- Email: [email protected]
Alternative proposal for the generation of the displaced number state
ARTICLE IN PRESS
Physica A 351 (2005) 251–259 www.elsevier.com/locate/physa
Alternative proposal for the generation of the displaced number state Guilherme C. de Oliveira, Agnaldo R. de Almeida, Iara P. de Queiro´s, Arthur M. Moraes, Ce´lia M.A. Dantas Instituto de Fı´sica, Universidade Federal de Goia´s, Caixa Postal 131, Goiaˆnia, GO 74. 001-970, Brazil Received 28 September 2004 Available online 19 January 2005
Abstract This work presents an alternative method for the generation of the displaced number state. The usual proposal found in the literature requires the generation of this state from a number state jni; which still have not been generated in laboratory and is difficult to obtain, because it demands properties of atom-field interactions in microwave cavities in processes with maximum reduction of quantum noise. Here we show that it is possible to generate the displaced number state from the coherent state, which is easily obtained, by means of the passage of this through a non-linear medium. The coefficients of the displaced number states obtained by our alternative definitions are calculated, showing that it is the same obtained by the usual definition in the literature. r 2005 Elsevier B.V. All rights reserved. PACS: 03.65.w; 42.50. Ct
1. Introduction Quantum optics deals with problems related to light with no classical analogy. Some of the many problems investigated by quantum optics refer to the definition and study of properties of new light field states, generation and production of new states, quantum noise and the implication of quantum properties of these states. In Corresponding author.
E-mail address: [email protected] (C.M.A. Dantas). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2004.11.066
ARTICLE IN PRESS 252
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the recent past, there has been a great interest in generating new states with nonclassical properties. The displaced number state (DNS) [1] is an example of such states. It was proposed by Oliveira et al. [1] and its statistical properties have been widely investigated, showing the relevance of this state in the context of quantum optics for its several purely quantum properties, such as the interference effects in the phase space [2]. In the past few years, many papers have referred to this state, especially because of the possibility of experimentally producing a class of quantum superposition and sub-Poissonian states where the one-photon-displaced number state is the archetype of this class of states [3,4]. Besides, generalisations of these states have also been proposed in the literature by Kra´l [5]. In Fig. 1 we show the pictorial representation of the DNS in the phase space: circles of intern radius ri ¼ ð2mÞ1=2 and extern radius re ¼ ½2ðm þ 1Þ 1=2 ; which represents the number states, displaced of the intensity jaj in the y direction. The DNS, as proposed by Oliveira et al. [1], is generated by applying Glauber’s displacement operator on a number state jni b ja; ni ¼ DðaÞjni ;
(1)
b where DðaÞ is defined as [6] b ¼ eaa^y a a^ : DðaÞ
(2)
Fig. 1. Pictorial representation of the displaced number state in the phase space: circles of internal radius ri ¼ ð2mÞ1=2 and external radius re ¼ ½2ðm þ 1Þ 1=2 ; which represents the number states, displaced of the intensity jaj in the y direction.
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From Eq. (1) we verify that the generation of ja; ni requires the passage of a classical current through a cavity which contains an initial number state jni [6]. However, an interesting question should be asked: Would this be the only way to generate the DNS? As the number states are difficult to be obtained experimentally due to the fact that they require the use of techniques of conditioned measurements or properties of atom-field interactions in microwave cavity [7] in process with maximum reduction of quantum noise, this question is in fact of fundamental relevance. In this paper we show that ‘‘no’’ is the answer to this question. We propose an alternative method to generate the DNS from the coherent state (CS), which can be easily obtained in laboratory.
2. The displaced number state Starting from the definition of the DNS given by Eq. (1), using the relationships of unitarity of Glauber’s operator displacement by ðaÞDðaÞ b ¼ DðaÞ b D by ðaÞ ¼ b D 1
(3)
and the definition of the number state jni; obtained by the successive application of the creation operator aby in the vacuum state j0i [8], ðb ay Þn jni ¼ pffiffiffiffi j0i ; n!
