V O L U M E 67, N U M B E R 24 P H Y S I C A L R E V I E W L E T T E R S 9 D E C E M B E R 1991

Enhancing the Sub-Poissonian Character of a Light Beam from the Down-Conversion Luminescence

Geraldo Alexandre Barbosa and Carlos Henrique Monken Departamento de Fisica, Instituto de Ciencia Exatas, Universidade Federal de Minus Gerais,

CP 702, CEP 30161, Belo Horizonte, Minus Geruis, Bruzi&^ (Received 11 January 1991)

Is it possible to number squeeze an already sub-Poissonian electromagnetic field? This work presents a system able to squeeze the sub-Poissonian photon statistics of a beam from the down-conversion luminescence produced in a nonlinear crystal by laser pumping. Parameters are shown to stress the ex­perimental difficulties inherent to the process. Even if this sub-Poissonian proposition for the down-converted light state turns out to be unrealistic, the proposed system is, in principle, a singular tool in noise reduction processes.

PACS numbers: 42.50.Dv, 03.65.-w, 42.65.Ky, 42.80.-f

Frequently, the knowledge of the distribution pin) of a given source of light is not a trivial problem. This difficulty is often due to the short coherence time of the light source, as samplings must be taken in a still shorter time. An even harder task is to manipulate or control p{n). This can be seen in the simple statement that even a field close to the Fock state of light has not been experi­mentally obtained with an arbitrary degree [1] of squeez­ing. The Fock state has stood just as a theoretical reali­zation.

The proposed experimental configuration consists of a laser operating well above threshold (photon energy HCOQ) with its output beam incident on a nonlinear crystal and the selected down-converted conjugated beams focused into a filled Fabry-Perot (FP) cavity [2] (Fig. 1). This down-converted luminescence [3] produced in the non­linear crystal, with wavelengths corresponding to Wa and (Oh, can always be chosen such that (OQ='(Oa~^(Ob and ko = k^+k/,. These frequencies are chosen such that (Oa is not absorbed and o)h is in resonance with the medium. Once the number of photons in both wavelengths fluctu­ates in a completely correlated way, by judiciously choos­ing the cavity length and the average incoming intensity, the dielectric fluctuation induced by the resonant wave­length will change the cavity transmittance in opposition to this fluctuation. In this way, the statistics of the transmitted beam of color (Oa may be modified, leading to a narrower pin). This is a purely dispersive effect with respect to the Wa photons, as explained throughout the paper. In short, an active quantum medium within a FP cavity senses the photon-number fluctuations in beam b,

nonlinear crystal MOW

/ ^ " ^ ^ - " ^ ' - - • - a - - ^ ^3. iiiiHiiiiiin b.s.p.

optical u u Isolotor i'*®'' 'ens ' coj,

FIG. 1. Basic experimental setup. A laser pumps a nonlinear crystal and the down-converted luminescence (type II matching conditions) interacts with the MQW cavity. Output beam with wavelength Xa has an enhanced number squeezing, under a set of specified conditions.

changing its refractive index. Consequently, the effective optical path for coa photons will be changed, and also the FP transmittance. This change in the transmittance of the FP cavity, if properly tuned to cOa (on the left-hand side of the (o transmittance curve, near 75% transmit­tance and for an appropriated cavity length L and ade­quate (A2>), will be in the opposite sense to the photon-number fluctuations.

The correct quantum state associated with the down-conversion process is not completely established yet. However, a solution |^,0) developed [4] for a slightly different situation describing down-converted lumines­cence in a stationary regime with nonlinear damping will be associated with these two colors as

ICO>=7Voi:-^h(.),«(M>, rt=o n\

(1)

where ^ is the eigenvalue of the product of the two an­nihilation operators a and b corresponding to the two wavelengths. This eigenvalue ^ is given by the physical process within the nonlinear crystal, specified by a gain G and a loss AT, as f=—2/G/Ar. A o is the normalization constant and is given by Ao = [/o(2/|C|)] ~''^^. This choice |f,0> also implies that the probability of finding n photons in mode a or 6 is

p{n)^N^\i;\^Vn\\ (2)

which is already sub-Poissonian. While our results in this work will be valid for a general distribution pin), the specific form (2) above can be compared to other more common statistics.

