Effective hamiltonians in two-photon laser theory

Volume 73, number 1

OPTICS COMMUNICATIONS

EFFECTIVE HAMILTONIANS I N T W O - P H O T O N

1 September 1989

LASER THEORY

A.W. B O O N E and S. S W A I N

Department of Applied Mathematics and Theoretical Physics. The Queen "s University of Belfast, Belfast BT7 INN. :Vorthern Ireland. UK Received 24 February 1989; revised manuscript received 1I May 1989

We show that different equations of motion for the field density matrix of the non-degenerate, two-photon laser are obtained depending upon whether one uses the full or the effective hamiltonian. Although the two models give the same equation for the diagonal elements under certain conditions, we have found no conditions under which they give the same results for the offdiagonal elements. This means that expressions for the linewidths, frequency shifts, etc. obtained using the effective hamiltonian are incorrect. The origin of the discrepancy is the neglect of Stark shifts in the effective hamiltonian approach.

1. Introduction The two-photon laser has been a topic o f considerable interest in theoretical q u a n t u m optics for some time (see e.g. [ 1 ] a n d references t h e r e i n ) , although there have been difficulties in its e x p e r i m e n t a l realisation as a conventional laser [ 2 ]. In the last couple o f years, interest has been rekindled by its experimental d e m o n s t r a t i o n as a m i c r o m a s e r [ 3 ], the theoretical prediction o f strong correlation between the m o d e s with its potential for noise reduction [ 1 ] and the p r e d i c t i o n o f strong squeezing u n d e r coherent p u m p i n g conditions [4]. Most previous theoretical t r e a t m e n t s have assumed an effective hamiltonian a p p r o a c h ( E H A ) although some recent papers [2,5] are exceptions. It was p o i n t e d out in ref. [ 5 ] that there are differences between results d e p e n d i n g on the diagonal field density matrix elements obtained using the E H A and full microscopic h a m i l t o n i a n a p p r o a c h ( F M H A ) . In the present paper we c o m p a r e the equations o f m o t i o n for the off-diagonal elements o f the field density matrix for the two-photon laser o b t a i n e d using each approach, and d e m o n s t r a t e that the usefulness o f the EHA approach is strictly limited. We also obtain new expressions for the linewidths and cross-correlation coefficient for the two modes o f the two-photon laser. We emphasize that the E H A gives quite incorrect resuits for such quantities as these, which d e p e n d upon

the off-diagonal elements o f the density matrix, and that it has been much used in the past to calculate these quantities. The calculations involved are quite lengthy, and we present only the m a i n results here. A full account will be published elsewhere. We consider a three level atomic system, with levels 10), II ) and 12) and corresponding energies hEo, hE, and hE2. A p h o t o n o f frequency Co, connects the lower transition I 0 ) ,--, I 1 ) and a photon frequency co2 connects the upper I 1 ) ,--* 12) transition, where we assume the non-degenerate case, col ~ Co,. The frequencies are d e t u n e d from the a t o m i c resonances by the detunings

c~, =El.o--Co, ,

c~2=E2., -Co2 •

( 1)

F o r simplicity, we henceforth assume two photon resonance: c ~ , = - ~ 2 = - c ~ . Each level undergoes spontaneous decay at the same rate y, and level 2 only is p u m p e d at the rate R. We a d o p t the full microscopic h a m i l t o n i a n for the two-photon non-degenerate laser,

Hnd=h

0 0 3 0 - 4 0 1 8 / 8 9 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )

li>(ilE,+a*la,co, +a~a2co2 t

)

+ [g, I I ) ( 0 1 a j + g 2 1 2 ) ( l l a 2 + h . c . ] ) .

