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arXiv:1703.11006 · APL Photonics

Exact solution to the steady-state dynamics of a periodically modulated resonator

Momchil Minkov, Yu Shi, Shanhui Fan

Published Jun 9, 2017• 60 citations
arXivHTMLPDF

Abstract

We provide an analytic solution to the coupled-mode equations describing the steady-state of a single periodically modulated optical resonator driven by a monochromatic input. The phenomenology of this system was qualitatively understood only in the adiabatic limit, i.e., for low modulation speed. However, both in and out of this regime, we find highly non-trivial effects for specific parameters of the modulation. For example, we show complete suppression of the transmission even with zero detuning between the input and the static resonator frequency. We also demonstrate the possibility for complete, lossless frequency conversion of the input into the sideband frequencies, as well as for optimizing the transmitted signal towards a given target temporal waveform. The analytic results are validated by first-principle simulations.

DOI
10.1063/1.4985381

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We provide an analytic solution to the coupled-mode equations describing the steady-state of a single periodically modulated optical resonator driven by a monochromatic input. The phenomenology of this system was qualitatively understood only in the adiabatic limit, i.e., for low modulation speed. However, both in and out of this regime, we find highly non-trivial effects for specific parameters of the modulation. For example, we show complete suppression of the transmission even with zero detuning between the input and the static resonator frequency. We also demonstrate the possibility for complete, lossless frequency conversion of the input into the sideband frequencies, as well as for optimizing the transmitted signal towards a given target temporal waveform. The analytic results are validated by first-principle simulations.
We present a general method to calculate the periodic steady state of a driven-dissipative system coupled to a transmission line (and more generally, to a reservoir) under periodic modulation of its parameters. Using Floquet's theorem, we formulate the differential equation for the system's density operator which has to be solved for a single period of modulation. On this basis we also provide systematic expansions in both the adiabatic and high-frequency regime. Applying our method to three different systems---two- and three-level models as well as the driven nonlinear cavity---we propose periodic modulation protocols of parameters leading to a temporary suppression of effective dissipation rates, and study the arising nonadiabatic features in the response of these systems.
We consider a two-level atom driven by modulated light and find that complete population inversion can be induced by light without any resonant frequency component. This is in contrast to the familiar case of monochromatic driving in which complete population inversion is only possible with resonant light. This result concerns experimentally realizable systems and hence the effect of spontaneous emission is also considered. We also relate our work to other recent works on quantum double-well. Analogies with the suppression of quantum tunneling and with the low-frequency radiation generation are discussed. By translating the results from the study of the two-level atom to the theory of optical waveguides, we introduce a new geometric representation for analysing the optical state of a single-mode optical coupler. We find that an optical coupler with out-of-phase index modulations can control and suppress the evanescent power transfer between cores. A new type of optical band-pass filter based on the modulated index coupler is proposed.
Using the density matrix formalism, we prove the existence of the periodic steady state for an arbitrary periodically driven system described by linear dynamic equations. This state has the same period as the modulated external influence, and it is realized as an asymptotic solution $(t\ensuremath{\rightarrow}+\ensuremath{\infty})$ due to relaxation processes. The presented derivation simultaneously contains a simple and effective computational algorithm (without using either the Floquet or Fourier formalisms), which automatically guarantees a full account of all frequency components. As a particular example, for three-level $\mathrm{\ensuremath{\Lambda}}$ system we calculate the line shape and field-induced shift of the dark resonance formed by the field with a periodically modulated phase. Also we have analytically solved a basic theoretical problem of the direct frequency comb spectroscopy, when the two-level system is driven by the periodic sequence of rectangular pulses. In this case, the radical dependence of the spectroscopy line shape on pulse area is found. Moreover, the existence of quasiforbidden spectroscopic zones, in which the Ramsey fringes are significantly reduced, is predicted. Our results have a wide area of applications in laser physics, spectroscopy, atomic clocks, and magnetometry. Also they can be useful for any area of quantum physics where periodically driven systems are considered.
