Spontaneous emission of an excited atom in a randomly occupied optical lattice

Optics Communications 252 (2005) 97–110 www.elsevier.com/locate/optcom

Spontaneous emission of an excited atom in a randomly occupied optical lattice Wei Guo

*

Department of Physics and Astronomy, The University of New Mexico, Albuquerque, NM 87131, USA Received 7 February 2005; received in revised form 31 March 2005; accepted 6 April 2005

Abstract The problem of spontaneous emission of an excited atom embedded inside a randomly occupied optical lattice is discussed herein by explicitly calculating the emission rate of the atom. After separately analyzing two properties of the lattice, a long range of regularity and randomness, we find that since the statistics of the density fluctuations of the trapped atoms on the lattice favor higher-order correlations whose numerical values are nevertheless rather small, the randomness merely plays a weak role, compared with the regularity, in determining the atomÕs spontaneous emission rate. We also find that the number of multiple scattering channels, previously defined [Phys. Rev. E 69 (2004) 036602] to describe light propagation inside a discrete medium with a long range of regularity, is still important in the atomÕs radiative evolution: As it changes from one value to another, the atomÕs spontaneous emission rate is brought to local extreme values.
2005 Elsevier B.V. All rights reserved. PACS: 42.50.Nn; 12.Ds; 32.80.
t Keywords: Spontaneous emission; Optical lattices

1. Introduction When other particles are brought to the vicinity of an excited atom, the atomÕs spontaneous emission is altered. This well-known phenomenon can be explained as: The presence of these external particles modifies the electromagnetic field operators and subsequently changes the structure of the fluctuating field of the *

Present address: Department of Physics and Optical Science, The University of North Carolina – Charlotte, 9201 University City Blvd., Charlotte, NC 28223, USA. Tel.: +1 704 364 2481; fax: +1 704 687 3160. E-mail address: [email protected]. 0030-4018/$ - see front matter
2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.04.013

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W. Guo / Optics Communications 252 (2005) 97–110

vacuum acting on the atom. Such a point of view was initiated by Feynman and later used by Power in 1966 to calculate the Lamb shift [1]. In the literature, these external particles are usually assumed to form regular, continuous structures, such as a perfect mirror [2–4], a dielectric sphere [5], a dielectric cylinder [6,7], and a photonic crystal [8,9], to mention a few, and can either enhance or inhibit spontaneous emission of the excited atom. For example, in an experiment, the natural lifetime of a Rydberg atom inside parallel conducting plates was observed to increase by a factor of 20 [10]. The external particles can also form irregular structures, one example of which is a continuous random medium. In a continuous random medium, where the constituents of the medium are allowed to reach any point in the space claimed by the medium, it was predicted that the excited atom will have a shorter lifetime due to the random correlations [11]. In addition to continuous random media, there are other types of random media, such as optical lattices, that possess a long range of regularity too. An optical lattice is a periodic arrangement of atoms confined in place by laser beams and is the direct result of atom cooling [12]. The randomness of an optical lattice comes from the often uncontrollable distribution of trapped atoms among the lattice sites. A significant difference accordingly exists between an optical lattice and a conventional continuous random medium: While the former simultaneously possesses both randomness and regularity, the latter merely owns randomness. Since spontaneous emission can be regarded to be caused by an interaction between an excited atom and the fluctuations of the ground state of the quantized electromagnetic field [13], and since the fluctuating fields of the vacuum are modified by an optical lattice and by a continuous random medium differently, it is highly possible that new features might occur in the radiative evolution of an excited atom embedded inside an optical lattice. Moreover, a prediction established in [15] that propagation inside an optical lattice of radiation already spontaneously emitted from the trapped atoms can be strongly influenced by the regularity and the randomness of the lattice also prompts us to ask if an optical lattice has any effects on light emission. In the present paper, we address theoretically the spontaneous emission of an excited atom embedded inside an optical lattice by computing the atomÕs spontaneous emission rate. One way to formulate spontaneous emission is to use FermiÕs golden rule. It then follows that, in the framework of the linear response theory [16], one can show the spontaneous emission rate C of an excited atom radiatively relaxing from an excited state to the ground state is proportional to the imaginary part (denoted as Im) of the response function of the system under study [3,17,18] by using the fluctuationdissipation theorem: C¼

