Collective generation of quantum states of light by entangled atoms

Collective generation of quantum states of light by entangled atoms

D. Porras    J. I. Cirac Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, Garching, D-85748, Germany.
July 5, 2019
Abstract

We present a theoretical framework to describe the collective emission of light by entangled atomic states. Our theory applies to the low excitation regime, where most of the atoms are initially in the ground state, and relies on a bosonic description of the atomic excitations. In this way, the problem of light emission by an ensemble of atoms can be solved exactly, including dipole-dipole interactions and multiple light scattering. Explicit expressions for the emitted photonic states are obtained in several situations, such as those of atoms in regular lattices and atomic vapors. We determine the directionality of the photonic beam, the purity of the photonic state, and the renormalization of the emission rates. We also show how to observe collective phenomena with ultracold atoms in optical lattices, and how to use these ideas to generate photonic states that are useful in the context of quantum information.

I Introduction

In the last years, the field of atomic, molecular and optical physics has witnessed an impressive advance in the development of setups to trap atoms under different conditions, like for example, ions in electromagnetic traps and ultracold atoms in optical lattices. Furthermore, the quantum state of these systems may be engineered by performing quantum operations such as as quantum gates between ions ions.gates (), or the excitation of neutral atoms under the dipole blockade LukinRydbergatoms (); exp.Rydberg (). In this way one can create deterministically collective entangled states, like the completely symmetric states with a single excited atom (-states) HÃ¤ffner (). In addition, those are systems where atoms can be coupled to light in a very controlled way. Since some of the atomic entangled states which may be created in those setups play an important role in the description of the interaction of atomic ensembles with light, an important question arises, namely, can we use our control on the states of trapped atoms to generate useful quantum states of light? If so, which are the properties of such states in terms of photon directionallity, purity, or photon entanglement?

A preliminary study of these ideas was recently presented in preprint (). The understanding of this problem requires a theoretical framework to describe the emission of light by collective atomic states under a variety of trapping conditions. Related problems have been indeed a subject of investigation since the seminal work by Dicke Dicke (). Nevertheless, the experimental systems that we have in mind share a few peculiarities which demand a new approach for their description. First, ultracold atoms are trapped in ordered arrangements such as Coulomb crystals or optical lattices, where the distance between atoms, , is comparable to the wavelength of the light, . This situation is very different from Dicke superradiance, where is larger than the size of the whole system. It also differs from the case of crystals, where as considered, for example, in Scully06 (). Also, we study a situation in which atoms are initialized in a collective state, which is not necesarilly created by the absorption of a photon. Thus, rather than the more traditional description in terms of light scattering, we need a theoretical model of the mapping between atomic collective excitations and photons.

In its more general form the above described situation poses a very complicated many-body problem. A crucial simplification is achieved by considering the low excitation sector of the atomic Hilbert space, which in the Holstein-Primakoff (HP) approximation can be described in terms of bosonic spin-waves HP (). This approach has been used in previous works, for example, to study Dicke superradiance Ressayre (), slow propagation of light Lukin00 (), and atom-light interfaces with atomic vapors Hammerer (). Also, the emission of light by ensembles of harmonic oscillators was studied in Zakowicz (), although in a different regime of trapping conditions than those considered here.

In this paper we make use of the HP approximation to describe the collective emission of light as a mapping between spin-wave excitations and photons. For atoms placed at fixed positions this mapping is a Gaussian completely positive map Giedke (), and we present a method to get explicit expressions of the photonic modes into which light is emitted. Furthermore, we extend this formalism to study the case of atomic vapors. We obtain the following results: (i) For atoms trapped in a regular lattice, there is a regime in which photons may be emitted in a collimated beam, which requires that . In this regime, there is a renormalization of the emission rates, leading to a classification of the low-excitation atomic Hilbert space in terms of superradiant and subradiant spin-waves. Superradiant states decay with a rate that is enhanced by a factor , which depends on the dimensionality of the lattice. In 1D and 3D we determine the values and , respectively, with the length of the lattice. This effect is related but not equivalent to Dicke superradiance. (ii) In the case of atomic vapors the collective characeter in the emission of light is determined by , where is the total number of atoms. The directional regime requires , and the emission rate is enhanced by . (iii) Some of these effects, like the renormalization of the emission rates and the directionality, could be observed in a relatively simple experiment with atoms in optical lattices. (iv) By making use of coherent effects, photonic entangled states could be generated by trapped ions or neutral atoms. Photons that are generated in this way could be collimated and in a pure state, and thus be useful in the context of quantum information.

