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Physical Review A

Multilevel systems: Simultaneously forbidden combination resonances

V. N. Smelyanskiy, G. W. Ford, R. S. Conti

Published Apr 1, 1996• 8 citations
Publisher

Abstract

A general Floquet approach is used to investigate the nonlinear response of a multilevel quantum system driven by a strong, in general off-resonance, pump field. The response to a weak probe field is studied near the quasienergy interval resonances. A general condition is derived for a phenomenon in which the transition matrix elements for a given pair of Floquet states simultaneously vanish for an infinite number of equally spaced resonances beyond a unique frequency edge. This phenomenon is illustrated with several models involving modulating interactions and possibilities for its observation are analyzed. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physreva.53.2598

Related papers

Simultaneously forbidden resonances in the Autler-Townes effect with a modulated pump

V. N. Smelyanskiy, R. S. Conti, G. W. Ford

It was shown recently that systems subject to a strong modulating interaction can exhibit a new property in their response to a probe field. Under certain conditions an infinite number of resonances are simultaneously forbidden [V. N. Smelyanskiy, G. W. Ford, and R. S. Conti, Phys. Rev. A 53, 2598 (1996)]. In the present paper we investigate this effect in the case of a three-state system in which a strong pump field with a periodic frequency modulation \ensuremath{\Omega} couples a pair of excited levels while the complex Autler-Townes spectrum is probed via a weak field that connects one of the coupled states to the ground state. Under certain conditions a half-infinite comb of spectral lines, spaced by \ensuremath{\Omega}, simultaneously disappear from the Autler-Townes spectrum. These lines are positioned above or below a unique edge frequency, which is that of the probe transition in the absence of the strong field. It is shown that the aforementioned effect results from a special factorization property of the corresponding Floquet Hamiltonian that describes the Autler-Townes spectrum. Detailed analysis of this property is presented. In particular, it is found that the subset of the parameter space of the system where the factorization occurs consists of an infinite number of quasiperiodic manifolds. These manifolds exhibit some universal features related to the degeneracy of the dressed states. The line shapes of the probe resonances near the degeneracy points are derived. The intensities of the probe resonances are investigated in the limit of \ensuremath{\Omega} small compared with the modulation depth and the strength of the pump field. In the latter case, effects related to the avoided crossings of the dressed-state levels are considered. The Floquet Hamiltonian that describes the Autler-Townes spectrum in the case considered, effectively corresponds to a special model of a periodically driven system (with period 2\ensuremath{\pi}/\ensuremath{\Omega}) in which an external perturbation has the form of an operator projecting onto a single-quantum state. We generalize this model to the case of an N-level periodically driven system where the simultaneous vanishing of a half-infinite number of the dressed-state Fourier harmonics are analyzed. Possible experimental tests of the effect are suggested.

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Multiphoton-resonance-induced fluorescence of a strongly driven two-level system under frequency modulation

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We study the fluorescence spectrum of a strongly driven two-level system (TLS) with modulated transition frequency, which is a bichromatically driven TLS and has multiple resonance frequencies. We are aiming to provide a reliable description of the fluorescence in a regime that is difficult to tackle with perturbation theory and the rotating-wave approximation (RWA), and illustrate the spectral features of the fluorescence under off- and multiphoton-resonance conditions. To go beyond the RWA, we use a semianalytical counter-rotating-hybridized rotating-wave method that combines a unitary transformation and Floquet theory to calculate the two-mode Floquet states and quasienergies for the bichromatically driven TLS. We then solve the master equation accounting for the spontaneous decay in the bases of the two-mode Floquet states, and derive a physically transparent fluorescence spectrum. In comparison with the numerically exact spectrum from the generalized Floquet-Liouville approach, the present spectrum is found to be applicable in a wide range of the parameters where the RWA and the secular approximation may break down. We find that the counter-rotating (CR) terms of the transverse field omitted in the RWA have non-negligible contributions to the spectrum under certain conditions. Particularly, at the multiphoton resonance the width of which is comparable with the Bloch-Siegert shift, the RWA and non-RWA spectra markedly differ from each other because of the CR-induced shift. We also analyze the symmetry of the spectrum in terms of the transition matrix elements between the two-mode Floquet states. We show that the strict symmetry of the spectrum cannot be expected without the RWA but the almost symmetric spectrum can be obtained at the single-photon resonance that takes the Bloch-Siegert shift into account if the driving is moderately strong and at the multiphoton resonance with a sufficiently weak transverse field.

