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22
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84926801183
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The modes at the input and output of the phase-shift box in Fig. 1 represent different physical locations in space, e.g., the input and output facets of an optical fiber. Spatiotemporal mode expansions of the Heisenberg field operators at these locations have the annihilation operators shown in this figure as coefficients. Thus, Eq. (1) is the natural—field based—description of the phase shift. However, when an output-mode measurement is specified in terms of a hat, it is convenient to derive its statistics, as a function of the input state, via Eq. (2).
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24
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84926844926
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Commuting observables have a common CON set of eigenkets. Thus, if A hat and B hat are commuting, discrete-spectrum observables, there is a CON set lcurl |A,B rangle: A member scrA ,B member scrB rcurl such that when A hat and B hat are measured simultaneously, the joint outcome A,B occurs with probability P(A,B| | ψ > ) = |< A,B | ψ > |2 for A member scrA, B member scrB.
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25
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84926801182
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In particular, because the Π hatn are Hermitian and positive semidefinite, Eq. (21) produces a non-negative probability for each n. Because of Eq. (20), these probabilities sum to 1.
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26
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84926866632
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This PDF is non-negative because d Π hat ( β ) is Hermitian and positive semidefinite. It integrates to unity because d Π hat ( β ) resolves the identity.
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28
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84926822938
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Overcompleteness is a vector-space property whose utility can be illustrated as follows. Let scrX be the two-dimensional (2D) Euclidean space of real-valued column vectors x vecT=(x1,x2), where T denotes transpose. The standard basis for scrX is e vec1T== (1,0) and e vec2T== (0,1). These are orthonormal vectors that resolve the 2 times 2 identity matrix, I vec = sumn=12 e vecne vecnT, producing the Cartesian representation of x vec via projection, x vec = Ix vec = sumn=12 e vecne vecnTx vec = sumn=12 xne vecn. Any vector x vec can be expressed as a linear combination of an arbitrary pair of linearly independent vectors in scrX. However, unless this pair comprises orthogonal vectors, the representation coefficients will not be obtainable from projection. Consider the overcomplete vectors c vec1T== (0,1), c vec2T== (cos ( π /6), - sin ( π /6) ), and c vec3T== (- cos ( π /6),- sin ( π /6) ). They form a symmetric, triangular constellation in the plane with π /3-rad angular separation between adjacent vectors. It is easily verified that these vectors resolve the identity, viz., I vec = case 2 over 3 sumn=13 c vecnc vecnT. Thus, even though any two of the lcurl c vecnrcurl are sufficient to represent an arbitrary x vec member scrX, the overcomplete set of all three permits the x vec coefficients to be found via projection, x vec = Ix vec = case 2 over 3 sumn=13 ( c vecnTx vec ) c vecn.
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29
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84926844925
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Figure 3 is a single-detector heterodyne configuration. Although much simpler to analyze, it is not as practical as the dual-detector, balanced-mixer, arrangement. The latter, however, leads to identical statistics (Ref. 14), and so will not be presented here.
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30
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84926844924
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It should be emphasized that we are concerned with the quantum description of heterodyne detection. The semiclassical model, in which heterodyne detection is limited by LO shot noise, gives correct quantitative results when the input field is a coherent state, but cannot be used for nonclassical-light experiments. Moreover, even within its limited region of quantitative validity, the semiclassical model is qualitatively incorrect (Ref. 14). The noise in heterodyne detection is the quantum noise of the signal and image modes; LO shot noise is a semiclassical fiction.
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31
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84926844923
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An example of such a phase POM is provided by the heterodyne measurement of phase (Ref. 13), d Π hat ( φ ) = tint0infd | α | | α | | α > < α | d φ ( 2 π )-1, where | α | exp (i φ ) is the polar decomposition of the complex number alpha. In addition to being a nonobservable POM for which there is a known realization, this example shows that d Π hat ( φ ) need not be a kettimesbra outer product.
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33
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84926822937
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It will turn out that small r gives better δ phi versus N behavior, but that larger r values may be of interest in that they postpone an exponential increase in photon-number fluctuations. The restriction to integer r values is inessential. It permits the key properties of the optimum state to be calculated by means of elementary techniques, but there are other approaches which can handle noninteger r.
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35
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84926844922
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A state is nonclassical if it is not a coherent state, or a classically random mixture of such states; nonclassical states lead to photodetection statistics which cannot be accounted for by the semiclassical shot-noise theories of direct detection, homodyne detection, and heterodyne detection (Ref. 14).
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36
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84926801181
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Equation (61) is usually obtained by a high-mean-field linearization of the Heisenberg limit, langle Δ N hat2> < Δ [ s i hat n ( φ )]2rangle >= case 1 over 4 | < c o hat s ( φ ) > |2, as elaborated in Sec. III, cf. R. Loudon, The Quantum Theory of Light (Oxford University, Oxford, 1973), or the heterodyne-detection linearization in Ref. 13.
