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Characteristic functions in .NET Integrating code128b in .NET Characteristic functions

3.8 Characteristic functions use vs .net code-128 creator todraw code-128 with .net SQL server characteristic function:. W ( ) 1 2 1 = 2 exp( ) CW ( )d 2 (3.136) exp( ) CN ( )e 2 /. 2. d 2 .. This de nition is equiv Code 128C for .NET alent to the previous one with the proper interpretation of variables. As an application of the characteristic function, we now derive the P function corresponding to the thermal, or chaotic, state of the eld.

Recall that the eld in this case is a mixed state given by density operator Th of Eq. (2.144).

We rst calculate the Q function for this density operator according to. Q( ) = . Th / 1 . 2 = e n e = (1 + n). m. Th n . ( )m n (m!n!)1/2 ( )n n! , (3.137). n 1+n 2 1 = exp (1 + n) 1+n where n for the thermal state is given by Eq. (2.141). Then from Eq. (3.132) we have CA ( ) =. Now letting = (q + i p)/ 2, = (x + i y)/ 2 where d 2 = dqdp/2 we have CA (x, y) = 1 2 (1 + n) exp (q 2 + p 2 ) 2(1 + n). 1 (1 + n). d 2 exp 2 1+n (3.138). exp[i(yq x p)]dqdp. (3.139). Using the standard Gaussian integral e as e s ds = 2 e a (3.140). we straightforwardly obtain CA ( ) = exp[ (1 + n). 2 ] . (3.141).

But from Eq. (3.129) we have CN ( ) = CA ( ) exp(.

2 ) and thus from Eq. (3.135) we nally obtain P ( ) = 1 exp( n 2 ) e d 2 2 2 1 . exp = n n (3.142). This is Gaussian so it may be interpreted as a true probability distribution. Coherent states 1 0 y 1. Q(x,y). 0.2 0 3 2 1. 5 2.5 0. 2.5 5 0.4 0.

3. Q(x,y). 0.2 0.1 0 5 2.

5 5 2.5 0. Fig. 3.9. Q function for (a) a coherent state with n = 10, (b) a number state with n = 3. 3.8 Characteristic functions 2 1 0. 1 2 1 0.75 0.5.

W(x,y). 0.25 0 4 3 2 1 0. 0.2 0. 0.2 0.4.

W(x,y). Fig. 3.10. Wigner function for (a) a coherent state with n = 10, (b) a number state with n = 3. Coherent states Finally, to conclude th is chapter, we examine the Q and Wigner functions for the most classical of the quantum states, the coherent state, and for the most quantum-mechanical state, the number state. For the coherent state = . one easily has the Q function Q( ) = 1 1 . 2 = exp( . 2 ), 1 1 . 2n . n . 2 = exp( . 2 ) . n! (3.143).

whereas for the number state = n n , . Q( ) = (3.144). Setting = x + i y, we .net framework Code 128A plot these as functions of x and y in Fig. 3.

9. The Q function of the coherent state is just a Gaussian centered at while the state for a number state is rather annular of center radius r n. Note how these functions correlate with the phase-space gures introduced in Section 3.

6. The corresponding Wigner functions, obtained from Eq. (3.

136), are. W ( ) = 2 exp( 2. 2 ) (3.145). for the coherent state , and for a number state n . W ( ) = 2 ( 1)n L n (4. 2 ) exp( 2. 2 ), (3.146). where L n ( ) is a Lag uerre polynomial. (The derivation of these functions is left as an exercise. See Problem 3.

12.) We plot these functions in Fig. 3.

10, again with = x + i y. Evidently, the Q and Wigner functions for the coherent state are identical apart from an overall scale factor. But for the number state we see that the Wigner function oscillates and becomes negative over a wide region of phase space.

The Q function, of course, can never become negative. It is always a probability distribution. But the Wigner function is not always positive, the Wigner function of the number state being a case in point, in which case it is not a probability distribution.

A state whose Wigner function takes on negative values over some region of phase space is nonclassical. However, the converse is not necessarily true. A state can be nonclassical yet have a non-negative Wigner function.

As we said earlier, for nonclassical states, the P function becomes negative or more singular than a delta function over some region of phase space. The squeezed states are strongly nonclassical in this sense, as we shall discuss in 7, yet their Wigner functions are always positive. Nevertheless, for these and other nonclassical states, the Wigner function is still of primary interest as the P function generally cannot be written down as a function in the usual sense, whereas this can always be done for the Wigner function.

Furthermore, the Wigner function can be more sensitive to the quantum nature of some states than can the Q function, as we have seen for the number state. More importantly, as we shall see in 7 (and below in Problem 12), the Wigner function can display the totality of interference effects associated with a quantum state [15]. Finally, it happens that it is possible to reconstruct the Wigner from experimental.

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