Shadowed negative binomial state

5 August 1996

PHYSICS

ELSEVIER

LETTERS

A

Physics Letters A 218 (1996) 151-156

Shadowed negative binomial state Rajani Srinivasan, Ching Tsung Lee

’

Deparment of Physics, Alabama A 8sM University, Normal, AL 35762, USA

Received I9 September 1995; accepted for publication 24 April 19% Communicated by P.R. Holland

Abstract Given a real photon-number distribution p’(n), a virtual distribution p”(n) - (- l)“p’(n) can be defined as its shadow. The combination of the two is called a shadowed state. This concept is illustrated by the negative binomial state. The nonclassical effects are studied by using the Q and R parameters and the nonclassical depth.

For a given

real photon state with photon-number distribution p’(n), let us define a virtual state with

“distribution” pS(n) = (- l)“p’(n),

(1)

which is unnormalized and has negative probability when n is an odd number. of course, the negative probabilities do not make any physical sense; that is why it is called a virtual distribution. When the virtual distribution is added to the real one, we obtain a new photon-number distribution p”(n)

= C[P’W

+P”Wl*

(2)

where -I c=

1+ !

&z) ?I==0

(3)

1

is the normalization constant. It is obvious that Eq. (2) represents a perfectly legitimate photon state, since the negative probabilities are always canceled out. The photon states represented by virtual photon-number distributions are called shadow states because, like a

shadow, they are not real, but they bear close resemblances to the real ones, and they cannot stand by themselves. Then the real states, attached with their shadow states, are called shadowed states. It should be

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152

Letters A 218 (1996) 151-156

pointed out that the summation in Eq. (2) represents an incoherent superposition; so the shadowed states are not pure states. The introduction of shadow states not only provides a more vivid picture of the subject states, but also makes it easier to identify or isolate the origin of nonclassicality in the class of states we call shadowed states. We can do all kinds of things with shadow states as if they were real states; we just need to keep in mind that they must be attached to their corresponding real states at the end. We believe that the shadowed states are very likely, if not always, nonclassical states. Therefore constructing shadowed states is a very simple and systematic way to create nonclassical states. In this Letter we shall illustrate the concept of shadow states and shadowed states by using the negafiue binomial state as the real state, which has been studied recently by Matsuo [l], by Gray et al. [2] and by Vourdas and Bishop [3]. We are particularly interested in exposing the nonclassical effects in this shadowed state. The photon-number distribution of the negative binomial state can be written as

APf3

Phr(n)=(l -t)

T(h)n!

t”*

with 1 s A < CQand 0 < t < 1. The corresponding shadow state has the photon-number distribution

‘,‘;*;;/(-I)“.

A

A(n)=(l-t)

Then the photon-number distribution for the shadowed ntgatiue binomial state can be written as p?(n) =C(l

-t)

^r,‘;,;3)

[l +(-l)“]t”,

(6)

with C= (1 +r”)_‘,

(7)

where r=(l

-t)/(l

+t)

(8)

and it is obvious that we have 0 < r < 1. We can easily calculate the factorial moments of these photon-number distribution as

(9) (10) and

(11) respectively. We can now use Eq. (11) to try various criteria to detect possible nonclassical effects in the shadowed negative binomial state. We first try Mandel’s Q parameter [4] Q~ (@‘>(n>

t [ Ar”(1 +r)2+(l

(nj2 =

(1 -t)(l

+rA)(l +r”)(1-rA+‘)

which does not indicate any nonclassical effects.

+rA+2)] >O,

(12)

R. Srinivasan,

Fig.

C.T. Lee/Physics

Letters A 218 (1996) 151-156

153

I. Plot ofR(2, 2) as a function of I/h and r with the positive part clipped off. A negative value implies nonclassicality.

Fig. 2. Plot of R(4.2)

as a function of I/h

and

t with the positive part clipped off. A negative value implies nonclassicality

We then try the generalized Q parameters defined as [5] Q’“‘E((~(~+‘))-(n(k))(n))/(n).

(‘3)

For k = 2 we have f2 Q’2’

=

[ Ar*(

1 - r)( 1 + r)’ + 2( 1 + r”)( 1 - r’+‘)] (1 -t)‘(l

+rA)(l

-r’+‘)

> 0,

(14)

which again fails to expose any nonclassical effects. We then try the R parameters defined as [6] R(I,

m) ~(n”+“Xn’m-“)/(n(“Xn(m))-

1,

12m.

