PHYSICAL REVIEW A 75, 043817 共2007兲

Quenching the collective effects on the two-photon correlation from two double-Raman atoms C. H. Raymond Ooi* Max-Planck-Institut für Quantenoptik, D-85748, Garching, Germany and Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea 共Received 26 January 2007; published 26 April 2007兲 We obtain an analytical expression for two-photon correlation G共2兲 from two atoms driven in a doubleRaman 共or ⌳兲 configuration where collective effects such as superradiant, subradiant, and dipole-dipole interaction are included. It is found that the collective effects on the G共2兲 can be quenched to some extent by a resonant control laser field. The collective effects provide features via G共2兲 that enable the two atoms to be resolved at subwavelength separation. We also identify an effect in the double Raman scheme due to the collective effects and the control field, i.e., the Stokes and anti-Stokes frequencies are increased by fourfold. DOI: 10.1103/PhysRevA.75.043817

PACS number共s兲: 42.50.Fx, 42.50.Ar, 32.80.⫺t, 42.65.Ky

II. TWO-ATOM DYNAMICS

I. INTRODUCTION

It is known that correlated multiphoton detection scheme enhances the resolving power of two point sources, for example, by using Glauber’s two-photon correlation G共2兲 关1兴. Subwavelength resolution using interference of classical thermal light has recently been reported 关2兴. Nonclassical light is also capable of providing an enhancement factor. In particular, the Raman-EIT 共electromagnetic induced transparency兲 scheme 共Fig. 1兲 which shows photon antibunching and quantum interference 关3兴 can be used to resolve two atoms as close as ␭ / 8 by measuring the two-photon correlation in an interferometric setup 关5兴. However, at subwavelength distance, the presence of collective phenomena 共superradiant and subradiant兲 via dipole-dipole interaction due to vacuum fields 关6兴 may invalidate the approximate analysis based on the summing of two-photon amplitudes ␺ j 共j = 1 , 2兲 of two independent atoms, i.e., ␺1 + ␺2. The twoatom system 共of Fig. 1兲 must be solved as a whole, taking into account the collective many-particle radiation states. The main purpose of this paper is to study the physics of the G共2兲 in the simplest driven collective many-body system that produces nonclassically correlated photon pairs. A particularly interesting question is, how does the coexistence of the dipole-dipole interaction and the control field in the Raman-EIT scheme affect the two-photon correlation? The G共2兲 for a single atom with Raman-EIT scheme has been studied by various methods 关7兴. It is convenient to use the Schrödinger’s equation approach 关4兴 to obtain an analytical expression for the G共2兲 that would facilitate physical interpretations. We shall focus our discussions of the physics around small interatomic distance r. We show how the collective effects of dipole-dipole interaction can be quenched by a control field. We also find that G共2兲 as a function of time delay ␶ between the Stokes and anti-Stokes photons contains features due to the dipole-dipole interaction that enable the two atoms at subwavelength distance to be resolved without using the interferometric setup.

*Email address; [email protected] 1050-2947/2007/75共4兲/043817共6兲

The interaction Hamiltonian for two independent 共noninteracting兲 Raman-EIT atoms in free space radiation and driven by two laser fields 共Fig. 1兲 in the interaction picture is simply the independent sum of two Hamiltonians, each for a single atom, Vˆ = − ប

冋兺

−i␦t Gk共j兲aˆk† 兩b j典具c j兩e−i⌬tei⌬kt + ⍀共j兲 兩a j典具b j兩 c e

j=1,2,k

册

+ 兺 gq共j兲aˆq† 兩c j典具a j兩ei⌬qt + adj . q

共1兲

Each Hamiltonian describes a single Raman-EIT atom 关3兴, the Gk共j兲 = gk共j兲⍀共j兲 p / ⌬ is the spontaneous Raman coupling, ⍀共j兲 共⬍ ⬍ ⌬兲 and ⍀共j兲 p c are the Rabi frequencies of the jth atom coupled to the far detuned weak pump laser that drives the c ↔ d transition and the control laser that drives the b ↔ a ␻k ជ · ␧ˆ k* 兲bd 2␧ បV e−ik·r j and gq共j兲 transition, respectively, gk共j兲 = −共㜷 o