(4)
we obtain 1 ay a Þn jai ja; ni ¼ pffiffiffiffi ðb n! ! n n 1 X ¼ pffiffiffiffi ðb ay Þnk ða Þk jai : n! k¼0 k
ð5Þ
To reach this result we have used the relation by ðaÞ ¼ ðb b ay a Þn DðaÞðb ay Þ n D
(6)
demonstrated in Appendix A. According to Eq. (5), the DNS, ja; ni; can be generated from the CS jai: The temporary evolution of the CS that leads to the DNS can be described in terms of the passage of the CS through a non-linear medium; the nonlinearity of this medium being associated to the number n, that one wants to displace, as shown in Eq. (5). For example, if one wants to obtain the number state n ¼ 3 displaced by a; one has to use a non-linear medium with a level of non-linearity, g; such that g follows the relation 0pgpn; with g and n 2 f0; 1; 2; . . .g: Thus, in our example for n ¼ 3; the medium must contain terms like ðb ay a Þ3 ¼ ðb ay Þ3 3a ðb ay Þ2 þ 3ða Þ2 aby ða Þ3 ;
(7)
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note that the medium must be proportional to ðb ay Þ3 and as mentioned above, must also have all the lower order terms. It should also be pointed out that in order to obtain the DNS for n ¼ 1; the medium takes the form of a linear term ðb ay a Þn ¼ ðb ay a Þ1 ¼ aby a
(8)
and for n ¼ 0 one obtains the CS itself, as shall be demonstrated in the following section, where we calculate and show that coefficients generated through the procedure described by Eq. (5) result in the DNS as obtained by Oliveira et al. [1]. Note that this process of generating states for the DNS requires a very specific non-linear medium containing all the orders of non-linearity 0pgpn: This differs, for example, of processes used for the generation of squeezed states which require a non-linear medium quadratic in ab and aby [9] and for the generation of number-phase squeezed states which can be achieved through a quartic non-linearity in ab and aby [10].
3. Coefficients of displaced number state The coefficients of the DNS can be calculated from Eq. (5). Thus, we shall calculate for each n value the corresponding coefficient for the expansion of the state ja; ni on a number-state basis jmi; i.e., C nm ¼ hmja; ni ;
(9)
for the displaced vacuum state ja; 0i; corresponding to n ¼ 0; we have 1 ja; 0i ¼ pffiffiffiffi ðb ay a Þ0 jai ¼ jai ; 0! which is as expected the CS, whose coefficients C nm ¼ C 0m are given by [2] rffiffiffiffiffiffi 0! ð1=2Þjaj2 m 0 C m ¼ hmja; 0i ¼ a : e m!
(10)
(11)
In the same way for n ¼ 1 we have 1 ja; 1i ¼ pffiffiffiffi ðb ay a Þ1 jai ¼ ða a Þjai 1! and the associated coefficients C 1m are given by rffiffiffiffiffiffi i 1! ð1=2Þjaj2 m1 hm 1 jaj2 : Cm ¼ a e m! 1!
(12)
(13)
Repeating this procedure for n ¼ 2; 3; . . . we can check if it is possible to identify a generalisation for the coefficients C nm for any n. Thus for n ¼ 2 and 3, we have rffiffiffiffiffiffi
2! ð1=2Þjaj2 m2 mðm 1Þ jaj4 2 mjaj2 þ Cm ¼ a e ; (14) m! 2! 2
ARTICLE IN PRESS G.C. de Oliveira et al. / Physica A 351 (2005) 251–259
C 3m
255
rffiffiffiffiffiffi
3! ð1=2Þjaj2 m3 mðm 1Þðm 2Þ mðm 1Þ 2 mjaj4 jaj6 e jaj þ ¼ a : m! 3! 2! 2! 3! (15)
Note that although the coefficients for n ¼ 2 and 3 are easily obtained, a generalisation for C nm is not entirely obvious. In order to illustrate how this generalisation can be done, let us take the coefficient C 3m as an example. For C 3m we verify that the first term within square brackets can be written as mðm 1Þðm 2Þ ð1Þk jaj2k m! ¼ : (16) 3! ðm 3Þ! 3! k¼0
