Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0512130 · New Journal of Physics

Nonlinear response of a driven vibrating nanobeam in the quantum regime

Vittorio Peano, Michael Thorwart

Published Feb 14, 2006• 52 citations
arXivHTMLPDF

Abstract

We analytically investigate the nonlinear response of a damped doubly clamped nanomechanical beam under static longitudinal compression which is excited to transverse vibrations. Starting from a continuous elasticity model for the beam, we consider the dynamics of the beam close to the Euler buckling instability. There, the fundamental transverse mode dominates and a quantum mechanical time-dependent effective single-particle Hamiltonian for its amplitude can be derived. In addition, we include the influence of a dissipative Ohmic or super-Ohmic environment. In the rotating frame, a Markovian master equation is derived which includes also the effect of the time-dependent driving in a non-trivial way. The quasi-energies of the pure system show multiple avoided level crossings corresponding to multiphonon transitions in the resonator. Around the resonances, the master equation is solved analytically using Van Vleck perturbation theory. Their lineshapes are calculated resulting in simple expressions. We find the general solution for the multiple multiphonon resonances and, most interestingly, a bath-induced transition from a resonant to an antiresonant behaviour of the nonlinear response.

DOI
10.1088/1367-2630/8/2/021

Related papers

Macroscopic quantum effects in a strongly driven nanomechanical resonator

Vittorio Peano, Michael Thorwart

We investigate the nonlinear response of a vibrating suspended nanomechanical beam on external periodic driving. The amplitude of the fundamental transverse mode behaves like a weakly damped quantum particle in a driven anharmonic potential. Upon using a Born-Markovian master equation, we calculate the fundamental mode amplitude for varying driving frequencies. In the nonlinear regime, we observe resonances which are absent in the corresponding classical model. They are shown to be associated with resonant multiphonon excitations. Furthermore, we identify resonant tunneling in a dynamically induced bistable effective potential.

arXiv:cond-mat/0407720•Physical Review B•Dec 2, 2004•55 citations

Dissipative dynamics of a qubit coupled to a nonlinear oscillator

Carmen Vierheilig, Johannes Hausinger, Milena Grifoni

We consider the dissipative dynamics of a qubit coupled to a nonlinear oscillator (NO) embedded in an Ohmic environment. By treating the nonlinearity up to first order and applying Van Vleck perturbation theory up to second order in the qubit-NO coupling, we derive an analytical expression for the eigenstates and eigenfunctions of the coupled qubit-NO system beyond the rotating wave approximation. In the regime of weak coupling to the thermal bath, analytical expressions for the time evolution of the qubit's populations are derived: they describe a multiplicity of damped oscillations superposed to a complex relaxation part toward thermal equilibrium. The long-time dynamics is characterized by a single relaxation rate, which is maximal when the qubit is tuned to one of the resonances with the nonlinear oscillator.

arXiv:0907.5535•Physical Review A•Nov 25, 2009•9 citations

Quantum Properties of a Nanomechanical Oscillator

Aziz Kolkıran, G. S. Agarwal

We study the quantum properties of a nanomechanical oscillator via the squeezing of the oscillator amplitude. The static longitudinal compressive force $F_0$ close to a critical value at the Euler buckling instability leads to an anharmonic term in the Hamiltonian and thus the squeezing properties of the nanomechanical oscillator are to be obtained from the Hamiltonian of the form $H= a^†a+β(a^†+a)^4/4$. This Hamiltonian has no exact solution unlike the other known models of nonlinear interactions of the forms $a^{†2}a^2$, $(a^†a)^2$ and $a^{\dag4}+a^4-(a^{\dag2}a^2+a^2a^{\dag2})$ previously employed in quantum optics to study squeezing. Here we solve the Schrödinger equation numerically and show that in-phase quadrature gets squeezed for both ground state and coherent states. The squeezing can be controlled by bringing $F_0$ close to or far from the critical value $F_c$. We further study the effect of the transverse driving force on the squeezing in nanomechanical oscillator.

arXiv:cond-mat/0608621•Aug 29, 2006•0 citations

Generalized master equation for driven quantum oscillators: microscopic origin of nonlinear dissipation and asymmetric resonances

