- Email: [email protected]
Phase properties of superposition of pair-coherent states
1 October 2001
Optics Communications 197 (2001) 363±373
www.elsevier.com/locate/optcom
Phase properties of superposition of pair-coherent states Faisal A.A. El-Orany a,b, J. Perina b,* a
Department of Mathematics and Computer Science, Faculty of Natural Science, Suez Canal University, Ismailia, Egypt b Department of Optics, Palack y University, 17. listopadu 50, 772 07 Olomouc, Czech Republic Received 7 June 2001; accepted 2 August 2001
Abstract In this article we discuss the phase properties of the superposition of pair-coherent states in the framework of Pegg± Barnett formalism. We also investigate the behaviour of Wigner and Weyl functions for the single-mode case. We show for speci®c modes and quantities that even-pair-coherent states can map onto odd-pair-coherent states and vice versa based on the values of the degeneracy parameter. Also we discuss the possibility of using the Weyl function to describe the phase distribution as a new tool in quantum optics. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 42.50.Dv Keywords: Phase properties; Pair-coherent states; Quasi-distributions; Phase space
1. Introduction
B n; q
Two modes of a quantized electromagnetic ®eld can become entangled showing many nonclassical eects compared to the decoupled modes. One of these two-mode optical cavities which involves strong entanglement is the pair-coherent states [1], which are de®ned as jf; qi
1 X
B n; qjn q; ni;
1
n0
where * Corresponding author. Address: Joint Laboratory of Optics of Palack y University and Institute of Physics of Academy of Sciences of Czech Republic, 17. listopadu 50, 772 07 Olomouc, Czech Republic. Tel.: +420-68-563-4263; fax: +42068-522-5246. E-mail address: [email protected] (J. Perina).
N qjfjn exp i/ p ; n! n q!
2
and N q is the normalization constant expressed as " N q
1 X n0
2n
jfj n! n q!
#
ijfj q Jq 2ijfj
1=2
1=2
;
3
where Jq is the ordinary Bessel function of order q; f jfj exp i/ and this makes the pair-coherent state as a partial phase state; q is the ``charge'' (degeneracy) parameter, which is a ®xed integer. The Fock state basis jn q; ni means that mode a includes n q photons, while mode b has n photons. In fact, these states jf; qi are eigenstates of both the pair-annihilation operator as well as the dierence in the number operators for the two modes, i.e.
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 4 7 2 - 9
364
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
^ qi fjf; qi; a^bjf; ^ ay a^
4
^ qi qjf; qi; b^y bjf;
5
where a^ and b^ are boson operators. It is worth mentioning that these states are related to the harmonic-oscillator coherent states jca ; cb i [1] by Z i q p E du hp jf; qi f exp iu f exp iu 2p a p E f exp iu exp jfjN q: 6 b
Further, the probability of ®nding n photons in mode b and n q photons in mode a in jf; qi is 2
P n q; n jhn q; njf; qij N 2 q
jfj2n ; n! n q! 7
which is not Poissonian. We proceed, pair-coherent states admit strong nonclassical properties such as various forms of squeezing, violations of the Cauchy±Schwarz and Bell inequalities [1] as well as many-photon antibunching [2]. Also their phase properties have been investigated in the view of Pegg±Barnett technique [3]. Moreover, evolution of pair-coherent states in two-level [4] and three-level [5] Jaynes±Cummings model as well as in a nondegenerate parametric ampli®er [6] have been discussed. These studies have been supported by several proposals for generation of these states. For example, they may be generated in a process involving the competition between nondegenerate parametric ampli®cation and nondegenerate two-photon absorption [1]. Also they can be generated in a two-dimensional trapped ion [7] in which the ion is excited bichromatically using three laser beams along dierent directions in its horizontal plane. If the initial vibrational state is a two-mode Fock state, the equilibrium motional state of the ion is the highly correlated pair-coherent state. On the other hand, the concept of superposition principle has been applied to the pair-coherent states aiming to produce new type of two-mode cat states, which can involve more pronounced nonclassical eects than the original states. Such superimposed states possess two mechanisms con-
trolling their behaviours, which are entanglement and interference in phase space. The superposition of pair-coherent states has been de®ned [8] in the Fock states basis as jf; qi
1 X
B n; q; jn q; ni;
8
n0
where n
B n; q;
N q; jfj exp i/ p 1 1n ; n! n q!
9
and the parameter takes on values 0, 1 and 1 associated to pair-coherent, even- and odd-paircoherent states, respectively; and N q; is the normalization constant, which reads ) 1=2 ( 1 X jfj2n 2 n N q; : 1 jj 2 1 n! n q! n0 10
Even and odd states of this type are orthonormalized states and they are eigenstates of the op^ 2 and a^y a^ b^y b, ^ i.e. erators ^ ab 1 1 hf; qjf; qi1 1
^ jf; qi ^ ab 1 2
^ ay a^
1
0;
f2 jf; qi1
^ qi b^y bjf; 1
1
11 1 ;
qjf; qi1
12 1 :
13
Further, the behaviour of the photon-number distribution of jf; qi is completely dierent from that of pair-coherent states as a result of the superposition principle. This is quite clear by comparing the following expression for superposition pair-coherent states: P n q; n N 2 q
2n
n 2
jfj 1 1 ; n! n q!
14
with Eq. (7) for the pair-coherent states, i.e. jf; qi possess nonclassical pairwise oscillations in the photon-number distribution which do not exist in the pair-coherent states. Moreover, jf; qi can produce nonclassical eects stronger than those of pair-coherent states, e.g. the violation of the Cauchy±Schwarz inequality for the superposition of pair-coherent states are always stronger than that for pair-coherent states of a nonzero q [8]. The
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
superposition of pair-coherent states can be generated by means of the competition between a nonlinear multiphoton parametric process and incoherent multiphoton absorption [8], or by using trapped ion technique as for pair-coherent states, however, in this case the trapped ion is excited bichromatically by ®ve laser beams along dierent directions in its horizontal plane [9]. Furthermore, this type of states can be generated via the atomic interference [10]. Finally, the concept of phase is an important as well as a controversional issue since the early days of quantum optics [11]. This importance has been motivated by the experimental realization of the phase in optical homodyne tomography [12]. Generally speaking, there are various methods for treating phase in quantum optics having advantage and disadvantage points. In this article we use the Pegg±Barnett formalism [13] to treat the phase distribution for the superposition of pair-coherent states. Actually, this technique is more convenient for this problem since it does not include such diculties as the others, i.e. the lack of unitarity. Further, we discuss the single-mode behaviour of both Wigner (W) and Weyl functions. Without any loss of generality we assume that q is nonnegative integer. The article will be organized in the following sequences: In Section 2 we discuss Wigner and Weyl functions. In Section 3 we demonstrate the phase properties by means of Pegg±Barnett formalism and also we compare the phase information included in both Wigner and Weyl functions. Finally we summarize the main conclusions in Section 4.