Ii Theoretical framework

Our first task is to describe the collective emission of light by an ensemble of atoms. First, we focus on the situation in which atoms are placed at fixed positions. At the end of the section we discuss the effect of atomic motion.

ii.1 The atom-photon map in the Holstein-Primakoff approximation

We consider an ensemble of atoms trapped by harmonic potentials with trapping frequency , and internal levels forming a -scheme, see Fig. 1. Two ground states , , are coupled by means of a laser with Rabi frequency and wavevector , through and auxiliary level . We define atomic operators , , which fulfill the commutation relations,

 ⟨[σj,σ+l]⟩=δj,l(1−2⟨σssj⟩). (1)

Under the condition that the excitation probability of each atom is low, , we replace atomic operators by HP bosons, .

The HP approximation allows us to recast the atom-light Hamiltonian as a bosonic quadratic Hamiltonian. After the adiabatic elimination of the upper level , our system is described by

 H=H0+Hlm+Hmot. (2)

, with . We consider, for the shake of clarity, a scalar model for the electromagnetic field, since our conclusions do not change when including the dipole pattern, as we show later. However, the photon polarization can be included straightforwardly in our formalism. is the atom-light interaction Hamiltonian in the rotating wave approximation note.rwa () (we set ),

 Hlm = ∑j,kgkb+jakei(k−kL)rj+iωLt+\textmdh.c. , gk = ΩL2Δℏωk2ϵ0Vdga. (3)

is the quantization volume of the electromanetic field, the vacuum permitivity, and the dipole matrix element of the - transition. is the Hamiltonian describing the motion of the atoms during the emission process, whose energy scale is given typically by the trapping frequency . Eq. (3) is obtained under the assumption , with the mean vibrational occupation number. In the resonant case, , the validity of the adiabatic elimination of requires that the decay rate from to , , satisfies , and a similar coupling as (3) is obtained. The spontaneous decay of back to is neglected in the forecoming analysis. This is justified either when collective effects enhance the - channel, or by choosing - to be a cycling transition, see Appendix D. We deal first with the simplest case, in which atoms are placed at fixed positions and the motion of the atoms can be neglected. This is a good approximation in the Lamb-Dicke regime, , where is the ground state size in the harmonic trap. The effect of the motion is considered in the last subsection.

This work relies on the observation that generates a beam-splitter transformation between atomic states and photons. The evolution of the atom-photon system can thus be understood in terms of a Gaussian completely positive map Giedke (). In the low excitation regime, any atomic state can be expressed in terms of bosonic excitations,

 |Ψ⟩at=∑n1,…,nNΨn1,…,nN(b+1)n1√n1!…(b+N)nN√nN!|0⟩at. (4)

We consider the following initial state,

 |ψ(0)⟩=|Ψ⟩at|0⟩ph, (5)

where is the photon vacuum. Our goal is to find the photon state at a time longer than the atomic decay time,

 |ψ(t)⟩=U(t)|ψ(0)⟩=|0⟩at|Φ⟩ph, (6)

where , and is the total atom-system Hamiltonian. Together with the beam-splitter form of (3) this implies that the problem is reduced to finding the exact form of the transformation,

 U(t)b+jU(t)†=∑kgjk(t)a+k+∑lhjl(t)b†l. (7)

Taking into account that , with , then can be found by using the relation,

 gjk(t)=⟨vac|akU(t)b+j|vac⟩=⟨vac|ak(t)b+j|vac⟩, (8)

where is the photon operator in the Heisenberg picture. Also,

 hjl(t)=⟨vac|blU(t)b+j|vac⟩=⟨vac|bl(t)b+j|vac⟩, (9)

which shows that for longer than the atomic decay time. is thus the only interesting term, since it determines the photon mode into which light is radiated. Its determination can be readly done by means of the Heisenberg equation of motion for the photonic operators, which yields

 gjk(t)= (10) −igke−iωkt∑l∫t0dτe−i(k−kL)rl+i(ωk−ωL)τ ⟨vac|bl(τ)b+j|vac⟩.

