PHYSICAL REVIEW A 75, 013820 共2007兲

Correlation of photon pairs from the double Raman amplifier: Generalized analytical quantum Langevin theory C. H. Raymond Ooi,1,2,3,4 Qingqing Sun,2 M. Suhail Zubairy,2 and Marlan O. Scully1,2,3 1

Max-Planck-Institut für Quantenoptik, D-85748, Garching, Germany Institute for Quantum Studies and Department of Physics, Texas A&M University, Texas 77843-4242, USA 3 Princeton Institute for the Science and Technology of Materials and Department of Mechanical & Aerospace Engineering, Princeton University, New Jersey 08544, USA 4 Department of Physics, KAIST, Guseong-dong, Yuseong-gu, Daejeon, 305-701 Korea 共Received 26 November 2006; revised manuscript received 27 December 2006; published 31 January 2007; publisher error corrected 5 February 2007兲 2

We present a largely analytical theory for two-photon correlations G共2兲 between Stokes 共s兲 and anti-Stokes 共a兲 photon pairs from an extended medium 共amplifier兲 composed of double-⌳ atoms in counterpropagating geometry. We generalize the parametric coupled equations with quantum Langevin noise given in a beautiful experimental paper of Balic et al. 关Phys. Rev. Lett. 94, 183601 共2005兲兴 beyond adiabatic approximation and valid for arbitrary strength and detuning of laser fields. We derive an analytical formula for cross correlation 共2兲 Gas = 具Eˆs†共L兲Eˆ†a共0 , ␶兲Eˆa共0 , ␶兲Eˆs共L兲典 and use it to obtain results that are in good quantitative agreement with the 共2兲 experimental data. Results for Gas obtained using our coupled equations are in good quantitative agreement with the results using the equations of Balic et al., while perfect agreement is obtained for sufficiently large 共2兲 detuning. We also compute the reverse correlation Gsa which turns out to be negligibly small and remains classical while the cross correlation violates the Cauchy-Schwartz inequality by a factor of more than a hundred. DOI: 10.1103/PhysRevA.75.013820

PACS number共s兲: 42.50.Dv, 42.50.Gy, 42.50.Lc, 03.67.Mn

I. INTRODUCTION

One of the amazing properties of the quantum world is quantum correlation. The quantum mechanical concept of photon-photon correlation introduced by Glauber has provided us with insights into the distinct quantum statistical nature of photons from various light sources such as lasers, the Sun, and resonance fluorescence. Spontaneous emission, being a quantum mechanical process, has an important role in establishing quantum correlation. Recently, various versions of double-⌳ schemes have been explored 关1兴 in regards to entanglement, but not so much in the interest of quantum correlation. Entanglement, the heart of quantum informatics, is related to quantum correlation, a concept which extends beyond the pure state. Furthermore, quantum correlation in the sense of Glauber’s two-photon correlation G共2兲 in an extended medium with propagation deserves proper theoretical studies, especially for the double-⌳ scheme. The scheme has remarkable features and has been widely studied in the context of quantum erasers 关2兴, quantum information 关3–5兴, nonlinear optics 关6,7兴, and subwavelength resolution microscopy 关8兴. A Stokes photon is generated via a spontaneous Raman process. It is possible to generate another photon, the antiStokes photon, which is strongly correlated to the Stokes photon by applying a strong resonant control field ⍀c 关Fig. 1共a兲兴. The laser field creates a dressed state 冑12 共兩a , nc − 1典 + 兩b , nc典兲 共with nc the photon number of the control field兲: a coherent superposition of state 兩b , nc典 following the emission of a Stokes photon and state 兩a , nc − 1典 from which an antiStokes photon would be emitted. We refer to the correlated photon pairs produced by this “spontaneous Raman-EIT 共electromagnetically induced transparancy兲” scheme as the 1050-2947/2007/75共1兲/013820共13兲

Raman emission doublet 共RED兲. The scheme exhibits nonclassical properties such as squeezing 关6兴, violation of the Cauchy-Schwartz inequality 关3,9兴, and antibunching with Rabi oscillations in G共2兲 for the single-atom case 关10兴. The RED scheme also enables efficient mapping of the quantum information of the input Stokes photon into the atomic ensembles and reading off the information as an anti-Stokes photon after a controllable time delay up to 2 ␮s 关5兴, much longer than those produced in cascade scheme and in parametric down-conversion 关11兴. In an extended medium where a large number of correlated photon pairs can be generated, application of the scheme in quantum lithography 关12兴 becomes more feasible. The correlation time can be increased via a slow light effect through the control field. Recently, nonclassical macroscopic photon correlation of the RED scheme has been demonstrated for many 共cold 87Rb兲 atoms in backward propagating geometry 关13兴 in a beautiful experiment by Balic et al. 关9兴. They obtained coupled parametric oscillator equations based on the adiabatic approximation which assumes that the entire population is in the ground level. Their experimental data were fitted remarkably well with the numerical solutions of the coupled equations. Motivated by their work, here we extend their coupled equations beyond the RED scheme. Without the adiabatic approximation, we derive the coupled parametric equations that are valid for any detuning and strength of the pump and control laser fields and arbitrary populations. We have obtained analytical expressions for the two-photon correlation in an extended medium with noise operators 共more specifi共2兲 共2兲 , reverse correlation Gsa , and cally the cross correlation Gas 共2兲 self-correlations G f f , with f = s , a兲 which seems to be a for共2兲 midable task so far. The results for Gas 关Eq. 共28兲兴 are in

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PHYSICAL REVIEW A 75, 013820 共2007兲

RAYMOND OOI et al.

FIG. 1. 共Color online兲 共a兲 The double Raman scheme produces correlated pairs of Stokes and anti-Stokes photons. The present four-level scheme 共instead of three levels兲 describes the experimental situation in 关9兴. The coupling of the pump and Stokes fields to level 兩a典 is negligible since the fields are far detuned from it. The Raman emission doublet 共RED兲 scheme corresponds to ⌬ p,s Ⰷ ␥ac, ⍀ p, ⌬c,a = 0, and ⍀c ⬎ ⍀ p where the Stokes photon is generated by the spontaneous Raman process and the anti-Stokes is by the resonant Raman process. Counterpropagating Stokes and anti-Stokes photons amplified into correlated macroscopic quantum fields Eˆs and Eˆa are generated by lasers propagating 共b兲 along the extended medium 共laser depletion may be significant兲 and 共c兲 perpendicular to the extended medium 共negligible laser depletion兲. Also shown are phase-matched diagrams of the four fields.

good agreement with the experiment 关9兴. On the other hand, 共2兲 we find that the Gsa is negligibly small for the RED scheme, as expected 关15兴. Even though we focus on the small signal regime and disregard laser field depletion, our theory yields a lot of new results for laser parameters in different limiting cases. Further results and analysis will be reported in a series of forthcoming papers. They include 共a兲 forward 共copropagating兲 geometry, 共b兲 the role of noise operators and the connection with group delay in signal propagation 关16兴, and 共c兲 dependence of nonclassicalness on laser parameters. In Sec. II, we present the coupled oscillator equations for counterpropagating 共backward geometry兲 laser fields with arbitrary strength and detuning. The solutions of the coupled equations in the frequency domain for the boundary parts and the noise parts are given in Sec. III. We show that they reduce to the known solutions in Ref. 关6兴 in the case of the RED scheme. In Sec. IV, we derive analytical expressions for 共2兲 共2兲 , Gsa , and G共2兲 Gas f f . Normalized correlations such as CauchySchwartz correlation are defined in Sec. V to quantify the degree of nonclassicalness. We also compute the quantitative 共2兲 detection rate of the experiment 关9兴 from Gas . Finally, in Sec. VI we compare the computed detection rates 共based on our coupled equations and analytical correlation兲 and the hybrid results 共the coupled equations of Balic et al. and our analytical correlation兲 with the experimental data 关9兴. II. GENERALIZED COUPLED EQUATIONS FOR EXTENDED MEDIUM

