Comparison of non classical effects: quantum phase fluctuation, antibunching and minimum total noise in various non linear optical processes

Priyanka; Gill Savita

NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2020, 11 (2), P. 161-170

Comparison of non classical effects: quantum phase fluctuation, antibunching and minimum total noise in various non linear optical processes

Priyanka, Savita Gill*

Department of Applied Science, University Institute of Engineering and Technology,

Kurukshetra 136119, India * [email protected]

We study various non classical effects of light like reduction of quantum phase fluctuation, antibunching and minimum total noise present in various nonlinear optical processes. Further, we have shown that depth of non-classicality can be directly measured using all these parameters. We have done the comparative study to correlate these non-classical effects in seven wave mixing, eight wave mixing and third harmonic generation. Keywords: phase fluctuation, photon number, antibunching, minimum total noise. Received: 26 January 2020 Revised: 20 February 2020

1. Introduction

For being a non-classical quantum state, negativities of Glauber-Sudarshan P-function may appear. However, this function may be strongly singular [1,2]. Generally we use standard deviation as the most natural measure to study quantum fluctuation [3]. Non classical effects of radiation field like reduction of quantum fluctuation, antibunching and minimum total noise were recently attracted a great interest in quantum optics. Quantum phase can be considered as a main feature to differentiate between classical and quantum physics. Antibunching and squeezing has no classical correlation and are called non classical states. In Sub-Poissonian (antibunching) photon statistics the variance of photon number is less than the mean value. The Hermitian quantum phase operators have some uncertainty [4-6] which leads to a lot of adverse formalism [7-9] of phase problem. Out of all these formalisms, Barnett-Pegg(BP) [8] and Susskind-Glogower(SG) [9] has contributed a considerable role in phase fluctuation and have been used by various researchers [10-18]. Study of phase fluctuation of coherent light coupled to a nonlinear medium and in intermediate state have already been studied [19,20]. The development of nanotechnology and nanoscience has provided new opportunities for nonlinear optics. In the last few decades, numerous studies on the nonlinear optical properties of novel materials have been performed because of the potential of these materials in optical device applications [21-25]. Strong optical nonlinearities in nanoparticles are observed due to quantum confinement, such as nonlinear optical absorption and second and third order optical nonlinearities; these can be studied for making optical modulators, optical limiters and laser second and third harmonic generators [23,24]. To study the optical nonlinearity of nanoparticles, there is a significant body of research concerning the measurements of third order nonlinear susceptibility x3 which can be used as a source for the generation of third harmonic generation process [26]. With the role of quantum phase fluctuation in quantum cryptography [27], super conductivity [28,29] and with the success in experimental study of phase fluctuation [30], there has been a significant increase in importance of study of non-classical parameters.

In the present work, Carruther and Nieto phase parameters [16] are used to study phase fluctuation parameters in nonlinear processes like seven and eight wave interaction process and in third harmonic generation using Barnett-Pegg (BP) formalism [7]. We have reported in the paper that reduction of phase parameters not only directly implies antibunching but also increases with an increase in antibunching. We have also found minimum total noise in terms of number operators and observed that the greater the sub-Poissonian (antibunched) state, the greater will be the minimum total noise in the system.

2. Measurement of quantum phase fluctuation parameters

Barnett and Pegg [7] defined the exponential of phase operator E and its Hermitian conjugate E^ as:

DOI 10.17586/2220-8054-2020-11-2-161-170

where Nis the mean photon number in the coherent state. Cosine and sine operators are defined as [7]:

C =2(E + Et), S = -2(E - Et),

and satisfy the following relations: and

and, we get

(C)2 + (S)2 = 1,

(AC)2 (AS)2 >

16 (N + 2 )'

(2)

(3)

(4)

(5)

To measure quantum phase fluctuation, Carruthers and Nieto [16] had introduced U, S and Q parameters in the following way:

U l«|2) = (AN)

2 (AS)2 + (AC)2 (S)2 + (C)2

and

S(0,t, |a|2) = (AN)2 (AS)2

nia + I ^^ |a| )

(C )2

(6)

(7)

(8)

where 6 is the phase of input Poissonian state, t is interaction time and |a|2 is mean photon number in coherent state.