The term jaj2k was included since it is present in all remaining terms within the square brackets in Eq. (15), the parameter 2k appears due to the expansion of even order terms and the parameter k is used as our attempt to identify a sum. It is also clear that ð1Þk is used to reproduce the signal change seen for the terms in this example. The remaining terms within square brackets in Eq. (15) can be further analysed. Thus for the second term we have mðm 1Þ 2 ð1Þk jaj2k m! jaj ¼ 2! ðm 2Þ! 2! k¼1 k 2k ð1Þ jaj m! ¼ ; ð17Þ ð3 kÞ! ðm 3 þ kÞ! k¼1
note that 2! was replaced by ð3 kÞ! and ðm 2Þ! by ðm 3 þ kÞ! because it is convenient to put the parameter n ¼ 3 in evidence for this example. Similar rearrangements can be done for Eq. (16) which becomes mðm 1Þðm 2Þ ð1Þk jaj2k m! ¼ : (18) 3! ð3 kÞ! ðm 3 þ kÞ! k¼0
The other remaining terms in this analysis are re-written as mjaj4 ð1Þk jaj2k m! ¼ 2!ðm 1Þ! 2! k¼2 k 2k ð1Þ jaj m! ¼ 2!ðm 3 þ kÞ! k¼2 k 2k ð1Þ jaj m! ¼ ; 2!ð3 kÞ!ðm 3 þ kÞ! k¼2
ð19Þ
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note that a factor 1=2! can be conveniently replaced by 1=k! in Eq. (19) and can also be included in Eqs. (17) and (18) which transforms them into the following set: mðm 1Þðm 2Þ ð1Þk jaj2k m! ¼ ; (20) 3! k!ð3 kÞ! ðm 3 þ kÞ! k¼0
mðm 1Þ 2 ð1Þk jaj2k m! jaj ¼ 2! k!ð3 kÞ! ðm 3 þ kÞ!
;
(21)
k¼1
mjaj4 ð1Þk jaj2k m! ¼ k!ð3 kÞ!ðm 3 þ kÞ! 2!
:
(22)
k¼2
Finally, the fourth remaining term can be easily written in the same format as the previous terms, though now for a value of k ¼ 3; jaj6 ð1Þk jaj3k m! ¼ : (23) k!ð3 kÞ!ðm 3 þ kÞ! 3! k¼3
Hence it becomes clear that Eq. (15) can be written in terms of a sum in k using the results shown from Eqs. (20)–(23): " # rffiffiffiffiffiffi n 3! ð1=2Þjaj2 m3 X ð1Þk jaj2k m! 3 Cm ¼ a e : (24) m! k!ð3 kÞ! ðm 3 þ kÞ! k¼0 This can be generalised to any n value, and thus, the coefficients of the DNS ja; ni in an expansion in a number-state basis can be written as follows: " # rffiffiffiffiffiffi n n! ð1=2Þjaj2 mn X ð1Þk jaj2k m! n Cm ¼ a e : (25) m! ðn kÞ!ðm n þ kÞ!k! k¼0 In the general expression given above one notice that the terms between square ðjaj2 Þ: Therefore, we can brackets are the Laguerre associated polynomials Lmn n write the coefficients of the DNS in the form rffiffiffiffiffiffi n! ð1=2Þjaj2 mn mn 2 n a Ln ðjaj Þ ; (26) Cm ¼ e m! which agrees with the expression displayed in Ref. [1].
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Fig. 2. Wigner function of the displaced number state, with initial number of photon n ¼ 1; displacement parameter a ¼ 3eip=4 and its contours curves.
The statistical properties of the DNS were already investigated thoroughly [1], showing that this state displays desirable quantum characteristics, among them we can mention the interference effects in the phase space [2], as shown in its Wigner function, which is given by 2 2 ð1Þn e2jbaj Ln ð4jb aj2 Þ ; (27) p where Ln are the Laguerre polynomial. In Fig. 2 we show the Wigner function of the DNS with initial number of photon n ¼ 1; displacement parameter a ¼ 3eip=4 and its contours curves.
W ðbÞ ¼
4. Comments and conclusion We have presented in this work an alternative method to generate the DNS starting from the CS, through the passage of latter for an non-linear medium, where the non-linearity of this is associated to the number n that one wants to displace. We showed that the resulting state is the same as the one obtained starting from the number state, already proposed in Ref. [1]. As the number states are difficult to
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obtain from the experimental point of view, our proposal offers a new and viable way to generate the DNS.