We study the dynamics of two ensembles of atoms (or equivalently, atomic clocks) coupled to a bad cavity and pumped incoherently by a Raman laser. Our main result is the nonequilibrium phase diagram for this experimental setup in terms of two parameters: detuning between the clocks and the repump rate. There are three main phases: a trivial steady state (phase I), where all atoms are maximally pumped, a nontrivial steady state corresponding to monochromatic superradiance (phase II), and amplitude-modulated superradiance (phase III). Phases I and II are fixed points of the mean-field dynamics, while in most of phase III stable attractors are limit cycles. Equations of motion possess an axial symmetry and a Z2{\mathbb{Z}}_{2}Z2​ symmetry with respect to the interchange of the two clocks. Either one or both of these symmetries are spontaneously broken in various phases. The trivial steady state loses stability via a supercritical Hopf bifurcation bringing about a Z2{\mathbb{Z}}_{2}Z2​-symmetric limit cycle. The nontrivial steady state goes through a subcritical Hopf bifurcation responsible for coexistence of monochromatic and amplitude-modulated superradiance. Using Floquet analysis, we show that the Z2{\mathbb{Z}}_{2}Z2​-symmetric limit cycle eventually becomes unstable and gives rise to two Z2{\mathbb{Z}}_{2}Z2​-asymmetric limit cycles via a supercritical pitchfork bifurcation. Each of the above attractors has its own unique fingerprint in the power spectrum of the light radiated from the cavity. In particular, limit cycles in phase III emit frequency combs---series of equidistant peaks, where the symmetry of the frequency comb reflects the symmetry of the underlying limit cycle. For typical experimental parameters, the spacing between the peaks is several orders of magnitude smaller than the monochromatic superradiance frequency, making the lasing frequency highly tunable.
A quantum theory describing N two-level atoms in a coherently driven single-mode optical ring cavity is presented. The usual approximation that one may adiabatically eliminate either the cavity or atomic variables is not made. Hence the results are valid for a range of relative cavity and atomic relaxation rates. The theory is based on the linearized equations derived in the positive P representation. A simple analytical solution is presented, including the effect of detunings, for the steady-state transmitted spectrum. The transmitted intensity and squeezing spectra are discussed.
Adiabatic frequency conversion (AFC) in microring resonators is a promising alternative for fully integrable and tunable frequency shifting of optical signals. Nonetheless, the magnitude of the frequency shift via simple AFC in a single ring is fundamentally limited by the material platform. To overcome this limitation, we propose and analyze a scheme to induce cascaded AFC (CAFC) along a chain of coupled, yet initially detuned rings, without requiring unloading the optical signal into a bus waveguide between successive modulations. For concreteness, we examine thoroughly the simplest nontrivial case of a chain of two rings and briefly discuss the generalization to a chain of an arbitrary number of rings. We analyze the temporal dynamics of this CAFC process using temporal coupled mode theory. We examine the transformation of the input into the frequency-shifted output as a rank-one linear operator in the vector space of finite-energy pulses. In this way we show that the energy efficiency of CAFC depends on the input pulse shape through a Schwarz inequality, just as in single-ring AFC. We propose a numerical scheme to maximize the CAFC efficiency with respect to the process's timescales and discuss the physics involved. We show that the resulting CAFC efficiency converges in a polynomial manner to a maximum as the process becomes progressively idealized. Furthermore, we show that this maximum efficiency is identical to that for single-ring AFC, e.g., 0.7951 for a symmetric, single-lobe input pulse. Thus, we show that CAFC can become more efficient than multiple instances of single-ring AFC. We explain the polynomial convergence of the CAFC efficiency as a consequence of its real analyticity as a function of the process's timescales under our scheme. We leverage this polynomial convergence to model the CAFC efficiency as a simple polynomial in few normalized timescales. We then utilize this polynomial model to predict optimal values for the remaining free parameters and the scaling of the CAFC with the interring detuning.