and is sensitive to interference in phase space. To study W function we start by calculating the single-mode symmetrical characteristic function de®ned as o n Ca m; Tr q^a expm^ ay m a^ ; 15 where q^a Trb q^ is the reduced density matrix for mode a. We restrict our formulae to those of the mode a, however, those of mode b can be easily obtained by some tricks. It can be shown that Tra q^2a < 1 for 0 < jfj < 1 [15], i.e. single-mode state resulting from the superposition of pair-coherent states is mixed. This is arising from the strong correlation between the two modes. For the superposition of pair-coherent states the symmetric characteristic function (15) reads 1 1 2 X 2 2 Ca m; exp jmj jB n; q; j Lnq jmj ; 2 n0 16 where Lm are Laguerre polynomials of order m. Employing Eq. (16) in the following relation Z 1 Ca m; exp bm b m d2 m; Wa b; 17 p and carrying out the integration, we arrive at the single-mode W function as Wa b;
1 X 4 2 2 exp 2jbj jB n; q; j f n q; b; p n0
18 where f n q; b 1
2. Single-mode Wigner and Weyl functions Recently, there is a lot of both theoretical and experimental work on quantum optical tomography whereby the W function of optical signals is reconstructed from a set of optical homodyne measurements (see Ref. [14] and references therein). This function is an eective tool in quantum optics, which can give a good indication about the possibility of occurrence of nonclassical eects, e.g. it can involve negative values, stretched contours, etc. Furthermore, it is well-behaved function
365
nq
2
Lnq 4jbj ;
19
and b x iy. In fact, the behaviour of the W function depends essentially on the function (19). Furthermore, expression (18) reveals that the behaviour of the W function of the single-mode case of the superimposed pair-coherent states is somewhat similar to that associated with the Fock states in such a way that the main contributions are localized at the origin. This situation is similar to binomial states [16] and geometric states [17]. Further, the information about the entanglement of the two subsystems is included in the coecient B n; q; . It is worth noting that the parameter q
366
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
plays a signi®cant role in the behaviour of W function. To be more speci®c, it can map the W function of the even-pair-coherent states onto that of odd-pair-coherent states or vice versa. This fact is clear from Eq. (19) where this expression for the even case f 2n q; b with q 1; 3; 5; . . . is equivalent to the odd case f 2n 1 q; b with q 0; 2; 4; . . ., correspondingly. Further, we have seen that this fact is established well in the behaviour of Mandel's Q parameter, too. It is worth reminding that Mandel's Q parameter is characterizing the departure from Poissonian photon statistics, i.e. if the variance of the photon number h D^ ay a^2 i is less than the mean photon number h^ ay a^i, then the state under consideration has subPoissonian statistics meaning that it has no classical description. More speci®cally, if Q 0, 0 > Q P 1 and Q > 0, then the state is called Poissonian, sub-Poissonian and super-Poissonian, respectively. We proceed, we have seen that when the value of the degeneracy parameter q increases, the oscillatory behaviour between positive and negative values in W function is more pronounced. Further the interference in phase space manifests itself by more pronounced nonclassical negative values of W function for even- and odd-paircoherent states compared to that for pair-coherent states. There is only one exceptional case to this rule which are the even-pair-coherent states for
q 0 and if jfj is small. Fig. 1 gives insight into this point where the W functions of pair-coherent (a) and even-pair-coherent states (b) are plotted for given values of the parameters. Comparison of Fig. 1(a) and (b) is instructive. From Fig. 1(b) we can see that W function is positive and has a Gaussian form. These characters indicate that nonclassical eects cannot be established (see Fig. 1 in Ref. [18]). Now if we turn our attention to the W function of pair-coherent states for this case (Fig. 1(a)) one can observe that it has a Gaussiantop-hole peak similar to speci®c type of geometric states [17]. Actually, this type of geometric states exhibits a maximum sub-Poissonian statistics and this is also valid to pair-coherent states [1]. Further, the comparison of Fig. 1(a) and (b) emphasizes that the interference in phase space can destroy the nonclassical eects inherent in the system, in particular, for speci®c quantities. This is in contrast with the superposition of coherent states [19], which can exhibit various nonclassical properties even if the original states are close to the classical states. We proceed, from the structure of Eq. (19) one can ®nd out that the negativity of W function relies on both the type of the states under consideration and the value of the degeneracy parameter q. Fig. 2 has been given for this purpose where W function of even-pair-coherent states have been plotted for dierent values of q and
Fig. 1. W function of the single-mode case for pair-coherent states (a) and for even-pair-coherent states (b) when jfj; q 0:75; 0.
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
367
Fig. 2. W function of the single-mode even-pair-coherent states when jfj 1:5 and for (a) q 0; (b) q 1; and (c) q 2.
®xed value of jfj. From these ®gures, the nonclassicality is more pronounced when q is odd. In other words, the strongest sub-Poissonian statistics of the even single-mode case occur when q is odd. Further, as is known the superposition of pair-coherent states cannot produce squeezing in the individual modes because of the strong correlation between modes and this manifests itself in an isotropic form W function. Moreover, from Fig. 2 we can see that when q is odd W function has downward peak and upward peak when q is even. This situation is in general valid for this case for all values of q. This feature (downward/upward peak) indicates that these two types of states are orthogonal [14]. Also comparison of Fig. 1(b) and Fig. 2(a) shows that when jfj increases the nonclassical eects start to appear [18]. In conclusion, the behaviour of W function is in a good agreement with the quantum features of these states. As we have mentioned earlier the behaviour of oddpair-coherent states can be obtained from that of even ones; for this reason we have not included the behaviour of odd-pair-coherent states. However, it is reasonable mentioning that there is a signi®cant dierence between these two states, i.e. for even (odd)-pair-coherent states even (odd) numbers of photons have nonzero probability of being observed. Such oscillations in the photon-number distribution may be interpreted as an interference in phase space [20]. Further, we noted that the oscillations in the photon-number distribution increase as the values of jfj increase, but in all cases the distribution exhibits Gaussian envelope similar to that of pair-coherent states [4].
Now we look at the interference in phase space from the point of view of Weyl function. In this treatment we show that this function can carry information similar to that included in W function [21,22]. Further, this function has been used as an auxiliary quantity in theoretical studies (e.g. in homodyne tomography [14]) and can be de®ned in many ways. For example, the single-mode Weyl function is just the symmetrical characteristic function (16), for convenience we can write X iY Wa X ; Y Ca m p p ; ; 20 2 2 where X ; Y represent the position and momentum increments [21]. Replacing m by its new argument in Eq. (16) we obtain 1 2 2 X Y Wa X ; Y exp 4 2 1 X X Y2 2 jB n; q; j Lnq : 2 2 n0 21 From Eq. (21) one can see that Weyl function has its maximum value at the origin, which is positive, equal to unity and also jWa X ; Y j 6 1. Further, comparison of this function with W function (18) shows that they can have similar behaviour nq when n q is even integer (i.e. 1 1). In this caseÐapart from an insigni®cant prefactor 4=pÐeach of which can transform pto the other via the transformation X ; Y $ 2 2 x; y. On the other hand, for the inverse situation (i.e. n q is odd integer) the locations of the cross-term and
368
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
Fig. 3. Weyl function for the single-mode case for q; ; jfj 0; 0; 0:75; 1; 0; 1:5 and 1; 1; 1:5 is shown in (a)±(c), respectively.