Acknowledgements This work was financed by the National Council of Technological Scientific Development (CNPq) and PRONEX/MTC/FINEP of Brazil.
Appendix A The equation y
b ðaÞ b An ¼ DðaÞðb ay Þ n D
(A.1)
can be written in the form y
b ðaÞ : b ay Þðb ay Þn1 D An ¼ DðaÞðb
(A.2)
Using the unitarity property of the displacement operator, in Eq. (A.2) we obtain by ðaÞ ; b by ðaÞDðaÞðb b An ¼ DðaÞðb ay Þ D ay Þn1 D
(A.3)
by ðaÞ ¼ aby a : b DðaÞðb ay Þ D
(A.4)
using
Eq. (A.3) becomes y
b ðaÞ : b ay a ÞDðaÞðb ay Þn1 D An ¼ ðb
(A.5)
Repeating the procedure we obtain by ðaÞ b An ¼ ðb ay a ÞDðaÞðb ay Þðb ay Þn2 D by ðaÞ b by ðaÞDðaÞðb b ay ÞD ay Þn2 D ¼ ðb ay a ÞDðaÞðb y
b b ðaÞ : ay Þn2 D ¼ ðb ay a Þ2 DðaÞðb
ðA:6Þ
If the process is repeated n times one obtains by ðaÞ ¼ ðb b An ¼ DðaÞðb ay Þ n D ay a Þn :
References [1] [2] [3] [4]
F.A.M. Oliveira, M.S. Kim, P.L. Knight, V. Buzek, Phys. Rev. A 41 (1990) 2645. W. Schleich, J.A. Wheeler, J. Opt. Soc. Am. B 40 (1987) 1195. A.F.R.T. Piza, N. Zagury, Phys. Rev. A 54 (1996) 3338. N. Zagury, A.F.R.T. Piza, Phys. Rev. A 50 (1994) 2908.
(A.7)
ARTICLE IN PRESS G.C. de Oliveira et al. / Physica A 351 (2005) 251–259 [5] [6] [7] [8] [9] [10]
259
P. Kra´l, J. Mod. Opt. 37 (1990) 889. J.R. Glauber, Phys. Rev. 131 (1963) 2766. M. Brune, S. Haroche, J.M. Raimonde, N. Zagury, Phys. Rev. Lett. 65 (1990) 976. W.H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973. R.E. Slusher, L.W. Holberg, B. Yurke, J.C. Mertz, J.F. Valley, Phys. Rev. Lett. 55 (1985) 2409. M. Kitagawa, Y. Yamamoto, Phys. Rev. A 34 (1986) 3974.
Physica A 351 (2005) 251–259 www.elsevier.com/locate/physa
Alternative proposal for the generation of the displaced number state Guilherme C. de Oliveira, Agnaldo R. de Almeida, Iara P. de Queiro´s, Arthur M. Moraes, Ce´lia M.A. Dantas Instituto de Fı´sica, Universidade Federal de Goia´s, Caixa Postal 131, Goiaˆnia, GO 74. 001-970, Brazil Received 28 September 2004 Available online 19 January 2005
Abstract This work presents an alternative method for the generation of the displaced number state. The usual proposal found in the literature requires the generation of this state from a number state jni; which still have not been generated in laboratory and is difficult to obtain, because it demands properties of atom-field interactions in microwave cavities in processes with maximum reduction of quantum noise. Here we show that it is possible to generate the displaced number state from the coherent state, which is easily obtained, by means of the passage of this through a non-linear medium. The coefficients of the displaced number states obtained by our alternative definitions are calculated, showing that it is the same obtained by the usual definition in the literature. r 2005 Elsevier B.V. All rights reserved. PACS: 03.65.w; 42.50. Ct
1. Introduction Quantum optics deals with problems related to light with no classical analogy. Some of the many problems investigated by quantum optics refer to the definition and study of properties of new light field states, generation and production of new states, quantum noise and the implication of quantum properties of these states. In Corresponding author.
