cos t sin t

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(1014)

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The corresponding covariance matrix is equal to ( X )2 {X Y } with ( E )2 = E 2

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10212 In a Pure Squeezed State A pure squeezed state is de ned as produced by the sole action of the unitary operator S( ) on the vacuum |0 :

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{X Y } ( Y )2

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(1015)

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= S( )|0 = e 2 (

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a a2 )

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|0

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(1016)

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102 Squeezed States in Quantum Optics

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In view of the computation of the mean value and the covariance matrix for the electric eld, we need to determine the unitary transport of the lowering and raising operators, S ( )aS( ) and S ( )a S( ) By applying the formula e A Be A = B + [A, B] + and, with = re i , one nds S ( )a S( ) = a cosh r + ae i sinh r 1 [A, [A, B]] + , 2!

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S ( )aS( ) = a cosh r + a e i sinh r ,

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Hence, with U + iV = (X + iY )e i /2 , we have S ( )(U + iV )S( ) = Ue r + iV e r There follows for the eld mean value 0, |E (t)|0, = 0, (1017)

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and for the covariance matrix ( X )2 {X Y } {X Y } ( Y )2 = 1 4 cosh 2r I 2 + sinh 2r cos sin sin cos (1018) In particular, we have for the product of variances ( X )2 ( Y )2 = or equivalently U V = 1 4 (1019) 1 (1 + sinh2 2r sin2 ) , 16

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In consequence, with pure squeezed states, there is saturation of the Heisenberg inequalities, but, contrarily to the coherent states, with U = V /

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10213 In a General Squeezed State In a general squeezed state, | , the same as in a coherent state:

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= D ( )S( )|0 , the average value of the eld is cos t sin t

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, |E (t)| ,

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= , 0|E (t)| , 0 = 2E R e i t = R( )

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I( )

On the other hand, the variance is the same as for a pure squeezed state: ( E )2 = , |E 2 E 2 | , = 0, |E 2 E 2 |0,

The same holds for the covariance matrix Note that these results are independent of the manner in which one would de ne the squeezed states, | ,

= D ( ) S( )|0 or | , = S( ) D ( )|0

10 Squeezed States and Their SU (1, 1) Content

10214 A General De nition of Squeezing One can now give a general de nition of squeezing with respect to a pair of quantum observables A and B with commutator [A, B] = iC = 0 When variances / are calculated in a generic state, one obtains from Cauchy Schwarz inequality A B v 1 | C | Now a state will be called squeezed (with respect to the pair (A, B)) 2 if ( A)2 (or ( B)2 ) < 1 | C | A state will be called ideally squeezed (with respect to 2 the pair (A, B)) if the equality A B = 1 | C | is reached together with ( A)2 (or 2 ( B)2 ) < 1 | C | Hence, the class of ideally squeezed states contains the set of pure 2 squeezed states 10215 Squeezing the Uncertainties To conclude this section, let us brie y explain the interest in squeezed states in various applications, particularly in the coding and transmission of information through optical devices From the above study of the electric eld, Glauber (standard) optical coherent states have, in the phase quadrature plane, circularly symmetric uncertainty regions, so the uncertainty relation dictates some minimum noise amplitudes, for instance, for the amplitude and phase (Figure 101) A further reduction in amplitude noise is possible only by squeezing the uncertainty region, reducing its width in the amplitude, that is, radial, direction while increasing it in the orthogonal, that is, angular or phase, direction, so that the phase uncertainty is increased Such light is called amplitude-squeezed (Figure 102) Conversely, phase-squeezed light (Figure 103) has decreased phase uctuations at the expense of increased amplitude uctuations There is also the so-called squeezed vacuum, where the center of the uncertainty region (corresponding to the average amplitude) is at the origin of the coordinate system, and the uctuations are reduced in some direction The mean photon number is larger than zero in this case; a squeezed vacuum is a vacuum only in the sense that the average amplitude (but

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Fig 101 Coherent light in the plane of phase quadratures The uncertainty region around one point of the plane is circularly symmetric