This equation is our starting point for the exact determination of the photonic modes.

ii.2 Photonic modes

To find the mapping between atomic and photonic modes, we need to solve the master equation, to calculate the time evolution of the atomic correlator in Eq. (10). In general, this is a difficult many-body problem, but in the bosonic limit it can be described exactly. In terms of HP bosons, the master equation which describes the atomic dynamics reads Lehmberg (),

 dρdt = ∑i,j(Jijbjρb+i−Jijb+ibjρ+\textmdh.c.), (11)

where the include multiple light scattering and dipole-dipole interactions,

 Jij=∑kg2k∫∞0ei(ωk−ωL)τ+i(k−kL)(ri−rj)dτ. (12)

This expression is evaluated by using the identity , which yields the result,

 Jii = 12¯Γ, (13) Jij = 12¯Γe−ikL(ri−rj) (sin(kL|ri−rj|)kL|ri−rj|−icos(kL|ri−rj|)kL|ri−rj|)(i≠j),

where

 ¯Γ=13π(ΩL2Δ)2ω3Lϵ0c3d2ga, (14)

is the single atom decay rate. The inclussion of the photon polarization would change the spatial dependence of the couplings . However, as we show later, the collective phenomena would be the same. Note that we are not including the single atom Lamb shift, which may be simply absorved into the laser frequency . Since Eq. (11) is quadratic in bosonic operators, it is readly solved by defining eigenmodes which diagonalize the atomic quantum dynamics in the low-excitation limit,

 bj=∑nMjnbn, (M−1JM)nm=Jnδn,m. (15)

The matrix is not hermitean, and thus canonical commutation relations are not conserved, . However, the evolution of averages takes a simple form given by

 ⟨bl(τ)⟩=∑nMlne−Jnτ⟨bn(0)⟩. (16)

Thus, the spin-wave dynamics is governed by the eigenvalues . The latter contain the collective decay rates, , and the collective energy shifts,

 Γn=2 Re(Jn),Δn=2 Im(Jn). (17)

Conservation of the trace under the transformation (15) leads to the following sum rules,

 ∑nΓn=N¯Γ,∑nΔn=0. (18)

Note that whenever collective effects induce a renormalization of , the sum rule implies the existence of super- and subradiant states.

By application of (15, 16), and the quantum regression theorem, we determine the two-time atomic average in (10). In the limit that we get

 gjk(t)=ie−iωktgk∑ln(M−1)nje−i(k−kL)rlMlni(ωk−ωL)−Jn, (19)

which yields the explicit form of the atom-photon mapping. By means of this relation it is possible to determine the many-photon state emitted by any initial atomic state like (4).

To get insight of the characteristics of the photonic states emitted by the collective atomic states, we focus from now on, on the mapping to a single photon. To clarify the notation, let us define , the -spin-wave state with a single excitation,

 |Ψn⟩at = 1Nn∑jMjnb+j|0⟩at, Nn = √∑jM∗jnMjn. (20)

has to be included due to the non-hermiticity of . We determine , the single photon state into which is mapped,

 |Φn(t)⟩ph = ∑kϕn,k(t)a+k|0⟩ph, ϕn,k(t) = igke−iωktNn∑le−i(k−kL)rlMlni(ωk−ωL)−Jn. (21)

Although these results allow one to solve exactly the problem of collective emission of light including the effects of reabsorption, aditonal insight can be gained by considering the case of a system with periodic boundary conditions. This will be a good approximation for a finite system, provided that the number of atoms in the volume is much larger than in the surface, that is, . In this case we get the matrices

