Nonclassical properties of coherent light in a pair of coupled anharmonic oscillators

Optics Communications 359 (2016) 221–233

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Nonclassical properties of coherent light in a pair of coupled anharmonic oscillators Nasir Alam, Swapan Mandal n Department of Physics, Visva-Bharati, Santiniketan 731235, India

art ic l e i nf o

a b s t r a c t

Article history: Received 10 July 2015 Received in revised form 3 September 2015 Accepted 10 September 2015

The Hamiltonian and hence the equations of motion involving the field operators of two anharmonic oscillators coupled through a linear one is framed. It is found that these equations of motion involving the non-commuting field operators are nonlinear and are coupled to each other and hence pose a great problem for getting the solutions. In order to investigate the dynamics and hence the nonclassical properties of the radiation fields, we obtain approximate analytical solutions of these coupled nonlinear differential equations involving the non-commuting field operators up to the second orders in anharmonic and coupling constants. These solutions are found useful for investigating the squeezing of pure and mixed modes, amplitude squared squeezing, principal squeezing, and the photon antibunching of the input coherent radiation field. With the suitable choice of the parameters (photon number in various field modes, anharmonic, and coupling constants, etc.), we calculate the second order variances of field quadratures of various modes and hence the squeezing, amplitude squared, and mixed mode squeezing of the input coherent light. In the absence of anharmonicities, it is found that these nonlinear nonclassical phenomena (squeezing of pure and mixed modes, amplitude squared squeezing and photon antibunching) are completely absent. The percentage of squeezing, mixed mode squeezing, amplitude squared squeezing increase with the increase of photon number and the dimensionless interaction time. The collapse and revival phenomena in squeezing, mixed mode squeezing and amplitude squared squeezing are exhibited. With the increase of the interaction time, the monotonic increasing nature of the squeezing effects reveal the presence of unwanted secular terms. It is established that the mere coupling of two oscillators through a third one does not produces the squeezing effects of input coherent light. However, the pure nonclassical phenomena of antibunching of photons in vacuum field modes are obtained through the mere coupling and hence the transfers of photons from the remaining coupled mode. & 2015 Elsevier B.V. All rights reserved.

Keywords: Squeezing antibunching vacuum field quantum state transfer coupled anharmonic oscillators

1. Introduction The availability of high power laser sources is found extremely useful in the development of nonlinear optics, laser spectroscopy and quantum optics. In particular, during the last three decades, we find tremendous progresses in the field of quantum optics. These include the generation of squeezed states [1,2] and photon antibunching of the radiation field [3], etc. According to Heisenberg, there are always some fluctuations involving the measurements of canonically conjugate variables and their product obeys the famous uncertainty relation named after him. Barring the ground state of the harmonic oscillator, the uncertainty product of canonically conjugate variables attains the minimum value for n

Corresponding author. E-mail address: [email protected] (S. Mandal).

http://dx.doi.org/10.1016/j.optcom.2015.09.034 0030-4018/& 2015 Elsevier B.V. All rights reserved.

coherent state only. It is established that the fluctuation of the dimensionless quadrature components involving the coherent state is equal and is termed as zero point fluctuations [4]. Before the discovery of squeezed states, the ZPF was regarded as the standard quantum limit (SQL). The squeezed state is a quantum state of the radiation field in which one of the quadrature fluctuations goes below the ZPF and hence the SQL [5]. Of course, the uncertainty relation due to Heisenberg is to be respected for the squeezed state as well. The abstract idea of squeezed state came into reality through the remarkable experiment in the degenerate four wave mixing of sodium atom [6]. Till now, we find lot of experimental and theoretical developments involving the researches in squeezed state [1,2,7–11]. It is because the squeezed state could be used in the detection of gravitational wave and noise free communication [1,2]. It is claimed that the sensitivity of the LIGO (Laser Interferometer Gravitational wave Observatory) detector is increased in an unprecedented way by using the

222

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

squeezed state. In addition to the applications of squeezed states in gravitational wave detection [9], it is further used in the demonstration of entangled state and hence the quantum teleportation [12] and in the high precision measurements and meteorology [13,14]. The essence of these researches is based on the nonlinear matter-field interactions. In the theoretical investigation of the squeezing effects of the radiation field, we calculate the second order variances involving the field (quadrature) operators. The non-vanishing second order variances (ZPF in terms of the initial coherent state) follow from the non-commutative nature of the quadrature operators. Therefore, it appears that the nonlinear interaction and hence the additional parameters are essential to manipulate the second order variances and hence the squeezed state. Keeping in mind of second order variances and hence the squeezed states, people have generalized for higher order variances and hence the higher ordered squeezed state. Higher order squeezing in terms of the field amplitude has also been developed [15]. A close look to these investigation of squeezing, higher order squeezing and in the higher ordered squeezing involving the amplitudes reveals that the interaction basically nonlinear in nature. As a matter of fact, it is normally believed that the nonlinear interaction involving field operators is necessary to invoke the squeezing effects. It may be accounted because the linear interaction does not provide any significant contribution to the second and higher order variances involving the quadrature operators. Of late, we observe the huge development in the investigation of the quantum properties of micro- or nano-mechanical systems [16–25]. It is because of the availability of ultra-cooled mechanical system which were otherwise impossible before the advent of laser cooling. Depending upon the various situations and purposes, these nanomechanical systems are fabricated. Because of the complex nature, it is not always possible to provide theoretical analysis of all these systems. We, therefore, seek some relevant simple nanomechanical systems which are very close to the physical situations. For example, the model of two anharmonic oscillators coupled through a third oscillator is found useful in the context of quantum entanglement between various modes and also from the quantum optical applications. The model is potentially useful for further investigation on squeezing, higher-ordered squeezing, mixed mode squeezing and the antibunching of various field modes of the radiation fields. The present paper is thus aimed for a complete presentation of the quantum optical properties of two anharmonic oscillators coupled through a third oscillator.

2. The model Hamiltonian We consider two two-photon anharmonic oscillators of field frequency ω0. These oscillators are coupled to a linear oscillator of field frequency ω . In order to simplify the situation, we consider that the anharmonic constant β of the two nano-anharmonic oscillators are same. Nevertheless, the anharmonic terms are chosen so as to remove the nonconserving energy terms. In other words, the anharmonic terms are diagonalizable in the Fock state basis. The coupling parameter k is of comparable magnitude with those of β. Therefore, the present investigation takes care the situation of a weak coupling condition. The field operators a1 (a1† ), a2 (a2† ) and a3 (a3† ) correspond to the annihilation (creation) operators involving the first anharmonic oscillator, second anharmonic oscillator and the linear oscillator respectively. Therefore, the Hamiltonian of the system follows as [22,23]

(

(

)

H = =ω 0 a1† a1 + a2† a2 + =ωa3† a3 + =β a1† 2 a12 + a2† 2 a22 + =k

(

a1†

+

a2†

) a3 +

=k ⁎

( a1 + )

a2 a3†

) (1)

It is clear that the Hamiltonian (1) is quite useful from the view point of nanomechanical coupled anharmonic oscillators one and hence it attracts potential applications in quantum information theory. In fact, the present model can be thought of as a cavity with two modes which is a special case of a cavity of several modes [26,27]. The Hamiltonian (1) in its present form or in a slightly different form [28] might be useful in chemistry since it is established that the two C–H bonds of dihalomethane lead to the model of two quartic anharmonic oscillators coupled through the Jaynes–Cummings [29] type interaction. Of late, we find that the chain of coupled oscillator could be of use in utilizing the quantum state transfer [30–32]. Hence, it will be an interesting problem to investigate the effects of the coupling term and hence the quantum state transfer on the squeezing effects of the input coherent light involving three coupled oscillator (i.e. β¼ 0 in (1)). In the absence of coupling (i.e. k ¼0), the Hamiltonian (1) corresponds three decoupled oscillators and hence an exactly solvable model. The presence of nonlinear terms involving β and the coupling constant k in the model Hamiltonian (1) makes the model more general and a realistic one. In particular, the most of the problems of coupled oscillators are basically a special case of the present one under consideration. Of course, the presence of nonlinearity and the coupling terms poses a serious problems for analytical investigation. As a matter of fact, the presence of anharmonic terms are on the way of getting closed form analytical solutions of the problem. Nevertheless, we believe that the presence of anharmonic term might be useful for investigating the squeezed states, amplitude-squared squeezed states, photon antibunching and nonclassical photon statistics of the input radiation field. Therefore, in the present investigation, we would like to address few of the nonclassical properties of the input coherent light. Of course, we will provide approximate analytical solution to the model Hamiltonian (1). 2.1. The solution of field operators The equations of motion involving the field operators corresponding to the Hamiltonian (1) are given by [23]

a1̇ = − iω 0 a1 − 2iβa1† a12 − ika3 a2̇ = − iω 0 a2 − 2iβa2† a22 − ika3 a3̇ = − iωa3 − ik (a1 + a2 )

(2)

where the coupling constant k is real. The solution of the above equations (2) is used in our recent article [23]. However, in the present communication, the detailed solutions are given. It will make the presentation self-consistent. Now, we have two special situations where these differential equations (2) and hence the model Hamiltonian (1) offers exact analytical solutions. First one, for k¼0 (i.e. there is no coupling), three differential equations in (2) are completely decoupled. The corresponding solutions for the field † † operators are a1 (t ) = e−i (ω0 + 2βa1 a1) t a1 (0) , a2 (t ) = e−i (ω0 + 2βa2 a2 ) t a2 (0) , − i ω t and a3 (t ) = e a3 (0) . Therefore, in case of mode a1 and a2, the field frequency ω0 gets modified by the presence of Kerr-type nonlinearities involving the nonlinear constant β. The second situation for which the set of differential equations are exactly solvable is for β¼ 0. The corresponding solutions may be obtained from our general solutions (3) by putting β¼0. Therefore, all the three field modes a1, a2, and a3 are decoupled and are hardly useful for the investigation of entanglement and other nonclassical properties of the radiation fields. On the other hand, the presence of both the nonlinear constant β and the coupling constant k make the equations (2) unsolvable in closed analytical forms. Therefore, one has to explore either numerical solutions or an approximate analytical solution for further investigations. It is true that the numerical

