Time evolution of the coherent state in a degenerate parametric amplifier

Accepted Manuscript Title: Time evolution of the coherent state in a degenerate parametric amplifier Author: Gang Ren Jian-ming Du Wen-hai Zhang Hai-jun Yu PII: DOI: Reference:

S0030-4026(16)30182-6 http://dx.doi.org/doi:10.1016/j.ijleo.2016.03.020 IJLEO 57414

To appear in: Received date: Accepted date:

28-1-2016 4-3-2016

Please cite this article as: G. Ren, J.-m. Du, W.-h. Zhang, H.-j. Yu, Time evolution of the coherent state in a degenerate parametric amplifier, Optik - International Journal for Light and Electron Optics (2016), http://dx.doi.org/10.1016/j.ijleo.2016.03.020 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Time evolution of the coherent state in a degenerate parametric amplifier

ip t

Gang Ren*, Jian-ming Du，Wen-hai Zhang, Hai-jun Yu

cr

Department of Physics, Huainan Normal University, Huainan 232001, People’s Republic of China

Corresponding author: [email protected]

us

*

In this paper, we study the time evolution of the coherent state in a degenerate parametric

an

amplifier by the technique of integration within an ordered product (IWOP) of operators. We

M

investigate some interesting and important physical properties of this state, such as the quantum statistical properties and squeezing effects, both analytically and numerically. Furthermore, the

d

fidelity between this state and the Schrödinger cat state is also given. It is interesting to find that

te

this quantum state has not only squeezing effect, but also rotating effect by appropriately choosing the related parameters.

Ac ce p

PACS:03.65.−w;02.30.Cj

Keywords: degenerate parametric amplifier ;normal ordering;The IWOP technique

1. INTRODUCTION

The fascinating coherent states have many important applications in the fields of quantum theory, especially in quantum optics and radiophysics[1-3].There are several different definitions of coherent states, such as Barut–Girardello coherent states[4], Perelomov coherent states[5,6] and intelligent coherent states[7]. Some generalized coherent states are labeled as nonclassical states in many literatures [8,9]. In the recent review, much information about the nonclassical states was given in quantum mechanics and mathematical physics [10-13]. 1

Page 1 of 28

There are large amounts of proposals for generating nonclassical states, such as photonadded thermal states [14], photon addition coherent states [15], photon subtraction squeezed coherent state states [16], etc. There is also much interest in the construction of coherent states

ip t

for time-dependent systems [17]. The time evolution of the coherent state (TECS) may be generated from the time-dependent harmonic oscillators [18-20]. A possible realization

cr

corresponds to the case of dynamic cavity with a resonantly oscillating boundary. This

us

realization can be called the dynamical Casimir effect), which is the creation of quanta from vacuum due to the motion of macroscopic neutral boundaries [21-23].

an

Many nonlinear optical systems have also been used to generate squeezed states. These include the degenerate four-wave mixing[24-26], resonance fluorescence[27-29], free-electron

M

lasers[30,31], Jaynes-Cumming model[32,33] and the degenerate parametric amplifier

d

(DPA)[34,35].These are the non-classical states of the quantum optics in which certain

te

observables exhibit fluctuations less than in the coherent state. These states are important because they can achieve lower quantum noise than the fluctuations of the coherent states.

Ac ce p

Thus an interesting and challenging problem naturally arises: although DPA has been employed extensively in nonlinear optics, however, from the point of view of quantum optics, how does squeezing mechanism happens in the DPA? To our knowledge, the exact solution of this problem without any approximation in one space dimension has not been reported in literature before. We approach this specific problem in a way decomposing the time evolution operator via the technique of integration within an ordered product of operators. The rest of the paper is organized as follows. In Sec.2, we shall derive explicit expression of the time evolution of the coherent state (TECS) in the DPA. In Sec.3, quantum statistics properties of the TECS, such as Q-function, distribution of the quadrature, Mandel’s Q-

2

Page 2 of 28

parameter, second-order correlation function and photon-number distribution, are studied analytically and numerically. Some interesting results which are caused by the DPA are given. In Sec.4, we consider squeezing properties of the TECS in the DPA by its normal squeezing and

ip t

amplitude squared squeezing. The degree of squeezing is also studied by the negative value of Sopt. Furthermore, we investigate the fidelity between the TECS in the DPA and Schrödinger’s

us

cr

cat state. Finally, the main conclusions are summarized in the last section.