auto-terms in W function and in Weyl function interchange. Generally, we have seen that the behaviour of Weyl function is in a good agreement with that of the W function. For the sake of comparison we have displayed the Weyl function in Fig. 3(a)±(c) for three dierent cases. From the above discussion it is obvious that the Weyl and W functions corresponding to Fig. 1(b) are similar, i.e. they are positive and have a Gaussian shape. This shape can be considered as the border line between classical and nonclassical behaviour regarding to Weyl and W functions as well. In other words, the state having this form will be close to the classical wave packet, otherwise there is a possibility to include nonclassical eects. Now comparison of Figs. 1(a) and 3(a) as well as of Fig. 3(b) and (c) is instructive. The former shows how the nonclassical eects manifest themselves in the behaviour of Weyl function, whereas the latter demonstrates the role of the interference in phase space in the behaviour of Weyl function. This means that Weyl function can be used as an additional tool in describing the phase space where the nonclassical eects can manifest themselves in this function as negative values, many-fold structure or stretched shape (for squeezed states). Actually, there are some weak points in using this function which are: it is in general complex function and in this sense there is no direct relation between this and the measurements. The ®rst point has been solved by analysing the behaviour of its imaginary and real parts individually as well as its modulus [21,23]. The second point can be overcome using the fact that this function can be de-
®ned in terms of the Fourier transformation of W function and the latter can be reconstructed in quantum tomography from optical homodyne measurements [14]. In other words, indirect measurement scheme can be arranged. Moreover, the connection between these two functions has been used in developing a ®ltering algorithm to the noisy Wigner function [22]. In this technique the Radon transform of the W function of a noisy signal has been taking and producing the W function of the clean (noiseless) signal. This process has been performed with the help of Weyl function. Actually, this technique is of practical interest since the quantum signals inevitably contain noise. So far we have discussed the behaviour of mode a only, however, the behaviour of mode b is quite dierent. This is evident from Eq. (8) where mode a possesses q photons more than those in mode b. Nevertheless, the behaviour of this mode can be easily understood from their own expressions. To show this we ®rst give expressions for the W and Weyl functions related to mode b. Careful examination of Eq. (8) shows that one can obtain the required expressions by simply setting q 0 in the quantities resulting from jn qi in the expressions for mode a. To be more speci®c, we set q 0 in both Eq. (19) (for W function) and in the index of the Laguerre polynomial in Eq. (21) (for Weyl function). From the formulae for mode b one can immediately conclude that the degeneracy parameter q has no signi®cant role in the behaviour of this mode. Also, for the even-pair-coherent states, the Weyl and Wigner functions possess similar
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
behaviour in phase space as we have discussed above. More illustratively, they are almost represented by a Gaussian peak similar to that for vacuum ®eld with negligible negative values, i.e. there is a negligible nonclassical eect [18]. On the other hand, the nonclassical negative values dominate in the form of these functions for the odd-pair-coherent states. This agrees with the fact that odd-pair-coherent states can exhibit more pronounced sub-Poissonian statistics.
3. Phase distribution In this section we investigate the phase distribution for the superposition of pair-coherent states using the Pegg±Barnett formalism [13]. Also we address the question: how far the Weyl function is relevant to describe the phase distribution? Basically, the Pegg±Barnett formalism is based on de®ning Hermitian-phase operator in a truncated Hilbert space, and after calculating all necessary expectation values for the target state, the dimension of this space extends to include the whole space by appropriate limit [13]. The details related to this technique have been extensively discussed in the literature [24,25] and then we do not repeat them here. The generalization of this method to two-mode case is a straightforward way [3]. Now the joint-phase distribution for the superposition of pair-coherent states reads 1 X 1 P H1 ; H2 12 B n; q; B k; q; 2 2p n>k #) " cos n
k H1 H2
:
369
squeezed states [26]. It is worth reminding that the single-mode version of these states has dissimilar quantum features, e.g. the two-mode squeezed states possess thermal properties, however, the pair-coherent states are pure nonclassical states. Moreover, when jfj ! 0, expression (22) gives a joint-uniform distribution corresponding to the two-modes Fock states jq; 0i. Also we note that the parameter q broadens the phase distribution, i.e. speeds up the randomization of the phase (classical phase). Actually, these two cases are established irrespective of which states has been considered, i.e. 0, 1 or 1. We proceed, for even (odd) states it is clear that P H1 ; H2 is a p-periodic function in both H1 and H2 owing to the argument that n k is always even integer. This makes that the distribution involves more nonclassical oscillations compared to that associated with paircoherent states [3] where, within the phase window, the distribution (22) produces maxima along ®ve parallel lines controlled by the equations H 0, (1/2)p, p, however, that of pair-coherent states gives maxima along three lines only (H 0, p). Fig. 4 sheds the light on the behaviour of distribution (22) for given values of the parameters. Comparison of this ®gure with Fig. 1 for paircoherent states given in Ref. [3] shows the role of interference in phase space. Moreover, from Fig. 4 one can observe that the distribution is symmetric around the diagonals of the square of the phase windows H1 ; H2 and it re¯ects the dependence of the distribution on the phase sum H only. Also
22
The phase / of the parameter f disappeared in Eq. (22) owing to the choice of the phase window, for more details about the derivation of Eq. (22) the reader can consult Ref. [3]. Expression (22) reveals that the distribution depends only on the phase sum H H1 H2 and this makes that the states jf; qi are uniformly distributed regarding to the individual phases as well as to the phase dierence H H1 H2 . Such behaviour is similar to that of pair-coherent states [3] and of two-mode
Fig. 4. Joint-phase distribution of the even-pair-coherent states for q; ; jfj 1; 1; 1.
370
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
we can see that the phase localization occurs for ®xed phase window. Now for the symmetrical case (q 0), i.e. when the photons in the system are equally distributed over the two modes, distribution (22) reduces to the following closed form: P H
N 2 0; 2p2 1
f 1 2 cosh 2jfj cos H
2 sinh 2jfj cos H
2 cos 2jfj sin H g;
23
all the notations have the same meaning as before. From this expression it is obvious that (for even case) the distribution possesses maximum N 2 0; 1=p2 cosh2 jfj at H mp, m 0, 1, 2 and minimum N 2 0; 1=p2 cos2 jfj at H m p= 2, m 1, 3. This shows that when jfj ``evolves'', the maxima of the distribution increase monotonically, while the minima of the distribution evolve periodically between 0 and N 2 0; 1=p2 . In other words, when jfj increases the distribution becomes narrower similarly as in pair-coherent states [3]. Narrower distribution means more wellde®ned phase. We conclude this part by referring that the joint-phase distribution can be re-expressed in terms of the parameters H after applying the casting procedure mod 2p [24]. ^ 2 i h U ^1 The sum-phase variance h DU 2 2 ^ ^ ^ U2 i hU1 U2 i , for the state under discussion, is given by ^ 2 i 2 p2 h DU 3
4
1 X B n; q; B k; q; n>k
n
k
2
:
24
The ®rst term in expression (24) is the sum of the individual phase variances of the single modes, however, the second term results from the correlation between the single-mode phases and this is a direct consequence of the quantum mechanical eect. In Fig. 5 we have displayed the sum-phase variance for dierent values of the degeneracy parameter q. From this ®gure one can observe that ^ 2 i is smooth, it starts from the behaviour of h DU 2p2 =3 of the two-modes Fock states jq; 0i, goes to its minimum values and eventually the phase locking occurs. In other words, as jfj increases, the sum-phase variance decreases, i.e. the sum of the two phases becomes less uncertain. The role of
Fig. 5. The phase variance of the even-pair-coherent states for q 0 (Ð), 5 (± ± ±) and 10 (- - -).