2.1. Seven wave interaction process

In seven wave mixing process, the interaction takes place in such a way that the absorption of two photons of frequency w each and the emission of two photons of frequency w2 each and other two photons of frequency w3. The Hamiltonian represents total energy for the system and is given as (taking h = 1):

H = w1a+a + w2b+b + w3c+c + g (a2b+2c+3 + a+2b2c3) , (9)

where g is a coupling constant, a (a+), b(b+), c(c+) are the annihilation(creation) operators respectively. A = a exp iw1t, B = b exp iw2t, C = c exp iw3t are slowly varying operators at frequencies w1, w2 and w3.

To study quantum phase fluctuation, a coherent state |a) is used as pump for mode A and before the interaction process there was no photon in signal mode B and stokes mode C i.e.,

|V> = |a>|0>|0>-

The Heisenberg equation of motion for fundamental mode A is given as (ft = 1):

dA dA ,[TT A.

dA = Tfc + * H'A]'

(10)

(11)

in this process, we assume the interaction time t to be very small. Using short time approximation technique, expand A (t) by using Taylor's series expansion and retain the terms up to second order in g2t2:

,, N t dA(0) t2 d2A(0) A(t) = A(0) + ïï 2! ^,

and we get:

A(t) = a - 2igtAtB2C3 + g2t2 (2AN|N§ - 9AfA2N|N,2 - 18AU2n%nc

AfA2NB NC - 72AtA2NB NC

-2At A2NC - 18AtA2NC - 36Ata2nc - 12AtA^, (12)

-6AtA2N2 - 4AtA2NBNC - 36AtA2NBNC - 72AtA2NBNC - 24AtA2NB

where NB = btb and NC = CtC.

2

From equation (12), the mean number of photons i.e. N(t) = x A(t) can be expressed as:

N(t) = AA - 2igt (At2B2C — A2Bt2Ct) + 4g2t2 (2AUn£Nc + N|Nc)

—2g2t2 (At2A2NB + 4At2A2NBNc + 4At2A2NB + 2At2A2Nc + 12At2A2), (13)

the expectation value of N(t) using condition (10) is:

(N> = |a|2 - 24g212 |a|4 . The squaring of expectation value of number operator is given as:

(N>2 = |a|4 - 48g2t2 |a|6 + 576g4t4 |a|8 , as the interaction time is small, we are taking terms up to g2t2 and above equation is written as:

(N>2 = |a|4 - 48g2t2 |a|6 , (N2) = |a|4 + |a|2 - g2t2 (48 |a|6 + 96 |a|4) . Using equation (15) and (16), we get:

(AN)2 = (N2) - (N>2, (AN)2 = |a|2 - g2t2 (96 |a|4) . The condition of sub-Poissonian photon statistics is given as:

d = (AN)2 - (N> < 0.

Using (14) and (17), we get:

d = -72g2t2 |a|4 .

(14)

(15)

(16)

(17)

(18)

(19)

We obtained a negative value, which shows that the photon statistics is sub-Poissonian or antibunched light. By substituting (13) in (2), we obtain:

C =1 (n +1 2 V +2

A + At — 2«gtAtB2C3 + 2«gtABt2Ct3 + g2t2 ( 2AN2 N3 — 9At A2N2 N2

— 18AtA2N2 Nc — 6At A2N2 — 4AtA2NB Ng — 36AtA2 NB Ng — 72At2A2NB Nc —24At2ANB — 2AtA2Ng — 18AtA2 Ng — 36AtA2NC — 12AtA2 + 2AtN| Ng —9At2AN| Ng — 18At2AN| NC — 6At2AN| — 4At2ANB Ng — 36At2ANB Ng

-72At2A2NRNc — 24AtA2NB - 2At2AN3 - 18At2AN2 - 36At2ANg - 12At2A

(20)

and

is = — - (n + 1 2 V +2

A - At - 2igtAtB2C3 - 2igtABt2Ct3 + g2t2 (2AN|Ng - 9AfA2N|Ng

+72At2A2NBNC + 24AtA2NB + 2At2AN3 + 18At2AN2 + 36At2ANC + 12At2 A

The expectation value of C and S operators of equation (20) and (21) by applying condition (10) are:

<c >=2

m = — 2

N + 1 ) " {a + — g2t2 (12 la|2 a +12 |a|2 a*) }

Then square of expectation value of C and S are:

(C)2 = 4 (n + 1 ^ "2 ■ ~*2

N + I a — a* — g2t2(12 |a|2 a — 12 |a|2 a*)}

C and S are:

a2 + a*2 + 2 |a|2 — 24g2t2 (|a|2 a2 + |a|2 a*2 + 2 |a|4)

(21)

(22)

(23)

(24)

