Fully quantum-mechanical analytic results for single-photon transport in a single-mode waveguide coupled to a whispering-gallery resonator interacting with a two-level atom
We analyze the single-photon transport in a single-mode waveguide coupled to a whispering-gallery-type resonator interacting with a two-level atom. The single-photon transport properties such as the transmission and reflection amplitudes, as well as the resonator and the atom responses, are solved exactly via a real-space approach. The treatment includes the inter-mode backscattering between the two degenerate whispering gallery modes of the resonator, and the dissipations of the resonator and the atom. We also show that a generalized critical coupling condition, that the single-photon transmission at the output of the waveguide goes to zero on resonance for a matched system, holds for the full coupled waveguide-ring resonator-atom system.
pacs:42.50.Ct, 42.79.Gn, 42.65.Wi
There has been a lot of recent interest in the systems of coupled whispering-gallery-type micro-resonators and atoms Thompson et al. (1992); Vernooy et al. (1998); Klimov et al. (1999); Rosenblit et al. (2004); Aoki et al. (2006); Srinivasan and Painter (2007a, b); Mazzei et al. (2007); Dayan et al. (2008) (a schematic configuration is shown in Fig. 1). This configuration is particularly explored in a variety of fundamental and applied studies such as cavity quantum electrodynamics (cavity QED), single-atom manipulation and detection, and biochemical sensing. In the context of cavity QED, this kind of system has been treated semiclassically by assuming a weak classical input Carmichael (2003). Recently a comprehensive study of this system based on the semiclassical approach has been carried out by Srinivasan and Painter Srinivasan and Painter (2007a). Such semiclassical treatment is of direct current experimental interest, since the incident state in most cavity QED experiments are indeed weak classical beam.
There are, however, several reasons for which the semiclassical treatment is not completely satisfactory. First of all, given that these systems would ultimately be used to process quantum states of photons, it would seem valuable to treat these systems with fully quantized input states as well. Secondly, even in the presence of a classical input beam, the output of this system is not a coherent state. The semiclassical treatment typically provides various correlation functions of the output state. For a complete description of the output, however, one might instead prefer a direct description of the output quantum state itself. Thirdly, the semiclassical treatment is typically carried out numerically, by setting up a set of master equations for the evolution of the density matrix for the atom-cavity system, and by truncating the density matrix assuming a maximum number of photons in the cavity. For very weak classical input, for example, the maximum number of photon is often taken to be 1. For a deeper understanding of the underlying physics, however, exact analytic results are quite valuable. Finally, the physics intuition related to a semiclassical treatment has to be always treated with caution. For example, the truncation procedure, as discussed above, yields a set of coupled nonlinear ordinary differential equations, even when the maximum photon number in the cavity is fixed to be 1. Based upon this set of coupled nonlinear differential equations, one may be tempted to discuss the so-called “single-photon nonlinearity”. However, it is worthwhile to emphasize that nonlinearity has to arise from photon-photon interaction.
Motivated by the discussions above, in this article, we consider the single-photon transport for the system of coupled whispering-gallery-type micro-resonators and atoms, assuming a single-photon Fock state input. In contrast to the semiclassical treatments, where in most cases only numerical results are obtained so far, here the use of a fully quantized formalism allows one to straightforwardly obtain analytic and exact results of the transport properties and the system responses, which has not been done before for this system. Our analytic results are in well agreement with the experimental data Kippenberg et al. (2004); Aoki et al. (2006) and with the numerical results in Srinivasan and Painter’s for the parameters considered in their studies Srinivasan and Painter (2007a). Using these analytic results, we systematically explore the parameter regime that is of direct experimental interests, to understand the delicate interplay between the various types of coupling and the intrinsic atom and resonator dissipations.
