It is well known that emission processes reflect certain inherent properties of atoms, but it has also been demonstrated, in both theory [1,2] and experimentation [3-8], that these same processes are also sensitive to incidental boundary conditions. One example is how they can be modified if contained inside a cavity of dimensions comparable to the emitted light wavelength. The modification can involve emission enhancement or inhibition and is a result of an alteration of field mode structure inside the cavity compared to free space, which can be explained in terms of an interaction between atom and cavity modes [9,10].
The density of states (DOS) can be interpreted as a probability density of exciting a single eigen-state of the electromagnetic (e.m.) field. When the plot of DOS vs. frequency over atomic transition spectral range is found to be smooth, then the rate of emission can be defined by Fermi’s golden rule. However, emission dynamics can be drastically modified by photon localization effects  and sudden changes in DOS. Such modifications can be interpreted as long-term memory effects and examples of non-Markovian atom-reservoir interactions.
Marked transformations can be induced in the DOS using photonic crystals. These dielectric materials exhibit very noticeable periodic modulations in their refractive indices which result in the formation of inhibited [12,13] frequency bands or photonic band gaps. The DOS inside a photonic band gap (PBG) is automatically zero. It is proposed in literature [14-17] that these conditions might result in classical light localization, inhibition of single-photon emissions, fractionalized single-atom inversions, photon-atom bound states, and anomalously strong vacuum Rabi splitting.
There have been rigorous investigations of spontaneous decay of two-level atoms coupled with narrow cavity resonance according to Hermitian "universal" modes as against the dissipative quasi-modes of the cavity by reference . These have concentrated in particular on cases in which the atomic line-width
The emission processes investigated in this chapter regard a 1D unenclosed cavity, analysed according to the theory of reference . There is specific discussion of the stimulated release of an atom under strong coupling regime inside a 1D-PBG cavity generated by two colliding laser beams. Atom-e.m. field coupling is modelled by quantum electro-dynamics, as per reference , with the atom considered as a two tier system, and the e.m. field as a superposition of normal modes. The coupling is in dipole approximation, Wigner-Weisskopf and rotating wave approximations are applied for the motion equations. An unenclosed cavity is conceived in the Quasi-Normal Mode (QNM), as in reference , and so the local density of states (LDOS) is defined as the local probability density of exciting a single cavity QNM. As a result the local DOS is effectively dependent on the phase difference between the two laser beams.
1.1. Quasi-Normal Modes (QNMs)
Describing a field inside an unenclosed cavity presents a problem that various authors have confronted, with references [19-22] proposing a QNM-based description of an electromagnetic field inside an open, one-sided homogeneous cavity. Because of the leakage, the QNMs exhibit complex eigen-frequencies as a consequence of leakage from the unenclosed cavity, with an orthogonal basis being assumed only inside the cavity and following a non-canonical metric.
It is only possible to quantize a leaky cavity , considered a dissipative system, if the container is viewed as part of the total universe, within which energy is conserved . A fundamental step towards the application of QNMs to the study of quantum electro-dynamics phenomena in cavities was already achieved in reference .
The second QNM-theory-based quantization scheme was extended to 1D-PBGs in reference . References [29,30] applied the second QNM quantization to 1D-PBGs, excited by two pumps acting in opposite directions. The commutation relations observed for QNMs are not canonical, while also depending on the phase-difference between the two pumps and the unenclosed cavity geometry. Reference  applies QNM theory in an investigation of stimulated emission from an atom embedded inside a 1D-PBG under weak coupling regime, with two counter-propagating laser beams used to pump the system. The most significant result in reference  is the observation that the position of the dipole inside the cavity controls decay-time. This means that the phase-difference of the two laser beams can be used to control decay-time, which could be applicable on an atomic scale for a phase-sensitive, single-atom optical memory device.
The present chapter discusses stimulated emission of an atom enclosed inside a 1D-PBG cavity under strong coupling regime, generated by two counter-propagating laser beams [32,33]. The principal observation is a demonstration that high LDOS values can be used as a definition for a strong coupling regime. Further observations agree with literature in stating that the atomic emission probability decays with an oscillating pattern, and the atomic emission spectrum split into two peaks, known as Rabi splitting. What makes the observations of this chapter unique compared to literature is that by varying the laser beam phase difference it is possible to effectively control both the atomic emission probability oscillations, and the characteristic Rabi splitting of the emission spectrum. Some criteria are proposed for the design of active cavities, comprising a 1D-PBG together with atom, as active delay line, when it is possible to achieve high transmission in a narrow pass band for a delayed pulse by applying suitably differing laser beams phases.
In section 2, quantum electro-dynamic equations are used to model the coupling of an atom to an e.m. field, as an analogy of the theory of an atom in free space. In section 3, the atom is also contained within an unenclosed cavity, and the local probability density of a single QNM being excited is considered a definition of LDOS probability density. The atomic emission processes are modelled in section 4 with the LDOS of the stimulated emission depending on the phase difference of two counter-propagating laser beams. Section 5 discusses the probability of atomic emission under strong coupling regime. In section 6, the atomic emission spectrum is defined on the basis of its poles. Some criteria for the design of an active delay line are proposed in section 7, while section 8 is dedicated to a final discussion and concluding remarks.