We proceed to the main focus of this paper: to study the two-photon correlation between the Stokes and anti-Stokes fields of an extended medium or amplifier composed of a macroscopic number of atoms with the double Raman scheme 关Fig. 1共a兲兴. Despite the appreciable propagation effect in the extended sample, the off-resonant and weak pump field in the RED scheme lead to a negligible depletion of the laser fields even for the scheme in Fig. 1共b兲. It is possible to minimize the depletion of laser fields via the configuration where the lasers are orthogonal to the Stokes and anti-Stokes fields 关Fig. 1共c兲兴. This enables us to neglect Maxwell’s equations for the laser fields which describe the depletion. The

resulting Maxwell’s equations for the macroscopic 关17兴 Stokes Eˆa = Aˆ / ga and anti-Stokes Eˆs = Sˆ / gs fields 共gs = 㜷db / ប and ga = 㜷ac / ប as the coupling strengths兲 together with linearized atomic equations 共Appendix A兲 lead to 关21兴 the coupled equations

冉

冊 冉

冊

冉

⳵ ˆ† ˆ ⳵ + Gs Sˆ + Ks − ␤s A = Fs , ⳵z ⳵z

−

⳵ ˆ ˆ† ⳵ + Ga Aˆ† + Ka + ␤a S = Fa , ⳵z ⳵z

冊 冉

冊

共1兲

共2兲

with the effective noise operators ˆ +X G ˆ ˆ Fˆs = XadG ad bc bc + XbdGbd ,

共3兲

ˆ +Y G ˆ ˆ Fˆ†a = Y adG ad bc bc + Y acGac ,

共4兲

where the coefficients G f , K f , ␤ f , Xx, and Y x 共x = ac , ad , bc , bd兲 are given by Eqs. 共B1兲–共B4兲 and 共B7兲–共B10兲 in Appenˆ = 兰⬁ ei␪xFˆ 共z , t兲ei␯tdt is the Foudix B, respectively, and G x x −⬁ rier transform of the noise operators in Eqs. 共A6兲–共A9兲 including the rapidly varying phases ␪x共z , t兲 = kxz − ␯xt. Equations 共1兲 and 共2兲 describe the dynamical evolutions of the macroscopic quantum fields for the Stokes and antiStokes photons in the spectral domain. By multiplying Eq. 共1兲 by ␤a and then subtracting it from Eq. 共2兲 and similarly multiplying Eq. 共2兲 by ␤s and subtracting it from Eq. 共1兲 we obtain the familiar parametric oscillator coupled equations 关9,22兴

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冉

冉

⳵ + Gs Sˆ + KsAˆ† = ¯Fs , ⳵z

冊

−

⳵ + Ga Aˆ† + KaSˆ = ¯F†a , ⳵z

冊

共5兲

共6兲

PHYSICAL REVIEW A 75, 013820 共2007兲

CORRELATION OF PHOTON PAIRS FROM THE DOUBLE…

since the coefficients in Eqs. 共1兲 and 共2兲 are related to that of Eqs. 共5兲 and 共6兲 by 共Gs − ␤sKa兲 Gs = , Isa Ks =

共Ks − ␤sGa兲 , Isa

ˆ† ˆ ¯F = Fs − ␤sFa , s Isa

共Ga − ␤aKs兲 Ga = , Isa

共7兲

共Ka − ␤aGs兲 , Isa

共8兲

Ka =

ˆ ˆ† ¯F+ = Fa − ␤aFs , a Isa

where coefficients for the boundary operators are 共12兲

¯␺s 共L, ␯兲 = Z共L, ␯兲, a

共13兲

¯␺a共L, ␯兲 = a

共9兲

with Isa = 1 − ␤s␤a. The two sets of coefficients are approximately equal when ␤s ⯝ ␤a ⯝ 0. Here, the Gs is the spontaneous Raman gain coefficient, Ga gives the EIT dispersion and absorption profiles modified in the presence of the pump laser, and Ks and Ka are the cross couplings. The coefficients G f and K f generalize those obtained by Refs. 关9,22兴 beyond the adiabatic approximation. The effective Fourier transforms of the noise operators ¯Fs and ¯F†a serve as the driving “forces” or seeds to both fields and their physical origin is the quantum vacuum fluctuations. Note that Eqs. 共5兲 and 共6兲 are equivalent to a driven oscillator equation with effective gain or damping 共Gs + Ga兲 and oscillation angular frequency 冑GaGs − KsKa. These coupled equations are also obtained for the case of the resonance fluorescence of two-level atoms in an extended medium, which will be a subject of future publications.

¯␺s共L, ␯兲 = ⌶ ¯ ¯ 共L, ␯兲 + ⌶ ¯ 共L, ␯兲兵G + Z共L, ␯兲K 其, q a a s

1 ¯ ¯ 共L, ␯兲 − ⌶ ¯ 共L, ␯兲G ⌶ q s

¯␺a共L, ␯兲 = s

Ka Ks

,

Z共L, ␯兲,

共14兲

共15兲

with Isa = 1 − ␤s␤a , Z共L, ␯兲 =

¯ 共L兲K ⌶ s ¯ ¯ 共L兲 − ⌶ ¯ 共L兲G ⌶ q s

共16兲 共17兲

,

the oscillatory functions ¯ ¯ 共x, ␯兲 = ¯q+e ⌶ q

¯ +x −q

− ¯q−e−q¯−x , ¯q+ − ¯q−

共18兲

¯ +x −q − e−q¯−x ¯ 共x, ␯兲 = e , ⌶ ¯q+ − ¯q−

共19兲

and effective wave vectors III. SOLUTIONS FOR COUNTERPROPAGATING (BACKWARD) GEOMETRY

The solutions of the generalized coupled equations for the Stokes field at z = L and the anti-Stokes field at z = 0 are composed of the boundary operators Bˆ f 共specifically the Stokes operator at z = 0 and the anti-Stokes operator at z = L兲 and ˆ 关specifically Fˆ 共z , ␯兲 at all points in the noise operators N f f medium兴—i.e.,

¯q± = − ¯␣ ± ¯␤ ,

共20兲

1 ¯␣ = 共Ga − Gs兲, 2

共21兲

¯␤ = 冑¯␣2 − 共K K − G G 兲. s a s a

共22兲

The kernels for the noise operators are ¯ s共␰, ␯兲 = 1 关⌶ ¯ ¯ 共␰兲共1 − Z␤ 兲 + ⌶ ¯ 共␰兲共G + ZK 兲兴, 共23兲 U q a a a s Isa ¯ s 共␰, ␯兲 = 1 关⌶ ¯ ¯ 共␰兲共Z − ␤ 兲 − ⌶ ¯ 共␰兲共K + ZG 兲兴, U q s s s a Isa

共10兲 ¯ a共 ␰ , ␯ 兲 = U a

¯ a共 ␰ , ␯ 兲 = − U s

共11兲

¯ ¯ 共␰兲 − ⌶ ¯ 共␰兲G ⌶ q s D共L兲

D共L兲

共25兲

,

¯ ¯ ¯ 共␰兲 − K ⌶ ␤ a⌶ q a 共␰兲

共24兲

,

共26兲

¯ ¯ 共L兲 − ⌶ ¯ 共L兲G 其. Equations 共10兲 and 共11兲 where D共L兲 = Isa兵⌶ q s are general solutions for backward geometry with arbitrary detuning and laser fields.

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The limit ␤s ⯝ ␤a Ⰶ 1 corresponds to the RED scheme ¯ a共␰ , ␯兲 → ¯␺a共␰ , ␯兲 but ¯ s共␰ , ␯兲 → ¯␺s共␰ , ␯兲 and U where we find U s s s s ¯ s 共␰ , ␯兲 y ¯␺s 共␰ , ␯兲 in contrast with ¯ a共␰ , ␯兲 y ¯␺a共␰ , ␯兲 and U U a a a a the case of forward geometry. However, in the limit of short ¯ ⯝ 0 and hence ¯ ¯ ⯝ 1, ⌶ samples, ¯q+x Ⰶ 1, we have ⌶ q s s a a ¯ 共␰ , ␯兲 → ¯␺ 共␰ , ␯兲; i.e., the coeffi¯ 共␰ , ␯兲 → ¯␺ 共␰ , ␯兲 and U U a a a a cients of the noise part are identical to that of the boundary part. This explains the correspondence of the results with boundary operators and that with noise operators. We have verified that by neglecting ␤ f our Eqs. 共10兲 and 共11兲 reproduce the solutions of Ref. 关6兴 共with the additional boundary operators兲 for the RED scheme.