ជ · ␧ˆ q* 兲ca = −共㜷

冑

␻q −iq·r j 2␧oបV e

冑

are the couplings of the radiation to the b ↔ d transition and the c ↔ a transition, respectively, r j denotes the position of the jth atom, and ⌬k共q兲 = ␯k共q兲 − ␻db共ac兲 with ⌬ = ␯ p − ␻dc are the detunings.

FIG. 1. 共Color online兲 Two atoms driven by a pump laser and a control laser with Rabi frequencies ⍀ p and ⍀c, respectively, in the Raman-EIT scheme. Correlated photon pairs, called Raman emission doublet 共RED兲 are emitted. The atoms are separated by a distance r which can be close to the emission wavelength ␭ ⬟ 2␲c / ␻ ⯝ 2␲c / ␯.

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©2007 The American Physical Society

PHYSICAL REVIEW A 75, 043817 共2007兲

C. H. RAYMOND OOI

We consider a sufficiently weak pump field such that states with two and more Stokes photons are negligible. Thus, the collective two-atom-radiation state vector is 兩⌿共t兲典 = C兩c1,c2,0典 + 兺 兵Bk共1兲兩b1,c2典 + Bk共2兲兩c1,b2典其兩1k典

cally give rise to the f共r兲 function 关see Eq. 共A8兲兴 which accounts for the dipole-dipole interaction, without introducing any Hamiltonian to describe the direct interaction between the two atoms. Here, the dipole-dipole interaction is indirect, mediated by the vacuum radiation.

k

III. COLLECTIVE TWO-PHOTON AMPLITUDE

+ 兺 兵Ak共1兲兩a1,c2典 + Ak共2兲兩c1,a2典其兩1k典 k

+ 兺 Ck,q共t兲兩c1,c2,1k,1q典,

共2兲

k,q

where C, Bk共j兲, Ak共j兲, and Ck,q are the probability amplitudes for the collective basis states: 共i兲 both atoms in c with no photon, 共ii兲 one of the atoms driven to b by spontaneous Raman process, and 共iii兲 coherently coupled to a with the emission of a Stokes photon k, and 共iv兲 the same atom that emits photon k decays to level c emitting an anti-Stokes photon q. Note that the intermediate states b and a are entangled; we know only one is excited but never know which one. The Schrödinger’s equation gives a set of coupled equations for the coefficients C, Bk共j兲, Ak共j兲, and Ck,q that are solved in Appendix using standard procedure 关4兴. As shown in the Appendix, the above Hamiltonian and state vector automati-

␺±共±兲共Bj,Aj兲 = −

兩⍀c兩2

1 − 关兵␺+共+兲共B2,A1兲 − ␺−共+兲共B2,A1兲其 2 − 兵␺+共−兲共B2,A1兲 − ␺−共−兲共B2,A1兲其 + 共1 − 2兲兴, 共3兲 where 共⫾兲 are the signs for superradiant 共⫹兲 and subradiant 共⫺兲 cases, with the partial amplitudes

冎

e−i␯␶Aje−i␻␶Bje−␥R␶Aj ,

共4兲

C21 ˜ 共±兲 ˜ 共±兲 共±兲 共±兲 ⌰共␶A1兲e−兵i共␯⬘⫿⍀ 兲+关␥R−共1/4兲⌫ 兴其␶A1e−关i共␻⬘±⍀ 兲+共1/4兲⌫ 兴␶B2⌰共␶B2 − ␶A1兲, ˜ 共±兲 2⍀