This article is organized as follows. In Sec. II we first introduce the system configuration and the Hamiltonian. In Sec. III, the single-photon transport properties and the system responses are then solved analytically. Sec. IV describes the general properties of the exact solutions, such as spectral symmetry of the transmission spectrum, and generalized critical coupling condition of the full coupled system. In Sec. V, we compare our results with some experimental results of waveguide-resonator systems, and with published numerical results of coupled waveguide-resonator-quantum dot systems. Sec. VI presents a systematic study of the single-photon transport of the full coupled system of waveguide-ring resonator-atom. Sec. VII discusses the effects of the intrinsic dissipations of the resonator and the atom. In Sec. VIII we investigate the effects of complex inter-mode backscattering. Sec. IX discusses the effects of detuning between the resonator and the atom. Finally, Sec. X sums up the article, pointing out some applications and generalizations of the formalism.
Ii The system and Hamiltonian
The system of interest in this article is schematically shown in Fig. (1): a whispering-gallery type resonator interacting with a two-level atom is side-coupled to a single-mode waveguide. A whispering-gallery type microresonator, such as a ring resonator, a microsphere, a microtoroid or microdisk, supports two degenerate whispering gallery modes (WGMs) that propagate around the resonator in opposite directions. (Throughout this paper, we will often use the term “ring resonator” to designate a whispering-gallery type microresonator). We also include the interactions of the atom and the resonator with the reservoirs. Such interactions with reservoirs give rise to intrinsic dissipations Scully and Zubairy (1997); Carmichael (2003).
The Hamiltonian of the composite system is :
is the Hamiltonian of the system of coupled waveguide-resonator-atom. This Hamiltonian includes the waveguide, the resonator, and the atomic part, as well as the interaction between the waveguide and the resonator, and the atom and the resonator.
The first line in of Eq. (1a) describes the propagating photon modes in the waveguide. is a bosonic operator creating a right/left-moving photon at . is a reference frequency, around which the waveguide dispersion relation is linearized Shen and Fan (2008). Its value does not affect the photon transport properties.
The second line in describes the modes in the resonators and the atomic states. For the resonator, is the creation operator for the counter-clockwise WGM mode and is the creation operator for the clockwise WGM mode, both of frequency . For the atom, () is the creation operator of the ground (excited) state, () is the atomic raising (lowering) ladder operator satisfying and , where describes the state of the system with propagating photons, photons in cavity mode, and the atom in the excited () or ground () state. is the atomic transition frequency.
The third line in describes the interactions between the waveguiding modes and the WGMs. is the waveguide-resonance coupling strength of each WGM. Here the right-moving(left-moving) mode only couples to the phase-matched counter-clockwise(clockwise) mode.
The fourth line in describes the interactions between the atom and the whispering gallery modes. and are the resonator-atom coupling strength for each respective WGM.
The last line in describes the inter-mode backscattering between the two degenerate WGMs, induced through imperfection of the resonator. is the inter-mode backscattering strength.
of Eq. (1b) describes the reservoir, which is composed of three subsystems: . , and are assumed to be independent. Each , and is modeled as a collection of harmonic oscillators with frequencies , , and , and with the corresponding creation (annihilation) operators (), (), and (), respectively.
of Eq. (1c) describes the interactions between the resonator and the atom with the reservoirs, respectively. The WGM couples to the th reservoir oscillator in , while the WGM couples to the th reservoir oscillator in . Both WGMs are assumed to couple with the reservoir with the same coupling constant . The atom couples to the th reservoir oscillator in with a coupling constant .
By incorporating the excitation amplitudes of the reservoir , it can be shown that the effective Hamiltonian of can be obtained and is given by Shen and Fan (2008):
where and are the intrinsic dissipation rates of the resonator WGM and the atom, respectively, due to coupling to the reservoir. We will call as in the following for brevity. Note that , , , , , and all have the same unit as frequency.
Here we make some remarks on the Hamiltonian of Eq. (II):
The coupling constants are determined by the underlying photonic and electronic states of each constituent of the system. The relations between these coupling constants are not arbitrary but rather are constrained by symmetry of the system. These symmetry properties in turn are reflected in the response spectra of the system. The relevant symmetry operations here are mirror and time-reversal symmetries. A discussion of the symmetry transformations is given in Appendix A.