2d i d j Im½Gij ðrA ; rA ;xA Þ; h 

ð1Þ

where the excited atom is assumed to be at rA, the atomic transition frequency is denoted as xA, and d is the dipole matrix element linking the excited and the ground states. The repeated indices in Eq. (1) have to be summed over the Cartesian coordinates. The response function G(r, r1) is, in fact, the familiar dyadic Green function in the classical theory of electromagnetic fields and is well known to be equal to the electric field produced at point r by a unit dipole at r1, subject to all boundary conditions presented in the problem. Our task of finding C is therefore reduced to evaluating the Green function inside an optical lattice. One advantage of using the fluctuation-dissipation theorem in the present paper is that this theorem relieves us from the burden of quantizing the electromagnetic fields inside an optical lattice; quantization of electromagnetic fields in irregular systems is always a nontrivial problem [19]. Since the purpose of this paper is to examine the effects of the regularity and the randomness of an optical lattice on the atomÕs spontaneous emission, we assume that the excited atom has been embedded inside the lattice by some mechanism that needs not be specified in advance. Through its expression in Eq. (1), the spontaneous emission rate C can be shown to be proportional to the local density of states of electromagnetic fields [20]. This conclusion implies that the modification of C by the presence of the external particles can also be interpreted as the particlesÕ control over the distribution of the electromagnetic fields spontaneously emitted from the excited atom, an effect known for a long time

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[21]. In photonic crystals, for example, a combination of Bragg scattering and Mie scattering causes light propagation to form band structures, in which light propagation is prohibited for a range of frequencies [22], and the spontaneous emission was predicted to be completely inhibited if the atomic transition frequency is inside any band gap. Measuring the spontaneous emission rate is one way to probe the local density of electromagnetic fields [23,24]. Since each lattice site in the lattice has only a small chance to be occupied by a trapped atom, usually around 10% for a typical near-resonance optical lattice [25], the microscopic density function of the trapped atoms becomes a random function [26] and can be separated into two parts representing, respectively, an optical lattice uniformly occupied by the trapped atoms and occupation fluctuations about that distribution [27]. In Section 2, such a separation of the density function is used to decompose the whole Green function G into a uniform Green function g, corresponding to wave propagation inside a uniformly occupied lattice, and corrections to g from the randomness of occupation. From the g function, one component of C, which is merely the spontaneous emission rate of an excited atom inside a uniformly occupied optical lattice, is obtained and denoted as Cuniv. We then fix the lattice constant, allowing the variations of Cuniv as a function of the atomic transition frequency xA to be studied to reveal how the radiative evolution of an excited atom inside the lattice is subject to the long range of regularity of the lattice. For the sake of simplicity, the optical lattice is assumed throughout to have a simple cubic structure with a lattice constant a and N sites. Corrections to Cuniv from the density fluctuations form the other component of C, which is denoted as DC. So that we have the relation C = Cuniv + DC. In Section 3 we compute DC by averaging over the random distribution of the trapped atoms. Thus, by separating the microscopic density function of the trapped atoms into a uniform part and a random part, we can separately analyze the effects of the regularity and the randomness of an optical lattice for their effects on an excited atom. We summarize the results in Section 4.

2. Spontaneous emission inside a uniformly occupied optical lattice For simplicity, the atoms trapped inside the lattice are treated as identical electric dipoles characterized by an isotropic polarizability a and by a diminishing physical size. Since the Green function G(r, r1) at an observation position r is formed by the fields emitted either directly from a source at r1 or first from the source and then multiply scattered by the trapped atoms, the Green function can be expressed in the following integral equation in Gaussian units: Z 2 2 Gðr; r1 Þ ¼ 4pk A G0 ðr; r1 ;k A Þ þ 4pk A l G0 ðr; r2 ;k A Þ  G0 ðr2 ; r1 ;k A Þnðr2 Þ dr2 Z Z þ 4pk 2A l2 G0 ðr; r2 ;k A Þnðr2 Þ dr2  G0 ðr2 ; r3 ;k A Þnðr3 Þ dr3  G0 ðr3 ; r1 ;k A Þ þ    Z 2 ð2Þ ¼ 4pk A G0 ðr; r1 ;k A Þ þ l G0 ðr; r2 ;k A Þ  Gðr2 ; r1 Þnðr2 Þ dr2 ; where kA = xA/c, l ¼ 4pk 2A a, and G0(r, r1; kA) is the dyadic Green function corresponding to the operator ðr  r 
k 2A Þ, ! 1 1 eikA jr
r1 j I þ 2 rr ; ð3Þ G0 ðr; r1 ;k A Þ ¼ 4p jr
r1 j kA with I denoting the unit dyadic. Throughout this paper, we shall formally refer to G0 as spontaneous radiation from the excited atom, although, in principle, spontaneous radiation is never monochromatic due to shifts in the atomÕs excited energy levels.

