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Phase fluctuations of coherent light coupled to a nonlinear medium of inversion symmetry
7 August 2000
Physics Letters A 272 Ž2000. 346–352 www.elsevier.nlrlocaterpla
Phase fluctuations of coherent light coupled to a nonlinear medium of inversion symmetry Anirban Pathak, Swapan Mandal ) Department of Physics, VisÕa-Bharati, Santiniketan 731235, India Received 27 July 1999; received in revised form 13 March 2000; accepted 28 June 2000 Communicated by A.R. Bishop
Abstract An analytical approach is being adopted for the calculation of phase fluctuations of coherent light interacting with a nonlinear medium of inversion symmetry. The useful parameters for such calculations are expressed in closed analytical forms Žup to the linear power of coupling constant.. It is found that the presence of non-conserving energy terms in the model Hamiltonian lead to enhancement and reduction of phases compared to their initial values. q 2000 Published by Elsevier Science B.V. PACS: 02.30.Tb; 42.50.Dv
1. Introduction The phase in quantum mechanics plays a crucial role in the understanding of basic physics. The examples are from double slit experiment w1x to the dark resonances w2x and from Aharonov–Bohm w3x effect to Bose–Einstein condensate w4x. The introduction of hermitian phase operators have some ambiguities Žsee for example a recent review by Lynch w5x.. People, however, use both the Susskind–Glowgower ŽSG. w6–9x and Barnett–Pegg ŽBP. w10–12x formalisms for the studies of phase properties and the phase fluctuations of various physical systems. For example, the phase fluctuations of coherent light interacting with a nonlinear nonabsorbing medium of inversion symmetry are reported in the recent past )
Corresponding author. Fax: q91-3463-52672. E-mail address: [email protected] ŽS. Mandal..
w9,10x. It is found that both the SG w9x and BP w10x formalisms lead to same type Žqualitatively. of phase fluctuations. The Hamiltonian considered by Gerry w9x and Lynch w10x is of the form H s " v a†a q lg a† 2 a 2
Ž 1.
where v and lg Žthe subscript g is used to differentiate between the two anharmonic constants used in this Letter. are frequency and anharmonic Žcoupling. constant respectively. The usual annihilation and creation operators are a and a† respectively. The interaction terms involving the gain or loss of photons are called energy non-conserving terms. The interaction involving the operator a 2 Ž a† 2 . may be viewed as the loss Žgain. of two photons. On the other hand, interaction involving a†a keeps the photon number conserved Ži.e one photon is created and one photon is destroyed.. So the terms proportional to a 2 , a† 2 , a†a 3 , a†3 a, a 4 , and a†4 are energy non-
0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 4 6 - 1
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
conserving in nature. The Hamiltonian Ž1. is based on the assumption that these non-conserving energy terms have no significant contribution in the time development of the field operators. This assumption may be realized in the following way. In absence of interaction, the time development of the annihilation operator is aŽ t . s aŽ0. eyi v t. Thus the terms proportional to a 2 , a† 2 , a†a 3, a†3 a, a 4 , and a†4 are rapidly oscillating compared to the terms a†a and a† 2 a 2 . These rapidly oscillating terms contribute little to the interaction Hamiltonian. The assumption is widely used and is termed as rotating wave approximation ŽRWA.. However, the RWA is hardly valid if the noncatalytic nonlinearities are present in the system w13x. For example, the term proportional to a†4 is responsible for higher harmonic generation and occurs naturally corresponding to the field interacting with a nonlinear medium. In fact, the terms proportional to a 4 and a†4 were taken into consideration for the calculation of squeezing of coherent light passing through a nonlinear medium w14x. Again the non-conserving energy terms are responsible for the well known Bloch–Siegert shift if the field frequency is low Že.g RF and MW. w15x. Therefore, in the present Letter, we take care all the nonconserving energy terms for the calculation of the phase fluctuations of coherent light interacting with a nonlinear medium of inversion symmetry. The electromagnetic field interacting with a nonlinear medium induces macroscopic polarization w13–16x P s x1 E q x2 E P E q x3 E P E P E q . . .
Ž 2.
where E is the electric field and x 1 is the linear susceptibility. The parameters x 2 and x 3 are second and third order nonlinear susceptibilities respectively. A medium is called inversion symmetric if the potential energy remains invariant under parity transformation i.e V Ž X . s V ŽyX ). Thus the even order susceptibilities Ž x 2 , x4 etc.. would vanish for an inversion symmetric medium. Hence the leading contribution to the nonlinear polarization due to an inversion symmetric medium comes through the third order susceptibility Ž x 3 .. The corresponding electromagnetic energy density is given by Hem s
1 8p
Ž E q 4p P . P E q B P B
Ž 3.
347
where B is the magnetic field. In Eq. Ž3. we neglect the macroscopic magnetization Žif any.. The leading contribution to the interaction part of the electromagnetic energy comes through the coupling of third order nonlinear susceptibility. Therefore, the interaction energy is proportional to the fourth power of the electric field Ži.e E P E P E P E .. Now we use the operator representations of the electric and magnetic fields. The corresponding field operators are proportional to the usual position Ž X . and momentum Ž X˙ . operators respectively. The quantized form of the Hamiltonian for a single mode is given by 4
H s " v a†a q l Ž a† q a . .
Ž 4.
The parameter l is the coupling constant and is proportional to the third order nonlinear susceptibility of the medium. In addition to the free field Hamiltonian, Eq. Ž4. contains quartic interaction term. Eq. Ž4. is identified as the Hamiltonian of a quantum quartic anharmonic oscillator ŽAHO.. The model Hamiltonian Ž1. is obtained if the off-diagonal terms Žnonconserving energy terms. are dropped from Ž4.. We use the BP formalism for the present investigation. We define the dimensionless field operators as Xs
1 2
Ž a q a† . ,
X˙ s y
i 2
Ž a y a† . .
Ž 5.
Thus we have following equal time commutation relation i
w X , X˙ x s , 2
Ž 6.
where w a,a† x s 1. We use the Heisenberg operator equation of motion to have the following differential equation X¨ q v 2 X q l X 3 s 0.
Ž 7.