Except obeying the symmetry transformations, the specific numerical values of the coupling constants depend upon each specific experimental realization. A detailed discussion is provided in Appendix B. Here, we just note that that in general , and both and can be complex numbers.
In the following, we first derive the exact solutions to the single-photon transport and the system responses (i.e., WGMs and atom excitations) described by the Hamiltonian of Eq. (II), without placing any constraints on the coupling constants. Later, we will specialize to specific choices of the coupling constants to compare with experimental data, and to some published numerical results.
Iii Exact Solutions for Single-Photon Transport
We consider the temporal evolution of an arbitrary single-photon state , as described by the Schrödinger equation
where is the Hamiltonian of Eq. (II). In general, can be expressed as
where is the vacuum, which has zero photon and has the atom in the ground state. is the single-photon wave function in the mode. is the excitation amplitude of the whispering gallery mode, and is the excitation amplitude of the atom. For this state, the Schrödinger equation (Eq. (3)) thus gives the following set of equations of motion:
For any given initial state , the dynamics of the system can be obtained directly by integrating this set of equations (Eqs. (5)). In this way, one could study the time-dependent transport of an arbitrary single-photon wave packet.
In the following, we concentrate on the steady state properties. When is an eigenstate of frequency , i.e., , Eq. (3) yields the time-independent eigen equation
and the interacting steady-state solution can be solved for. Here is the total energy of the system.
For an input state of one-photon Fock state, the most general time-independent interacting eigenstate for the Hamiltonian of Eq. (II) is:
where we denote the time-independent amplitudes by the corresponding untilded symbols, e.g. , etc. The connection between the interacting eigenstate and a scattering experiment is described by the Lippmann-Schwinger formalism Taylor (1972); Huang (1998); Shen and Fan (2007).
with , and . Our aim is to solve for the transmission and reflection amplitudes for an incident photon. For this purpose, we take , and , where is the transmission amplitude, and is the reflection amplitude Shen and Fan (2005a, b). The set of equations of motion, Eqs. (8a)-(8e) now read:
which can be solved straightforwardly for , , , , and :
where we have used , , , and , all are real numbers. Notice that is the external linewidth of the WGM’s due to waveguide-cavity coupling. These analytic expressions of the amplitudes, Eq. (10b)-(10f), provide a complete description on the single-photon transport properties. These equations are applicable to arbitrary two-level systems having any orientation of the electric dipole moment, specified through and .
Iv General Properties of The Transmission Spectrum
The expressions of the amplitudes (Eqs. (10)) allow us to make exact statements on the general properties of the spectrum of the full coupled system, such as the spectral symmetry properties and the generalized critical coupling conditions. These statements are valid even in the presence of resonator and atom dissipations. We will exemplify these general properties with concrete examples in the following sections.
iv.1 Spectral Symmetry Properties of The Transmission Spectrum
It is straightforward to show that the transmission amplitude of Eq. (10b) satisfies the following spectral symmetry condition, regardless of the values of the cavity and atom dissipations:
where , is the frequency detuning between the resonator and the atom, and is the phase mismatch. This condition states that the transmission spectrum of one system with parameters is mirror-imaged with respect to to that of another system with parameters .
In particular, for a system with and , the transmission spectrum is symmetric with respect to : . The former condition says the resonator and the atom are in-tuned (); while the latter condition requires to be purely imaginary so that . The same spectral symmetry property holds for other amplitudes of Eq. (10c)-(10f).
iv.2 Generalized Critical Coupling Condition
Critical coupling is defined for a system of coupled waveguides and resonators when transmission of the input signal goes to zero at the output port at resonance. The existence of critical coupling is of direct interest to quantum optics experiments where one often desires to eliminate single-photon transmission so that the signatures of two-photon transmission are distinct from single-photon effects. Here we present the condition under which critical coupling can be achieved for single-photon transport in the full coupled waveguide-ring resonator-atom system.