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Nonlinear Resonance Effects in Multilevel Quantum Systems Weakly Coupled to a Thermal Reservoir

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Calculations are made to determine the response (power absorbed, magnetization, etc.) of a multilevel quantum system under the influence of any number of classically described driving fields (well defined in amplitude and phase). A nonperturbative theory is developed to account for those fields that are in resonance with the system. The nonresonant effects, such as virtual energy shifts, are included in the theory. We limit ourselves to a discussion of quantum systems that are in weak contact with a thermal reservoir. The procedure is the following A projection-operator formalism is developed which separates the density matrix of a system into two parts. One part is related to those portions of the driving fields that interact quite strongly with the system. The other part is the nonresonant or weakly interacting portion. An exact equation of motion, which includes the effects of the nonresonant part, is derived for the strongly interacting part of the density matrix. It is then shown that when relaxation can be represented by phenomenological decay constants, a unitary transformation can be found which reduces the equation of motion for the strongly interacting part of the density matrix so that an exact solution is possible. This solution is shown to give the major effects of the driving fields and can be used as the zeroth-order solution in a perturbation treatment of the whole density matrix. Frequency shifts such as the Bloch-Siegert shift and shifts due to couplings between the levels of interest (resonance levels) and other levels (nonresonance levels) all arise in the same natural way in our treatment. Our results in many instances are the extensions of the well-known results for a two-level quantum-mechanical systemm to an $N$-level one, and include resonant multiple quantum effects. As an example, a straightforward application of the method is carried out in order to compute the shifts in the resonance frequencies of a four-level system, driven by two oscillating fields connecting three of the levels, due to the counter-rotating components of the driving fields and the presence of the fourth level, which is not coupled to the other levels by the driving fields.

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Resonance and Transition-Channel Competition in Cavity-Assisted Floquet-State Stabilization

Zhuocheng Ma, Hongzheng Zhao, Liang-You Peng

Static coupling to a dissipative cavity has been proposed as a route to autonomously stabilize Floquet eigenstates. The cavity provides an energy-selective relaxation channel: when its photon energy matches a quasienergy difference modulo drive quanta, it mediates transitions between Floquet states, followed by irreversible photon loss. Here, we show that at intermediate driving frequencies, this resonance condition alone is not sufficient for stabilization. Instead, the steady state is controlled by the interplay between resonance selection and the cavity-induced transition strengths between Floquet states, which can either cooperate or compete. This interplay leads to successful stabilization, mixed steady states despite resonance, and complete breakdown near Floquet-zone hybridization. We further show that the same mechanism extends to multilevel spin-S systems, providing a general framework for understanding when static dissipation can reliably stabilize driven quantum states.