Hence, Eq. Ž7. contains cubic nonlinearity in field operator X. The exact analytical solution of Eq. Ž7. is, however, not available. Recently, Bender and Bettencourt have given a solution to the AHO by using multiscale perturbation theory w17x. A simple minded Taylor series solution for the same problem has been reported by one of us w18x.
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
348
2. Time evolution of useful operators
D5 s i
The solution of Eq. Ž7. up to the linear power of l is given by w18x X Ž t . s cos v tX Ž 0 . q
1
v
sin v tX˙ Ž 0 . y
l 32 v 2
= cos v t y cos3 v t q 12 v tsin v t 4 X 3 Ž 0 .
l q
32 v 3
sin3 v t y 7sin v t q 4v tcos v t 4
8v2
sin v t and
v texp Ž yi v t . .
Ž 10 .
The time evolution of the creation operator a† Ž t . is the Hermitian conjugate of Ž9.. Thus the number operator is defined as N Ž t . s a† Ž t . a Ž t . s a† Ž 0 . a Ž 0 . q Ž D 1) D 2 a† 2 Ž 0 . q h.c. . y Ž D 1) D4 a†4 Ž 0 . q h.c. .
l
y Ž D 1) D5 a†3 Ž 0 . a Ž 0 . q h.c. .
32 v 4
y Ž D 1) D6 a† 2 Ž 0 . a 2 Ž 0 . q h.c. . ,
= cos3 v t y cos v t q 4v tsin v t 4
Ž 11 .
where h.c. stands for the Hermitian conjugate. The parameters D 1) , D 2) , D 3) , D4) , D5) and D6) are the complex conjugate of D 1 , D 2 , D 3 , D4 , D5 and D6 respectively. Eq. Ž11. is used to obtain
= X˙ 2 Ž 0 . X Ž 0 . q X˙ Ž 0 . X Ž 0 . X˙ Ž 0 . q X Ž 0 . X˙ 2 Ž 0 . y
8v2 3l
y Ž D 1) D 3 a† Ž 0 . a 3 Ž 0 . q h.c. .
= X˙ Ž 0 . X 2 Ž 0 . q X Ž 0 . X˙ Ž 0 . X Ž 0 . q X 2 Ž 0 . X˙ Ž 0 . y
D6 s i
3l
l 32 v 5
= sin3 v t q 9sin v t y 12 v tcos v t 4 X˙ 3 Ž 0 . Ž 8.
N 2 Ž t . s Ž a† 2 Ž 0 . a 2 Ž 0 . q a† Ž 0 . a Ž 0 . . q 2 D 1) D 2 Ž a†3 Ž 0 . a Ž 0 . q a† 2 Ž 0 . . q h.c.
Now X˙ may simply be obtained by differentiating Eq. Ž8. with respect to time t. Eq. Ž8. along with its time derivative X˙ obey equal time commutation relation Ž6. for linear power of l. Hence, the time evolution of the annihilation operator aŽ t . may simply be obtained as
y 2 D 1) D 3 Ž a† 2 Ž 0 . a 4 Ž 0 . q 2 a† Ž 0 . a 3 Ž 0 . .
a Ž t . s X Ž t . q iX˙ Ž t . s D 1 a Ž 0 . q D 2 a† Ž 0 .
q2 a†3 Ž 0 . a Ž 0 . . q h.c.
qh.c. x y 2 D 1) D4 Ž a†5 Ž 0 . a Ž 0 . q2 a†4 Ž 0 . . q h.c. y 2 D 1) D5 Ž a†4 Ž 0 . a 2 Ž 0 .
y D 3 a3 Ž 0 . q D4 a†3 Ž 0 . q D5 a† 2 Ž 0 . a Ž 0 .
y 2 D 1) D6 Ž a†3 Ž 0 . a 3 Ž 0 .
qD6 a† Ž 0 . a2 Ž 0 .
q2 a† 2 Ž 0 . a 2 Ž 0 . . q h.c. .
where the parameters
ž
D1 s 1 y i D 2 s yi
3l 8v2
3l 2
Ž 9.
Ž 12 .
The terms beyond the linear power of l are neglected in Eq. Ž12..
/
v t exp Ž yi v t . , 3. Phase fluctuations
sin v t ,
8v l D3 s i sin v texp Ž y2 i v t . , 8v2 l D4 s i sin2 v texp Ž i v t . , 16 v 2
The purpose of the present section is to calculate the phase fluctuations of coherent light interacting with the nonlinear medium of inversion symmetry. The corresponding Hamiltonian is given by Eq. Ž4.. The coherent state < a : is defined as the right eigenket of the annihilation operator aŽ0. corresponding to
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
the eigenvalue equation aŽ0.< a : s < a < a : w19x. The eigenvalue a is in general complex and may be written as a s < a
1 y1 r2 2
.
†
aŽ t . ,
E s Ž Nq
1 y1r2 2
.
1 2
y2
i
²C : s
1 2
Ž Nq 2 .
D 1 q D 2)
Ž
qŽ
D 1) q D 2
. a yŽ
yŽ
D 3) q D4
)3
yŽ
D5) q D6
² S: s y
i
)
.a
.a
yŽ
D5 q D6)
2 Ž DC . s 14 Ž N q 12 .
D 1 y D 2)
Ž Nq .
yŽ
D 1) y D 2
. a yŽ
qŽ
D 3) y D4
)3
)
.a
D 3 y D4)
yŽ
.a
D5 y D6)
2
.
2 Ž DN . s< a < 2 q
)
Ž 15 .
Ž D12 q 2 D1 D 2) . a 2 q c.c. 4
q2 < D 1 < 2 q Ž D 1 D 2 q c.c. . 4 < a < 2 ) 4
žŽ D D qD D 1
3
1
q Ž
y1
y< D1 < 2 q Ž D1 D2 q c.c. . 4
q< a < 2 sin2 v tsin2 Ž v t y 2 u . .
Ž 20 .
The usual parameters for the purpose of calculation of the phase fluctuations are defined by w9,10x 2
2
² S :2 q ² C :2 2
S Ž u , v t , < a < 2 . s Ž D N . Ž DS .