(S)2=- 4 (n+2

a2 + a*2 - 2 | a |2 - 24g2t2 ( | a |2 a2 + | a |2 a*2 - 2 | a |4)

(25)

Similarly:

<c2>=J (n+2

+ a*2 + 2 |a|2 + 1 - 12g2t2 (a2 + a*2 + 2 |a|2 a2

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+2 |a|2 a*2 + 4 |a|4 + 4 |a|2 )

(26)

(S 2

- 4 (n+2

i

a2 + a*2 - 2 |a|2 - 1 - 12g2t2 (a2 + a*2 +2 |a|2 a2

+2 |a|2 a*2 - 4 |a|4 - 4 |a|2 )

Using equations (24)-(27), second order variances is expressed as:

and

(AC)2 = (C2) - (C)2 = 4 (N + £) 1 - 12g2t2 (a2 + a*2 +4 |a|2) (S2) - (S)2 = -4 (N +2) -1 [ - 1 - 12g2t2 (a2 + a*2 - 4 |a|2)

(AS)2

Now equations (6)-(8) can be expressed as:

U (M,|a|2) = 1 ( , , | | ) 2 1 1 - 24g2t2 |a|2

S(0,t, |a|2) = 4 (|a|2 - 24g2t2 |a|4 + 2

-1

|a|2 + 24 |a|4 g2t2(cos20 - 6)

and

2 1 + 24 |a|2 g2t2(cos20 - 6)

Q(0,t, |a|2) =

. (27)

(28)

(29)

(30)

(31)

(32)

2(cos 26 + 1)(1 - 24 |a|2 g2t2)' From equation (19) and (30), it is clear that the reduction of phase parameter U increases withan increase in the number of photons and is associated with antibunching.

2.2. Eight wave interaction process

In an eight wave mixing process, the interaction takes place in such a way that the absorption of two photons of frequency w1 each and the emission of four photons of frequency w2 each and another two photons of frequency w3. The Hamiltonian for the given process is:

H = ^a+a + w2b+b + w3c+c + g (a2b+4c+2 + a+2b4c2) , (33)

where g is a coupling constant, a (a+), b (b+), c (c+) are the annihilation(creation) operators, respectively. A = a exp iw1t, B = b exp iw2t, C = c exp iw3t are slowly varying operators at frequencies w1, w2 and w3. Now, using the short time approximation technique, Taylor's series expansion of A (t) can be expressed as:

A(t) = A - 2«gtAtB4C2 + g2t2(2AN|N£ - 16AtA2N|n£ - 72AtA2N|NC

-192AtA2NB - 48AtA2, (34)

where NB = BtB and NC = CtC.

From equation (33), mean number of photons using condition (10) can be written as:

N(t) = AtA - 96g2t2(At2A2).

The expectation value of N(t) is:

(N) = |a|2 - 96g2t2 |a|4 .

(35)

(36)

1

Using equation (35), we get:

(AN)2 = |a|2 - 192g2t2 (2 H4) . (37)

The condition of sub-Poissonian photon statistics is given as:

d = (AN)2 - (N) < 0. (38)

Using equations (36) and (37), we get:

d = —288g2t2 |a|4 , (39)

as d is negative, and thus, we achieve an antibunched state.

Now, by substituting equation (34) in equation (2), we obtain:

C= 1 N+i

_ 1

^ 2 A + A - 2igtAtB4C2 + 2igtABt4Ct2 + g2t2 (2AN|N2

2 V 2^

-96At2ANC - 2At2AN| - 32At2AN| - 144At2AN| - 192At2ANB - 48At2A^

and

S = - + Г - 2

A - At - 2«gtAtB4C2 - 2«gtABt4Ct2 + g2t^2AN|NC - 16AtA2N|NC -72AtA2N2 N2 - 96AtA2NBN2 - 24AtA2N2 - 4AtA2N4 NC — 64AtA2N|NC

+96At2ANc + 2At2AN| + 32At2AN| + 144At2AN2 + 192At2ANB + 48At2 A

Using equations (40) and (41), second order variance is expressed as

-1

(AC)2 = (C2) - (C)2 = 1 (N + £) [1 - 48g2t2(a2 + a*2 +4 |a|2) (AS)2 = (S2) - (S)2 = -1 (V +2) 1 [ - 1 - 48g2t2(a2 + a*2 - 4 |a|2)

42

Now, equations (6)-(8) can be expressed as:

|2. 1f 1 - 576g2t2 |a|2"

2\ Ml |2 nc 2,2 \ |4 1

1

S(0,i, |a|2) = ^ ( |a|2 - 96g2t2 |a|4 + ^ ) |a|2 + 96 |a|4 g2t2(cos20 - 6)

and

_ 1 + 96 |a|2 g2t2(cos20 - 6) 2(cos 20 + 1)(1 - 96 |a|2 g2t2) From equation (38) and (44), we can correlate reduction of U with sub-Poissonian state.