(2)

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OPTICS COMMUNICATIONS

The corresponding effective hamiltonian is

Bed =h{ 10) ( 0 l E o + 12) (2lE2+a~a,co, +a~a2co2 + [g' 12) ( 0 l a , a 2 + h . c . ] } ,

(3)

where a,* is the creation operator for a photon in mode i and the g, and g' are the usual coupling constants. We calculate the contributions to the field density matrix equations of motion due t o p u m p i n g using standard techniques [6 ]. Introducing the notation m, =n, +k, and pn,.. ...... . (t) =p.~2(k~, k2; t), we find. using the full hamiltonian (2) •

a2pmn 2 ( n~+. I n2"3L 1 + k 2 / 2

Pm.2lpump-- d.,+,

A202(n2+ 1 )(n2+ 1 +k2) F,,, + l ,,~+ ,'~

8D,,,,,2d,,,,,2

X F.,.~o.,_, .~_, ,

(4)

where aj=41gjl2/~ 2,

& =all4,, &=6/7, d.,,,2 ( k, . k2 ) =f,,,,2f~,,, +k.... k2 , ~V'(kl, k2 ) =0"1

kl "l-t72k2

and D .... (k,, k 2 ) = 1 +c~ +S.,,,,2 +..~2/16,

(5)

S.,,,~(k,, k2) =a, n, +a2n2 + Z / 2 .

(6)

f . , . . ( k , , k2)= 1 +i~-+ (a,n, +a2n2)/4+Z2/16, (7) F,,,.2(k,, k2 ) =6+ 3S.,.:/2-ic~Z/4+ Z.2/16+ 2o~ .

(8)

In eq. (4), it is understood that every density matrix element is a function of the kt. k2, which are the same for every element. The Aj and Bj are the usual Scully-Lamb linear and nonlinear gain parameters 48

- C 2 ( n 2 +k2/2)p,,,2,

(9)

where the C~ are constants. The pumping contributions to the field density matrix obtained using the effective hamiltonian (3) can be found in a similar way. In this case we find Pnln2

X [A(nl +l)(n2

+

I)+K/4+BK2/8] Phi--In2-1 ,

(10)

+ A2al [nl n2(n, +k, ) (n2+k2) ],/2

Aj=2RIgjl2/~ 2,

+ C 2 [ ( n 2 + 1)(n2+k2 + l)]'/2p,,n2+,

e Dn,n2

Pn, n2- I

D m + I ;12

- - G ( n l +k,/2)pn,~2

+ A[njn2(nl +kL)(n2+k2)]l/2

)

A2[n2(n2+k2) ]t/2 +

P~,n2 ] l o s s = C I [ ( h i + I ) ( n t + k l + 1 ) ] ] / 2 p m + l , ~ 2

D,~,+, ,~2+~

+ (n2+ 1 +k2) [a, (n, + I ) +a2(n2+ 1 ) ]/8+i~k2/2

8Din + ~ n2+ I

evaluated for the two-photon laser• To obtain the full equation of motion for the density matrix, we must add the loss terms to eq. (4). These are given by

,Dn~n2 ] pump = --

+ (n2+ 1) [tr, ( n , + 1 + k , ) +or2(n2+ 1 +k2) ]/8

i

1 September 1989

where D~,,,: = I + a ( n , n2 + K / 2 ) + ( a K / 4 ) 2 and K = n, k2 + n2 k, + k~ k2. The quantities A, B and tr are as defined above eq. (5) but with g~ and 82 replaced by g'. The damping terms are the same as for the FMHA. We might expect EHA to bc a good approximation when 13-1>> 1, for then the transition from 12) to 10) would be predominantly two-photon in character. On the other hand, for 151 << 1, we would expect profound differences between the microscopic and effective models, for in this range of detunings the stepwise process 12) ~ I 1 ) --. I0 ) will be as important as the two-photon one. Such small detunings may be realisable in microlasers. Important quantities for which the microscopic and effective hamiltonian models lead to different estimates in general are the mean photon numbers in the steady-state, which are determined by the diagonal elements Pn, n2(0, 0; t--. 0o ) only. Henceforth we assume that all the laser parameters are equal i.e. A, =A2, B~ =B2 and C, =6"2. This is not essential, but is done to avoid unnecessarily complicated expressions. There are several limiting cases where expressions for the mean photon numbers are particularly simple. We define/~=A,/C~ =A2/C2 as a measure of the

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I September 1989

pumping rate, and giving consideration first of all to the case I~1 >> 1, we define the quantity x=#/413-1 as a measure of how far the laser is operating above threshold. ( x = 1 is the condition for threshold.) We find

effectively zero. Under condition (B), the power broadening equals the detuning. In the opposite limit, I~-I-.0, we find

/~a( I~1 + 2 a x ) atria= 2 ( l ~ l + 3 a x ) '

Well-above threshold,/t >> 1, we again have a2 r~2 = 2a~ rh as in condition (C), confirming that in case (C) the detuning is effectively zero. It can be shown directly that in the limit (16), the diagonal elements of the density matrix obtained from the EHA approach satisfy the same equation of motion as those obtained from the microscopic hamiltonian. This property does not hold for the off-diagonal elements. Quantities such as the linewidths and cross-correlation coefficient which depend upon the off-diagonal elements may be calculated as in ref. [ 1 ] by evaluating the time derivative of the quantity

/zce( I~-I+ 4 a x ) cr2r~2= 2 ( l ~ l + 3 o t x ) ' (ll)

where a = 1 + ( 1 -I¢ -2) ~/2 varies from 1 at threshold to 2 far above the threshold. The limiting value of these expressions varies according to the relative values of I~1 and x, for (A)

1
I~'1 ,

(12)

(71 ?~"~ 0"2~2 " ~ / ~ / 2 ,

for(B)

1 << x = Ib~l , alril "--5/z/7,

for(C)

(13) o'2ri2"--9/t/7,

1 ,m I~-I << x ,

(14)

a~ff, = 2 ( / ~ - 3 ) / 3 ,

a2~2=2(21t-3)/3.

y ( t ) = ~ pn,n2(k~,k2)

(17)

(18)

rt I n2

alrT, "-2/~/3,

a2ri2~4/t/3 .

For the EHA (with i~e=A/C), eqs. (9) and (10) give r~, =a2 = [/re+ (lt~ - 4 a ) ' / 2 ] / 2 0 .

(15)

The condition for real mean photon numbers is /4, >t x/~-a. Thus the EHA predicts that r~, = r~2 under all conditions (as must be the case for a purely twophoton transition). Well above threshold, eq. (15) implies an, =ar~2 = m . O f the FMHA results ( 12)(14), this agrees only with eq. ( 12 ) in the limit x : ~ 1, when ot ~ 2. Thus for the EHA to be a good approximation, it is not sufficient for the single photon detuning I~1 to be large: we must also require that the laser be operating above threshold, but not too far above threshold. Precisely, we require that x and I~1 satisfy the conditions l , m x ~ < Ib'l. We may use the definitions of x and Ib~l to write this condition as 1612:~ Igl 2r7>> 1617.

(16)

The first inequality may be interpreted as stating that the power broadening (or the Rabi splitting) must be small compared with the detuning and the second is the condition for the laser to be operating wellabove threshold. When condition (C) holds, the power broadening dominates the detuning i.e. I~1 is

using eq. (4). By manipulating the sums on the right hand side it may be written in the form f = u(k~, k2)y where R e [ u ( l , 0)] determines the linewidth of mode 1 ( I m [ v ( 1, 0) ] determines the frequency shift), Re[u(0, 1 ) ] the linewidth of mode 2, and R e [ u ( l , 1 ) - v ( l , 0 ) - u(0, 1 ) ] / 2 is the crosscorrelation coefficient. The expressions we obtain are complicated, and we look for some simple limits. First we consider case (A). In terms of the quantities A and a of the EHA we find for the linewidths b~ and b2 and cross-correlation coefficient b,2 of the two lasing modes

b, =A( 2fl2 + f l - 2 ) /8fl2 + C/8ri , b2 =A ( 2fl 2+ fl+ 2 )/832 + C/ 8a , b~2 =A ( 2 f l - l ) /gfl ,

(19)

where fl= l + a r i 2 and a~ =ri2-ri... For the EHA

b]=~(A+C/a)=b~,

b~,2=A/8.

(20)

It is clear that from these expressions that well-above threshold, ,8>> 1 (but 1 <
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OPTICS COMMUNICATIONS

Another limit which gives reasonably simple expressions is ~=0. In this regime, the EHA approach is obviously inappropriate. We find when n~ and n2 are sufficiently large, that we again have near critical cross-correlation, b,-~o~K 3 ,

b,_~-a~K.~, b,2~-a,a~.K3.

(21)

It can also be shown by direct calculation that the off-diagonal density matrix elements obtained from the EHA do not tend to those obtained from the FMHA, even in the limit (16). As indicated, we would expect the EHA to be a good approximation under conditions ( 16 ). The fact that it fails for the off-diagonal elements is at first surprising. The origin of the discrepancy may be traced to the neglect of Stark shifts in the EHA. In a three-level atomic model, the two ouler levels have their energies shifted by interaction with the intermediate state. These shifts are small, and indeed they may be neglected when considering properties dependent on the diagonal elements, but they are crucial in calculating such properties as the linewidths and frequency pushing/pulling. If the Stark shifts are correctly built into the EHA, then this approach is adequate to describe the two-photon laser under conditions (16). These shifts have been neglected in previous applications of the EHA to the two-photon laser. In summary, we have considered a three-level model for the two-photon laser under conditions of two-photon resonance. Its properties have been investigated under conditions where the stepwise process is important, and where the two-photon process dominates. We have shown that the EHA yields a good approximation for the diagonal elements of the field density matrix when the laser is operating well-

50

1 September 1989

above threshold and the intermediate level detuning is much larger than the power broadening of the transitions. It is however, in its standard formulation, a poor approximation for the off-diagonal elements even under these conditions. For example, it yields expressions for the laser linewidths which underestimate the correct values by a factor of two (well-above threshold). However, both models predict near critical cross-correlation.

Acknowledgement One of us (AWB) wishes to thank the Department of Education for Northern Ireland for financial support.

References [ 1 ] S. Swain, J. Mod. Optics 35 (1988) 103. [2] B. Nikolaus, D.Z. Zhang and P.E. Toschek. Phys. Rev. Lelt. 47 (1981) 171: G. Grynberg, E. Giacobino and F. Biraben, Optics Comm. 36 ( 1981 ) 403: J.Y. Gao, W.W. Eidson, M. Squicciarini and L.M. Narducci, J. Opt. Soc. Am. B 1 ( 1984 ) 606. [ 3 ] M. Brune, J.M. Raimond and S. Haroche, Phys. Rev. A 35 (1987) 154; L. Davidovich, J.M. Raimond, M. Brune and S. Harochc, Phys. Rev. A 36 (1987) 3771; M. Brune, J.M. Raimond, P. Goy, L. Davidovich and S. Haroche, Phys. Rev. Lett. 59 (1987) 1899. [4] M.O. Scully, K. Wodkiewicz, M.S. Zubairy, J. Bergou, Ning Lu and J. Meyer ter Vehn, Phys. Rcv. Left. 60 ( 1989 ) 1832. [ 5 ] Shi-Yao Zhu and Xiao-Shen Li, Phys. Rev. A 36 ( 1987 ) 3889: Shi-Yao Zhu and M.O. Scully, Phys. Rev. A 38 ( 1988 ) 5433. [6] M. Sargent, M.O. Scully and W.E. Lamb, Laser physics (Addison-Esley, Reading, MA, 1974). [7] B.J. Dalton and P.L. Knight, J. Phys. B 15 (1982) 3997: Optics Comm. 42 (1982) 411.