共5兲

共

1 2␦

+

1 i 4 ⌫共±兲

˜ 共±兲 ⫿⍀

兲

where ␯⬘ ⬟ ␯ + 21 ␦ and ␻⬘ ⬟ ␻ − 21 ␦ are the effective Stokes and anti-Stokes frequencies with ␯ ⬟ ␯ p − ␻bc , ␻ ⬟ ␯c + ␻bc = ␦ + ␻ac; ␶BjAj = ␶Bj − ␶Aj; and ␶Aj = tA − rAj / c , ␶Bj = tB − rBj / c are the retarded times. The physical significance of each term in Eq. 共3兲 is illustrated in Fig. 2. The coefficients are

C21 = −

关兵␺+共+兲共Bj,Aj兲 − ␺−共+兲共Bj,Aj兲其 兺 j=1,2 − 兵␺+共−兲共Bj,Aj兲 − ␺−共−兲共Bj,Aj兲其兴

再

i⌫f共r兲

Cj = −

␺RED共B,A兲 =

1 Cj ˜ 共±兲 ˜ 共±兲 共±兲 共±兲 ⌰共␶Aj兲 共±兲 e−兵i共␯⬘⫿⍀ 兲+关␥R−共1/4兲⌫ 兴其␶Aje−关i共␻⬘±⍀ 兲+共1/4兲⌫ 兴␶Bj⌰共␶BjAj兲 共±兲 2 ˜ 2⍀

+

␺±共±兲共B2,A1兲 =

In order to compute the two-photon correlation G共2兲 = 兩␺RED共B , A兲 + ␺RED共A , B兲兩2, the steady state solution for Ck,q, i.e., Eq. 共A9兲 is used. After lengthy calculations involving pole integrations 关9兴, we obtain the two-photon amplitude as the main result

共␯␻/c2兲3 ⍀ p ⍀cKAjQBjei共kc+kp兲·r j , 共4␲␧o兲2 ⌬

共6兲

共␯␻/c2兲3 ⍀ p ⍀cKA1QB2eikc·r2eikp·r1 , 共4␲␧o兲2 ⌬

共7兲

and the effective complex Rabi frequencies that contain the collective effects 共via ⌫共±兲兲 and the ac Stark shift 共via ⍀c兲 are ˜ 共±兲 = 冑⍀2 − 共 1 ⌫共±兲 − i 1 ␦兲2 . ⍀ c 2 4

共8兲

The coherent phases kq · r j = kqr j cos ␣q 共q = p , c兲 give rise to an additional interference feature shown in Fig. 3. The ⌫共±兲 = ⌫关1 ± f共r兲兴 in Eqs. 共4兲, 共5兲, and 共8兲 are complex; where f共r兲 = g共r兲 + ih共r兲 is the dimensionless collective parameter that includes the modified collective 共subradiant and superradiant兲 decay rate factor g共r兲 and the vacuum-induced coherent dipole-dipole interaction factor h共r兲. Full expressions for g共r兲 , h共r兲 are well known and can be found, for example in 关6,8兴, but we reproduce them in Fig. 4共c兲 for convenience. For sufficiently large interatomic distance: ␻r / c , ␯r / c Ⰷ 1, we have negligible collective effects since f共r兲 → 0. Note that for sufficiently weak field ⍀c Ⰶ ⌫, Eq. 共8兲 gives 1 共±兲 ˜ ⍀ ⯝ i 4 ⌫共±兲 + 21 ␦. Here, the real part 21 ␦ ⫿ 41 ⌫h simulates the role of Rabi frequency. This indicates the contributions of the collective 共two atom兲 effect, namely the vacuum induced coherent dipole-dipole interaction, and finite laser detuning ␦ to the coherent quantum phenomena such as Rabi oscillations,