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PN By writing the microscopic density function of the trapped atoms nðrÞ ¼ i bi dðr
Ri Þ as the sum of its PN spatially averaged value n0 ¼ b0 i dðr
Ri Þ and fluctuations about that value dnðrÞ ¼ P N i ðbi
b0 Þdðr
Ri Þ, we can use a linear-operator method to transform G into a series in ascending powers of dn [27]: Z Z Gðr; r1 Þ ¼ 4pk 2A gðr; r1 Þ þ 4pk 2A l gðr; r2 Þdnðr2 Þ  gðr2 ; r1 Þ dr2 þ 4pk 2A l2 gðr; r2 Þdnðr2 Þ dr2 Z  gðr2 ; r3 Þdnðr3 Þ dr3  gðr3 ; r1 Þ þ    ð4Þ with g obeying the following integral equation: Z gðr; r1 Þ ¼ G0 ðr; r1 ;k A Þ þ l G0 ðr; r2 ;k A Þ  n0 ðr2 Þgðr2 ; r1 Þ dr2 .

ð5Þ

Note, by writing the density function n(r) as an explicit function of the lattice sites, we have built into our formulation an approximation that the trapped atoms are tightly bound to the lattice sites through d functions. Known as Lamb–Dicke limit, this approximation is supported by the experimental evidence that, under typical experimental conditions, the deviation of an atom from the lattice site to which it is trapped is very small [28]. The random distribution of the trapped atoms among the lattice sites is described by each variable bi, which takes on the value 1 when the ith site is occupied and 0 when that site is vacant. The average occupation probability of each site is denoted as b0. Therefore, with the help of the variables bi and the d functions, that the lattice possesses both regularity and randomness is established mathematically through the density function n(r). As in [27], an assumption is made in the present paper that density fluctuations at two different sites are uncorrelated: Æbib j æ = Æ bi æ Æ bjæ if i 6¼ j; thus, the statistics of the density fluctuations established there can be applied directly to the present discussion. The reasoning behind this assumption is that the lattice is so sparsely occupied – an occupation fraction of 10% is assumed throughout the paper – that one lattice site is vacant or not does not depend on whether there is one atom on another site. The separation of the density function n(r) into the uniform distribution n0 and the density fluctuations dn allows us to regard formally the propagation of the spontaneous radiation G0 inside the randomly occupied lattice, see Eq. (4), as propagation inside a uniform lattice plus multiple scattering from the density fluctuations. Note even inside the uniform lattice, the propagation of G0 is a multiple scattering process (represented by g), since, as Eq. (5) shows, G0 can either arrive at r directly from r0, see the first term on the right-hand side (RHS) of Eq. (5), or first experience multiple scattering from the uniformly distributed atoms and then reach r, see the second term. The physics conveyed by the various terms in Eq. (4) can be understood as follows. The first term on the RHS of Eq. (4) simply means that the spontaneous radiation propagates inside the uniform lattice. The second and latter terms illustrate the first- and higher-order scatterings from the density fluctuations of the radiation propagating inside the uniform lattice. Before undergoing its first scattering from the density fluctuations, the radiation propagates inside the uniform lattice; between two such scatterings and after the last such scattering, the radiation still propagates inside the uniform lattice. It was revealed in [15] that light propagation inside an optical lattice is largely determined by a parameter called the number of multiple scattering channels (NMSC). This parameter, denoted as Nb in the present paper, is a stepwise function of the lattice constant and the wavelength of a wave traveling inside the lattice and can be regarded as characterizing the scattering ability of the lattice on the wave. When the wavelength is larger than twice the lattice constant, Nb approaches its minimum value 1, and the wave then cannot discern the structure of the lattice. To fully exhibit the effects of the long-range regularity on atomic lifetimes, we devote the present paper to a discussion in which the lattice constant is assumed to be larger than the wavelength k0 of the spontaneous radiation. It subsequently becomes valid to approximate light waves at

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each scattering step as scalar waves. The scalar approximation has been used by many authors in describing light scattering from random media both with [27] and without a long range of regularity [29,30]. It is worth noting, however, that when magnetoactive media are under investigation (not the present case), it is no longer accurate to neglect the vector nature of the radiation fields [31]. The scalar approximation enables us to express the function g in a serial form by reference to Eq. (5): " X gðr; r1 Þ ¼ G0 ðr; r1 ;k A Þ þ I lb0 G0 ðr; Ri ;k A ÞG0 ðRi ; r1 ;k A Þ i