4
q Ž D1 D5) q D 1 D6 . < a < 2a 2 q c.c. 4
8v2
= Ž 3 q 4 < a < 2 . sin v tsin Ž v t y 2 u .
. a 4 q c.c. 4
. q c.c. 4 < a <
Ž 19 .
4l < a < 2
U Ž u , v t , < a < 2 . s Ž D N . Ž D S . q Ž DC .
q Ž D 1 D 3) q D 1 D4 . < a < 2a ) 2 q c.c. 4 D 1 D5 q D 1 D6)
Ž 18 .
2
q Ž D 1 D5) y D 1 D6 . a 2 q c.c. 4 ,
where Eqs. Ž9. – Ž11. and Eqs. Ž13. – Ž14. are used. Again, the square of the averages are
y2
< D1 < 2 q Ž D1 D2 q c.c. . 4
3
q Ž D5) y D6 . < a < a
y1
y1
y2 Ž D 1 D5 y D 1 D6) . < a < 2 q c.c. . 4
2
² C :2 s 14 Ž N q 12 .
Ž 17 .
q 3 Ž D 1 D 3) y D1 D4 . a ) 2 q c.c. 4
2
.a
/
y Ž D 1 D5) q D1 D6 . a 2 q c.c. 4 ,
.
. a 4 q c.c. 4
y2 Ž D 1 D5 q D 1 D6) . < a < 2 q c.c. . 4
2
1 y1 r2 2
1
y3 Ž D 1 D 3) q D 1 D4 . a ) 2 q c.c. 4
s y 14 Ž N q 12 .
. < a < 2a )
3
where c.c. stands for the complex conjugate. Using Eqs. Ž15. – Ž17. the second order variances of C, S and N can be written as
Ž DS .
3
.a
D 3 q D4)
1
y Ž D 1 D5) y D 1 D6 . < a < 2a 2 q c.c. 4
The expectation values of the operators C and S are given by 1 y1 r2
) 4
žŽ D D yD D
q Ž D 1 D5 y D 1 D6) . q c.c. 4 < a < 4
Ž 14 .
2
Ž D12 y 2 D1 D 2) . a 2 q c.c. 4
y Ž D 1 D 3) y D 1 D4 . < a < 2a ) 2 q c.c. 4
a Ž t. . Ž 13 .
Ž E q E† . , S s y Ž E y E† . .
y1
y2 < D 1 < 2 y Ž D1 D 2 q c.c. . 4 < a < 2
†
N is the average number of photons present in the radiation field after interaction. The usual cosine and sine of the phase operator are defined in the following way Cs
² S :2 s y 14 Ž N q 12 .
349
2
2
Ž 21 . Ž 22 .
and
/
Ž 16 .
Q Ž u , v t , < a < 2 . s S Ž u , v t , < a < 2 . ² C :2
Ž 23 .
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
350
The above relations hold for a fixed value of l. Now one can, in principle, calculate the above parameters analytically by using Eqs. Ž16. – Ž20.. The average number of photons after interaction is given by Ns< a <2 1q
l 8v2
and QŽu ,v t ,< a < 2 . 1 s 2 4cos Ž v t y u .
l
6 Ž1 q 2 < a < 2 . sin2v t 8v2 q4 Ž 3 q 4 < a < 2 . sin v tsin Ž v t y 2 u . = w1 q
2
= Ž 2 Ž 3 q 2 < a < . sin v tsin Ž v t y 2 u . q< a < 2 sin2 v tsin2 Ž v t y 2 u . . .
Ž 24 .
where Eq. Ž11. is used. Interestingly, N depends on the interaction coupling l, phase angle u and on the free evolution time v t. Thus the number of photons are not conserved. The result is not surprising since the nonconserving energy terms are included in the model Hamiltonian Ž4.. However, in earlier studies w9,10x the photon numbers were conserved. Actually, N is the most important physical parameter which causes differences between our calculation and the calculations reported else where in connection with the phase fluctuations of coherent light interacting with the same physical system. Now Eqs. Ž21. – Ž23. assume the following forms, UŽ u , v t ,< a < 2 . s
l
1
q< a < 2 6 v tsin2 Ž v t y 2 u . q3sin2 u sin2 v t q4sin2 v tsin2 Ž v t y 2 u .
4
l y
2
2
16 v cos Ž v t y u . = y6 v tsin2 Ž v t y u . q Ž 6 q 4 < a < 2 . =sin v tsin Ž v t y 2 u . y 6 Ž 1 q < a < 2 . sin2v t y2 < a < 2 sin v tsin Ž 3 v t y 4u . q< a < 2 sin2 v tsin2 Ž v t y 2 u . y< a < 2 sin2 v tsin2 u y 6 < a < 2v tsin2 Ž v t y u . 4 Ž 27 .
= 6 Ž 1 q 2 < a < 2 . sin2v t q 6 < a < 2v tsin2 Ž v t y u .
Hence Eqs. Ž25. – Ž27. are our desired results. In the derivation of Eq. Ž27., we assume < a < 2 / 0. y1 Now, U0 s 12 ,S0 s 14 < a < 2 Ž < a < 2 q 12 . and Q 0 2 s 1 4cos Ž v t y u . are the initial Ži.e l s 0. values of U, S and Q respectively. Thus U0 , S0 and Q0 signify the information about the phase of the input coherent light. The suitable choice of v t may cause the enhancement and reduction of all the above parameters compared to their initial values. It is to be noted that the parameters S and Q contain the secular terms proportional to v t. However, it is not a serious problem since the product vl2 v t is small w18x. Eqs. Ž25. – Ž27. are good enough to have the flavor of analytical results. Now we give two interesting special cases.
q3 < a < 2 sin2 u sin2 v t
3.1. The Õacuum field effect
1q
2
8v2
= Ž 6 Ž 1 q 2 < a < 2 . sin v tsin Ž v t y u . q3 < a < 2 sin2 v tsin2 Ž v t y 2 u . .
Ž 25 . SŽ u , v t ,< a < 2 .
