Non-Markovian Decay of a Three Level Cascade Atom in a Structured Reservoir

We present a formalism that enables the study of the non-Markovian dynamics of a three-level ladder system in a single structured reservoir. The three-level system is strongly coupled to a bath of reservoir modes and two quantum excitations of the reservoir are expected. We show that the dynamics only depends on reservoir structure functions, which are products of the mode density with the coupling constant squared. This result may enable pseudomode theory to treat multiple excitations of a structured reservoir. The treatment uses Laplace transforms and an elimination of variables to obtain a formal solution. This can be evaluated numerically (with the help of a numerical inverse Laplace transform) and an example is given. We also compare this result with the case where the two transitions are coupled to two separate structured reservoirs (where the example case is also analytically solvable).


I. INTRODUCTION
The quantum optical behaviour of atomic systems coupled to a continuum of the quantized electromagnetic field modes has been studied since the early days of quantum physics.
The quantum electromagnetic field is a large system, which can be treated as a bath or reservoir.In most cases the atom-field coupling constants and the electromagnetic field mode density are slowly varying functions of frequency, and the dynamics of the atomic system can be treated via Markovian master equations [1,2], or equivalent methods such as quantum Langevin equations (see e.g.[2] for details of these standard methods).These techniques are based on quantum electromagnetic field states with no special distinction for any particular mode in terms of photon occupation number, such as thermal states or broad-band squeezed states.Naturally, if one mode of the electromagnetic field was in a special state, such as a large amplitude coherent state (as in the case where the atom is also coupled to a laser field), then this special mode and the atomic system would be treated as a small quantum system with the remaining modes constituting the reservoir, so that Markovian behaviour would still apply for the small system.
In certain cases however, such as for atoms in high Q cavities or in photonic band gap materials, either the coupling constants or the mode density (or both) are no longer slowly varying functions and standard Markovian master equation methods are no longer valid (see [3] for a recent review).A number of non-Markovian methods have been formulated, see, for example, references in Ref. [4].These include non-Markovian master equations [5,6,7], the time-convolutionless projection operator master equation [8], Heisenberg equations of motion [9,10], stochastic wave-function methods for non-Markovian processes [11,12,13,14,15,16], methods based on the essential states approximation or resolvent operators [3,17,18,19], the pseudo-mode approach [20,21], Fano diagonalization [22] (and [4,23]), and various short-time approximations [24,25].The last four approaches are easiest to apply, providing clear physical insight into the processes involved.
One such method is that of pseudo-mode theory [20,21].This method was developed for the case of a two level atom coupled to a structured electromagnetic field reservoir in the vacuum states and was then restricted to single photon excitations of the reservoir.The treatment started from the time dependent state vector for the atom-field system, written as a linear combination of one photon states with the atom in the ground state and vacuum states with the atom in the excited state.The basis of the method was that the atomic dynamics only depended in this case on the behaviour of a single function, the reservoir structure function, defined as the product of the mode density and the square of the coupling constant.The complex frequencies and residues of the poles of this function in the lower half complex frequency plane enabled so called pseudo-modes to be introduced, one for each of the finite number of poles.The non-Markovian equation for the complex amplitude of the state with the atom excited (and the field in the vacuum state) could then be replaced by Markovian equations involving the finite number of pseudo-mode amplitudes together with the amplitude for the excited atomic state.The pseudo-modes are originally related mathematically to the reservoir structure function, but in some cases their physical origin can be explained.For the case of the atom in a high Q cavity, where the coupling constants vary rapidly near the cavity resonance frequencies whilst the mode density is slowly varying, the pseudo-modes can be interpreted [4] in terms of the cavity quasimodes [26].For the case of an atom in a photonic band gap system, no pseudo-mode theory is yet available, though a treatment in terms of quasimodes [27] can be used to account for the frequency dependence of the coupling constants and mode densities.A treatment of superradiance in a photonic band gap continuum [28] is based on the idea of replacing the photonic band gap system by a pair of degenerate cavity modes coupled to the multi-atom system and with each other, one of the modes being also coupled to a Markovian bath.In terms of the treatment in [4], such a case would produce a Fano-profile reservoir structure function, with the Fano window representing the photonic band gap.The two cavity modes would correspond to two pseudo-modes.The problem for photonic band gap situations is that the mode density is actually a discontinuous function of the frequency, and thus the reservoir structure function would not have a finite number of simple poles, though approximate representations of the reservoir structure function in such a form might be found.