 Mjn=1√NeiKnrj. (22)

The spin-wave state is then defined like

 |Ψn⟩at=1√N∑jeiKnrjb+j|0⟩at. (23)

The vector is the momentum of the collective atomic state. The resulting photonic mode is defined by the following expression,

 ϕn,k(t)=igke−iωkt1√N∑le−i(k−kL−Kn)rli(ωk−ωL)−Jn, (24)

such that collective effects and dipole-dipole interactions enter through the dependence of the collective emission rate on the mode number, . Note that by using periodic boundary conditions, the matrix define an unitary transformation, and thus the sets , and , form an orthogonal basis of spin-waves, and photonic modes, respectively.

ii.3 Angular photon number distribution

We determine now the properties of the emitted photonic modes. In particular, let us define

 I(Ω)=V(2π)3∫∞0⟨a+kak⟩k2dk, (25)

the average photon number per solid angle. Consider an initial atomic state with a single excitation,

 |ψ⟩at=∑jψjb+j|0⟩at. (26)

The emitted photon distribution is

 I(Ω) = ∑j,j′ψ∗jIjj′(Ω)ψj′, Ijj′(Ω) = V(2π)3∫∞0g∗j,kgj′,kk2dk. (27)

Upon substitution of (19) in the expression for , and under the condition that (see Appendix A), we get,

 Ijj′(Ω) = ¯Γ4π∑n,n′B(Ω)jn1¯J∗n+¯Jn′B(Ω)j′n′, B(Ω)jn = (M−1)∗nj∑lM∗lnei(kLnΩ−kL)rl, (28)

where is a unit vector pointing in the direction of the solid angle . In spherical coordinates, is determined by , such that .

The general recipe for calculating the photon distribution probability involves the following steps: (i) Calculate the coefficients of the master equation and find the eigenvalues and eigenvectors of . (ii) Use Eq. (28) to calculate . (iii) Use the latter to calculate the emission spectrum with the wavefucntion of any given initial atomic state expressed in terms of bosonic spin-waves. In the planewave approximation (22) we can get close expressions for the photon distribution. We focus again in the single photon case, and define as the photon number distribution corresponding to the photonic mode emitted by . By using Eq. (28) we get

 In(Ω)=14π¯ΓΓn∑j,j′ei(kΩ−kL−Kn)(rj−rj′). (29)

This result has a clear interpretation in terms of interference of light emitted by the atomic system Jackson (). Eq. (29) not only allows one to calculate the angular emission probabilty, but also, due to the normalization condition ,

 Γn¯Γ=∫dΩ14π∑j,j′ei(kΩ−kL−Kn)(rj−rj′). (30)

This expression provides us with a simple way to determine the collective rates under the planewave approximation.

ii.4 Effects of the atomic motion

Finally, we discuss the effect of the atomic motion on the light emission. In the most general case, the inclusion of the motional degrees of freedom poses a very complicated problem which goes beyond the scope of this work. Two time scales determine this problem. First, , the time scale of the motion of atoms in the trap. In the case of trapped particles, , with the trapping frequency. We can extend this discussion to the case of atomic vapors, and consider that in this case, , with the length of the sample, and the atom velocity. is to be compared with the radiative decay time, , or more specifically, the set of collective decay times . Based on the comparison between these time scales we define two limits in which the application of our theoretical framework is particularly straightforward.

(i) Slow motion limit, . Since the emission process is much faster than the motion of the particles, we can assume that atomic positions are frozen. The system is in the initial state

 |ψ(0)⟩=|Ψ⟩at|0⟩ph|Ψm⟩mot, (31)

where is the initial wavefunction in terms of the atomic positions, which in this limit does not evolve during the emission time. The atom-photon mapping can be still applied to this system by solving it for each value of the atomic positions,

 |Ψ⟩at|0⟩ph→|0⟩at|Φr1,…,rN⟩ph, (32)

where is the photonic state obtained under the assumptions that ions are located at positions . Any photonic observable, , is then obtained upon averaging with respect to the wavefunction , for example,

 ⟨O⟩=∑r1,…,rN⟨Φr1,…,rN|O|Φr1,…,rN⟩|Ψ(r1,…,rN)|2. (33)

This method can be readly extended to the case of a mixed motional state.