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

solutions are exact. However, the numerical solutions are not the good option when we are looking for minute physical insights of the problem. It is because, the numerical artifact may kill the minute effects during the calculations. One more point has to be remembered that the coupled nonlinear differential equations (2) are involving the non-commuting operators. The possibilities of getting the numerical solutions of the coupled differential equations involving noncommuting operators are definitely remote if not impossible. Therefore, in spite of the inexactness, the approximate analytical solutions are preferred to the exact numerical solutions for the coupled differential equations (2). In the present investigation, we shall keep ourselves confined to the solutions up to the second orders in β and k which in our opinion are fair enough to dealt with the physical problems. Therefore the approximate solutions up to the second order in β and k are chosen as

f1̇ a1 + f2̇ a2 + f3̇ a3 + f4̇ a1† a12 + f5̇ a1† a1a1† a12 + f6̇ a1† a1a3 + f7̇ a3† a12

(

= − iω 0 f1 a1 + f2 a2 + f3 a3 + f4 a1† a12 +

f5 a1† a1a1† a12

+ f6 a1† a1a3 + f7 a3† a12 )

(

− 2iβ|f1 |2 f1 a1† a12 + f3 a1† a2 a1 + f4 a1† 2 a13 + f3 a1† a1a3 +

f4 a1† a1a1† a12

) − 2iβ ( f

2 ⁎ † 2 f a a 1 3 3 1

+ f 12 f4⁎ a1† 2 a13

)

− ik ( h1a1 + h2 a2 + h3 a3 )

(4)

At this point, we once again reiterate that the above equation is valid up to the second orders in the coupling constant k and the anharmonic constant β . Now, we collect the coefficients of a1, a2, a3, a1† a12, a1† a1a1† a12, a1† a1a3, and a3† a12 of both sides of Eq. (4). Hence, we have

f j̇ = − iω 0 f j − ikhj ,

j = 1, 2 and 3

f4̇ = − iω 0 f4 − 2iβ|f1 |2 (f1 − f4 ) + 2iβf 12 f4⁎

a1 (t ) = f1 a1 (0) + f2 a2 (0) + f3 a3 (0) + f4 a1† (0) a12 (0) + f5 a1† (0) a1 (0) a1† (0) a12 (0) + f6 a1† (0) a1 (0) a3 (0) + f7 a3† (0) a12 (0)

f5̇ = − iω 0 f5 − 4iβ|f1 |2 f4 − 2iβf 12 f4⁎ f6̇ = − iω 0 f6 − 4iβ|f1 |2 f3 f7̇ = − iω 0 f7 − 2iβf 12 f3⁎

a2 (t ) = g1a1 (0) + g2 a2 (0) + g3 a3 (0) + g4 a2† (0) a22 (0) + g5 a2† (0) a2 (0) a2† (0) a22 (0) + g6 a2† (0) a2 (0) a3 (0) + g7 a3† (0) a22 (0)

(3)

In the subsequent development of the present investigation we shall simply write a1, a2, and a3 instead of a1 (0) , a2 (0) and a3 (0) respectively. The similar notation will be adopted for their creation operators counterpart. The assumed solutions (3) are completely known provided the functional forms of the time dependent parameters fi (t ) , gi(t) and hi(t) are obtained. The above choice of the trial solutions (3) could easily be substantiated by the mathematical and physical reasoning. For example, the solution a1 (t ) may be derived from the mathematical expression of the temporal evolution of the operator a1 (t ) = exp (iHt /= ) a1 (0) exp ( − iHt /= ). Now, in the absence of the nonlinear term and the interaction term, the operators a1 (t ) , a2 (t ) , and a3 (t ) evolve freely. Therefore, the parameters f1 , g2 and h3 are the free evolution terms corresponding to the field modes a1 (t ) , a2 (t ) and a3 (t ) respectively. On the other hand the coupling of the oscillator corresponding to the field operator a1 (t ) with those of the other two oscillators corresponding to the field modes a2 (t ) and a3 (t ) is manifested through the terms proportional to f2 and f3respectively. The parameters f5 , f6 and f7 appearing because of the involvement of the nonlinear parameter β. We observe that the parameter f4 is responsible for simultaneous creation and annihilation of one photon and two photon respectively of the field mode a1. These types of nonlinear generation of photons are attributed by the presence of the nonlinear term β. Interestingly, the parameter f6 corresponds the creation and annihilation of one photon each in the field mode a1 and the annihilation of one photon corresponding to the field mode a3. Therefore, the parameter f6 signifies the presence of nonlinear and coupling phenomenon between the modes a1 and a3 as well. In a similar fashion the remaining parameters may also be interpreted physically. With the mathematical and physical justifications of the solutions (3), now, we go ahead for the evaluation of the analytical expressions of these expansion coefficients fi , gi , and hi . In order to do so, we put these solutions (3) in the differential equations (2). Therefore, we have

(5)

In a similar manner, we obtain the differential equations involving the parameters gi and hi . These are

gj̇ = − iω 0 gj − ikhj ,

a3 (t ) = h1a1 (0) + h2 a2 (0) + h3 a3 (0) + h4 a1† (0) a12 (0) + h5 a2† (0) a22 (0)

223

j = 1, 2 and 3

g4̇ = − iω 0 g4 − 2iβ|g2 |2 (g2 − g4 ) + 2iβg22 g4⁎ g5̇ = − iω 0 g5 − 4iβ|g2 |2 g4 − 2iβg22 g4⁎ g6̇ = − iω 0 g6 − 4iβ|g2 |2 g3 g7̇ = − iω 0 g7 − 2iβg22 g3⁎

(6)

and

hj̇ = − iωhj − ik (f j + gj ),

j = 1, 2 and 3

h4̇ = − iωh4 − ikf4 h5̇ = − iωh5 − ikg4

(7)

It is clear that the differential equations involving fi (t ), gi(t) and hi(t) are coupled to each other. However, these coupled differential equations are involving the c-numbers which are opposed to the coupled nonlinear differential equations (2) involving noncommuting operators. In order to solve these coupled differential equations (5)–(7) up to the second order in nonlinear constants and in the coupling constants, we require the relevant initial conditions. A close look of the analytical solution of the field operator a1 (t ) in Eq. (3) gives rise to the initial conditions for f1 (0) = 1 and fi (0) = 0, for i¼ 2,3,4,5,6, and 7. In a similar manner the remaining initial conditions for gi and hi are easily obtained from the solutions of the field operators a2 (t ) and a3 (t ) respectively in Eq. (3). Interestingly, the coupled differential equations involving f1 , f2 , f3 , g1, g2, g3, h1, h2 and h3 are easily decoupled and are solved in an exact way. Therefore, by using the initial conditions, we obtain the solutions for f1 , f2 , g1, g2, h1, and h2. These are given by

⎞ e−iω 0 t e − i Ωt ⎛ iΩ′ ⎜ cos b′t + sin b′t ⎟ + ⎝ ⎠ ′ b 2 2 − i Ω t ⎞ e−iω 0 t ⎛ e iΩ′ ⎜ cos b′t + g1 = f2 = sin b′t ⎟ − ⎠ b′ 2 ⎝ 2 f1 = g2 =

h1 = h2 =

−ike−iΩt sin b′t b′

where the parameters b′, Ω , and Ω′ follow as

(8)

224

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

ω + ω0 = ω − Ω′ 2 ω − ω0 Ω′ = = Ω − ω0 2

Ω=

2

b′ =

2k2

+ Ω′

F (t ) =

−i

(9)

2

1⎡⎛ e−2ib1t − 1 e−ib1t − 1 ⎫ β ⎞⎧ ⎬ + 2i ⎢ ⎜1 − ⎟⎨ t + i 4⎣⎝ b1 ⎠ ⎩ 2b1 b1 ⎭ b1 ⎧ e−2ib2 t − 1 e−ib2 t − 1 e−2iβt − ⎨ +2 + b2 ⎩ b2 2b2 2β

⎧ ⎫ ⎪ 2β ⎪ ⎨ sin b1t + b1t ⎬ 2⎪ ⎪ b1 ⎩ ⎭

Now, the solutions for f3 , g3, and h3 are also obtained. These are

+

e−iΩt sin b′t b′ e−iΩt sin b′t g3 = − ik b′ ⎛ ⎞ iΩ′ h3 = e−iΩt ⎜ cos b′t − sin b′t ⎟ ⎝ ⎠ b′

⎧ ⎪ e−2ib1t − 1 + 8 e−ib1t − 1 + iβ ⎨ ⎪ 4b12 ⎩

f3 = − ik

(

{ 1 + A (t ) − A (0) } e−iω t + { −1 + B (t ) − B (0) } e−i (ω − 2β) t

f4 = g4 =

0

0

(11)

where β|f1 |2 ∼ β ,

A (t ) =

βe−ib1t − iβt b1

B (t ) =

−βe−ib2 t 1 − e−2iβt b2 2

(12)

b1 = Ω′ − b′, and b2 = Ω′ − b′ + 2β are used. Interestingly, the parameter A(t) increases indefinitely with the increase of dimensionless interaction time βt and hence the divergent nature of the solution (12). These type of terms for which the solutions exhibit divergent nature are called secular terms. Now, the solution for f4 is utilized to obtain the solution for h4 and h5. The corresponding solutions follow immediately

h4 = h5 =

⎡⎛ ⎞ iβtk −iω 0 t 1 β β ⎟ e−iω 0 t − 1 e + k ⎢ ⎜⎜ − + − b1 2Ω ′ 2Ω′ ⎟⎠ 2Ω ′ ⎣⎢ ⎝ 2