quant-pharXiv:2610.39169•Oct 30, 2026
A general Floquet approach is used to investigate the nonlinear response of a multilevel quantum system driven by a strong, in general off-resonance, pump field. The response to a weak probe field is studied near the quasienergy interval resonances. A general condition is derived for a phenomenon in which the transition matrix elements for a given pair of Floquet states simultaneously vanish for an infinite number of equally spaced resonances beyond a unique frequency edge. This phenomenon is illustrated with several models involving modulating interactions and possibilities for its observation are analyzed. \textcopyright{} 1996 The American Physical Society.
It was shown recently that systems subject to a strong modulating interaction can exhibit a new property in their response to a probe field. Under certain conditions an infinite number of resonances are simultaneously forbidden [V. N. Smelyanskiy, G. W. Ford, and R. S. Conti, Phys. Rev. A 53, 2598 (1996)]. In the present paper we investigate this effect in the case of a three-state system in which a strong pump field with a periodic frequency modulation \ensuremath{\Omega} couples a pair of excited levels while the complex Autler-Townes spectrum is probed via a weak field that connects one of the coupled states to the ground state. Under certain conditions a half-infinite comb of spectral lines, spaced by \ensuremath{\Omega}, simultaneously disappear from the Autler-Townes spectrum. These lines are positioned above or below a unique edge frequency, which is that of the probe transition in the absence of the strong field. It is shown that the aforementioned effect results from a special factorization property of the corresponding Floquet Hamiltonian that describes the Autler-Townes spectrum. Detailed analysis of this property is presented. In particular, it is found that the subset of the parameter space of the system where the factorization occurs consists of an infinite number of quasiperiodic manifolds. These manifolds exhibit some universal features related to the degeneracy of the dressed states. The line shapes of the probe resonances near the degeneracy points are derived. The intensities of the probe resonances are investigated in the limit of \ensuremath{\Omega} small compared with the modulation depth and the strength of the pump field. In the latter case, effects related to the avoided crossings of the dressed-state levels are considered. The Floquet Hamiltonian that describes the Autler-Townes spectrum in the case considered, effectively corresponds to a special model of a periodically driven system (with period 2\ensuremath{\pi}/\ensuremath{\Omega}) in which an external perturbation has the form of an operator projecting onto a single-quantum state. We generalize this model to the case of an N-level periodically driven system where the simultaneous vanishing of a half-infinite number of the dressed-state Fourier harmonics are analyzed. Possible experimental tests of the effect are suggested.
A multilevel system driven strongly by several near-resonant and arbitrary modulated optical fields is considered. The orthonormal basis of the time-dependent quasienergy states is introduced to represent in a matrix form the density operator and the operators describing atom-field interaction and relaxation. The above formalism simplifies the density-matrix equations of motion and provides a uniform treatment of resonance phenomena such as probe wave resonances, parametric resonances, and quantum interference. Special consideration is given to the steady-state and transient-time evolution of two- and three-level systems.
We study the fluorescence spectrum of a strongly driven two-level system (TLS) with modulated transition frequency, which is a bichromatically driven TLS and has multiple resonance frequencies. We are aiming to provide a reliable description of the fluorescence in a regime that is difficult to tackle with perturbation theory and the rotating-wave approximation (RWA), and illustrate the spectral features of the fluorescence under off- and multiphoton-resonance conditions. To go beyond the RWA, we use a semianalytical counter-rotating-hybridized rotating-wave method that combines a unitary transformation and Floquet theory to calculate the two-mode Floquet states and quasienergies for the bichromatically driven TLS. We then solve the master equation accounting for the spontaneous decay in the bases of the two-mode Floquet states, and derive a physically transparent fluorescence spectrum. In comparison with the numerically exact spectrum from the generalized Floquet-Liouville approach, the present spectrum is found to be applicable in a wide range of the parameters where the RWA and the secular approximation may break down. We find that the counter-rotating (CR) terms of the transverse field omitted in the RWA have non-negligible contributions to the spectrum under certain conditions. Particularly, at the multiphoton resonance the width of which is comparable with the Bloch-Siegert shift, the RWA and non-RWA spectra markedly differ from each other because of the CR-induced shift. We also analyze the symmetry of the spectrum in terms of the transition matrix elements between the two-mode Floquet states. We show that the strict symmetry of the spectrum cannot be expected without the RWA but the almost symmetric spectrum can be obtained at the single-photon resonance that takes the Bloch-Siegert shift into account if the driving is moderately strong and at the multiphoton resonance with a sufficiently weak transverse field.