# X þ l2 b20 G0 ðr; Ri ;k A ÞG0 ðRi ; Rj ;k A ÞG0 ðRj ; r1 ;k A Þ þ    ;

ð6Þ

i6¼j

where G0 ðr; r1 ;k A Þ ¼ eikA jr
r1 j =ð4pjr
r1 jÞ. The dyadic characters of G0 have been kept for the first term on the RHS of Eq. (6), because when computing Cuniv, we need to take the limit of r ! r1 ! rA, which evidently makes the scalar approximation that has been used to simplify Eq. (5) inapplicable to that G0(r, r1;kA). Eq. (6) establishes the fact that, at any point inside a uniformly occupied optical lattice, the total field is formed by summing up the fields emitted both directly from their sources and initially from their sources and then multiply scattered by the trapped atoms. The series in the square brackets in Eq. (6) is mathematically equivalent to another process: The radiation is multiply scattered to each lattice site with subsequent re-emission from these lattice sites directly to where it is measured. By following the approach used in [27], we can partially sum up the serial expression of the g function into the following form: pffiffiffiffiffiffiffiffiffiffiffi2ffi lb0 X eikA 1þt3 =kA jr
Ri j gðr; r1 Þ ¼ G0 ðr; r1 ;k A Þ þ G0 ðRi ; r1 ;k A Þ. I ð7Þ w1 4pjr
Ri j i See Appendix A for a detailed derivation. The polarizability a of the trapped atoms is obtained by using the classical theory of electrodynamics [32] as 3x0 s ;  ð8Þ a¼  2  3 2k 30 1
xxA0
i xxA0 x0 s where x0 denotes the internal oscillation frequency of the trapped atoms, and k0 = x0/c. The reactive effects of radiation are taken into account in the preceding expression through a parameter s, often referred to as the characteristic time. For the purpose of exhibiting those variations in C that might otherwise be entirely swamped by the resonant peak, we shall choose the value of x0s in the following numerical analysis of Cuniv and DC to be 0.01. Such a choice is allowed by the classical model of the dipoles [32]. Replacing G in Eq. (1) with its uniform component g, we express the spontaneous emission rate Cuniv of an excited atom embedded inside a uniformly occupied lattice in terms of the vacuum emission rate C0 ¼ ð4jdj2 k 3A Þ=ð3 hÞ [27,18]: 9 8 > > p ffiffiffiffiffiffiffiffiffiffiffi ffi > > > > = < ik A 1þt3 =k 2A jrA
Ri j X Cuniv 0.09b0 a 1 e   ; ð9Þ G  ¼1þ Im ðR ;r Þ 0 i A 2  3  3 > > C0 4pjrA
Ri j kA ð1.4Þ > > a a i > > ; :w1 1
1.4kA
i 1.4kA  0.01 where k0 has been set to a/1.4. The variations of Cuniv as a function of the wavelength kA are graphically illustrated in Fig. 1 for rA = 0.5a(1,1,1) and b0 = 0.1. The prominent feature of Fig. 1 is a dip located at a = 1.4kA, whose origin is easily understood to come from the resonant enhancement by the trapped atoms of the fluctuations of

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W. Guo / Optics Communications 252 (2005) 97–110 10

Normalized spontaneous emission rate

5

0

-5

-10

-15

-20

-25 0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Normalized transition wavelength

Fig. 1. Normalized spontaneous emission rate (Cuniv/C0
1) · 103 versus normalized transition wavelength a/kA for rA = 0.5a(1,1,1).

the ground state of the quantized electric field, as an examination of the expression of a in Eq. (8) reveals. The resonance alone, however, cannot determine the behavior of Cuniv around a = 1.4kA: If the excited atom is shifted to another point, rA = a(0.5,0.5,0.7) for example, a peak is produced in the neighborhood of resonance instead (refer to Fig. 2). Spontaneous emission of an excited atom is, as expected, sensitive to the environment of the atom. In photonic crystals, spontaneous emission was also pointed out to depend critically on the position of an excited atom [9]. Photonic crystals are artificial materials with periodic dielectric elements whose size is comparable to optical wavelengths [22]. Although inside a photonic crystal, light propagation is characterized by both Bragg scattering from planes of the dielectric elements and Mie scattering from the individual dielectric elements and thus lacks the properties of multiple scattering studied here, results from the studies of spontaneous emission inside a photonic crystal are still valuable in aiding the interpretation of the present discussion. It was predicted in [15] that Nb is a function of a/kA and changes its values three times: from 26 to 32 at a/kA = 1.5, from 32 to 56 at a/kA = 1.585, and from 56 to 74 at a/kA = 1.66 for the range of kA considered in Figs. 1 and 2. The variations of Cuniv in this range in these figures clearly exhibit that spontaneous emission inside a uniformly occupied optical lattice can be either enhanced or inhibited, depending on the transition frequency xA. No similar properties were found in the studies of spontaneous emission inside a uniform dielectric medium [17]. An examination of Figs. 1 and 2 additionally reveals that, at the points where Nb changes its values, there exist two peaks and one dip in Fig. 1

W. Guo / Optics Communications 252 (2005) 97–110

103

10

Normalized spontaneous emission rate

8

6

4

2

0

-2

-4 0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Normalized transition wavelength