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QUENCHING THE COLLECTIVE EFFECTS ON THE TWO-…

and nonclassical effect 共photon antibunching兲, as shown in Fig. 3. In the limit of large atomic separation, the cooperative effects vanish since f共r兲 → 0 and taking ␦ = 0 we find ⍀±共±兲 ˜ where ⍀ ˜ = 冑⍀2 − 共⌫ / 4兲2. Then, ␺共−兲共Bj , Aj兲 ⯝ −␺共+兲 ⯝ ±⍀ c ± ± ⫻共Bj , Aj兲 and Eq. 共3兲 reduces to the sum of amplitudes of two independent atoms 关3,9兴,

␺RED共B,A兲 ⯝

兺 j=1,2

C j i⍀˜ 共␶ −␶ 兲 −i⍀˜ 共␶ −␶ 兲 Bj Aj 兴 关e Bj Aj − e ˜ 2⍀

⫻⌰共␶Bj − ␶Aj兲e−共i␯⬘+␥R兲␶Aje−i␻⬘␶Bje−共1/4兲⌫共␶Bj−␶Aj兲 . 共9兲 Here, Eq. 共9兲 corresponds to the asymmetrical emission paths shown in Fig. 2共a兲. The symmetrical paths of Fig. 2共b兲 do not contribute since the paths that correspond to the subradiant 关superscript 共⫺兲兴 cancel those paths for the superradiant 关superscript 共⫹兲兴. IV. COHERENT CONTROL VIA COLLECTIVE EFFECTS

The essence of this paper can be found in the complex ˜ 共±兲 and we discuss how the field depenRabi frequency ⍀ dence modifies the frequency components, Rabi oscillations and decay rates. A. Multiple frequencies

The amplitude Eq. 共9兲 for large r has simple meaning: the Stokes photon with frequency ␯⬘ is emitted at an emission time ␶Aj with probability exp兵−␥R␶Aj其 from the jth atom and goes to detector A, the anti-Stokes photon with frequency ␻⬘ is emitted at an emission time ␶Bj with the probability exp兵− 41 ⌫共␶Bj − ␶Aj兲其 from the jth atom and goes to detector B. The general amplitude equations 共3兲–共5兲 can be inter˜ 共±兲 and preted similarly. Thus, we find that the real part Re ⍀ 共±兲 the imaginary part of ⌫ give the eight possible frequency ˜ 共±兲共⫿兲 1 h⌫ for Stokes photon and components: ␯⬘ ⫿ Re ⍀ 4 ˜ 共±兲共±兲 1 h⌫ for anti-Stokes photon. These frequen␻⬘ ± Re ⍀ 4 cies give rise to the various possible interfering paths shown in Fig. 2. Clearly, the frequency sidebands are due to Rabi splitting and dipole-dipole interaction. As found in Fig. 4共a兲, the distinction between these lines increase for small r and large ⍀c. These are real frequencies that can be obtained by calculating the power spectrum via the field correlation 具E†共t + ␶兲Eˆ共t兲典. Of course, they would not show up in the correlated measurement of the G共2兲. Only the beats among the frequencies are found if we make a Fourier transformation of the G共2兲. It is essentially the beats 共due to the dipoledipole interaction兲 that were used for subwavelength measurement of two atoms in Ref. 关10兴. We expect that a larger number of atoms would give a larger number of sidebands. This could be an alternative method to nonlinear optical processes for the generation of multiple new frequencies. We shall not pursue this further here since it is not the main focus and beyond the present scope. Further analysis will be reported elsewhere.

FIG. 2. 共Color online兲 Possible 共a兲 asymmetrical and 共b兲 symmetrical emission paths based on the result of two photon amplitude, Eq. 共3兲 关red 共dark兲 lines: for Stokes; green 共light兲 lines: for ˜ 共±兲¯, dashed lines: for ¯ anti-Stokes; solid lines: for ¯ + Re ⍀ 共±兲 ˜ −Re ⍀ ¯.兴 Note that Eq. 共3兲 also predicts the existence of eight frequency sidebands, i.e., a fourfold increase in the number of frequencies due to Autler-Townes splitting by the control field ⍀c and dipole-dipole interaction, as shown below 共b兲.