the superposition of the states manifests itself as an oscillatory behaviour over small values of jfj, in particular, for q 0 and also in the sense that sum-phase variance cannot approach zero. This situation is in contrast with that of pair-coherent states where when jfj is large the variance approaches zero and the classical situation of perfectly de®ned phase sum dominates. Further, we can observe that when q increases and for ®nite ^ 2 i goes to randomly distributed values of jfj, h DU phase more rapidly. This fact can be realized by comparing the behaviour of the dierent curves in Fig. 5. From this discussion we can conclude that the phase information involved in the superposition of pair-coherent state is more pronounced than that included in pair-coherent states. We have seen in the preceding section that Wigner and Weyl functions can carry similar information about the nonclassical eects of the ®eld amplitudes, so it is convenient to follow the same line to see the situation regarding to phase distribution. Firstly, it is worth mentioning that the phase distribution resulting from the integration of W function over the radial part can give similar conclusions as given here. More illustratively, the distribution of the single-mode case is uniformly distributed and that of the two-mode case depends
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
only on the sum of the phases [6], as we see in the following. On the other hand, it is dicult to use Weyl function as a standard technique for analysing the phase in quantum systems, however, for particular states and after some arrangements this function can achieve this goal. For example, intuitively, the states whose Weyl and Wigner functions have similar structures can provide similar phase information. For the states under discussion this can be achieved when n q (q) is even integer for the single-mode (two-mode) case. Further, according to the lack of normalization of the Weyl function (this fact is clear from Eq. (20)) we have to suggest the de®nition of the uniformly distributed phase in the view of this function. This can be easily done by calculating the phase distribution related to the Weyl function for the Fock state jmi, m which is 2 1 and taking its absolute value we obtain the uniformly distributed phase with respect to this function, which is 2. In other words, factor 2 for the phase resulting from this function is equivalent to 1= 2p of the uniformly distributed phase from the other techniques. This should be used with a caution to get equivalence between all these techniques. In other words, we should apply such a transformation to the prefactor of the distribution (resulting from Weyl function) only, which will be 2 for the single-mode case and 4 for the two-mode case. We now write down the jointphase distribution for the superposition of paircoherent states regarding to two-mode Weyl and Wigner functions as P H1 ; H2
h1 2p
f1 2 2
1 X
G n; k
n>k
G n q; k qB n; q; B k; q; #) " cos n
k H1 H2
;
25
where r n! 1k 2 1=2 n k n k 1 G n; k C k! 2 2 n k! n k F k; 1; n k 1; 2 ; 2 2 26
371
where C and F are the Gamma and the con¯uent hypergeometric functions, respectively; and q h1 1 and 1 for the distribution resulting from Weyl and W functions, respectively. The derivation of Eq. (25) has been done by evaluating ®rstly the two-mode Weyl and Wigner functions and then by applying the standard technique of obtaining the phase distribution from the quasiprobability functions. From Eqs. (22) and (25) we can conclude that the three distributions can give similar behaviours in a qualitative way and for speci®c values of the parameters. Further, from Eq. (25) it is clear that when q is even integer the Weyl-phase and Wigner-phase distributions are typical, i.e. they carry the same phase information. The origin of this is that the two-mode Weyl and Wigner functions are similar. On the other hand, when q is odd integer, they give opposite behaviours. In fact, the Wigner and Weyl techniques can include negative values making them less favourite to be used in the phase analysis. Nevertheless, these negative values should be interpreted as a signature of the quantum nature of the ®eld state rather than some peculiarity of its phase properties. For more details related to the analysis of such phenomenon the reader can consult Ref. [27] (and references therein). Further, for some states the Wigner-phase technique can give similar behaviour as the Pegg±Barnett one, e.g. for the displaced number states whose phase distribution has been interpreted in terms of the area of overlap in phase space [28]. We conclude this section by showing that the W function is more suited to describe the phase than the Weyl function. This can be easily obtained by comparing the way by which the single-mode Weyl (Wigner) function can be obtained from its joint function. For example, the single-mode Weyl function results from its joint one by simply setting the parameters of the absent modes by zeros, however, the integration has to be performed for the joint-W function to get the single-mode W function. Such a distinction exists also in the phase distribution. More illustratively, the single-mode phase distribution derived from the Weyl-jointphase distribution by integration is dierent from that evaluated directly from the single-mode Weyl function. We emphasize this via analysing the
372
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
phase distribution of single-mode case with respect to Weyl function for the state under consideration. Before starting this we should stress that this distribution is uniformly distributed regarding to W function or Pegg±Barnett techniques irrespective of the type of states. Now applying the standard technique, the single-mode phase distribution resulting from the single-mode Weyl function reads 1 1 X n 2 27 P Hj 1 jB n; q; j ; 2p n0 however, that resulting from the Weyl-joint-phase distribution by integration over H1 or H2 is 1= 2p. Actually, these two results can coincide for even (odd)-pair-coherent states. So we can conclude that in general when the two-mode Weyl and Wigner functions are similar it is sucient to deal with the joint-phase distribution of the Weyl function to obtain all relevant results and to exclude those results coming from direct calculation of the single-mode Weyl function.
4. Conclusion In this article we have investigated the phase properties of the superposition of pair-coherent states in the view of Pegg±Barnett formalism. Also we have discussed the possibility of using the Weyl function to investigate such eects. Further, we have analysed the behaviour of the single-mode Wigner function and Weyl function for the states under consideration. It is important to stress that these two functions can provide similar information about the nonclassical eects. For the present states, we have shown that these functions can exhibit more pronounced nonclassical eects resulting from the strong correlation between the two modes. Also we have shown that for a speci®c mode, even states can carry information about odd states and vice versa meaning that one can enhance the nonclassical sub-Poissonian statistics based on adjusting the dierence between the mean number of photons in the two modes. Further, it is worth mentioning that this situation is also valid for the two-mode squeezed number states suggested in
Ref. [29]. It should be borne in mind that Weyl and Wigner functions (besides the others, e.g. Husimi Q- and Glauber P-functions) can be used as qualitative tools in quantum optics and their irregularities are necessary to indicate the existence of the nonclassical eects but they are not sucient. On the other hand, we have shown that the joint-phase distribution can exhibit more pronounced nonclassical oscillations arising from the superposition between the dierent components of the states. Further, we have shown that the three types of the distributions (Weyl, Wigner and Pegg± Barnett) can lead to this result. However, the single-mode phase is always uniformly distributed regarding to Wigner and Pegg±Barnett techniques. Moreover, we have shown that Weyl function can carry similar information about the phase as Wigner function for speci®c types of states. The ®nal remark is that the Wigner and Weyl functions are problematic to be used in analysing the phase because they can include negative values and also for some states the Weyl function is complex; so one has to deal with its absolute value, which can lead to missing information about the phase.
Acknowledgements J.P. and F.A.A.E.-O. acknowledge the partial support from the project LN00A015 of Czech Ministry of Education.
References [1] G.S. Agarwal, Phys. Lett. 57 (1986) 827; G.S. Agarwal, J. Opt. Soc. Am. B 5 (1988) 1940. [2] C.T. Lee, Phys. Rev. A 41 (1990) 1569. [3] Ts. Gantsog, R. Tanas, Opt. Commun. 6 (1991) 145. [4] S.-C. Gou, J. Mod. Opt. 37 (1990) 1469. [5] R. D'Souza, R.R. Puri, J. Opt. Soc. Am. B 11 (1994) 1108. [6] C.C. Gerry, J. Mod. Opt. 42 (1995) 585. [7] S.-C. Gou, J. Steinbach, P.L. Knight, Phys. Rev. A 54 (1996) R1014. [8] C.C. Gerry, R. Grobe, Phys. Rev. A 51 (1995) 1698; X.-M. Liu, Phys. Lett. A 279 (2001) 123. [9] S.-C. Gou, J. Steinbach, P.L. Knight, Phys. Rev. A 54 (1996) 4315.