(ii) Fast motion limit, . In this case, we are assuming that trapped atoms move along the sample in a time that is smaller than the emission time. This case is particularly relevant, since it describes hot atomic vapors. We notice first that condition , implies that the atomic positions are not correlated with the atomic operators in Eq. (10). Thus, we can describe the atomic radiative decay indepently of the evolution of atomic positions in (10). This can be done by using a master equation with averaged coefficients,

 Jij=π∑kg2k δ(ωk−ωL)⟨ei(k−kL)(ri−rj)⟩mot, (34)

In the case of an harmonic trap, are position operators, and is the average with the atomic motional state. This situation can be extended to describe hot atomic vapors, by replacing the atomic positions, , by a set of random variables with a given probability distribution , and performing the corresponding average to get .

Once the atomic dynamics is solved, our results on the atom-photon mapping can be used to describe the emitted photons. A general atomic state is mapped now into a mixed photonic state. For example, consider the atomic state , defined by Eq. (20), which is obtained by the diagonalization of the master equation with (61). The atom-photon mapping yields the following photon density matrix after the emission process,

 ρph(t)=∑k,k′⟨ϕn,k(t)ϕ∗n,k′(t)⟩mota+k|0⟩ph⟨0|ak, (35)

and this expression is easily generalized to the multiphoton case. Note that the solution of the fast motion limit seems similar to the case of slow motion. However, the crucial difference is that in the fast case, we are allowed to solve the radiative emission problem, and to perform subsequently the spatial average in (35). We will study in more detail this situation later in the case of an the collective emission properties of atomic ensambles.

Iii Atom-Photon mapping in a square lattice

The situation in which atoms are arranged in a crystal is found in experimental setups such as ultracold atoms in optical lattices and Coulomb crystals of trapped ions. The results presented in the previous section are applied here to study the collective emission process in these systems. We obtain analytical results by using the plane-wave approximation.

iii.1 General Discussion

Let us study for concreteness the case of one (1D) or three dimensional (3D) square lattices, although our results are easily generalized to different lattice geometries. Assuming periodic boundary conditions, the allowed wavevectors are

 Kn=2πd0∑αnαNα^α. (36)

In the 1D case we consider that the atom chain is aligned in the direction. Thus, in (36) and the forecoming expressions, runs over in 1D, and in 3D. is a unit vector in the direction , and is the number of atoms along . Considering, for concreteness, the case of even , each wavevector is determined by the set of integers . By applying Eq. (29) we determine the photon number angular distribution for the photonic state emitted by a spin-wave excitation with momentum ,

 In(Ω)=14π¯ΓΓn (37) 1N∏αsin2((kLuαΩ−kαL−Kαn)d0Nα/2)sin2((kLuαΩ−kαL−Kαn)d0/2).

shows a series of diffraction maxima at solid angles at which the -function in the denominator vanishes. Note that by including the photon polarization, we would have got an additional function of multiplying the photon distribution , which would correspond to the single atom dipole pattern. The latter would induce the suppression of diffraction peaks, if they are in a direction forbidden by the dipole pattern. Since we are specifically interested on collective effects we do not consider this effect in the discussion that follows.

To get a quantitative description of the photon distribution we notice first that

 (38)

where the function describes the shape of each of the diffraction peaks,

 fN(x) = Nsinc2(Nx),  −π/2

Thus in the limit , one can approximate the emission probability as a sum over Bragg scattering contributions,

 In(Ω) = ∑mI[m]n(Ω), I[m]n(Ω) = 14π¯ΓΓn ∏αfNα(kLd0nαΩ2−kαLd02−nαNαπ+mαπ).

