Complex Representation of Dynamics of Coupled Nonlinear Oscillators

Brad R. Trees, Elizabeth Segelken

L.S. Lima, L. G. de Almeida Arruda

Abstract The Feynman-Vernon influence functional formalism which is adequate in the study of the dissipative dynamics, is used to describe weakly coupled circuits. The classical behavior of the system formed by resistance, electric and magnetic polarizations is characterized by the influence functional approach whose behavior although is linear in some sense, is not representable by systems of perfect oscillators. In addition, linear combinations of complex q-exponentials are used in representation of nonlinear circuits in which the solution of the correspondent nonlinear differentials equations are mapped in modified differentials equations for a new deformed variable.

Victor Nikolaevich Popov, Vladimir Sergeevich Yarunin

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.

Chi‐Fai Lo

Kazi Tahsin Mahmood, M. Afridi Hasan, Abrar N.-E. Faiaz +4 more

Abstract Vibration responses from nonlinear mechanical systems exhibit rich dynamical structure that can be utilized for information encoding and processing. We demonstrate that such structures can be used to encode and manipulate information in a manner analogous to multi-qubit systems. By using a coupled mass and conical spring oscillator, we reveal that distinct harmonic segments of the nonlinear response can be projected onto modal eigenstates to form two-level elastic-bit subsystems, which are analogous to qubits. These bits arise from measurable amplitudes and phase relationships across the Fourier spectrum and evolve deterministically under steady-state excitation. By combining multiple spectral segments within a single oscillator, we achieve two-bit and three-bit states that occupy four- and eight-dimensional Hilbert spaces, respectively. The time dependence of the complex modal coefficients yields intrinsic transformations that act as phase and rotation type gates. The temporal evolution of the complex modal coefficients results in phase accumulation and a rotation-like evolution within this state space. To characterize how the system moves between experimentally observed logical states at different times, we derive a Householder reflection that yields the exact Hermitian and unitary operator connecting these states. This unitary transformation is subsequently decomposed into sequences of analogous quantum gates, providing a representation of the observed modal evolution in terms of familiar multi-qubit logic primitives. This spectral-encoding approach enables scalable state construction within a single mechanical platform, establishing a pathway toward room-temperature mechanical computation based on deterministic nonlinear dynamics.