Leaving aside the difficulties associated with the pseudo-mode theory for photonic band gap systems, it would be desirable to see if pseudo-mode theory could be extended to cases where multiple photon excitation of the structured reservoir is involved, as the original treatment [20] only covers single photon excitation.The limitation of current treatments to the single photon excitation case has been noted in Ref. [3], but some work has been carried out on cases of multiphoton excitation of the reservoir.Such a multiphoton situation would apply if the two level atom was replaced by a three level cascade (or ladder) system, with an initial condition of the atom in the uppermost state and no photons present in the electromagnetic field.The two-step decay will generate electromagnetic field states with two photons present.Another case of multiphoton excitation occurs for an excited two level atom coupled to a defect mode containing one photon, the atom also being coupled to a photonic band gap continuum.Beginning with the essential states approximation, a numerical method based on replacing the density of modes by a discretised model has been used in this latter situation [29,30] and in the case of the cascade system [31].Similar numerical methods have also been used to treat stimulated emission in a photonic crystal [32].The cascade system case with one transition coupled near-resonantly to the edge of a photonic gap (and the other coupled to a flat continuum) has also been treated via the resolvent operator method in Ref. [33].Although the treatment is analytic, this feature results from being able to ignore processes in which the two emitted photons are produced in a different sequence-a reasonable approximation if the two transition frequencies are very different.However, a more general analytical method would be desirable, and therefore we aim to see if pseudomode theory can be extended to treat the multiphoton excitation case without having to make assumptions about the order in which the photons are produced.Whether an extension is possible involves first showing that the atomic dynamics only depends on the behavior of reservoir structure functions-in a cascade system we would expect there to be more than one reservoir structure function, since two coupling constants are present.A next step would be to then introduce suitable pseudo-mode amplitudes, based on the poles of the reservoir structure functions and to show that Markovian equations apply to these pseudo-mode amplitudes.
The present paper shows (following the approach of Ref. [19]), that in the case of a three-level atomic system the dynamics is completely controlled by the reservoir structure functions, and gives several methods for determining the atomic and field behaviour.These methods could be applied both to photonic band-gap and high Q cavity cases, since the general equations (14)(15)(16)(17)(18)(19)(20)(21)(22)(23) defining the solution only depend on the reservoir structure functions and not on the specific type of structured reservoir involved.However, as a test, in this paper we only apply the methods to a situation involving a single Lorentzian reservoir structure function.This situation could apply when both cascade transitions have the same frequency and are equally coupled to a single high Q cavity mode.We also are able to interpret the results via an equivalent pseudo-mode model.Situations involving photonic band gaps could be modelled by appropriate reservoir structure functions (see, for example, [20,28]).
Section II of this paper sets out the theory of non-Markovian dynamical behaviour for the three level cascade system where both transitions are coupled to a single structured reservoir.The state amplitudes are determined from solutions to certain integral equations.
Approaches to solving the dynamical equations, including a numerical determination of the excited state probability for a simple case (and its pseudo-mode theory interpretation) is presented in section III.Section IV deals with the simpler case of non-Markovian dynamics for the cascade system with the two transitions coupled to two separate reservoirs, again with numerical results presented for comparison to the single reservoir case.An alternative approach to solving the dynamical equations based on non-orthogonal eigenfunction methods is set out in Appendices A-D.The paper is summarised in section V.

II. DYNAMICAL THEORY FOR A SINGLE RESERVOIR
A. The Hamiltonian The model system we will consider has a three level atom, with states denoted |0 , |1 and |2 , coupled to a reservoir of electromagnetic radiation modes (or heatbath) which is to be at effectively zero temperature.The bath modes will be described by a density ρ λ , frequency ω λ , and raising and lowering operators â † λ and âλ .The Hamiltonian for the system is given (in the rotating wave approximation) by where the atomic transition frequencies are ω 1 for 0 ←→ 1 (i.e. between atomic states |0 and |1 ) and ω 2 for 1 ←→ 2 (i.e. between atomic states |1 and |2 ); see Fig. 1.The coupling of electromagnetic radiation with frequency ω λ to the transition 0 ←→ 1 involves the frequency dependent coupling constant g λ1 .Likewise, g λ2 represents the coupling of the electromagnetic radiation field to the 1 ←→ 2 transition.Real coupling constants will be chosen.It is these frequency dependent coupling constants combined with the mode density which define the reservoir structure.