This is the second in a series of three papers on nonlinear resonance effects in multilevel quantum systems in which quantum-mechanical systems are considered that are under the influence of any number of classically described driving fields (well defined in amplitude and phase) and are in weak contact with a thermal reservoir, such that relaxation may be described by means of phenomenological decay constants. In the first paper, referred to as I, a projection-operator formalism was developed for separating the equation of motion for the density operator into a part which may be treated by conventional time-dependent perturbation techniques and another part which, because it would give rise to secular terms using a time-dependent perturbation theory, must be treated by more exact means. In this paper three applications of the formalism developed in I are considered. The first application is to a system in which a multiple-quantum transition is taking place between two levels of the system which contains many levels between the two levels under consideration. In particular, the transition probability is calculated for these two levels, which are not directly connected by matrix elements of the interaction but are connected indirectly through intermediate states. The next application considered is the Raman effect in a four-level system. As a final application of the formalism developed in I an expression is derived for the magnetization M+(t)=Mx(t)+iMy(t){M}_{+}(t)={M}_{x}(t)+i{M}_{y}(t)M+​(t)=Mx​(t)+iMy​(t) of a system of NNN noninteracting spins under the influence of a static field in the zzz direction and an arbitrary number of rf fields in the xxx direction. As in I, a phenomenological form is assumed for the relaxation in all of these applications. It is shown in this problem that fields which are not involved in directly causing transitions in this system give rise to frequency shifts in the resonance conditions, which are accounted for in this treatment in a very natural and straightforward manner. For these fields, the rotating as well as the counter-rotating fields give shifts in the resonance condition.
Calculations are made to determine the response (power absorbed, magnetization, etc.) of a multilevel quantum system under the influence of any number of classically described driving fields (well defined in amplitude and phase). A nonperturbative theory is developed to account for those fields that are in resonance with the system. The nonresonant effects, such as virtual energy shifts, are included in the theory. We limit ourselves to a discussion of quantum systems that are in weak contact with a thermal reservoir. The procedure is the following A projection-operator formalism is developed which separates the density matrix of a system into two parts. One part is related to those portions of the driving fields that interact quite strongly with the system. The other part is the nonresonant or weakly interacting portion. An exact equation of motion, which includes the effects of the nonresonant part, is derived for the strongly interacting part of the density matrix. It is then shown that when relaxation can be represented by phenomenological decay constants, a unitary transformation can be found which reduces the equation of motion for the strongly interacting part of the density matrix so that an exact solution is possible. This solution is shown to give the major effects of the driving fields and can be used as the zeroth-order solution in a perturbation treatment of the whole density matrix. Frequency shifts such as the Bloch-Siegert shift and shifts due to couplings between the levels of interest (resonance levels) and other levels (nonresonance levels) all arise in the same natural way in our treatment. Our results in many instances are the extensions of the well-known results for a two-level quantum-mechanical systemm to an NNN-level one, and include resonant multiple quantum effects. As an example, a straightforward application of the method is carried out in order to compute the shifts in the resonance frequencies of a four-level system, driven by two oscillating fields connecting three of the levels, due to the counter-rotating components of the driving fields and the presence of the fourth level, which is not coupled to the other levels by the driving fields.
Static coupling to a dissipative cavity has been proposed as a route to autonomously stabilize Floquet eigenstates. The cavity provides an energy-selective relaxation channel: when its photon energy matches a quasienergy difference modulo drive quanta, it mediates transitions between Floquet states, followed by irreversible photon loss. Here, we show that at intermediate driving frequencies, this resonance condition alone is not sufficient for stabilization. Instead, the steady state is controlled by the interplay between resonance selection and the cavity-induced transition strengths between Floquet states, which can either cooperate or compete. This interplay leads to successful stabilization, mixed steady states despite resonance, and complete breakdown near Floquet-zone hybridization. We further show that the same mechanism extends to multilevel spin-S systems, providing a general framework for understanding when static dissipation can reliably stabilize driven quantum states.