The surfaces of the FP cavity can be made with high reflectivity for Xa and low reflectivity for Xt- For exam­ple, a multiple-quantum well (MQW) could be sand­wiched between these mirrors. Let us consider that the first excitonic level is set at 1.513 eV (a typical GaAs value), matching the photon energy in beam b iXb =8195 A) . With h(Oo = 3.0 eV, the energy of photon a is defined and is equal to h(Oa = \ASS eV (X«=8332 A) . This choice for hcoo (Xo==4131 A) coincides with a known las-ing line of krypton, while the wavelength Xa is then in the band gap of the MQW, with negligible absorption. The

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V O L U M E 67, N U M B E R 24 P H Y S I C A L R E V I E W L E T T E R S 9 D E C E M B E R 1991

absorption [5] at Xa, through the entire MQW length, is much smaller than 1% and so the resulting squeezing ab­sorptive degradation is also very small. Of course, the MQW samples must be carefully constructed to achieve this small degree of absorption.

The active layers are supposedly separated by inactive layers that are transparent to both wavelengths with a band gap at a higher energy (1.8 eV for AlGaAs). The cavity and MQW length L is set such that the average transmittance is 0.75 for A, .

Excitons will be created by light absorption and sub­jected to several interactions besides light-stimulated pro­cesses. Vacuum fluctuations will produce spontaneous decay (10 ~^ s) of excitons while interaction with the strong electric field associated with the longitudinal opti­cal phonons might ionize [6] them, that is to say, break the electron-hole bound state feeding the conduction band with the corresponding free charges. This ionization is a very fast optical process ( ^ 1 0 " ' ^ s) [5] and is highly temperature dependent. At low temperatures this process is absent as thermal phonons are not created.

In the vicinity of the first excitonic peak at ft>i a semi-classical susceptibility x can be written as [7,8]

d^ (7V|-7Vi)/K

£o2h co — co\-\-iy (3)

where the normal mode of the FP cavity was averaged to I/V2. d is the exciton transition electric dipole moment, (TV I —N\)/V gives the density of excitonic population in­version in the MQW, and (0\=o)b is the exciton lowest frequency. / is the damping coefficient at the frequency CO, which will be set equal to zero, since the absorption at (Oa along the entire MQW length is negligible [5].

Nonlinear effects arise when N\—T^\, coi, or 7 is changed by the optical excitation. It has been established [8] that a saturation density Ns of excitons within one layer exists for a MQW and is given by Ns^nre) of the order of 10 ~^ where ;rr/ gives the effective exciton area. Within an area ^ = 20 jum^ corresponding to a beam di­ameter of 5 jum and for an exciton radius of = 1 0 0 A, the maximum allowed number of excitons is then ANs ^NQ ^10"* excitons. A high optical excitation with frequency co=(Ob'^co\ can create an appreciable number of exci­tons, that is to say, can change 7V| — TVj correspondingly. This source of nonlinearity is utilized in this work. It has been experimentally studied [5] by optical femtosecond techniques. Several physical mechanisms are responsible for it, including Pauli's exclusion principle and charge screening effects [8].

The choice for the mirror transmissivity at Xb, riXb) = 1, and the mirror reflectivity at Xa, p(Xa) — \y will let the Xb photons interact with the MQW for a time of the order of one flight time T\ while the Xa photons will be interacting with the cavity for an average classical time [7] Tf=T\/Ta, much longer than T\.

The interaction of the Xb photons will actually take into account the absorption in order to create excitons, while

the Xa photons will be considered as just suff'ering phase changes within the cavity, that is, an approximation con­sistent with the absorption spectrum.

To deal with the quantum aspects of the interaction of the cavity with the Xa photons, an idealized plane Fabry-Perot symmetric cavity 'Vacuum-medium-vacuum" is as­sumed with a reflectivity p(Xa)=p associated with the surfaces. These idealized surfaces are seen as lossless beam splitters [9], while the medium is the MQW.

Assuming that the light has a phase shift d due to propagation between mirrors, a final matrix transforma­tion for the filled cavity results:

(4)

where ar and dp represent annihilation operators for the transmitted and reflected Xa photons, while ar represents vacuum fluctuations entering the exit port, r ^\ —p and // and V are given by

re'' ^(e^''-\) .^. = v = — ^ r- . (5)

^ r

.^P.

