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arXiv:2207.04517 · Physical Review Research

Propagating single photons from an open cavity: Description from universal quantization

Astghik Saharyan, Benjamin Rousseaux, Z. Kis, Sergiy Stryzhenko, S. Guérin

Published Jul 26, 2023• 6 citations
arXivHTMLPDF

Abstract

Over the last decades, quantum optics has evolved from high-quality-factor cavities in the early experiments toward new cavity designs involving leaky modes. Despite very reliable models, in the concepts of cavity quantum electrodynamics, photon leakage is most of the time treated phenomenologically. Here, we take a different approach, and starting from first principles, we define an inside-outside representation which is derived from the original true-mode representation, in which one can determine the effective Hamiltonian and Poynting vector. Unlike the phenomenological model, they allow a full description of a leaking single photon produced in the cavity and propagating in free space. This is applied for a laser-driven atom-cavity system. In addition, we propose an atom-cavity nonresonant scheme for single-photon generation, and we rigorously analyze the outgoing single photon in time and frequency domains for different coupling regimes. Finally, we introduce a particular coupling regime ensuring adiabatic elimination for which the pulse shape of the outgoing single photon is tailored using a specifically designed driving field envelope.

DOI
10.1103/physrevresearch.5.033056

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Over the last decades, quantum optics has evolved from high-quality-factor cavities in the early experiments toward new cavity designs involving leaky modes. Despite very reliable models, in the concepts of cavity quantum electrodynamics, photon leakage is most of the time treated phenomenologically. Here, we take a different approach, and starting from first principles, we define an inside-outside representation which is derived from the original true-mode representation, in which one can determine the effective Hamiltonian and Poynting vector. Unlike the phenomenological model, they allow a full description of a leaking single photon produced in the cavity and propagating in free space. This is applied for a laser-driven atom-cavity system. In addition, we propose an atom-cavity nonresonant scheme for single-photon generation, and we rigorously analyze the outgoing single photon in time and frequency domains for different coupling regimes. Finally, we introduce a particular coupling regime ensuring adiabatic elimination for which the pulse shape of the outgoing single photon is tailored using a specifically designed driving field envelope.
Effective models to describe the dynamics of an open cavity have been extensively discussed in the literature. In many of these models the cavity leakage to the outside is treated as a loss introduced phenomenologically. In contrast to these, we focus here on characterizing the outgoing photon using a novel approach where the outside is treated as part of the system. In such a global system, in order to separately characterize the photon inside and outside cavity, we demonstrate a first-principle derivation of a coherent cavity-reservoir coupling function for cavities with mirrors consisting of a stack of dielectric layers. In particular, we show that due to the effects induced by the multilayer nature of the cavity mirror, even in the standardly defined high-finesse cavity regime, the cavity-reservoir system description might differ from the one where the structure of the mirror is neglected. Based on this, we define a generalized cavity response function and a cavity-reservoir coupling function, which account for the longitudinal geometric structure of the cavity mirror. This allows us to define an effective reflectivity for the cavity with a multilayer mirror as if it was sitting in a well-defined effective mirror plane. We estimate the error of such a definition by considering cavities of different lengths and mirror structures. Finally, we apply this model to characterize the dynamics of a single photon produced in such a cavity and propagating outside.
We analyse the system of Λ-atoms in a cavity QED of semi-transparent mirror and driven by laser fields. We derive effective models and connect concepts (photonic flux, input-output operators, photonic state) characterizing the propagation of the resulting leaking photons. We propose an atom-cavity non-resonant scheme for single-and 2-photons generation. The pulse shapes of outgoing single photons are tailored using a specifically designed driving field envelope. For the production of 2-photon states, two trapped atoms are used with two driving pulses. Their pulse shapes are characterized and it is shown that the multiphoton outgoing photonic states cannot be Fock states, since the photons are not generated strictly simultaneously.
A single two-level emitter coupled to a single mode of an electromagnetic resonator constitutes the most fundamental instance of light--matter interaction. Realizations of this type of system are investigated in the field of cavity quantum electrodynamics (cavity QED). Particularly important are schemes in which the coupling is strong enough to exceed the spontaneous emission rate as well as cavity damping. In this case, the coherent exchange of excitation between source and photon is the dominating process, providing the unique possibility to deterministically control the quantum evolution of the system. Great progress has recently been accomplished, technologically as well as conceptually, and the focus section in this issue of Journal of Optics B: Quantum and Semiclassical Optics features five articles highlighting important recent developments in the field. Strong coupling between single atoms and the electromagnetic cavity field have been achieved in two distinct regimes, the optical and the microwave region. The latter is represented by the micromaser, a high-Q microwave cavity pumped by a beam of highly excited Rydberg atoms. It has proved to be an almost ideal system for the generation of non-classical field states. A constraint in these experiments is the fact that the field inside the cavity can only be examined indirectly, by analysing the occupation of atoms leaving the cavity. While only two atomic states were previously identified for this purpose, V A Reshetov shows in his article (pages 119--126) that more information on the field may be extracted by taking advantage of the Zeeman substructure of the atomic levels. In the optical domain of cavity QED, by contrast, the state of the resonator field may be probed directly by measuring the light leaking from the cavity using photodetectors. In order to minimize damping, high-quality cavities must be employed. In most optical experiments these are Fabry--Perot resonators with ultra-high reflectivity mirrors, with losses as low as a few parts per million being state of the