(40)

(41)

(42)

(43)

u^ a | 2)=21 - 96g2t2| a | 2 , (44)

(45)

nia , I |2ч 1 + 96 |a| (coo2^ - 6)

|a| )= ——^ ^^—^ ,2 „,„ч. (46)

2.3. Third harmonic generation process

In third harmonic generation process, the interaction takes place in such a way that the absorption of three photons of frequency w1 each and the emission of one photon of frequency w2 such that w2 = 3w1. The Hamiltonian for the given process is:

H = ^a+a + w2b+b + g (a3b+ + a+3b) , (47)

where g is a coupling constant, a (a+), b (b+) are the annihilation(creation) operators, respectively. A = a exp iw1t, B = b exp iw2t are slowly varying operators at frequencies w1, w2.

By using the same techniques used in above interaction process, we get;

3 ( )

A(t) = A - 3«gtAt2B + 3g2t2 (6AtA2NB + 6ANB - At2A3),

where NB = BtB and NC = CtC.

From equation (48), the mean number of photons using condition (10) can be simplified as:

N(t) = AtA - 3g2t2 (At3A3) .

The expectation value of N(t) is:

(N) = |a|2 - 3g2t2 |a|6 .

Using equation (48), we get:

(AN)2 = |a|2 - 18g2t2 (|a|6) . The condition of sub-Poissonian photon statistics is given as:

d = (AN)2 - (N) < 0.

Using (50) and (51), we get:

d = -15g2t2 |a|6 ,

as d is negative and thus we achieve a non-classical state. By substituting (48) in (2), we obtain:

(48)

(49)

(50)

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(51)

(52)

(53)

and

C =2 N ^ - 2

S=- 2(N ^- "

A + At - 3igtAt2B + 3igtA3Bt - 3g2t2 (At2A3 + At3A2)

A - At - 3igtAt2B - 3igtA2Bt - 3g2t2 (At2A3 - At3A2)

Using equations (54)-(55), second order variance is expressed as:

and

(AC)2 = (C2) - (C)2 = 1 (n +2) - [1 - 2g2t2 (2 |a|2 a2 +2 |a|2 a*2 + 6 |a|4) (AS)2 = (S2) - (S)2 = -1 (n +2) -1 [ - 1 - 2g2t2 (2 |a|2 a2 + 2 |a|2 a*2 - 6 |a|4)

Now equations (6)-(8) can be expressed as:

U(0,t, |a|2) = —

2, 1 Î1 - 27g2t2 |a|

2 I 1 - 3g2t2 |a|4

(54)

(55)

(56)

(57)

(58)

and

S(0,t, |a|2) = 1 (|a|2 - 3g2t2 |a|6 + ^ |a|2 + 3 |a|6 g2t2(2cos20 - 9)

n(flt||2) 1 + 3 |a|4 g2t2(2cos 20 - 9)

g(0,t, |a| ) -

2(cos 26 + 1)(1 - 3 |a|4 g2t2) equations (53) and (58) indicate the presence of non-classicality.

(59)

3. Minimum total noise in terms of number operator

The minimum total noise of a single mode with density matrix p can be stated in terms of A and A as [31]:

Xi = 1 [A + Af] and X2 = ^[A - A*]. (61)

2 2«

From equation (60), we get:

1 ' (62)

(X2) + (X2)2 = (N + 0 .

2

In order to find out the minimum total noise of state in terms of number operator we use Schwartz inequality:

((X - (X2)) (N - (N))) |2 <((X - (X2))2) ((N - (N))2) < (AX2)2 (AN)2. (63)

Equation (63) is simplified as:

-,2 , A nr\2 > 1

4

where [X2, N] = ¿X1.