2 β 2 e − i (b 2 + ω 0 ) t − 1 ⎤ b e−i (ω 0 − 2β) t − 1 ⎥ + 1 − ⎥⎦ b2 b3 b1b2 b3 4

(

(13)

⎡⎧ ⎛ ⎞ ⎛ ⎫⎛ ⎪ ⎪ t b − 2β ⎞ ⎟ + i ⎜ 2β − b2 ⎬ ⎜ e2iβt − 1⎟ f5 = g5 = − 4iβe−iω 0 t ⎢ ⎨ ⎜⎜ 1 ⎟ ⎟ ⎜ ⎪⎜ ⎢⎪ 2 ⎠ ⎣ ⎩ ⎝ b1 ⎠ 2β ⎝ 2b2 ⎭ ⎝ ⎤ ⎞ 2iβ2 e−ib1t − 1 ⎟ 1 iβ − t2 + + F (t ) ⎥ ⎟ 2 ⎥ b1b2 b3 2 ⎦ ⎠ (14) where

)

⎞⎫ ⎛ 2 ⎪ ite−ib1t ⎜ −ib1t ⎟ ⎬ + iβt e + 4 ⎟⎪ 2b1 ⎜⎝ 2 ⎠⎭

−

⎫ 2β ⎧ ⎨ sin b1t + b1t ⎬ ⎩ ⎭ b1b2

−

⎤ ⎫ i ⎧ −2ib1t ⎨ e − 1 + 4 e−ib1t − 1 − 2ib1t ⎬ ⎥ ⎭ ⎥⎦ 4b1 ⎩

(

)

(

)

(15)

In the symbolic calculation, we will use the analytical expressions for f5 and g5 exhibited through Eq. (14). Eqs. (12)–(14) exhibit the presence of secular term which are proportional to βt, β2t 2, kt, k2t 2, etc. These terms rise indefinitely with the increase of time t and are responsible for divergent nature of the equation involving them. Undoubtedly, the presence of secular terms in the perturbative solutions of the nonlinear coupled differential equation are inevitable. Therefore, the present solutions are also restricted by the secular terms. The secular terms can also be tackled with the help of multiscale perturbation theory [33,34] and Tucking-in techniques [35]. Unfortunately, these techniques are successfully used for a single system. To the best of our knowledge, these techniques are yet to use in the many particle system. Therefore, the present investigation assumes the small interaction time so as to ensure the contribution of secular terms is less harmful. Now, the solutions for f6 and g6 follow as

f6 = g6 =

⎤ 2β − i ω t iΩ′ 2β − i Ω t ⎡ e e 0 sin b′t ⎥ − ⎢⎣ cos b′t + ⎦ k b′ k

(16)

Finally, the solutions for f7 and g7 are obtained. These are given by

f7 = g7 = +

iβk ⎧ sin b1t ⎫ −iω 0 t ⎨t − ⎬e b1 ⎭ 2b′ ⎩

βk −iω 0 t ⎡ e−i (b1− 2b′) t − 1 e2ib′t − 1 eib3 t − 1 ⎤ − − e ⎢ ⎥ ⎣ ⎦ b1 − 2b′ b′ b3 4b′

(17)

Now, we check the equal time commutation relation (ETCR) for field operators a1 and a1†. It can easily be verified that

(18)

is indeed satisfied. In a similar manner it is possible to establish that

)

}

(

[a1 (t ), a1† (t )] = 1

where b3 = Ω′ + b′. The solutions for f5 and g5 are now in order. These are given by

{

)

+

(10)

For small coupling constant k, we may assume that Ω′ = b′ and hence |f1 |2 = 1, |f2 |2 = 0, and |f3 |2 = 0. Hence, the problem reduces to the problems of three uncoupled harmonic oscillators evolving with their characteristic frequency. Up to this point the solutions for f1 , f2 , f3 , g1, g2, g3, h1, h2, and h3 are exact for β¼0. It is because, the differential equations involving those parameters will contain β if the higher order terms are taken care of. In other words, the differential equations involving those coefficients are exact if β¼0. Now, for β ≠ 0, the remaining coefficients f4 , f5, etc. will be nonvanishing. We observe that the solutions for f4 and g4 are quite complicated. However, the removal of the terms beyond second orders in β and k are found useful to obtain the following solutions:

1⎫ ⎬ ⎭

[aj (t ), al† (t )] = δjl,

j and l = 1, 2 and 3.

(19)

The analytical solutions for the field operators involving two quartic oscillators coupled through a linear oscillator are provided in this communication. It is true that the present solutions are approximate and are valid up to the second orders in β (anharmonic constant) and k (coupling constant). In spite of that, we hope that the present solutions will be of immense helpful in investigating the quantum statistical properties of the radiation filed. In particular, the squeezing, higher order squeezing and the photon antibunching may easily be investigated through the present model. Nevertheless, the present model involves three oscillators and is a potential model for investigating the quantum entanglement, etc. These require detailed systematic studies in each occasion.

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

3. Squeezed states in pure and in mixed modes In the present section, we investigate the possibilities of getting the squeezed states in pure and in mixed modes of the input coherent light involving two nano-anharmonic oscillators coupled through an oscillator. In order to do so, we define the quadrature operators involving various field modes a1, a2, and a3. These are

1 Xj = (aj + a†j ) 2 i Ẋj = − (aj − a†j ) 2

(20)

where j¼ 1, 2 and 3. In terms of the initial coherent state, the second order variances of the quadrature Xj (0) and Xj̇ (0) are equal and are termed as the zero the point fluctuations (ZPF). Therefore, the quadrature Xj (or Xj̇ ) is said to be squeezed if the corresponding variance goes below the ZPF. The corresponding squeezing is termed as pure mode squeezing. In order to respect the Heisenberg uncertainty relation, the simultaneous squeezing of both the quadratures is completely ruled out. In other words, the squeezing in one of the quadrature components automatically prohibits the squeezing of the conjugate quadrature. Hence, the Xj (Xj̇ ) quadrature is said to be squeezed at the cost of the conjugate 1 quadrature Xj̇ (Xj ) if the second order variances (ΔXj )2 < 2

1

((ΔXj̇ )2 < 2 ) . In order to calculate the second order variances of the quadrature operators (20), we assume that the initial composite coherent state |Ψ 〉 = |α1〉|α2 〉|α3 〉 obeys the following eigenvalue equations:

a l |Ψ 〉 = α l |Ψ 〉 ,

l = 1, 2 and 3

(21)

where α1, α2 and α3 are complex. The parameters |α1 |α2 and |α3 |2 are the number of photons present in the field mode involving a1, a2, and a3 respectively. In terms of the initial composite coherent state, we calculate the second order variances of the quadrature Xj and Xj̇ . These are given by

|2 ,

⎛ ΔX 2 ⎞ 1 ⎜ ( 1) ⎟ 1 = + |f4 |2 5|α1|4 + 2|α1|2 ⎜ 2 ̇1 2 ⎟ 2 Δ X ⎝ ⎠

(

(

)

|2

)

±

1⎡ 2 2 2 ⁎ 2 2 2 2 2 ⎣ 2f 4 |α1| α1 + f1 f4 α1 ± 2f1 f4 |α1| + f1 f5 2|α1| α1 + α1 2

±

f1 f5⁎

±

2f1 f7⁎ α1⁎ α3

{

(

( 3|α1

|4

+ 2|α1

|2

) + f1 f6 α1α3 ±

)

f1 f6⁎ α1α3⁎

} + c. c.]

⎛ ΔX 2 ⎞ 1 1 ⎜ ( 2) ⎟ 1 = + |g4 |2 5|α2 |4 + 2|α2 |2 ± ⎡⎣ 2g42 |α2 |2 α22 + g2 g4 α22 ⎜ ̇2 2 ⎟ 2 2 2 Δ X ⎝ ⎠

(

(

)

)

(

{

± 2g2 g4⁎ |α2 |2 + g2 g5 2|α2 |2 α22 + α22

(

)

)

± g2 g5⁎ 3|α2 |4 + 2|α2 |2 + g2 g6 α2 α3 ± g2 g6⁎ α2 α3⁎

}

± 2g2 g7⁎ α2⁎ α3 + c . c .]

