Squeezed States in Quantum Optics in .NET

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102 Squeezed States in Quantum Optics
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Fig 103 Phase-squeezed light in the plane of phase quadratures The uncertainty ellipse is stretched in the radial direction
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not the average photon number) is zero For example, an optical parametric ampli er with a vacuum input can generate a squeezed vacuum with a reduction in the noise of one quadrature components on the order of 10 dB In summary, the main feature of the squeezed states lies in the fact that they offer the opportunity of deforming (squeezing!) in a certain direction the circle of quantum uncertainty that so becomes an uncertainty ellipse Hence, they provide a way of reducing the quantum noise for one of the two quadratures
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1022 Algebraic (su(1, 1)) Content of Squeezed States
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We have seen that squeezed states are obtained by forcing two-photon processes with time-dependent classical sources (from this the appellation two-photon coherent
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Fig 102 Amplitude-squeezed light in the plane of phase quadratures The uncertainty ellipse is stretched in the angular direction
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10 Squeezed States and Their SU (1, 1) Content
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states) The most general form of the corresponding Hamiltonian reads H = a a + 1 2 + f 2 (t) a + f 2 (t) a 2 + f 1 (t) a + f 1 (t) a
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(1020)
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The production of coherent states or of squeezed states will depend on the respective importance granted to the factors f 1 (t) and f 2 (t) Now, the Hamiltonian in (1020) has clearly the form of an element in the two-photon Lie algebra, denot2 ed by h6 , generated by the set of operators {a, a , I d , N = a a, a 2 , a }, and already mentioned in Section 224 From the (nontrivial) commutation rules, [a, a ] = I d , [a 2 , a ] = 2a , [a, N ] = a ,
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[a , N ] = a , [a , N ] = 2a ,
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[a , a] = 2a , [a 2 , N ] = 2a 2 ,
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(1021)
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we see that h6 is a representation of the semidirect sum of su(1, 1) with the Weyl Heisenberg algebra The corresponding group, denoted by H 6 , is the semidirect product H 6 = W SU (1, 1) The existence of such an algebraic tool in the construction of squeezed states is very useful in solving problems involving Hamiltonians such as (1020) Suppose we have to deal with an evolution equation of the type i U (t, t 0 ) = H(t) U (t, t 0 ) , t U (t 0 , t 0 ) = I d , (1022)
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where the mathematical objects have a h6 nature: the time-dependent Hamiltonian H is an element of the Lie algebra h6 , whereas the evolution operator U (t, t 0 ), as a solution to (1022), should be an element of a unitary representation of the Lie group H6 The trick is of disentangling nature [9], like in (874) It amounts to solving the equation by choosing among linear faithful representations of H6 or h6 the simplest one, namely, the four-dimensional one in which group and algebra elements are realized as 4 ~ 4 matrices Let us associate with a generic element X in h6 , written as X = N+
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+ I d + Ra + La 2 + ra + la ,
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, , R, L, r, l C ,
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(1023)
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the following matrix in M(4, C): 0 r M(X ) = l 2
0 2L l
0 2R r
0 0 0 0
(1024)
This representation of X is made possible because of the Lie algebra isomorphism between basic operators de ning h6 and elementary projectors E i j with matrix elements E i j kl = ik j l generating the Lie algebra M(4, C): N+
1 E 22 E 33 , 2
I d 2E 41 , (1025)
a 2E 23 , a 2 2E 32 ,