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A BOX OF PHOTONS
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Figure 4.5 Schematic representation of a coherent state. The noise is represented by a circle, because any quadrature has the same variance of 1/4.
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is always 1/2. Then
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(4.114)
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Before we can say that this is the variance of ii, let us think about what we cannot trust and should not take seriously in such a calculation. There is a very good reason why terms involvin products of noise ellipse projections should not be taken seriously. As all and all do not commute, the operators associated with their variances do not commute either, so that the expectation values of mixed products of them will generally be different from the products of their expectation values. Thus, we should throwaway the 1/2 in (4.114) and write
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(4.115)
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Surprisingly enough, this is the exact result! Let us see why. For a state where (all) = A, we can always write all = A where .6.0.11
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+ .6.0.11 ,
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(4.116) (4.117)
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= all - (all)'
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Writing all in terms of 0. 1 (15) and 0. 2(15), we find that all = A
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+ .6.0.1 (15) + i .6.0.2(15).
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(4.118)
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In this example, we can obtain the correct values for ";(.6.0. 1 (J)2) and ..; (.6.0. 2(J)2) from the projections of the noise ellipse parallel and perpendicular to the phasor,
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SQUEEZED STATES
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respectively, because {~(h ~ii2 + ~ii2 ~iil) vanishes. This, however, does not explain why we can obtain an exact result for {( ~n )2) by discarding terms involving products of fluctuations. To understand this point, let us look into the quantum term corresponding to the 1/2 in the last line of (4.114). Using (4.116), we write the number operator as (4.119) The last two terms in (4.119) correspond to the operator ~ndescribing the fluctuations in the photon number. The reader can easily verify that this operator can also be written as
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~n = 2A~iil(8)
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+ [~iil(8) - i ~ii2(8)1 [~iil(8) + i~ii2(8)1.
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(4.120)
The term [~iil(8) - i~ii2(8)1 [~iil(8) +i~ii2(8)1 corresponds to the product of the two noise ellipse projections, (1/2 - i/2)(1/2 + i/2), that gave rise to the 1/2 in the last line of (4.114). Caves's diagrams cannot reproduce any quantum correlations between these two noise operators. However, if the average amplitude A is large enough, this term will be negligible in comparison with the first term on the righthand side of (4.120). Then the calculation in terms of Caves's diagrams will be a good approximation to the exact result. The coherent-state example we have considered is the best of both worlds, because for a coherent state the last term in (4.120) does not contribute to the final result. The variance of n is given by
((~n)2)
= 4A2{~iil(8)2) + {~iirl ~iill ~iirl ~iill
+ 2A {~iil (8) ~iirl ~iill +
~iirl ~iill ~iil (8)}),
(4.121)
and for a coherent state the last term on the right-hand side of (4.121) vanishes, yielding (4.122) which coincides with (4.115). The variance (~iil(8 ) is related to amplitude fluctuations, whereas (~ii2(8)2) is related to phase fluctuations. Equation (4.122) shows that for a coherent state, photon number fluctuations are connected only to amplitude fluctuations. So the Caves's diagram technique has given us some insight into the nature of coherent states. In the next subsection it will help us to understand a method of detecting squeezed states. 4.2.4 Observing squeezed states: Homodyne detection Homodyne detection was first developed for radio [460] but is now widely used in optics. In this section we follow closely the treatment of homodyne detection presented in [416].
A BOX OF PHOTONS
Figure 4.6 The four ports of a beam splitter.
Consider a beam splitter with two plane waves Es and Eo incident on each of its faces, respectively, as in Figure 4.6. If the complex amplitude reflection and transmission coefficients are rand t, respectively, we can write
El = rEo +tEs
(4.123)
(4.124) Or in a more compact form, (4.125) For a lossless beam splitter, energy conservation implies that the 2 x 2 matrix in (4.125) is unitary: (4.126) and