Each term in the sum is labeled by the vector , and corresponds to a different diffraction peak. The probability that the spin-wave emits a photon in the diffraction peak is given by

 p[m]n=∫dΩ I[m]n(Ω). (41)

Eq. (III.1) has a clear interpretation in terms of momentum conservation. has a maximum whenever there is a value of such that

 kLuαΩ=kαL+nα2πd0Nα+mα2πd0,  ∀ α. (42)

That is, the linear momentum of the emitted photon has to match the sum of three contributions: the momentum of the incident laser, ; the initial momentum of the spin-wave, , and the contribution from the lattice periodicity, which enters through the reciprocal wavector . In 1D, condition (42) has to be satisfied only by the component of these vectors, that is, only the projection of the momentum on the chain is conserved. In 3D, on the contrary, the equality has to be satisfied by all the vector components. The relation (42) determines the maxima in the emission pattern depending on and , but it also determines the collective rates, through the normalization condition on , see Eq. (30).

Since , if there is at most a single value of , for which condition (42) can be fulfilled for some vector . Thus, is the directional regime in the emission of photons, whereas in the case , we cannot ensure that photons are collimated in a single direction. In the following two subsections, we will study these regimes in 1D and 3D. In particular, we will be interested in determining , the angular width of the photon beam in the directional regime, and , the collective emission rates. These quantities will be studied as a function of dimensionality, , , and .

iii.2 Atom chains

We consider for concreteness that points in the direction, parallel to the chain axis. The vectors , are reduced now to scalars , , that correspond to the projections on the chain axis. Due to the symmetry of the problem, the photon distribution emitted by a spin-wave depends only on the angle , through ,

 In(Ω)= 14π¯ΓΓn1Nsin2((kLd0(uzΩ−1)−n2πN)N/2)sin2((kLd0(uzΩ−1)−n2πN)/2). (43)

Taking the limit and using (38),

 In(Ω) = ∑mI[m]n(Ω), (44) I[m]n(Ω) = 14π¯ΓΓnfN(kLd02(uzΩ−1)−nπN+mπ).

The peaks in the photon distribution correspond to the emission of photons such that the component of momentum is conserved,

 uzΩ−1=λd0nN−λd0m, (45)

that is, there is a maximum whenever there is a value of which satisfies this relation. In the directional regime, , there is at most a single value of which satisfies (45). However, in this case, momentum conservation only determines the value of at the emission maximum, which implies that, in general, photons are emitted in cones spanned by different values of . Only when the maximum happens at , or , photons are collimated in the forward- or backward-scattering directions, respectively (see Fig. 2).

Let us study first the case , that is, the emission properties of the completely symmetric state. The forward-scattering contribution, , has an angular width given by . The contribution to the emission pattern from each of the Bragg terms is

 p[0]0 = ∫dΩI[0]0(Ω)=¯ΓΓ0λ4d0, p[m]0 = 2p[0]0,m≠0. (46)

Condition (45) leads to the result that the number of emission cones with is given by . A calculation made without resorting to the planewave approximation, yields the same results, even for relatively small atom numbers (), see Fig. 3. However, the shape of the photon angular distribution in the exact calculation shows a departure from the -shape predicted by Eq. (III.2), see Fig. 4. By using the normalization condition for , we determine the probability of emission in the forward-scattering direction,

 p[0]0=11+2 int(2d0λ). (47)

Thus, in the directional regime, , all the diffraction peaks but the forward-scattering one are suppressed. The collective emission rate is given by

 Γ0¯Γ = λ4d0+λ2d0int(2d0λ). (48)

Finally, we calculate the emission rate for all the 1D spin-wave states. They can be written as an integration over the contributions coming from different diffraction peaks, through the normalization condition, we have

 Γn¯Γ=14π∑m∫1−1dxfN(kLd02(x−1)−nπN+mπ). (49)

The collective rates are thus determined by the number of diffraction peaks which appear in the emission pattern of each collective state . Considering the limit in which , that is, the delta limit for each of these peaks, we get the following distribution of emission rates,