B. Coupled amplitude equations
The Schrödinger picture state vector for the three level cascade atom coupled to the quantum electromagnetic field may be written as: where c 2 , c 1λ , c 0λλ and c 0λµ are the amplitudes of the various states in the interaction picture.
The radiation states included are: |0 λ in which all the bath modes are in the vacuum state; |1 λ in which the mode with frequency ω λ has a single excitation, with other modes being in the vacuum state; |2 λ in which the mode with frequency ω λ has been raised to the second excitation, with other modes being in the vacuum state; and, |1 λ 1 µ in which the modes with frequencies ω λ and ω µ have a single excitation, with other modes being in the vacuum state.A convention for an ordered listing of the modes λ for the quantum electromagnetic field is assumed, so that double sum over λ, µ does not lead to a specific state |1 λ 1 µ being included twice.
The initial state vector is assumed to be of the form which allows us to explore the non-trivial case of two photons appearing in the reservoir as a result of the interaction with the excited atom.
Substitution of Eq. ( 2) into the time dependent Schrödinger equation gives a closed set of coupled equations for the amplitudes in the situation where the initial state is given by Eq. ( 3).If we then take Laplace transforms of the the coupled amplitude equations we obtain the algebraic equations where the Laplace transforms of the amplitudes are c2 , c1λ , c0λλ and c0λµ and the transform variable is s.These equations for a cascade system coupled to a single structured reservoir, are equivalent to those in Ref. [31].It should be noted that similar equations are given in Ref. [33], for the case where the two transitions are coupled to two separate reservoirs.This case is treated below in section IV.The two separate reservoirs case leads to simpler equations-firstly, because it is assumed that the λ, µ photons are produced in just one sequence (for example , and secondly, because states of the form |0 |2 λ are not present.Whilst these may be a good approximation for the single reservoir case when the transition frequencies are very different, the other pro- ) would need to also be included when the transition frequencies are similar, such as in a quantum harmonic oscillator or a Rydberg atom.

C. Reservoir structure functions and integral equation to determine amplitudes
Next we eliminate b0λµ , b0λλ by substitution of ( 8) and ( 9) in (7).This gives Together with Eq. ( 6) we now have a set of coupled equations for the b2 (s) and b1λ (s).If we eliminate b2 (s) we obtain an equation for the b1λ alone: It is useful to rewrite this by by dividing by s, using the properties of the Kronecker delta function, and substituting from Eqs. ( 10) and (12) to obtain We note that the terms involving the quantity α λµ are absent in similar equations in Ref. [33], resulting in their equations for b1λ being easily solvable.As mentioned earlier, the additional terms we have result from allowing for photons to be emitted into the single reservoir in two different sequences, an effect not present in the two separate reservoir case treated in Ref. [33].In our case, we next convert the sums to integrals, i.e. µ −→ dω µ ρ(ω µ ), where ρ(ω µ ) is the mode density, so that Eq. ( 13) can be written in the form of an integral equation with The quantities f (ω λ ), A(ω λ ), B(ω λ , ω µ ) and C are of course all functions of the Laplace variable s, but for simplicity of notation s is left implicit.The integral equation ( 14) is a Fredholm integral equation of the second kind (see e.g., [34]).Methods of solution for such equations include replacing the frequency spaces by grids of points, thereby converting the integral equation into matrix equations that could be solved numerically for each value of s.
We will discuss one such approach in section III A. In Appendix D we also discuss a more formal method of solving the integral equation, based on the eigenfunctions of the kernel B(ω λ , ω µ )/A(ω λ ) and of its adjoint.
We also find it convenient to write the integral equation in the form where We note that the coupling constants and mode density appear in the integral equation only in the form "ρg 2 ".These forms are called reservoir structure functions, as they contain all the essential features of the reservoir and its coupling to the atomic system.Specifically, the reservoir structure functions that appear in Eqs. ( 16) and ( 17) are As the coupling constants are proportional to dipole matrix elements multiplied by the square root of the angular frequency, it is clear that the factors g µ2 /g µ1 and g λ1 /g λ2 in Eq. ( 17) are independent of the frequencies ω λ and ω µ .Hence a third reservoir structure function involving the factor α λµ is not needed.As the dipole matrix elements would essentially cancel out, the factor α λµ is of order unity.
