Quantum statistics of multimode parametric amplification

Volume 48, number 2

OPTICS COMMUNICATIONS

15 November 1983

QUANTUM STATISTICS OF MULTIMODE PARAMETRIC AMPLIFICATION A. LANE, P. TOMBESI 1, H J . CARMICHAEL 2 and D.F. WALLS

Physics Department, University of lCaikato, Hamilton, New Zealand Received 26 August 1983

The effect of a multimode interaction on the squeezing attainable in parametric amplification is investigated. In the non degenerate case the maximum squeezing attainable is reduced from the single mode case, whereas in the degenerate case the squeezing is essentially unaffected by the multimode nature of the interaction.

Present theoretical efforts to find suitable schemes to produce squeezed states o f light have in the main been restricted to considering single mode devices. For example it has been predicted that squeezed states may be produced in such nonlinear interactions as four wave mixing [ 1], parametric amplification [ 2 - 4 ] and second harmonic generation [5,6]. While a single mode analysis is suitable for intracavity configurations it has been shown that the squeezing in such cases is restricted to values of the order of two due to vacuum fluctuations entering the cavity [2 ,3]. In order to avoid this limitation to the squeezing an extracavity nonlinear optical interaction may be preferable. In this situation one has a travelling wave device and the single mode approximation is no longer valid. It is the aim of this paper to calculate the effect o f a multimode interaction on the squeezing obtained in the output light. We consider the case of parametric amplification though similar considerations would hold for four wave mixing. We shall extend methods used by Tucker and Walls [7] to calculate the mean field in a travelling wave frequency converter to calculate the quantum statistics of the output field in parametric amplification. We consider the following hamiltonian describing 1 On leave from Istituto di Fisica G. Marconi, Universita di Roma, 00185 Roma, Italy. 2 Present address: Physics Department, University of Arkansas, FayetteviUe, AR 72701, USA. 0 030-4018/83/0000-0000/$ 03.00 © 1983 North-Holland

multimode parametric amplification in a crystal with a X(2) optical nonlinearity pumped by a laser with frequency o t passing along the principal axis (which we take as the z axis) o f the crystal

H = ~ ~oka~a k -- h ~ k

k

[K(OL, Ok, Ok, )

X exp (--ioLt)a?ka~,gkL_kgc, -- h.c.] .

(1)

The laser has been assumed to have linear polarization and to be sufficiently intense to be undepleted by the interaction and is treated classically. We also assume the interaction volume is large compared to the wavelength of the laser. This yields the momentum matching condition

k L =k+k'.

(2)

This condition does not, in general, allow energy matching as well, although the use o f birefringence can enable simultaneous momentum and energy matching for one pair of modes. We will denote these modes by the superscript °. The coupling constant K(OL, Ok, o k,) is given by K(OL, Ok, Ok, ) = i(EL(O , 0)>

X 3X(2):.(~kL~k~k,)[(ok/ek)(Ok,/ek,)]l/2 ,

(3)

where
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OPTICS COMMUNICATIONS

redefining the mode function to include the change in electric permeability along the (principal) z axis. We restrict our considerations to a one dimensional electric field travelling in the z direction.

E(z, t) = i ~ (hcok/2Lek ) 1/2 k X [ak(t)e ikz - a~(t)e-ikz I .

(4)

We may solve the time evolution o f an arbitrary pair o f coupled modes with detuning dxco C°L = c°k + cok' + &co •

(5)

The solutions are

ak(t) = f l (cok, Oak(O) + fZ(cok, t)a?k'(O) ,

(6)

with k' = k L - k, and fl (cok, t) = exp [-i(co k + Aco/2)t] X [cosh A t + (iAco/2A) sinh At] ,

f2(cok, t) = exp [-i(co k + dxco/Z)t] (iK/A)sinhAt, where A = [[K[2 - (Aco/2)2] 1/2 . we will assume that we have phase matched two o frequencies cos , co~ to the laser pump frequency so that both momentum and energy are conserved. o

o

COL =cOs +cOi'

kL = k ~ + k ~ "

(7)

kL = k(co) + k (co') - ks +

o

o

Note that when coi ~ COs ~ C°L/2 the detuning vanishes to first order and one will be able to find a small frequency band around COL/2 which has approximate energy matching. Therefore to first order in the detuning the statistics of the multimode degeerate parametric amplifier will resemble the single mode case. The solutions (6) may be used to construct a multimode wavepacket. The mean field in a travelling wave frequency converter was calculated in this manner by Tucker and Walls [7]. We shall calculate the variance in the quadrature phases of the field after interaction with the medium for a time t. The appropriate physical quantities are the norreally ordered variances since these are what are measured in a homodyne or heterodyne detection scheme [8,91 . The presence of squeezing is indicated by a normally ordered variance less than zero. We assume the modes are initially in coherent states which have a normally ordered variance equal to zero (: V(E(z,O): = 0). We shall consider the electric field operator for a wavepacket [10] E(+)(z,t) = i ~ (hcok/2Lek)l/2ak(t)eikZF(cok), k