E-mail address: [email protected] (C.M.A. Dantas). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2004.11.066
ARTICLE IN PRESS 252
G.C. de Oliveira et al. / Physica A 351 (2005) 251–259
the recent past, there has been a great interest in generating new states with nonclassical properties. The displaced number state (DNS) [1] is an example of such states. It was proposed by Oliveira et al. [1] and its statistical properties have been widely investigated, showing the relevance of this state in the context of quantum optics for its several purely quantum properties, such as the interference effects in the phase space [2]. In the past few years, many papers have referred to this state, especially because of the possibility of experimentally producing a class of quantum superposition and sub-Poissonian states where the one-photon-displaced number state is the archetype of this class of states [3,4]. Besides, generalisations of these states have also been proposed in the literature by Kra´l [5]. In Fig. 1 we show the pictorial representation of the DNS in the phase space: circles of intern radius ri ¼ ð2mÞ1=2 and extern radius re ¼ ½2ðm þ 1Þ 1=2 ; which represents the number states, displaced of the intensity jaj in the y direction. The DNS, as proposed by Oliveira et al. [1], is generated by applying Glauber’s displacement operator on a number state jni b ja; ni ¼ DðaÞjni ;
(1)
b where DðaÞ is defined as [6] b ¼ eaa^y a a^ : DðaÞ
(2)
Fig. 1. Pictorial representation of the displaced number state in the phase space: circles of internal radius ri ¼ ð2mÞ1=2 and external radius re ¼ ½2ðm þ 1Þ 1=2 ; which represents the number states, displaced of the intensity jaj in the y direction.
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253
From Eq. (1) we verify that the generation of ja; ni requires the passage of a classical current through a cavity which contains an initial number state jni [6]. However, an interesting question should be asked: Would this be the only way to generate the DNS? As the number states are difficult to be obtained experimentally due to the fact that they require the use of techniques of conditioned measurements or properties of atom-field interactions in microwave cavity [7] in process with maximum reduction of quantum noise, this question is in fact of fundamental relevance. In this paper we show that ‘‘no’’ is the answer to this question. We propose an alternative method to generate the DNS from the coherent state (CS), which can be easily obtained in laboratory.
2. The displaced number state Starting from the definition of the DNS given by Eq. (1), using the relationships of unitarity of Glauber’s operator displacement by ðaÞDðaÞ b ¼ DðaÞ b D by ðaÞ ¼ b D 1
(3)
and the definition of the number state jni; obtained by the successive application of the creation operator aby in the vacuum state j0i [8], ðb ay Þn jni ¼ pffiffiffiffi j0i ; n!
(4)
we obtain 1 ay a Þn jai ja; ni ¼ pffiffiffiffi ðb n! ! n n 1 X ¼ pffiffiffiffi ðb ay Þnk ða Þk jai : n! k¼0 k
ð5Þ
To reach this result we have used the relation by ðaÞ ¼ ðb b ay a Þn DðaÞðb ay Þ n D
(6)
demonstrated in Appendix A. According to Eq. (5), the DNS, ja; ni; can be generated from the CS jai: The temporary evolution of the CS that leads to the DNS can be described in terms of the passage of the CS through a non-linear medium; the nonlinearity of this medium being associated to the number n, that one wants to displace, as shown in Eq. (5). For example, if one wants to obtain the number state n ¼ 3 displaced by a; one has to use a non-linear medium with a level of non-linearity, g; such that g follows the relation 0pgpn; with g and n 2 f0; 1; 2; . . .g: Thus, in our example for n ¼ 3; the medium must contain terms like ðb ay a Þ3 ¼ ðb ay Þ3 3a ðb ay Þ2 þ 3ða Þ2 aby ða Þ3 ;
(7)
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note that the medium must be proportional to ðb ay Þ3 and as mentioned above, must also have all the lower order terms. It should also be pointed out that in order to obtain the DNS for n ¼ 1; the medium takes the form of a linear term ðb ay a Þn ¼ ðb ay a Þ1 ¼ aby a
(8)
and for n ¼ 0 one obtains the CS itself, as shall be demonstrated in the following section, where we calculate and show that coefficients generated through the procedure described by Eq. (5) result in the DNS as obtained by Oliveira et al. [1]. Note that this process of generating states for the DNS requires a very specific non-linear medium containing all the orders of non-linearity 0pgpn: This differs, for example, of processes used for the generation of squeezed states which require a non-linear medium quadratic in ab and aby [9] and for the generation of number-phase squeezed states which can be achieved through a quartic non-linearity in ab and aby [10].