ž
Nq
1 2
y1
/
l 1q
8v2
q4 Ž 3 q 2 < a < 2 . sin v tsin Ž v t y 2 u . q4 < a < 4 sin2 v tsin2 Ž v t y 2 u . 4
Ž 26 .
The radiation field with zero photon is termed as the vacuum field. The interaction of vacuum field with the medium gives rise to some interesting quantum electrodynamic effects w20x. Now the purpose of
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
this subsection is to study the effects of the vacuum field on the phase fluctuation of input coherent light. Eqs. Ž25. and Ž27. reduce to
ž
U Ž u , v t . s U0 1 q Q Ž u , v t . s Q0 1 q
3l 4v 2
sin v tsin Ž v t y u .
/
Ž 28 .
3l
A huge reduction of U is possible with the increase of the photon number < a < 2 . However, care should be taken about the condition of the solution during such increase. The circular nature of the trigonometric function ensures the occurrence of Eq. Ž30. for other values of v t s 2 mp q p4 , where m is an integer Now for v t s pr2, we have
4v 2 U
=sin v t sin v t q 2sin Ž v t y 2 u . 4
ž
p p 3'2 l , , < a < 2 s U0 1 q 1q2< a <24 . 2 4 2 8v
8 v 2 cos 2 Ž v t y u .
= yv tsin2 Ž v t y u . qsin v tsin Ž v t y 2 u . y sin2v t 4 ,
Ž 29 .
™
where < a < 2 s 0 Žin case of Eq. Ž29., the limiting value of < a < 2 0 is used.. Interestingly, the vacuum field itself couples with the medium and gives rise to the condition l / 0. Thus the phase fluctuations for vacuum field are of purely quantum electrodynamic in nature. Now for u s 0, the parameters U and Q are enhanced compared to U0 and Q 0 respectively. The corresponding maximum fluctuation of U is obtained if v t is an odd multiple of pr2. The value of Q is infinite if v t becomes an odd multiple of pr2. For u / 0, the parameters U and Q may be reduced or enhanced by the suitable choices of v t. It is to be noted that the parameter U is 0.5 and is independent of u in the earlier occasions w9,10x. In case of the present Letter, however, the parameter U depends on u and v t. The parameter S is identically zero and coincides exactly with the earlier results w9,10x. 3.2. u s p4 Eqs. Ž25. – Ž27. are still complicated even for u s pr4. A further simplification is made with the choice v t s pr4. Eq. Ž25. reduces to a simple form U
ž
p p 3 l< a < 2 , , < a < 2 s U0 1 y . 4 4 8v2
/
ž
/
/
ž
/
Ž 31 .
3l y
351
Ž 30 .
Eq. Ž31. clearly shows the enhancement of U parameter as the photon number increases. Thus we conclude by noting that the parameter U may decrease Ž30. and increase Ž31. with the increase of < a < 2 by suitable choices of v t. Similarly, one can obtain the reduction and enhancement of the remaining two parameters Ž Q and S . by suitable manipulations of free evolution time. In earlier works w9,10x, the parameter Q is found to decrease with the increase of photon number till a minimum is reached. Subsequent increase of < a < 2 causes the increase of Q. However, the parameters U and S are enhanced compared to their initial values as < a < 2 increases. Hence, the present results are in sharp contrast with those of the earlier studies w9,10x. The above differences are attributed due to the fact that the nonconserving energy terms are included in the model Hamiltonian.
4. Conclusion The phase fluctuation of coherent light interacting with a nonlinear medium of inversion symmetry is carried out by using the BP formalism. The usual parameters for this purpose are U, Q and S. The calculated parameters Ž25. – Ž27. depend on u , < a < 2 and v t. It is interesting to note that the free evolution time v t is absent in the earlier works w9,10x. However, the presence of v t is automatic in the present calculation. It accounts for the fact that the nonconserving energy terms are present in our model Hamiltonian. The effect of vacuum field on the phase fluctuation parameters are expressed in closed analytical
352
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
forms. It is found that the enhancement of U and Q are possible when the phase angle u s 0. However, for u / 0, both reduction and enhancement of U and Q are possible by suitable choices of v t. The observed results are in sharp contrast with those of the earlier studies w9,10x. However, the parameter S reduces to zero for vacuum field and agrees exactly with those of the earlier works w9,10x. Apart from the general expressions for U, Q and S, we also made a qualitative comparison between the present results and the results obtained earlier for the identical physical system. For u s pr4, we obtain the reduction and enhancement of phase parameters ŽU, Q, and S . by suitable manipulation of free evolution time v t. Those results are in sharp contrast with the results already obtained for the same physical system w9,10x. Clearly, the free evolution time v t makes the differences. Of late the preparation of quantum states have been reported by several laboratories w21–23x. These production of quantum states have opened up the possibilities of experimental studies on quantum phase and hence the verification of the present results.
Acknowledgements A.P. thanks the CSIR, Government of India for providing him with a Junior Research Fellowship. SM thanks the DST ŽNo. SRrSYrP-10r92. and CSIR ŽNo. 03r0798r96 EMR-II. for financial support.
