Spontaneous Photon Emission in Cavities

G. Alber1 and N. Trautmann1

1Institut fur Angewandte Physik, Technische Universitat Darmstadt,D-64289, Germany(Dated: December 4, 2014)

We investigate spontaneous photon emission processes of two-level atoms in parabolic and ellipsoidal cavitiesthereby taking into account the full multimode scenario. In particular, we calculate the excitation probabilities ofthe atoms and the energy density of the resulting few-photon electromagnetic radiation field by using semiclassi-cal methods for the description of the multimode scenario. Based on this approach photon path representationsare developed for relevant transition probability amplitudes which are valid in the optical frequency regimewhere the dipole and the rotating-wave approximations apply. Comparisons with numerical results demonstratethe quality of these semiclassical results even in cases in which the wave length of a spontaneously emittedphoton becomes comparable or even larger than characteristic length scales of the cavity. This is the dynamicalregime in which diffraction effects become important so that geometric optical considerations are typically notapplicable.

I. INTRODUCTION

The investigation of resonant light-matter interaction hasreceived considerable attention during the last decades and re-markable experimental progresses in this field has opened thedoor to whole new experimental opportunities [1–3]. Thesedevelopments are not only interesting from a fundamentalpoint of view but also from practical points of view for possi-ble applications in the area of quantum information process-ing, for example. In this latter context methods for achiev-ing an optimal transfer of quantum information between sin-gle photons (flying qubits) and elementary material systems(stationary qubits) over large distances by using the resonantstrong coupling, as required for the realization of a quantumrepeater, are of particular interest. Therefore, recently con-siderable efforts [4–6] have been devoted to investigate theinteraction of matter qubits with few modes of the radiationfield within the framework of Jaynes-Cummings-Paul mod-els [7]. Recent developments stimulated extensions of thiswork to extreme multimode scenarios or to scenarios involv-ing structured continua of electromagnetic field modes whichare characteristic for half open cavities of a parabolic shape[8, 9], for example.

In the following we review some of these recent researchactivities focusing on the resonant matter-field interaction oftwo-level atoms located around the focal point of a parabolicshaped cavity and generalize them to prolate ellipsoidal cav-ities which include parabolic cavities as limiting cases. Aparticularly interesting elementary quantum electrodynamicalsituation arises in a prolate ellipsoidal cavity if two two-levelatoms are trapped in the two foci of this cavity and exchange asingle photon which is emitted spontaneously by one of thesetwo-level atoms.

II. THE QUANTUM ELECTRODYNAMICAL MODEL

In our model we investigate the dynamics of n ∈ {1, 2}identical two-level atoms with transition frequencies ωeg anddipole matrix elements ~da = 〈e|a da |g〉a. These two-levelatoms are assumed to be situated in the focal points of a pro-

late elliptical cavity ~xa and to interact with the quantized ra-diation field inside this cavity. Furthermore, we assume thatthe sizes of the relevant electronic states two-level atoms aresmall compared to the wavelengths λeg = 2πc/ωeg of theatomic transition so that the dipole approximation is appli-cable (c is the speed of light in vacuum.). Moreover, as we areconsidering atomic transitions in the optical frequency regimethe relevant coupling (Rabi-) frequencies are small in com-parison with ωeg so that the rotating-wave approximation isapplicable. Thus, the quantum electrodynamical interactionbetween n = 2 two-level atoms and the quantized electromag-netic field inside the cavity can be described by the Hamil-tonian H = Hatoms + Hfield + Hi with Hfield = ~

∑i ωia

†

i ai,

Hatoms = ~ωegn∑

a=1|e〉a 〈e|a , Hi = −

n∑a=1

E+⊥(~xa) · d−a + H.c. and

with the dipole operator d−a = ~d∗a |g〉a 〈e|a of the two-level atoma . Thereby, the energies of the atomic ground states |g〉a havebeen set equal to zero. In the Schrodinger picture the posi-tive frequency part of the (transversal) electromagnetic fieldoperator is given by

E+⊥(~r) := i

∑i

√~ωi

2ε0~gi(~r)a†i

and ~gi(~r) denote the normalized transversal electric modefunctions which fulfill the boundary conditions for an ideallyconducting cavity.