3. Coefficients of displaced number state The coefficients of the DNS can be calculated from Eq. (5). Thus, we shall calculate for each n value the corresponding coefficient for the expansion of the state ja; ni on a number-state basis jmi; i.e., C nm ¼ hmja; ni ;
(9)
for the displaced vacuum state ja; 0i; corresponding to n ¼ 0; we have 1 ja; 0i ¼ pffiffiffiffi ðb ay a Þ0 jai ¼ jai ; 0! which is as expected the CS, whose coefficients C nm ¼ C 0m are given by [2] rffiffiffiffiffiffi 0! ð1=2Þjaj2 m 0 C m ¼ hmja; 0i ¼ a : e m!
(10)
(11)
In the same way for n ¼ 1 we have 1 ja; 1i ¼ pffiffiffiffi ðb ay a Þ1 jai ¼ ða a Þjai 1! and the associated coefficients C 1m are given by rffiffiffiffiffiffi i 1! ð1=2Þjaj2 m1 hm 1 jaj2 : Cm ¼ a e m! 1!
(12)
(13)
Repeating this procedure for n ¼ 2; 3; . . . we can check if it is possible to identify a generalisation for the coefficients C nm for any n. Thus for n ¼ 2 and 3, we have rffiffiffiffiffiffi
2! ð1=2Þjaj2 m2 mðm 1Þ jaj4 2 mjaj2 þ Cm ¼ a e ; (14) m! 2! 2
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C 3m
255
rffiffiffiffiffiffi
3! ð1=2Þjaj2 m3 mðm 1Þðm 2Þ mðm 1Þ 2 mjaj4 jaj6 e jaj þ ¼ a : m! 3! 2! 2! 3! (15)
Note that although the coefficients for n ¼ 2 and 3 are easily obtained, a generalisation for C nm is not entirely obvious. In order to illustrate how this generalisation can be done, let us take the coefficient C 3m as an example. For C 3m we verify that the first term within square brackets can be written as mðm 1Þðm 2Þ ð1Þk jaj2k m! ¼ : (16) 3! ðm 3Þ! 3! k¼0
The term jaj2k was included since it is present in all remaining terms within the square brackets in Eq. (15), the parameter 2k appears due to the expansion of even order terms and the parameter k is used as our attempt to identify a sum. It is also clear that ð1Þk is used to reproduce the signal change seen for the terms in this example. The remaining terms within square brackets in Eq. (15) can be further analysed. Thus for the second term we have mðm 1Þ 2 ð1Þk jaj2k m! jaj ¼ 2! ðm 2Þ! 2! k¼1 k 2k ð1Þ jaj m! ¼ ; ð17Þ ð3 kÞ! ðm 3 þ kÞ! k¼1
note that 2! was replaced by ð3 kÞ! and ðm 2Þ! by ðm 3 þ kÞ! because it is convenient to put the parameter n ¼ 3 in evidence for this example. Similar rearrangements can be done for Eq. (16) which becomes mðm 1Þðm 2Þ ð1Þk jaj2k m! ¼ : (18) 3! ð3 kÞ! ðm 3 þ kÞ! k¼0
The other remaining terms in this analysis are re-written as mjaj4 ð1Þk jaj2k m! ¼ 2!ðm 1Þ! 2! k¼2 k 2k ð1Þ jaj m! ¼ 2!ðm 3 þ kÞ! k¼2 k 2k ð1Þ jaj m! ¼ ; 2!ð3 kÞ!ðm 3 þ kÞ! k¼2
ð19Þ
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note that a factor 1=2! can be conveniently replaced by 1=k! in Eq. (19) and can also be included in Eqs. (17) and (18) which transforms them into the following set: mðm 1Þðm 2Þ ð1Þk jaj2k m! ¼ ; (20) 3! k!ð3 kÞ! ðm 3 þ kÞ! k¼0
mðm 1Þ 2 ð1Þk jaj2k m! jaj ¼ 2! k!ð3 kÞ! ðm 3 þ kÞ!