= 兩Iba共L, ␶兲 + Ina共L, ␶兲兩2 + 兵Iba共L兲 + Ina共L兲其2 . 共30兲 Similarly, the reverse correlation for backward geometry is 共2兲 共␶兲 = 具Eˆ†a共0,t兲Eˆs†共L,t + ␶兲Eˆs共L,t + ␶兲Eˆa共0,t兲典 Gsa

= 兩具Bˆs共L,t + ␶兲Bˆa共L,t兲典 + 具Nˆs共L,t + ␶兲Nˆa共L,t兲典兩2 + 兵Isb共L兲 + Isn共L兲其兵Iba共L兲 + Ina共L兲其.

We now use the solutions for the field operators and the Glauber’s two-photon correlation G共2兲 to compute the cross 共2兲 correlations between the Stokes and anti-Stokes fields, Gas 共2兲 共2兲 共2兲 and Gsa , and the self-correlations Gss and Gaa . One way to derive the two-photon correlation is to note that it can be expressed as decorrelated 关24兴 paired products 共2兲 共L, ␶兲 = 兩具Eˆa共t + ␶兲Eˆs共t兲典兩2 Gas

A. Noise products

共27兲

2␲ 具Nˆa共L, ␶兲Nˆs共L兲典 = ei⌬kL AN

a

x

x⬘

2␲ 具Nˆs共L, ␶兲Nˆa共L兲典 = ei⌬kL AN

ˆ 共L,t + ␶兲N ˆ 共L,t兲典兩2 = 兩具Bˆa共L,t + ␶兲Bˆs共L,t兲典 + 具N a s +

兵Isb共L兲

+

Isn共L兲其兵Iba共L兲

+

Ina共L兲其,

共28兲

ˆ 共L , ␶兲N ˆ 共L兲典, Ib共L , ␶兲, and where 具Bˆa共L , t + ␶兲Bˆs共L , t兲典, 具N a s f Inf 共L , ␶兲 are given below and are evaluated in a similar fashion in Appendix C for Eq. 共C4兲. The superscripts “b” and “n” indicate that the terms are evaluated with boundary operators and noise operators, respectively. For an inseparable two-photon pure state 兩⌿典, the 共2兲 共L , ␶兲 = 兩具0兩Eˆa共L , t correlation can be described by Gas + ␶兲Eˆs共L , t兲兩⌿典兩2. Thus, the first term in Eq. 共28兲 describes the correlation of the two-photon state while the second term describes the stimulated quantum fields corresponding to the uncorrelated states of two photons. For long time delay, the 共2兲 would take a contwo photons become uncorrelated and Gas stant value given by the second term of Eq. 共28兲 which is simply a direct product of the intensities of the Stokes and anti-Stokes photons. The corresponding self-correlations are

=

+

Isn共L, ␶兲兩2

+

兵Isb共L兲

+

Isn共L兲其2 ,

冕

⬁

共32兲

e−i␯␶⌿共L, ␯兲d␯ ,

共33兲

−⬁

2␲ Isn共L, ␶兲 = 具Nˆs†共L, ␶兲Nˆs共L兲典 = AN ˆ †共L, ␶兲Nˆ 共L兲典 = 2␲ Ina共L, ␶兲 = 具N a a AN

冕

⬁

ei␯␶S共L, ␯兲d␯ , 共34兲

−⬁

冕

⬁

e−i␯␶A共L, ␯兲d␯ ,

−⬁

共35兲 with the spectral functions of the noise products in above equations defined as ⌽共L, ␯兲 = 兺

冕

⌿共L, ␯兲 = 兺

冕

S共L, ␯兲 = 兺

冕

A共L, ␯兲 = 兺

冕

x,x⬘

x,x⬘

x,x⬘

L

˜ n 共z兲Ca*共␰, ␯兲Cs 共␰, ␯兲dz, 2D x x⬘ x,x⬘

共36兲

˜ an 共z兲Cs 共␰, ␯兲Ca*共␰, ␯兲dz, 2D x x,x⬘ x⬘

共37兲

˜ n 共z兲Cs*共␰, ␯兲Cs 共␰, ␯兲dz, 2D x x,x⬘ x⬘

共38兲

˜ an 共z兲Ca共␰, ␯兲Ca*共␰, ␯兲dz, 2D x x,x⬘ x⬘

共39兲

0

x,x⬘

L

0

L

0 L

0

n共an兲

˜ 共z兲 are the normal 共anwhere x , x⬘ = ac , ad , bc , bd and D x,x⬘ tinormal兲 ordered diffusion coefficients defined in Appendix E and the coefficients are defined as

共2兲 共␶兲 = 具Eˆs†共L,t兲Eˆs†共L,t + ␶兲Eˆs共L,t + ␶兲Eˆs共L,t兲典 Gss

兩Isb共L, ␶兲

ei␯␶⌽共L, ␯兲d␯ ,

−⬁

s

a

⬁

with the phase mismatch ⌬k. The self-correlation amplitudes due to the noise terms are

⬀ 具Fˆx共␯兲Fˆx⬘共␯⬘兲典 = 0. Thus, the correlation for backward geometry can be computed as 共2兲 共␶兲 Gas

冕

and similarly the reverse-correlation amplitude is

where the terms 具Eˆs†共t兲Eˆa共t + ␶兲典 and 具Eˆ†a共t + ␶兲Eˆs共t兲典 vanish ˆ 共t + ␶兲典 ⬀ 具Fˆ†共␯兲Fˆ† 共␯⬘兲典 = 0 and 具N ˆ †共t + ␶兲Nˆ 共t兲典 since 具Nˆ†共t兲N s

共31兲

The cross-correlation amplitude in Eq. 共28兲 due to noise is derived in Appendix C as

IV. ANALYTICAL TWO-PHOTON CORRELATION

+ 具Eˆs†共t兲Eˆs共t兲典具Eˆ†a共t + ␶兲Eˆa共t + ␶兲典,

共2兲 Gaa 共␶兲 = 具Eˆ†a共0,t兲Eˆ†a共0,t + ␶兲Eˆa共0,t + ␶兲Eˆa共0,t兲典

共29兲 013820-4

Csx共␰, ␯兲 =

1 ¯s ¯ s 共␰, ␯兲Y 共␯兲兴, 关U 共␰, ␯兲Xx共␯兲 + U x a gs s

共40兲

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CORRELATION OF PHOTON PAIRS FROM THE DOUBLE…

Cax 共␰, ␯兲 =

1 ¯a ¯ a共␰, ␯兲X 共␯兲兴, 关Ua共␰, ␯兲Y x共␯兲 + U x s g*a

共41兲

with Xx and Y x given by Eqs. 共B9兲 and 共B10兲. The adjoint of Eq. 共32兲 can also be expressed in an alter2␲ei⌬kL ⬁ ˜ a* L ˜n ˜s native form AN 兺x,x⬘兰0 2Dx,x⬘共z兲兰−⬁Cx 共␰ , t⬘兲Cx⬘共␰ , ␶ − t⬘兲dt⬘dz using the convolution theorem. If the spatial variations of the populations and the coherences are much slower ˜ n共an兲 can be sepathan G f and K f , the diffusion coefficients D x,x ⬘

具Bˆa共L,t + ␶兲Bˆs共L,t兲典 =

具Bˆs共L,t + ␶兲Bˆa共L,t兲典 =

冕冋

冕冋

B. Boundary products

Analytical expressions for the products of boundary operators can be obtained by using the commutation relation derived in Appendix D 关Eˆ f 共0 , ␯兲 , Eˆ†f 共0 , ␯⬘兲兴 ⯝ C f ␦共␯ − ␯⬘兲 ប␯ f ␲ , with C f = ␧0Ac

册 册 册 册

g* g*a a* d␯ ␺a 共L, ␯兲␺sa共L, ␯兲 + Cs¯ns s ␺sa*共L, ␯兲␺ss共L, ␯兲 ei␯␶ , ga 2␲ gs

共42兲

g*a a* g* d␯ ¯ s + 1兲 s ␺sa*共L, ␯兲␺ss共L, ␯兲 e−i␯␶ , ␺a 共L, ␯兲␺sa共L, ␯兲 + Cs共n gs 2␲ ga

共43兲

¯ a + 1兲 Ca共n

Ca¯na

rated from the functions Csx and Cax and Eqs. 共36兲–共39兲 can be integrated analytically.