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373 [10] S.-B. Zheng, G.-C. Guo, Opt. Commun. 137 (1997) 308. [11] See the special issue of, Phys. Scripta T 48 (1993); R. Lynch, Phys. Rep. 250 (1995) 367. [12] D.T. Smithey, M. Beck, J. Cooper, M.G. Raymer, Phys. Rev. A 48 (1993) 3159; M. Beck, D.T. Smithey, M.G. Raymer, Phys. Rev. A 48 (1993) 890; M. Beck, D.T. Smithey, J. Cooper, M.G. Raymer, Opt. Lett. 18 (1993) 1259; D.T. Smithey, M. Beck, J. Cooper, M.G. Raymer, M.B.A. Faridani, Phys. Scripta T 48 (1993) 35. [13] S.M. Barnett, D.T. Pegg, J. Phys. A: Gen. Math. 19 (1986) 3849; S.M. Barnett, D.T. Pegg, Phys. Rev. A 39 (1989) 1665; S.M. Barnett, D.T. Pegg, J. Mod. Opt. 36 (1989) 7. [14] U. Leonhardt, Measuring the Quantum State of Light, Cambridge University Press, Cambridge, 1997. [15] C.C. Gerry, A. Benmoussa, Phys. Rev. A 61 (2000) 033812. [16] D. Stoler, B.E.A. Saleh, M.C. Teich, Opt. Acta 32 (1985) 345; A. Vidiella-Barranco, J.A. Roversi, Phys. Rev. A 50 (1994) 5233; M.S. Abdalla, M.H. Mahran, A.-S.F. Obada, J. Mod. Opt. 41 (1994) 1889; A.-S.F. Obada, M.H. Mahran, F.A.A. El-Orany, M.S. Abdalla, Int. J. Th. Phys. 35 (1996) 1393; M.H. Mahran, M.S. Abdalla, A.-S.F. Obada, F.A.A. ElOrany, Nonlinear Opt. 19 (1998) 189;
[17]
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
373
F.A.A. El-Orany, M.H. Mahran, A.-S.F. Obada, M.S. Abdalla, Int. J. Th. Phys. 38 (1999) 1493; F.A.A. El-Orany, M.S. Abdalla, A.-S. Obada, G. Abdl Kader, J. Mod. Phys. B 15 (2001) 75. A.-S.F. Obada, S.S. Hassan, R.R. Puri, M.S. Abdalla, Phys. Rev. A 48 (1993) 3174; H.A. Batar®, M.S. Abdalla, A.-S.F. Obada, S.S. Hassan, Phys. Rev. A 51 (1995) 2644. C.C. Gerry, R. Grobe, Phys. Rev. A 51 (1995) 1698. E. Schr odinger, Nature 23 (1935) 44. W. Schleich, Quantum Optics in Phase Space, Wiley-VCH, Berlin, 2001. S. Chountasis, A. Vourdas, Phys. Rev. A 58 (1998) 848. S. Chountasis, L.K. Stergioulas, A. Vourdas, J. Mod. Opt. 46 (1999) 2131. S. Chountasis, A. Vourdas, C. Bendjaballah, Phys. Rev. A 60 (1999) 3467. R. Tanas, A. Miranowicz, Ts. Gantsog, in: E. Wolf (Ed.), Progress in Optics, vol. 35, Elsevier, Amsterdam, 1996. V. Perinova, A. Luks, J. Perina, Phase in Optics, World Scienti®c, Singapore, 1998. S.M. Barnett, P.L. Knight, J. Opt. Soc. Am. B 2 (1985) 467. B.M. Garraway, P.L. Knight, Phys. Scripta 48 (1993) 66. R. Tanas, B.K. Murzakhmetov, Ts. Gantsog, A.V. Chizhov, Quant. Opt. 4 (1992) 1. A.V. Chizhov, B.K. Murzakhmetov, Phys. Lett. A 176 (1993) 33.
Optics Communications 197 (2001) 363±373
www.elsevier.com/locate/optcom
Phase properties of superposition of pair-coherent states Faisal A.A. El-Orany a,b, J. Perina b,* a
Department of Mathematics and Computer Science, Faculty of Natural Science, Suez Canal University, Ismailia, Egypt b Department of Optics, Palack y University, 17. listopadu 50, 772 07 Olomouc, Czech Republic Received 7 June 2001; accepted 2 August 2001
Abstract In this article we discuss the phase properties of the superposition of pair-coherent states in the framework of Pegg± Barnett formalism. We also investigate the behaviour of Wigner and Weyl functions for the single-mode case. We show for speci®c modes and quantities that even-pair-coherent states can map onto odd-pair-coherent states and vice versa based on the values of the degeneracy parameter. Also we discuss the possibility of using the Weyl function to describe the phase distribution as a new tool in quantum optics. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 42.50.Dv Keywords: Phase properties; Pair-coherent states; Quasi-distributions; Phase space
1. Introduction
B n; q
Two modes of a quantized electromagnetic ®eld can become entangled showing many nonclassical eects compared to the decoupled modes. One of these two-mode optical cavities which involves strong entanglement is the pair-coherent states [1], which are de®ned as jf; qi
1 X
B n; qjn q; ni;
1
n0
where * Corresponding author. Address: Joint Laboratory of Optics of Palack y University and Institute of Physics of Academy of Sciences of Czech Republic, 17. listopadu 50, 772 07 Olomouc, Czech Republic. Tel.: +420-68-563-4263; fax: +42068-522-5246. E-mail address: [email protected] (J. Perina).
N qjfjn exp i/ p ; n! n q!
2
and N q is the normalization constant expressed as " N q
1 X n0
2n
jfj n! n q!
#
ijfj q Jq 2ijfj
1=2
1=2
;
3
where Jq is the ordinary Bessel function of order q; f jfj exp i/ and this makes the pair-coherent state as a partial phase state; q is the ``charge'' (degeneracy) parameter, which is a ®xed integer. The Fock state basis jn q; ni means that mode a includes n q photons, while mode b has n photons. In fact, these states jf; qi are eigenstates of both the pair-annihilation operator as well as the dierence in the number operators for the two modes, i.e.
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 4 7 2 - 9
364
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
^ qi fjf; qi; a^bjf; ^ ay a^
4
^ qi qjf; qi; b^y bjf;
5
where a^ and b^ are boson operators. It is worth mentioning that these states are related to the harmonic-oscillator coherent states jca ; cb i [1] by Z i q p E du hp jf; qi f exp iu f exp iu 2p a p E f exp iu exp jfjN q: 6 b
Further, the probability of ®nding n photons in mode b and n q photons in mode a in jf; qi is 2
P n q; n jhn q; njf; qij N 2 q
jfj2n ; n! n q! 7
which is not Poissonian. We proceed, pair-coherent states admit strong nonclassical properties such as various forms of squeezing, violations of the Cauchy±Schwarz and Bell inequalities [1] as well as many-photon antibunching [2]. Also their phase properties have been investigated in the view of Pegg±Barnett technique [3]. Moreover, evolution of pair-coherent states in two-level [4] and three-level [5] Jaynes±Cummings model as well as in a nondegenerate parametric ampli®er [6] have been discussed. These studies have been supported by several proposals for generation of these states. For example, they may be generated in a process involving the competition between nondegenerate parametric ampli®cation and nondegenerate two-photon absorption [1]. Also they can be generated in a two-dimensional trapped ion [7] in which the ion is excited bichromatically using three laser beams along dierent directions in its horizontal plane. If the initial vibrational state is a two-mode Fock state, the equilibrium motional state of the ion is the highly correlated pair-coherent state. On the other hand, the concept of superposition principle has been applied to the pair-coherent states aiming to produce new type of two-mode cat states, which can involve more pronounced nonclassical eects than the original states. Such superimposed states possess two mechanisms con-
trolling their behaviours, which are entanglement and interference in phase space. The superposition of pair-coherent states has been de®ned [8] in the Fock states basis as jf; qi
1 X
B n; q; jn q; ni;
8
n0
where n
B n; q;
N q; jfj exp i/ p 1 1n ; n! n q!
9
and the parameter takes on values 0, 1 and 1 associated to pair-coherent, even- and odd-paircoherent states, respectively; and N q; is the normalization constant, which reads ) 1=2 ( 1 X jfj2n 2 n N q; : 1 jj 2 1 n! n q! n0 10
Even and odd states of this type are orthonormalized states and they are eigenstates of the op^ 2 and a^y a^ b^y b, ^ i.e. erators ^ ab 1 1 hf; qjf; qi1 1
^ jf; qi ^ ab 1 2
^ ay a^
1
0;
f2 jf; qi1
^ qi b^y bjf; 1
1
11 1 ;
qjf; qi1
12 1 :
13
Further, the behaviour of the photon-number distribution of jf; qi is completely dierent from that of pair-coherent states as a result of the superposition principle. This is quite clear by comparing the following expression for superposition pair-coherent states: P n q; n N 2 q
2n
n 2
jfj 1 1 ; n! n q!
