PHYSICAL REVIEW A
VOLUME 54, NUMBER 5
NOVEMBER 1996
COMMENTS Comments are short papers which criticize or correct papers of other authors previously published in the Physical Review. Each Comment should state clearly to which paper it refers and must be accompanied by a brief abstract. The same publication schedule as for regular articles is followed, and page proofs are sent to authors.
Comment on ‘‘Theorem on nonclassical states’’ A. F. de Lima* and B. Baseia† ˜o Paulo, Caixa Postal 66318, Codigo de Enderecamento Postal 05389-970, Sa ˜o Paulo(SP), Brazil Instituto de Fı´sica, Universidade de Sa ~Received 29 November 1995! In a recent paper, C. T. Lee @ Phys. Rev. A 52, 3374 ~1995! # showed that if rˆ 5 ( n,m r (n,m) u n &^ m u is the density operator describing an arbitrary state, with r (0,0)5 ^ 0 u rˆ u 0 & 50, the state is as nonclassical as possible. Here we exhibit an example in apparent contradiction with the above theorem. @S1050-2947~96!09211-6# PACS number~s!: 42.50.Dv, 42.50.Ar
The generation and detection of nonclassical states of the light field, as well as their striking properties, constitute one of the main subjects of quantum optics. In these studies, a point deserving attention is, ‘‘How much is a nonclassical state?’’ Or, in other words, ‘‘What is the greatest degree of nonclassicity that a state describing a system may exhibit?’’ In preliminary papers Lee @1# established a criterion defining a measure of nonclassicity of a state, and based on this result he was able to prove an interesting theorem on nonclassical states @2#, by establishing the condition under which a system should obey in such a way that its state is as nonclassical as possible, according to the parameter introduced in Ref. @1# ~an equivalent parameter has been discussed also by Lutkenhaus and Barnett @3#!. The result of the theorem is that if rˆ 5 ( n,m r (n,m) u n &^ m u is the density operator describing the state of a system, then if rˆ is such that r (0,0)5 ^ 0 u rˆ u 0 & 50 the state of the system is nonclassical as possible when one removes its vacuum component, as shown in the Ref. @2#. As one example, among others, the author showed that by removing the vacuum component of the coherent state u a & , namely, 2
u c & 5N@ u a & 2e 2 u a u /2u 0 & ]
~1!
2
with N5(12e 2 u a u ) 21/2 and `
u a & 5e 2 u a u
2 /2
(
n50
an
An!
un&,
~2!
where ¯ n 5 u a u 2 , Qc is Mandel’s Q parameter calculated in the state u c & , and a is taken to be real. Note that Qc →21 if a →0 and Qc →0 if a →`; hence, nonclassical effects are appreciable only for small values of a 5 A¯ n ( a ,1). This calls our attention to an apparently contradictory result, as follows: one way to remove the vacuum component is through its displacement, namely, ˆ ~ a !u 0 & 5 u a & , D †
ˆ 5e ( a a 2 a * a) is the displacement operator. In this where D procedure we have that ˆ ~ a ! u 0 &^ 0 u D ˆ †~ a ! rˆ 5 u a &^ a u 5D
¯ 2n e ¯n 21
n
54
4589
On leave from UFPB, Campina Grande ~PB!, Brazil. Electronic address:
[email protected]
1050-2947/96/54~5!/4589~2!/$10.00
2
~3!
* †
~5!
and
one obtains a ~nonclassical! sub-Poissonian statistics Qc 5
~4!
© 1996 The American Physical Society
COMMENTS
4590
Then, e.g., following Ref. @4#, for a 5103 @hence ¯ n 5106 #, the normalization condition becomes
(
n50
r ~ n,n ! >
(
¯210A¯n n5n
r ~ n,n !
~10!
~8!
with ¯ n 610A¯ n 510 610 @1, showing that the contribution of the vacuum component is effectively zero, for a . a 0 , a 0 >103 . Alternatively, note that we have in the Eq. ~7!, for m5n, 6
r ~ 0,0! 6 >e 2 ¯n 5e 210 !1, a 5103 , r max ~ n,n !
¯110A¯n n5n
`
15
54
4
r ~ n,n ! 5e 2 ¯n ~¯ n ! n /n!,
¯ n 5uau2,
~9!
which gives
@1# C. T. Lee, Phys. Rev. A 44, R2775 ~1991!; 44, 6586 ~1992!. @2# C. T. Lee, Phys. Rev. A 52, 3374 ~1995!. @3# N. K. Lutkenhaus and S. M. Barnett, Phys. Rev. A 51, 3340
¯,n ¯). This shows again the negligible with r max (n,n)> r (n contribution of the vacuum component in comparison with the n components around the maximum of the ~Poissonian! line shape describing the function r (n,n), for a . a 0 , a 0 >103 . This result rests as a point to be circumvented in the theorem of Ref. @2#. This work was partially supported by CAPES and CNPq of Brazil.
~1995!. @4# See, e.g., H. M. Nussenzveig, Introduction to Quantum Optics ~Gordon and Breach, New York, 1973!, p. 155.