Isb共L, ␶兲 = 具Bˆs†共L,t + ␶兲Bˆs共L,t兲典 =

冕冋 ⬁

−⬁

Iba共L, ␶兲 = 具Bˆ†a共L,t + ␶兲Bˆa共L,t兲典 =

冕冋 ⬁

冏 冏

¯ a + 1兲 Ca共n

¯ s + 1兲 Cs共n

−⬁

where ¯n f = 共e␤ប␯ f − 1兲−1 are the mean photon numbers for f = s , a. Equations 共42兲–共45兲 give the parts of the correlations in terms of the coefficients of the boundary operators Eˆs共0 , ␯兲 and Eˆa共0 , ␯兲. The spatially dependent Stokes and anti-Stokes intensities ˆ †共L , t兲N ˆ 共L , t兲典 and Ib共L兲 = 具Bˆ†共L , t兲Bˆ 共L , t兲典 are obInf 共L兲 = 具N f f f f f tained from the correlated intensities, Eqs. 共34兲, 共35兲, 共44兲, and 共45兲, by setting ␶ = 0 and will be used to compute the normalized correlations defined in the following section. Thus, Eqs. 共28兲–共31兲 together with the analytical expressions 共36兲–共39兲 and 共42兲–共45兲 constitute the main results of this paper. Note that the correlations depend only on the relative time delay ␶ and are independent of the absolute time t even though the solutions Eˆ f 共L , t兲 depend on t. Since Xx and Y x are proportional to N, the correlations are proportional to N2, showing a collective effect. V. DEGREE OF NONCLASSICALNESS AND QUANTITATIVE JOINT DETECTION RATE

The finite value of Glauber’s two-photon correlation G共2兲 does not necessarily imply that the correlation is quantum mechanical or nonclassical. Let us recall that g共2兲共0兲 ⬍ g共2兲 ⫻共␶兲 共antibunching兲 and g共2兲共␶兲 ⬍ 1 共sub-Poissonian兲 correspond to nonclassical correlation, with g共2兲 = 1 for the coherent state and g共2兲 = 2 for the thermal state. The photon statis-

冏 冏

g*a s ␺ 共L, ␯兲 gs a

gs a ␺s 共L, ␯兲 g*a

2

2

+ Cs¯ns兩␺ss共L, ␯兲兩2 ei␯␶

d␯ , 2␲

共44兲

d␯ , 2␲

共45兲

+ Ca¯na兩␺aa共L, ␯兲兩2 e−i␯␶

tics of the Stokes and anti-Stokes photons can be determined from the normalized self-correlations 关23兴 共2兲 2 g共2兲 f f 共L, ␶兲 ⬟ G f f 共L, ␶兲/I f 共L兲 .

共46兲

共2兲 The existence of a second order of coherence gas = 1 and the 共2兲 degree of correlation 共gas − 1兲 between the Stokes and antiStokes fields can be seen from the normalized cross 共reverse兲 correlation 共2兲 共2兲 共L, ␶兲 ⬟ Gas共sa兲 共L, ␶兲/Is共L兲Ia共L兲. gas共sa兲

共47兲

The degree of nonclassical correlation can be quantified by defining the Cauchy-Schwartz cross 共reverse兲 correlation 共2兲 共2兲 共2兲 CS 共L, ␶兲 = Gas共sa兲 共L, ␶兲/冑Gss 共L, ␶兲Gaa 共L, ␶兲, gas共sa兲

共48兲

CS where the nonclassical regime gas ⬎ 1 corresponds to violaCS tion of the Cauchy-Schwartz inequality. A larger value of gas indicates a large degree of nonclassical correlation. This is justified, for example, in the case of sub-Poissonian correla共2兲 共2兲 tion 关gss 共0兲 ⬍ 1兴 and large cross correlation gas ⬎ ⬎ 1. We now relate our results for the two-photon correlation with the experimental detection rate. The detection rate of Stokes or anti-Stokes photons can be expressed as

Rf =

n f 4I f 2␧0Vdet = = 8␧0I f Adetc/ប␯ f , tdet ប␯ f tdet

共49兲

where Adet = 41 ␲d2 is the area of the detector and we have used Vdet = Adetzdet and c = zdet / tdet. The Stokes and anti-Stokes

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FIG. 2. 共Color online兲 Comparison of the joint detection results computed from the cross correlation, Eqs. 共28兲 and 共51兲. 共a兲 Experimental results of Ref. 关9兴. 共b兲 Hybrid theory based on the coupled equations of Balic et al. 关9兴 and our analytical approach to calculate G共2兲. Quantitative agreement is obtained when the coefficients ␰s and ␰as in 关9兴 are multiplied by 冑N␴db and ␴ is replaced by ␴ac as in Ref. 关25兴 to obtain the correct units. 共c兲 Our complete theory. The curves in 共b兲 and 共c兲 are plotted using the experimental parameters of the decay rates 兩㜷兩2␻ac and dipole moments 关26兴 for 87Rb with optical depth N␴acL = 2共ga␬a兲L / ␥ac = 11 共where we define the cross sections ␴ac = បc␧o␥ac , ␴db 兩㜷兩2␻db

= បc␧o␥db 兲, weak pump ⍀ p = 0.8␥ac / 2, detuning ⌬ = −7.5␥ac, decoherence ␥bc = 0.6␥ac, ⍀c / ␥ac = 23.4/ 2, 16.8/ 2, 8.4/ 2, 6 / 2, and 4 / 2. The factor of 1 / 2 is due to a different definition of the Rabi frequency in our case. Quantitative agreement is obtained by multiplying Rc by 16 due to the fact that the electric field defined is twice the physical field. We take the effective transverse cross section as A = 共␲w2o兲冑3 / 4 where wo = w1 / 冑2 is the waist and 2w1 = 280 ␮m is the diameter at 1 / e2 intensity 关9兴 of the pump laser. The factor of 冑3 / 4 leads to a smaller effective area by taking into account the transverse variation of the laser beam.

wavelengths are ␭s = 780 nm and ␭a = 795 nm. According to 关9兴, the correlated detection rate 关as reproduced in Fig. 2共a兲兴 is related to the normalized correlation as ¯ 共2兲⑀2⌬T = g共2兲R R ⑀2⌬T, Rc = G as s a

共50兲

¯ 共2兲 is an integrated correlation and has units of s−2, ⑀ where G is the detection efficiency, and ⌬T is the bin width. From 共2兲

Gas

共2兲 =˙ IsIa we have a useful relation between the correlated gas 共2兲 detection rate and Glauber’s absolute correlation Gas :

Rc =

冉

共2兲 Gas G共2兲 8⑀␧0Adetc RsRa⑀2⌬T = as I sI a ␯ a␯ s ប

冊

2

⌬T.