( ΔX3 )2 = ( ΔX3̇ )2 =

1 2

(22)

where c.c stands for the complex conjugate, 2 (ΔXj )2 = 〈Ψ |X2j |Ψ 〉 − 〈Ψ |Xj |Ψ 〉2 and (ΔXj̇ )2 = 〈Ψ |Ẋ j |Ψ 〉 − 〈Ψ |Xj̇ |Ψ 〉2 . The expressions for (ΔX1)2 and (ΔX1̇ )2 are obtained by taking the upper and the lower sign respectively on the right-hand side of the first equation in (22). In the derivation of Eq. (22), we have made use of the solutions for the field operators available through Eq. (4). At

225

t¼0, it is clear that the variances (22) reduce to their minimum 1 values (ΔXj (0))2 = (ΔXj̇ (0)) = 2 and hence the uncertainty product 1 reaches the minimum value (ΔXj (0))2 (ΔXj̇ (0))2 = corresponding 4

to the coherent state. In the absence of photons in the field mode a1, and a2, the initial coherent state remains coherent and hence no 2 squeezing. Interestingly, ΔX2 = ΔẊ = 1 up to the second orders in 3

3

2

β and k and hence the squeezing of the field mode a3 is completely ruled out. However, the possibility of squeezing in a3 is not ruled out if the higher order (beyond second order) corrections for field operators are taken care off. For β¼0, the parameters f4 , f5 , f6 and f7 are identically zero and the possibility of getting squeezed state is washed out. Therefore, the presence of coupling alone (i.e. k ≠ 0) is unable to produce squeezed state of the input coherent light. In other words, the quantum state transfer [29–31] realized in a chain of coupled oscillator does not affect the squeezing effects of the input coherent light. In spite of the analytical expressions (22) for the second order variances of various field modes, it is very difficult to predict the squeezing behavior out of it. In order to have some feelings of the above analytical expressions (22), we consider the special case for which αi are real and α3 = 0. Therefore, we have (for a1-mode only)

⎛ ΔX 2 ⎞ 1 1 ⎜ ( 1) ⎟ 1 = + |f4 |2 5α14 + 2α12 ± ⎡⎣ ⎜ ̇ 2⎟ 2 2 2 Δ X 1 ⎝ ⎠

(

(

)

(

)

{ ( 2f

2 4

) } + c. c⎤⎦

+ f1 f4 + f1 f5 ± 2f1 f4⁎ ± 2f1 f5⁎ α12

)

+ 2f1 f5 ± 3f1 f5⁎ α14 (23)

Apart from the temporal dependance, the second order variances (23) are extremely sensitive to the photon number α12, the coupling constant k and nonlinear constants β. It is clear that the nonvanishing values of f4 and/or f5 are necessary for the existence of squeezing and hence the anharmonic constant β is nonzero. Now, for small values of α1, the amount of squeezing will be small and will be controlled by the photon number α12 of the coherent field mode a1. On the other hand, for reasonably high values of α1, the corresponding squeezing will be controlled by the square of the photon number α14. In accordance with the requirements of the physical situations, the further simplifications of Eq. (23) are possible. However, in the present investigation, we shall use Eq. (22) for investigating the squeezing effects of the pure mode. With the suitable choice of the physical parameters, the analytical expressions for the second order variances (22) could be used for investigating the squeezing of the input coherent light. In order to give some flavor of the analytical expressions of (ΔX1)2 and (ΔX1̇ )2 , we give some numerical estimates of Eq. (22) and are exhibited in Figs. 1 and 2. In these estimates, we assume αi as real. In addition to the photon numbers and the phases in three coherently prepared field modes a1, a2, and a3, we chose the dimensionless interaction time kt, anharmonic constant β, and the field frequencies ω0 and ω. In Fig. 1, the second order variances in X1 (dotted curve) and in X1̇ (solid line) quadrature are plotted as a function of dimensionless interaction time kt for ω = 1.1 × 1014 Hz, ω0 = 1.0 × 1014 Hz , α1 = 0.1, α2 = 1.5, and α3 = 0.01. The straight line parallel to the kt axis corresponds the variances involving the coherent state. The squeezing in X1 and X1̇ quadrature (dotted line) are obtained when the corresponding variances go below the value corresponding to the coherent state. It is clear that the simultaneous squeezing in both the quadratures are ruled out and are consistent with the Heisenberg uncertainty relation. The collapse and revival of squeezing in both the quadrature components are exhibited. Interestingly, the amount of squeezing is increased with the increase of kt. In Fig. 1, for example, after two complete evolutions we obtain 0.02% squeezing of the X1 quadrature for kt ∼ 0.007115. However, the amount of squeezing of that quadrature is markedly increased to 13.48% when α1 = 1.5 and

226

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

0.50015

0.50010

X12 and X 1

2

0.50005

0.50000

0.49995

0.49990

0.49985

0.000

0.002

0.004

kt

0.006

0.008

0.010

2

Fig. 1. Plot of (solid curve) as a function of dimensionless interaction time kt for ΔX1̇ k = 1011 s−1, β = 0.5 × 1011 s−1, ΔX12 (dotted curve) and α1 = 0.1, α 2 = 1.5, α3 = 0.01, ω0 = 1014 Hz and ω = 1.1 × 1014 Hz . Within the axes range, the maximum squeezing after two complete evolutions in X1 and X1̇ are indicated by arrow marks.

2

X21 and X 1

0.55

0.50

0.45

0.000

0.002

0.004

0.006

0.008

0.010

kt Fig. 2. Same as in Fig. 1 with α1 = 1.5 instead of α1 = 0.1.

0.8

X21 and X 1

2

0.7 0.6 0.5 0.4 0.3 0.2 0.0

0.5

1.0

1.5

2.0

2 Fig. 3. Plot of ΔX12 (dotted curve) and ΔX1̇ (solid curve) as a function of photon number for kt ¼0.01, k = β = 0.5 × 1011 s−1, α 2 = 1.5, α3 = 0.01, ω0 = 1014 Hz and ω = 1.1 × 1014 Hz .

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

227

2

2 X13 and X 13

0.50005

0.50000

0.49995

0.000

0.002

0.004

0.006

0.008

0.010

kt 2 ̇2 Fig. 4. Plot of (dotted curve) and (solid curve) as a function of dimensionless interaction time kt for ΔX13 ΔX13 k = 1011 s−1, β = 0.5 × 1011 s−1, α1 = 0.1, α 2 = 1.5, α3 = 0.01, ω0 = 1014 Hz and ω = 1.1 × 1014 Hz . Within the axes range, the maximum squeezing after two complete ̇ evolutions in X13 and X13 are indicated by arrow marks.

kt ∼ 0.007093 (Fig. 2). It is found that the squeezing keep on increasing with the increase of kt (Figs. 1 and 2). By keeping the dimensionless interaction time kt = 0.009, we plot the ΔX12 and 2 ΔẊ as a function of the photon number α1 in Fig. 3. In Figs. 1–3, 1

the second order variances of quadrature components involving pure field mode a1 and hence the squeezing effects are investigated. The squeezing of the pure field mode a2 will be identical with those of the field mode a1 and hence are not exhibited. The monotonic increasing nature of the second order variances with the increase of kt in Figs. 1 and 2 reveals the perturbative/ approximate nature of the solution. In Fig. 3, the monotonic nature of the second order variances is also exhibited with the increase of α1 and hence the photon number in a1 mode. The secular nature of the solution is inherent for the perturbative treatment and hence the monotonic increasing/decreasing nature of the second order variances. Now, we investigate the squeezing for compound (mixed) modes. In order to do so, we define the mixed mode quadrature operators involving a1 and a3. The corresponding operators are given by

1 a1 + a1† + a3 + a3† 2 ̇ = − i a1 − a † + a3 − a † X13 1 3 2 X13 =

(

)

(

)

(24)

The second order variances are calculated in terms of the initial coherent state

⎛ ΔX 2 ⎞ 1 ⎜ ( 13 ) ⎟ 1 = + ( f1 h1⁎ + f3 h3⁎ + c . c ) ⎜ ̇ 2⎟ 2 4 Δ X 13 ⎝ ⎠

(

)

±

1⎡ 2 2 2 2 ⎣ 2f 4 |α1| α1 + f1 f4 + f1 h4 + h1f4 α1 4

{

(

)

( ) } { ( ± f5⁎ ( 3|α1|2 + 2) |α1|2 + f6 α1α3 ± f6⁎ α1α3⁎ ± 2f7⁎ α1⁎ α3 }

± 2 f1 f4⁎ + f1 h4⁎ + h1f4⁎ |α1|2 + f1 f5 2|α1|2 + 1 α12

(

± h3 f6⁎ |α1|2 ± f7 α12

) ) + c. c ]

(25)

It is clear from Eq. (25) that the vacuum field mode a1 is not suitable for getting the mixed mode ( a1 − a3 mode) squeezing of the

input coherent light. However, the possibility of getting the mixed mode squeezing governed by Eq. (25) is not ruled out for nonzero photon numbers in the field mode a1 (Figs. 4 and 5). The fundamental nature of the squeezing pattern in the mixed modes is identical with those of the pure mode squeezing. In addition to these, the percentage of squeezing hardly differs from those of the normal squeezing counterpart. These manifest the fact that the squeezing is predominantly controlled by nonlinearities not by the coupling between the oscillators. These are exhibited in the tabular form (Table 1). 3.1. Higher order squeezing: amplitude squared squeezing The squeezing in the normal sense employs second order moments of the field operators. However, it is natural to explore the squeezing of higher orders which employ higher order moments of the field operators. With the experimental observation of higher order moments available in the literature [36,37], the idea of higher (beyond quadratic in field quadrature) order squeezing is no more an abstract idea. By using the homodyne technique, a proposal for realizing the higher-ordered amplitude squeezing is available in the literature as well [38]. The idea of higher order squeezing originated from the work of Hong and Mandel [39] who generalized the concept of normal squeezing. The possibilities of realizing the higher order squeezing are proposed in various quantum optical phenomena [39–46]. For example, in the second harmonic generation the fundamental mode gets squeezed to the second order [39]. Interestingly, Hillery proposed a new way of finding the squeezing for the higher orders in amplitude [15,40]. It is argued that if the second harmonic is originally in a coherent state, then the square of the amplitude of the fundamental is squeezed [15,40]. The concept of higher order squeezing due to Hillery is used to investigate the amplitude squared squeezing inside and outside of a degenerate parametric amplifier [41]. The direct observation of higher ordered amplitude and/or quadrature squeezing is not available till to-date. However, the recent observation of higher ordered photon correlation function [36,37] has boost up the expectation for realizing the higher ordered amplitude/quadrature squeezing. Now, it is our academic interest to check the possibilities of amplitude squared squeezing in the field mode a1. In order to do this, we follow the prescription