In principle, we can solve the integral equation and thus determine the b1λ (s+i(ω λ −ω 2 )).
Furthermore, the solutions obtain their particular form from just the reservoir structure functions, rather than the density of states or coupling constants alone.
Next we see that in the new notation Eq. ( 6) becomes b2 and again the step to obtaining b2 (s) just involves using the reservoir structure function R 2 (ω λ ).Note again that f (ω λ ) is a function of the Laplace variable s, so the decay of the initial atomic state |2 described by b2 (s) is non-exponential in general.
Finally, we note Eqs. ( 8) and ( 9) imply that b0λµ and b0λλ are fully determined once b2 , b1λ are known, and α λµ (Eq.( 10)) introduces no new frequency dependence as it is independent of frequency.Thus all the reduced amplitudes b2 , b1λ , b0λµ , and b0λλ can be determined in principle from reservoir structure functions.As we will see next, this is sufficient to determine the reduced density operator describing the atomic system.
Note that the non-Markovian methods could be applied both to photonic band-gap and high Q cavity cases, since the general equations (14-23) defining the solution only depend on the reservoir structure functions and not on the specific type of structured reservoir involved.
Markovian results can be obtained under conditions where the reservoir structure functions ρ(ω λ )g 2 λ1,λ2 are slowly varying functions of ω λ .Certain integrals give a constant term whose imaginary parts are the (formally divergent) frequency shifts and whose real parts are the decay rates for the states |1 and |2 .

D. Atomic density operator
The atomic density operator is defined by and it is not difficult to show that Thus we see that the atomic operator only depends on the reduced amplitudes , and the reservoir structure functions.As the former can be determined, in principle, from the reservoir structure functions, we see that the behaviour of the cascade atom in the structured reservoir is completely determined by the reservoir structure functions (for the initial state given in Eq. ( 3)).
On the basis of this key result, it would follow that any existing system could be replaced by an equivalent system, provided that the reservoir structure functions were the same in both models.This is the basis of the treatment of superradiance in a photonic band gap continuum [28], where the photonic band gap system is replaced by a pair of degenerate cavity modes coupled to the multi-atom system and with each other, one of the modes being also coupled to a Markovian bath.In terms of the treatment in [4], such a case would produce the required Fano-profile reservoir structure function, with the Fano window representing the photonic band gap.The two cavity modes would correspond to two pseudo-modes.
The absence of any coherence terms in the atomic density operator is a consequence of the choice of initial state, Eq. ( 3).The choice of a more general initial state (even with no photons present) of the form would require the introduction of a more general time dependent state vector |Ψ(t) than that given in Eq. ( 2), to include additional states of the form |0 |0 λ , |1 |0 λ and |0 |1 λ .The amplitudes for these additional states are not coupled to those for the other states included in Eq. ( 2).Again, the solutions for these amplitudes just involve reservoir structure functions and are analogous to those already discussed in Ref. [20] for the simpler case of a two level atom coupled to a structured reservoir.However, as indicated above, the atomic density operator would then include coherence terms.

III. SOLUTIONS FOR THE STATE AMPLITUDES
The integral equation ( 14) can be solved in different ways.These include: (a) numerical methods based on converting the integral equation to a matrix equation, (b) expansions using biorthogonal eigenfunctions and (c) expansions such as the Fredholm expansion [34].Only the first of these methods will be used here, but as the second approach using biorthogonal eigenfunctions may be used in later work and has not been used previously in Quantum Optics problems, it is included here in Appendices A -D for completeness.

A. Numerical solution of the integral equation: case of Lorentzian reservoir structure function
As an illustration we consider a greatly simplified example of a three-level system coupled to a reservoir with structure.The simplest possible case is that for the same Lorentzian reservoir structure function associated with both transitions, with all the couplings and transition frequencies equal to each other.That is, we choose a single coupling constant g λ such that which amounts to both the dipole moment matrix elements for the transitions being equal.
