Statistical-noise properties of an optical amplifier utilizing two-beam coupling in atomic-potassium vapor

Transcription

1 PHYSICAL REVIEW A VOLUME 53, NUMBER 5 MAY 1996 Statistica-noise properties of an optica ampifier utiizing two-beam couping in atomic-potassium vapor W. V. Davis, A. L. Gaeta,* R. W. Boy Institute of Optics, University of Rochester, Rochester, New York G. S. Agarwa Physica Research Laboratory, Ahmeaba 389, Inia Receive 2 March 1995; revise manuscript receive 31 January 1996 We have measure the gain statistica-noise properties of a weak probe beam ampifie through twobeam couping in atomic-potassium vapor for both the Rabi stimuate Rayeigh gain features. The probe-beam gain was observe to be as arge as 125 for the Rabi feature, as the gain was typicay ess than 2 for the Rayeigh feature. The rms noise at the Rabi gain feature was approximatey three times greater than the iea ampifier quantum-noise imit. For the Rayeigh gain feature, the rms noise was 12 to 15 times greater than the iea ampifier quantum-noise imit. We present a fuy quantum-mechanica theory of twobeam couping in a system of two-eve atoms which incues the effects of atomic motion pump-beam absorption. The preictions of this theory are in goo agreement with the experimenta ata. PACS number s : 42.5.Lc I. INTRODUCTION Whie the semicassica theory of the interaction of matter with an eectromagnetic fie is often sufficient, a fuy quantum-mechanica treatment is usuay require when the statistica properties of the fie are of interest 1,2. Unike semicassica theory, which can reaiy expain effects such as photon bunching 3, a fuy quantum-mechanica theory is abe to expain a whoe other cass of noncassica effects such as photon antibunching 4, sub-poissonian photon statistics 5, the quantum-noise properties of optica ampifiers attenuators 6. A etaie unersting of the quantum noise of the fie after it interacts with matter is important, since the resuting statistica properties of the fie wi imit the accuracy of optica measurements. It has been shown that even an iea phase-preserving optica ampifier egraes the signa-to-noise ratio of the output fie by at east a factor of 2 reative to that of the input 6,7. In aition, it has been shown in the iterature that noncassica features ike squeezing of a fie are ost when the intensity of the fie is ampifie with a phaseinsensitive ampifier by more than a factor of 2 8. Since the two-beam couping gain can be as arge as 1, the statistica-noise properties of the transmitte probe beam are expecte to be cassica. This concusion is consistent with the preictions of the fuy quantum-mechanica theory of two-beam couping in a homogeneousy broaene system of two-eve atoms 9,1. For the case of gain through the Rabi feature, the theory preicts that the ampifier can operate at the iea ampifier quantum-noise imit when the atomic system is raiativey broaene. The minimum noise figure for the Rayeigh gain feature is equa to 4, occurs when the atomic system is preominanty coisionay broaene. The *Permanent aress: Schoo of Appie Physics, Corne University, Ithaca, New York experimenta measurements of the noise properties of a probe beam ampifie through two-beam couping in atomic vapors, however, cannot be aequatey expaine by this theory without moifying it to incue the effects of atomic motion. In this paper, a fuy quantum-mechanica theory of twobeam couping, incuing the effects of atomic motion, is presente 11. This theory is use to cacuate the noise factors neee to etermine the quantum-mechanica noise properties of the transmitte probe beam. The preictions of the theory are compare with the experimenta measurements of the gain noise of the transmitte probe beam. II. EXPERIMENTAL RESULTS The