Quantum electrodynamics (QED) experiments in the optical region started in the 1990s with the aim of achieving strong coupling between atoms and the cavity field. Experiments involved atomic beam transits, through the mode of a high finesse cavity (Miller et al., 2005; Thompson, Rempe, and Kimble, 1992). The flux of atoms was small enough to make the average intercavity number approximately one. Since then, developments have brought about interesting cavity structures, which, in turn present new possibilities. Great potential to the field of quantum information processing is offered by cavity QED in semiconductors in combination with quantum dots or semiconductor microcavity structures.
Whispering Gallery Modes microcavities on a Si-chip: SEM micrographs of a silica microdisk on a Si-chip with a radius of 40 µm, a microtoroid with a radius of 25 µm, and a silica microsphere with a radius of 11.5 µm.
Image source/credit. .
An optical cavity, also referred to as a microresonator facilitates the formation of a standing wave cavity that is resonant for electromagnetic radiation in the optical wavelength via a construction of mirrors in an assembly. Multiple reflection events take place when light is confined to within the cavity, which brings about the emergence of standing wave modes at certain resonant frequencies. Longitudinal modes vary in frequency, whilr transverse modes vary both in frequency and intensity characteristics throughout the cross section of the beam. Resonator variations are differentiated between by the focal lengths of and separations of their constituent mirrors. Optical cavities are designed such that their Q-factor is maximised, which makes possible the repeated reflection of a beam with little attenuation.
The resonant frequency spectrum of an optical microcavity is dependent upon size (Vahala, 2003), in the same sense as the size dependence exhibited by its acoustic counterpart, the tuning fork. The microscale volumes of these cavities gives way to resonant frequency spectrums that are more scarcely separated than in their macroscale counterparts. An ideal microcavity is characterised by an ability to indefinitely contain light, without loss, along with the scope to facilitate the emergence of resonant frequency modes at precise values. The quantity that describes the deviation from this ideal condition is the cavity Q-factor.
The irreversible process whereby a photon is emitted into the continuum of radiation modes, as a result of the transition of an electron within an atom, from an excited state to a ground state, is an example of weak coupling. If this atom is inserted into a microcavity, and the strong-coupling conditions are met, then a coherent reaction with the cavity mode may occur, over a non trivial time interval. We now shift our attention to the strong-coupling conditions that act as the barrier to this effect.
Consider, for a moment, the situation characterised by the existence of an atom, initially in the excited state of a dipole transition, which a microcavity of volume V , that is free from loss. In this situation, it is the case that the ground state of a single mode is resonant with the transition. The phenomenon whereby a quantum of energy shifts between the atom and mode at the Rabi frequency arises due to the fact that the atom and vacuum field yield to coupling. Large Rabi frequencies are generated within small cavity volumes. This is because the strength of the interaction between atom and cavity mode is linear in the field, so it is the case that smaller cavity volumes concentrate the vacuum field of the mode. Now, if the system is isolated then this fundamental dynamic of the atom-field system is reversible. In real systems, the cavity photon lifetime is finite which gives rise to the leaking of energy irreversibly into the continuum, and the atom transition will likewise couple to the continuum radiation modes. After this coupling is established, it will experience a spontaneous decay of its population along with polarisation dephasing. This mechanism limits the formation of Rabi oscillations.
System that are in the strong-coupling regime are characterised by the existence of the aforementioned Rabi dynamic, which is of potential brevity, however, and subject to the eventuality of certain dissipation. The conditions for strong-coupling require that the atom-field coupling strength, g is greater than any low lying dissipative rate, and greater than 1/T, where T is the interaction time (Kimble, 1998). It is the case that the atom-field coupling strength is equivalent to half the Rabi frequency, and that two spectral transition peaks are uncovered under these requirements as a response to weak optical probing in close proximity to the resonant frequency of the microcavity.