共51兲

Using the values of Ref. 关9兴, d = 5.6 ␮m 共for single-mode fiber兲, ⑀ = 0.3, and ⌬T = 1 ns in Eq. 共51兲 and using Eq. 共28兲 共including boundary operators兲, we compute the theoretical rates shown in Fig. 2共b兲 using the coupled equations of Balic et al. and Fig. 2共c兲 using our coupled equations. We obtain good quantitative agreement with the experimental data 关9兴 reproduced in Fig. 2共a兲. We find that results with the boundary operators alone do not give good quantitative agreement

with experimental data. This also shows that both the boundary operators and the noise operators are necessary to provide a correct quantitative description of the two-photon correlation. The comparison between noise and boundary operators will be elaborated on in a subsequent paper. VI. DISCUSSIONS AND CONCLUSIONS

We have presented a complete analytical quantum theory 共2兲 共2兲 of the two-photon correlations Gas , Gsa , and G共2兲 f f for the macroscopic Stokes and anti-Stokes fields generated in an amplifier 共extended coherent medium兲 in a double-⌳ scheme. We have derived coupled parametric oscillator equations valid for arbitrary detuning and intensity of the lasers and analytical expressions for the correlations. The hybrid results 关Fig. 2共b兲兴, obtained by using our analytical expression of the correlations along with the coupled equations of Balic et al. 关9兴 for the RED scheme give good quantitative agreement with the experimental data 关Fig. 2共a兲兴 without any fitting parameter for both high- and low-control ⍀c fields. Similarly, the results obtained using our more general coupled equations 关Fig. 2共c兲兴 correspond quite well with the hybrid results.

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PHYSICAL REVIEW A 75, 013820 共2007兲

FIG. 3. 共Color online兲 Comparison of the real 共solid line兲 and imaginary 共dotted line兲 pairs of the gain and loss Gs,a共␯兲 and coupling Ks,a共␯兲 coefficients for 共a兲 ⌬ = −7.5␥ac 共as used in experiment兲 and 共b兲 ⌬ = −75␥ac 共larger detuning兲. Other parameters follow from the experiment 关9兴: ⍀ p = 0.8␥ac / 2, ⍀c = 8.4␥ac / 2, and N␴acL = 11. Cases 共i兲 coefficients of Balic et al. 关9兴, 共ii兲 our coefficients, Eqs. 共7兲 and 共8兲, and 共iii兲 our coefficients but i␯ / c is neglected. 013820-7

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tive. This can be verified by comparing with the case when ␯ i c is neglected: namely, Figs. 3共aii兲 and 3共aiii兲 or 3共bii兲 and ␯ 3共biii兲. However, i c is an odd function which integrates to give zero 共when ␤s,a ⬍ ⬍ 1兲 and therefore does not affect the results. If the detunings and Rabi frequencies of the two lasers are the same, we have Gs = Ga and Ks = Ka. To summarize, the correctness of our generalized coefficients have been verified and they include the coexistence of the pump and control fields and the coherence between upper levels a and d, which give rise to additional resonant features in the spectra, particularly of Gs. In Fig. 4, we plot the normalized self-correlations, cross correlation, reverse correlation, and Cauchy-Schwartz correlation defined in Sec. V. The self-correlations change from 共2兲 thermal 共g共2兲 f f = 2兲 to coherent 共g f f = 1兲 nature as the time delay increases, but remain classical. The cross correlation 共2兲 CS , gas ⬎ ⬎ 1兲. The shows a large nonclassical correlation 共gas 共2兲 reverse correlation is negligible 关15兴 共gsa ⬃ 1兲 and remains 共2兲 CS ⬍ 1 which is consistent with bunching in gsa . classical gsa Finally, we conclude that the correspondence of our generalized coupled equations and the ability of our analytical expression for the correlation to provide good quantitative agreement with experimental data allow us to forecast new results in different regimes of laser parameters in a series of subsequent papers. 共2兲n

FIG. 4. 共Color online兲 Normalized self-correlations g f f , cross 共2兲n 共2兲n correlation gas , and reverse correlation gsa with CauchyCS CS Schwartz cross correlation gsa and reverse correlation gas for the experimental parameters N␴L = 11, ⍀ p = 0.8␥ac / 2, ⌬ = −7.5␥ac, ␥bc = 0.6␥ac, and ⍀c / ␥ac = 8.4/ 2.

The above correspondence can be understood through the following analysis. Figure 3共aii兲 shows that all our coefficients 共Ga, Ka, and Ks兲 except Gs agree almost perfectly with the coefficients of Balic et al. given in Eqs. 共B11兲–共B14兲. By st st st Gs from Eq. 共7兲 with wbb ⯝ ˜␳ba ⯝ 0 and ˜␳cd ⯝ − T1dc i⍀*p, then Tad共␯兲

ACKNOWLEDGMENTS

We would like to thank Professor S. E. Harris and Professor K. Wodkiewicz for helpful feedback and stimulating discussions. We gratefully acknowledge support of the Air Force Office of Scientific Research, the Office of Naval Research 共Grant No. N00014-02-1-0478兲, and the Robert A. Welch Foundation 共Grant No. A-1261兲. R.O. would like to thank MPQ, particularly Professor Herbert Walther for support, and Brain Korea 21 for funding, and P. Kolchin for information on experimental data.

applying the conditions for the RED scheme, 关 Tbc共␯兲 ⬎ ⬎ 1, 兩⍀ 兩2

兩⍀ 兩2

* 共␯兲 ⬎ ⬎ Tadc共␯兲 兴 and neglecting the term Tac共␯兲 ⬎ ⬎ Tadp共␯兲 , Tdb

with

␯ ic,

APPENDIX A: CLOSED SET OF HEISENBERGLANGEVIN-MAXWELL EQUATIONS

we obtain Gs ⯝ −

gs␬sTac共␯兲 , * TdcTdb共␯兲 Tbc共␯兲Tac共␯兲 + 兩⍀c兩2 兩⍀ p兩2

共52兲

which reduces exactly to the expression of Balic et al., Eq. 共B11兲 if we use ⌬ ⬎ ⬎ ␥dc , ␥db. Thus, our analytical results are consistent with the theory of Balic et al. even the though our coupled equations depend on additional parameters not found in the coupled equations of Balic et al. 关12兴—i.e., the decoherence rates ␥dc, ␥db, and ␥ab. The asymmetric feature of our Gs 关Figs. 3共aii兲 and 3共aiii兲兴 across ␯ = 0 is due to the quasiresonant effect of the pump and the coexistence with the control field. For larger detuning, Fig. 3共biii兲 shows that Gs becomes symmetric and agrees very well with the coefficient of Balic et al., Fig. 3共bi兲, while we confirm that the corresponding correlations agree perfectly. The divergence of the imaginary component of our Gs ␯ as 兩␯兩 increases is due to i c corresponding to the time deriva-

The Hamiltonian for the four-level system 关Fig. 1共a兲兴 in the Schrödinger picture is H = 兺 j兵Hoj + V j其 where H0j = and Vj = − ប

兺 ប␻␵兩␵ j典具␵ j兩 + 兺 ␵=a,b,c,d k,␭

冋兺

冉

† aˆk␭ aˆk␭ +

冊

1 ប␯k␭ 共A1兲 2

共gka␵兩a j典具␵ j兩 + gkd␵兩d j典具␵ j兩兲aˆk共t兲eik·r j + ⍀ p兩d j典

k,␵=b,c

⫻具c j兩ei共kp·r j−␯pt兲 + ⍀c兩a j典具b j兩ei共kc·r j−␯ct兲 + gsEˆs兩d j典

册

⫻具b j兩ei共ks·r j−␯st兲 + gaEˆa兩a j典具c j兩ei共ka·r j−␯at兲 + adj

共A2兲

are the free Hamiltonian and interaction Hamiltonian in the Schrödinger picture for single particles in the medium and gs = 㜷db / ប, ga = 㜷ac / ប, gka␵ = 㜷a␵Ek / ប, and Ek = 冑ប␯k / 2␧oV.