We shall expand the detuning to first order around these matched frequencies. Consider an arbitrary pair of coupled modes o f frequency co, co' near cos and w L respectively

o

15 November 1983

where F(cok) is a filter function which may represent a frequency selection due to detector bandwidth. Following Mandel [9] we shall introduce a photon flux operator. This may be done by noting that the mean energy flux through the medium is given by the time averaged Poynting vector. For a monochromatic wave travelling along the principal axis (and polarised along principal axes), the mean energy flux is [11 ]

E × It= (u/2 )e [/:'12 ,

~dk (w-oa~)+.o co=w~_

(12)

~dk (oa'-c~])

gi +

] co'=w~

o

Denoting 8 s = co - COs,8 i = c o expansion in frequency gives

(11)

"

(8)

O

--

coi our first order

8 s/Vs ~- --6 i/Vi ,

where o is the velocity through the medium. An appropriate flux operator for each mode is then

N k = (vk/2h) (ek/co k) (1:'~~ )E)~+)) (9)

= (c~o,,k/2hcok)<~

~F.~+~>,

(13)

where v s and vi are the group velocities at frequencies o o cos and coi respectively. The deturting Aco is given by

using v k = c(eo/ek ) 1/2 = c /n k . The photon flux operator may then be written as

ACO= --(8 s + 8i)

N = (/?()/~(+))~,

[(O i -

156

Vs)/Vs] 6 s

~ [(O s - - V i ) / o i ] 8 i .

(10)

where ~(-), ~-(+) are scaled field operators

(14)

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(+) = ½i(c/L)l/2 ~a [F(Wk)/X/'n-kk] ak(t)e ikz , k

(15)

15 November 1983

c G F2(cok) (if?(-), E(+)) = - ~ k nk tf2(wk' 0[2 c ~F(Wk)g(WkL-k)

k

Two quadratures of the field may be defined as follows E 1 =E (+)+E(-),

E 2 = i ( E (+)- E ( - ) ) .

(16)

4hk,kL_

X [--eikLZfl (cok, t)fl (WkL-kt)]"

(21)

Thus we arrive at the following result for the variance

We shall calculate the variance in the E 1 quadrature

:V(EI(Z't)):

:V(E 1 ): = 2 [(E(-)(z, t),E(+)(z, t)) + Re (E(+)(z, t),E(+)(z, t))],

(17)

where ~a, b) = (ab) - (a)(b). We split the field operator into two frequency bands denoted by the regions Rs and R i (we choose s, i so that w s < w~) where

kERs=~kKkL]2,

kCRi~k>kL]2.

(18)

We expand the terms in eq. (17) in this manner and obtain (for p, p' ~ s, i) P

v

C

~

F(wk)F(Wk')(ai(t),ak,(t)) ' 4L kRp~'np, X/nkn k, ~

× [-ei(k~')Z(ak(t),ak,(t)) ] .

(19)

Using the solutions (6) we find

(ark(t), ak'(t)) = f~(cok, t)f2 (Wk', t)fkk' ,

I

+ Re ( F ( W k t - k ) F ( W k ) i~1 exp [i(kLZ - WEt + 4)] \

X --

- i

2A

'

where q~ is the phase of the coupling constant (K = ]KleiO). The variance in the fluctuations of the electric field clearly oscillate with time and space. In order to demonstrate the squeezing in one o f the quadrature phases we beat the field with a coherent reference beam of frequency w L and wavevector k L. This effectively transforms to a frame of stationary phase. Thus

¢~+ kLZ - colt + 7r]2 = 0 .

F(wk)F(Wk') ' P "- 4L kRp,k'Rp, ~/rtkn k,

(k~(p+) ~ ( + ) ~ _ c

2L k [ nk

(23)

The variance in the other quadrature phase may be detected by changing the phase of the reference beam by 7r/2. We shall convert the sum in eq. (22) to an integral, which for one dimension is given by

1

1f

1 [ dw

(24)

(20)

(ak (t), ak'(t)) = f l (Wk, t) f 2(Wk', t)6k, kL -k' • We note that the magnitude of the coherent state amplitudes does not affect the variance of the electric field. Thus one cannot discriminate between modes by exciting some as was the case for the mean field. Only the explicit filter function and implicit detuning and coupling constant variation will favour certain modes over others. From eqs. (19) we Find

where v(w) is the group velocity dw/dk. The quadrature variance becomes

- I , I F(wkL-k)F(--~wk) [sin~--2at]l

d'k'kL-k

(25)