228

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

0.54

2

X213 and X 13

0.52

0.50

0.48

0.46 0.000

0.002

0.004

0.006

0.008

0.010

kt Fig. 5. Same as in Fig. Fig. 4 with α1 = 1.5 instead of α1 = 0.1.

suggested by Hillery [15]. Therefore, we define the amplitudesquared quadrature operators Y1 and Y2

1 2 a1 (t ) + a1† 2 (t ) 2 i 2 Y2 = − a1 (t ) − a1† 2 (t ) 2 Y1 =

(

⎛ (ΔY1)2 ⎞ 1 ⎡ ⎜⎜ ⎟⎟ = ⎢ |f1 |4 4|α1|2 + 2 + 4|f1 |2 |f3 |2 |α1|2 + |α3 |2 + 1 ⎝ (ΔY2 )2 ⎠ 4 ⎢⎣

(

(

)

(

(

1 2

)

(27)

)

(

(

) (

+ 2 f1 f3 a1a3 ± f1⁎ f3⁎ a1† a3† + f 32 a32 ± f3⁎ 2 a3† 2

(

)

⎪

⎪

(

± f 12 f3⁎ f4⁎ 2|α1|2 α1α3⁎ + α1α3⁎

(

⎪

± f1⁎ f5⁎ 2a1† 4 a12 + 4a1† 3 a1 + a1† 2

(

± f1⁎ f6⁎ 2a1† 2 a1a3† + a1† a3†

)

⎫ ⎬+ ⎭ ⎪

(

)

⎫ ⎪ ⎬ c⎪ ⎭

† 2 1 1 3

(

⎫

⎪

) ) + c. c⎬⎭ ⎪

⎫ ⎪ ⎛ 7 1 ⎞⎞ ⎬ ± 8|f1 |2 f1 f5⁎ ⎜ 2|α1|6 + 6|α1|4 + |α1|2 + ⎟ ⎟ + c . c ⎪ ⎝ 2 4 ⎠⎠ ⎭

)

)

{ f f ( 2a a a 1 6

⎪

⎧ ⎪ ⎛ ⎛ ⎞ ⎨ ⎜ 4f5 f13 ⎜ 2|α1|2 α14 + 3α14 ⎟ ±⎪ ⎝ ⎠ ⎩⎝

(

⎪

⎫

⎪

) ) + c. c⎬⎭

(

)

⎫ ⎧ ± f4⁎ 2 a1† 4 a12 + 2a1† 3 a1 ⎬ + ⎨ f1 f5 2a1† 2 a14 + 4a1† a13 + a12 ⎭ ⎩

)

(

⎧ ⎪ ⎨ 4f 12 f4⁎ 2 2|α1|6 + 6|α1|4 + 3|α1|2 + c . +⎪ ⎩

)

(

⎪

)

(

⎫ ⎧ ± f1⁎ f4⁎ 2a1† 3 a1 + a1† 2 ⎬ + ⎨ f 42 a1† 2 a14 + 2a1† a13 ⎭ ⎩

)

⎪

⎧ ⎪ ⎨ 4f13 f4 α14 ± 2f1⁎ f4⁎ f 12 6|α1|4 + 8|α1|2 + 1 ±⎪ ⎩

⎧ + 2 f3 f4 a1† a12 a3 ± f3⁎ f4⁎ a1† 2 a1a3† + ⎨ f1 f4 2a1† a13 + a12 ⎩

(

)

⎧ ⎨ f 12 f3 f4 α13 α3 + 4 |f1 |2 f1 f3⁎ α1α3⁎ + c . c ± 4 ⎪ ⎩

⎛ Y1 ⎞ 1 ⎡ ⎜ ⎟ = ⎢ f 2 a12 ± f1⁎ 2 a1† 2 + 2 f1 f2 a1a2 ± f1⁎ f2⁎ a1† a2† ⎝ − iY2 ⎠ 2 ⎣ 1

(

)

)

(

Now, we express the operators Y1 and Y1̇ in terms of the initial annihilation and creation operators of three oscillator modes. Therefore, we have

(

(

)

+ 2|f1 |2 |f4 |2 20|α1|6 + 48|α1|4 + 26|α1|2 + 1

(26)

Therefore, we have

⎡⎣ Y1, Y2 ⎤⎦ = 2i a † a1 + 1

(

⎧ ⎫ ⎪ ⎪ ⎨ 2f 12 f 42 8|α1|2 + 5 α14 + c . c ⎪ ⎬ + 4 |f1 |2 f2⁎ f1 α1α2⁎ + c . c ± ⎪ ⎩ ⎭

)

(

)

⎧ ⎪ ⎨ 4f13 f6 α13 α3 ± 2|f1 |2 f1 f6⁎ 4|α1|2 α1α3⁎ + 3α1α3⁎ ±⎪ ⎩

(

)

+ a1a3

) ±

(28)

In terms of the initial coherent state, we calculate the second order variances of the amplitude squared operators Ya and Yȧ involving the filed mode a. After few algebraic steps, we end up with the following results:

⎫

⎪

) ) + c. c⎬⎭ ⎪

⎧ ⎫ ⎪ ⎪ ⎨ 12|f1 |2 f1 f7⁎ |α1|2 α1⁎ α3 + α1⁎ α3 + c . c ⎪ ⎬ +⎪ ⎩ ⎭

(

⎤

) } + 2 ( f1 f7 a3† a13 ± f1⁎ f7⁎ a3 a1†3 ) ⎥⎦

(

{ ( 4f

2 3 1 f3 f4 α1 α 3

)

(

± 4|f1 |2 f3 f4⁎ 3|α1|2 α1⁎ α3 + α1⁎ α3

⎤

) ) + c. c}⎥⎥⎦

(29)

In order to make the investigation consistent, the above analytical results of the second order variances involving the amplitude squared operators (29) are taken care up to the second orders in the interaction coupling and anharmonic constants. Now, we define two parameters P1 and Q1

No squeezing in X˙13 quadrature No squeezing in X13 quadrature

No squeezing in Y2 quadrature No squeezing in Y1 quadrature

No squeezing in Y2 quadrature No squeezing in Y1 quadrature

No squeezing in Y2 quadrature No squeezing in Y1 quadrature

6.86% (after two evolutions)

5.17% (after six evolutions) 4.83% (after five evolutions)

18.61% (after six evolutions) 17.12% (after five evolutions)

35.78% (after six evolutions) 33.55% (after five evolutions)

8.36% (after two evolutions)

No squeezing in X˙13 quadrature No squeezing in X13 quadrature 0.02% (after two evolutions)

0.02% (after two evolutions)

No squeezing in X˙1 quadrature No squeezing in X1 quadrature 15.96% (after two evolutions)

13.48% (after two evolutions)

No squeezing in X˙1 quadrature No squeezing in X1 quadrature 0.02% (after two evolutions)

0.034% (after two evolutions)

Remarks Squeezing/mixed mode/amplitude squared squeezing

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

Q 1 = (ΔY1)2 − n^1 +

1 2

P1 = (ΔY2 )2 − n^1 +

1 2

229

(30)

According to Mark Hillery [15], the amplitude squared squeezing is obtained if P1 < 0 or Q 1 < 0. Now, for instance, we assume that all the field modes a1, a2 and a3 are in vacuum state. Therefore, we have α1 = 0, α2 = 0 and α3 = 0. Hence, we do not have amplitude squared squeezing for this particular case. Obviously, the analytical expressions (30) for investigating the amplitude squared squeezing of the field mode a1 look very complicated. Therefore, we give some numerical estimates of the above expressions by using the parametric values. These results are available in Figs. 6 and 7. The divergent, collapse and revival nature of the amplitude squared squeezing in Figs. 6 and 7 have resemblance with those of the normal (Figs. 1 and 2) and mixed mode squeezing (Figs. 4 and 5) counterparts. Apart from the improvement of the percentage of squeezing, we observe that the collapse and revival phenomena occur more rapidly compared to those of the normal squeezing. The percentage of squeezing is greatly increased (approximately 7 times) in Fig. 8 if the nonlinear constant β gets doubled compared to those of Fig. 6.

In Fig. 8, β = 1011 HZ instead of 0.5 × 1011 for remaining Figs. 1, 2, 4, 5, 6 and 7.

In addition to the standard definition of squeezing, Luks et al. [47] proposed geometrical (elliptical) representation of variances and hence the Principal Squeezing [47–51]. The geometrical representation of squeezed state received further advancement by Loudon [48] who represented the quadrature variances in terms of the Booth elliptical lemniscate. Interestingly, the principal squeezing is discussed in the anharmonic oscillator model [50]. Miranowicz et al. [51] established that the results of Luks et al. [47] are indeed a special case for their general results involving the nonclassicality of multimode fields. In previous sections, we already discussed the quadrature squeezing and the amplitude squared squeezing of coherent light interacting with a pair of anharmonic oscillator coupled through a linear one. In order to detect the squeezing phenomena of the coherent light through a homodyne detection technique, a strong coherent local oscillator is used. Finally, the quadrature variances are obtained as a function of the local oscillator phase. Therefore, the minimum value of the variance and hence the maximum squeezing is involving the phase angle of the local oscillator. In order to explore the normal and principal squeezing for the mode a1, we define the fluctuations of the operators a1 and a1†

Δa1 = a1 − 〈a1〉 Δa1† = a1† − 〈a1† 〉 Now, the correlation and second order variance of the operator a1 are calculated in terms of the initial coherent state. These are given by

(

)

〈Δa1† Δa1〉 = |f4 |2 |α1|4 〈(Δa1)2〉 = f1 f4 α12 + 2 f 42 + f1 f5 |α1|2 α12 +

f1 f5 α12

+ f1 f6 α1α3

(31)

Now, we follow the definition of principal and normal squeezing due to Luks et al. [47]. Accordingly, the standard definition of squeezing [47]

〈Δa1† Δa1〉 − R〈(Δa1)2〉 < 0

(32)

where R corresponds the real part. On the other hand, the principal squeezing occurs for the following condition:

a

Squeezing in Y1 quadrature Squeezing in Y2 quadrature −0.9841 −0.9228 8a

1.5

0.009801 0.00901

Squeezing in Y1 quadrature Squeezing in Y2 quadrature −0.5119 −0.4707 7

1.5

0.009793 0.009019

Squeezing in Y1 quadrature Squeezing in Y2 quadrature −0.07755 −0.07248 6

1.0

0.009808 0.009006

Squeezing in X13 quadrature

Squeezing in X˙13 quadrature 0.4582

Squeezing in X˙13 quadrature

0.4657

0.00865

0.007088 1.5 5

Squeezing in X13 quadrature

0.4999 0.00868

0.4999 4

0.1

0.007089

Squeezing in X1 quadrature

Squeezing in X˙1 quadrature 0.4202

Squeezing in X˙1 quadrature

0.4326

0.00866

0.007093 1.5 2

Squeezing in X1 quadrature

0.49983 0.00867

0.4999 1

0.1

0.007115

Nature of the squeezing Variances as indicated by arrow kt

α1 Figs. no.