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39
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84926844921
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We limit our attention to product states on the signaltimesapparatus space only because we want these modes to be quantum-mechanically independent. The statistics of the Y hat measurement can also be developed for correlated signal and apparatus modes, cf. the quantum correlations studied for heterodyne detection in Ref. 13.
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40
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84926844920
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In fact, once we study the wave-function interpretation of the number and phase representations of an arbitrary state, it will become clear that it is ei φ mdashnot phi itself—that is complementary to photon number.
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41
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84926801180
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The coherent state | α rangle, with | α | >> 1, is a high-mean-field state. Squeezed states can also satisfy Eq. (100). Number states violate this condition, as does the optimal state from Sec. II.
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42
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84926866631
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The linearized result cannot be saved by appeal to a 0 times inf indeterminate form on the left in Eq. (99)—the phase of a single-mode field is confined to a 2pi-rad interval, hence its variance can never exceed pi.
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43
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84926801179
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We do not put a caret on the phase variance, to emphasize that it is not the variance of an observable on scrH.
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44
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84926866630
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Interestingly, the number state |n rangle is the only state which achieves equality in Eq. (117). equality holds in Eqs. (115) and (116) if and only if PSI (ei φ )=C1exp [-C2( phi - φ bar )2] exp (-in̄φ ), where C1 and C2 are constants, the latter real valued. These constants cannot be arbitrary; the phase representation must be properly normalized, and, more importantly, its inverse transform from Eq. (112) must be a proper number representation. The second of these conditions—which we will call number-ket causality when we confront it in Sec. V—can only be ensured by taking C2=0, reducing the preceding PSI (ei φ ) to the number-ket result.
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47
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84926844919
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They focused their attention on the ordinarymdash lcurl a hatj: j=1,2 rcurl mdashsqueezing exhibited by coherent phase states. Thus, with the exception of the A=0 case—in which the coherent phase state is the vacuum state |0 > mdashthe coherent phase states are all nonclassical.
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48
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84926801178
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T. A. Brennan, S. B. thesis, MIT, 1989.
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49
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84926822936
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The squeezed vacuum state also has even photon numbers only. However, the squeezed vacuum and the squeezed phase vacuum are not the same state, cf. Ref. 45 for the photon counting distribution of ordinary squeezed states.
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50
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84926822935
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Equations (163) and (165) presume a unit-length ket, | ψ rangle. The condition of finite average photon number must be applied explicitly.
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51
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84926844918
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There would, in general, be a nontrivial phase shift associated with each frequency.
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52
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84926801177
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The Hilbert-transform construction is said, in linear-system theory, to lead to a minimum phase arg[ PSI (ei φ )] for a given | PSI (ei φ )|, although a better term would be minimum delay. Because the time parameter of system theory becomes the photon-number index in our quantum context, minimum delay translates into minimum photon number.
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53
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84926866629
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The zeros of PSI (z) can lie on or outside the unit circle. Pole and zero multiplicities greater than 1 are permitted.
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54
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84926844917
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The case of zero multiplicity greater than 1 can easily be handled; this includes having zeros on the unit circle.
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55
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84926801176
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This is the rational z-transform version of the Hilbert-transform process. For a pair of phase-PDF zeros on the unit circle, one zero must be assigned to PSI (z); the resulting state is still one of minimum average photon-number with the desired phase statistics.
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56
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84926866628
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We have that Hi( ei φ )= sumn=-inf inf H(i φ +i2 π n) relates the impulse-invariant, digital filter's frequency response to that of the analog filter (Ref. 18). The desired term in this sum is the n=0 term. The other, frequency-translated H( ), terms represent aliasing. Because we are only concerned with - π < φ <= pi, the effects of the n != 0 terms will be negligible if the region of appreciable |H(i φ )| values is confined to | φ | << 1.
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57
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84926801175
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Using the bilinear-transformation method, we map an analog filter of frequency response H(i ω ) into a digital filter of frequency response Hb(ei φ )=H(i2 tan ( φ /2) ) (Ref. 18). This tangent-function mapping is a warping of the frequency axis in going from the analog to the digital filter. Because we are only concerned with - π < φ <= pi, the effects of the tangent-function mapping will be negligible if the region of appreciable |H(i ω )| values is confined to | ω | << 1.
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58
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84926844916
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This truncation is not the same as the one we used in conjunction with our optimum state in Sec. II. In particular, all our measurements have been established on the full state space scrH.
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59
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84926866627
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Equation (196) does define an observable on the full state space scrH. However, for states which are not confined to the truncated space scrHM, this observable does not conform to the behavior expected of a phase measurement. For example, if | ψ > =|n rangle, with n > M, then the φ hatM measurement gives outcome 0 with probability 1, not the uniformly distributed random variable on (- π , π ] that should result from measuring the phase of a number-ket field.
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60
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84926866626
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Probability-theory purists will recognize that convergence in distribution is necessitated by the fact that the PB approach yields a discrete random variable for all finite M, whereas the SG-POM yields a continuous random variable.
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