Similarly,we can achieve for X1 is:

2 2 1 2 1, l2

(AX2)2 (AN)2 > 1 ([X2,N]) > ^ (Xi) |2, (64)

(AX1)2 (AN)2 > 11 (X2) |2. (65)

Using equation (64) and (65), we get:

(X1)2 + (X2)2 < 4 (AN)2 (AX1)2 + (AX2)2 Using equation (62), we obtain:

N+2> ^

4 (AN)2 + 1 [(AX1)2 + (AX2)2

thus minimum total noise in terms of number operator is:

Tmin = [(AX1)2 + (AX2)2]. Substituting equation (67) in above relation, we get:

T > ^N + 2)

Tmin > [4(AN)2 + 1]' equation (68) shows that as the value of (AN)2 decreases, minimum total noise increases. 3.1. Seven wave mixing process

Minimum total noise in seven wave interaction process can be obtained by substituting equation (14) and (17) in equation (68):

22

(66)

(67)

(68)

,a|2 - 24g2t2 |a|4 + 2

Tn > -rr|2-g-| 2 2, 24 . (69)

|2 - 24g2t2 |a|4 + 2 4 |a|2 + 1 - 384g2t2 |a|

3.2. Eight wave mixing process

In this case we substitute equation (36) and (37) in equation (68) to obtain the expression for minimum total noise and is given as:

|a|2 - 96g2t2 |a|4 + 2

Tn > -JLJ2---LJ-^. (70)

" 4 |a|2 + 1 - 1536g2t2 |a|4

3.3. Third harmonic generation process

After substituting equation (50) and (51) in equation (68), we get expression of minimum total noise as:

|a|2 - 3g2t2 |a|6 + 2

Tn > ---LJ-^, (71)

> 4 |a|2 + 1 - 72g2t2 |a|6

thus equations (69)-(71) show that for a fixed value of (N) as the value of [AN]2 decreases, minimum total noise increases which indicates the presence of quantum state.

4. Conclusion

Comparison of non-classical parameters like phase fluctuation, antibunching and minimum total noise have been done in seven and eight wave mixing interaction and third harmonic generation nonlinear optical processes. Figs. 14 show that reduction of phase fluctuation parameters U, S, Q and antibunching increases with increasing mean photon number |a|2. After comparing Figs. 1-4, we can conclude that rate of reduction of parameters U, S, Q and antibunching is maximum in third harmonic generation as compared to eight wave mixing and seven wave mixing. Equations (30)-(32), (44)-(46), (58)-(60) show that the parameter U is independent of в while S and Q parameters can be tuned by varying the values of t and в. The result also indicates that higher the rate of reduction of phase fluctuation parameter, higher will be the antibunching.

Fig. 1. Variation of phase fluctuation parameter U with |a|2 for seven wave mixing, eight wave mixing and for third harmonic generation (taking g2t2 « 10-4)

0.5 r-1-1-1-1-1-1-1-1-r

.4 5__I_____i_____I___I__I __L,____L__

0 5 10 15 20 25 30 35 4D 45 SB -laf

Fig. 2. Variation of phase fluctuation parameter S with |a|2 for seven wave mixing, eight wave mixing and for third harmonic generation (taking g2t2 « 10-4 and в = 0)

Figures 4 and 5 show the variation of antibunching and minimum total noise with |a|2 and we observe that more is the antibunched state (sub-Poissonian), more will be the minimum total noise present in the system.

Thus we can conclude that the depth of non-classicality can be directly measured by using the quantum parameters i.e. phase fluctuation, antibunching and minimum total noise present in the system.

D.5-1-i-i-i-i-i-i-1-г

0 S 1Q 15 20 25 30 Э5 40 45 EQ

-lal3

Fig. 3. Variation of phase fluctuation parameter Q with |a|2 for seven wave mixing, eight wave mixing and for third harmonic generation (taking g2t2 « 10-4 and 0 = 0)

■200 -1-1-1-1-1-'-1-1-1-

0 5 10 IS 20 25 30 35 40 45 50 -|CL|2

Fig. 4. Variation of d with |a|2 for seven wave mixing, eight wave mixing and third harmonic generation (taking g2t2 « 10-4)

-k-lnl2

Fig. 5. Variation of Tm;n in antibunched (sub-Poissonian) state with |a|2 for seven wave mixing, eight wave mixing and for third harmonic generation (taking g2t2 « 10-5)

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Comparison of non classical effects: quantum phase fluctuation, antibunching and minimum total noise in various non linear optical processes Текст научной статьи по специальности «Физика»

Аннотация научной статьи по физике, автор научной работы — Priyanka, Gill Savita

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Текст научной работы на тему «Comparison of non classical effects: quantum phase fluctuation, antibunching and minimum total noise in various non linear optical processes»