2. Coupling of an atom to an electromagnetic (e.m.) field
The present case study examines an atom coupled to an e.m. field at a point inside a one-dimensional (1D) universe with refractive index
The atom is quantized into two levels, with an oscillating resonance of
At start time (
when denotes the upper state of the atom, without any photons in all the e.m. modes; and denotes the lower state of the atom, with one photon in the
2.1. Quantum electro-dynamic equations
If an initial condition is assumed, then the atom-field system state at time instant
introducing the probability amplitudes
the second of these can be formally integrated producing a time evolution equation for the probability amplitude
It is possible to establish a correspondence between the discrete modes and continuous modes of, respectively, a 1D cavity of length
the kernel function
As emerges from Equation (8), there is a marked dependence of the kernel function on the LDOS through
2.2. Atom in free space
The probability of atomic emission decays exponentially in free space,
Free space is an infinitely large photon reservoir (a flat spectrum), and so it should respond instantaneous, with any memory effects associated to emission dynamics being infinitesimally short relative to any time intervals of interest. According to the so-called Markovian  interactions, an excited state population gradually decays to ground level in free space, regardless of any driving field strength. This result is generally valid for almost any smoothly varying broadband DOS.
The following parameter is now introduced as a step for the analysis of the next section,
interpretable as the degree of atom-field coupling, and with the possibility of expressing Equation (8) as:
3. Atom inside an unenclosed cavity
3.1. Density Of States (DOS) as the probability density to excite a singleQNM
By filtering two counter-propagating pumps at an atomic resonance
which is related directly to the (integral) probability density
For the investigation of spontaneous emissions, the two pumps are modelled as fluctuations of vacuum, based on the e.m. field ground state (for examples, see references [26,28-30]). The (integral) probability density that the
It is possible to deduce a normalization constant
From Equation (16) it was deduced that the probability density due to fluctuations in vacuum for the
Stimulated emissions are considered by modelling the two pumps as a pair of laser beams in a coherent state (see references [26,28-30] for examples). When the symmetry property is achieved by the refractive index
Equation (18) shows that the phase-difference Δ
4. Atomic emission processes
With an atom at point
4.1. Spontaneous emission: DOS due to vacuum fluctuations
If the unenclosed cavity is only affected by fluctuations of vacuum filtered at the atomic resonance
The resulting signal
Applying Equations (20) and (21) for the kernel function of the spontaneous emission process gives:
and deriving Equation (22) again, sampled at point
which after some algebraic transformations produces a second order differential equation in time for spontaneous emission probability:
4.2. Stimulated emission: DOS dependant on the phase difference of a pair of counter-propagating laser-beams
If the unenclosed cavity is pumped coherently by two counter-propagating laser beams with a phase difference Δ
The quantity (
5. Atomic emission probability
The initial conditions being the same as Equation (25), the algebraic equation associated with the Cauchy problem (27) can be recast as:
This is solved with two roots,
which permit expression of the particular integral of the differential Equation (27) as:
The atom and the
5.1. Strong coupling regime
Spontaneous emission is examined in order to assess the atom -
there is a strong coupling regime if :
Equation (32) shows that a strong coupling regime is present when the probability density (16) inside the unenclosed cavity, sampled at atomic resonance in units of DOS (9) with reference to free space, is in excess of the inverse of the atomic parameter
and the behaviour of the particular integral (30) is oscillatory:
In reality [see Equation (22)] . It is possible to interpret the oscillatory behaviour as emission re-absorption of a single photon and so the net decay rate can thus be determined from the rate of photon leakage, which is
In the case of stimulated emissions, the coupling between atom and the
when is linked to the phase difference ∆
is satisfied if the atom is coupled to an odd QNM, i.e.
6. Atomic emission spectrum
An atom located at point
Applying the Laplace transform for probability coefficient
Equation (38) can thus be re-formulated as:
If decay has occurred (
If sums over discrete quantities are converted to integrals over continuous frequencies, using Equation (6), then the Dirac delta properties can be used to reduce the emission spectrum (42) to:
Given that most optical experiments apply a narrow band source , it is possible to extend the frequency range from
which derives directly from the interpretation of emission spectrum probability (43). Assuming 0
when is the normalization constant that satisfies Equation (45).
The atomic emission processes exhibit a characteristic kernel function
The emission spectrum
6.1. Poles of the emission spectrum
The two poles that solve Equation (28),
Considering stimulated emission processes, the paired counter-propagating laser beams are set to a phase difference ∆
If the operative condition is close to that defined by Equation (36), such that
when is the normalization constant that satisfies condition (45). The two poles,
7. Criteria for designing an active delay line
In references [23,24] and subsequent papers, the QNM theory was applied to a photonic crystal (PC) as a symmetric Quarter-Wave (QW) 1D-PBG cavity. The present study considers a symmetric QW 1D-PBG cavity with parameters
The active cavity consists of the 1D-PBG cavity containing one atom, and it is characterized by a
It is possible to define the active cavity's “density of coupling” (DOC)
when . If the active cavity is to be designed as an ideal delay line, then the pulsed input needs to be retarded and highly amplified, but free of any distortion. For a narrow pass band the global transmission (52) needs to be very high, with a quasi-constant acceleration of coupling (53).