= a

ar

A^ = , 2iS ' .2 /5 \-pe'" \-pe' From expression (4) the annihilation operator a can be

written as

a = -jiar van

(6)

The difference in interaction time of the Xa and Xb pho­tons suggests, as a simplified procedure, that a separate theoretical treatment can be done for each of them. For the Xb photons, with short interaction time (low-g cavity) but a strong interaction, a thoroughly quantum treatment will be applied while for Xa photons (high-g cavity) a re­fractive index derived from Eq. (3) will be considered. The superposed interaction time when both photons are present will be neglected, considering first the short in­teraction time with the arriving Xb photons and then the much longer interaction time with the Xa photons.

The existence of a saturation density [8] TV for a layer allows one to associate to every area of size A/NQ the numbers one or zero to represent a created exciton or a vacancy, respectively; that is to say, to each site is associ­ated a two-level system. Of course, N]+Ni=NoNi, where N] and 7V| give the number of occupied and unoc­cupied areas, respectively, and Ni gives the number of layers in the sample.

The interaction of n photons with all sites in the volume also will be simplified in a statistical approach where a typical /th site in the volume will be interacting with k photons and k is sampled within n. The sampling procedure establishes that when k photons are interacting with the /th site, n — k photons will be interacting with the remaining sites. A desired statistical result for the to­tal number of sites will then be taken as NQN/ times the result of the /th site, under the additional hypothesis that n^NoNj, which assures that absorption of the Xb beam throughout the MQW is not appreciable.

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VOLUME 67, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 1991

The simplified character of this approach can be seen in the association of an annihilation operator bi to the ith site instead of an operator b associated with the whole cavity volume or NQN/ sites.

Assuming light propagating normal to the MQW lay­ers, in the x-y plane, and the electric field operator given by E=Eo/y(6^ + Z?), where Eo = (hcob/eoV)^^^ and fikz) is the Fabry-Perot normal (lowest) mode con­sidered, the Hamiltonian for the ith site in the rotating

wave approximation (RWA) is

H^\s/\=^h(Obb^bi-\- J h(0\Oz+ jihqia^b^ — biO-) ,

(7)

where q =dEo/h. Hamiltonian (7) has a famous exact solution [10]

given in terms of the occupancy number k (for the pho­tons) and m = ± J (for the excitons), specified at the be­ginning of the interaction time:

< T C J . ( / ) ) / = < y c T . ( 0 ) > / l -2qHk-\-m+{)

A^-^q^ik-hm-hj) sinHn^t^ (8)

where ft^^, =A -\-q^(k-\-m + y ) and A=a)\ —cob. f The number of photons k striking a given site in the

area A is given by the probability law connected with the generation of n photons by the down-conversion lumines­cence together with their division into No identical sites. The probability of an interaction between one photon and one of the exciton sites in the active volume will be given approximately by I/NQN/ and the probability of k pho­tons to interact with this /th site will be given by the bi­nomial distribution:

p,(n,k) = n\ k\(n-k)\

1 NoNi

k r

1 -1

/VoTV/

In this way, the statistical value obtained at the /th site IS

A=0

TVt-A^I (9)

The phase shift 3 imposed on the XQ photons is given by

i5(0=A:(OL=(2;rLAJ>/iT^(7T, (10) and with x given by formula (3) with 7 = 0 and by Eq. (9), this value will be determined at desired time values.

For a rough estimate of q ^^dEo/h one uses

fi^= (electron charge) x (exciton radius) = 1.6x 10 ~^^ Cm

(|c7,(r,)) = - ^ i : 2k-

Iq^k n A^-\-q^k sm

k,-\ nT\

and Eo = ( f tWe()^^) '^^ = 3.0x10^ V/m, obtaining q ^ 4 . 5 x 1 0 " rad/s.

The coherence time of the down-conversion lumines­cence [1,3] generated within the crystal is supposedly r , . > 1 0 " ' ^ s. One transit time is T\=nL/c^XQ'^"^ s (for a 3-jum cavity) and gives approximately the lifetime of the Xb photons within the cavity. The Xa photons ar­riving with the Xb photons will interact with the cavity during Tf^(nL/c)/Ta^XO~^'^ s, which is long enough to feel the variations induced by the Xb photons. There­fore, the corrections in the Xa beam due to Xb photons are made within one coherence time. The total number NQN/ of exciton sites is estimated to be ~ 10^.