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CORRELATION OF PHOTON PAIRS FROM THE DOUBLE…

The subscripts p, s, q, and a stand for pump, Stokes control and anti-Stokes fields, respectively. The number density N in the extended medium of volume V is assumed to be sufficiently large n = VN ⬎ ⬎ 1 such that the discrete operators can be converted into continuous variables operators:

pˆbd = ␴ˆ bde−ikstei␯st. The “⫺” is for counterpropagating geometry, and the complex decoherences are

n

共2␲兲2 兺 ␴ˆ j 共t兲␦共z − z j兲 → ␴ˆ ␤␣共z,t兲, AN j=1 ␤␣

共A3兲

e−ik␤␣·r j → e−ik␤␣z ,

共A4兲 共A5兲

j 共t兲 are the quantum noise operawhere ␴ˆ ␤␣共t兲 = 兩␤ j典具␣ j兩, Fˆ␤␣ tors, and A is the effective transverse interaction area covered by the laser beams. However, the interparticle distance d ⬃ N−1/3 is larger than the optical wavelength ␭ so that the dipole-dipole interaction can be neglected. Hence, we obtain a set of 16 Heisenberg-Langevin atomic operator equations in continuous variables coupled to the 4 共including the adjoints兲 propagation equations for the Stokes Eˆs and antiStokes Eˆs. The above equations can be simplified to a closed set of equations that can be solved exactly for the field operators Eˆs,a if pˆxx and pˆab , pˆcd are taken as c numbers, corresponding to the steady-state density matrix elements. The coupled equations are

d pˆac = − Tac pˆac − ig*aEˆ†a共pˆcc − pˆaa兲 − i共⍀*c pˆbc − ⍀*p pˆad兲 dt + eikaze−i␯atFˆac ,

共A6兲

共A7兲

d pˆbc = − Tbc pˆbc − i共gs pˆdcEˆs − g*aEˆ†a pˆba兲 − i共⍀c pˆac − ⍀*p pˆbd兲 dt + e−ikqazei␯qatFˆbc ,

冉 冉

冊 冊

共A13兲

Tbc = i共␯ p − ␯s − ␻bc兲 + ␥bc ,

共A14兲

Tdb = i共␯s − ␻db兲 + ␥db ,

共A15兲

1 ¯ db + 1兲 + ⌫dc共n ¯ dc + 1兲 + ⌫ab共n ¯ ab + 1兲 ␥ad = 兵⌫db共n 2 dep ¯ ac + 1兲其 + ␥ad + ⌫ac共n ,

共A8兲

1 dep ¯ db + 1兲 + ⌫dc共n ¯ dc + 1兲其 + ␥db ␥db ⬟ 兵⌫db共2n , 共A19兲 2 where ⌫␣␤ are the spontaneous emission rates and ¯n f dep = 共e␤ប␯ f − 1兲−1 and ␥␣␤ are the dephasings due to phonons in the condensed phase or atomic collisions in gas. Here, we consider a cold sample with negligible inhomogeneous broadening 共1 / T*2 → 0兲. APPENDIX B: COEFFICIENTS FOR COUPLED PARAMETRIC EQUATIONS

We focus on the quasistatic regime which enables us to solve either Eqs. 共1兲 and 共2兲 or Eqs. 共5兲 and 共6兲 by Fourier transforming the t variable to ␯. The resulting equations can then be solved algebraically by Laplace transforming the z variable to q. The inverse Laplace transform gives the solutions 共10兲 and 共11兲. The gain and loss coefficients are

再 再

共A9兲

1⳵ ⳵ ˆ + Es共z,t兲 = i␬s pˆbd共z,t兲, c ⳵t ⳵z

共A10兲

1⳵ ⳵ ˆ† ± E 共z,t兲 = − i␬*a pˆac共z,t兲, c ⳵t ⳵z a

共A11兲

† = Fˆbd and the slowly varying atomic operators pˆac with Fˆdb ik z = ␴ˆ ace a e−i␯at, pˆad = ␴ˆ adeikqste−i␯qst, pˆbc = ␴ˆ bceikaqze−i␯aqt, and

共A17兲

1 dep ␥bc = 兵⌫ab¯nab + ⌫db¯ndb + ⌫ac¯nac + ⌫dc¯ndc其 + ␥bc , 2 共A18兲

d * pˆbd = − Tdb pˆbd + igs共pˆbb − pˆdd兲Eˆs + i共⍀ p pˆbc − ⍀c pˆad兲 dt † , + e−ikszei␯stFˆdb

Tad = i共␯c − ␯s − ␻ad兲 + ␥ad ,

1 dep ¯ ac + 1兲 + ⌫ab共n ¯ ab + 1兲其 + ␥ac , 共A16兲 ␥ac ⬟ 兵⌫ac共2n 2

d pˆad = − Tad pˆad + i共gs pˆabEˆs − g*aEˆ†a pˆcd兲 + i共⍀ p pˆac − ⍀*c pˆbd兲 dt + eikqsze−i␯qstFˆad ,

共A12兲

with the decoherence rates

n

共2␲兲2 兺 Fˆ j 共t兲␦共z − z j兲 → Fˆ␤␣共z,t兲, AN j=1 ␤␣

Tac = i共␯a − ␻ac兲 + ␥ac ,

冎 冎

st + Gs = − ␣s wbb

␯ i⍀c st i⍀ p st ˜␳ + ˜␳ −i , c Tad共␯兲 ba Tbc共␯兲 cd

共B1兲

st Ga = − ␣a wcc −

i⍀*c st i⍀*p st ␯ ˜␳ab − ˜␳dc − i , c Tbc共␯兲 Tad共␯兲

共B2兲

where the superscript “st” implies steady state and Tx共␯兲 st st st st st = Tx − i␯. The ˜␳␤␣ = 具pˆ␣␤ 共⬁兲典 and inversions wcc = ␳aa − ␳cc , st st st wbb = ␳dd − ␳bb are the steady-state solutions of the density st st and ˜␳cd matrix equations. Note that the imaginary parts of ˜␳ba lead to amplification Re兵Gs , Ga其 ⬍ 0 even without inversion st st , wcc ⬍ 0. wbb ␯ The term i c is due to the time derivatives of the field operators. It will not affect the final result for the correlation.

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precise form of Ref. 关25兴 in terms of our notation ⍀c = ⍀Bc / 2:

This has been verified numerically and can be understood analytically if we use Eqs. 共A6兲–共A11兲 in the retarded frame through the new variables t⬘ = t − z / c. The cross couplings are Ks = i␣s

再 再

冎 冎

␯ ⍀ p st ⍀c st ˜␳ab + ˜␳dc − i ␤s , c Tbc共␯兲 Tad共␯兲

Ka = − i␣a

Gs = − Tac共␯兲

共B3兲

⍀*c st ⍀*p st ␯ ˜␳ba + ˜␳cd − i ␤a , c Tad共␯兲 Tbc共␯兲

冉 冊 ⍀p ⌬

2

1 N␴db␥db 2 , D *共 ␯ 兲

共B4兲

1 N␴ac␥ac 2 , Ga = Tbc共␯兲 D *共 ␯ 兲

共B5兲

1 N␴ ␥ ⍀*p⍀*c 2 db db , Ks = − i ⌬ D *共 ␯ 兲

共B11兲

共B12兲

with the effective propagation coefficients g*a␬*a ␣a = Tac共␯兲 + ␦a

g s␬ s ␣s = * , Tdb共␯兲 + ␦s

共B13兲

and the power broadening frequencies 兩⍀ p兩2 兩⍀c兩2 + , ␦s = Tad共␯兲 Tbc共␯兲

兩⍀c兩2 兩⍀ p兩2 + , ␦a = Tad共␯兲 Tbc共␯兲

共B6兲

Ka = i

and the self-coupling coefficients appearing in Eqs. 共5兲 and 共6兲 are

␤s =

␤a =

冉 冉

冊 冊

1 1 g s␬ s ⍀ c⍀ p + , * * * ga␬a Tdb共␯兲 + ␦s Tbc共␯兲 Tad共␯兲 g*a␬*a

⍀*c ⍀*p

gs␬s Tac共␯兲 + ␦a

Y ad = ␣a

⍀c , Tad共␯兲

⍀*p , Tad共␯兲

Y bc = − ␣a

⍀p , Tbc共␯兲

Y bc = − ⍀*c

冕

ei␯2共t+␶兲

−⬁

= ei⌬kz

冕

⬁

−⬁

d␯2 2␲

冕

ei␯2共t+␶兲

−⬁

d␯2 2␲

e−i␯1t

冕

⬁

−⬁

N冑␴db␴ac␥db␥ac , D*

共B17兲

N␴ac␥ac , D 兩㜷兩2␻

Y ac = − i␣a .