Table 1 Percentage of maximum squeezing in pure, mixed mode and amplitude squared modes.

3.2. Principal squeezing

〈Δa1† Δa1〉 − |〈(Δa1)2〉| < 0

(33)

230

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

1.0

0.05

Q1 and P1

Q1 and P1

0.5

0.00

0.5

0.05

0.000

0.002

0.004

0.006

0.008

1.0 0.000

0.010

kt

Now, for 〈(Δa1)2〉 real, both the normal squeezing and principal squeezing governed by Eqs. (32) and (33) are same. Therefore, it is also clear that the principal squeezing is independent of pump phases. Because of the dependence of phase, the normal squeezing is not unique [49]. Now, we express Eqs. (32) and (33) in terms of the known parameters. Hence, we have

− R〈(Δa1 +2

(

f 42

)2〉

=

|f4 |2 |α1|4

− R ⎡⎣ f1 f4 α12

+ f1 f5 |α1|2 α12 + f1 f5 α12 + f1 f6 α1α3 ⎤⎦

)

(34)

and

〈Δa1† Δa1〉 − |〈(Δa1)2〉| = |f4 |2 |α1|4 −

(

)

0.002

0.004

0.006

(f f α

2 1 4 1

+ 2 f 42 + f1 f5 |α1|2 α12 + f1 f5 α12 + f1 f6 α1α3

)

(35)

In Fig. 9, we plot Sn = 〈Δa1† Δa1〉 − R〈(Δa1)2〉 as a function of the dimensionless time kt. The normal squeezing is obtained for Sn < 0. From Fig. 9, we observe the revival and collapse of the squeezing many times within the axis range. On the other hand, the principal squeezing Sp = 〈Δa1† Δa1〉 − |〈(Δa1)2〉| < 0 is exhibited (Fig. 10) throughout the axis range. In Fig. 10, it appears that the

squeezing is decreased monotonically with the increase of kt. As a matter of fact, the decrease of squeezing continues till kt ∼ 0.24. For kt > 0.24, the squeezing increases again. However, these results are not exhibited. It is to be remembered that we cannot keep on increasing the value of kt since it will violate the condition of the solution.

4. Antibunching of photons In the regime of linear optics, people were interested in the correlation of the field amplitudes and hence the diffraction and interferences. The remarkable experiment by Hanbury-Brown and Twiss [52] is the first one who considered the correlation of intensities even for an incandescent light source to conclude that the photons come together. The phenomenon in which the photons come together is known as photon bunching. There are light sources where the photons do not come in clusters and hence the photon antibunching [1,3,53–67]. In spite of the natural tendency of bunching of photons, there are several predictions where the photons show the antibunching effects. For example, degenerate parametric amplifier [53], quartic anharmonic oscillator [54], twophoton processes [55], damped harmonic oscillator [56], nonlinear optical couplers [57], the resonance fluorescence from a two level

0.2

Q1 and P1

0.010

Fig. 8. Same as in Fig. 7 with β = 1011 s−1 instead of β = 0.5 × 1011 s−1.

0.4

0.0

0.2

0.4

0.000

0.008

kt

1 1 Fig. 6. Plot of Q1 = (ΔY1)2 − 〈n^1 + 〉 (dotted curve) and P1 = (ΔY2 )2 − 〈n^1 + 〉 (solid 2 2 curve) as a function of dimensionless interaction time kt for and k = 1011 s−1, β = 0.5 × 1011 s−1, α1 = 1.0, α 2 = 1.5, α3 = 0.01, ω0 = 1014 Hz ω = 1.1 × 1014 Hz . Within the axes range, the maximum squeezing after five and six complete evolutions in Y1 and Y2 respectively are indicated by arrow marks.

〈Δa1† Δa1〉

0.0

0.002

0.004

0.006

kt

Fig. 7. Same as in Fig. 6 with α1 = 1.5 instead of α1 = 1.0 .

0.008

0.010

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

g 2 (0) =

2

Sn

〈A†A†AA〉 , 〈A†A〉2

0 1 2 0.002

0.004

0.006

0.008

0.010

kt Fig. 9. Plot Sn (34) as a function of dimensionless interaction time kt for and k = 1011 s−1, β = 0.5 × 1011 s−1, α1 = 0.1, α 2 = 1.5, α3 = 0.01, ω0 = 1014 Hz ω = 1.1 × 1014 Hz .

where A is the usual annihilation (creation) operator of the radiation field. Now, for thermal/chaotic light g 2 (0) > 1 and hence the photons are bunched. Interestingly, the radiation field prepared in coherent state gives rise to g 2 (0) = 1 and hence the photons are unbunched (i.e. neither bunching nor antibunching). It is natural to expect that there should be some radiation sources with g 2 (0) < 1 and hence the photon antibunching. Interestingly, the bunching of photons could be explained by using both the wave and particle nature of the light and, therefore, it is regarded as the classical phenomenon. On the other hand, the antibunching of photons could only be explained by using a quantum (photon) nature of light. For this reason the antibunching of photons is regarded as a purely quantum mechanical phenomenon without classical analogue. Now, for field mode i (i = a, b and c ) Eq. (36) is casted in the following form:

gi2 (0) − 1 = 2.2470 2.2475

Sp

2.2480 2.2485 2.2490 2.2495 2.2500 0.000

(36)

(A† )

1

0.000

231

0.002

0.004

0.006

0.008

0.010

kt Fig. 10. Plot Sp (35) as a function of dimensionless interaction time kt for and k = 1011 s−1, β = 0.5 × 1011 s−1, α1 = 0.1, α 2 = 1.5, α3 = 0.01, ω0 = 1014 Hz ω = 1.1 × 1014 Hz .

atom [58], the two photon Dick model [59], and the coherent light interacting with a two-photon medium [60] could be the possible sources for producing antibunched light. The fluorescence spectrum of a single two level atom shows the antibunching and the squeezing as well [58]. There are lots of experiments available where the antibunched lights are produced in the laboratory [61–67]. In most of these experiments [62–65], the resonance fluorescence from a small number of ions [62] atoms or molecules [63–65] are exploited to obtain the antibunching effects. The basic physics behind these experiments are easily understood. The atoms (or molecules) emit radiation and go to a ground state from where no subsequent radiation is possible and hence the photons are antibunched. The resonance fluorescence field from manyatom source is not suitable for antibunching of photons since the photons emitted are highly uncorrelated. However, a suitable phase matching condition similar to that of four-wave mixing leads to the photon antibunching even in the resonance fluorescence of a multi atomic system [66]. In addition to the usual photon-bunching and photon antibunching, the quantum statistical properties (QSP) are useful to derive more information about the detailed nature of the radiation field. For example, the nature of the photon number distribution could be obtained from the knowledge of QSP of the radiation field. In order to study the QSP, we calculate the second order correlation function for zero time delay [1,2]

(Δni )2 − 〈ni 〉 〈ni 〉2

(37)

where 〈ni 〉 = 〈ai† ai 〉 is the average number of photons present in the radiation field. The usual second order variance in photon number operator (n1) for the field mode a1 is (Δn1)2 = 〈n12 〉 − 〈n1〉2. Identical expressions for the second order correlation functions g22 (0) and g32 (0) follow for field modes a2 and a3 respectively. We write the numerator for the field mode ai as Di = (Δni )2 − 〈ni 〉. The denominator on the right-hand side of Eq. (37) is always positive and hence the QSP are determined solely by the parameter Di . For Di = 0 (i.e. gi2 (0) = 1), the corresponding photon number distribution (PND) is called Poissonian and hence the photons are unbunched. In most of the situations, for Di < 0 ( Di > 0), the corresponding PNDs follow the sub-Poissonian (super-Poissonian) photon statistics. It is normally seen that the photon antibunching comes along with the sub-Poissonian photon statistics. However, it does not imply that the photon antibunching (bunching) will show the sub-Poissonian (super-Poissonian) photon statistics [62]. Now, in terms of the initial coherent state we calculate the average photon number in various modes. These are

(

)( |α1|6 + |α1|4 ) + ( f1⁎ f2 α1⁎ α2 + f1⁎ f3 α1⁎ α3 + c . c ) + ( f1⁎ f4 + f1 f4⁎ ) |α1|4 + ( f1⁎ f6 |α1|2 α1⁎ α3 + c . c ) + ( f1⁎ f7 |α1|2 α1α3⁎ + c . c ) + ( f3⁎ f4 |α1|2 α1α3⁎ + c . c )