 Γn¯Γ= χ1D((θ(−nN)θ(2d0λ+nN)+int(2d0λ+nN)), (50)

with , and is the Heaviside function. This expression describes quite well the emission rates calculated without assuming periodic boundary conditions, that is, by diagonalizing the matrix given by (14), see Fig. 5. In the limit , we recover . On the contrary if is comparable to , emission rates are renormalized. In the directional regime, , some of the states have an enhanced rate, (superradiant), whereas there are states for which (subradiant) note.finite (). Due to the relation between the photon distribution and the emission rates (49), subradiant states correspond to spin-waves whose emission pattern is not peaked at a given value of . On the contrary, superradiant states are spin-waves whose emission pattern does contain a maximum as a function of . Note that Eq. (50) does not describe the case of Dicke superradiance, which would predict a single superradiant state, with . The reason is that we have assumed condition , that is, where are always in the regime in which the light wavelength is much smaller than the size of the chain. Our results for 1D are consistent with Ref. Carmichael (), where the collective light emission form an atomic chain was studied with the quantum jump formalism.

To summarize the situation in 1D, in the directional regime , superradiant spin-waves emit photons into a single emission cone, with a rate that is enhanced by . This effect can be used to generate photons that are collimated along the chain axis. Note that, in general, the atom-photon mapping induced with the -scheme of Fig. 1 may compete with other radiative processes, such as the radiative decay from back to , or the radiative decay from to other atomic levels that are not included in the -scheme. This problem can be solved by enhancing the atom-photon mapping rate, choosing . Since does not depend on the chain size, this implies to choose . Another way out of this problem is to use a cycling transition (see Appendix D).

iii.3 3D Atom lattices

Contrary to the 1D case, in 3D it is not simple to obtain closed expressions to describe the emission in the planewave approximation. In order to get a simpler picture, we replace the -function in the definition (38) by a gaussian which is normalized in the same way,

 fN(x)≈√πNxe−x2N2. (51)

This approximation is justified in the limit in which the diffraction peaks are narrow enough, such that they do not overlap, that is, . Assuming that the length of the lattice is the same in any spatial direction, , then the angular photon distribution for a state is given by

 In(Ω) = ∑mI[m]n(Ω), (52) I[m]n(Ω) = 14π¯ΓΓnπ3/2Ne−(kLd02uΩ−kLd02−πnN−πm)2N2x.

A given term , has a non-negligible contribution only if

 |kLd0+2πn/N+2πm|kLd0=1, (53)

since otherwise, there are no photons which satisfy the energy-momentum conservation.

Let us study first the directional regime, . In this case we find the following two possible situations:

(i) Superradiant states – Spin-waves for which there exists a value of , say , such that (53) is satisfied. The condition ensures that there is a single value , such that the photon distribution can be simplified,

 In(Ω)=I[mc]n(Ω). (54)

Thus, there is a single emission peak in the direction,

 uΩmax=kLd0+2πn/N+2πmckLd0. (55)

Photons are collimated in a beam with width . From (54), and the normalization condition, we can determine the emission rate,

 Γn¯Γ = (56) N4π∫dΩ π3/2e−(kLd02nΩ−kLd02−πnN−πmc)2N2x= 1√πNx(λ2d0)2=χ3D.

Thus, superradiant states have a decay rate that is enhanced by the optical thickness, .

(ii) Subradiant states – Spin-waves for which there is no value of which satisfies (53). In this case, there is not a single dominant contribution , such that, directionality in the emission of photons is not guaranteed. Also, the normalization condition on leads directly to the result note.finite ().