2. Explicit Expression of the TECS in the DPA

an

In a DPA, a pumping field of frequency 2 interacts with a nonlinear medium and gives rise

approximation, by the Hamiltonian [34]

M

to a field frequency  . This process is described at resonance and with rotating-wave

(1)

te

d

i H  a † a    a 2 ei  a †2 e i   2

Ac ce p

where a  a †  are the annihilation (creation) operators for the single field,  is an appropriate coupling constant and  are the real amplitude and the phase of the pump field. In this section, we shall give the explicit expression of the evolution of an initial coherent state  in the DPA based on the technique of IWOP of quantum operators. Let us first present the formula for calculating the evolution of the coherent at any time point in the Schrödinger representation,

t

DPA

 exp  iHt   .

(2)

To derive the explicit expression of Eq. (2), we use the concise operator identity [36,37]

3

Page 3 of 28



ga† 2

f

exp  fa † a  ga †2  ka 2   e 2 e D coth D f e

 a a   ln †

1 2

DsechD D  f tanh D

ka2

e D coth D f 

(3)

D

ip t

where

f 2  4kg 

us

cr

it follows

(4)

(5)

an

 it 1   e iHt  e 2 exp  H1a †2  exp  a † a   H 2  exp  H 3a 2   2     it  H 2  †2  H2 † 2    exp   exp  H1a   exp  e  1 a a   exp  H 3 a  ,  2 

 e  i t  2  D1 coth D1  it 

H 2  ln

d

H1 

M

where

D2 sechD1  D1  it tanh D1

(6)

te

 ei t  D1  t 4 2  2  1 H3  2  D1 coth D1  it 

Ac ce p

and in the last step we have used

e a a  exp  e  1 a † a    †

(7)

Using Eq.(5), Eq.(2) may be written as

t

DPA

 it  H 2   exp   H 3 2  H1a †2   e H 2  1 a †   .  2 

(8)

It is obvious to see that the initial state in the DPA is the coherent state  at time t  0 as expected.

4

Page 4 of 28

3. QUANTUM STATISTICS PROPERTIES

ip t

3A. Q-function Some measures concerning the quantum degree of squeeze have been proposed [38-40] and here

cr

we consider the measure based on the Q-function because it is always positive definite at any

us

point of the phase space for any quantum state, being a truly probability distribution obtained by projection on the coherent states. The Q-function is defined as 1     

an

Q 

(9)





d

2 1 2 2 exp      Re H 2 exp  H 3 2  H1 2  e H 2     

(10)

te

Q 

M

Using Eqs.(8) and(9) and after some simple calculations, we have

Ac ce p

Now we examine the behavior of the Q-function for the present state. Our results presented in Fig.1 show the Q-function against the parameter  ,  and t .From Figs.1(a) and 1(b), we can see the shapes of the Q-function is rotated in phase space when the phase of the pump field  increases. According Figs.1 (a), 1(c) and 1(d), it shows that the squeezing is enhanced as  decreases and t increases.

5

Page 5 of 28

(b)

(d)

te

d

M

an

(c)

us

cr

ip t

(a)

Ac ce p

Fig.1. The Q-function for the TECS in the DPA against the phase coupling constant



and the phase of the pump field

  4  (b)   08 t  06

and



with

q and p for different values of t ,

  12 

  34  (c)   08 t  12

and

3 2

i : (a)   08 t  06 and

  4  (d)   05 t  06

and

  4 

3B. The distribution of the quadrature The coherent state in the coordinate representation is

 x  

 14

  1 2  exp      x 2   2   2 x     2 

6

(11)

Page 6 of 28

Thus the distribution of the quadrature x for the TECS in the DPA is 2 DPA



(12)

Using Eqs.(8) and (11) and after some algebra, we have

 14



 14



1 H 2  x2    2



exp 

2

 H  3

 2  



 1 1 exp  H 2  x 2    2 1  2 H1

2

 d  exp 2





    

 1 2    2 x  e H 2     2  H1 2   2 



 

1 2 H 2 2   1  H2 2 e     2 xe   2 H1 x  1  2 H   2   1  

 H 3 2  exp  

(13)

us



cr

2   1  d  exp H   x  exp  H 3 2  H1a †2   e H 2  1 a†     2 DPA 2    

an

 x  t

ip t

Px   x  t

M

where in the last step we have used the integral formula





Re



convergent

condition

is

Re   f  g   0 Re



 2  4 fg   f g

  0

or

  

(14)

Re   f  g   0

Ac ce p

whose

te

d

    2 g   2 f d 2z 1 2   z   z  fz 2  gz2   exp exp z     2  4 fg  2  4 fg 

 2  4 fg   f g

  0

From the numerical results depicted in Fig.2, one can find that the width of the distribution becomes narrower as  increases. Thus, we can assume that the field in DPA does show squeezing properties of the quadrature.

7

Page 7 of 28

ip t cr us an

Fig.2. The q-quadrature distributions of TECS in the DPA as a function of the with 

 12 

i t  15   4 .

te

3C. Mandel’s Q-parameter

3 2

d

M

  02 04 06 08

x for

The Mandel’s Q-parameter is the most popular way to see the non-classicality of a quantum

Ac ce p

state[41], which is defined as

QM 

a †2 a 2 †

aa

 a† a 

(15)

Using the commutation relation  a a †   1 we have 2

QM 

a 2 a †2  2 aa†  aa†  1 aa †  1

8



(16)

Page 8 of 28

QM is a natural measure of the departure of the variance of the photon number n from the variance of a Poisson distribution. The quantum state such as the coherent state has a Poisson distribution for which Q  0 . QM becomes positive for any classical state which has photon

ip t

number variance larger than the average photon number. Specifically, QM becomes negative if

cr

the photon number variance is less than the average photon number and the corresponding state has been classified as the nonclassical quantum state [42].

us

In order to derive the Mandel’s Q-parameter for the TECS in the DPA, we first calculate the

an

ensemble average a n a † m appearing in Eq. (10). Using the completeness relation of the coherent

  12     

n  

m  min n  m 

g   f  i i A   A 

 k 0

k

    2 g     2 f    nm H nk  H mk      2i gA   2i fA  fg  k n  k   m  k     

 W  e 

(17)

te

A

d 2 z n m 2   z z exp   z   z   z   fz 2  gz 2    

d

I 

M

state and the integral formula [43]

Ac ce p

where

A    4 fg  2

  g 2  f  2  W A

(18)

we obtain

9

Page 9 of 28

An m  a n a † m





   1 d 2z   exp  H 2  H 3 2  H1 a 2  e H 2  1 a   a n z  z    2 

 k 0

m

 2   i K1   i K1  exp  Re  H 2  2 H 3 2          n

k

 1  nm H n  k  K 2  H m  k  K 2  e K3     H1  k m  k   n  k  

us



min m  n 

 12   

cr

 1  4  H1 2 

ip t

1  a † m exp  H 2  H 3 2  H1a†2   e H 2  1 a†   2  2 (19)  d z n m 2 2    exp  Re  H 2  2 H 3 2      z z exp   z  e H 2   z  e H 2  z   H1 z 2  H1 z 2      

H1  1  4  H 1 2 

2i H1 1  4  H1 2  1

  2 

te

K3 

2e H 2   H 1  e H 2 

d

K2 

M

K1 

an

where

Ac ce p

1  4 H1



2

(20) 





e H 2   Re  H1e2 H 2  2    



Thus Mandel’s Q-parameter can be obtained by substituting Eq.(19) into (16) as

QM 

A22  2 A11  A121  1 A11  1



(21)

For some different values of the coupling constant  , we plot the Q-parameter for the TECS in the DPA in Fig.2. As we can see, the Q-parameter starts with a decrease to QM  0 and then increases to QM  0 . The coupling constant  plays an important role in this evolution.

10

Page 10 of 28

ip t cr 3 2

us

t with   12 

i  4 for   02 04 06 08

an

Fig. 3. The Q-parameter as a function of

M

3D. The second-order correlation function



a †2 a 2

te

 2

d

The second-order correlation function is defined by

a†a

2







aa †  1

2



(22)

Ac ce p

g

a 2 a †2  4 aa†  2

It reflects the bunching or anti-bunching effects of the photons in the quantum optical state. The relation g  2  1 indicates that the state is non-classical, wherein photons of optical field present anti-bunching effects, while g  2  1 suggests that the optical field is classical (or random) and the relevant photons are bunching. Using Eqs.(19) and(22), we have

g   2

A22  4 A11  2

A

11  1

11

2



(23)

Page 11 of 28

The second-order correlation function g(2) is plotted against time parameter for different values

an

us

cr

ip t

of  in Fig.4. This figure indicates that the state is non-classical till t  1.8 .

M

Fig. 4. The second-order correlation function g(2) as a function of

t with   12 

3 2

i  4 for

te

d

  02 04 06 08

Ac ce p

3F. Photon-number distribution

The photon-number distribution (PND) is an important part of the modern description of quantum state[44]. PND may be measured by a photon detector based on the photoelectric effect. The PND in single modes is given by





P  n   Tr  n  n 

(24)

where n is the Fock state. Noticing the unnormalized coherent state z   exp  za †  0 and a † m n 

 m  n  n

m  n  one can

find 12

Page 12 of 28

1 dn z n n dz

n 



(25)

z 0

polynomials

an

the probability of finding n photons is derived as P  n    n  t

2 DPA

M

1 dn  it  H 2    z exp   H 3 2  H1a †2   e H 2  1 a†   n dz n 2   

(26)

us

cr

n n  A    2   i B  H exp   At Bt   n  n     t  2i B  t 0

ip t

Substituting Eqs.(8) and (25) into (24) and using the generating function of the Hermite

2

  dn 1 1 2 it  H 2 exp      H 3 2  n exp  e H 2  z   H1 z 2    dz n 2 2   z 0

te

d

   eH2 n  1 2 H 1  exp    exp  2  H 3 2   i H1  H n    2i H  2  n  2    1 

(27)

z  0 2

2

   

Ac ce p

Thus, the PND of TECS in the DPA is the Hermite polynomial. Using Eqs. (6) and (27), the PND of the TECS a function of n for different values of  ,  and t with   12 

3 2

i   1.5 is plotted in Fig.5. We can see the position and values of peak depend

on the coupling constant  [see Figs.5 (a) and 5(b)], the phase of the pump field [see Figs.5 (a) and 5(c)] and the evolution time t [see Figs.5 (a) and 5(d)].

13

Page 13 of 28

(a)

cr

ip t

(b)

(d)

d

M

an

us

(c)

n with  

te

Ac ce p

Fig. 5. PND of the TECS as a function of

(b)

  0.9, t  0.5, 

 4

;(c)

1 3   i and (a)   0.8, t  0.5,   ; 2 2 4

  0.8, t  0.5, 

3  ;(d)   0.8, t  0.9,  . 8 4

3G. Wigner distribution function As another indicator of non-classicality, one may refer to the negativity of Wigner distribution function [45]. For a single-mode density operator  , its Wigner function can be expressed in the coherent state representation as

W       

2 z   z  2 2 2 d 2z e   z  z e   

14

(28)

Page 14 of 28

Then we can have

  

e2   

2

d

2

z



 zexp  H1a †2  H 3 2   e H 2  1 a †    

 exp  H1 a 2  H 3 2   e H 2  1 a †  z e



2 z   z



ip t

W     

1 1 1 2 2  exp  2   H 2  H 2  H 3 2  H 3 2    2 2    2    d z 2    exp   z   e H 2    2   z   2  e H 2   z   H1 z 2  H1 z 2       1 2 2  exp  2   Re  H 2  2 H 3 2       2  1  4 H1 1 4 H

  2  1 

  2  e    2 Re e   2     

2

H 2



an

1

H2

   

2

H

     1      

.

M

 exp

     

(29)

us

cr



Using Eq. (29), the Wigner functions are depicted for several different values of  ,  and t in

d

Fig.6. It is easy to see the squeezing is enhanced as  decreases [see Figs.6 (a) and 6(c)] and t

te

increases [see Figs.6 (a) and 6(d)]. Furthermore, the shapes of the Wigner function rotates in

Ac ce p

phase space when the phase of the pump field  increases [see Fig.6 (a) and 6(b)], which indicates that the DPA not only produces the squeezing effect but also the rotating effect in phase space.