experimenta setup use to measure the statisticanoise properties of the transmitte probe beam is shown in Fig. 1. An argon-ion aser was use to pump two Coherent continuous-wave frequency-stabiize ring-ye asers, both ineary poarize out of the page operating at a waveength of approximatey 767 nm 4 2 S 1/2 4 2 P 3/2 atomic transition of potassium. The pump beam was focuse into a 5-mm-ong potassium vapor ce with a 5-mm ens. The weak-probe beam was focuse into the vapor ce with a 4-mm ens was mae to overap the pump beam in the ce. The ange between the wave vectors of the pump probe beams was approximatey 2.5. The transmitte probe beam was coecte irecte onto etector 3, the signa was ampifie anayze with an rfspectrum anayzer. The refection off of the winow of etector 3 was use to measure the gain of the transmitte probe beam with etector 1. The gain noise properties of the transmitte probe beam were measure as a function of probe etuning from the atomic resonance frequency. Figure 2 shows a typica pot of the gain normaize rms noise of the transmitte probe for a spectrum anayzer frequency of 1 MHz as a function of probe etuning for the case in which no buffer gas was present in the potassium /96/53 5 / /$ The American Physica Society

2 3626 W. V. DAVIS, A. L. GAETA, R. W. BOYD, AND G. S. AGARWAL 53 FIG. 3. Experimenta measurement the corresponing theoretica preictions of the gain rms noise of the transmitte probe beam as a function of probe etuning for the case in which approximatey 5 Torr of heium buffer gas was present in the potassium ce. The rms noise is normaize to the iea ampifier quantum-noise imit shown by a ashe ine in the wings. FIG. 1. Experimenta setup use to measure the gain quantum-noise properties of the transmitte probe beam. vapor ce. Gains as arge as a factor of 125 were observe uner somewhat ifferent conitions. The power of the pump beam was 14 mw the atomic number ensity of the potassium in the vapor ce was approximatey atoms/cm 3. The preictions for an iea optica ampifier see Sec. III C are aso shown on the graph. The rms noise is normaize to the iea ampifier quantum-noise imit when the frequency of the probe aser is tune far off resonance i.e., the gain g 1). The rms noise of the Rabi feature is typicay times greater than the iea ampifier quantum-noise imit. Figure 3 shows a typica pot of the gain normaize rms noise of the transmitte probe for a spectrum anayzer frequency of 1 MHz as a function of probe etuning for the case in which approximatey 5 Torr of heium buffer gas was present in the potassium vapor ce. The power of the pump beam was 13 mw, the atomic number ensity of the potassium in the vapor ce was approximatey atoms/cm 3. The rms noise of the Rayeigh feature is typicay times greater than the iea ampifier quantumnoise imit. The ratio of the rms noise to the iea ampifier quantumnoise imit at the peak of the Rabi gain feature is potte as a function of potassium number ensity in Fig. 4. The ratio increases sighty as the number ensity increases. This observation is consistent with the noise preictions of Gaeta, Boy, Agarwa 9 for two-beam couping in a homogeneousy broaene system of two-eve atoms. III. QUANTUM THEORY OF TWO-BEAM COUPLING FIG. 2. Experimenta measurement corresponing theoretica preictions of the gain rms noise of the transmitte probe beam as a function of probe etuning for the case in which no buffer gas was present in the potassium ce. The rms noise is normaize to the iea ampifier quantum-noise imit shown by a ashe ine in the wings. The quantum theory of two-beam couping in a two-eve atomic system is presente in this section. From this theory, the Langevin operator for the interaction can be etermine. In aition, this theory can be use to preict the quantumnoise properties of a beam of ight ampifie or attenuate through two-beam couping in an atomic vapor. The preictions of this theory are compare with the experimenta resuts. A. Derivation of the Langevin equation In the quantum theory of two-beam couping 9, the interaction between the two-eve atoms with an energy eve