III. MODE FUNCTIONS OF THE CAVITY AND THEIRSEMICLASSICAL APPROXIMATION

For arbitrary boundary conditions the calculation of themode functions ~gi(~r) is generally difficult. However, in casesin which the wave lengths of relevant field modes are small incomparison with all other length scales of the problems thistask may be facilitated by semiclassical methods. In order toput the problem into perspective let us consider cases in whichthe dipole matrix elements ~da of the atomic transitions are ori-ented along the symmetry axis, i.e. the z-axis, of a parabolic orellipsoidal cavity. Thus, we can write ~da = D~ez. This symme-

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try can be exploited as it is possible to show that in these casesthe only field modes which couple to the two-level atoms inthe dipole approximation are of the form [10]

~gi(ϕ, ξ, η) = ∇ × H, H = ~eϕFi(ξ)Gi(η).

Thereby, ϕ, ξ and η are the symmetry adapted parabolic co-ordinates in the case of a parabolic cavity and the prolate el-lipsoidal coordinates in the case of an ellipsoidal cavity. Fora parabolic cavity the functions Fi(ξ) and Gi(η) turn out tobe Coulomb functions and for an ellipsoidal cavity they turnout to be prolate spheroidal wave functions. In both caseswe can use semiclassical methods to express the quantizationconditions resulting from the boundary conditions of an ide-ally conducting wall, for example, semiclassically in a simpleway by taking advantage of the separability of the Helmholtzequation in these coordinates. This way one obtains a singlequantization function for one of the coordinates, say η, in thecase of a parabolic cavity and two quantization conditions inthe case of an ellipsoidal cavity. Furthermore, these semiclas-sical methods are also well suited for normalizing these modefunctions appropriately.

IV. PHOTON PATH REPRESENTATION OFPROBABILITY AMPLITUDES

We can now use these results to calculate the time evolu-tion of the system. Our Hamiltonian has the property that the

number of excitations in the system (son∑

a=1|e〉a 〈e|a +

∑i

a†i ai)

is a conserved quantity. If we assume that initially only onetwo-level atom is in an excited state and that the field is inthe vacuum state we know that for all later times the numberof excitations in the system equals unity. This leads to thefollowing ansatz for the wave function

|ψ(t)〉 = b1(t) |e, g〉A |0〉P + b2(t) |g, e〉A |0〉P

+∑

i

fi(t) |g, g〉A |1〉Pi

in the case of two two-level atoms in the cavity. (The super-scripts A and P refer to atoms and photons, respectively.) It isa straight forward task to generalize the following calculationto an arbitrary number of atoms. The Schrodinger equationleads to a coupled system of linear differential equations. Inorder to solve this system we apply the Laplace transform inorder to obtain a system of linear algebraic equations. Dueto basic properties of our Hamiltonian it is straight forward toeliminate the Laplace transforms of the photonic excitationsfi(s) (s denotes the parameter of the Laplace transform) in or-der to obtain a linear system of equation for b1(s) and b2(s).This system of equations is given by(

b1(0)b2(0)

)=

[(s + iωeg)1 +

(A1,1 A1,2

A2,1 A2,2

)] b1(s)b2(s)

with

Aa,b := |D|2∑

j

ω j

2ε0~

(~g j(~xa)

)z

(~g j(~xb)

)∗z

s + iω ja, b ∈ {1, 2} .