Jakob Wagner, Jeff Maki, Oded Zilberberg +1 more

Driven nonlinear quantum oscillators are a central platform for quantum technologies, yet their dissipative dynamics are typically described using Lindblad or Caldeira-Leggett master equations derived under assumptions that exclude nonlinearities and driving. Here, we derive a generalized Caldeira-Leggett master equation for driven nonlinear oscillators by retaining the full nonlinear and time-dependent system dynamics in the construction of the dissipator. For position- and momentum-dependent system-bath coupling, the dissipator itself becomes dynamically dressed, generating nonlinear and drive-dependent dissipative channels beyond conventional fixed-dissipator approaches. This produces nonlinear damping together with dissipation-induced corrections to the effective drive. The resulting dissipative dynamics suppress large-amplitude excitations and reduce phase-space fluctuations. For a driven Kerr oscillator, this leads to the suppression of bistability, asymmetric resonance responses, and strongly modified fluctuation distributions. More broadly, our results establish a microscopic framework in which nonlinear dynamics and driving directly reshape the dissipative sector of driven open quantum systems.

quant-pharXiv:2605.08021•May 11, 2026•0 citations

Damping and decoherence of a nanomechanical resonator due to a few two-level systems

Laura G. Remus, M. P. Blencowe, Yukihiro Tanaka

We consider a quantum model of a nanomechanical flexing beam resonator interacting with a bath comprising a few damped tunneling two-level systems. In contrast with a resonator interacting bilinearly with an ohmic free oscillator bath (modeling clamping loss, for example), the mechanical resonator damping is amplitude dependent, while the decoherence of quantum superpositions of mechanical position states depends only weakly on their spatial separation.

arXiv:0907.0431•Physical Review B•Nov 4, 2009•57 citations

Amplitude Noise Reduction in a Nano-Mechanical Oscillator

Aziz Kolkıran

We study the quantum properties of a nano-mechanical oscillator via the squeezing of the oscillator amplitude. The static longitudinal compressive force 0 F close to a critical value at the Euler buckling instability leads to an anharmonic term in the Hamiltonian and thus the squeezing properties of the nano-mechanical oscillator are to be obtained from the Hamiltonian of the form H = a+a + β(a+ + a) / 4 . This Hamiltonian has no exact solution unlike the other known models of nonlinear interactions of the forms a+2 a2, (a2a)2 and a+4 + a4 - (a+2 a2 + a2 a+2 ) previously employed in quantum optics to study squeezing. Here we solve the Schrodinger equation numerically and show that in-phase quadrature gets squeezed for both vacuum and coherent states. The squeezing can be controlled by bringing F0 close to or far from the critical value Fc. We further study the effect of the transverse driving force on the squeezing in nano-mechanical oscillator.