3 53 STATISTICAL-NOISE PROPERTIES OF AN OPTICAL the effects of atomic reaxation are containe in the term ( ˆ/ t) reax. The equation of motion for the fie ensity operator ˆ F is etermine from the tota ensity operator by tracing ˆ over the atomic variabes, that is, ˆ F Tr A ˆ. The etais of this cacuation are given in Refs. 9 1 are summarize in the Appenix. Now that the equations for the expectation vaue of the annihiation operator, its ajoint, the photon number operator have been etermine, the Langevin equation for the annihiation operator can be etermine from the Einstein reation. The resuting Langevin equation is â t T 2 C i â fˆ t. 4 FIG. 4. Ratio of the rms noise to the iea ampifier imit at the peak of the Rabi gain potte as a function of potassium number ensity. The ine shows the preictions of the theory. separation ba ) the fies are treate in the eectricipoe approximation. A fies are assume to propagate in the positive x irection are ineary poarize in the z irection. The interaction Hamitonian Vˆ is given by Vˆ P ˆ r E r,t 3 r P ˆ r E ˆ1 r,t 3 r, 1a the pump fie E at a frequency ba given by E r,t ẑe e ik x i t c.c. is treate cassicay, the probe fie E ˆ 1 given by 1b The properties of the Langevin operator fˆ(t) are fˆ t fˆ t, fˆ t 1 fˆ t 2 2D t 1 t 2, fˆ t 1 fˆ t 2 2D t 1 t 2, 2D T 2 Re Q i C i, 2D T 2 Re Q i C i, 5a 5b 5c 5 5e E ˆ 1 r,t ẑ L x L y L z 1/2 âe ik 1 x i 1 t h.a. 1c is quantize in the voume L x L y L z. The probe fie osciates at the frequency 1 is associate with an annihiation operator â. The normaization constant is equa to i 2 1. The poarization operator P ˆ (r ) is reate to the ipoe moment operator ˆ (q) for an atom at the position R (q) through the expression P ˆ r q r R q ˆ q, 2 with the summation extening over a of the atoms in the interaction region. The tota Hamitonian Ĥ is given by Ĥ Ĥ A Ĥ F Vˆ, Ĥ A Ĥ F are the unperturbe Hamitonian for the atomic system the fie, respectivey. The equation of motion for the tota ensity operator ˆ is etermine from the Heisenberg equation ˆ t i ˆ Ĥ, ˆ, 3 t reax 4 N ba 2 T 2 / c is the weak-fie ine-center intensity absorption coefficient. For the case when the probe fie propagates a istance through the interaction region, the equation of motion for its annihiation operator etermine from Eq. 4 can be repace with an equation escribing its spatia evoution by setting the time t equa to n 1 x/c, n 1 is the inex of refraction of the probe fie. Furthermore, the correation functions C (i ) Q (i ) wi in genera be functions of position. That is, the operating conitions of the reservoir change with position. In the case of two-beam couping, the spatia variation of the correation functions is a irect consequence of pump-beam absorption. The spatia variation of the pump-beam intensity I (x) is etermine from I x D pump I, 6a D pump ( ) is the Dopper-average vaue of the pump absorption coefficient pump ( ) given by pump n ba 1 T 2 2 I /I s. 6b