By requiring the transmission amplitude of Eq. (10b) to be zero, one can show that the critical coupling is reached at if the following criterions are satisfied:
The resonator and the atom are in-tuned, i.e., ; and
V Numerical Validation
Before we proceed to understand the detailed predictions of Eqs. (10), here we first compare Eq. (10b)-(10f) to experimental data of waveguide-resonator systems Aoki et al. (2006); Kippenberg et al. (2004), and to numerical results of Srinivasan and Painter’s Srinivasan and Painter (2007a) on the waveguide-microdisk-quantum dot system. In all cases the agreements are excellent.
v.1 Fitting to Experimental Data of Waveguide-Resonator Systems
To demonstrate the validity of the exact solutions, we apply these expressions to fit two of the recent experimental data of waveguide-microtoroidal resonator systems Aoki et al. (2006); Kippenberg et al. (2004). The results are plotted in Fig. (2). The amplitudes for the waveguide-resonator system can be deduced from those of the full coupled waveguide case by decoupling the atom. In particular, the transmission amplitude is given by
The details are given in Appendix C. Note only the magnitude of appears in the amplitude, its phase does not.
Fig. (2)(a) shows transmission spectra of a waveguide-microtoroidal resonator system Aoki et al. (2006). In the experiments, the waveguide-resonator coupling strength is varied by adjusting the distance between the waveguide and the microtoroid. The lower trace is taken such that the critical coupling condition is approximately satisfied: ; while the upper trace is for conditions of under-coupling: . By fitting to the spectra using Eq. (13), one can extract the numerical values of , , and . Our numerical results show that .
When the same values of and determined from fitting to the lower trace are used to fit the upper curve, the width of the spectrum is narrower than that of the experimental data, and the minimum region has significant deviation. If, however, somewhat larger values of and are used instead to fit the data, the spectrum can be fitted reasonably well, as shown by the upper blue curve in Fig. (2)(a). This indicates that both the inter-mode backscattering and the intrinsic loss of the resonator are slightly suppressed when the coupling between the waveguide and the resonator is strong.
As another validation, in Fig. (2)(b), we use Eq. (13) to fit the transmission spectrum of another waveguide-microtoroidal resonator system Kippenberg et al. (2004), which is in the deep under-coupling regime such that . The numerical fitting indicates that so the inter-mode backscattering dominates. In this regime, the transmission spectrum shows a doublet structure. The doublet structure in the transmission spectrum induced by the inter-mode backscattering for ultra-high-Q resonator in the regime is well-known Weiss et al. (1995); Kippenberg et al. (2002, 2004).
In both cases above, the extracted numerical values agree reasonably well with those reported in the experiments.
v.2 Comparing to Numerical Results of Waveguide-Resonator-Atom Systems
We next compare our analytic results of the amplitudes of Eqs. (10b)-(10f) to the numerical results on coupled waveguide-microdisk-quantum dot system in Srinivasan and Painter Srinivasan and Painter (2007a), where by properly choosing the orientation of the quantum dot dipole polarization, and the azimuthal origin. From Eqs. (10b)-(10f), one immediately sees that all amplitudes, except , depend on the magnitude of only but not its phase. The atom excitation amplitude is proportional to , the spectrum however also depends on only.
Fig. 3 plots the transmission and reflection spectra using the parameters corresponding to those in Fig. 5 of Ref. [Srinivasan and Painter, 2007a]. (The relation between the parameters used in this article and Ref. [Srinivasan and Painter, 2007a] is: , , , , and ). In Fig. 3, the upper panel plots the spectra when the quantum-dot dephasing is zero; while the lower panel plots the spectra when . Each panel plots the in-tuned () and de-tuned () cases. The results of both approaches are in excellent agreement. We were able to obtain excellent agreements with all results in Ref. [Srinivasan and Painter, 2007a] (not shown here).
In Fig. 3 we also plot the atom excitation. In general, there are only two resonances in the atom excitation, in contrast to the three resonances in the transmission spectrum. Since the atom is an intermediary for correlating photons, one expects the photon-photon correlations are qualitatively different at the three resonances of the transmission spectrum. This is supported by comparing the atomic excitation spectra with the photon correlation function in Fig. 12 of Ref. [Srinivasan and Painter, 2007a].