References w1x A. Goswami, Quantum Mechanics, Brown Publishers, Dubque, 1992, p. 21. w2x M.O. Scully, Phys. Rev. Lett. 67 Ž1991. 1855. w3x S. Gasiorowicz, Quantum Physics, Wiley, New York, 1974, p. 222. w4x J. Javanainen, Y.S. Mi, Phys. Rev. Lett. 76 Ž1996. 161. w5x R. Lynch, Phys. Rep. 256 Ž1995. 367. w6x H.-Y. Fan, H.R. Zaidi, Opt. Commun. 68 Ž1988. 143. w7x B.C. Sanders, S.M. Barnett, P.L. Knight, Opt. Commun. 58 Ž1986. 290. w8x D. Yao, Phys. Lett. A 122 Ž1987. 77. w9x C.C. Gerry, Opt. Commun. 63 Ž1987. 278. w10x R. Lynch, Opt. Commun. 67 Ž1988. 67. w11x J.A. Vaccaro, D.T. Pegg, Opt. Commun. 70 Ž1989. 529. w12x R. Lynch, J. Opt. Soc. Am. B 4 Ž1987. 1723. w13x F.A. Hopf, G.I. Stegeman, Applied Classical Electrodynamics, vol. II, Wiley, New York, 1986. w14x P. Tombesi, A. Mecozzi, Phys. Rev. A 37 Ž1988. 4778. w15x F. Bloch, A.J.F. Siegert, Phys. Rev. 57 Ž1940. 522. w16x S.L. Braunstein, R.I. McLachlan, Phys. Rev. A 35 Ž1987. 1659. w17x C.M. Bender, L.M.A. Bettencourt, Phys. Rev. Lett. 77 Ž1996. 4114. w18x S. Mandal, J. Phys. A 31 Ž1998. L501. w19x W.H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973, p. 104. w20x P.W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics, Academic Press, New York, 1993. w21x D.T. Smithney, M. Beck, M.G. Raymer, A. Faridani, Phys. Rev. Lett. 70 Ž1993. 1244. w22x D. Leibfried, D.M. Meekhof, B.E. King, C. Monroe, W.M. Itano, D.J. Wineland, Phys. Rev. Lett. 77 Ž1996. 4281. w23x S. Schiller, G. Breitenbach, S.F. Pereira, T. Muller, J. Mlynek, Phys. Rev. Lett. 77 Ž1996. 2933.
Physics Letters A 272 Ž2000. 346–352 www.elsevier.nlrlocaterpla
Phase fluctuations of coherent light coupled to a nonlinear medium of inversion symmetry Anirban Pathak, Swapan Mandal ) Department of Physics, VisÕa-Bharati, Santiniketan 731235, India Received 27 July 1999; received in revised form 13 March 2000; accepted 28 June 2000 Communicated by A.R. Bishop
Abstract An analytical approach is being adopted for the calculation of phase fluctuations of coherent light interacting with a nonlinear medium of inversion symmetry. The useful parameters for such calculations are expressed in closed analytical forms Žup to the linear power of coupling constant.. It is found that the presence of non-conserving energy terms in the model Hamiltonian lead to enhancement and reduction of phases compared to their initial values. q 2000 Published by Elsevier Science B.V. PACS: 02.30.Tb; 42.50.Dv
1. Introduction The phase in quantum mechanics plays a crucial role in the understanding of basic physics. The examples are from double slit experiment w1x to the dark resonances w2x and from Aharonov–Bohm w3x effect to Bose–Einstein condensate w4x. The introduction of hermitian phase operators have some ambiguities Žsee for example a recent review by Lynch w5x.. People, however, use both the Susskind–Glowgower ŽSG. w6–9x and Barnett–Pegg ŽBP. w10–12x formalisms for the studies of phase properties and the phase fluctuations of various physical systems. For example, the phase fluctuations of coherent light interacting with a nonlinear nonabsorbing medium of inversion symmetry are reported in the recent past )
Corresponding author. Fax: q91-3463-52672. E-mail address: [email protected] ŽS. Mandal..
w9,10x. It is found that both the SG w9x and BP w10x formalisms lead to same type Žqualitatively. of phase fluctuations. The Hamiltonian considered by Gerry w9x and Lynch w10x is of the form H s " v a†a q lg a† 2 a 2
Ž 1.
where v and lg Žthe subscript g is used to differentiate between the two anharmonic constants used in this Letter. are frequency and anharmonic Žcoupling. constant respectively. The usual annihilation and creation operators are a and a† respectively. The interaction terms involving the gain or loss of photons are called energy non-conserving terms. The interaction involving the operator a 2 Ž a† 2 . may be viewed as the loss Žgain. of two photons. On the other hand, interaction involving a†a keeps the photon number conserved Ži.e one photon is created and one photon is destroyed.. So the terms proportional to a 2 , a† 2 , a†a 3 , a†3 a, a 4 , and a†4 are energy non-
0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 4 6 - 1
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
conserving in nature. The Hamiltonian Ž1. is based on the assumption that these non-conserving energy terms have no significant contribution in the time development of the field operators. This assumption may be realized in the following way. In absence of interaction, the time development of the annihilation operator is aŽ t . s aŽ0. eyi v t. Thus the terms proportional to a 2 , a† 2 , a†a 3, a†3 a, a 4 , and a†4 are rapidly oscillating compared to the terms a†a and a† 2 a 2 . These rapidly oscillating terms contribute little to the interaction Hamiltonian. The assumption is widely used and is termed as rotating wave approximation ŽRWA.. However, the RWA is hardly valid if the noncatalytic nonlinearities are present in the system w13x. For example, the term proportional to a†4 is responsible for higher harmonic generation and occurs naturally corresponding to the field interacting with a nonlinear medium. In fact, the terms proportional to a 4 and a†4 were taken into consideration for the calculation of squeezing of coherent light passing through a nonlinear medium w14x. Again the non-conserving energy terms are responsible for the well known Bloch–Siegert shift if the field frequency is low Že.g RF and MW. w15x. Therefore, in the present Letter, we take care all the nonconserving energy terms for the calculation of the phase fluctuations of coherent light interacting with a nonlinear medium of inversion symmetry. The electromagnetic field interacting with a nonlinear medium induces macroscopic polarization w13–16x P s x1 E q x2 E P E q x3 E P E P E q . . .
Ž 2.
where E is the electric field and x 1 is the linear susceptibility. The parameters x 2 and x 3 are second and third order nonlinear susceptibilities respectively. A medium is called inversion symmetric if the potential energy remains invariant under parity transformation i.e V Ž X . s V ŽyX ). Thus the even order susceptibilities Ž x 2 , x4 etc.. would vanish for an inversion symmetric medium. Hence the leading contribution to the nonlinear polarization due to an inversion symmetric medium comes through the third order susceptibility Ž x 3 .. The corresponding electromagnetic energy density is given by Hem s
1 8p
Ž E q 4p P . P E q B P B
Ž 3.
347
where B is the magnetic field. In Eq. Ž3. we neglect the macroscopic magnetization Žif any.. The leading contribution to the interaction part of the electromagnetic energy comes through the coupling of third order nonlinear susceptibility. Therefore, the interaction energy is proportional to the fourth power of the electric field Ži.e E P E P E P E .. Now we use the operator representations of the electric and magnetic fields. The corresponding field operators are proportional to the usual position Ž X . and momentum Ž X˙ . operators respectively. The quantized form of the Hamiltonian for a single mode is given by 4
H s " v a†a q l Ž a† q a . .