The saturation photon number, n_0 , and critical atomnumber, N_0 , quantify the degree to which the atom and cavity mode are strongly-coupled, given modal and atomic dissipation. The quantities are dependent upon parameters that describe the reversible dynamic along with the various decay rates (Kimble, 1998),
where γ_|| ,γ_⊥ are the atomic dissipation rates and τ is the cavity lifetime. The atomic dissipation rates are characterised as population relaxation and atomic dephasing, respectively. Unity n_0 and N_0 were achieved as of 1992, at wavelengths in the optical regime (Thompson, Rempe, and Kimble, 1992), however, recently demonstrated strongly-coupled systems have generated quantities of n_0 and N_0 far below this value (Hood et al., 2000; Buck and Kimble, 2003), giving rise to a Rabi dynamic for which the lifetime survives many cycles.
The transmission spectrum of the cavity is revealed through use of a weak optical probe to be split by the presence of the atom into two two distinct peaks. Now, these peaks correspond to eigen frequencies of the entangled atom-cavity states (Thompson, Rempe, and Kimble, 1992). Transmission is blocked in the case where the probe frequency is aintained at the original resonant frequency of the cavity and a single atom is inserted (Mabuchi et al., 1996). Great research effort has been devoted to the study of ultra-high Q, small-volume resonant cavities (Kimble, 1998; Rempe et al., 1992; Lefevre-Seguin and Haroche, 1997; Gorodetsky, Savchenkov, and Ilchenko, 1996; Vernooy et al., 1998b; Vuckovic et al., 2001) in order to increase strong coupling effects (quantified by the reduction of n_0 and N_0 ). The earliest demonstrations of single-atom vacuum Rabi splitting at optical frequencies yielded n_0 and N_0 values near unity, however, recent advances have facilitated the lowering of these values to (Hood et al., 2000; Buck and Kimble, 2003) n_0 = 2.82 × 10 −4 and N_0 = 6.1 × 10 −3. Microcavities in these experiments are characterised by Fabry—Perot style resonators, with reflectance mirror technology (Rempe et al., 1992). A cavity finesse of 1.9 × 10 6 has been obtained using these mirrors (Rempe et al., 1992).
While it is the case that great effort has been dedicated to the study of ultrahigh-finesse Fabry—Perot microcavities, the same is true of the Whispering gallery modes of silica and quartz microspheres (Lefevre-Seguin and Haroche, 1997; Gorodetsky, Savchenkov, and Ilchenko, 1996; Vernooy et al., 1998b; Knight et al., 1995; Vernooy et al., 1998a). Now, these Whispering gallery resonators are usually characterised as a spherical dielectric structure, which operates on the basis of continued total internal reflection. The robust ultrahigh-Q microresonator that manifests via silica microspheres were first studied by Braginsky, Gorodetsky, and Ilchenko (1989). (Knight et al., 1995) reported that the spheres comprise an atomic-like mode spectrum in which high principal angular index and low radial number modes execute orbits near the surface of the sphere. In order to achieve a maximal Q, it is the case that an excellent surface finish is necessary, with spheres generated via surface tension giving way to a close to atomically smooth surface. Bulk optical loss from silica is extremely low, and exceptionally high Q factors 8×10 9 (and finesse of 2.3×10 6 (Vernooy et al., 1998b)) have been obtained. As reported by Vernooy et al. (1998b), it is the case that for these measurements, dependence of Q on sphere diameter is consistent with the limitation of Q via losses due to surface roughness.
Photonic crystal based microcavities have emerged with the potential to comprise volumes on an extremely small scale (Painter et al., 1999); indeed, Vuckovic et al. (2001) showed that donor-mode cavity constructions could be modelled with a neutral atoms suspended within a hole. It should be noted, however, that theoretically optimal Q-factor olumes have not yet been breached, with the implication that the potential strong-coupling cavity phenomena has not yet been experimentally realised. A fibre optic cable is considered a likely medium along which quantum information will be transported, however for many applications in quantum information studies, the notorious difficulty is parasitic loss in coupling to and from microresonators. To this end, fibre optic tapers were reported by Cai, Painter, and Vahala (2000), providing ultra-low loss, direct coupling to ultrahigh-Q spheres and have been proposed as a means to couple quantum states to or from a resonator onto a fibre (Cai, Painter, and Vahala, 2000; Spillane, Kippenberg, and Vahala, 2002).
