. Even though the expression for the fields in terms of annihilation and creation operators is different before and after the sudden change in cavity length, the field operators, like q and p for the harmonic oscillator, remain unchanged. 12 Using this, we can work out how the annihilation and creation operators after the sudden change relate to the annihilation and creation operators before. Assuming that L f :::; L i , we multiply the electric field expressions in (4.88) and (4.89) by J2/Lfsin(n7rx/Lf) and integrate over x from 0 to Lf. Equating the results, we find that
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l_~~(bm L ~ i m=l n
t -bm )
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--Lf Li m
Lf )
(4.90)
Analogously, multiplying the magnetic field expressions in (4.88) and (4.89) by J2/ L f cos(n7rx/ Lf), integrating over x from 0 to Lf, and equating the results, we obtain
(4.91)
12We are working in the SchrOdinger picture.
SQUEEZED STATES
Adding (4.90) and (4.91), we get13
(4.92) As the quantum state of the field remains unchanged for a sudden change in the Hamiltonian, (4.92) implies that the initial vacuum becomes squeezed. But as we had already foreseen, the new annihilation operators are related not only to the annihilation and creation operators of the corresponding oscillator (mode) before the sudden change, but also to those of every other oscillator (mode) as well.
4.2.3 A geometrical picture for squeezed states
Any reader who has ever browsed through the literature on squeezing has probably come across pictures of noise ellipses and circles. In this subsection we explain how this geometrical picture of quantum noise works. In his pioneering paper on squeezed states, Yuen [660] defined his two-photon coherent states as in (4.79): (4.93) except that rather than demanding real JL and
his only requirement on them was (4.94)
otherwise leaving them free to assume any complex value. Apart from a global arbitrary phase multiplying bl',v, Yuen's requirement on JL and II allows us to write
= e-iO coshr,
e sinh r,
(4.95) (4.96)