;
(21)
k¼1
mjaj4 ð1Þk jaj2k m! ¼ k!ð3 kÞ!ðm 3 þ kÞ! 2!
:
(22)
k¼2
Finally, the fourth remaining term can be easily written in the same format as the previous terms, though now for a value of k ¼ 3; jaj6 ð1Þk jaj3k m! ¼ : (23) k!ð3 kÞ!ðm 3 þ kÞ! 3! k¼3
Hence it becomes clear that Eq. (15) can be written in terms of a sum in k using the results shown from Eqs. (20)–(23): " # rffiffiffiffiffiffi n 3! ð1=2Þjaj2 m3 X ð1Þk jaj2k m! 3 Cm ¼ a e : (24) m! k!ð3 kÞ! ðm 3 þ kÞ! k¼0 This can be generalised to any n value, and thus, the coefficients of the DNS ja; ni in an expansion in a number-state basis can be written as follows: " # rffiffiffiffiffiffi n n! ð1=2Þjaj2 mn X ð1Þk jaj2k m! n Cm ¼ a e : (25) m! ðn kÞ!ðm n þ kÞ!k! k¼0 In the general expression given above one notice that the terms between square ðjaj2 Þ: Therefore, we can brackets are the Laguerre associated polynomials Lmn n write the coefficients of the DNS in the form rffiffiffiffiffiffi n! ð1=2Þjaj2 mn mn 2 n a Ln ðjaj Þ ; (26) Cm ¼ e m! which agrees with the expression displayed in Ref. [1].
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Fig. 2. Wigner function of the displaced number state, with initial number of photon n ¼ 1; displacement parameter a ¼ 3eip=4 and its contours curves.
The statistical properties of the DNS were already investigated thoroughly [1], showing that this state displays desirable quantum characteristics, among them we can mention the interference effects in the phase space [2], as shown in its Wigner function, which is given by 2 2 ð1Þn e2jbaj Ln ð4jb aj2 Þ ; (27) p where Ln are the Laguerre polynomial. In Fig. 2 we show the Wigner function of the DNS with initial number of photon n ¼ 1; displacement parameter a ¼ 3eip=4 and its contours curves.
W ðbÞ ¼
4. Comments and conclusion We have presented in this work an alternative method to generate the DNS starting from the CS, through the passage of latter for an non-linear medium, where the non-linearity of this is associated to the number n that one wants to displace. We showed that the resulting state is the same as the one obtained starting from the number state, already proposed in Ref. [1]. As the number states are difficult to
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obtain from the experimental point of view, our proposal offers a new and viable way to generate the DNS.
Acknowledgements This work was financed by the National Council of Technological Scientific Development (CNPq) and PRONEX/MTC/FINEP of Brazil.
Appendix A The equation y
b ðaÞ b An ¼ DðaÞðb ay Þ n D
(A.1)
can be written in the form y
b ðaÞ : b ay Þðb ay Þn1 D An ¼ DðaÞðb
(A.2)
Using the unitarity property of the displacement operator, in Eq. (A.2) we obtain by ðaÞ ; b by ðaÞDðaÞðb b An ¼ DðaÞðb ay Þ D ay Þn1 D
(A.3)
by ðaÞ ¼ aby a : b DðaÞðb ay Þ D
(A.4)
using
Eq. (A.3) becomes y
b ðaÞ : b ay a ÞDðaÞðb ay Þn1 D An ¼ ðb
(A.5)
Repeating the procedure we obtain by ðaÞ b An ¼ ðb ay a ÞDðaÞðb ay Þðb ay Þn2 D by ðaÞ b by ðaÞDðaÞðb b ay ÞD ay Þn2 D ¼ ðb ay a ÞDðaÞðb y
b b ðaÞ : ay Þn2 D ¼ ðb ay a Þ2 DðaÞðb
ðA:6Þ
If the process is repeated n times one obtains by ðaÞ ¼ ðb b An ¼ DðaÞðb ay Þ n D ay a Þn :
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(A.7)
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