14
with Eq. (7) for the pair-coherent states, i.e. jf; qi possess nonclassical pairwise oscillations in the photon-number distribution which do not exist in the pair-coherent states. Moreover, jf; qi can produce nonclassical eects stronger than those of pair-coherent states, e.g. the violation of the Cauchy±Schwarz inequality for the superposition of pair-coherent states are always stronger than that for pair-coherent states of a nonzero q [8]. The
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
superposition of pair-coherent states can be generated by means of the competition between a nonlinear multiphoton parametric process and incoherent multiphoton absorption [8], or by using trapped ion technique as for pair-coherent states, however, in this case the trapped ion is excited bichromatically by ®ve laser beams along dierent directions in its horizontal plane [9]. Furthermore, this type of states can be generated via the atomic interference [10]. Finally, the concept of phase is an important as well as a controversional issue since the early days of quantum optics [11]. This importance has been motivated by the experimental realization of the phase in optical homodyne tomography [12]. Generally speaking, there are various methods for treating phase in quantum optics having advantage and disadvantage points. In this article we use the Pegg±Barnett formalism [13] to treat the phase distribution for the superposition of pair-coherent states. Actually, this technique is more convenient for this problem since it does not include such diculties as the others, i.e. the lack of unitarity. Further, we discuss the single-mode behaviour of both Wigner (W) and Weyl functions. Without any loss of generality we assume that q is nonnegative integer. The article will be organized in the following sequences: In Section 2 we discuss Wigner and Weyl functions. In Section 3 we demonstrate the phase properties by means of Pegg±Barnett formalism and also we compare the phase information included in both Wigner and Weyl functions. Finally we summarize the main conclusions in Section 4.
and is sensitive to interference in phase space. To study W function we start by calculating the single-mode symmetrical characteristic function de®ned as o n Ca m; Tr q^a expm^ ay m a^ ; 15 where q^a Trb q^ is the reduced density matrix for mode a. We restrict our formulae to those of the mode a, however, those of mode b can be easily obtained by some tricks. It can be shown that Tra q^2a < 1 for 0 < jfj < 1 [15], i.e. single-mode state resulting from the superposition of pair-coherent states is mixed. This is arising from the strong correlation between the two modes. For the superposition of pair-coherent states the symmetric characteristic function (15) reads 1 1 2 X 2 2 Ca m; exp jmj jB n; q; j Lnq jmj ; 2 n0 16 where Lm are Laguerre polynomials of order m. Employing Eq. (16) in the following relation Z 1 Ca m; exp bm b m d2 m; Wa b; 17 p and carrying out the integration, we arrive at the single-mode W function as Wa b;
1 X 4 2 2 exp 2jbj jB n; q; j f n q; b; p n0
18 where f n q; b 1
2. Single-mode Wigner and Weyl functions Recently, there is a lot of both theoretical and experimental work on quantum optical tomography whereby the W function of optical signals is reconstructed from a set of optical homodyne measurements (see Ref. [14] and references therein). This function is an eective tool in quantum optics, which can give a good indication about the possibility of occurrence of nonclassical eects, e.g. it can involve negative values, stretched contours, etc. Furthermore, it is well-behaved function
365
nq
2
Lnq 4jbj ;
19
and b x iy. In fact, the behaviour of the W function depends essentially on the function (19). Furthermore, expression (18) reveals that the behaviour of the W function of the single-mode case of the superimposed pair-coherent states is somewhat similar to that associated with the Fock states in such a way that the main contributions are localized at the origin. This situation is similar to binomial states [16] and geometric states [17]. Further, the information about the entanglement of the two subsystems is included in the coecient B n; q; . It is worth noting that the parameter q
366
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
plays a signi®cant role in the behaviour of W function. To be more speci®c, it can map the W function of the even-pair-coherent states onto that of odd-pair-coherent states or vice versa. This fact is clear from Eq. (19) where this expression for the even case f 2n q; b with q 1; 3; 5; . . . is equivalent to the odd case f 2n 1 q; b with q 0; 2; 4; . . ., correspondingly. Further, we have seen that this fact is established well in the behaviour of Mandel's Q parameter, too. It is worth reminding that Mandel's Q parameter is characterizing the departure from Poissonian photon statistics, i.e. if the variance of the photon number h D^ ay a^2 i is less than the mean photon number h^ ay a^i, then the state under consideration has subPoissonian statistics meaning that it has no classical description. More speci®cally, if Q 0, 0 > Q P 1 and Q > 0, then the state is called Poissonian, sub-Poissonian and super-Poissonian, respectively. We proceed, we have seen that when the value of the degeneracy parameter q increases, the oscillatory behaviour between positive and negative values in W function is more pronounced. Further the interference in phase space manifests itself by more pronounced nonclassical negative values of W function for even- and odd-paircoherent states compared to that for pair-coherent states. There is only one exceptional case to this rule which are the even-pair-coherent states for
q 0 and if jfj is small. Fig. 1 gives insight into this point where the W functions of pair-coherent (a) and even-pair-coherent states (b) are plotted for given values of the parameters. Comparison of Fig. 1(a) and (b) is instructive. From Fig. 1(b) we can see that W function is positive and has a Gaussian form. These characters indicate that nonclassical eects cannot be established (see Fig. 1 in Ref. [18]). Now if we turn our attention to the W function of pair-coherent states for this case (Fig. 1(a)) one can observe that it has a Gaussiantop-hole peak similar to speci®c type of geometric states [17]. Actually, this type of geometric states exhibits a maximum sub-Poissonian statistics and this is also valid to pair-coherent states [1]. Further, the comparison of Fig. 1(a) and (b) emphasizes that the interference in phase space can destroy the nonclassical eects inherent in the system, in particular, for speci®c quantities. This is in contrast with the superposition of coherent states [19], which can exhibit various nonclassical properties even if the original states are close to the classical states. We proceed, from the structure of Eq. (19) one can ®nd out that the negativity of W function relies on both the type of the states under consideration and the value of the degeneracy parameter q. Fig. 2 has been given for this purpose where W function of even-pair-coherent states have been plotted for dierent values of q and
Fig. 1. W function of the single-mode case for pair-coherent states (a) and for even-pair-coherent states (b) when jfj; q 0:75; 0.
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
367
Fig. 2. W function of the single-mode even-pair-coherent states when jfj 1:5 and for (a) q 0; (b) q 1; and (c) q 2.
®xed value of jfj. From these ®gures, the nonclassicality is more pronounced when q is odd. In other words, the strongest sub-Poissonian statistics of the even single-mode case occur when q is odd. Further, as is known the superposition of pair-coherent states cannot produce squeezing in the individual modes because of the strong correlation between modes and this manifests itself in an isotropic form W function. Moreover, from Fig. 2 we can see that when q is odd W function has downward peak and upward peak when q is even. This situation is in general valid for this case for all values of q. This feature (downward/upward peak) indicates that these two types of states are orthogonal [14]. Also comparison of Fig. 1(b) and Fig. 2(a) shows that when jfj increases the nonclassical eects start to appear [18]. In conclusion, the behaviour of W function is in a good agreement with the quantum features of these states. As we have mentioned earlier the behaviour of oddpair-coherent states can be obtained from that of even ones; for this reason we have not included the behaviour of odd-pair-coherent states. However, it is reasonable mentioning that there is a signi®cant dierence between these two states, i.e. for even (odd)-pair-coherent states even (odd) numbers of photons have nonzero probability of being observed. Such oscillations in the photon-number distribution may be interpreted as an interference in phase space [20]. Further, we noted that the oscillations in the photon-number distribution increase as the values of jfj increase, but in all cases the distribution exhibits Gaussian envelope similar to that of pair-coherent states [4].