The functions Aa,b encode all the properties of the cavity. Itis a non trivial task to evaluate these functions because theyare defined by an infinite sum over all modes j which cou-ple to the atomic dipoles. However, we can evaluate thesefunctions with the help of the semiclassical methods devel-oped previously. With these functions at hand we can solvethe linear system of equations and apply the inverse Laplacetransform in order to obtain expressions for the time evolu-tion of the probability amplitudes of observing atoms 1 and 2in the excited states, i.e. b1(t) and b2(t). However, evaluatingthe inverse Laplace transform is still a non trivial task. For thispurpose it is convenient to solve the linear system of equationsby applying the Neumann series in order to obtain an infinitesum of simple expressions for b1(s) and b2(s) and then ap-ply the inverse Laplace transform to each of these terms. TheNeumann series is given by

b1(s)b2(s)

=

∞∑n=0

[−

(T 1,1

1 T 1,21

T 2,11 T 2,2

1

)]n

(s +Γfree

2 + iωeg)n+1·

(b1(0)b2(0)

).

Thereby, we have defined T a,b1 := Aa,b − δa,bΓfree/2 and Γfree

denotes the spontaneous decay rate of each two-level systemoriginating from spontaneous emission of a photon in freespace, i.e. in the absence of any boundary conditions for theradiation field. Of course, at first sight it may appear as a ma-jor complication to apply the Neumann series in order to in-vert a 2 by 2 matrix. However, this expansion leads to a semi-classical photon path representation of the relevant probabilityamplitudes [11, 12] so that we are able to interpret each termof this expansion as a contribution of a photon path. Thereby,each photon path involves a sequence of spontaneous emis-sion and absorption processes. In this expansion the functionsAa,b encode the possible paths a photon can take to reach atomb starting from atom a. In case of a parabolic cavity we canperform the same calculations but instead of a two by two ma-trix of the functions Aa,b we only have to consider the singleterm A1,1.

V. MODIFICATION OF THE SPONTANEOUS DECAYRATE BY BOUNDARY CONDITIONS

We can use our results of the preceding chapter to calcu-late the influence of the Purcell effect [13] on the sponta-neous decay rate. In the pole or Weisskopf-Wigner [14] ap-proximation the spontaneous decay rate of atom a is given byΓ = 2Re [Aa,a]

∣∣∣∣s=−iωeg

. Thus, for a parabolic cavity we obtain

the result [12]

Γ

Γfree= 1 + 2

∞∑M=1

3 cos [2M(u − π/2)]

×2MS(u) coth [2MS(u)] − 1

sinh2 [2MS(u)]

∣∣∣∣u=2π f

λeg

with S(u) :=u∫

0sin2(y)/y and with f denoting the focal length

of the parabola. Each summand in the expression for Γ/Γfree

3

0 2 4 6 8 10 12 140.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.5

1.0

1.5

2.0

FIG. 1. Scaled spontaneous decay rate Γ/Γfree as a function of thescaled focal length 2π f /λeg of the parabola: Numerically exact re-sults (dots), semiclassical results (solid). We also indicate the free-space value as a horizontal line.

is connected with a possible photon path inside the cavity, i.e.with bouncing back and forth of a photon (counted by M) be-tween the atom and the vertex of the parabola. These contri-butions to the decay rate are connected with diffraction effectswhich are suppressed for large values of f � λeg. Similar butmore complicated results are obtained for an ellipsoidal cav-ity. The main difference between both cases is that for an ellip-soidal cavity we obtain a double sum which can be traced backto the existence of two quantization conditions. Of course, theresults for a parabolic cavity can be obtained from the moregeneral results for an ellipsoidal cavity in the limit in whichthe distance between the two focal points tends to infinity. Infigure 1 the semiclassical analytical results are compared withexact numerical results. This numerical comparison demon-strates the quality of the semiclassical results even in cases inwhich the wave length of the spontaneously emitted photonbecomes comparable to the characteristic length scales insidethe cavity. Thus, the previously discussed semiclassical ap-proximation also enables us to investigate diffraction effectswhich become relevant for f . λeg.