Fig. 2. Normalized spontaneous emission rate (Cuniv/C0
1) · 103 versus normalized transition wavelength a/kA for rA = a(0.5,0.5,0.7).

and three dips in Fig. 2 where the deviations of Cuniv away from C0 reach local extreme values. To understand this phenomenon, we remember that Nb characterizes the wave scattering ability of the lattice: As Nb increases its values, the frequency component (xA) of the ground state of the quantized electromagnetic fields that induces the spontaneous emission is scattered more and more intensively by the lattice, leaving the fields acting on the excited atom stronger and stronger for the present locations of the atom. Consequently, the atomÕs radiative evolution is changed by these fields. It is helpful to refer to the expression of g in Eq. (6) for an appreciation of the close relationship between the latticeÕs scattering ability and the fieldsÕ magnitudes. Although the spontaneous emission rate Cuniv is noted from Eq. (7) to be a function of the NMSC, its curves remain continuous (see Figs. 1 and 2). One explanation is that in the present discussion, the stepwise variations of Nb are averaged out in the summation over the lattice sites. As illustrated in Figs. 1 and 2, the deviations of Cuniv with respect to C0, except at a = 1.4kA, are in fact very small in magnitude. One possible explanation to this observation is that optical lattices are not strong scattering media (unless the spontaneous radiation is in resonance with the trapped atoms) such that the spontaneously emitted radiation, when scattered back to the emitting atom by the trapped atoms, becomes negligible in magnitude and has little effect on the radiative evolution of the emitting atom. Experiments on fluorescence lifetimes in photonic crystals, on the other hand, reported various degrees of influence of the photonic band structures on atomic lifetimes [23,24].

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3. Contributions from density fluctuations A specific form for dnÕs in Eq. (4) corresponds to a specific distribution of the trapped atoms among the lattice sites. Since whether a lattice site is occupied or not can only be stated statistically, it is more reasonable to compute the spontaneous emission rate DC corresponding to an ensemble average over the random distribution of the trapped atoms, rather than to a particular configuration of the same atoms. The statistics of the density fluctuations of the trapped atoms on the lattice sites are studied in detail in [27]. From its definition, we know Ædn(r)æ = 0, and the contribution to DC from the first order density fluctuation correlation is then null. The second- and the third-order density fluctuation correlations make no contribution to DC either. To see this, we note that, from their expressions given below, both of these correlations require the density fluctuations involved to be at common lattice sites: hdnðrÞdnðr1 Þi ¼ b0 ð1
b0 Þdðr
r1 Þ

N X

dðr
Ri Þ;

ð10Þ

i

3

hdnðrÞdnðr1 Þdnðr2 Þi ¼ hðb
b0 Þ i

N X

dðr
Ri Þdðr1
Ri Þdðr2
Ri Þ;

ð11Þ

i

where in Eq. (11) and in the following expressions the notation Æ(b
b0)næ denotes the nth moment of site occupation fluctuations. When the preceding relations are used to compute the ensembly averaged G with the help of Eq. (4) [33], they produce, respectively, the first- and the second-order radiation reactions: interaction between the radiation from an atom and the atom itself. Higher orders of such interactions occur in higher-order correlation functions. Quantum mechanically, the radiation reactions are known to be responsible for effects, such as atomic energy shifts, spontaneous emission, and so on [14]; classically, they amount to a correction to the polarizability of the dipoles [32]. In the classical theory of Green function discussed in the present paper, we shall assume that such a correction has already been made in our definition of a [refer to Eq. (8)]. As a result of this assumption, any further account of the radiation reactions in the following discussion becomes redundant. The first nonzero contribution to the ensembly averaged Green function ÆGæ in fact comes from the fourth-order correlation function, even though this correlation function still requires its four density fluctuations to be at either a single lattice site or a pair of lattice sites: hdnðrÞdnðr1 Þdnðr2 Þdnðr3 Þi ¼ hðb
b0 Þ4 i

N X

dðr
Ri Þdðr1
Ri Þdðr2
Ri Þdðr3
Ri Þ

i 2 2

þ hðb
b0 Þ i

N X

dðr
Ri Þdðr1
Ri Þdðr2
Rj Þdðr3
Rj Þ

i6¼j

þ hðb
b0 Þ2 i2

N X

dðr
Ri Þdðr2
Ri Þdðr1
Rj Þdðr3
Rj Þ

i6¼j

þ hðb
b0 Þ2 i2

N X

dðr
Ri Þdðr3
Ri Þdðr1
Rj Þdðr2
Rj Þ.