Vi Single-Photon Transport of The Coupled Waveguide-Resonator-Atom Systems
We now turn the attentions to the single-photon transport of the full coupled waveguide-resonator-atom system and the system responses. The analytic results enable us to present a systematic parametric study of the transport properties by continuously varying the parameters. Such a systematic study facilitates the understanding of the underlying physics in this complicated coupled systems. As a concrete example, we will take , and let throughout the discussions hereafter. This choice of coupling constants correspond to that considered in Ref. [Srinivasan and Painter, 2007a]. Other choices of the coupling constants could be investigated in the same manner.
In this section the inter-mode backscattering strength is taken as real and the system is assumed lossless (). The atom and the resonator are also assumed to be in-tune (). The effect of dissipation, complex , and atom-cavity detuning will be considered in later sections.
We consider the spectra of transmission (Fig. 4), the group delay (Fig. 5), the atom excitation (Fig. 6), and the phase matched WGM excitation (Fig. 7) for different value of and . (For completeness, we also plot the spectra of reflection and the counter-propagating WGM excitation in Appendix D). To facilitate visualization, these spectra are presented in a matrix form with different rows or columns corresponding to different and , respectively. Comparing this set of spectra provides very useful information to understand the full coupled system, since the atom and the WGM excitation determine the nature of the transmission resonances. Below, we refer to a resonance with large (small) atom excitation as a resonance of atom (cavity) nature.
vi.1 First column: the atom decoupled
The leftmost column describes the waveguide-resonator case with the atom decoupled:
When , the resonator acts as an all-pass filter with transmission for all frequency. The group delay shows a Lorentzian peak.
When is slightly increased but small than , a transmission dip develops at the resonance frequency , with
Also, the transmission has a non-Lorentzian lineshape.
When , the transmission spectrum has a flat bottom centered at where the transmission becomes zero. The transmission has a maximally-flat 2nd-order Butterworth filter lineshape given by
with a full width at half maximum (FWHM) equal to . The group delay however shows two splitted peaks.
When , the transmission shows two resonance dips at which the transmission is zero. The spectral separation between the two dips is , which approaches when .
Thus the qualitative behavior of the transmission spectrum is determined by the ratio of . For an ultra-high resonator with very weak external coupling to the waveguide, i.e., very small value of , even a weak mode cross-talking would qualitatively change the transmission spectrum.
Since the atom is decoupled, all the transmission dips(resonances) have a cavity nature, i.e., the atom excitation is zero.
vi.2 First row: no inter-mode backscattering in the resonator
The uppermost row describes an ideal resonator without mode cross-talking, i.e., :
When , the atom is decoupled, and the resonator is an all-pass filter as discussed above.
When is slightly increased from zero such that (shown in figure: ), a transmission dip down to zero occurs at frequency . This transmission dip is induced by the atomic resonance and is in contrast to that induced by small above, which does not go to zero. In this regime, the poles of the transmission amplitude are , , and . The transmission spectrum can be well approximated by a Lorentzian with FWHM equal to . Small deviation of the spectrum from a Lorentzian occurs only when , where the poles with larger imaginary parts become important in the spectral response. Moreover, the atom excitation has a strong peak at , thus the transmission dip has an atomic nature. The group delay shows a single peak. Notice that the group delay in this case is much larger than case B1 above.
When is further increased such that , the transmission spectrum has a flat bottom centered at where the transmission is zero. The transmission has a maximally-flat 3rd-order Butterworth filter lineshape given by
with a full width at half maximum (FWHM) equal to . Both the group delay and the atom excitation however shows two splitted peaks.
When is further increased such that (shown in figure: ), three transmission dips develop. This is because the two counter-propagating WGMs of the resonator, through linear superposition, can alternatively be described as two standing-wave modes. One of the standing-wave modes has zero amplitude at the atom location, and corresponds to the middle transmission dip at . This resonance is of cavity nature. The other mode has non-zero amplitude at the atom location, and experiences Rabi-splitting through interacting with the atom. The transmission spectra exhibits three dips when such Rabi splitting is large enough compared with . The two side dips, which result from the Rabi splitting, have a mixture of cavity and atom nature, as confirmed by examining the atom excitation plot. The spectral separation between the two side dips is . In addition, the two side-dips have a larger group delay than that of the middle peak, since the photon experiences delay from both the cavity and the atom.