The atom will also have two equally spaced transitions which are resonant with the reservoir structure, We refer only to ω 0 in the following.Thus for the single reservoir structure function, we have as in Ref. [20].The parameter Ω represents the strength of the coupling and Γ represents the width of the reservoir structure function.This situation would apply for identical cascade transitions coupled to a single high Q cavity mode.Cascade transitions in a photonic band gap reservoir would be treated via a different choice of the reservoir structure functions.
Using this expression for the reservoir structure function we can determine the functions A(ω λ ), B(ω λ , ω µ ) and C in Eqs.(16)(17)(18) and then the kernel, Eq. ( 21), becomes where Q(ω) is a quadratic polynomial such that For this model we thus have an analytic form for the kernel, but to go further it appears that we need to use a numerical method.We could utilize an eigenfunction method, such as that of Appendix D, but choose a very simple approach to solve Eq. ( 19).The process is simply to represent Eq. ( 19) as a matrix equation where K and I are matrices and f and d are vectors.We then invert (K + I) to solve for f.Thus K(ω λ , ω µ ) is represented at discrete frequency points, in effect a discrete basis of spatial delta-functions, e.g., K ω λ ,ωµ = K(ω λ , ω µ ).Similarly, f (ω λ ) and d(ω λ ) are represented at discrete frequency points.From the definition in Eq. ( 15), we see that if we introduce the function f (ω λ , t) (which we denote as f (t) for short) via then f (t) is the function whose Laplace transform is f (s) ≡ f (ω λ , s).However, in order to obtain the real and imaginary parts of f (t), we will need the separate inverse Laplace transforms f r (s), f i (s) which are the Laplace transforms of the real and imaginary parts of For complex s the latter Laplace transforms cannot be obtained by just writing f (s) as the sum of its real and imaginary parts.However, the Laplace transform f r (s) of the real part f r (t) is real (and similarly the Laplace transform f i (s) of the imaginary part f i (t) is real), if the Laplace transform parameter s is real.Hence, the real and imaginary parts of f (s) are equal to the Laplace transforms of the real and imaginary parts of f (t) for s on the real axis, so Ref (s) = f r (s), Imf (s) = f i (s) for s real.As f (s) is an analytic function of s, the analytic continuation of f r (s) + if i (s) from the real axis will determine f (s) everywhere.
In this example, if we discretise K(ω λ , ω µ ) on an N ×N grid we define the N ×N complex matrices K r and K i from the real and imaginary parts of Eq. ( 30) on the real s-axis, and then Eq. ( 32) becomes for s on the real axis.The formal solution for f r = f r (s) and f i = f i (s) for real s then generates the solution for f (s) everywhere.Because of this (and having first identified K r , K i , d r and d i for real s using Eq. ( 21) and Eq. ( 20)), we can now regard Eq. ( 34) as applying for all s.This approach could not be used if the the real and imaginary parts of K(ω λ , ω µ ) and d(ω λ ) on the real s-axis are not analytic.The matrix inversion step thus involves a matrix with 4N 2 elements compared to, say, O(N 4 ) elements represented by Eqs.(6 -9) in an equivalent discretised form.
Thus we solve for f r and f i in Eq. ( 34), and hence determine the b1λ (s + i(ω λ − ω 2 )) of Eq. ( 15).We then find the b2 (s) from the scalar product form Eq. ( 23) obtained from Eq. ( 6) so that where r ≡ {ρ λ g 2 λ2 }.Finally, b 2 (t) is determined by a numerical inverse Laplace transform.Figure 2 shows some results for this numerical matrix approach with the kernel given in Eq. ( 30), which was derived from the reservoir structure function in Eq. ( 29).The three curves show the upper state population for three different sizes of matrix which were used to discretize the integral equation.Each case used the same parameters Ω=1 and Γ=1, where there is a distinct non-Markovian evolution that could not be treated perturbatively because of the strong coupling to the reservoir structure.The solid curve in Fig. 2 shows a good result that was obtained with a matrix of size 150×150 for this problem.Reducing the matrix size to 100×100 (dashed) results in only a slight degradation of the result.However, further reduction of the matrix size affects the numerical result quite badly.