〈n1〉 = |f1 |2 |α1|2 + |f3 |2 |α3 |2 + |f4 |2 + f1⁎ f5 + f1 f5⁎

(

)( |α2 |6 + |α2 |4 ) +( + + c. c) + ( + g2 g4⁎ ) |α2 |4 + ( g2⁎ g6 |α2 |2 α2⁎ α3 + c . c ) + ( g2⁎ g7 |α2 |2 α2 α3⁎ + c . c ) + ( g3⁎ g4 |α2 |2 α2 α3⁎ + c . c )

〈n2 〉 = |g2 |2 |α2 |2 + |g3 |2 |α3 |2 + |g4 |2 + g2⁎ g5 + g2 g5⁎ g2⁎ g1α2⁎ α1

g2⁎ g3 α2⁎ α3

g2⁎ g4

〈n3 〉 = |h1|2 |α1|2 + |h2 |2 |α2 |2 + |h3 |2 |α3 |2 + ( h1⁎ h2 α1⁎ α2 + h1⁎ h3 α1⁎ α3 + h2⁎ h3 α2⁎ α3 + h3⁎ h4 |α1|2 α1α3⁎ + h3⁎ h5 |α2 |2 α2 α3⁎ + c . c )

(38)

The field with zero photons is termed as the vacuum field. The concept of vacuum field is purely a quantum electrodynamic idea of having no classical analogue. Now, we assume that the field mode a1 is in the vacuum state (i.e. |α1|2 = 0) initially. In the post interaction scenario, we obtain the average photon number 〈n1〉 = |f3 |2|α3 |2 . The result is quite surprising and is unexplainable

232

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

in the classical domain since we are getting photons in spite of the zero photons initially. Interestingly, this type of generation of photons in the field mode a1 requires nonzero coupling constant not the anharmonic constant. Hence, the mere coupling between the oscillators produces the photons in a vacuum field mode (a1). The result is quite new since the photons in field mode a1 is generated due to the photons in the field mode a3 through the mere linear coupling. The result substantiate the phenomena of quantum state transfer which is possible in a chain of harmonic oscillator [29–31]. On the other hand, the initial vacuum field in all the three modes remains in the vacuum in spite of the presence of coupling. These results corroborate the facts that the nonlinear interaction with fast rotating term (energy nonconserving terms) is responsible for the generation of photons. Now, in terms of the initial coherent state, we calculate the second order variances of the number operator. The corresponding variances for field mode a1 are given by

(Δn1)2 = |f1 |2

2

1

)

+ |f3 |2 |α1|2 + |f3 |2 |α3 |2

) ( ) ⁎ ⁎ 2 + 4( + ) |α1 + 2 ( |f4 | + f1 f5 + f1 f5 ) 6 4 ( 3|α1| + 2|α1| ) + 3|α1|2 ( f1⁎ f6 α1⁎ α3 + f1 f6⁎ α1α3⁎ ) + 3|α1|2 ( f1⁎ f7 α1α3⁎ + f1 f7⁎ α1⁎ α3 ) + 3|α1|2 ( f3⁎ f4 α1⁎ α3 + f3 f4⁎ α1α3⁎ ) } +

(

{ ( |f |

f1⁎ f2 α1⁎ α2 f1⁎ f4

+ f1 f2⁎ α1α2⁎ + f1⁎ f3 α1⁎ α3 + f1 f3⁎ α1α3⁎ |4

f1 f4⁎

(

+ 2|α1|2 f1⁎ 2 f3 f4 α1⁎ α3 + f 12 f3⁎ f4⁎ α1α3⁎ +2

(

f1⁎ 2 f 42

+

f 12 f4⁎ 2

+

|f1 |2 |f4 |2

)

) ( 2|α1|6 + |α1|4 )

(Δn3 )2 = |h3 |2 ( h1⁎ h3 α1⁎ α3 + h2⁎ h3 α2⁎ α3 + c . c )

(40)

Therefore, the numerator D1 for field mode a1 follows as

{ ( |f | 1

(

+ |f3 |2 − 1 |α1|2 + |f3 |2 |α3 |2 + f1⁎ f2 α1⁎ α2 + f1 f2⁎ α1α2⁎

)

) + 4 ( + ) |α1|4 + 2 ( |f4 + + )( 3|α1|6 + 2|α1|4 ) + 3|α1|2 ( f1⁎ f6 α1⁎ α3 + f1 f6⁎ α1α3⁎ ) + 3|α1|2 ( f1⁎ f7 α1α3⁎ + f1 f7⁎ α1⁎ α3 ) + 3|α1|2 ( f3⁎ f4 α1⁎ α3 + f3 f4⁎ α1α3⁎ ) } + 2|α1|2 +

(

)

2

f1⁎ f3 α1⁎ α3

|2

+

f1 f3⁎ α1α3⁎

f1⁎ f5

f1⁎ f4

f1 f4⁎

f1 f5⁎

( f1⁎2 f3 f4 α1⁎ α3 + f12 f3⁎ f4⁎ α1α3⁎ ) + 2 ( f1⁎2 f 42 + f12 f4⁎2 + |f1 |2|f4 |2) ( 2|α1|6 + |α1|4 ) − ( |f3 |2|α3 |2 + ( |f4 |2 + f1⁎ f5 + f1 f5⁎ ) ( |α1|6 + |α1|4 ) + ( f1⁎ f2 α1⁎ α2 + f1⁎ f3 α1⁎ α3 + c. c) + ( f1⁎ f4 + f1 f4⁎ ) |α1|4 ) − ( f1⁎ f6 |α1|2 α1⁎ α3 + c . c ) − ( f1⁎ f7 |α1|2 α1α3⁎ + c . c ) − ( f3⁎ f4 |α1|2 α1α3⁎ + c . c )

(41)

The numerator D3 corresponding to the field mode a3 follows as

D3 =

{ h h ( |h | ⁎ 1 3

−

3

|h1|2 |α1|2

2

}

− 1) α1⁎ α3 + h2⁎ h3 ( |h3 |2 − 1) α2⁎ α3 + c . c |h2 |2 |α2 |2

2

and the number of photons in the field mode a1 compared to those in Fig. 11, we obtain the antibunching of photons in the field mode a1 for small interaction time (kt ) till the bunching of photons for higher values of kt (Fig. 12).

(39)

The second order variances of the number operator corresponding to the field mode a2 can easily be obtained. The second order variance for the number operator corresponding to the field mode a3 assumes an extremely simple form and is given by

D1 = |f1 |2

mode a1 in spite of the fact there is no photons present. It became possible since the presence of photons in the field mode a1 is assured by the nonzero number of photons in the field mode a3 through the coupling. Therefore, the antibunching of photons of the vacuum field mode a1 is possible through the mere coupling with the field mode a3 and does not require nonlinear interaction, etc. It is exactly the situation where a quantum state transfer is responsible for the antibunching of photons in vacuum field mode a1. An identical situation will happen for the mode a2. On the other hand the vacuum field mode a3 has the signature of antibunching controlled by the presence of photons in field mode a1 and a2. The analytical expression (42) demands that the antibunching in field mode a3 is quite strong and can be improved by increasing the photon numbers in the field modes a1 and a2. The detailed behavior of antibunching of the field mode a1 as a function of interaction time is exhibited in Figs. 11 and 12. In Fig. 11, the monotonic decrease of the parameter D1 is exhibited with the increase of dimensionless interaction time kt. Therefore, the antibunching of photons in the field mode a1 becomes stronger with the increase of kt. The undesirable monotonic nature of D1 is attributed by the presence of the secular terms in the perturbative solutions. Now, the decrease of coupling constant by a factor of 1

|h3 |2 |α3 |2

− − − ⎡⎣ ( h1⁎ h2 α1⁎ α2 + h3⁎ h4 |α1|2 α1α3⁎ + h3⁎ h5 |α2 |2 α2 α3⁎ + c . c ) ⎤⎦

(42)

Now, we are in a position to give a special case for which the field mode a1 is in vacuum with no photons (i.e. |α1|2 = 0). In that case, the corresponding D1 = − |f3 |2|α3 |2 (1 − |f1 |2) is negative definite and hence the photon antibunching. The quantitative behavior of photon antibunching will be improved if the number of photons in field mode a3 is increased considerably. The result is quite interesting since we are getting the antibunching of photons in the field

5. Conclusion In the present investigation, we obtain the Heisenberg equation of motion for various field operators involving two quantum anharmonic oscillators coupled through a linear one. These differential equations are quite complicated and are coupled to each other. Moreover, the non-commuting nature of the field operators is definitely a nontrivial problem for getting the numerical solutions of the coupled differential equations. The small anharmonic constant along with the small coupling constant provides the basis of the approximate analytical solutions for the field operators involving the model Hamiltonian (1). In the present investigation, we do not use numerical solutions of the coupled differential equations involving the noncommuting operators. It is because of getting the numerical solutions for the noncommuting operators involving a nonlinear differential equation is really a nontrivial problem if not impossible. Moreover, we believe that the analytical solution (even approximate) is more useful in getting the physical insights of the problem compared to those of the numerical solution (if at all available) counterpart. It is because, the numerical artifact may kill the subtle effects in the physical model. The approximate analytical solutions of the filed operators are provided up to the second orders in anharmonic and coupling constants [23]. These solutions are used to investigate the squeezing, mixed mode squeezing, amplitude squared squeezing of the initial composite coherent state. It is found that the squeezing, mixed mode squeezing, and amplitude sugared squeezing are increased with the increase of dimensionless interaction time when the photon numbers in various modes remain constant. The squeezing in a particular quadrature collapses and revives many times as the interaction time increased. The identical nature are also exhibited for mixed mode and amplitude squared squeezing. The percentage of squeezing for a particular interaction time is increased with the increase of photon numbers in the corresponding mode. It is found that the squeezing, mixed mode squeezing and amplitude squared squeezing are monotonically increased with the increase of interaction time kt (Figs. 1–2 and 4–8). These monotonic increasing

N. Alam, S. Mandal / Optics Communications 359 (2016) 221–233

References

0.000

0.004

[1] [2] [3] [4]

0.006

[5] [6]

D1

0.002

[7] [8] [9] [10]

0.008 0.010 0.000

0.002

0.004

0.006

0.008

0.010

kt Fig. 11. Plot of D1 as a function of dimensionless interaction time kt for k = 1011 s−1, β = 0.5 × 1011 s−1, α1 = 1.5, α 2 = 1.5, α3 = 0.01, ω0 = 1014 Hz and ω = 1.1 × 1014 Hz .