4 3628 W. V. DAVIS, A. L. GAETA, R. W. BOYD, AND G. S. AGARWAL 53 The Rabi frequency is reate to the pump intensity through the expression I /I s 2 T 1 T 2, I s is the x exp x x x * x ine-center saturation intensity. Aowing for pump-beam absorption, the equation for the x spatia variation of the probe-fie annihiation operator is x * x exp x x * x. given by â x x â fˆ x, 7a The resuting expression for this expectation vaue is 11 the quantity x T 2 C i,x 7b can be shown to be ientica to the moifie probe-beam absorption coefficient erive from the semicassica theory of two-beam couping. The soution to Eq. 7a can be easiy etermine by using an integrating factor, the resut is â gâ Lˆ, 8a â() is the annihiation operator of the probe fie before entering the interaction region, â() is the annihiation operator after exiting the interaction region, g given by g exp x x 8b is the gain or oss experience by the probe-fie ampitue, Lˆ given by x Lˆ g x fˆ x exp x x 8c is the Langevin or noise operator associate with the twobeam couping process. It can be shown that the Langevin operator Lˆ satisfies the conition Lˆ,Lˆ 1 g 2 for an arbitrary state. Thus the commutator itsef is equa to Lˆ,Lˆ 1 g 2. Note that this conition is ientica to the conition erive by requiring that the output operator â() obey the Bosonic commutator reation. Furthermore, the expectation vaues Lˆ Lˆ Lˆ Lˆ can be erive from the genera expression for Lˆ Eq. 8c its commutation reation. Consier the expectation Lˆ Lˆ given by Lˆ Lˆ g 2 x fˆ x fˆ exp x x * x exp. 9 1 This expectation vaue wi be of great importance in the foowing since it is the quantity neee to cacuate the noise properties of the photocurrent. Equation 1 can be further simpifie by using Eqs. 5b 5 the ientity Lˆ Lˆ g T 2 g 2 x Re Q i,x exp x x x * x. 12 For an iea optica ampifier with g 2 1), it can be shown that Lˆ Lˆ is equa to ( g 2 1). Furthermore, uner iea conitions Lˆ Lˆ is equa to zero when g 2 1. B. Effects of atomic motion Atomic motion affects the two-beam couping process in two ways: through Dopper shifts, grating washout effects. Consier first the affects of atomic motion on the popuation gratings. Note that the pump fie with a frequency wave vector k ) the probe fie with a frequency 1 wave vector k 1 ) set up a grating in the meium through the interference term exp i(k 1 k ) r i t. The pump fie can then scatter off of this grating, proucing raiation with a frequency 1 wave vector k 1. That is, energy from the pump fie is coherenty ae to the probe fie. Atomic motion wi cause these gratings to isperse or wash out, thus reucing the energy transfer efficiency of the pump beam into the probe beam. Grating washout processes are incue phenomenoogicay for each atomic veocity group v. As mentione above, atomic motion aso prouces Dopper shifts in the frequency of the pump probe. This effect is accounte for by performing Dopper averages. Thus a proper escription of the process of two-beam couping in atomic vapors requires the incusion of atomic motion. In our theory, the effects of atomic motion are introuce by first mutipying grating terms by an efficiency factor S(v ) then performing a Dopper average. The grating efficiency factor S(v ) is cose to unity when an atom moves ony a sma fraction of the grating perio in a time T 1 before making a transition back to the groun state. However, S(v ) wi be approximatey zero if an atom moves a istance comparabe to the grating perio in a time T 1. A convenient choice for the grating efficiency factor is S v cosx v for X v 13a for X v,

5 53 STATISTICAL-NOISE PROPERTIES OF AN OPTICAL X v 1 2 k g v T1, 13b k g k 1 k is the grating wave vector, is a grating parameter which has a vaue greater than zero. When is equa to unity, the grating wi be competey washe out for atoms which have move a istance greater than one grating perio in a time T 1. The parameter serves as a free parameter when comparing the experimenta resuts the theoretica preictions. As wi be shown beow, the gratings terms are containe in the correation functions C (i ) Q (i ). The popuation grating terms can be isoate by examining the Boch equations for the atoms in the presence of both the pump probe fies. In a frame rotating with the