B. Rabi oscillations and threshold

The Rabi oscillation period is governed by the real part of 共±兲 ˜ ⍀ , particularly by the control field ⍀c, g, and h. Figures ˜ 共±兲 as a function of r in two perspec4共a兲 and 4共b兲 show ⍀ tives. There is a threshold ⍀thr for transition between overdamp 共⍀c ⬍ ⍀thr兲 and oscillatory 共⍀c ⬍ ⍀thr兲 regimes. For large r, the threshold is clearly seen around 41 ⌫ 共as in single atom case兲 in Fig. 4共b兲. If h can be neglected and ␦ = 0, we have ⍀thr ⯝ 41 ⌫共1 + g兲. The overdamp and oscillatory regimes are well defined by hyperbolic and trigonometric functions, ˜ 共±兲 are either real or purely imagirespectively, only when ⍀ nary. As r decreases h becomes significant, the argument inside 冑¯ of Eq. 共8兲 becomes complex and there is no simple analytical expression for the threshold.

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C. H. RAYMOND OOI

FIG. 3. 共Color online兲 Plots of two-photon correlation 共normalized兲 in the regime where dipole-dipole interaction is significant for ␣c = ␣ p = 90° with different control fields 共a兲 ⍀c = 0.2⌫ corresponds to overdamp regime in the limit of large r, 共b兲 ⍀c = 3⌫, and 共c兲 ⍀c = 7⌫, and for ␣c = ␣ p = 0° 共with the same set of control fields兲 which gives the interference effect of the coherent phase factor ei共kp+kc兲·r j. Note that the onset to a more rapid Rabi oscillation in each plot is dependent on ⍀c. This feature can be understood by looking at Fig. 4. We have considered parallel transitions ⌬M = 0 共with identical results for ⌬M = ± 1兲. For simplicity, we take ␦ = 0. The results are independent of the angle ␣Dj between the photon from atom j to detector D and the interatomic axis. C. Multiple decay rates

Since ⌫共±兲 is complex, Eq. are eight effective decay ˜ 共±兲 − ␥ 其␶ ) exp(兵 41 ⌫关1共±兲g兴 ⫿ Im ⍀ R Aj 1 ˜ 共±兲其␶ ) and exp(兵 ⌫关1共±兲g兴 ⫿ Im ⍀

共2兲 shows that there rates governed by for the Stokes photon for the anti-Stokes phoBj 4 ton. At r = n␭ / 2 where n = 1 , 2 , 3 , . . .. we find that the value of ˜ 共±兲 jumps from positive to negative or otherwise for ⍀ Im ⍀ c

˜ 共±兲 also depends on below ⍀thr. Since the imaginary part of ⍀ ⍀c, the laser field also affects the multiple decay rates. Thus, the laser field not only modifies the Rabi oscillations in the G共2兲 but also its damping rate as a function of ␶. V. RESULTS OF THE CORRELATION

By using filters at the detectors such that detector A detects photon ␯ and B detects photon ␻, the term ␺RED共A , B兲

~ Re W ( + ) / G

a)

c)

b)

˜ 共+兲 关from Eq. 共8兲兴 for ⍀ from 0.05⌫ to 3⌫ for ⌬M = 0. The dashed line estimates the turning FIG. 4. 共Color online兲 共a兲 The real part of ⍀ c ˜ 共+兲 versus r and ⍀ . The kink points close to r = ␭, ␭ / 2, and 3␭ / 2 correspond to the points rt of the curves with different ⍀c. 共b兲 The Re ⍀ c ˜ 共+兲 = 0 correspond to the transition between overdamp and oscillatory regimes. Similar features zeros of the h function. The points where Re ⍀ ˜ 共−兲. 共c兲 g共r兲 and h共r兲 for parallel transition ⌬M = 0 共solid line兲 and perpendicular ⌬M = ± 1 共dashed-dotted line兲, ⌬M is the are found for ⍀ change in magnetic quantum number, jn is spherical Bessel of order n. 043817-4