It is the case that a two-level system will, at some point, undergo a spontaneous decay event via interaction with a vacuum continuum. whereby at the transition frequency, the rate is proportional to the spectral density of modes per volume. The density of modes within a given cavity is subject to variation, with the potential for large fluctuations in amplitude to occur. From the viewpoint of the cavity modes, the maximal mode density occurs atquasi-mode resonant frequencies, with potential to generate frequencies that are far greater than those cosponsoring to free space density. Purcell (1946) reached this conclusion by noting that a single (quasi) mode occupies a spectral bandwidth ν/Q within a cavity volume V. Normalising a resulting cavity-enhanced mode density per unit volume to the mode density of free space yields the ‘Purcell’ spontaneous emission enhancement factor (Purcell, 1946; Haroche and Kleppner, 1989)
where refractive index, n is a recent addition to the expression; the role of which is to account for circumstances in which emission takes place within a dielectric, as outlined by Gerard and Gayral (1999). It is the case that for an atom whose transition falls within the mode linewidth, an enhancement to its spontaneous decay rate will be present, as given by the Purcell factor. As reported by Yamamoto and Slusher (1993), the spontaneous emission is directed to a quasi-mode of the resonator since it is only in this situation where enhancement comes about from coupling to the continuum modes, since it is only these quasi-modes that yield to coupling.
As established by Chang and Campillo (1996); Haroche and Kleppner (1989), it is the case that in spectral locations that are intermediate to modal resonance frequencies, the modal density may be susceptible to a decrease in density, far below that exhibited by free space. The design of microcavities with the observation of the Purcell effect as an object must take into account the relevant atomic transition characteristics. Enhancements that are driven by the manipulation of Q, only, are limited by the spectral width of the transition, so it is the case that a small volume must be implemented. In an ideal situation, it is entirely possible for narrow atomic transitions to be enhanced by the Purcell effect as high Q-values become more attainable. As reported by Gerard and Gayral (2001), quantum dots are emerging as a significant vessel through which to study the Purcell effect, due to their narrow transition widths.
Weak coupling is prevalent when the fundamental Rabi dynamic becomes overshadowed by dissipative effects. The control of spontaneous emission in this regime, via the Purcell effect has emerged as a key application of microcavities (Yamamoto, Tassone, and Cao, 2000). Since it is a given that all cavities are subject to some degree of loss, cavity modes are more precisely identified as quasi-modes, with the spectral linewidth function of a discrete mode being proportional to a continuum density of modes function. This modal duality has manifested in applications of the Purcell effect which utilise local enhancement of the continuum density of modes function in order to influence spontaneous emission while at the same time exploiting the quasi mode of the resonator in the form of a target for the emission process. This leads to an increase in atomic decay rates, and the establishment of directional dependence.
The critical breakthrough in the study of the Purcell effect arose when highly luminescent InAs quantum dots emerged via the study of III-V semiconducting materials. These structures have the potential to stand as a local probe of the field within in a III-V microcavity, while also capturing electrons and holes into a regime in which they are confined. As reported by Gerard et al. (1996), it is the case that this mechanism results in the decrease of electron/hole susceptibility to semiconductor surface effects that come about when the cavity dimension decreases. Quantum dots have also been identified as a promising and well suited candidate for further study of the Purcell effect since it is the case that their spectral lineshape character is relatively narrow, and fits well within a high Q cavity mode (Geerard et al., 1998; Gerard and Gayral, 2001).
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