ð12Þ

i6¼j

To be out of this apparent dilemma, it is essential to realize that the orders of integration in Eq. (4) actually define formal radiation scattering sequences. For example, the third term on the RHS of Eq. (4) specifies that a radiation field is scattered along the route r1 ! r3 ! r2 ! r. This interpretation suggests that two density fluctuations at a common lattice site, depending on the radiation scattering sequence, can represent that

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the radiation fields are scattered to the common site twice via another scatterer. When the formula in Eq. (12) is used, together with the relation in Eq. (4), to compute ÆGæ, it turns out that only one (out of four) of the d product series on the RHS of Eq. (12) survives and generates a third-order scattering event between two atoms at Ri and Rj. More specifically, the contribution to ÆGæ from the fourth-order correlation function reads 2 2

4pk 2A l4 hðb
b0 Þ i I

N X

gðr; Ri Þg3 ðRi ; Rj ÞgðRj ; r1 Þ;

ð13Þ

i6¼j

where the scalar approximation has been applied to the g functions. Other d product series in Eq. (12) simply reproduce the radiation reactions and should be ignored accordingly. Similarly, we find that the only possible way to distribute the five density fluctuations in the fifth-order density fluctuation correlation function is to set two fluctuations at one site Ri and three at another site Rj. Thus, the contribution to ÆGæ from this correlation function takes the following form: 4pk 2A l5 hðb
b0 Þ2 ihðb
b0 Þ3 iI

N X

gðr; Ri Þg4 ðRi ; Rj ÞgðRi ; r1 Þ;

ð14Þ

i6¼j

which formally represents a fourth-order scattering event between Ri and Rj. The density fluctuations in the sixth-order correlation function can be grouped in two ways, one of which is to set three fluctuations at one lattice site and the remaining three at another, and a contribution to ÆGæ that resembles the expression in Eq. (13) results, 3 2

4pk 2A l6 hðb
b0 Þ i I

N X

gðr; Ri Þg5 ðRi ; Rj ÞgðRj ; r1 Þ.

ð15Þ

i6¼j

Alternatively, we can set two fluctuations at one site, two at another site, and the remaining two at a third site, so that the second possible contribution to ÆGæ from the sixth-order correlation function yields 2 3

4pk 2A l6 hðb
b0 Þ i I

N X

gðr; Ri Þg2 ðRi ; Rj ÞgðRi ; Rl Þg2 ðRj ; Rl ÞgðRl ; r1 Þ.

ð16Þ

i6¼j;j6¼l

Since formally in the first arrangement of the density fluctuations, see Eq. (15), the fluctuations are restricted to be at two lattice sites Ri and Rj, while in the second one, see Eq. (16), the fluctuations are allowed to reside on three, the statistics of the randomness in these two arrangements are not identical. For a typical near-resonance optical lattice with b0 . 0.1, the magnitude of Æ(b
b0)3æ2 in Eq. (15) is approximately equal to [b0(1
b0)3]2, whereas the magnitude of Æ(b
b0)2 æ 3 . [b0(1
b0)2]3 in Eq. (16). The latter number is only 10% of the former one. This is a direct exhibition of the conclusion that the occupation fluctuations are only correlated with themselves under the assumption that the density fluctuations at different sites are uncorrelated [27]. In the following, our discussion is limited to the case where the density fluctuations only reside on two (the minimum number allowed without regenerating radiation reactions) different lattice sites. This approximation not only simplifies the mathematical calculations but also gives the most important contributions to DC from the randomness. The expression in Eq. (16) will, therefore, be ignored in the subsequent analysis. To give one more example of evaluating correlation functions, we find that, by following the same argument as that leading to the expression in Eq. (15), the seventh-order correlation function gives a contribution to ÆGæ in a form that formally contains a sixth-order scattering event: 3

4

4pk 2A l7 hðb
b0 Þ ihðb
b0 Þ iI

N X i6¼j

gðr; Ri Þg6 ðRi ; Rj ÞgðRi ; r1 Þ.

ð17Þ

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W. Guo / Optics Communications 252 (2005) 97–110

Two series are, respectively, formed and summed up for even orders of density fluctuation correlations N X G4 ðr; r1 Þ 2 2 4 ¼ l hðb
b Þ i I gðr; Ri Þg3 ðRi ; Rj ÞgðRj ; r1 Þ 0 4pk 2A i6¼j

þ l6 hðb
b0 Þ3 i2 I

N X

gðr; Ri Þg5 ðRi ; Rj ÞgðRj ; r1 Þ

i6¼j 4 2

þ l8 hðb
b0 Þ i I

N X

gðr; Ri Þg7 ðRi ; Rj ÞgðRj ; r1 Þ þ   

i6¼j 4

’ l4 b20 ð1
b0 Þ I

N X

gðr; Ri Þ

i6¼j

g3 ðRi ; Rj Þ 2

1
l2 ð1
b0 Þ g2 ðRi ; Rj Þ

gðRj ; r1 Þ;