From this analysis, we conclude that superradiant spin-waves are interesting in the context of quantum information, since they emit collimated photons with an enhanced emission rate. However the emission direction is determined by a value of which has to be calculated for each particular spin-wave. There are two relevant cases in which this situation becomes simpler, since energy-momentum conservation ensures that the only Bragg contribution is . The first one is the case , because

 |kLuαΩ−kαL| < 2kL, |nαN2πd0+mα2πd0| > 122πd0  (if  m≠0), (57)

and thus for values the energy-momentum conservation condition (42) is only fulfilled by . Second, assume that the spin-wave is created by the absorption of a photon within the -configuration of Fig. 1, such that the spin-wave linear momentum can be written like , with . Then, the energy-momentum condition is now

 q+k=m2πd0, (58)

which is satisfied only by if . Indeed, the spin-wave created in this way is always superradiant, since automatically satisfies Eq. (53). A similar result is obtained in Ref. Scully08 () for the case of a cloud of atoms, where the rate of spontaneous emission of a state created by the absorption of a photon, is shown to increase with the optical thickness.

To summarize the situation in 3D, in the directional regime , superradiant spin-waves emit collimated photons with a rate enhanced by . Contrary to the situation found in 1D, in 3D lattices , and thus it can be increased by increasing the size of the system. In this way, the atom-photon mapping could be much faster than other competing decay channels.

Iv Emission of light by hot atomic ensembles

The formalism presented in the previous section can be extended to situations in which atoms are not at fixed positions in space. In this case, one expects the photonic state not to be a pure state, but a mixed one. This is relevant to the description of hot atomic vapors. We follow the discussion presented in subsection II.4, and consider the case of fast motion (). The results presented here are in agreement with Ref. duan.pra (), where the photonic mode emitted by an atomic ensemble is studied. Our first task is to reformulate the problem of photon emission by finding the master equation that describes this situation.

iv.1 Master Equation for atomic ensembles

In the fast motion limit, particle positions may be described as a set of independent random variables, . We choose for simplicity a gaussian distribution probability,

 ρ(r)=1π3/2L3e−(r/L)2. (59)

The particle positions fulfill the following identity, which will turn out to be the basis for the following calculations,

 ⟨e−iq(rj−rl)⟩mot=δjl+(1−δjl)e−q2L2/2, (60)

where is an average over the atomic positions.

To calculate the coefficients of the master equation we start from Eq. (12) and perform the following average,

 Jij=π∑kg2k δ(ωk−ωL)⟨ei(k−kL)(ri−rj)⟩mot, (61)

were we have neglected the Cauchy principal value contribution, since it leads to an energy shift that does not play any role in the discussion that follows. After using (60), integrating over , and taking the limit , we get

 Jii = ¯Γ/2, Jij = ¯Γ4(kLL)2.  (i≠j). (62)

One can readly diagonalize the matrix and obtain the eigenspaces of the master equation. The first is the completely symmetric state (),

 Mj0=1√N,J0=¯Γ2(χen+1), (63)

where is the optical thickness of the atomic ensemble. Note that in the limit , and defining the atom density, , the optical thickness can be recast in the more familiar form . The second eigenspace is spanned by the spin-waves orthogonal to ,

 ∑jMjn=0,Jn=¯Γ2(1−12(kLL)2),if n≠0. (64)

Collective effects happen if . In this case, there is a single superradiant spin-wave mode, corresponding to the completely symmetric state, and states which decay with the single atom emission rate, .

iv.2 Single photon state

The spin-wave mode is the only one to show collective effects, and it is the collective state which can be created in experiments with atomic vapors. For these reasons we focus on the following on its properties. First, we consider the emission of light by the initial atomic state with a single excitation,

 |Ψ0⟩at=1√N∑jb+j|0⟩at. (65)

The atom-photon mapping can be extended to this situation by considering first that the atomic state is mapped into a given photonic state, and then by performing the average on the atomic positions. This gives as a result a photon density matrix,

 |Ψ0⟩at|0⟩ph⟨0| → |0⟩atρ, ρ = ∑k,k′ρkk′a+k|0⟩ph⟨0|ak′. (66)

Note that the mapping is now form pure to mixed states. Unfortunately, with the statistical properties of the atomic positions considered here, it is not possible to ensure that this is still a gaussian map. We can, however, use Eqs. (24,60) to get

 ρkk′ = ⟨ϕ0,kϕ∗0,k′⟩mot=(1−ϵ)¯ϕ0,k¯ϕ0,