anguar frequency of the pump see Eq. A3b, the Boch equations can be written in the form t M M *M 1 e i k 1 k r i 1 t M 1 *e i k 1 k r i 1 t M, 14 the expressions for the matrices M M are easiy obtaine from Eq. A3, 1 2 E 1 / is the Rabi frequency associate with the probe fie which is assume to be much smaer in magnitue than. The ipoe moment at the probe frequency 1 etermines the two-beam couping efficiency. Since the cacuation is performe in a rotating frame, the ipoe moment inversion at the frequency 1 are etermine from the quantity ( ). Soving Eq. 14 to owest orer in 1, the part of the Boch vector osciating at the probe-pump etuning is equa to i I M 1 1 M e i k 1 k r, 15 () given by Eq. A5 is the steay-state vaue of the Boch vector for the atom in the presence of the pump ony, M M M *M. The grating terms in Eq. 15 can be isoate by examining the expe form of Eq. 14 for 2. The equation for 2 becomes 2 t 1 T 2 i 2 i 3 2 NG 1 1 T 2 i i i 1 3 e i k 1 k r G T 2 i i i 3 18b 18c are the nongrating grating terms, respectivey. These quantities can be rewritten in the form of Eq. 15 through use of i I M 1 2 j i i 1 1 T 2 2 j, i I M 1 i I M 1 i I M 1 19a M *M i I M 1, 19b 2 NG i I M 1 1 M 2 e i k 1 k r, 19c 2 G i I M 1 i I M 1 1 M 2 e i k r 1 k r r r. 19 Comparing Eqs. 19 with Eq. 15, the foowing proceure can be use to isoate the nongrating grating parts. The nongrating terms are obtaine by repacing M by M in the ynamica matrix U. That is, the matrix U Eq. A6 is repace by the matrix V V U. The grating terms are obtaine by repacing the ynamica matrix U by (U V). Note that the steay-state vaues of the Boch vector in the presence of the pump fie are sti use. As mentione earier, the quantity C is reate to the noninear susceptibiity of the meium. Thus this proceure can be use to isoate the nongrating grating terms of C. A simiar assignment appies to the quantity Q the noise terms. This can be unerstoo by reaizing that in the fuy quantum-mechanica theory, the Boch equations become operator Langevin equations with the probe fie now being repace by the mutimoe vacuum of the raiation fie, i.e., i 1 e i k 1 k r i 1 t e ik 1 r i 1 t 3 k e ik r i k t S z â k. 2 Note further that for a weak probe fie, 3 can be expe in the form e i 1 t c.c., 17 3 () is given by Eq. A7c, 3 ( ) is at east of the orer 1 as can be seen from Eq. 15. The quantity 3 ( ) is ceary the grating term in the meium. Thus the soution to Eq. 16 to owest orer in 1 is ceary given by Such noise terms can be he using star methos. The grating contributions in these noise terms wou again arise from terms ike 3 or S z in Eq. 16. In ight of the above iscussion, the effects of atomic motion are incue in the foowing way. The correation functions C (i ) Q (i ) are separate into a nongrating C NG (i ) Q NG (i )] a grating C G (i ) Q G (i )] part, that is, 2 2 NG 2 G, 18a C i C NG i C G i S v 21a

6 363 W. V. DAVIS, A. L. GAETA, R. W. BOYD, AND G. S. AGARWAL 53 C. Photocurrent fuctuations Q i Q NG i Q G i S v, 21b The soution for the output annihiation operator â() Eq. 8a can now be use to cacuate the quantum-noise properties of the transmitte probe beam. The expectation vaue of the output photon-number operator is given by C NG i 2 T 2 V 2 i, 2 3, 1, 21c nˆ g*â Lˆ gâ Lˆ. 22 Q NG i 2 T 2 V 2 i 2 1 1, , Using the fact that the Langevin operators are Gaussian in nature, this expression can be simpifie to 2 1 3, 21 nˆ g 2 nˆ Lˆ Lˆ, 23 C G i 2 T 2 U V 2 i, 2 3, 1, 21e Q G i 2 T 2 U V 2 i 2 1 1, , 2 1 3, 21f S(v ) is the grating efficiency factor. Finay, the Dopper-average correation functions are cacuate by performing a two-imensiona Dopper integration on Eqs. 21. g Lˆ Lˆ are given by Eqs. 8b 12, respectivey. Simiary, the variance of the photon number for a highy excite coherent-state input is given by nˆ 2 g 2 nˆ 2 Lˆ Lˆ 1. Then the Fano number is given by Z 2 Lˆ Lˆ 1, it is by efinition equa to the ratio of the power fuctuations to the shot noise. Uner iea operating conitions, the Fano number is given by Z 2G 1 for G 1 iea ampifier quantum-noise imit 1 for G 1 iea attenuator quantum-noise imit, 26 G g 2. In aition, the noise figure efine by 2 2 F (S/N) input /(S/N) output is equa to F Z G for the two-beam couping process. 