(b)

(a)

15

Page 15 of 28

(d)

cr

ip t

(c)

  0.5, t  0.6, 

us

1 3   i and (a)   0.5, t  0.6,  ; 2 2 3

4   ;( c)   0.9, t  0.6,  ;( d)   0.8, t  1.5,   . 3 3 3

M

(b)



an

Fig. 6. Wigner function in phase space with 

te

4A.Normal Squeezing

d

4. Squeezing properties of the TECS in the DPA

Ac ce p

The investigation of normal squeezing is based the definition of the two quantum quadrature operators [46]

X1 

1 1 † † a  a  X2  a  a     2 2i 

(30)

which corresponds to the real and imaginary parts, respectively, of the mode amplitude. For X 1 and X 2 satisfy the following commutation relation

 

i X 1  X 2    2

(31)

the uncertainty relation of the two quantum quadrature operators is given by 16

Page 16 of 28

 X 1   X 2  2

2



1  16

(32)

 X i  2 

ip t

where the quadrature variances is defined 2

The quantum state is said to be squeezed if X i2  1  4  i  1 or 2  

(33)

cr

X i2  X i  i  1 2

us

Using Eq.(26), the squeezed conditions of a quantum state can also be rewritten as V  X 1   a 2  a†2  a  a†  2 a a†  2 a† a  0 2

M

an

2

V  X 2    a 2  a †2  a  a†  2 a a†  2 a† a  0 2

te

Using Eq.(19), we have

(34)

d

2

V  X 1   A20  A0 2  A120  A021  2 A10 A01  2  A11  1 

(35)

Ac ce p

V  X 2    A20  A02  A120  A021  2 A10 A01  2  A11  1 

According to Eq.(35), we present the quadrature variance V  X 1  and V  X 2  of the TECS against the time parameter t for different values of   02 04 06 0.8 with   12 

2 2

i in

Figs.7(a) and 7(b)  It is visible that the squeezing in the X 1 is enhanced with the decreasing of the parameter  in the initial time, with no squeezing in the X 2 .

17

Page 17 of 28

us

V  x2  as a function of t with   02 04 06 0.8 for   12 

2 2

2 2

i

i.

an

(b) Quadrature variance

V  x1  as a function of t with   02 04 06 0.8 for   12 

cr

Fig.7. (a) Quadrature variance

(b)

ip t

(a)

4B. Amplitude Squared Squeezing

M

To extend our discussion on squeezing, let us take another example of a non-classical effect, i.e. the higher order squeezing, namely the amplitude squared squeezing (ASS) [47]. This type of

d

squeezing arises in a natural way in second-harmonic generation and in a number of nonlinear

te

optical processes. In this subsection, we discuss the amplitude-squared squeezing (ASS) of the

Ac ce p

TECS. The ASS arises in a natural way in second-harmonic generation. In a similar method as Eq. (30), one can define operators

Y1 

1 2 1 2 †2  †2   a  a   Y2  a  a      2 2i

(36)

which represent the real and imaginary parts of the square of the amplitude. Heisenberg uncertainty relation of these two conjugate operators in Eq. (36) is given by

 Y1   Y2  2

2



2 1 Y Y    1 2  4

(37)

It follows that the quantum state is amplitude squared squeezing if

18

Page 18 of 28

 Y1 

2



1  1 2   Y1  Y2   Y1  Y2  or  Y2  2 2

(38)

With the help of Eqs. (36) and (38), one can introduce the following squeezing conditions

cr

2 2  1  4 1 a  a †4  a †2 a 2  a 2 a†2  a 2  a†2  2 a 2 a†2   a† a   0   4 2

(39)

us

I1

ip t

corresponding to Y1 and Y2  respectively

and

an

2 2  1  1  a 4  a †4  a †2 a 2  a 2 a †2  a 2  a†2  2 a 2 a †2   a† a   0   4 2

(40)

M

I2 

Using Eqs.(19) ,(39) and (40), the I1 and I 2 are given by

d

1 1 2 2    A4 0  A0  4  2 A2 2  4 A11  A2 0  A0 2   A11    4 2 1 1 I 2    A40  A04  2 A22  4 A11  A220  A022   A11   4 2

(41)