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PHYSICAL REVIEW A 75, 043817 共2007兲

obtained by interchanging A with B in Eq. 共3兲 can be disregarded. Figure 3 shows the variations of G共2兲共r , ␶兲 = 兩␺RED共r , ␶兲兩2 as a function of detection time delay ␶ = tB − tA between Stokes and anti-Stokes and interatomic distance r. Collective effects on G共2兲. The collective effect on the G共2兲 profile becomes significant when the period of oscillations varies appreciably around some transition 共or turning兲 point referred to as rt, which can only be estimated subjectively. Typically rt ⬍ ␭. For r ⬍ rt, the dipole-dipole interaction dominates and causes the Rabi oscillations to become more rapid. In this regime, the correlation can be oscillatory even when ⍀c ⬍ ⌫ / 4 共overdamp for large r兲; as in the case of ⍀c = 0.3⌫ in Fig. 3共a兲. However, the magnitude of the correlation decreases for r ⬍ rt, since the atoms are more likely to exchange photons between each other 共Rabi oscillations via emission and reabsorption兲 instead of emitting photons to the environment 共dissipation兲. Quenching effect of control field. By comparing the plots for several values of the control field ⍀c in Fig. 3, we find that rt becomes smaller as ⍀c increases. This shows the quenching of the effect of dipole-dipole interaction by the ˜ 共+兲 control field. This feature is supported by Fig. 4共a兲 共Re ⍀ versus r兲 which shows the variation of the estimated turning points rt for different ⍀c 共visually guided by the dashed line兲. ˜ 共+兲 increases drastically around For larger ⍀c, the Re ⍀ smaller values of rt, which corresponds to Fig. 3共c兲. For small ⍀c the Rabi oscillation period begins to change significantly only at the larger rt, which corresponds to Fig. 3共a兲. For ⍀c = 3⌫ Fig. 3共b兲 shows that the sum of amplitudes of two independent atoms G共2兲 ⯝ 兩␺RED1 + ␺RED2兩2 is a good approximation down to ␭ / 8, thus the analysis of Ref. 关5兴 which employs this approximation to achieve a resolution up to ␭ / 8 is valid. Coherent phase of lasers. When ␣ p,c ⫽ ␲ / 2, the coherent phase factors eikp,c·r j in Eqs. 共6兲 and 共7兲 give richer features shown in Figs. 3共d兲–3共f兲. For large r, the oscillations across r become more rapid. Similarly for small r, but the oscillations also become more rapid across ␶. The wave vectors of the lasers introduce a relative phase 共that depends on r兲 between the two atoms such that their transient dynamics evolve out of phase between each other and this creates the beating. Subwavelength resolution. The variation of the Rabi oscillations period in the G共2兲 with r 共due to collective effect兲 is a feature that can be used to measure the interatomic distance r below the diffraction limit of the photons. One should realize that this feature is in time domain, in contrast to the approach using the frequency domain 关10兴 although the physics is the same. Besides, the resolution may be limited by the vanishingly small correlation for r ⱗ ␭ / 20. Further analysis of the G共2兲 as a function of ␶ and its Fourier transform for subwavelength resolution will be reported elsewhere. In conclusion, the significance of the above results is that the collective effects at small interatomic distance give rise to distance dependent two-photon correlation that can be “quenched” by a strong resonant control field ⍀c. This provides a possibility of coherently controlling the collective photon statistics and multiple frequency generation in the many-body system.