[11] [12] [13] [14] [15] [16]

D1

[17] [18] [19]

1. 10

8

8. 10

9

6. 10

9

4. 10

9

2. 10

9

[20] [21] [22] [23] [24] [25] [26]

0 2. 10

233

[27]

9

0.000

0.002

0.004

0.006

0.008

0.010

kt Fig. 12. Plot of D1 as a function of dimensionless interaction time kt for k = β = 0.5 × 1011 s−1, α1 = 0.1, α 2 = 1.5, α3 = 0.01, ω0 = 1014 Hz and ω = 1.1 × 1014 Hz .

nature are attributed by the presence of secular terms in the solutions (4) of the field operators. The occurrence of the undesirable secular terms is certainly a great problem in the perturbation theory. However, the small values of anharmonic and the coupling constant ensure that the secular terms are not too harmful in the present investigation. By using the prescription of Luks et al. [47], we also investigated the principal and normal squeezing. The antibunching of photons in the field mode a1 and a3 is also investigated for various values of α1 as a function of dimensionless interaction constant kt. The vacuum field mode a1 exhibits the antibunching of photons. These nonclassical effects are attributed by the coupling constant not by the anharmonic constant. On the other hand, the vacuum field mode a3 exhibits antibunching of photons through the anharmonic constant not by the coupling constant. It is exactly the quantum state transfer which is responsible for the availability of photons in the vacuum field mode a1 and hence the antibunching. The behavior of field mode a2 is expected to be identical with those of the field mode a1.

Acknowledgment One of the authors (S.M.) thanks the University Grants Commission, New Delhi for financial support through a major research project (F.No. 42-852/2013(SR)). S.M. also thankful to the Council of Scientific and Industrial Research (CSIR), Government of India for financial support (03(1283)/13/EMR-II). We are thankful to Professor Dr. Patrik Öhberg for useful discussions.

[28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67]

R. Loudon, P.L. Knight, J. Mod. Opt. 34 (1987) 709. D.F. Walls, Nature 306 (1983) 141. D.F. Walls, Nature 280 (1979) 451. R. Loudon, The Quantum Theory of Light, 2nd ed., Oxford University Press, New York, 1983. M.T. Jaekel, S. Reynaud, Europhys. Lett. 13 (1990) 301. R.E. Slusher, L.W. Hollberg, B. Yurke, J.C. Mertz, J.F. Valley, Phys. Rev. Lett. 55 (1985) 2409. G. Breitenbach, S. Schiller, J. Mlynek, Nature 387 (1997) 471. H. Vahlbruch, et al., Phys. Rev. Lett. 97 (2006) 011101. J. Aasi, et al., Nat. Photon. 7 (2013) 613. A. Ourjoumtsev, A. Kubanek, M. Koch, C. Sames, P.W.H. Pinkse, G. Rempe, K. Murr, Nature 474 (2011) 623. Tobias Eberle, et al., Phys. Rev. Lett. 104 (2010) 251102. A. Furusawa, et al., Science 282 (1998) 706. H. Vahlbruch, et al., Phys. Rev. Lett. 95 (2005) 211102. N. Treps, et al., Science 301 (2003) 940. M. Hillery, Phys. Rev. A 36 (1987) 3796. G. Anetsberger, O. Arcizet, Q.P. Unterreithmeier, R. Rivière, A. Schliesser, E. M. Weig, J.P. Kotthaus, T.J. Kippenberg, Nat. Phys. 5 (2009) 909. H. Ritsch, P. Domokos, F. Brennecke, T. Eslinger, Rev. Mod. Phys. 85 (2013) 553. G.J. Milburn, M.J. Woolley, Contemp. Phys. 49 (2008) 413. I. Wilson-Rae, N. Nooshi, W. Zwerger, T.J. Kippenberg, Phys. Rev. Lett. 99 (2007) 093901. F. Marquardt, J.P. Chen, A.A. Clerk, S.M. Girvin, Phys. Rev. Lett. 99 (2007) 093902. Y.-H. Ma, E. Wu, J. Mod. Opt. 58 (2011) 839. C. Joshi, M. Jonson, E. Andersson, P. Öhberg, J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 245503. N. Alam, S. Mandal, P. Öhberg, J. Phys. B: At. Mol. Opt. Phys. 48 (2015) 045503. C. Joshi, J. Larson, M. Jonson, E. Andersson, P. Öhberg, Phys. Rev. A 85 (2012) 033805. C. Joshi, A. Hutter, F.E. Zimmer, M. Jonson, E. Andersson, P. Öhberg, Phys. Rev. A 82 (2010) 043846. F. Massel, S.U. Cho, J.-M. Pirkkalainen, P.J. Hakonen, T. Heikkilä, M.A. Sillanpää, http://dx.doi.org/10.1038/ncoms1993, 2012. J.-M. Pirkkalainen, S.U. Cho, J. Li, G.S. Paraoanu, P.J. Hakonen, M.A. Sillanpää, Nature 494 (2013) 211. S. Sivakumar, J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 095502. E.T. Jaynes, F.W. Cummings, Proc. IEEE 51 (1963) 89. D.E. Chang, C.A. Regal, S.B. Papp, D.J. Wilson, J. Ye, O. Painter, H.J. Kimble, P. Zoller, Proc. Natl. Acad. Sci. 107 (2010) 1005. D. Portes Jr., H. Rodrigues, S.B. Duarte, B. Baseia, Eur. Phys. J. D 67 (2013) 156. C. Chudzicki, F.W. Strauch, Phys. Rev. Lett. 105 (2010) 260501. C.M. Bender, L.M.A. Bettencourt, Phys. Rev. Lett. 77 (1996) 4114. M. Janowicz, Phys. Rep. 375 (2003) 327. R. Bellman, Methods of Nonlinear Analysis, vol. 1, Academic Press, New York, 1970, p. 198. A. Allevi, M. Lamperti, M. Bondani, J. Perina Jr., V. Michalek, O. Haderka, R. Machulka, Phys. Rev. A 88 (2013) 063807. A. Allevi, S. Olivares, M. Bondani, Int. J. Quant. Inf. 10 (2012) 1241003. R. Prakash, A.K. Yadav, Opt. Commun. 285 (2012) 2387. C.K. Hong, L. Mandel, Phys. Rev. Lett. 54 (1985) 323. M. Hillery, Opt. Commun. 62 (1987) 135. D. Yu, Phys. Rev. A 45 (1992) 2121. H. Fan, Z. Zhang, Quant. Opt. 6 (1994) 411. C.S. Sudheesh, S. Lakshmibala, V. Balakrishnan, J. Opt. B: Quant. Semiclass. Opt. 7 (2005) S728. X. Yang, X. Zhang, J. Mod. Opt. 36 (1989) 607. A. Kumar, P.S. Gupta, Quant. Semiclass. Opt. 7 (1995) 835. D.K. Giri, P.S. Gupta, J. Mod. Opt. 52 (2005) 1769. A. Luks, V. Perinova, J. Perina, Opt. Commun. 67 (1988) 149. R. Loudon, Opt. Commun. 70 (1989) 109. V. Perinova, J. Krepelka, J. Perina, Opt. Acta 33 (1986) 1263. R. Tanas, A. Miranowicz, S. Kielich, Phys. Rev. A 43 (1991) 4014. A. Miranowicz, M. Bartkowiak, X. Wang, Y.-Xi Liu, F. Nori, Phys. Rev. A 82 (2010) 013824. R. Hanbury-Brown, R.Q. Twiss, Nature 177 (1956) 27. D. Stoler, Phys. Rev. Lett. 33 (1974) 1397. A. Pathak, S. Mandal, Mod. Phys. Lett. B 17 (2003) 225. S. Mandal, Mod. Phys. Lett. B 16 (2002) 963. S. Mandal, Opt. Commun. 240 (2004) 363. S. Mandal, J. Perina, Phys. Lett. A 328 (2004) 144. H.J. Carmichael, D.F. Walls, J. Phys. B 9 (1976) L43. C.C. Gerry, J.B. Togeas, Opt. Commun. 69 (1989) 263. C.C. Gerry, S. Rodrigues, Phys. Rev. A 36 (1987) 5444. H.J. Kimble, M. Dagenais, L. Mandel, Phys. Rev. Lett. 39 (1977) 691. F. Diedrich, H. Walther, Phys. Rev. Lett. 58 (1987) 203. G. Rempe, R.J. Thompson, W.D. Lee, H.J. Kimble, Phys. Rev. Lett. 67 (1991) 1727. R. Short, L. Mandel, Phys. Rev. Lett. 51 (1983) 384. T. Basche, W. Moerner, M. Oritt, H. Talon, Phys. Rev. Lett. 69 (1992) 1516. P. Grangier, G. Roger, A. Aspect, A. Heidmann, S. Reynaud, Phys. Rev. Lett. 57 (1986) 687. S.L. Mielke, G.T. Foster, L.A. Orozco, Phys. Rev. Lett. 80 (1998) 3948.