Ac ce p

te

I1 

Employing Eqs. (19), (20) and (41), the ASS of the TECS are depicted as a function of time parameter t for different values of   02 04 06 08 with   12 

3 2

i in Fig.8. It is

clearly to that the quadrature variances of I1 and I 2 have a complicated behavior depending on the time parameter t and the coupling constant   Figs.8(a) and 8(b) show that ASS of the TECS in the Y1 and Y2 occurred for the TECS in the initial time. This tendency is enhanced with the deceasing the coupling constant  

19

Page 19 of 28

(b)

cr

4C. Degree of Squeezing effects

an

Y2 as a function of t with   02 04 06 08 for  

M

(b) The ASS in

Y1 as a function of t with   02 04 06 08 for  

2

2 min

 1 with respect to  or by the

 1 , where X  is an appropriate quadrature defined as

te

 X  

 X  

1 3  i,    2 2 4

d

The squeezing properties can be characterized by normal ordering form

1 3  i,   ;  2 2 4

us

Fig.8. (a) The ASS in

ip t

(a)

min

Ac ce p

X   ae  i  a † ei [48, 49]. By expanding

 X  

2 min

 its value can be obtained over the whole

angle   which is given by

Sopt  2 a †2  a †

2

2

 2 a † a  2 a† 

(42)

The negative value of Sopt in the range  1 0  shows the squeezing properties of the quantum state. Sopt   A0 2  A021  2  A11  1  2 A01 . 2

20

(43)

Page 20 of 28

The degree of squeezing of the TECS in the DPA is plotted against t for different values of  in Fig.9. It shows that the degree of squeezing increases with t , and quickly approaches to the

an

us

cr

ip t

maximum degree of squeezing −1 with the decreasing of the coupling constant  .

1 1   i,   ,   0.5. 2 2 8

d



M

Fig.9. Degree of squeezing of the TECS in the DPA as the function of the parameter t with

Ac ce p

CAT STATE

te

6. THE FIDELITY BETWEEN TECS IN THE DPA AND SCHRÖDINGER’S

The fidelity is an important quantity in describing the transmission of quantum information through quantum channels. The fidelity can be able to quantify the non-Gaussian character of a quantum state [50].

As a kind of non-Gaussian state, Schrödinger’s cat states attracted much attention due to their strong nonclassical properties. One of these states is defined as [51]:



S

 N m      

21

(44)

Page 21 of 28

   

   

where N m  2 1  e

2 

2

   

 12

is a normalized constant.

To calculate the fidelity between the TECS in DPA and Schrödinger’s cat state, we evaluate S

 t



2 DPA



 N m    exp  H1a†2   e H 2  1 a†   2



(45)



1  exp 2  

2



2

 . 

cr



 N m2 exp H 2  H 3 2  H1 2   e H 2  1 

2

ip t

F

us

We plotted the fidelity in Eq. (45) as a function of  and t for different values of  in Figs.

an

10(a) and 10(b), respectively. The fidelity has a periodical variation as a function of the phase of the pump field  [see Fig. 10(a)]. The fidelity decreases with the increasing of the coupling

te

d

M

constant  , which can be seen from Fig. 10(b).

Ac ce p

(a)

(b)

1 3  i and t  0.5 ; 2 2 1 3 2 (b) Fidelity as a function of t for different values of  with    i and   . 2 2 3

Fig.10. (a) Fidelity as a function of



for different values of



with



7. CONCLUSIONS

In summary, we have shown how the effect of the DPA on the coherent state by the technique of IWOP of operators. Significantly, the degree of squeezing not only depends on the evolution time but also on the parameters of the DPA, such as coupling constant  and the phase of the pump field.

22

Page 22 of 28

From its quantum distribution function and squeezing properties, it is interesting to find that the effect of the DPA not only produces the squeezing effect but also the rotating effect in phase space. These methods will be developed and put on a firm theoretical foundation in the quantum

ip t

optics.

cr

ACKNOWLEDGMENT

us

This work is supported by the Natural Science Foundation of the Anhui Higher Education

an

Institutions of China (Grant No. KJ2015A268 and KJ2014A236).

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M

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te

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ip t

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cr

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te

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ip t

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ip t

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