Adopting a recursive time development for the process, which includes the random arrival of photons at every r^, first the number of excitons created by the Xb photons is calculated, from 0 to T\. The random arrival of photons has to obey pin) given by formula (2). From T\ to 7 / the evolution of the exciton number with no Xb photons present is then calculated. Within this same time interval the phase change presented to the Xa photons is defined as given by expressions (3) and (10), with (o='(Oa.

Starting with no excitons present, the initial condition is ( y (TziO)) = — y and, after T\, one has

_ 1 Pi(n,k) = | w + ( r , ) ~ ^ > v - ( r , ) .

Of course, w-\-(t)-\-w-it) = \ and also w + (/) = ^-l-<^(j,(r)> and w-(t) = { - < | c T , ( r ) ) .

w-\-(T\) and w-(T\) define the initial conditions for the next time interval, from T\ to Tf.

i\o,{Tf r,)> = ^ w + ( r , ) iq:

A^ + q' -sm no,i/2(7/-r,)

y w - ( r i ) ,

generating a recursive process. The nonabsorptive character of the MQW for the Xa

photons establishes that when n (Xa) photons are incident upon it, transmitted rir and reflected ftp photons obey the condition n = fir-^ ftp.

A general incident state \X) can be written as

| A ' ) = i c „ | « ) = i c „ - ^ | 0 ) .

The operator a^ can be developed by taking Eq. (6), and by application of the binomial theorem it results that

p(ftr;ft) = \inp,nr;n\X)\^

= P ( A Z ) | A X | ' ' ^ 1 V | ' % ! / « , ! V - 01 ) As discussed above, p and v are n dependent and « is a

random variable that follows the statistical distribution pin), n will be sampled at every T/^^Zc producing a se-

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V O L U M E 67, N U M B E R 24 P H Y S I C A L R E V I E W L E T T E R S 9 DECEMBER 1991

16-

14-

O 12-

X 1 0 -

JsJ CL 6 -

4 ~

2 -

0 -

it V

1 ' 2785 2786 2787 2788

FIG. 2. Normalized photon count histogram over 2.8x10^ samplings for the output beam (dots) showing ((A«)^>/(«> ^ 0 . 2 8 . Solid lines indicate Poisson (broader) and Agarwal [formula (2)] distribution functions for the same kn) for com­parison purposes.

quence of numbers feeding Eq. (11). This step-by-step recursive procedure will give the final desired result,

cx>

p^n^:) = 2 pirir'.n) .

Along with these ideas a quite restricted set of parame­ters can be found such that the proposed system achieves the desired squeezing operation. A single choice will be presented here to show the feasibility of the system to squeeze an input field, even if this input field is already a squeezed one.

Figure 2 shows the normalized histogram for the out­put beam with enhanced number squeezing {(An)^)/{n) — 0.28. This result was obtained through a computer simulation starting from an input field with in) =3.1 x 10^ per sampling time of ^ 5 . 4 x 10~ '^ s and a total number of sites of 1.2x10^. Random number generators were used to produce the input n and the output rir according to formulas (2) and (11), respectively. Cavity and MQW parameters were those described throughout the text. The result obtained from this simplified or skeleton model indicates the system's adequacy to produce or enhance a sub-Poissonian state of light.

Several approximations were employed in this work but we believe that none of them would destroy the main corrective property of this system. However, they indi­cate that improvements could be made, for example, tak­ing a more complete solution for two colors and many two-level systems into account. Exciton cooperation numbers [11] and interaction among sites could also be included. A diagram with a complete set of adequate pa­rameters could be generated in order to give an overall view of possible experimental conditions. Nevertheless, our particular result is quite sufficient to establish the

ability of the proposed system to act as a singular tool in noise reduction processes.

This research was partially supported by the Brazilian agencies CNPq (Conselho Nacional de Desenvolvimento Cientifico e Technologico) and FAPEMIG (Funda9ao de Amparo a Pesquisa do Estado de Minas Gerais). C. H. Monken is a graduate student supported by a CNPq fel­lowship.

^^'>Electronic address: 5531.4481372 (FAX); op t ika^ brufmg (bitnet).

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