27 IV. NUMERICAL RESULTS The theoretica preictions for the gain noise of the transmitte probe beam were generate numericay by integrating the Dopper-average pump propagation equation Eqs. 6 using this resut to cacuate the Dopperaverage atomic-poarization correation functions. These correation functions were use to cacuate the gain g Lˆ Lˆ. The Fano number of the transmitte probe fie was then etermine from Eq. 25. Finay, the normaize rms noise was cacuate by taking the square root of the prouct of the Fano number at the etector, the probe transmittance through the ce. Figure 2 shows a typica experimenta measurement of the gain normaize rms noise for the Rabi gain feature with corresponing theoretica preictions for the case in which no buffer gas present in the ce. The pump power entering the ce was equa to 14 mw. The best agreement between the preictions of the theory the experimenta resuts was obtaine for an entering pump intensity of 115 W/cm 2, a crossing ange of 2.5, a pump etuning of.7 GHz, a grating parameter of 1 3. The agreement between the theory experimenta resuts is quite goo. Figure 3 shows a typica experimenta measurement of the gain normaize rms noise for the Rayeigh gain feature with the corresponing theoretica preictions for the case in which approximatey 5 Torr of heium buffer gas is present in the ce. The pump power entering the ce was equa to 13 mw. The best agreement between the preictions of the theory the experimenta resuts was obtaine for an entering pump intensity of 8 W/cm 2, a crossing ange of 2.5, a pump etuning of 1.5 GHz, a grating parameter of 1 3. The agreement between the theory experimenta resuts is goo, but not as goo as that for the Rabi feature. The increase sensitivity to the crossing ange buffer gas pressure for the Rayeigh gain feature makes it very ifficut to etermine the precise experimenta conitions. The ratio of the rms noise to the iea ampifier quantumnoise imit at the peak of the Rabi gain feature is potte as a function of potassium number ensity in Fig. 4. The curve shows the preiction of the theory. This ratio increases sighty as the number ensity increases. The theoretica preictions ispay the correct quaitative behavior.

7 53 STATISTICAL-NOISE PROPERTIES OF AN OPTICAL V. CONCLUSIONS In concusion, a theoretica experimenta investigation of the quantum-noise properties of a probe beam ampifie through two-beam couping in atomic-potassium vapor has been presente. For a spectrum anayzer frequency of 1 MHz, the rms noise at the Rabi gain feature is typicay times greater than that of the iea ampifier quantum-noise imit. For the Rayeigh gain feature, the rms noise at is typicay times greater than that of the iea ampifier quantum-noise imit, ecreases as the heium buffer gas pressure increases. These resuts are in goo agreement with the preictions of a fuy quantummechanica singe-moe theory of two-beam couping, incuing the effects of atomic motion pump-beam absorption. ACKNOWLEDGMENTS The authors gratefuy acknowege iscussions of the contents of this paper with M. Kauranen V. Iruvanti. This research was supporte by the U.S. Army Research Office through a University Research Initiative Center by NSF Grant No. INT APPENDIX In this appenix we outine some of the intermeiate