ð18Þ

and for odd orders of correlations N X G5 ðr; r1 Þ 2 3 ¼ l5 hðb
b0 Þ ihðb
b0 Þ iI gðr; Ri Þg4 ðRi ; Rj ÞgðRi ; r1 Þ 2 4pk A i6¼j 3

4

þ l7 hðb
b0 Þ ihðb
b0 Þ iI

N X

gðr; Ri Þg6 ðRi ; Rj ÞgðRi ; r1 Þ

i6¼j

þ l9 hðb
b0 Þ4 ihðb
b0 Þ5 iI

N X

gðr; Ri Þg8 ðRi ; Rj ÞgðRi ; r1 Þ þ   

i6¼j

’ l5 b20 ð1
b0 Þ5 I

N X i6¼j

gðr; Ri Þ

g4 ðRi ; Rj Þ 1
l2 ð1
b0 Þ2 g2 ðRi ; Rj Þ

gðRi ; r1 Þ.

ð19Þ

In deriving the last two expressions, an approximation has been made: In the nth moment of the site occupation fluctuation Æ(b
b0)næ, only its main component b0(1
b0)n is kept. This approximation is valid as long as b0 is much smaller than 1 and is consistent with the choice of a near-resonance optical lattice in the present discussion. The general expression of Æ(b
b0)næ is presented in [27]. In the preceding two expressions, g(r, Ri) represents multiple scattering of G0 between r (different from any lattice site) and one lattice site Ri, whose approximate expression is given in Appendix A, whereas g(Ri, Rj) represents multiple scattering of G0 between two different lattice sites Ri and Rj. The serial expression of g(Ri,Rj) was summed in [27] as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 0 gðRi ; Rj Þ ¼ ei kA þw2 N b =ð1
w2 s ÞjRi
Rj j ; ð20Þ 2 4pð1
w2 s0 Þ jRi
Rj j with w2 = lb0/a3 and  a 3 Z X 1 1
; s0 ¼ dk 2 2 2 2p k
k 2A fGs g Gs
k A where we have used {Gs} to represent a set of reciprocal vectors that satisfy the relation |Gs
kA| æ p/a. After G4 in Eq. (18) and G5 in Eq. (19) are substituted into Eq. (1), we find that the component of C that accounts for the density fluctuations becomes DC=C0 ¼

6p Im½G4 ðrA ; rA Þ þ G5 ðrA ; rA Þ. kA

ð21Þ

W. Guo / Optics Communications 252 (2005) 97–110

107

With the same parameters as those used in Figs. 1 and 2 in Section 2, we graphically represent DC as a function of the wavelength kA in Fig. 3. It is evident that, as in the case of a uniformly occupied lattice discussed in the preceding section, the dominant contributions to DC from the density fluctuations occur when the spontaneous radiation from the excited atom is in resonance with the trapped atoms. The resonance at a/kA = 1.4 is so strong, see the coefficients l4 in Eq. (18) and l5 in Eq. (19), that other variations away from the resonance are entirely swamped. As the radiation fields are multiply scattered between the density fluctuations located at two lattice sites, the amplitudes of the fields are modified, and a denominator containing a parameter proportional to l2 is produced [refer to the expressions in Eqs. (18) and (19)]. Consequently, when the distribution of the density fluctuations are ensembly averaged over the lattice sites, we see more variations occurring in Fig. 3 than in Figs. 1 and 2 in the neighborhood of the resonance. In Section 2, the emission rate Cuniv as a function of xA was shown to have a strong dependence on Nb. No such dependence, however, is found in DC. One possible explanation to the weak dependence of DC on Nb is that after the ensemble average over the density fluctuations the influence from the discontinuous dependence of Nb on kA is wiped out. It is also evident that DC is smaller than Cuniv by a few orders of magnitude, even around a/kA = 1.4 where resonant scattering takes place. This is because that, as formulas in Eqs. (18) and (19) illustrate, the statistics governing the density fluctuations of the trapped atoms rule that only the fourth and higher

1

Normalized spontaneous emission rate

0.5

0

-0.5

-1

-1.5 0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Normalized transition wavelength Fig. 3. Normalized spontaneous emission rate DC/C0 · 105 versus normalized transition wavelength a/kA for rA = 0.5a(1,1,1).