Now we look at the interference in phase space from the point of view of Weyl function. In this treatment we show that this function can carry information similar to that included in W function [21,22]. Further, this function has been used as an auxiliary quantity in theoretical studies (e.g. in homodyne tomography [14]) and can be de®ned in many ways. For example, the single-mode Weyl function is just the symmetrical characteristic function (16), for convenience we can write X iY Wa X ; Y Ca m p p ; ; 20 2 2 where X ; Y represent the position and momentum increments [21]. Replacing m by its new argument in Eq. (16) we obtain 1 2 2 X Y Wa X ; Y exp 4 2 1 X X Y2 2 jB n; q; j Lnq : 2 2 n0 21 From Eq. (21) one can see that Weyl function has its maximum value at the origin, which is positive, equal to unity and also jWa X ; Y j 6 1. Further, comparison of this function with W function (18) shows that they can have similar behaviour nq when n q is even integer (i.e. 1 1). In this caseÐapart from an insigni®cant prefactor 4=pÐeach of which can transform pto the other via the transformation X ; Y $ 2 2 x; y. On the other hand, for the inverse situation (i.e. n q is odd integer) the locations of the cross-term and
368
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
Fig. 3. Weyl function for the single-mode case for q; ; jfj 0; 0; 0:75; 1; 0; 1:5 and 1; 1; 1:5 is shown in (a)±(c), respectively.
auto-terms in W function and in Weyl function interchange. Generally, we have seen that the behaviour of Weyl function is in a good agreement with that of the W function. For the sake of comparison we have displayed the Weyl function in Fig. 3(a)±(c) for three dierent cases. From the above discussion it is obvious that the Weyl and W functions corresponding to Fig. 1(b) are similar, i.e. they are positive and have a Gaussian shape. This shape can be considered as the border line between classical and nonclassical behaviour regarding to Weyl and W functions as well. In other words, the state having this form will be close to the classical wave packet, otherwise there is a possibility to include nonclassical eects. Now comparison of Figs. 1(a) and 3(a) as well as of Fig. 3(b) and (c) is instructive. The former shows how the nonclassical eects manifest themselves in the behaviour of Weyl function, whereas the latter demonstrates the role of the interference in phase space in the behaviour of Weyl function. This means that Weyl function can be used as an additional tool in describing the phase space where the nonclassical eects can manifest themselves in this function as negative values, many-fold structure or stretched shape (for squeezed states). Actually, there are some weak points in using this function which are: it is in general complex function and in this sense there is no direct relation between this and the measurements. The ®rst point has been solved by analysing the behaviour of its imaginary and real parts individually as well as its modulus [21,23]. The second point can be overcome using the fact that this function can be de-
®ned in terms of the Fourier transformation of W function and the latter can be reconstructed in quantum tomography from optical homodyne measurements [14]. In other words, indirect measurement scheme can be arranged. Moreover, the connection between these two functions has been used in developing a ®ltering algorithm to the noisy Wigner function [22]. In this technique the Radon transform of the W function of a noisy signal has been taking and producing the W function of the clean (noiseless) signal. This process has been performed with the help of Weyl function. Actually, this technique is of practical interest since the quantum signals inevitably contain noise. So far we have discussed the behaviour of mode a only, however, the behaviour of mode b is quite dierent. This is evident from Eq. (8) where mode a possesses q photons more than those in mode b. Nevertheless, the behaviour of this mode can be easily understood from their own expressions. To show this we ®rst give expressions for the W and Weyl functions related to mode b. Careful examination of Eq. (8) shows that one can obtain the required expressions by simply setting q 0 in the quantities resulting from jn qi in the expressions for mode a. To be more speci®c, we set q 0 in both Eq. (19) (for W function) and in the index of the Laguerre polynomial in Eq. (21) (for Weyl function). From the formulae for mode b one can immediately conclude that the degeneracy parameter q has no signi®cant role in the behaviour of this mode. Also, for the even-pair-coherent states, the Weyl and Wigner functions possess similar
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
behaviour in phase space as we have discussed above. More illustratively, they are almost represented by a Gaussian peak similar to that for vacuum ®eld with negligible negative values, i.e. there is a negligible nonclassical eect [18]. On the other hand, the nonclassical negative values dominate in the form of these functions for the odd-pair-coherent states. This agrees with the fact that odd-pair-coherent states can exhibit more pronounced sub-Poissonian statistics.
3. Phase distribution In this section we investigate the phase distribution for the superposition of pair-coherent states using the Pegg±Barnett formalism [13]. Also we address the question: how far the Weyl function is relevant to describe the phase distribution? Basically, the Pegg±Barnett formalism is based on de®ning Hermitian-phase operator in a truncated Hilbert space, and after calculating all necessary expectation values for the target state, the dimension of this space extends to include the whole space by appropriate limit [13]. The details related to this technique have been extensively discussed in the literature [24,25] and then we do not repeat them here. The generalization of this method to two-mode case is a straightforward way [3]. Now the joint-phase distribution for the superposition of pair-coherent states reads 1 X 1 P H1 ; H2 12 B n; q; B k; q; 2 2p n>k #) " cos n
k H1 H2
:
369
squeezed states [26]. It is worth reminding that the single-mode version of these states has dissimilar quantum features, e.g. the two-mode squeezed states possess thermal properties, however, the pair-coherent states are pure nonclassical states. Moreover, when jfj ! 0, expression (22) gives a joint-uniform distribution corresponding to the two-modes Fock states jq; 0i. Also we note that the parameter q broadens the phase distribution, i.e. speeds up the randomization of the phase (classical phase). Actually, these two cases are established irrespective of which states has been considered, i.e. 0, 1 or 1. We proceed, for even (odd) states it is clear that P H1 ; H2 is a p-periodic function in both H1 and H2 owing to the argument that n k is always even integer. This makes that the distribution involves more nonclassical oscillations compared to that associated with paircoherent states [3] where, within the phase window, the distribution (22) produces maxima along ®ve parallel lines controlled by the equations H 0, (1/2)p, p, however, that of pair-coherent states gives maxima along three lines only (H 0, p). Fig. 4 sheds the light on the behaviour of distribution (22) for given values of the parameters. Comparison of this ®gure with Fig. 1 for paircoherent states given in Ref. [3] shows the role of interference in phase space. Moreover, from Fig. 4 one can observe that the distribution is symmetric around the diagonals of the square of the phase windows H1 ; H2 and it re¯ects the dependence of the distribution on the phase sum H only. Also
22
The phase / of the parameter f disappeared in Eq. (22) owing to the choice of the phase window, for more details about the derivation of Eq. (22) the reader can consult Ref. [3]. Expression (22) reveals that the distribution depends only on the phase sum H H1 H2 and this makes that the states jf; qi are uniformly distributed regarding to the individual phases as well as to the phase dierence H H1 H2 . Such behaviour is similar to that of pair-coherent states [3] and of two-mode
Fig. 4. Joint-phase distribution of the even-pair-coherent states for q; ; jfj 1; 1; 1.
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F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
we can see that the phase localization occurs for ®xed phase window. Now for the symmetrical case (q 0), i.e. when the photons in the system are equally distributed over the two modes, distribution (22) reduces to the following closed form: P H
N 2 0; 2p2 1
f 1 2 cosh 2jfj cos H
2 sinh 2jfj cos H
2 cos 2jfj sin H g;
23
all the notations have the same meaning as before. From this expression it is obvious that (for even case) the distribution possesses maximum N 2 0; 1=p2 cosh2 jfj at H mp, m 0, 1, 2 and minimum N 2 0; 1=p2 cos2 jfj at H m p= 2, m 1, 3. This shows that when jfj ``evolves'', the maxima of the distribution increase monotonically, while the minima of the distribution evolve periodically between 0 and N 2 0; 1=p2 . In other words, when jfj increases the distribution becomes narrower similarly as in pair-coherent states [3]. Narrower distribution means more wellde®ned phase. We conclude this part by referring that the joint-phase distribution can be re-expressed in terms of the parameters H after applying the casting procedure mod 2p [24]. ^ 2 i h U ^1 The sum-phase variance h DU 2 2 ^ ^ ^ U2 i hU1 U2 i , for the state under discussion, is given by ^ 2 i 2 p2 h DU 3
4
1 X B n; q; B k; q; n>k
n
k
2
:
24
The ®rst term in expression (24) is the sum of the individual phase variances of the single modes, however, the second term results from the correlation between the single-mode phases and this is a direct consequence of the quantum mechanical eect. In Fig. 5 we have displayed the sum-phase variance for dierent values of the degeneracy parameter q. From this ®gure one can observe that ^ 2 i is smooth, it starts from the behaviour of h DU 2p2 =3 of the two-modes Fock states jq; 0i, goes to its minimum values and eventually the phase locking occurs. In other words, as jfj increases, the sum-phase variance decreases, i.e. the sum of the two phases becomes less uncertain. The role of
Fig. 5. The phase variance of the even-pair-coherent states for q 0 (Ð), 5 (± ± ±) and 10 (- - -).