Ž 4.
The parameter l is the coupling constant and is proportional to the third order nonlinear susceptibility of the medium. In addition to the free field Hamiltonian, Eq. Ž4. contains quartic interaction term. Eq. Ž4. is identified as the Hamiltonian of a quantum quartic anharmonic oscillator ŽAHO.. The model Hamiltonian Ž1. is obtained if the off-diagonal terms Žnonconserving energy terms. are dropped from Ž4.. We use the BP formalism for the present investigation. We define the dimensionless field operators as Xs
1 2
Ž a q a† . ,
X˙ s y
i 2
Ž a y a† . .
Ž 5.
Thus we have following equal time commutation relation i
w X , X˙ x s , 2
Ž 6.
where w a,a† x s 1. We use the Heisenberg operator equation of motion to have the following differential equation X¨ q v 2 X q l X 3 s 0.
Ž 7.
Hence, Eq. Ž7. contains cubic nonlinearity in field operator X. The exact analytical solution of Eq. Ž7. is, however, not available. Recently, Bender and Bettencourt have given a solution to the AHO by using multiscale perturbation theory w17x. A simple minded Taylor series solution for the same problem has been reported by one of us w18x.
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
348
2. Time evolution of useful operators
D5 s i
The solution of Eq. Ž7. up to the linear power of l is given by w18x X Ž t . s cos v tX Ž 0 . q
1
v
sin v tX˙ Ž 0 . y
l 32 v 2
= cos v t y cos3 v t q 12 v tsin v t 4 X 3 Ž 0 .
l q
32 v 3
sin3 v t y 7sin v t q 4v tcos v t 4
8v2
sin v t and
v texp Ž yi v t . .
Ž 10 .
The time evolution of the creation operator a† Ž t . is the Hermitian conjugate of Ž9.. Thus the number operator is defined as N Ž t . s a† Ž t . a Ž t . s a† Ž 0 . a Ž 0 . q Ž D 1) D 2 a† 2 Ž 0 . q h.c. . y Ž D 1) D4 a†4 Ž 0 . q h.c. .
l
y Ž D 1) D5 a†3 Ž 0 . a Ž 0 . q h.c. .
32 v 4
y Ž D 1) D6 a† 2 Ž 0 . a 2 Ž 0 . q h.c. . ,
= cos3 v t y cos v t q 4v tsin v t 4
Ž 11 .
where h.c. stands for the Hermitian conjugate. The parameters D 1) , D 2) , D 3) , D4) , D5) and D6) are the complex conjugate of D 1 , D 2 , D 3 , D4 , D5 and D6 respectively. Eq. Ž11. is used to obtain
= X˙ 2 Ž 0 . X Ž 0 . q X˙ Ž 0 . X Ž 0 . X˙ Ž 0 . q X Ž 0 . X˙ 2 Ž 0 . y
8v2 3l
y Ž D 1) D 3 a† Ž 0 . a 3 Ž 0 . q h.c. .
= X˙ Ž 0 . X 2 Ž 0 . q X Ž 0 . X˙ Ž 0 . X Ž 0 . q X 2 Ž 0 . X˙ Ž 0 . y
D6 s i
3l
l 32 v 5
= sin3 v t q 9sin v t y 12 v tcos v t 4 X˙ 3 Ž 0 . Ž 8.
N 2 Ž t . s Ž a† 2 Ž 0 . a 2 Ž 0 . q a† Ž 0 . a Ž 0 . . q 2 D 1) D 2 Ž a†3 Ž 0 . a Ž 0 . q a† 2 Ž 0 . . q h.c.
Now X˙ may simply be obtained by differentiating Eq. Ž8. with respect to time t. Eq. Ž8. along with its time derivative X˙ obey equal time commutation relation Ž6. for linear power of l. Hence, the time evolution of the annihilation operator aŽ t . may simply be obtained as
y 2 D 1) D 3 Ž a† 2 Ž 0 . a 4 Ž 0 . q 2 a† Ž 0 . a 3 Ž 0 . .
a Ž t . s X Ž t . q iX˙ Ž t . s D 1 a Ž 0 . q D 2 a† Ž 0 .
q2 a†3 Ž 0 . a Ž 0 . . q h.c.
qh.c. x y 2 D 1) D4 Ž a†5 Ž 0 . a Ž 0 . q2 a†4 Ž 0 . . q h.c. y 2 D 1) D5 Ž a†4 Ž 0 . a 2 Ž 0 .
y D 3 a3 Ž 0 . q D4 a†3 Ž 0 . q D5 a† 2 Ž 0 . a Ž 0 .
y 2 D 1) D6 Ž a†3 Ž 0 . a 3 Ž 0 .
qD6 a† Ž 0 . a2 Ž 0 .
q2 a† 2 Ž 0 . a 2 Ž 0 . . q h.c. .
where the parameters
ž
D1 s 1 y i D 2 s yi
3l 8v2
3l 2
Ž 9.
Ž 12 .
The terms beyond the linear power of l are neglected in Eq. Ž12..
/
v t exp Ž yi v t . , 3. Phase fluctuations
sin v t ,
8v l D3 s i sin v texp Ž y2 i v t . , 8v2 l D4 s i sin2 v texp Ž i v t . , 16 v 2
The purpose of the present section is to calculate the phase fluctuations of coherent light interacting with the nonlinear medium of inversion symmetry. The corresponding Hamiltonian is given by Eq. Ž4.. The coherent state < a : is defined as the right eigenket of the annihilation operator aŽ0. corresponding to
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
the eigenvalue equation aŽ0.< a : s < a < a : w19x. The eigenvalue a is in general complex and may be written as a s < a
1 y1 r2 2
.
†
aŽ t . ,
E s Ž Nq
1 y1r2 2
.
1 2
y2
i
²C : s
1 2
Ž Nq 2 .