steps in the cacuation of the Langevin equation escribing the quantum ynamics of the probe fie. The master equation for the fie ensity operator is given by 9 1, ˆF t 2 N 2 2 Q i â, â, ˆF C i â, â, ˆF H.a., A1 N is the atomic number ensity. The quantities C (i ) Q (i ) are the Lapace transforms of specific inear combinations of two-time atomic-poarization correation functions 9. Physicay, C (i ) is proportiona to the noninear susceptibiity that appears in the semicassica theory of two-beam couping. The quantity Q (i ), however, has no counterpart in semicassica theories, it represents quantum fuctuations of the atomic system. For the case of a two-eve atom, the correation functions C (i ) Q (i ) are etermine from the optica Boch equations. First, the poarization operator is rewritten in the form the components of the Boch vector the vector are 1 S e i k R t, 2 1 *, 3 S z 1 2, 3 1 2T 1, respectivey. The matrix M is equa to A3b A3c M 1/T 2 i i * 1/T 2 i i i /2 i */2 1/T 1, A3 ba is the etuning of the pump aser from the atomic resonance, T 1 is the popuation ecay time, T 2 is the ipoe ephasing time, the Rabi frequency is efine by 2 z E (r )/. The correation functions C (i ) Q (i ) can then be cacuate in terms of the soution of Eq. A3a by using the quantum regression theorem. The resuts given in terms of the steay-state soution of the Boch equations a matrix U are 9 C i 2 T 2 U 2 i, 2 3, 1 A4a Q i 2 T 2 U 2 i 2 1 1, , 2 1 3, A4b the efinition z has been use to simpify the notation. The steay-state soution of the Boch equations in the presence of the pump () the matrix U are etermine from M 1 T 2 U i i I M 1, A5 A6 I is the ientity matrix. The resuts of the cacuations are P ˆ r Ŝ r R H.a., A2 Ŝ, its ajoint Ŝ, Ŝ z 2 Ŝ 1,Ŝ obey the commutation reations for a spin- 1 2 system. The quantity is efine by. Then the matrix form of the Boch equations for an atom ocate at the position R interacting with the pump fie are given by t M, A3a 1 *T 2 T 2 i, A7a 2P 2 1 * T 2 T 2 i, A7b 2P 3 1 T 2 2 2P A7c

8 3632 W. V. DAVIS, A. L. GAETA, R. W. BOYD, AND G. S. AGARWAL 53 U 21 i 2 T 1 T 2 2P, A7 U 22 i 2 T 2 i T 2 i 2 T 1 T 2, 2P A7e U 23 i T 1 T 2 i, A7f P P 1 i T 2 1 i T 2 2 T i T 2 2 T 1 T 2 A7g T 1 /T 2. The master equation Eq. A1 can be converte into an equation for the tempora evoution of the expectation vaue of the moments of the probe fie operator. In the Schröinger picture, it can be shown that the erivative of the expectation vaue of a fie operator Ĝ is given by 12 t Ĝ t Tr Ĝ ˆ F Tr Ĝ ˆF t. Furthermore, with the hep of the ientities A8 the evoution of the expectation vaue of a fie operator Ĝ is etermine from the equation t Ĝ 2 N 2 2 ˆQ i â, â,ĝ C i â, Ĝ,â Q i * â, â,ĝ C i * â, Ĝ,â. A1 The equations of motion for the expectation vaue of the annihiation operator â, its ajoint â, the photon number operator nˆ â â are t â T 2 C i â, t â T 2 C i * â, A11a A11b t nˆ T 2 C i C i * nˆ 1 2 C i C i * Q i Tr Ĝ Â, Bˆ, ˆ F Bˆ, Â,Ĝ A9a Q i *, A11c Tr Ĝ Â, Bˆ, ˆ F Bˆ, Ĝ,Â, A9b respectivey. Equations A11 are use in Sec. III A to erive the Langevin equation 4. 1 W. H. Louise, Quantum Statistica Properties of Raiation Wiey, New York, R. Louon, The Quantum Theory of Light Carenon, Oxfor, F. T. Arecchi, E. Gatti, A. Sona, Phys. Lett. 2, H. J. Kimbe, M. Dagenais, L. Me, Phys. Rev. Lett. 39, D. F. Was, Nature 36, C. M. Caves, Phys. Rev. D 26, Y. Yamamoto T. Mukai, Opt. Quantum Eectron. 21, S C. K. Hong, S. Friberg, L. Me, J. Opt. Soc. Am. B 2, A. L. Gaeta, R. W. Boy, G. S. Agarwa, Phys. Rev. A 46, G. S. Agarwa, Phys. Rev. A 34, W. V. Davis, Ph.D. thesis, University of Rochester, See, for exampe, C. Cohen-Tanouji, J. Dupont-Roc, G. Grynberg, Atom-Photon Interactions: Basic Processes Appications Wiey, New York, 1992.