As described above, an atom embedded in the centre of a symmetric QW 1D-PBG cavity with
If spontaneous emission occurs, assuming perfect tuning so that Δ
The two spontaneous emission spectrum poles, shifted by the atomic resonance
An example of stimulated emission is now considered, with the atom inside the symmetric QW 1D-PBG cavity being excited by a pair of counter-propagating laser beams. The phase difference ∆
Using stimulated emission to obtain an ideal delay line, requires that the paired laser beams have higher quadrature, so ∆
The two stimulated emission spectrum poles are shifted by the resonance
Finally, stimulated emission is considered in the presence of a degree of detuning, when the atom inside the symmetric QW 1D-PBG cavity remains coupled to the (
when detuning is maximum:
Detuned in this way, the two stimulated emission spectrum poles are shifted by the resonance
The result is the design of a close to ideal delay line, with an input pulse being retarded, amplified and only slightly distorted. The plots of Figures 5.c and 5.d show that compared to stimulated emission, the detuned example has an even narrower pass band, at
If a pair of counter-propagating laser beams are tuned to the resonance
8. Final discussion and concluding remarks
This chapter discussed atomic stimulated emission processes, under strong coupling, inside a one dimensional (1D) Photonic Band Gap (PBG) cavity, which is pumped by a pair of counter-propagating laser beams [32,33]. The atom-field interaction was modelled by quantum electro-dynamics, with the atom considered as a two level system, the electromagnetic (e.m.) field as superposition of its normal modes, and applying the dipole approximation, the Wigner-Weisskopf equations of motion, and the rotating wave approximations. The unenclosed cavity example under investigation was approached applying the Quasi-Normal Mode (QNM), while the local density of states (LDOS) was interpreted as the local probability density of exciting a single QNM within the cavity. In this approach, the LDOS depends on the phase difference of the paired laser beams, and the most significant result is that the strong coupling regime can occur with high LDOS values. The investigation also confirms the well known phenomenon [39-41] that atomic emission probability decays with oscillation, causing the atomic emission spectrum to split into two peaks (Rabi splitting). The novelty that emerged in this chapter is that it appears to be possible to coherently control both the atomic emission probability oscillations and the Rabi splitting of the emission spectrum using the phase difference of the paired laser beams. Finally, some criteria were proposed for the design of an active cavity comprising a 1D-PBG cavity plus atom, to serve as an active delay line. It is seen that suitable phase differences between the paired laser beams make it possible to achieve high delayed pulse transmission in a narrow pass band.
The issue of e.m. field interaction with atoms when the e.m. modes are conditioned by the environment (inside a cavity, or proximal to walls) can be approached in several ways. For example, the dynamics of the e.m. field can first be established inside and outside the cavity (or proximal/distant from walls), and then atomic coupling with the normal modes (NMs) of the combined system [42-46] can be considered. An alternative approach applies the discrete (dissipative) QNMs of the unenclosed cavity in place of the continuous (Hermitian) NMs. When applying the QNMs, the internal field cavity is coupled to the external e.m. fields (beyond the two cavity limits) by boundary conditions [47-50].
A third approach is proposed in the present chapter, combining both those described above with the aim of merging their analytic potentials. Canonical quantum electro-dynamics is applied for the definition of an e.m field as a superposition of NMs, while an unenclosed cavity is defined adopting a QNM approach, when LDOS is interpreted as the local probability density of exciting a single QNM of the cavity. The DOS is linked to the cavity boundary conditions. The e.m. field satisfies incoming and outgoing wave conditions on the cavity surfaces, and so the DOS depends on the externally pumped photon reservoir. When the cavity is excited by paired counter-propagating pumps, the DOS expresses the probability distribution of exciting a single QNM of the cavity.
In the case of spontaneous emission, the paired pumps are modelled as vacuum fluctuations from the ground state of the e.m. field, while the DOS is construed simply as a feature of cavity geometry. Instead, in the case of stimulated emission, the paired pumps are modelled as two laser beams in a coherent state, so that the DOS depends on the cavity geometry and can be controlled by the phase difference of the paired laser beams. These results clearly highlight how the DOS of an unenclosed cavity is determined by the cavity excitation state.
The author, Dr. Alessandro Settimi, is extremely grateful to Dr. Sergio Severini for his outstanding support and friendship, to Prof. Concita Sibilia and Prof. Mario Bertolotti for their interesting pointers to literature regarding photonic crystals, and to Prof. Anna Napoli and Prof. Antonino Messina for their valuable discussions regarding stimulated emission.
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