art. At the same time, the coupling of atoms and photons is enhanced by reducing the length of the cavity, so that strong coupling conditions are reached. A very attractive application of this type of cavity QED setup is the possibility of exerting control over the quantum dynamics of the system. An impressive example is the implementation of quantum feedback, reported in the paper by W P Smith and L A Orozco (pages 127--134). Using information gained from detecting a single photon escaping from the cavity, active feedback is applied to halt the dynamical evolution of the system. This is the first time active feedback has been used to modify the quantum evolution of a system at the single-photon level. The authors discuss how detunings between the subsystems affect the quantum feedback, an issue particularly relevant for ensembles of atoms with a spread in detuning. Perfect quantum feedback in this case requires the use of additional parameters to control the system. The majority of optical cavity QED experiments performed so far have used as a medium atoms traversing the cavity on random trajectories, such as, for example, thermal atomic beams. In this case, the observation of quantum effects is degraded in two ways. The Doppler effect and the finite transit time lead to line broadening, while the spatial structure of the cavity mode results in atoms in different locations experiencing different coupling with the photonic field. J E Reiner et al (pages 135--142) investigate the impact of these effects on quantum fluctuations in their cavity QED system. In order to overcome line broadening, they have developed a continuous cold atomic beam, extracted from a magneto-optical trap. An even better control of broadening effects and fluctuating coupling strength is provided by trapping the atoms inside the cavity and cooling them to temperatures at which their position spread is smaller than the optical wavelength. Much recent activity has been devoted to developing such an ultimate cavity QED system. A Vukics et al (pages 143--153) present a theoretical proposal for dynamically trapping and cooling atoms inside a cavity. By driving the atoms with a standing wave injected from the side of the cavity and by weakly exciting the cavity field itself, they predict three-dimensional trapping of the atom for periods around 1 second, an extremely long time on the typical scale of cavity QED dynamics. The essential component of their scheme is the interference between the two driving fields, determining the temperature of the particle and the stable trapping positions. In the quest for better confinement of radiation, lately, an appealing alternative to the Fabry--Perot cavity has been investigated -- the optical microsphere resonator. A sphere with a diameter of a few tens of microns, it supports so-called whispering gallery modes near its circumference, for which quality factors up to 10 10 have been measured. In order to exploit the correspondingly low field damping rates for cavity QED, a method must be found to couple atomic or solid state emitters to these modes. S Göotzinger et al (pages 154--158) report the coupling of single nanocrystals on the surface of a microsphere through the evanescent field of the mode. Selective excitation of a single nano-emitter was accomplished with a scanning confocal microscope. The system is a promising candidate for cavity QED with single-quantum emitters. The ability to fully control the quantum dynamics of atoms and light is beginning to spawn early applications. The most notable example is quantum information processing. This requires the coherent manipulation of single-quantum systems, which is ideally provided by cavity QED interaction in the strong-coupling limit. The development of practical devices will be among the principal future tasks for research in the field. This will include physical systems, which have not been covered in this issue, such as semiconductors, quantum dots as emitters, or photonic crystals as resonator structures. Cavity QED is therefore guaranteed to remain a vigorous field in the years to come.
Within the framework of exact quantum electrodynamics in dispersing and absorbing media, we have studied the quantum state of the radiation emitted from an initially in the upper state, prepared two-level atom in a high-Q\mathit{Q}Q cavity, including the regime where the emitted photon belongs to a wave packet that simultaneously covers the areas inside and outside the cavity. For both continuing atom-field interaction and short-term atom-field interaction, we have determined the spatiotemporal shape of the excited outgoing wave packet and calculated the efficiency of the wave packet to carry a one-photon Fock state. Furthermore, we have made contact with quantum noise theories where the intracavity field and the field outside the cavity are regarded as approximately representing independent degrees of freedom such that two separate Hilbert spaces can be introduced.
For single-photon generation based on cavity quantum electrodynamics, we investigate a practical model assuming a Gaussian wave packet. This model makes it possible to comprehensively analyze the temporal dynamics of an atom-cavity system with both adiabatic and nonadiabatic conditions using analytical expressions. These results enable us to clarify the relationship between pulse width and maximum success probability over the full range of coupling regimes. We demonstrate how to achieve a high success probability while keeping a short pulse width by optimizing the cavity transmittance parameter and the time-controlled external field. Our formulations provide a practical tool for efficient single-photon generation in a wide variety of experimental platforms.
In contrast to the free-space evolution of an atom governed by a multimode interaction with the surrounding electromagnetic vacuum, the evolution of a cavity-QED system can be characterized by just three parameters: (i) atom-cavity coupling strength ggg, (ii) cavity relaxation rate $\ensuremath{\kappa}$, and (iii) atomic decay rate into the noncavity modes $\ensuremath{\gamma}$. In the case of an atom inserted into a planar resonator with an input beam coupled from the outside, it has been shown by Koshino [Phys. Rev. A 73, 053814 (2006)] that these three parameters are determined not only by the atom and cavity characteristics, but also by the spatial distribution of the input pulse. By an ab initio treatment, we generalize the framework of Koshino and determine the cavity-QED parameters of a coupled system of atom, planar (leaky) resonator, and input single-photon pulse as functions of the lateral profile of the pulse and the length of resonator. We confirm that the atomic decay rate can be suppressed by tailoring appropriately the lateral profile of the pulse. Such an active suppression of atomic decay opens an attractive route towards an efficient quantum memory for long-term storage of an atomic qubit inside a planar resonator.