Mathematical and Computational Applications•Apr 1, 2011•2 citations
We analytically investigate the nonlinear response of a damped doubly clamped nanomechanical beam under static longitudinal compression which is excited to transverse vibrations. Starting from a continuous elasticity model for the beam, we consider the dynamics of the beam close to the Euler buckling instability. There, the fundamental transverse mode dominates and a quantum mechanical time-dependent effective single-particle Hamiltonian for its amplitude can be derived. In addition, we include the influence of a dissipative Ohmic or super-Ohmic environment. In the rotating frame, a Markovian master equation is derived which includes also the effect of the time-dependent driving in a non-trivial way. The quasi-energies of the pure system show multiple avoided level crossings corresponding to multiphonon transitions in the resonator. Around the resonances, the master equation is solved analytically using Van Vleck perturbation theory. Their lineshapes are calculated resulting in simple expressions. We find the general solution for the multiple multiphonon resonances and, most interestingly, a bath-induced transition from a resonant to an antiresonant behaviour of the nonlinear response.
We investigate the nonlinear response of a vibrating suspended nanomechanical beam on external periodic driving. The amplitude of the fundamental transverse mode behaves like a weakly damped quantum particle in a driven anharmonic potential. Upon using a Born-Markovian master equation, we calculate the fundamental mode amplitude for varying driving frequencies. In the nonlinear regime, we observe resonances which are absent in the corresponding classical model. They are shown to be associated with resonant multiphonon excitations. Furthermore, we identify resonant tunneling in a dynamically induced bistable effective potential.
We consider the dissipative dynamics of a qubit coupled to a nonlinear oscillator (NO) embedded in an Ohmic environment. By treating the nonlinearity up to first order and applying Van Vleck perturbation theory up to second order in the qubit-NO coupling, we derive an analytical expression for the eigenstates and eigenfunctions of the coupled qubit-NO system beyond the rotating wave approximation. In the regime of weak coupling to the thermal bath, analytical expressions for the time evolution of the qubit's populations are derived: they describe a multiplicity of damped oscillations superposed to a complex relaxation part toward thermal equilibrium. The long-time dynamics is characterized by a single relaxation rate, which is maximal when the qubit is tuned to one of the resonances with the nonlinear oscillator.
We study the quantum properties of a nanomechanical oscillator via the squeezing of the oscillator amplitude. The static longitudinal compressive force F0F_0F0​ close to a critical value at the Euler buckling instability leads to an anharmonic term in the Hamiltonian and thus the squeezing properties of the nanomechanical oscillator are to be obtained from the Hamiltonian of the form H=a†a+β(a†+a)4/4H= a^†a+β(a^†+a)^4/4H=a†a+β(a†+a)4/4. This Hamiltonian has no exact solution unlike the other known models of nonlinear interactions of the forms a†2a2a^{†2}a^2a†2a2, (a†a)2(a^†a)^2(a†a)2 and a†4+a4−(a†2a2+a2a†2)a^{\dag4}+a^4-(a^{\dag2}a^2+a^2a^{\dag2})a†4+a4−(a†2a2+a2a†2) previously employed in quantum optics to study squeezing. Here we solve the Schrödinger equation numerically and show that in-phase quadrature gets squeezed for both ground state and coherent states. The squeezing can be controlled by bringing F0F_0F0​ close to or far from the critical value FcF_cFc​. We further study the effect of the transverse driving force on the squeezing in nanomechanical oscillator.
Driven nonlinear quantum oscillators are a central platform for quantum technologies, yet their dissipative dynamics are typically described using Lindblad or Caldeira-Leggett master equations derived under assumptions that exclude nonlinearities and driving. Here, we derive a generalized Caldeira-Leggett master equation for driven nonlinear oscillators by retaining the full nonlinear and time-dependent system dynamics in the construction of the dissipator. For position- and momentum-dependent system-bath coupling, the dissipator itself becomes dynamically dressed, generating nonlinear and drive-dependent dissipative channels beyond conventional fixed-dissipator approaches. This produces nonlinear damping together with dissipation-induced corrections to the effective drive. The resulting dissipative dynamics suppress large-amplitude excitations and reduce phase-space fluctuations. For a driven Kerr oscillator, this leads to the suppression of bistability, asymmetric resonance responses, and strongly modified fluctuation distributions. More broadly, our results establish a microscopic framework in which nonlinear dynamics and driving directly reshape the dissipative sector of driven open quantum systems.
We study the quantum properties of a nano-mechanical oscillator via the squeezing of the oscillator amplitude. The static longitudinal compressive force 0 F close to a critical value at the Euler buckling instability leads to an anharmonic term in the Hamiltonian and thus the squeezing properties of the nano-mechanical oscillator are to be obtained from the Hamiltonian of the form H = a+a + β(a+ + a) / 4 . This Hamiltonian has no exact solution unlike the other known models of nonlinear interactions of the forms a+2 a2, (a2a)2 and a+4 + a4 - (a+2 a2 + a2 a+2 ) previously employed in quantum optics to study squeezing. Here we solve the Schrodinger equation numerically and show that in-phase quadrature gets squeezed for both vacuum and coherent states. The squeezing can be controlled by bringing F0 close to or far from the critical value Fc. We further study the effect of the transverse driving force on the squeezing in nano-mechanical oscillator.