The effect of changing the coupling strength Ω is shown in Fig. 3.The probability of finding the atomic system in the highest atomic state is shown.For strong coupling (Fig. 3(a)) we see damped oscillations that are a typical manifestation of non-Markovian processes.As the coupling is reduced, (Fig. 3(b)), the oscillations weaken and then further reductions in the coupling strength Ω, (Fig. 3(c)) result in no oscillations and decay that is closer to exponential and on a longer time-scale than the strong coupling cases.Fig. 3(c) still shows some visible initial quadratic behaviour because of the relatively high value of Ω/Γ.

B. Equivalent pseudo-mode model
The reservoir structure function given in Eq. ( 29) is extremely simple and as result we can reproduce the results of Fig. 3, i which is given in the interaction picture with the atom-'field' coupling term In this master equation we have introduced a single oscillator, or pseudo-mode [20], which is represented by the harmonic oscillator operators â and â † .In this approach (see [20]) pseudomodes are introduced as assumed bosonic entities, rather than via constructing pseudo-mode amplitudes.A cascade atom resonantly coupled to a damped high-Q cavity mode, which is also coupled to a Markovian bath of vacuum modes, is an example of a physical system which has the same master equation as (36).Such a model was considered in our earlier work [4], where we showed that multiple excitations of a structured reservoir could be treated for reservoir structure functions such as Eq. ( 29).To utilize the present pseudo-mode model we solve the master equation ( 36) with the initial condition of an empty pseudo-mode and the atom in the state |2 .On tracing out the pseudo-mode, to obtain atomic properties alone, we can reproduce the results of the matrix method used with the kernel of Eq. ( 30).It should be emphasised that it does not appear to be easy to find such a simple master equation for more complex reservoir structures such as photonic band gap models with branch cuts in the reservoir structure function.In such a case the approach outlined in this paper (which only depends on the reservoir structure functions) may be useful instead.For the present Lorentzian model, the agreement between the matrix method given earlier in this section and the master equation ( 36) is excellent.

IV. DYNAMICAL THEORY FOR TWO SEPARATE RESERVOIRS
In this paper we have commented in several places that there are differences in our single reservoir treatment from the simpler case of separate reservoirs coupled to the two transitions in our model system.In our model, the two photons may be emitted in either order, whereas with the distinguishable photons in the two reservoir model, only one order of emission is involved.So, with the formalism now complete, it is instructive to look at the explicit differences between our model and the simpler two separate reservoirs model of the kind considered in Ref. [33].In this case the Hamiltonian in Eq. ( 1) is replaced by where the bath operators â † λ and âλ for the first bath now couple only to the 1 ←→ 2 transition, and the new bath operators b † µ and bµ for the second bath couple only to the 0 ←→ 1 transition.For the initial state vector, Eq. (3), the state vector analogous to Eq.
(2) no longer contains a term involving c 0λλ , and there is no restriction over the double sum λ, µ, since the two types of bath modes are now distinct.We can write involving product states of: the atom, and one or zero excitation states of the two baths.
The equations for the Laplace transforms of the reduced amplitudes, Eqs.(6)(7)(8)(9) are then replaced by We note that at this point the differences are that, as well as the absence of the b0λλ terms, there are no terms involving α λµ (as in Eqs. ( 7) and ( 9)) and there are no restrictions over the sum over µ (as in Eq. ( 7)) These equations are equivalent to those in Ref. [33].
As in the case of both transitions coupled to one single reservoir, the dynamical behaviour only depends on reservoir structure functions, and following the same approach as in section II D it is easy to see that the atomic density operator is also determined from these functions.
If we now follow the elimination procedure of section II C we find the same equations (14-18) for f , A, B and C except that the consequence of no α λµ term being present in Eq. ( 42) is that the quantity B becomes Crucially B no longer depends on ω λ as previously.Expressions for f , A and C are otherwise unchanged.
The integral equation then simplifies to the easily solvable form for which the solution is In this case the equivalent kernel is separable: We can apply our results to the situation analogous to that treated in section III A, where both reservoirs, though now separate, have identical coupling constants and reservoir structure functions, and the two atomic transitions are equally spaced and resonant with the reservoir structures.We utilize Eqs.(27)(28)(29) and, for this simple model, the kernel can be easily obtained as: This result may be compared to the previous expression in Eq. ( 30) for the case of a single reservoir.
