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arXiv:cond-mat/0608621

Quantum Properties of a Nanomechanical Oscillator

Aziz Kolkıran, G. S. Agarwal

Published Aug 29, 2006• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0608621

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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.
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.
Mechanical qubits offer unique advantages over other qubit platforms, primarily in terms of coherence time and possibilities for enhanced sensing applications, but their potential is constrained by the inherently weak nonlinearities and small anharmonicity of nanomechanical resonators. We propose overcoming this shortcoming by using squeezed-Fock states of phonons in a parametrically driven nonlinear mechanical oscillator. We find that, under two-phonon driving, squeezed-Fock states become eigenstates of a Kerr-nonlinear mechanical oscillator, featuring an energy spectrum with exponentially enhanced and tunable anharmonicity, such that the transitions to higher energy states are exponentially suppressed. This enables us to encode the mechanical qubit within the ground and first excited squeezed-Fock states of the driven mechanical oscillator. This kind of mechanical qubit is termed mechanical squeezed-Fock qubit. We also show that our mechanical qubit can serve as a quantum sensor for weak forces, with its resulting sensitivity increased by at least one order of magnitude over that of traditional mechanical qubits. The proposed mechanical squeezed-Fock qubit provides a powerful quantum phonon platform for quantum sensing and information processing.
Quantum squeezing in macroscopic oscillator systems plays a critical role in bridging quantum mechanics with classical-scale phenomena, enabling high-precision measurements and fundamental tests of quantum physics. In this work, we investigate the effect of squeezing on the phonon state in a hybrid macroscopic mechanical system consisting of an ensemble of Rydberg atoms coupled to two macroscopic mechanical oscillators. We notice that the dipole–dipole coupling between atoms and mechanical oscillators can be transferred to the indirectly coupled mechanical interaction, and the nonlinear effective Hamiltonian can be solved to generate a squeezed effect on the mechanical mode. We also discuss the noise effects induced by amplitude and phase fluctuations on the squeezed quadratures of the system.
We present a comprehensive quantum framework to analyze the dynamics of a nanomechanical cantilever subjected to combined additive and parametric noise. Using the Wigner phase-space representation, we derive and numerically solve c-number stochastic differential equations that accurately capture the evolution of the quantum state under realistic noise conditions. This operator-to-c-number mapping makes the role of quadrature noise sources explicit and provides a transparent route to evaluating squeezing. Our simulations reveal distinct dynamical regimes, demonstrating a critical trade-off determined by the parametric drive strength g0. In the stable regime, parametric modulation suppresses quantum fluctuations, achieving squeezing of about −1.7 dB relative to the vacuum noise limit. For stronger parametric driving, transient squeezing levels approaching −5 dB are observed, but these gains are accompanied by rapid growth of the conjugate quadrature and eventual instability. We further connect these dynamics to effective quality factor enhancement, clarifying the implications for nano-cantilever sensor performance. An explicit check of the Heisenberg uncertainty relation confirms that our simulations remain physically consistent. Together, these results establish a unified framework that links fundamental quantum noise dynamics with practical benchmarks for quantum-enhanced sensing.
The quantum dynamics of a coherently driven nonlinear oscillator are studied by solving the Schrödinger equation numerically. The results are used to examine the effect of increasingly strong driving on the mean number, squeezing, the second-order correlation function and the general behaviour of the Q function. For comparison, the corresponding quantities for the classical Liouville dynamics are also computed. It is found that while the dynamics do maintain some quantum aspects as the strength of the driving is increased, the spectacular quantum features of the undriven nonlinear oscillator are in fact progressively lost.
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.