2A ] J"

At this stage we must choose some explicit form for the filter. The •ter may represent the detector frequency response or may model the frequency dependence of the nonlinear coupling which has been 157

Volume 48, number 2

OPTICS COMMUNICATIONS

assumed constant. We also note that the squeezing (which is given by the (/~(-), E(+)) contribution) is eliminated if the filter is around only one of the frequency bands. We choose the filter to have the form of a double o o lorentzian peaked around cos and col

[

3`2

1,/2

F(cog) = 3`2 + (COg _ co~)2 + 3,2 + (cok - co~)2J The natural symmetry o f the problem is brought out if we choose the lorentzian linewidths so that each pair o f coupled modes (cok, C°kL -k) are equally damped. Hence from eq. (9) we choose 3`s/Vs = 3`i/vi. We define the geometric means

3` ='@Ys3`i,

o = ~/OsVi,

n = ~ ,

(27)

and change the variables in the integral as follows o

cok = COp + (Up/V)6 ,

(28)

where p = s or i depending on whether cox is closer to o o cos or col respectively. If we assume that the width o f these lorentzians (given by 7s, 3`i) are small compared o to the separation o f the frequencies co~ and cos , then F(cok) ~- 0 unless COg ~- cop. For cog ~, COp one may show F2(cOg) = F(cok)F(cokL_k) = 72/(3` 2 + 62) .

15 November 1983

: V(/~ 1 (r)): = 1 7(c/vn){½ [(ns+ni)/nl [coshr - 1 + ½/3nr [L 1(r)lo(r) - L 0 ( r ) l 1(r)] + ½/32(r sinhT - 2coshr + 2)] - [sinhr-13rll(r)+½/32(rcoshr-sinhr)] } ,

(31)

where/3 = (7/LKI)[vs - oil/2o is our series expansion coefficient and r = 21Ktt isthe scaled time. L 0 , L 1 are modified Struve functions and Io, 11 are modified Bessel functions. The presence of squeezing may be detected by the deviation of photon counting statistics from poissonian in a homodyne experiment. In the case o f a strong local oscillator it may be shown that [8,9]

Q _ V(n) - (n)_ aT: V(/~ 1 ): = ~ a v T G ( r ) (n)

(32)

where a is the quantum efficiency of the detector and T is the counting time. From eq. (31) we plot G('r) in fig. 1 for/3 = 0,/3 = 0.05,13 = 0.2. We note that the second order term in/3 in eq. (31) gives a negligible contribution. The case 13= 0 corresponds to the degenerate parametric amplifier (o i = Vs) and perfect squeezing may be obtained for large r. There will be corrections to this result if higher order terms in the expansion (8) are included. In the non degenerate

(29)

We may also approximate n(cog), v(cok) by np, Vp since the refractive index and hence the group velocities are slowly varying functions of frequency. The integral may be split into two parts with cog < coL/2 and cog > COL/2 corresponding to the two Filter peaks. We may also extend the integral limits to infinity in each case, as the linewidths are negligible compared to the optical frequencies involved. The symmetry of the problem then allows us to add the two integrals together using only the geometric means defined in eq. (27) to give

G('6 1.5

1.0

0.S

0,0 0.

c :V(EI

):

? d8

3/2

= q.Zrr)a on (3`2+62)

(3o)

-0. B

-1.0

This integral has been evaluated in the Appendix by a power series method. The result is as follows. 158

Fig. 1. Variance : V(/~I ): versus time for/3 = 0 (----),/3 = 0.05 (- - - ) , / 3 = 0 . 2 ( . . . . . ).

case (v i ~ Vs) however the variance initially decreases

with time, reaches a minimum, then increases for large r. The maximum squeezing attained is less than for a single mode device. An explanation for this behaviour goes as follows. Let us assume we have transformed to a frame in which the axis of squeezing o f the resonant mode is stationary. The axis of squeezing of any detuned mode will rotate with time in this frame [ 12 ]. A multimode parametric amplifier therefore consists of an ensemble of detuned modes whose axes of squeezing rotate with time with respect to the resonance mode. Consequently the squeezing degrades and finally disappears for large times. We conclude therefore that the multimode character of the interaction will not significantly affect the squeezing in the case of degenerate parametric amplification (or degenerate four wave mixing). In the nondegenerate case multimode effects will reduce the maximum squeezing attainable.

t

II2(t ) = 2

Introduce the dimensionless variables

x = r/a/IKI, /3 = r/3'/IKI,

(m.5)

where

oill2o ,

77 = Ivs -

tells us how much the detuning is scaled by the move away from resonance (A~o = 2r/6). It is also convenient to introduce the scaled time r = 21• I t .

