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with |0, 0 = |0 The squeezing operator S( ) is unitary since it is the exponential of the anti-Hermi2 tian combination a a 2 Note that its inverse is given, like for the displacement 1 operator D ( ), by S ( ) = S ( ) = S( ) Let us examine more closely the interest in such states before pursuing the development of the formalism The elliptic deformation of the Gaussian results from the construction of the squeezed states There follows a decreasing value for one of the two variance factors in the Heisenberg uncertainty inequalities This fact can be exploited in the detection of weak signals; see, for instance, [139] Examples drawn from quantum optics are standard in regard to this notion of squeezing; as a matter of fact, squeezed light can be generated from light in a coherent state or a vacuum state by using certain optical nonlinear interactions For the sake of simplicity, let us consider a one-mode electromagnetic eld propagating in onedimensional volume L, with vector potential A(x, t) = i 4 c 2 2V

ae i(kx t) a e i(kx t)

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Neglecting spatial dependence, which is not important for our purpose, the electric eld operator reads as E (x, t) = 1 A(x, t) = E ae i t + a e i t , c t

where E = 4 2L Let us now introduce the following two self-adjoint observables X = 1 (a + a ) , 2 Y = 1 (a a ) , 2i (109)

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Some cannibalization of the core will happen with most extensions, but the level can be minimized and need not be life threatening Launching extensions that add real value is one way of doing this, as discussed earlier However, perhaps the biggest mistake to avoid is eating into the human and nancial resources of the core product

which would stand, in the context of one-particle quantum mechanics and up to the 1 factor 2 and appropriate units, for position and momentum operator, respectively Here, they describe the two components in phase quadrature of the eld They can, in principle, be measured separately through homodyne detection [43], but

not simultaneously, of course Indeed, the commutation rule and the Heisenberg inequality read, respectively, [X , Y ] = i Id , 2 X Y v 1 4 (1010)

117 119 121 121 126 129

In optical interferometry, the term homodyne signi es that the reference radiation (the local oscillator ) is derived from the same source as the signal before the modulating process For example, in a laser scattering measurement, the laser beam is split into two parts One is the local oscillator and the other is sent to the system to be probed The scattered light is then mixed with the local oscillator at the detector In terms of these quadratures, the electric eld reads E (t) = 2E(X cos t + Y sin t) Its mean value and variance in a certain state are given by E (t) = 2E( X cos t + Y sin t) , (1012) (1011)

( X )2 {X Y }

{X Y } ( Y )2

, (1013)

132 137 146

1 where {X Y } = 2 (X Y + Y X ) X Y The Hermitian square matrix in (1013) is called a covariance matrix It is an important object for the statistical description of the quantum eld Let us now compare these expressions when we choose a coherent state, a pure squeezed state, or a mixed (coherent + squeezed) state

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