⬁

共B16兲

2

共B18兲 兩㜷兩2␻

= 2gs␬s, N␴ac␥ac = 2ga␬a, and ␴db = បc␧o␥dbdb , ␴ac = បc␧o␥acac . Note that our definition of the Fourier transform is opposite to that of Balic et al. 关9兴. APPENDIX C: DERIVATION OF Eq. (32)

We derive the cross correlation of the noise terms in Eq. 共28兲 using the solutions in frequency space, Eqs. 共10兲 and 共11兲, as follows:

For comparison with our coefficients, we rewrite the coefficients of Balic et al. 关9兴 共Superscript B兲 based on a more

⬁

N␴ac␥ac , D*

⍀p ⌬

where D共␯兲 = 兩⍀c兩2 − 共i␥ac − ␯兲共i␥bc − ␯兲 = DB / 4 with N␴db␥db

共B10兲

ˆ 共t + ␶兲N ˆ 共t兲典 = 具N a s

冉 冊

Xac = − iTbc共␯兲

Xbd = i␣s , 共B9兲

⍀*c , Tbc共␯兲

共B14兲

共B15兲

Xac = − ⍀*c 共B8兲

1 N␴ac␥ac 2 , D *共 ␯ 兲

⍀*p N冑␴db␴ac␥db␥ac , D* ⌬

共B7兲

1 1 + , Tbc共␯兲 Tad共␯兲

Xbc = − ␣s

⌬

Xbc = − iTac共␯兲

with the propagation constants ␬s = N㜷bdc␮o␯s / 2 and ␬a = N㜷cac␮o␯a / 2. The real parts of Gs,a are even functions across ␯ and the imaginary parts are odd functions and viceversa for Ks,a. The coefficients in Eqs. 共3兲 and 共4兲 are Xad = ␣s

⍀*p⍀*c

d␯1 ˆ 具Na共z, ␯2兲Nˆs共z, ␯1兲典 2␲

e−i␯1t

d␯1 兺 2␲ x,x

⬘

冕 冕 z

z

dz2

0

0

s ˆ† ˆ dz1Ca* x 共␰2, ␯2兲Cx⬘共␰1, ␯1兲具Gx 共z2, ␯2兲Gx⬘共z1, ␯1兲典.

n The noise products in the frequency domain are related to the diffusion coefficients 2Dx,x defined in Appendix E, ⬘

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ˆ †共z , ␯ 兲G ˆ 共z , ␯ 兲典 = 具G x⬘ 1 1 x 2 2 =

冕 冕

⬁

dt2e−i␯2t2

−⬁ ⬁

冕 冕

⬁

−⬁

dt2e−i␯2t2

−⬁

⬁

dt1ei␯1t1e−i␪xei␪x⬘具Fˆ†x 共z2,t2兲Fˆx⬘共z1,t1兲典 dt1ei␯1t1

−⬁

˜ n 共z , t 兲 = 2Dn e−i␪xei␪x⬘ are the slowly varying where 2D x,x⬘ 1 1 x,x⬘ ˜ n vary much slower than diffusion coefficients. Since 2D x,x⬘ −i␯2t2 i␯1t1 the exponentials e and e , we can take their steady ˜ n 共z 兲 but keep the spatial dependence. We state-values 2D x,x⬘ 1 then have a ␦ function in frequency that enables further simplification of the number of integrals:

共2␲兲2 ˜ n 2Dx,x⬘共z1,t1兲␦共t2 − t1兲␦共z2 − z1兲, AN

APPENDIX E: DIFFUSION COEFFICIENTS

The quantum noise correlation for two discrete particles is j ␦ correlated in time 具Fˆ jx共t兲Fˆkx⬘共t⬘兲典 = 2Dx,x 共t兲␦共t − t⬘兲␦ jk. So, ⬘ in one spatial dimension, n

共2␲兲4 ˆj ˆk 具Fˆx共z,t兲Fˆx⬘共z⬘,t⬘兲典 = 2 兺 具Fx共t兲␦共z − z j兲Fx⬘共t⬘兲 共AN兲 j,k=1

3 ˜ n 共z 兲␦共z − z 兲␦共␯ − ␯ 兲. ˆ 共z , ␯ 兲典 = 共2␲兲 2D ˆ †共z , ␯ 兲G 具G x⬘ 1 1 2 1 1 2 x 2 2 x,x⬘ 1 AN

⫻␦共z⬘ − zk兲典

共C3兲

n

共2␲兲4 兺 2Dx,xj ⬘共t兲␦共t − t⬘兲␦共z − z j兲 共AN兲2 j=1

=

Finally, we have ˆ 共t兲典 = ei⌬kz 2␲ 兺 具Nˆa共t + ␶兲N s AN x,x ⫻

冕

⬘

冕

⬁

⫻␦共z⬘ − z j兲

e i␯␶

−⬁

⯝

z

0

共C2兲

˜ n 共z兲Ca*共␰, ␯兲Cs 共␰, ␯兲dzd␯ , 2D x x,x⬘ x⬘

共E1兲

共C4兲 which is identical to the first term of Eq. 共32兲. APPENDIX D: COMMUTATION RELATION FOR BOUNDARY FIELD OPERATORS

共2␲兲2

j 共t兲␦共z − z j兲. where Dx,x⬘共z , t兲 = AN 兺nj=1Dx,x ⬘ Hence, the normal-ordered noise correlations are related to the diffusion coefficients as

共2␲兲2 n 具Fˆ†x 共z,t兲Fˆx⬘共z⬘,t⬘兲典 = 2Dx,x⬘␦共z − z⬘兲␦共t − t⬘兲 共E2兲 AN

The Stokes and anti-Stokes operators at boundary are ˜E 共0,t兲 = E 兺 aˆ e−i␯kteikxx+ikyy , f f k

共2␲兲2 2Dx,x⬘共z,t兲␦共t − t⬘兲␦共z − z⬘兲, AN

共D1兲

and similarly for the antinormal-ordered noise correlations

k

Eˆ f 共0, ␯兲 = E f 兺 aˆk2␲␦共␯ − ␯k兲eikxx+ikyy ,

共2␲兲2 an † 具Fˆx共z,t兲Fˆx⬘共z⬘,t⬘兲典 = 2Dx,x⬘␦共z − z⬘兲␦共t − t⬘兲. AN

共D2兲

共E3兲

k

where E f = 冑ប␯ f / 2␧oV. Using

关aˆk , aˆk† ⬘兴 = ␦k , k⬘

we find

关Eˆ f 共0, ␯兲,Eˆ†f 共0, ␯⬘兲兴 = 共2␲E f 兲2 兺 ␦共␯ − ␯k兲␦共␯⬘ − ␯k兲. k

共D3兲 The conversion to integration L A 2 L → 共2␲兲2 兰⬜d k 2␲ 兰 dk . . . ⯝ 2␲ 兰 dk¯ gives ប␯ f 2␲L 关Eˆ f 共0, ␯兲,Eˆ†f 共0, ␯⬘兲兴 = 2␧0V c ⯝

冕

using

兺k . . .

␦共␯ − u兲␦共␯⬘ − u兲du

ប␯ f ␲ ␦共␯ − ␯⬘兲, ␧0Ac

共D4兲

where we have used the identity 兰f共u兲␦共␯ − u兲␦共␯⬘ − u兲du = f共␯兲␦共␯ − ␯⬘兲 and V = AL.

n an and antinormal-ordered 2Dx,x coThe normal-ordered 2Dx,x ⬘ ⬘ efficients are calculated using Einstein’s relation and the atomic equations for continuous variables. For the purpose of evaluating the correlations, these coefficients are expressed in terms of the steady-state matrix elements ␳stij and the thermal photon number ¯n f . Note that when the thermal temperature is zero, the st st ⯝ ␳aa ⯝ 0. If the excited populations are negligible, ␳dd dep dep dep dep dep + ␥bc , the dephasings are such that ␥dc = ␥ad + ␥ac = ␥db n st ˜ , only finite diffusion coefficients are 2Dac,ac = 2␥ac␳cc an st an st n st ˜ ˜ ˜ = 2 ␥ ␳ , 2D = 2 ␥ ␳ , and 2D = 2 ␥ ␳ . 2D db bb bc cc bc bb bc,bc bd,bd bc,bc This means that the correlations due to noise operators are governed by the decoherence rates ␥db and ␥ac of the Stokes and anti-Stokes transitions as well as the decoherence ␥bc between the ground states.