(A.6)

Rewriting VII :

0

/32

~ X2~

dx sinh[r\/1

Vl1(r) = J

r/X/q" --x2

(A.7)

t32 + X2

Now,

dy e-y~cosCvx),

(A.8)

0

and thus

°

/

sinh [ r ~ ]

n109 = I~ o

o

, ~

The equation to be evaluated is

7

c / d6 ,),2 2(2rr)_= on (72 +62)

-21xI \

(A.4)

0

/32 + x 2

Evaluation of the integral o f equation (30).

:V(E1):

fdt' H 1 ( t ' ) ,

=/3

Appendix

_

15 November 1983

OPTICS COMMUNICATIONS

Volume 48, number 2

f dy e-eY dx cos(~y)

(A.91

The Fourier Cosine Transform is tabulated in Erdelyi [13], p. 26, no. 30 using 2A

]J "

lo(x ) = J 0 ( i x ) .

(A.10)

Then

The following integrals must be evaluated:

7"

rr3" f dy e-OYlo [ ~ l .

oo

I11 ( t ) = f d6 sinh2At 3'2 _~o 2A q¢2 + 82 ' oo

(s )2 ,,,2

_oo

n 1 (~) = 2-71 o

(A.2)

To solve this,it is necessary to adopt a power series approach. Transforming to polar coordinates (y = r cos O) and expanding the exponential as a power series:

(a.3)

Ill(r)

T 2 + 62

Using

G dt

A

'

we may express II2(t ) in terms o f l l I (t). Thus

(A.11)

(A.1)

z_, ( - 1 ) " 21KIn=0

(~ry' n!

./2 X f d 0 sin0 cosn0 1o [r sin0] .

(A.12)

"o 159

Volume 48, number 2

OPTICS COMMUNICATIONS

This integral is tabulated in Abramowitz and Stegun [14], eq. (11.4.10).

iir)(n+l)/2

qr)

(n+l)/2t

.

(A.13) This work has been supported in part by the United States Army through its European Research Office.

2v E Z ,

References

- n < argz ~< 7r/2 .(A'14)

Thus oo

Ih (t)

2-~t 0

~n r(n+l)/2 2(n_1)/2

X F((n + 1)/2)I(n+l)/2(r).

(A.15)

Provided ~3is fairly small (which it will be normally), the series will converge fairly rapidly and we may truncate it. We will retain the first three terms in the series. (The half integer equivalents for Iv may be found in Abramowitz and Stegun [14], eqs. (10 2.13). We have then ll 1(t) = (Try/2 [KI) × [sinhr - fJrI 1 (r) + ½/32(rcoshr- sinhr)]. (A.16) These terms may be integrated in time to solve eq. (A.2). Integration by parts and use of Abramowitz

160

II2(7") = coshr - 1 + ½/3Trr[L 1(r)lo(r) - Lo(r)ll (r)]

These two results (A.16), (A.17) may be substituted into eq. (30) to give eq. (31).

F r o m Abramowitz and Stegun [14], equations (10.1.1), ( 1 0 2 2 ) , ( 9 . 6 3 ) it follows that (Z (integers)):

Jv(iz) = (i)Vlo(z),

and Stegun [14], eqs. (9.6 2 7 ) , (11.1.8) gives us

+ ½/32(rsinhr - 2coshr + 2) .

~/)d0 sin 0 cosn 010 [r sin 0 ] 0 = 2(n_1)/2 F ( ( n + l ) / 2 ) j

15 November 1983

[1 ] H.P. Yuen and J.H. Shapiro,Optics Lett.4 (1979) 334. [2] G J. Milburn and D.F. Walls, Optics Comm. 39 (1981) 401. [ 3 ] L.A. Lugiato and G. Strini, Optic~ Comm. 41 (1982) 67. [4] G.J. Milburn and D.F. Walls, Phys. Rev. 27A (1983) 392. [5] L. Mandel, Optics Comm.42 (1981) 437. [6] L.A. Lugiato, G. Strini and F. de Martini, Optics Lett. 8 (1983) 256. [7] J. Tucker and D.F. Wails, Phys. Rev. 178 (1969) 2036. [8] H.P. Yen and J.H. Shapiro, IEEE Trans. on Inform. Theory, IT 26 (1980) 78. [9] L. Mandel, Phys. Lett. 49 (1982) 136. [10] R.J. Ghuber, in: Quantum optics, eds. S.M. Kay and A. Maitland (Academic Press, 1970) p. 117. [11 ] A. YarN, Introduction to optical electronics (Holt, Rinehart and Winston, 1971 ) p. 11. [12] tt.J. Carmichael, G.J. Milburn and D.F. Walls, J. Phys. A, in press. [13] A. Erdelyi, Tables of integral transforms (McGraw-Hill, 1954). [14] M. Abramowitz and I.A. Stegun, Handbook of mathematical functions (Dover, 1964).