ACKNOWLEDGMENT

The author thanks Professor Iwo Bialynicki Birula for helpful feedback that led to this work. APPENDIX: COUPLED EQUATIONS AND SOLUTIONS

The coupled equations for two RED atoms obtained from Eqs. 共1兲 and 共2兲 are d C = i 兺 共Gk共1兲*˜Bk共1兲 + Gk共2兲*˜Bk共2兲兲, dt k

冉 冉

冊 冊 冊

共A1兲

d ˜ 共j兲 + iDk ˜Bk共j兲 = iGk共j兲C + i⍀共j兲* c Ak , dt

共A2兲

d 共j兲* ˜ ˜ 共j兲 + i共Dk − ␦兲 ˜Ak共j兲 = i⍀共j兲 c Bk + i 兺 gq Ck,q , dt q

共A3兲

冉

d ˜ = ig共1兲˜A共1兲 + ig共2兲˜A共2兲 , + iDk,q C k,q k q k q dt

共A4兲

where j = 1 , 2 and we define the transformation ˜B共j兲 = B共j兲ei⌬te−i⌬kt , k k ˜A共j兲 = A共j兲ei⌬te−i⌬ktei␦t , k k ˜ = C ei⌬te−i⌬ktei␦te−i⌬qt C k,q k,q with the detunings Dk = ⌬k − ⌬,Dk,q = ⌬k + ⌬q − ⌬ − ␦ = ␯k + ␯q − ␯ p − ␯c and ␦ = ␯c − ␻ab. By using the Weisskopf-Wigner approximation, we obtain the set of linear d ˜ 共1兲 ˜ 共1兲 A ⯝ 关− i共Dk − ␦兲 − 21 ⌼共1兲兴˜Ak共1兲 + i⍀共1兲 c Bk dt k −

1 2

冑⌫共1兲⌫共2兲 f共r兲A˜k共2兲共t兲,

共A5兲

d ˜ 共2兲 ˜ 共2兲 A ⯝ 关− i共Dk − ␦兲 − 21 ⌼共2兲兴˜Ak共2兲 + i⍀共2兲 c Bk dt k −

1 2

冑⌫共1兲⌫共2兲 f共r兲*˜Ak共1兲共t兲,

where

兺q 兩gq共j兲兩2 兺q gq共1兲*gq共2兲

冕

t

0

冕

t

e−iDk,q共t−t⬘兲 ⯝ 21 ⌼共j兲 ,

共A6兲

共A7兲

0

e−iDk,q共t−t⬘兲 ⯝

1 共1兲 共2兲 冑⌫ ⌫ f共r兲, 2

共A8兲

with ⌼共j兲 = ⌫共j兲 − i␰共j兲, the Lamb shift ␰共j兲 , f共r兲 = g共r兲 + ih共r兲, where g共r兲 gives the modified decay rates and h共r兲 is the level shift factor 关8兴 due to the collective effects. The coupled equations can be solved by Laplace transform method. Assume that initially C共0兲 = 1 and other coef-

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C. H. RAYMOND OOI

ficients are zero, after a lengthy calculations, we obtain the steady state solution

Ck,q共⬁兲 = i

冢

−

Ck,q共⬁兲 =

共2兲 共1兲 i共␦ − ⌬q兲 21 冑⌫共1兲⌫共2兲 f共r兲⍀共1兲 c gq Gk

再

M q共␥R − iDk,q兲 i共␦ − ⌬q兲

冉

冊

共2兲

冎

关i共

− i⌬q兲共␦ − ⌬q兲 + 兩⍀c兩2兴共iDk,q − ␥R兲

.

The exact coefficient in Eqs. 共6兲 and 共7兲 are

冣

储

KAj = 2㜷ba,Aj

共A9兲

⬜ + 㜷ba,Aj

= 冑兩⍀c兩2 − 共 41 ⌫共±兲 − i 2 ␦兲 , ␥R = 21 共␥R共1兲 + ␥R共2兲兲 is the collective Ra2

−

共j兲

兩Gk 兩2 1 共j兲 2 ␥R = 兺k s+iDk .