When (shown in figure: ), the spectral separation between the two dips approaches . The maximum transmission between each side-dip and the middle dip approaches 1 as . Moreover, the width of the middle dip is the sum of the widths of the two side-dips.
The case of represents the ideal case for WGM resonators and therefore is of fundamental importance. In Appendix E, we summarize all analytic results related to this important case.
vi.3 General case: non-zero and
We now discuss the general features of the cases when both and are non-zero.
Comparing the leftmost two columns in Fig. 4, where we increase slightly from zero to a small non-zero value ( in the figures). For all values of , the effect of introducing an atomic resonance by a small is to create a resonance of atomic nature at , resulting in an asymmetrical Fano-type lineshape in transmission Fan et al. (2003), and a large group delay at resonance. The rest of the spectrum is not significantly perturbed.
Comparing the uppermost two rows in Fig. 4, where we increase slightly from zero to a small non-zero value ( in the figures). For all values of , the effect of introducing an inter-mode backscattering by is to slightly distort the transmission spectrum to be asymmetrical.
The cases with intermediate values of and can be obtained from perturbing from its counterpart in each direction. The exact lineshapes have to be computed using the analytic expressions.
One important feature is the anti-crossing between the atomic and cavity resonances as one varies the values of or . We will demonstrate this point using the column of . Fig. 8 plots the real part of the poles of the transmission amplitude , which indicate the spectral locations of the resonances. At , the transmission spectrum starts with three resonances with the middle dip of cavity nature, and the two side-dips mixture of both cavity and atom nature, as previously explained. When is increased (from to ), the left two dips anticross. During the anti-crossing process, the nature of the resonance of the two dips is exchanged, as can be seen by examining the weight of the atomic excitation along the evolution process. In the end, when becomes , the left resonance is of purely cavity nature, while the middle resonance is of purely atomic nature, and the right resonance is of largely cavity nature with a little atomic mixture. Similar anti-crossing behavior can be observed when is continuously varied instead. Thus, it is difficult to ascertain the nature of a resonance judging from the position of the resonance alone, in the case where significant back scattering occurs.
Vii Effects of Dissipations
The effects of intrinsic dissipations on the transmission resonances strongly depend upon the nature of the resonances. When a resonance is of purely cavity nature, the transmission at the resonance frequency is insensitive to the atom dissipation, but only to the cavity dissipation. Similarly, when a resonance is of purely atomic nature, the transmission at the resonance frequency is insensitive to the cavity dissipation, but only to the atom dissipation. For a resonance of mixed nature, the transmission is affected by either type of dissipation.
This is clearly seen in all cases in Fig. 9, where we introduce dissipations to all cases considered in Fig. 4, which are lossless. We choose dissipations that are either purely from the atoms (, ), or from the cavities (, ).
As a more detailed example, Fig. 10 plots the effects of increasing dissipations for the same case of and . The left resonance, which is of purely cavity nature, is essentially unaffected by even very large atom dissipation. The right resonance, which has a little atomic nature, has its transmission minimum gradually lifted and its width slightly broadened at large atom dissipation. The cavity dissipation, on the other hand, strongly affects all three resonances, since the phase matched WGM has weights at each resonance. Among the three resonances, the middle one is least affected, since it is primarily atomic. At large cavity dissipations, it still leaves a small signature in the transmission spectra while the contributions from the other two resonances are no longer visible. Thus, the nature of the resonance influences the properties of the transmission spectrum even in the regime of large dissipation.
Viii The effects of complex
When , all the amplitudes in Eq. (10b)-(10f) depend upon the phase of only but not of . In the above discussion, for concreteness, we have assumed to be real. Here we consider the effects of a complex on the transport properties.
where . The spectral symmetry is determined by the phase of only.