D 1 q D 2)
Ž
qŽ
D 1) q D 2
. a yŽ
yŽ
D 3) q D4
)3
yŽ
D5) q D6
² S: s y
i
)
.a
.a
yŽ
D5 q D6)
2 Ž DC . s 14 Ž N q 12 .
D 1 y D 2)
Ž Nq .
yŽ
D 1) y D 2
. a yŽ
qŽ
D 3) y D4
)3
)
.a
D 3 y D4)
yŽ
.a
D5 y D6)
2
.
2 Ž DN . s< a < 2 q
)
Ž 15 .
Ž D12 q 2 D1 D 2) . a 2 q c.c. 4
q2 < D 1 < 2 q Ž D 1 D 2 q c.c. . 4 < a < 2 ) 4
žŽ D D qD D 1
3
1
q Ž
y1
y< D1 < 2 q Ž D1 D2 q c.c. . 4
q< a < 2 sin2 v tsin2 Ž v t y 2 u . .
Ž 20 .
The usual parameters for the purpose of calculation of the phase fluctuations are defined by w9,10x 2
2
² S :2 q ² C :2 2
S Ž u , v t , < a < 2 . s Ž D N . Ž DS .
4
q Ž D1 D5) q D 1 D6 . < a < 2a 2 q c.c. 4
8v2
= Ž 3 q 4 < a < 2 . sin v tsin Ž v t y 2 u .
. a 4 q c.c. 4
. q c.c. 4 < a <
Ž 19 .
4l < a < 2
U Ž u , v t , < a < 2 . s Ž D N . Ž D S . q Ž DC .
q Ž D 1 D 3) q D 1 D4 . < a < 2a ) 2 q c.c. 4 D 1 D5 q D 1 D6)
Ž 18 .
2
q Ž D 1 D5) y D 1 D6 . a 2 q c.c. 4 ,
where Eqs. Ž9. – Ž11. and Eqs. Ž13. – Ž14. are used. Again, the square of the averages are
y2
< D1 < 2 q Ž D1 D2 q c.c. . 4
3
q Ž D5) y D6 . < a < a
y1
y1
y2 Ž D 1 D5 y D 1 D6) . < a < 2 q c.c. . 4
2
² C :2 s 14 Ž N q 12 .
Ž 17 .
q 3 Ž D 1 D 3) y D1 D4 . a ) 2 q c.c. 4
2
.a
/
y Ž D 1 D5) q D1 D6 . a 2 q c.c. 4 ,
.
. a 4 q c.c. 4
y2 Ž D 1 D5 q D 1 D6) . < a < 2 q c.c. . 4
2
1 y1 r2 2
1
y3 Ž D 1 D 3) q D 1 D4 . a ) 2 q c.c. 4
s y 14 Ž N q 12 .
. < a < 2a )
3
where c.c. stands for the complex conjugate. Using Eqs. Ž15. – Ž17. the second order variances of C, S and N can be written as
Ž DS .
3
.a
D 3 q D4)
1
y Ž D 1 D5) y D 1 D6 . < a < 2a 2 q c.c. 4
The expectation values of the operators C and S are given by 1 y1 r2
) 4
žŽ D D yD D
q Ž D 1 D5 y D 1 D6) . q c.c. 4 < a < 4
Ž 14 .
2
Ž D12 y 2 D1 D 2) . a 2 q c.c. 4
y Ž D 1 D 3) y D 1 D4 . < a < 2a ) 2 q c.c. 4
a Ž t. . Ž 13 .
Ž E q E† . , S s y Ž E y E† . .
y1
y2 < D 1 < 2 y Ž D1 D 2 q c.c. . 4 < a < 2
†
N is the average number of photons present in the radiation field after interaction. The usual cosine and sine of the phase operator are defined in the following way Cs
² S :2 s y 14 Ž N q 12 .
349
2
2
Ž 21 . Ž 22 .
and
/
Ž 16 .
Q Ž u , v t , < a < 2 . s S Ž u , v t , < a < 2 . ² C :2
Ž 23 .
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
350
The above relations hold for a fixed value of l. Now one can, in principle, calculate the above parameters analytically by using Eqs. Ž16. – Ž20.. The average number of photons after interaction is given by Ns< a <2 1q
l 8v2
and QŽu ,v t ,< a < 2 . 1 s 2 4cos Ž v t y u .
l
6 Ž1 q 2 < a < 2 . sin2v t 8v2 q4 Ž 3 q 4 < a < 2 . sin v tsin Ž v t y 2 u . = w1 q
2
= Ž 2 Ž 3 q 2 < a < . sin v tsin Ž v t y 2 u . q< a < 2 sin2 v tsin2 Ž v t y 2 u . . .
Ž 24 .
where Eq. Ž11. is used. Interestingly, N depends on the interaction coupling l, phase angle u and on the free evolution time v t. Thus the number of photons are not conserved. The result is not surprising since the nonconserving energy terms are included in the model Hamiltonian Ž4.. However, in earlier studies w9,10x the photon numbers were conserved. Actually, N is the most important physical parameter which causes differences between our calculation and the calculations reported else where in connection with the phase fluctuations of coherent light interacting with the same physical system. Now Eqs. Ž21. – Ž23. assume the following forms, UŽ u , v t ,< a < 2 . s
l
1
q< a < 2 6 v tsin2 Ž v t y 2 u . q3sin2 u sin2 v t q4sin2 v tsin2 Ž v t y 2 u .
4
l y
2
2
16 v cos Ž v t y u . = y6 v tsin2 Ž v t y u . q Ž 6 q 4 < a < 2 . =sin v tsin Ž v t y 2 u . y 6 Ž 1 q < a < 2 . sin2v t y2 < a < 2 sin v tsin Ž 3 v t y 4u . q< a < 2 sin2 v tsin2 Ž v t y 2 u . y< a < 2 sin2 v tsin2 u y 6 < a < 2v tsin2 Ž v t y u . 4 Ž 27 .
= 6 Ž 1 q 2 < a < 2 . sin2v t q 6 < a < 2v tsin2 Ž v t y u .