冋冉

+ sin2 ␣Aj

where M q = 兿4j=1共⌬q − x j兲 with 兵x j其 = 兵a+共+兲 , a−共+兲 , a+共−兲 , a−共−兲其, 1 ˜ 共±兲 ˜ 共±兲 a±共±兲 = 21 ␦ − i 4 ⌫共±兲 ± ⍀ are the roots with ⍀ 1

j=1,2

1 2⌫

共A10兲

⌫ 共1兲 共1兲 − i⌬q + 兩⍀兩2 ⍀共1兲 c gq Gk 2 M q共␥R − iDk,q兲

+ 共1 ↔ 2兲,

共j兲 共j兲 i 兺 ⍀共j兲 c gq Gk

冋

1 ix3Aj

冉

−

3 2x2Aj

1 x2Aj −

冊

3 2ix3Aj

+

1 2ixAj

1 1 1 + 2 − 3 ixAj xAj ixAj

冉

1 3 1 3 sin2 ␣Aj 2 − 3 + 2 xAj ixAj ixAj

冊册

冊册

,

共A11兲

man decay rate with The complex rates that depend on interatomic distance are ⌫共±兲 = ⌫关1 ± f共r兲兴, where f共r兲 is defined in the text. For identical atoms, ⌫共2兲 = ⌫共1兲. By neglecting the Lamb shifts ⌼共1兲 ⯝ ⌼共2兲 ⯝ ⌫. We have verified that Eq. 共A9兲 gives the known result 关3兴 in the limit f共r兲 → 0, for two independent atoms

for dipole moment 㜷 parallel 共储兲 and orthogonal 共⬜兲 to the quantization axis, with xAj = ␯rAj / c and ␣Aj as the angles between A-j and the interatomic axis. The same expression for 储 ,⬜ QBj by the replacements xAj → y Bj = ␻rBj / c, 㜷ba,Aj 储 ,⬜ → 㜷ca,Bj共B-j兲 and ␣Aj → ␣Bj. The emission coupling factors KAj, QBj are valid for both far and near fields 关9兴.

关1兴 M. D‘Angelo, M. V. Chekhova and Y. Shih, Phys. Rev. Lett. 87, 013602 共2001兲. 关2兴 J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. Wang, Phys. Rev. Lett. 94, 173601 共2005兲. 关3兴 M. O. Scully and C. H. Raymond Ooi, J. Opt. B: Quantum Semiclassical Opt. 6, S816 共2004兲. 关4兴 M. O. Scully and M. S. Zubairy, Quantum Optics 共Cambridge University Press, Cambridge, 1997兲. 关5兴 M. O. Scully, Concepts Physics 2, 261 共2005兲. 关6兴 A very good description of dipole-dipole interaction can be found in the book by Z. Ficek and S. Swain, Quantum Interference and Coherence 共Springer, Berlin, 2005兲; and monograph by G. S. Agarwal, in Quantum Statistical Theories of

Spontaneous Emission and Their Relation to Other Approaches, edited by G. Hohler, Springer Tracts in Modern Physics, Vol. 70 共Springer, Berlin, 1974兲. For quantum regression approach, see A. K. Patnaik, G. S. Agarwal, C. H. Raymond Ooi, and M. O. Scully, Phys. Rev. A 72, 043811 共2005兲; for Schrödinger’s approach, see 关9兴 or 关3兴; Heisenberg-Langevin approach will be presented elsewhere. R. H. Lehmberg, Phys. Rev. A 2, 889 共1970兲. C. H. Raymond Ooi, A. K. Patnaik, and M. O. Scully, Proc. SPIE 5846, 1 共2005兲. J.-T. Chang, J. Evers, M. O. Scully, and M. S. Zubairy, Phys. Rev. A 73, 031803共R兲 共2006兲; J.-T. Chang, J. Evers, and M. S. Zubairy, ibid. 74, 043820 共2006兲.

关7兴

关8兴 关9兴 关10兴

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