the superposition of the states manifests itself as an oscillatory behaviour over small values of jfj, in particular, for q 0 and also in the sense that sum-phase variance cannot approach zero. This situation is in contrast with that of pair-coherent states where when jfj is large the variance approaches zero and the classical situation of perfectly de®ned phase sum dominates. Further, we can observe that when q increases and for ®nite ^ 2 i goes to randomly distributed values of jfj, h DU phase more rapidly. This fact can be realized by comparing the behaviour of the dierent curves in Fig. 5. From this discussion we can conclude that the phase information involved in the superposition of pair-coherent state is more pronounced than that included in pair-coherent states. We have seen in the preceding section that Wigner and Weyl functions can carry similar information about the nonclassical eects of the ®eld amplitudes, so it is convenient to follow the same line to see the situation regarding to phase distribution. Firstly, it is worth mentioning that the phase distribution resulting from the integration of W function over the radial part can give similar conclusions as given here. More illustratively, the distribution of the single-mode case is uniformly distributed and that of the two-mode case depends
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
only on the sum of the phases [6], as we see in the following. On the other hand, it is dicult to use Weyl function as a standard technique for analysing the phase in quantum systems, however, for particular states and after some arrangements this function can achieve this goal. For example, intuitively, the states whose Weyl and Wigner functions have similar structures can provide similar phase information. For the states under discussion this can be achieved when n q (q) is even integer for the single-mode (two-mode) case. Further, according to the lack of normalization of the Weyl function (this fact is clear from Eq. (20)) we have to suggest the de®nition of the uniformly distributed phase in the view of this function. This can be easily done by calculating the phase distribution related to the Weyl function for the Fock state jmi, m which is 2 1 and taking its absolute value we obtain the uniformly distributed phase with respect to this function, which is 2. In other words, factor 2 for the phase resulting from this function is equivalent to 1= 2p of the uniformly distributed phase from the other techniques. This should be used with a caution to get equivalence between all these techniques. In other words, we should apply such a transformation to the prefactor of the distribution (resulting from Weyl function) only, which will be 2 for the single-mode case and 4 for the two-mode case. We now write down the jointphase distribution for the superposition of paircoherent states regarding to two-mode Weyl and Wigner functions as P H1 ; H2
h1 2p
f1 2 2
1 X
G n; k
n>k
G n q; k qB n; q; B k; q; #) " cos n
k H1 H2
;
25
where r n! 1k 2 1=2 n k n k 1 G n; k C k! 2 2 n k! n k F k; 1; n k 1; 2 ; 2 2 26
371
where C and F are the Gamma and the con¯uent hypergeometric functions, respectively; and q h1 1 and 1 for the distribution resulting from Weyl and W functions, respectively. The derivation of Eq. (25) has been done by evaluating ®rstly the two-mode Weyl and Wigner functions and then by applying the standard technique of obtaining the phase distribution from the quasiprobability functions. From Eqs. (22) and (25) we can conclude that the three distributions can give similar behaviours in a qualitative way and for speci®c values of the parameters. Further, from Eq. (25) it is clear that when q is even integer the Weyl-phase and Wigner-phase distributions are typical, i.e. they carry the same phase information. The origin of this is that the two-mode Weyl and Wigner functions are similar. On the other hand, when q is odd integer, they give opposite behaviours. In fact, the Wigner and Weyl techniques can include negative values making them less favourite to be used in the phase analysis. Nevertheless, these negative values should be interpreted as a signature of the quantum nature of the ®eld state rather than some peculiarity of its phase properties. For more details related to the analysis of such phenomenon the reader can consult Ref. [27] (and references therein). Further, for some states the Wigner-phase technique can give similar behaviour as the Pegg±Barnett one, e.g. for the displaced number states whose phase distribution has been interpreted in terms of the area of overlap in phase space [28]. We conclude this section by showing that the W function is more suited to describe the phase than the Weyl function. This can be easily obtained by comparing the way by which the single-mode Weyl (Wigner) function can be obtained from its joint function. For example, the single-mode Weyl function results from its joint one by simply setting the parameters of the absent modes by zeros, however, the integration has to be performed for the joint-W function to get the single-mode W function. Such a distinction exists also in the phase distribution. More illustratively, the single-mode phase distribution derived from the Weyl-jointphase distribution by integration is dierent from that evaluated directly from the single-mode Weyl function. We emphasize this via analysing the
372
F.A.A. El-Orany, J. Perina / Optics Communications 197 (2001) 363±373
phase distribution of single-mode case with respect to Weyl function for the state under consideration. Before starting this we should stress that this distribution is uniformly distributed regarding to W function or Pegg±Barnett techniques irrespective of the type of states. Now applying the standard technique, the single-mode phase distribution resulting from the single-mode Weyl function reads 1 1 X n 2 27 P Hj 1 jB n; q; j ; 2p n0 however, that resulting from the Weyl-joint-phase distribution by integration over H1 or H2 is 1= 2p. Actually, these two results can coincide for even (odd)-pair-coherent states. So we can conclude that in general when the two-mode Weyl and Wigner functions are similar it is sucient to deal with the joint-phase distribution of the Weyl function to obtain all relevant results and to exclude those results coming from direct calculation of the single-mode Weyl function.
4. Conclusion In this article we have investigated the phase properties of the superposition of pair-coherent states in the view of Pegg±Barnett formalism. Also we have discussed the possibility of using the Weyl function to investigate such eects. Further, we have analysed the behaviour of the single-mode Wigner function and Weyl function for the states under consideration. It is important to stress that these two functions can provide similar information about the nonclassical eects. For the present states, we have shown that these functions can exhibit more pronounced nonclassical eects resulting from the strong correlation between the two modes. Also we have shown that for a speci®c mode, even states can carry information about odd states and vice versa meaning that one can enhance the nonclassical sub-Poissonian statistics based on adjusting the dierence between the mean number of photons in the two modes. Further, it is worth mentioning that this situation is also valid for the two-mode squeezed number states suggested in
Ref. [29]. It should be borne in mind that Weyl and Wigner functions (besides the others, e.g. Husimi Q- and Glauber P-functions) can be used as qualitative tools in quantum optics and their irregularities are necessary to indicate the existence of the nonclassical eects but they are not sucient. On the other hand, we have shown that the joint-phase distribution can exhibit more pronounced nonclassical oscillations arising from the superposition between the dierent components of the states. Further, we have shown that the three types of the distributions (Weyl, Wigner and Pegg± Barnett) can lead to this result. However, the single-mode phase is always uniformly distributed regarding to Wigner and Pegg±Barnett techniques. Moreover, we have shown that Weyl function can carry similar information about the phase as Wigner function for speci®c types of states. The ®nal remark is that the Wigner and Weyl functions are problematic to be used in analysing the phase because they can include negative values and also for some states the Weyl function is complex; so one has to deal with its absolute value, which can lead to missing information about the phase.
Acknowledgements J.P. and F.A.A.E.-O. acknowledge the partial support from the project LN00A015 of Czech Ministry of Education.
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