VI. TIME EVOLUTION OF ATOMIC EXCITATIONPROBABILITIES

We can now investigate the time evolution of atomic ex-citations. In this chapter we are going to present results for aprolate ellipsoidal cavity. For this purpose we need some addi-tional quantities for describing the cavity, namely the distanced between the focal points and the shortest distance f betweenone of the vertices and its closest focal point. In the follow-ing we concentrate on the regime of short wavelengths, i.e.d, f � λeg, in order to obtain a simple picture of the dynam-ics. If we start with an initial excitation of atom 1 we observe a

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Atom 2Atom 1

a)

0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

Atom 2Atom 1

b)

FIG. 2. Atomic excitation probabilities for Γfree · τ = 16 and f /d =

1/2 (a) and f /d = 9/2 (b). The excitation probability of atom 1 (2)is represented by the blue (red) curve.

bouncing back and forth of the atomic excitation between thetwo atoms inside the cavity by exchange of a spontaneouslyemitted photon. The time τ the excitation needs to travel fromone atom to another is given by τ = (d + 2 f )/c. The cor-responding behavior of the atomic excitation probabilities isdepicted in figures 2 a and b. We observe that the probabilityof a successful transfer of the excitation to the second atomdepends significantly on the shape of the cavity. If f � d,i.e. the cavity is almost spherically symmetric, the excitationprobabilities associated with repeated returns of a photon toan atom are significantly larger than for cases with d � f .This can be observed by comparing the excitation probabili-ties depicted in figures 2 a) and 2 b). This phenomenon maybe traced back to the fact that both atomic dipole matrix el-ements are oriented along the z-axis. In the case of d � fthe radiation field of the photon emitted spontaneously by thefirst atom has to propagate almost along the z-axis in orderto approach atom 2. Therefore, its electrical field is almostperpendicular to the z-axis and its coupling to the dipole ofatom 2 is small. As a consequence the excitation probabilityis also small. In the opposite limit of an almost sphericallysymmetric cavity such a confinement of the photonic wavepacket along the symmetry axis due to the shape of the cavitydoes not occur and the excitation probabilities are larger.

VII. TIME EVOLUTION OF THE ENERGY DENSITY OFTHE ELECTROMAGNETIC RADIATION FIELD

We can also calculate the energy density of the electromag-netic radiation field in the case f , d � λeg by using multidi-mensional semiclassical methods [15]. Let us investigate thedynamics of a photonic wave packet which carries the exci-tation from one atom to the other inside a prolate ellipsoidalcavity. Similar to the bouncing back and forth in the time evo-

4

FIG. 3. Time evolution of the (normally ordered) energy density (in units of 3π~ωegΓfree/(80d2c)) of a spontaneously emitted photon in aprolate ellipsoidal cavity: The parameters are Γfreeτ = 16 and f /d = 9/2; t = 0.2τ (a),t = 0.8τ (b), t = 1.4τ (c), and t = 2.4τ (d). The red dotsindicate the focal points.

lution of the atomic excitation probabilities we can observe abouncing back and forth of the photonic wave packet in thecavity. The dynamics of repeated absorption and spontaneousemission events also leads to a change of the shape of the pho-tonic wave packet. This process is illustrated in figures 3 a-dfor the electromagnetic energy density of a single photon. Thecorresponding atomic excitation probabilities are depicted infigure 2 b. Figure 3 a illustrates the energy density producedby the spontaneous emission of a single photon. The cav-ity guides the one-photon wave packet to the second focus asillustrated in figure 3 b. Figure 3 c shows the wave packetdirectly after the interaction with the second atom. Figure 3d illustrates the wave packet after its interaction with the firstatom.

VIII. CONCLUSION

We have investigated the dynamics of spontaneous photonemission of two-level systems in parabolic and prolate ellip-

soidal cavities assuming that the atoms are situated in the focalpoints of these cavities. Our theoretical approach is based ontreating the multimode aspects of this problem with the help ofone-dimensional semiclassical methods by exploiting the sep-arability of the relevant Helmholtz equation. This way we areable to improve straight forward multidimensional semiclas-sical approaches considerably. With the help of semiclassi-cal photon path representations of relevant probability ampli-tudes we have explored the intricate dynamical interplay be-tween spontaneous photon emission and absorption processeson the one hand and reflections of the photon wave packet atthe boundary of the cavity on the other hand.

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