Fig. 11 shows a series of transmission spectrum with , and and , i.e., the cavity and the atom are in-tuned:
When , two resonances of very different quality factor are close to each other at .
As , the spectrum has three dips and is asymmetric with respect to .
When , satisfying the condition that with an integer, the spectrum is symmetric with respect to .
The spectrum of is the mirror image of the spectrum of with respect to .
Finally, the spectrum of is the mirror image of that of with respect to .
A general case is plotted in Fig. 12, with atom-cavity detuning , and with finite atom and resonator dissipations. The transmission spectra with () and with () are mirror-imaged with respect to .
Ix Effects of Detuning ()
In this section, we discuss the effects of detuning between the resonator and the atom, with . Fig. 13 plots the transmission spectrum for the lossless case as or is varied. We note that the spectrum is always asymmetrical with respect to when both and are non-zero. We now discuss these spectra plots, as organized in a matrix form.
ix.1 The first row: large detuning with
When ( as plotted), the atom is decoupled from the system and creates a narrow atomic resonance at . The background is the transmission of waveguide-ring resonator subsystem that has unity transmission.
When is increased ( as plotted), the atom becomes weakly coupled to the waveguide-ring resonator subsystem. Analytically, such a weak coupling regime occurs in the parameter range , where is the Rabi frequency, and . In this weak coupling regime, the transmission spectrum exhibits an atomic resonance dip at . Also, the weak coupling of the atom and the cavity results in a small scattering between the two WGM, and consequently a small dip is present at the resonant frequency . Notice that the transmission dip at the cavity resonant frequency does not reaches zero at its minimum.
For large such that and , the atom is strongly coupled to the waveguide-resonator subsystem, and the transmission spectrum develops into three resonance dips, each reaching zero at its minimum. .
ix.2 Second column: decoupled and weakly coupled atom
For decoupled () and weakly coupled (, ) atom, the atom creates a narrow atomic resonance at , and the location of the resonance moves with . At , the weakly coupled atom mixes with the cavity resonance so the line width is slightly increased, compared with that of the cases.
For intermediate values of and , the exact lineshape and the weight of the nature of a resonance has to be computed using the exact expressions. Fig. 14 plots the effects of intrinsic losses of the resonator and the atom to the transmission spectrum. Again, the resonator loss strongly suppresses a resonance of cavity nature, and the atomic loss strongly suppresses a resonance of atom nature.
We have provided a full quantum mechanical approach to treat the coupled waveguide-ring resonator-atom system, and derived the analytic solutions for the single-photon transport. The real-space approach outlined in this article can be generalized straightforwardly to treat cases such as cascaded multi-ring resonator, multi-atom, or multi-port configuration that is relevant to applications of add-drop filter, single-photon switching and delay lines. Our formalism can also provide a starting point for treatment of pulse propagation in these more complicated systems.
Acknowledgements.J.-T. Shen acknowledges informative discussions with K. Srinivasan at NIST, and S. Chiow at Stanford. S. Fan acknowledges financial support by the David and Lucile Packard Foundation.
Appendix A Symmetry transformations of the Hamiltonian
The relations between the coupling constants are not arbitrary but rather are constrained by the symmetry of the Hamiltonian. The symmetries of interest for our configurations are mirror symmetry and the time-reversal symmetry. In this section, we consider these symmetries that constrain the form of the Hamiltonian, and establish the relations between the coupling constants. Note that the specific form of the relations depend upon the choice of the representation of the fields.
a.1 Waveguide mode operators
a.1.1 Mirror symmetry
We start with the waveguide mode operators. The following general considerations apply to any single-mode waveguide that obeys mirror symmetry. For such a waveguide, one can always choose a mirror plane perpendicular to the waveguide such that the dielectric function satisfies
where the -axis is along the direction of the waveguide. From the Maxwell’s equations
it is straightforward to show that if the fields of the following form
is a solution of the Maxwell’s equations for the waveguide, then the following fields are also a solution: