Quantum noise of lasers with multi-photon absorption and reduce pump noise

OPTICS COMMUNICATIONS

Optics Communications85 ( 199t ) 275-282 North-Holland

Full length article

Quantum noise of lasers with multi-photon absorption and reduced pump noise Ulrike Herzog Central Institute of Optics and Spectroscopy, R udower Chaussee 6, O-1199Berlin, Germany

Received 2 January 1991;revised manuscript received 22 April 1991

The internal and external quantum noise of a laser whose resonator contains an unsaturated multi-photon absorber is investigated in dependence on the degree of pump noise suppression by means of Langevin operator equations. It turns out that the quantum noise of the laser output can be diminished due to multi-photon absorption if the reduced pump noise still exceeds a certain sub-poissonian value. At a definite ratio of the multi-photon absorption losses and the single-photonresonator losses the external quantum noise reduction normalized with respect to the shot noise level reaches its largest possible value.

1. Introduction The internal quantum noise of lasers is determined by the time-dependent correlation of intracavity photon number fluctuations or, if one is not interested in its spectral properties, by the photon statistics inside the resonator. From the practical point of view, instead of the internal noise one has to consider the external quantum noise of the laser output, which depends on the internal noise and on the statistics connected with the transmission through the output mirror. Far above threshold the quantum noise of usual lasers with classical (poissonian) pump is characterized by poissonian intracavity photon statistics and shot noise limited external noise resulting in a flat spectrum of photocurrent fluctuations detected by a broadband photodetector. The problem of reducing the laser quantum noise below these classical limits has been an object of much interest for many years. For the case of large intracavity photon numbers two basically different methods have been discussed for that purpose: Firstly it has been shown theoretically that the quantum noise can be diminished by two- and multi-photon absorption or other nonlinear processes inside the laser resonator [ 1-4]. The second method of laser quantum noise reduction consists in pump noise suppression [ 5-16 ], which could be achieved by pe-

riodic pumping of atomic lasers with separate time scales of excitation, emission and photon lifetime inside the resonator [ 5-7 ], by pumping of lasers and masers with a regular beam of excited particles being distributed more regularly than in the poissonian case [9-11 ], by high-impedance suppression of laser diodes resulting in reduced fluctuations of the pumping current [ 12-14 ], or by laser pumping with squeezed light [ 15 ]. For the case of semiconductor lasers, quantum noise reduction below the shot noise level has already been verified experimentally [13,14]. It is the aim of the present paper to discuss the combination of both methods of noise reduction. For this purpose the quantum noise of a laser resonator containing an unsaturated broadband multi-photon absorber is studied in dependence on the degree of pump noise suppression. The laser is assumed to be either a semiconductor diode laser or an atomic laser with very short lifetime of the transition dipole moment and of the lower state of the laser transition. The latter condition prevents radiation reabsorption and subsequent spontaneous emission, which would diminish the effect of pump noise suppression [ 16 ]. We use the Langevin operator equations for the photon number and also for the population inversion, which need not be adiabatically eliminated in our approach as it is in the extended Scully-Lamb

0030-4018/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

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laser theory [ 3 ]. Linearizing the Langevin equations we are able to derive analytical results corresponding to those obtained in ref. [ 3 ] in the appropriate limiting cases. In contrast to ref. [ 3 ] we put special emphasis on the external quantum noise properties that are revealed in the noise spectrum and in the photon statistics of the outgoing light detected by a broadband photodetector. Especially we derive the conditions under which it is possible to reduce the external quantum noise resulting from incomplete pump noise suppression still further with the help of additional intracavity multi-photon absorption. Moreover, we give an analytical expression for the smallest possible amount of normalized external quantum noise achievable by intracavity multi-photon absorption at a given level of pump noise suppression.

1 September 1991

spondence between the description of semiconductor lasers and of atomic lasers with fast relaxation of the lower laser level has been discussed in ref. [ 16 ]. F, and FN are the fluctuation operators of the usual Langevin laser equations, Fk is the additional fluctuation operator resulting from k-photon absorption, and P and 7 denote the pumping rate and the longitudinal relaxation rate of the laser transition. We neglected spontaneous emission into the lasing mode supposing that ( a + a ) >> 1 and we used the abbreviation G(N) for the laser gain G ( N ) = w2~ (N) - w,2(N), where w2~ and w~2 are the rates of stimulated emission and absorption. In order to introduce the k-photon absorption term into eq. ( 1 ) we made use of the density-matrix master equation of unsaturated k-photon absorption [ 17 ], which can be written as

j b = - ~ Xk(a+kakp--akpa+k+h.a.) ,

(3)

2. Basic equations We consider a single resonator mode with annihilation and creation operators a and a ÷ which is in resonance with the active laser medium and with a broadband unsaturated k-photon absorber ( k = 2 , 3 .... ) characterized by the loss constant Xk. Furthermore we phenomenologically take into account radiation losses that are due to transmission through the output mirror with the output mirror loss rate x and also to possible additional single-photon losses x' caused by intracavity absorption or scattering and by unwanted transmission through the back mirror. When we assume that the transition dipole moments can be adiabatically eliminated we obtain the following Langevin equations for the photon number operator n=a +a and for the operator N which refers to the number of charge carriers in semiconductor lasers or to the population inversion of atomic lasers with fast relaxation of the lower level of the laser transition:

and which is valid when the longitudinal and transverse relaxation times of the k-photon absorber are small in comparison to the inverse of the cavity bandwidth x. The master equation yields the expectation value rate equations

(h)=--xk(n(n--l)...

[n-(k-l)])=--(O)

(4)

( h 2 ) = - - x k ( n ( n - - 1 ) ... [ n - ( k - 1 ) ] ( 2 n - k ) )

N=P-TN-G(N)n+FN.

(1) (2)

Without the last terms of eq. ( 1 ) describing k-photon absorption these equations are well known from the literature (see, e.g., refs. [6,16]). The corre276

,

(5) from which the drift term and the diffusion coefficient of the contribution of k-photon absorption to the Langevin equation ( 1 ) can be derived. Applying the Einstein relation valid for stationary markovian processes (see, e.g., ref. [18]) we find the correlation function

( Fk(t)Fk(t' ) ) = ( ( h 2) - 2 ( n O ) ) 8 ( t - t ' ) = kKk(n(n--1)... [ n - ( k - 1 ) ] ) 8 ( t - t ' ) .

(6)

h= - Oc+x' ) n + G ( N ) n + F,, -x,n(n-l)...[n-(k-l)]+Fk,

,

The other nonvanishing correlation functions of the fluctuation operators occurring in eqs. ( 1 ) and (2) are given by (see refs. [6,19] )

(F,(t)F,(t')) =((r+x')n+(2nsp-1)G(N)n)6(t-t'),

(7)

(Fu(t)FN(t') )

(An)'= -(k-I)xka~-~An+TzaAN+Fn+Fk,

= (apP+TN+ (2nsp- 1 )G(N)n)O(t-t' ),

(AN)'= -(x+x'+xkak-~)An--y(I+za)AN+FN. (16)

= - (2nsp - 1 ) ( G ( N ) n ) 6 ( t - t ' ) ,

(9)

where we introduced the population inversion parameter nsp.~(w21)/((w21)--(Wl2)) [12]. In atomic lasers with fast relaxation of the lower level the absorption rate (WI2) c a n be neglected and n~p is equal to unity, whereas in semiconductor lasers n~p~ 2 [ 12 ]. The parameter ap occurring in eq. (8) characterizes the statistics of the laser pumping process. The value ap= 1 corresponds to classical (poissonian) pumping, ap< 1 describes sub-poissonian pumping, i.e. pump noise reduction, and ap=0 is valid for complete pump noise suppression. To investigate the steady-state fluctuation properties we linearize the Langevin operator equations with the help of the Ansatz N=N+AN,

where r i = ( n ) and AT=(N) are the steady-state mean values of the photon number operator n and of the operator N for the population inversion or charge carrier number, respectively. Since the photon number distribution of the laser is expected to have a marked peak at a certain large photon number we suppose that t~>>l ,

( ( A n ) 2 ) - = A - ~ < < f i 2.

(15)

(8)

( F,(t)FN(t' ) ) = ( FN(t)F,(t' ) )

n=~q+An,

3. The intracavity quantum noise Taking into account the fluctuation operator correlation functions ( 6 ) - ( 9 ) we are able to calculate the two-time intracavity photon number correlation function (An(O)An(r)) by solving eqs. (15) and ( 16 ) by means of Fourier transformation. Using the Ansatz

(An(O)An(r))=-~

G(N)=zT(N-NT) ,

where Z is the saturation parameter and NT denotes the number of charge carriers that are necessary to reach transparency in semiconductor lasers. For atomic lasers, the gain is proportional to the population inversion N, and NT therefore has to be put equal to zero. Making use of eqs. ( 10 ) - ( 12 ) we obtain from eqs. ( 1 ) and (2) the stationary operation conditions (13) ,

(17)

and taking into consideration the stationary operation conditions ( 13 ) and (14), after some algebra we arrive at 2 (An2)~° = ~2 1 n

(

+ m 2 1+

~ ( 1+ x a ) ]

7(l+z~) 0.) 2

(11)

(12)

x+x' + Xk~k-' = Z T ( ~ r - N v ) ,

dm cos ogz(An2)o~, o

Furthermore we assume that the laser gain can be written as

1+ZrT= ( P - - ? N T ) / ~ ( N ' - - N T )

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Volume 85, n u m b e r 2,3

(14)

and the linearized Langevin operator equations

[ ×

l+ap ap+ Za

)V

+ 2 n s p ' l ( 1+ t°2)]} 7 ' (18)

where the abbreviation Za ( x + x ' ) + ( k X= Z--------~ 1+ _

1 "~ -k-, l----~Zajxkn

(19)

has been used. The first term on the right-hand side of eq. (18) reveals the resonance character of the photon number fluctuation spectrum, which plays an important role in semiconductor lasers due to the large value of the resonator loss constant x. The resonance character disappears when ~ << y( 1 +Xr~), i,e., either when the losses are small enough as is usually 277

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true for atomic lasers or when the laser is operated far enough above threshold resulting in a sufficiently large value of the saturation. In these cases the population inversion can be adiabatically eliminated from eqs. (15) and (16). Putting (AN)" equal to zero, solving eq. ( 16 ) for AN and inserting the result into eq. (15) we arrive at the Langevin equation •

Zr~

(An) = - ~ A n + F, + Fk + l ~ z a F x ,

(20)

which is valid for ~<< y( 1 +zri). In the following we consider atomic lasers where NT=0, nsp= 1 and ~<< ~,. Eq. (18) then can be approximated to yield a lorentzian frequency dependence of (An2),o, from which we find by Fourier transformation r/ f { %r/ ~21l+ap+ ( A n ( 0 ) A n ( r ) ) = 2 - ~ l - - + Z a ) Lap+ xh

1 S e p t e m b e r 1991

mechanisms remains unchanged. When r7 is large enough to ensure that single-photon losses are negligible in comparison to the k-photon losses, i.e. for ( x + x ' ) <
4. T h e e x t e r n a l q u a n t u m n o i s e

1 J

× ( x + x ' +Xka k-~ ) +K+X' + kxkrik- ~t exp( --a:l rl ).

(21) The same result could have been derived in a direct way by integration of eq. (20) making use of the correlation functions ( 6 ) - ( 9 ) and of the stationary operation conditions (13) and (14). In general, intracavity quantum noise results from fluctuations due to the pumping process, the loss mechanism and the interaction process itself. The contribution of the latter vanishes when the stimulated transition probability is so large that each excited atom will deliver a photon immediately after the onset of the interaction thus transmitting the pump statistics to the field without bringing about additional fluctuations. For the purpose of producing nonclassical light we have to consider just this case, which requires that Zri>> 1, x/7 [cf. eqs. ( 1 ), (2) and (12)]. Fromeqs. (21) and (19) we then obtain the approximation

When the outgoing laser light is detected by a broadband photodetector with efficiency ~/, the normalized mean photocurrent (in units of the elementary charge) is given by ( j ) = qxr/.

(23)

The connection between correlations of the photocurrent and of the intracavity field is described by the equation [ 19 ]

(j(O)j(r) ) =q2x2 ( a + (0)a + ( r)a( r)a( O) ) + qx~6( r) . (24) Since a + ( v ) a ( v ) = t/+ an(O, we find the following expression for the stationary photocurrent fluctuation spectrum: oo

( 8J 2 ) o~= _t dr cos o)r [ ( j ( 0 ) j ( z ) ) - ( j ) 2 ] 0

= i dr cos ~or o

(An(O)An(r))=2(1

+

K+K'+tCk~qk-' ~ ap x + x' "+ k l f k n k -- l ,]

× [q2x2(a+(O)An(r)a(O)> +rlxa6(r) ] . (25)

X e x p [ - (x+x' +kxkak-~)lrl] .

(22)

Obviously, the contribution of pump fluctuations to the intracavity photon statistics is reduced by k-photon absorption whereas the contribution from the loss 278

To evaluate the two-time correlation function occurring in the above equation we restrict ourselves to the case ~<
1 September 1991

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Volume 85, number 2,3

•

An..._.z

1

\\

"\\\

\ ~"

\

'

'

"\--

\ \"\4"

"'..

",,.

a

c.0.01

"/:....

,.oa

0.8

0.6

i

O.i,

\\

2<~2>='o

q>

5'0

~o

\

X\ I

i ~oo

~

b

\,,,

---_~.____~,~~

.....

o.,

0.4

~

0~

'c-0.01 1 ~o

g

i

si~

loo

x~

Fig. I. (a) Normalized intracavity photon number variance An2/a and (b) zero-frequency value 2(6fl)<~=o/(./) of the normalized external noise spectrum with t/= l, x' = 0 for intracavity two-photon absorption in dependence on the degree of laser saturation Xr7for complete pump noise suppression (full line) and classical pumping (dashed line) for different values of the parameter c=x2/Xx.

quantum fluctuation-regression theorem valid for stationary markovian processes we can write

used. For atomic lasers with ~<
( a + (O)An(z)a(O) )

2 <8J2>°" - 1 + 2 qx

= (A-n-~ - a ) ( An(0)An ~ - ~ (z) > ,

~

-I

(D 2-l- ~ 2 '

(27)

(26)

where the commutation relation [a, a ÷ ] = 1 has been

where 17 is defined by eq. (19). The above expression describes the external quantum noise spectrum, 279

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OPTICS COMMUNICATIONS

which is normalized with respect to the shot noise level. For sub-poissonian intracavity statistics the external quantum noise is reduced below the shot noise level and takes on its smallest possible value for o9=0. In fig. lb, for intracavity two-photon absorption the zero-frequency value of the normalized external laser noise spectrum is depicted for ideal detection (q = 1 ) and vanishing additional single-photon losses ( x ' = 0 ) in dependence on the degree of saturation zn for classical pumping (ap= 1) and complete pump noise suppression (ap=0). With growing degree of saturation or laser excitation, respectively, the noise first goes down since the intracavity quantity ~ / ~ is diminished (see fig. la). On the other hand, the noise reduction is proportional to the term x/~ occurring in eq. (27), which for sufficiently large values of vi decreases with growing values of fi [see eq. (19)] and therefore the external noise starts growing again. Physically, this increase of external fluctuations is due to the fact that a growing fraction of photons will be removed by twophoton absorption in a stochastic way before the light can leave the resonator. Let us now consider the limiting case xn >> 1, which is most interesting from the point of view of nonclassical light production. For k-photon absorption we then obtain from eqs. (27), (19) and (22)

1 September 1991

0.8[

v

0.01

005 0.1

(j)

--1 '

{

X+X'+Xk~ k-l

tq~apx+x,÷kxknk_

x(x+x'

X t_o2+ (/¢_l_x, .1_kK.k/~k_1) 2 .

5

10

At g+

( M ) = ~ d t ( j ( t ) ) =qmqAt,

(29)

0

At

=

2

At

d, f d,' t- 21, 0

(30)

0

(28)

Without any intracavity losses (x' =0, Xk=0), for complete pump noise suppression ( a p = 0 ) and r/= 1 the external noise spectrum goes to zero in the lowfrequency limit, whereas it is always larger than zero in the presence of additional intracavity k-photon absorption. On the other hand, for classical pumping (ap= 1 ) the external noise is reduced by intracavity k-photon absorption. In fig. 2 the zero-frequency value of the normalized external noise spectrum following from eq. (28) in the ideal case r/= 1, r ' = 0 is shown for Xa>> 1 in dependence on the relative strength of intracavity k-photon absorption (k = 2, 3, 5 ) for the case of classical pumping. For each value of k the noise reduction takes o n its largest possible 280

1

value for a definite relative k-photon absorption strength rk~ k- ~/X. If the external noise is measured by direct photon counting instead of spectrally analysing the photocurrent the detected photon statistics crucially depends on the length of the counting interval At. Since we consider stationary laser action we can write

, --1 ]

+ k x k a k - ~)

0.5

Fig. 2. Zero-frequency value 2 ( ( 3 j z ) , o = o / ( j ) of the normalized external noise spectrum with q = 1, x' = 0 in the high-saturation limit X~ >> 1 in dependence on the relative strength xka k- ~/x of intracavity k-photon absorption for k = 2 , 3, 5, and ap= 1.

( A M 2 ) = ( M 2 ) -- ( M )

2 (5J2)'°

,

with M being the operator for the number of photons detected during the time interval At. Making use of eqs. (24), (26) and (21 ) we find

_l+2-f,

-

-1. (31)

In the limiting approximation

case

(AM2-----~) ( M ) ~ 1 + ( M~ ) ( ~

-

At << if- J we

-1 ) ,

get

the

(32)

which gives the same connection between internal and external photon statistics as would be obtained by describing the outcoupling mechanism and the

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OPTICS COMMUNICATIONS

detection losses as single-mode one-photon absorption processes. Since (M)<> ~ - ' ) eq. (31 ) yields the approximation (AMZ-------~)- l + 2 q x ( A ~ n n_ - 1 ) = 2 (m) ~

(~J2)°J=°

(j) (33)

Obviously, in this case the degree of suppression of the external photon number variance ( AM 2) below the poissonian level ( A M 2) = ( M ) is the same as that of zero-frequency noise suppression below the shot noise level [cf. eq. (27) ]. From eqs. (33), (19) and (22) we conclude that in the special case r/= 1, x' = xk = 0, a o = 0 and Za >> 1, when the outcoupling process is the only noise source, all photon number fluctuations are smoothed away at counting intervals At >> K- '.

5. Discussion

With the help of our formulae we are able to investigate the conditions under which additional intracavity k-photon absorption reduces the external quantum noise observed for a definite degree of pump noise suppression in the limit Za>> 1. Comparing the result ofeq. (28) for Kk=0 with that for Kk~ 0 we find after some algebra that intracavity kphoton absorption can only cause further noise reduction if ap>k/(2k-

1 ) =apo.

(34)

It is interesting to note that the limiting parameter of pump noise suppression avo defined in the above expression is the same as the asymptotic value which would be obtained without a resonator after sufficiently strong k-photon absorption for the relative photon number variance ~ / a referred to the mode volume [ 20 ]. From eq. (28) we furthermore derive the fact that, even if the pump is still noisy enodgh for the condition (34) to be valid, additional external noise reduction actually occurs only in the case that the k-photon absorption is not too strong. The relative k-photon absorption strength has to obey the inequality

- -

tc+x'

<

1 September 1991

-

-

k(1 - a p )

--ap-1

(35)

Supposing the conditions (34) and (35) to be fulfilled, the question arises as to what is the smallest possible amount of external noise achievable at a definite pump noise level. By differentiation of eq. (28) we find that maximum noise reduction occurs if x+x'

ap-1

- k-ap

.

(36)

The resulting minimum zero-frequency quantum noise is given by 2 (8J2)'°=° - 1 (j)

r/x

(k-ap) 2

x--------7 +x 4k(k-

1 )ap "

(37)

From eq. (36) we conclude that without pump noise suppression (for ap= 1 ) maximum external noise reduction is achieved i f X k a k - ' / ( X + X ' ) = I l k , i.e., if the single-photon losses of the field are k times as large as the losses due to intracavity k-photon absorption (cf. fig. 2 ). According to eqs. ( 33 ) and ( 37 ), in this case under the ideal conditions r/= 1, x' = 0 the zero-frequency external quantum noise (~j2)w= 0 and the photon number variance ( A M 2) detected for large counting intervals are reduced below the shot noise level and below the poissonian value (AM 2) / ( M ) = 1, respectively, by a factor of (3k+ 1)/4k. This is in accordance with the result derived in ref. [4] for a classically pumped laser containing a two-photon absorber ( k = 2 ) . Fig. 3 shows the normalized external noise spectrum for the ideal case q= 1, ~' = 0 for classical pumping (ap= 1 ), pump noise reduction with ap=0.9 and total pump noise suppression for different values of the relative strength of the intracavity two-photon absorption. Maximum zero-frequency noise reduction is achieved when x 2 a / x is equal to 0.5 for ap= 1, to 0.32 for ap=0.9 and to 0 for ap=0.

6. Summary

It has been shown that noise reduction for the laser output can be achieved by a multi-photon absorber placed inside the laser resonator provided that the 281

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2<8~2>w q> 1

1 September 1991

of the multi-photon absorption losses and the singlephoton losses of the field inside the laser resonator. The latter are always present at least due to the outcoupling process. The multi-photon absorption losses depend on the mean intracavity photon number and can be made to fulfill the condition for maximum external noise reduction by properly choosing the above threshold ratio of the laser.

b-0

0.95 -0.5 "b . t ,

O,9 0.85

o

2

3

6

Acknowledgement

=/=¢

The author is indebted to Prof. H. Paul for stimulating discussions.

1 b°0

q>

o.9s

b'3

References 0,9

b=l

O.85

0

i

3

1

~j>

0.8

b.1

0.6

0,~ 0.2 0

0

1

2

3

~

5

Fig. 3. Normalized external noise spectrum 2(gj2)~/(j) with q= 1, pc'= 0 in the high-saturation limit xn>> 1 (a) for classical pumping (ap= 1 ), (b) pump noise reduction with ap=0.9 and (c) complete pump noise suppression (ap=0), for different values of the relative strength b= K2fi/K of intracavity two-photon absorption.

pump noise of the laser exceeds a certain sub-poissonian value. Maximum noise reduction below the shot noise level occurs at a definite value of the ratio

282

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