108

W. Guo / Optics Communications 252 (2005) 97–110

orders of correlation functions can have an effect on spontaneous emission. When the average occupation fraction is low, 10% in the present study, the numerical values of these high-order correlation functions are subsequently small, making DC insignificant compared with Cuniv. As it was pointed out in [27], the density fluctuations inside an optical lattice are more related to the departure from the regularity of the lattice based medium than to the actual occupation fraction. See, for an illustration, the term b0(1
b0) in the second-order correlation function in Eq. (10). Therefore, the small magnitudes of the correlation functions that appear in the expressions of G4 and G5 cannot be increased by increasing the occupation fraction of the lattice. As a result, the conclusion stated at the beginning of this paragraph – DC is much smaller than Cuniv – remains accurate even when the occupation fraction b0 approaches unit. Nevertheless, it was found that density fluctuations of a continuous random medium play a dominant role in determining the radiative lifetime of an excited atom [11], even though in that case the correlation functions involved are all with orders higher than or equal to four. This is because the density fluctuations in a continuous random medium follow a different statistical law, which permits the density correlation functions to increase with the average density of the medium [30].

4. Conclusion We have presented in this paper a theoretical study of the spontaneous emission of an excited atom embedded inside a randomly occupied optical lattice. By decomposing the microscopic density function of the trapped atoms, we resolve the Green function into two parts with respect to a uniformly occupied lattice and to the density fluctuations about the uniform lattice, respectively. Since the two parts of the Green function are related to the atomic spontaneous emission rate through the fluctuation-dissipation theorem, we are able to examine separately the regularity and the randomness of the lattice for their effects on the radiative evolution of the excited atom. We find that the atomÕs spontaneous emission rate is largely determined by the regularity, and that when the NMSC changes its values the spontaneous emission rate achieves local extreme values. Also identified is that the atomÕs spontaneous emission is sensitive to the atomÕs locations inside the lattice.

Appendix A. Partial summation of Eq. (6) The series P in the square brackets in Eq. (6) can be rearranged as " # X X P¼ G0 ðr; Ri ;k A Þ þ lb0 G0 ðr; Rj ;k A ÞG0 ðRj ; Ri ;k A Þ þ    G0 ðRi ; r1 ;k A Þ. i

ðA:1Þ

i6¼j

Following the approach used in [27], we can first Fourier transform each G0 function in the brackets in the last expression into a spatial frequency domain and then change the order of integration and summation. The following summation over the lattice sites produces a series of d functions, which in turn convert the integrals in the spatial frequency domain into sums over reciprocal vectors with respect to the optical lattice. It turns out that a geometrical series in the fundamental Brillouin zone X3 is formed and enables us to get P iðks þGÞr Z e
iks Ri G ðkeþGÞ2
k2 1 X s A h  P i G0 ðRi ; r1 ;k A Þ; P¼ dks ðA:2Þ R 3 lb0 2p 3 1 1 ð2pÞ i X3 1
ð2pÞ
dk 2 k 2
k 2 3 G1 ðk þG Þ2
k 2 a s

1

A

2

A

where {G} and {G1} are two identical sets of reciprocal lattice vectors. We then integrate over the orientation of ks and extend the range of ks to (
1, 1). With the help of the residual theorem, we have

W. Guo / Optics Communications 252 (2005) 97–110

pffiffiffiffiffiffiffiffiffi X ei k2A þt3 jr
Ri j G0 ðRi ; r1 ;k A Þ. P¼ 4pw1 jr
Ri j i

109

ðA:3Þ

In the preceding expression, w1 ¼ 1
lb0 t1 =ð8p3 Þ;  t1 ¼

2p a

3 X G2

1
2 G2
k 2A

Z dk2

k 22

1 ;
k 2A

and t3 ¼ lb0 N b 8p3 =ða3 8p3
lb0 t1 a3 Þ; where {G2} is such a set of reciprocal vectors that satisfies, for any ks in X3, the relation jks þ G2 j2 6¼ k 2A . A term proportional to ðk 2s
k 2A Þ=N b has been neglected in Eq. (A.3) for the reason that Nb is usually big and only those ksÕs that satisfy ks . kA contribute.

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[27]

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[28] C.I. Westbrook, C. Jurczak, G. Birkl, B. Desruelle, W.D. Phillips, A. Aspect, J. Mod. Opt. 44 (1997) 1837. [29] L. Tsang, A. Ishimara, J. Opt. Soc. Am. A 2 (1985) 1331; M.B. van der Marker, M.P. van Albada, A. Lagendijk, Phys. Rev. B 37 (1988) 3575. [30] W. Guo, S. Prasad, Opt. Commun. 212 (2002) 1. [31] G.L.J.A. Rikken, B.A. Tiggelen, Nature (London) 381 (1996) 54. [32] J.D. Jackson, Classical Electrodynamics, second ed., Wiley, New York, 1975. [33] In the calculation of the ensembly averaged Green function ÆGæ, the first term on the RHS of Eq. (4) is ignored, because this term does not contain density fluctuations and has already been accounted for in Cuniv.