102
P. Vasa
An emitter in an excited state relaxing via spontaneously emitting a photon in
free space is an irreversible process that is the most elementary light-matter interaction [1, 2, 4]. Its correct modelling is possible only within quantum mechanical
framework by considering coherent and incoherent additions. The model presented
by Victor Weisskopf and Eugene Wigner to explain spontaneous emission is based
on the interaction between a single emitter and a continuum of vacuum field fluctuations of the environment. The interaction between an quantum emitter and a single
electromagnetic field mode is a coherent process, in which the phase relationship is
preserved [2, 4, 5]. Hence it is a reversible process and the energy can flow back and
forth between the emitter and the field mode without any losses. However, when an
emitter (or a quantum mechanical 2-level system) interacts simultaneously with a
continuum of quantum-mechanical harmonic oscillator modes, it becomes an incoherent and irreversible process. The phase relationship here is lost due to the time
averaging over the continuum of vacuum states characterized by different frequencies
and phases without preserving any correlation [1, 2, 4, 5]. The energy once released
to the environment consisting of continuum of modes via emission of photon cannot
return to the emitter in a coherent fashion. In other words, the energy is lost to the
surroundings.
Optical resonators can modify the frequency distribution of vacuum field fluctuations by selectively enhancing those matching the resonance frequency of the
resonator. Thus, a resonator prevents loss of coherence by avoiding simultaneous
interaction with the continuum of modes. Therefore, if a two-level system or an emitter characterized by emission frequency ω emi is now placed within a lossless resonator
having the identical resonance frequency, ω res = ω emi , the rate of spontaneous emission can be drastically altered. It can be suppressed, enhanced and most interestingly
can even be made a reversible process [4–9]. Thus, optical resonators provide an
opportunity to investigate several coherent quantum mechanical effects, which otherwise are masked due to the presence of continuum of modes in free space. In more
advanced quantum optical model, known as the Jaynes-Cummings model [6, 8–10],
the coherent interaction between the emitter and the resonator is described as a twolevel system interacting with a quantum harmonic oscillator. Here, the time dependent
probability of the emitter being either in the excited (e) state or in the ground (g) state
is given by P e (t) = cos
2
[ R
√
(n + 1)t] and P g (t) = sin
2
[ R
√ (n + 1)t], where n
is the number of photons in a resonator mode. If there are no photons present, i.e.
in absence of the applied field the probability dynamics is governed by the vacuum
field fluctuations associated with the resonator, having maximum probability at ω res .
In this case, the oscillatory dynamics of the probability (as in case of two coupled
oscillators) occurs at the frequency 2 R , where R = μE is known as the vacuum
Rabi frequency. Here, μ represents the dipole moment of the emitter and E is the
strength of the vacuum field [6, 8–10]. Thus, the optical response of the emitter is
drastically different from that in the free space. Most importantly, there is a possibility to tailor it by placing it within a resonator. Another intriguing result is the
discrete photon number dependence of Rabi frequency, which is fundamentally different from the classical framework of light-matter interaction. This model can be
generalized for an ensemble of emitters interacting with multiple modes. However,
P. Vasa
An emitter in an excited state relaxing via spontaneously emitting a photon in
free space is an irreversible process that is the most elementary light-matter interaction [1, 2, 4]. Its correct modelling is possible only within quantum mechanical
framework by considering coherent and incoherent additions. The model presented
by Victor Weisskopf and Eugene Wigner to explain spontaneous emission is based
on the interaction between a single emitter and a continuum of vacuum field fluctuations of the environment. The interaction between an quantum emitter and a single
electromagnetic field mode is a coherent process, in which the phase relationship is
preserved [2, 4, 5]. Hence it is a reversible process and the energy can flow back and
forth between the emitter and the field mode without any losses. However, when an
emitter (or a quantum mechanical 2-level system) interacts simultaneously with a
continuum of quantum-mechanical harmonic oscillator modes, it becomes an incoherent and irreversible process. The phase relationship here is lost due to the time
averaging over the continuum of vacuum states characterized by different frequencies
and phases without preserving any correlation [1, 2, 4, 5]. The energy once released
to the environment consisting of continuum of modes via emission of photon cannot
return to the emitter in a coherent fashion. In other words, the energy is lost to the
surroundings.
Optical resonators can modify the frequency distribution of vacuum field fluctuations by selectively enhancing those matching the resonance frequency of the
resonator. Thus, a resonator prevents loss of coherence by avoiding simultaneous
interaction with the continuum of modes. Therefore, if a two-level system or an emitter characterized by emission frequency ω emi is now placed within a lossless resonator
having the identical resonance frequency, ω res = ω emi , the rate of spontaneous emission can be drastically altered. It can be suppressed, enhanced and most interestingly
can even be made a reversible process [4–9]. Thus, optical resonators provide an
opportunity to investigate several coherent quantum mechanical effects, which otherwise are masked due to the presence of continuum of modes in free space. In more
advanced quantum optical model, known as the Jaynes-Cummings model [6, 8–10],
the coherent interaction between the emitter and the resonator is described as a twolevel system interacting with a quantum harmonic oscillator. Here, the time dependent
probability of the emitter being either in the excited (e) state or in the ground (g) state
is given by P e (t) = cos
2
[ R
√
(n + 1)t] and P g (t) = sin
2
[ R
√ (n + 1)t], where n
is the number of photons in a resonator mode. If there are no photons present, i.e.
in absence of the applied field the probability dynamics is governed by the vacuum
field fluctuations associated with the resonator, having maximum probability at ω res .
In this case, the oscillatory dynamics of the probability (as in case of two coupled
oscillators) occurs at the frequency 2 R , where R = μE is known as the vacuum
Rabi frequency. Here, μ represents the dipole moment of the emitter and E is the
strength of the vacuum field [6, 8–10]. Thus, the optical response of the emitter is
drastically different from that in the free space. Most importantly, there is a possibility to tailor it by placing it within a resonator. Another intriguing result is the
discrete photon number dependence of Rabi frequency, which is fundamentally different from the classical framework of light-matter interaction. This model can be
generalized for an ensemble of emitters interacting with multiple modes. However,