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具Bs†共t兲B†a共t + ␶兲Ba共t + ␶兲Bs共t兲典 ⬀ 具共aˆs† + aˆa兲共aˆ†a + aˆs兲

APPENDIX F: DECORRELATION FOR GAUSSIAN NOISE

⫻共aˆa + aˆs†兲共aˆs + aˆ†a兲典,

The solutions of the Stokes and anti-Stokes operators are composed of the boundary and the noise parts

ˆ 共L,t兲, Eˆs共L,t兲 = Bˆs共L,t兲 + N s

共F1兲

共F3兲 where coefficients do not affect our argument and are left out. Straightforward expansion gives 2具nˆs典2 + 4具nˆs典 + 4具nˆs典具nˆa典 + 2具nˆa典2 + 4具nˆa典 + 2 = 具nˆs2典 + 3具nˆs典 + 4具nˆs典具nˆa典 + 具nˆ2a典 + 3具nˆa典 + 2,

ˆ †共L,t兲, Eˆ†a共0,t兲 = Bˆ†a共L,t兲 + N a

共F2兲

共F4兲

where the second line follows for the thermal state, 具nˆs2典 = 2具nˆs典2 + 具nˆs典. If we use Gaussian decorrelation on Eq. 共3兲, we have 具共as† + aa兲共a†a + as兲典具共aa + as†兲共as + a†a兲典

ˆ 共L , t兲, and N ˆ †共L , t兲 are the inverse where Bˆs共L , t兲, Bˆa共L , t兲, N s a Fourier transforms of the terms in Eqs. 共10兲 and 共11兲. Since the noise operators in vacuum are well known to satisfy Gaussian decorrelation and the odd products of noise operators vanish, the noise part satisfies the Gaussian decor共2兲 relation. Thus, we only need to verify one term in Gas 共␶兲 † † ˆ ˆ ˆ ˆ = 具Es 共t兲Ea共t + ␶兲Ea共t + ␶兲Es共t兲典—i.e., the term with all boundary operators. Since the boundary parts Bˆ f contain Eˆ f 共0 , ␯兲 ⬀ aˆ f , we can write

which proves that the Gaussian decorrelation applies to operators in the thermal state, as well as the vacuum state: when 具nˆs典 = 具nˆa典 = 0. Thus, the decorrelation of Eq. 共27兲 is justified.

关1兴 S. A. Moiseev and B. S. Ham, Phys. Rev. A 71, 053802 共2005兲; Zhuan Li, De-Zhong Cao, and Kaige Wang, Phys. Lett. A 341, 366 共2005兲; Xiao-xue Yang and Ying Wu, J. Opt. B: Quantum Semiclassical Opt. 7, 5456 共2005兲; L. Deng and M. G. Payne, Phys. Rev. A 71, 011803共R兲 共2005兲; Ryan S. Bennink, Alberto M. Marino, Vincent Wong, Robert W. Boyd, and C. R. Stroud, Jr., ibid. 72, 023827 共2005兲; Yang Xiao-Xue and Wu Xuan, Chin. Phys. Lett. 22, 2816 共2005兲; A. André, L.-M. Duan, and M. D. Lukin, Phys. Rev. Lett. 88, 243602 共2002兲. 关2兴 M. O. Scully and K. Drühl, Phys. Rev. A 25, 2208 共1982兲. 关3兴 A. Kuzmich et al., Nature 共London兲 423, 731 共2003兲; C. W. Chou, S. V. Polyakov, A. Kuzmich, and H. J. Kimble, Phys. Rev. Lett. 92, 213601 共2004兲. 关4兴 C. H. van der Wal et al., Science 301, 196 共2003兲. 关5兴 Wei Jiang, C. Han, P. Xue, Z. M. Duan, and G. C. Guo, Phys. Rev. A 69, 043819 共2004兲. 关6兴 M. D. Lukin, A. B. Matsko, M. Fleischhauer, and M. O. Scully, Phys. Rev. Lett. 82, 1847 共1999兲. 关7兴 M. D. Lukin, P. R. Hemmer, M. Loeffler, and M. O. Scully, Phys. Rev. Lett. 81, 2675 共1998兲. 关8兴 M. O. Scully, in Proceedings of the Conference on Effects of Atomic Coherence and Interference in Quantum Optics, Crested Butte, Colorado, 1993 共IOP, Bristol, 1994兲. 关9兴 V. Balic, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, Phys. Rev. Lett. 94, 183601 共2005兲; Pavel Kolchin, Shengwang Du, Chinmay Belthangady, G. Y. Yin, and S. E. Harris, ibid. 97, 113602 共2006兲. 关10兴 Marlan O. Scully and C. H. Raymond Ooi, J. Opt. B: Quantum Semiclassical Opt. 6, S816 共2004兲. 关11兴 C. Kurtsiefer et al., J. Mod. Opt. 48, 1997 共2001兲; Ling-An Wu, Min Xiao, and H. J. Kimble, J. Opt. Soc. Am. B 4, 1465

共1987兲. 关12兴 A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 共2000兲; Milena D’Angelo, Maria V. Chekhova, and Yanhua Shih, ibid. 87, 013602 共2001兲. 关13兴 The backward geometry has been shown to exhibit amplified reflection, phase conjugation and oscillations without a cavity 关14兴 and perfect squeezing 关6兴. 关14兴 Amnon Yariv and David M. Pepper, Opt. Lett. 1, 16 共1977兲. 关15兴 S. E. Harris 共private communication兲 expects the reverse correlation would be negligibly small. 关16兴 A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, Phys. Rev. Lett. 86, 3925 共2001兲. 关17兴 Many previous works have used the macroscopic quantum field Eˆ to describe the spatial-temporal evolution of the Stokes and anti-Stokes emissions in an amplifier. Glauber and Haake 关18兴 used it to describe the initiation of superflourescence in a thin sample. Later, Mostowski and Raymer 关19兴 developed the formalism by including the Langevin operators to study stimulated Raman scattering 共SRS兲. A semiclassical approach 共without quantum noise兲 employing the macroscopic fields has also been used 关20兴. 关18兴 R. J. Glauber and F. Haake, Phys. Lett. 68A, 29 共1968兲. 关19兴 J. Mostowski and M. G. Raymer, Opt. Commun. 36, 237 共1981兲. 关20兴 S. A. Akhmanov, Mater. Res. Bull. 4, 455 共1969兲; N. I. Shamrov, Opt. Spectrosc. 93, 95 共2002兲. 关21兴 Equations 共A6兲–共A11兲 are reduced to the coupled equations 共1兲 and 共2兲 by first taking the pˆ f f and pˆab, pˆdc to be the steady-state density matrix elements: linearization. The coupled equation Efor fields are derived by solving the linearized equations for

+ 具共as† + aa兲共aa + as†兲典具共a†a + as兲共as + a†a兲典 + 具共as† + aa兲共as + a†a兲典具共a†a + as兲共aa + as†兲典 = 2共具nˆs典 + 具nˆa典 + 1兲2 ,

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standard procedure for Gaussian operators, as for the radiation fields and the noise operators described in terms of the creation and annihilation bosonic operators. Detailed proof is given in Appendix F. 关25兴 Danielle A. Braje, Vlatko Balic, Sunil Goda, G. Y. Yin, and S. E. Harris, Phys. Rev. Lett. 93, 183601 共2004兲. 关26兴 We have used ⌫db = ⌫ac = ⌫0 = 2␲5.89⫻ 106 s−1 and asssumed ⌫dc = ⌫ab = 0 for the decay rates and 㜷dc = 㜷db = 㜷ac = 㜷ab = 㜷0 = 2.568⫻ 10−29 C m for the transition dipole moments.

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