The integral dω µ K(ω µ , ω µ ) can be performed by using a contour in the lower-half plane, and we obtain We may now find from Eq. ( 16) that so the solution for f (ω λ ) can be obtained from Eq. (45).We find that: A numerical inversion of f (ω λ ) can be performed to obtain b 2 (t) using the same approach as in section III A. However, The reservoir structure, Eq. ( 29), is sufficiently simple that a solution for b 2 (t) can be found from Eq. (49).We first need to perform the integral in Eq. ( 23) which is facilitated by the fact that Eq. (49) has no poles in the lower-half complex plane (for Re(s) > 0), while the factor ρ(ω λ )g 2 λ2 in Eq. ( 23) has only a single pole in the lower-half complex plane if we use the example given in Eq. (29).Then if we perform the integral of Eq. ( 23) around the single lower-half plane pole we find that V. CONCLUSION The dynamical behaviour of a three level atom in a cascade configuration in which both transitions are coupled to a single structured reservoir of electromagnetic field modes, and initially in the upper state, has been analysed via Laplace transform methods.This situation involves a two photon excitation of the reservoir, and our equations take into account the two possible sequences in which these two photons are emitted.We have shown that the atomic density operator is determined from the solutions of integral equations, in which the properties of the structured reservoir only appear via so-called reservoir structure functions, all essentially given by the product of the mode density times the square of coupling constants.
In the cascade system two distinct reservoir structure functions are involved since there are two transitions.The dependence of the dynamics solely on reservoir structure functions is the necessary condition for treating structured reservoir problems via pseudo-mode theory, so our results suggest that it may be possible to extend pseudo-mode theory to problems involving more than a single photon excitation of the reservoir.
This result also shows that any existing system could be replaced by an equivalent system, provided that the reservoir structure functions were the same in both models.This is the basis of the treatment of superradiance in a photonic band gap continuum [28] and the general treatment of multiphoton excitation in terms of quasimodes given in our earlier work [4].
In addition, a similar treatment of the dynamical behaviour of a three level atom in a cascade configuration coupled to two separate structured reservoirs of electromagnetic field modes, and initially in the upper state, has been carried out.One reservoir is coupled to the upper transition, the other to the lower transition.This situation again involves a two photon excitation of the reservoir, but now only one possible photon emission sequence is involved.
In this situation, the equations are simpler, and the integral equation for the amplitudes can be solved analytically.Again, the dynamical features only depend on reservoir structure functions.
A numerical method of solving the integral equations based on discretising the frequency space has also been obtained, and which can be applied to various structured reservoir situations -such as for high Q cavities and photonic band gap systems.Here we have applied this method in a numerical test for a high Q cavity situation, where the same Lorentzian reservoir structure function applies for both transitions, showing the non-Markovian decay of the excited state.Results for both the single structured reservoir case and the two separate reservoirs case have been obtained, showing the different behaviour in the two cases.This difference is to be expected, as the two photons emitted should have similar frequencies, and only in the two separate reservoirs cases should it be possible to distinguish which order the photons were emitted.In this latter case we were able to solve the model problem analytically.Finally, a formal solution of the integral equations based on the biorthogonal left and right eigenfunctions of the non-Hermitean kernel has been presented for completeness in the Appendices.
Our treatment of the cascade system coupled to a structured reservoir may be compared to those of Ref. [33] in the two separate reservoirs case and to Ref. [31] in the single reservoir case.Both these papers also demonstrate non-Markovian decay of the excited state.Our fundamental amplitude equations in sections II B and IV agree with those of these authors.
The work in Ref. [31] differs from our treatment, being based on replacing the structured reservoir with discrete modes and then using numerical methods.The work in Ref. [33] is analytic.However, a direct comparison of the numerical results is not yet possible with either Ref. [31] or Ref. [33], since both applied their theory to a photonic band gap system whereas our present application is for the equally important situation of a high Q cavity.Further applications of our theory involving good analytic approximations to the reservoir structure functions for photonic band gap systems will, however, enable more detailed comparisons to be made.
After multiplying the first equation by θ * m (ω λ ), the second by φ n (ω λ ) and then integrating over ω λ we find that Using the biorthogonality result for the eigenvalue ξ n we see that so that provided ξ n = −1 which gives the solutions for the expansion coefficients for f (ω λ ) in terms of known quantities.
The quantities f n , φ n (ω λ ), K(ω λ , ω µ ) and ξ n are of course all functions of the Laplace variable s, but for simplicity of notation s is left implicit.