(15)

It has been shown in Ref. [6] that R(1, m) must be positive for classical radiation; therefore R(1, m) < 0 implies nonclassicality. We have used Eq. (11) in Eq. (15) and carried out the numerical evaluation of R(2, 2) and R(4, 2) as functions of l/A and t; the results are shown as three dimensional plots in Figs. 1 and 2, respectively. In order to see the negative-valued parts of these plots more clearly, the positive-valued parts are clipped off. Both plots indicate nonclassicality only for small values of t. The negative-valued part narrows as we go from R(2, 2) to R(4, 2); so we are discouraged to go any higher. We believe that the R parameters still do not tell the whole story about the nonclassical nature of the shadowed negative binomial state. So we try next the nonclassical depth as defined by Lee [7]. A quantum state is defined as a nonclassical state whenever its P function is a singular distribution or not a positive definite, normalizable, regular function. Using Eq. (9) and the formula derived by Lee [81, we can determine the P function [9] of the real negative binomial state to be

(‘6) where ak+

s(k.O(

z,

z)

c

I

-6(z, dZk&

i)

(17)

R. Sriniuasan, C.T. Lee/Physics

154

Letters A 218 (19%) 151-1.56

is the derivative of the delta function in the complex domain with respect to z and its complex conjugate. Similarly, the P function of the shadow state can be obtained as

--c

O1 r(k+A)

Ph( z,

2) = r* C k_()

k

(‘8)

S(k,k)( z, 2). I-( A)k!k! ( 1 + r 1

It is shown in Ref. [3] that the singular distribution of Eq. (16) can be converted into a regular function by way of Fourier transformations, back and forth, as 1 PAYz, F) --- r(*)

(

l-t - f

1-t 1z12(A-l) exp - -lz12 I ( 1 h

1

(‘9)

But it is not possible to do so with the singular distribution of Eq. (18). This is the origin of the nonclassicality in the shadowed state. However, the P function can always be smoothed by the following convolution transformation into the R function defined as P( w, W) d*w,

where T is a continuous parameter. If T is large enough, the R function can become positive definite and acceptable as a classical distribution function. The smallest T, denoted by T,,,, that can achieve this goal can serve as a measure of how nonclassical a quantum state is; hence, it is called the nonclassical depth. The range of T,,, can be specified to be 0s 7,s

1.

(21)

A very interesting physical meaning of T,,, is that it is the minimum expectation value of the thermal photon number necessary to wash out any nonclassical effects in a quantum state. We now try to determine the nonclassical depth of the shadowed negative binomial state. We can obtain directly the R function from given factorial moments of the photon-number distribution by using the following formula,

(22) Using Eq. (9) in Eq. (22), changing the order of summations and carrying out one of them, we obtain

(23) where ,F,(cu, /3; X) is the confluent hypergeometric function, G,(I~1*,T,t)~(7+t--7t)-~exp

(1-t>Izl’ r+t-r7t

(24)

and we have used Kummer’s first formula [lo] to arrive at the last expression. Similarly, using Eq. (10) in IQ. (221, we obtain Rj;( I zI*, T) = ~*+(l

- t)“G( 1z] *, T, -t).

(25)

R. Srinivasan, C.T. Lee/Physics

Fig. 3. Plot of the nonclassical

Letters A 218 (1996) 151-156

depth 7” as a function of

155

t with A as a parameter.

We can combine Eqs. (7), (8), (231, and (24) to obtain the R function for the shadowed negative binomial state as

R;(~z~~,T)=

TA_‘( 1 -

t’)” [G,(IzIZ,~,~)+G~(Iz12,~,

(1 +?)*+(I-+

When A = I+ 1 is an integer, the confluent hypergeometric polynomial. We then have G,+,( I z12,T,

t) = (T+r-

Tr)-‘-I

exp

-t>].

(26)

function in Eq. (24) reduces to a Laguerre

(27)

We can now use Eq. (26) to determine the nonclassical depth of the shadowed state. For the special case of A = 1, the negative binomial state reduces to a thermal state, and we have 7m =r/(l

+r).

(28)

Unfortunately, except for this special case, it seems to be impossible to determine 7, analytically. So we have carried out a numerical evaluation of T,,, for the cases of A = 2,5, 10, and 20; the results, together with the case of A = 1, are presented in Fig. 3. From this figure we can see that Eq. (21) is always satisfied, which implies that the shadowed negative binomial state is always nonclassical. We can also see clearly that the nonclassicality of the state increases both as r increases and as A increases. Strangely, we might drew just the opposite conclusion if we examine the R parameters presented in Figs. 1 and 2. We believe that the nonclassical depth, which is more difficult to determine, is always more faithful in exposing the nonclassical nature of a quantum state than any of the other criteria available. In conclusion, we have used the negative binomial state as the real state to illustrate the concept of shadow and shadowed states as a simple way to construct nonclassical states. We have used the Q parameters, the R parameters, and the nonclassical depth to examine the nonclassical effects in the shadowed negative binomial state; and we have found that only the last one is reliable for this purpose.

References [l] (21 13) [4]

K. C. A. L.

Matauo. Phys. Rev. A 41 (1990) 519. Gray, R. Srinivasan and C.T. Lee, Inverse Problems 10 (1994) L35. Vourdas and R.F. Bishop, Phys. Rev. A 51 (1995) 2353. Mandel, Opt Lett. 4 (1979) 205.

156

R. Sriniuasan. C.T. Lee/Physics

Letters A 218 (1996) 151-156

[s] C.T. Lee, Quantum Opt. 6 (1994) 27. [6] CT. Lee, Phys. Rev. A 41 (1990) 1721.

[7] [8] [9] [IO]

C.T. Lee, Phys. Rev. A 44 (1991) R2775. C.T. Lee, Phys. Rev. A 45 (1992) 6586. E.C.G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277. G. Arfken, Mathematical methods for physicists, 3rd Ed. (Academic Press, Orlando, 1985) p. 754