Hence Eqs. Ž25. – Ž27. are our desired results. In the derivation of Eq. Ž27., we assume < a < 2 / 0. y1 Now, U0 s 12 ,S0 s 14 < a < 2 Ž < a < 2 q 12 . and Q 0 2 s 1 4cos Ž v t y u . are the initial Ži.e l s 0. values of U, S and Q respectively. Thus U0 , S0 and Q0 signify the information about the phase of the input coherent light. The suitable choice of v t may cause the enhancement and reduction of all the above parameters compared to their initial values. It is to be noted that the parameters S and Q contain the secular terms proportional to v t. However, it is not a serious problem since the product vl2 v t is small w18x. Eqs. Ž25. – Ž27. are good enough to have the flavor of analytical results. Now we give two interesting special cases.
q3 < a < 2 sin2 u sin2 v t
3.1. The Õacuum field effect
1q
2
8v2
= Ž 6 Ž 1 q 2 < a < 2 . sin v tsin Ž v t y u . q3 < a < 2 sin2 v tsin2 Ž v t y 2 u . .
Ž 25 . SŽ u , v t ,< a < 2 .
ž
Nq
1 2
y1
/
l 1q
8v2
q4 Ž 3 q 2 < a < 2 . sin v tsin Ž v t y 2 u . q4 < a < 4 sin2 v tsin2 Ž v t y 2 u . 4
Ž 26 .
The radiation field with zero photon is termed as the vacuum field. The interaction of vacuum field with the medium gives rise to some interesting quantum electrodynamic effects w20x. Now the purpose of
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
this subsection is to study the effects of the vacuum field on the phase fluctuation of input coherent light. Eqs. Ž25. and Ž27. reduce to
ž
U Ž u , v t . s U0 1 q Q Ž u , v t . s Q0 1 q
3l 4v 2
sin v tsin Ž v t y u .
/
Ž 28 .
3l
A huge reduction of U is possible with the increase of the photon number < a < 2 . However, care should be taken about the condition of the solution during such increase. The circular nature of the trigonometric function ensures the occurrence of Eq. Ž30. for other values of v t s 2 mp q p4 , where m is an integer Now for v t s pr2, we have
4v 2 U
=sin v t sin v t q 2sin Ž v t y 2 u . 4
ž
p p 3'2 l , , < a < 2 s U0 1 q 1q2< a <24 . 2 4 2 8v
8 v 2 cos 2 Ž v t y u .
= yv tsin2 Ž v t y u . qsin v tsin Ž v t y 2 u . y sin2v t 4 ,
Ž 29 .
™
where < a < 2 s 0 Žin case of Eq. Ž29., the limiting value of < a < 2 0 is used.. Interestingly, the vacuum field itself couples with the medium and gives rise to the condition l / 0. Thus the phase fluctuations for vacuum field are of purely quantum electrodynamic in nature. Now for u s 0, the parameters U and Q are enhanced compared to U0 and Q 0 respectively. The corresponding maximum fluctuation of U is obtained if v t is an odd multiple of pr2. The value of Q is infinite if v t becomes an odd multiple of pr2. For u / 0, the parameters U and Q may be reduced or enhanced by the suitable choices of v t. It is to be noted that the parameter U is 0.5 and is independent of u in the earlier occasions w9,10x. In case of the present Letter, however, the parameter U depends on u and v t. The parameter S is identically zero and coincides exactly with the earlier results w9,10x. 3.2. u s p4 Eqs. Ž25. – Ž27. are still complicated even for u s pr4. A further simplification is made with the choice v t s pr4. Eq. Ž25. reduces to a simple form U
ž
p p 3 l< a < 2 , , < a < 2 s U0 1 y . 4 4 8v2
/
ž
/
/
ž
/
Ž 31 .
3l y
351
Ž 30 .
Eq. Ž31. clearly shows the enhancement of U parameter as the photon number increases. Thus we conclude by noting that the parameter U may decrease Ž30. and increase Ž31. with the increase of < a < 2 by suitable choices of v t. Similarly, one can obtain the reduction and enhancement of the remaining two parameters Ž Q and S . by suitable manipulations of free evolution time. In earlier works w9,10x, the parameter Q is found to decrease with the increase of photon number till a minimum is reached. Subsequent increase of < a < 2 causes the increase of Q. However, the parameters U and S are enhanced compared to their initial values as < a < 2 increases. Hence, the present results are in sharp contrast with those of the earlier studies w9,10x. The above differences are attributed due to the fact that the nonconserving energy terms are included in the model Hamiltonian.
4. Conclusion The phase fluctuation of coherent light interacting with a nonlinear medium of inversion symmetry is carried out by using the BP formalism. The usual parameters for this purpose are U, Q and S. The calculated parameters Ž25. – Ž27. depend on u , < a < 2 and v t. It is interesting to note that the free evolution time v t is absent in the earlier works w9,10x. However, the presence of v t is automatic in the present calculation. It accounts for the fact that the nonconserving energy terms are present in our model Hamiltonian. The effect of vacuum field on the phase fluctuation parameters are expressed in closed analytical
352
A. Pathak, S. Mandalr Physics Letters A 272 (2000) 346–352
forms. It is found that the enhancement of U and Q are possible when the phase angle u s 0. However, for u / 0, both reduction and enhancement of U and Q are possible by suitable choices of v t. The observed results are in sharp contrast with those of the earlier studies w9,10x. However, the parameter S reduces to zero for vacuum field and agrees exactly with those of the earlier works w9,10x. Apart from the general expressions for U, Q and S, we also made a qualitative comparison between the present results and the results obtained earlier for the identical physical system. For u s pr4, we obtain the reduction and enhancement of phase parameters ŽU, Q, and S . by suitable manipulation of free evolution time v t. Those results are in sharp contrast with the results already obtained for the same physical system w9,10x. Clearly, the free evolution time v t makes the differences. Of late the preparation of quantum states have been reported by several laboratories w21–23x. These production of quantum states have opened up the possibilities of experimental studies on quantum phase and hence the verification of the present results.
Acknowledgements A.P. thanks the CSIR, Government of India for providing him with a Junior Research Fellowship. SM thanks the DST ŽNo. SRrSYrP-10r92. and CSIR ŽNo. 03r0798r96 EMR-II. for financial support.
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