Generation of coherence in an exactly solvable nonlinear nanomechanical system
At a Glance
Section titled “At a Glance”| Metadata | Details |
|---|---|
| Publication Date | 2020-03-16 |
| Journal | Physical review. B./Physical review. B |
| Authors | Abhayveer Singh, L. Chotorlishvili, Saurabha Srivastava, I. Tralle, Z. Toklikishvili |
| Institutions | Banaras Hindu University, Tbilisi State University |
| Citations | 14 |
| Analysis | Full AI Review Included |
Executive Summary
Section titled “Executive Summary”This theoretical study investigates the quantum dynamics and coherence generation in a hybrid system comprising a Nitrogen-Vacancy (NV) center spin coupled to a driven, nonlinear nanomechanical oscillator (NEMS cantilever).
- Core Achievement: The system’s complex nonlinear dynamics were transformed into an exactly solvable model based on the Mathieu-Schrödinger equation, allowing for analytical solutions of the spin dynamics.
- Coherence Generation: Unitary evolution protocols were analyzed, demonstrating that coherence generation is maximized when the system is initially prepared in a mixed state corresponding to the classically chaotic region (the vicinity of the homoclinic tangle).
- Quantum Chaos Signature: The study proves that quantum chaos, characterized by minimal quantum distance from the homoclinic tangle, favors the efficient generation of quantum coherence.
- Dissipation Analysis: Both Markovian (Lindblad master equation) and non-Markovian (13C nuclear spin bath) dissipation effects were modeled, showing that even low decoherence rates significantly affect long-term dynamics, purity, and entropy.
- Hybrid System Control: The model provides a framework for controlling the NV spin state (switching between |0> and |1>) by steering the parameters of the mechanical driving term.
- Engineering Relevance: This work establishes a theoretical basis for designing robust quantum hybrid devices that leverage controlled chaotic dynamics to enhance quantum information processing capabilities, particularly coherence.
Technical Specifications
Section titled “Technical Specifications”| Parameter | Value | Unit | Context |
|---|---|---|---|
| NV Center Frequency (ωr/2π) | 5 | MHz | Intrinsic spin frequency. |
| Rabi Frequency (ΩR/2π) | 0.1 - 10 | MHz | Range of microwave driving frequency. |
| Detuning (δ/2π) | 1 | kHz | Detuning between microwave and intrinsic spin frequency. |
| Cantilever Mass (m) | 6 x 10-17 | kg | Mass of the nanomechanical cantilever. |
| Coupling Constant (k/2π) | 100 | kHz | Cantilever stiffness/spring constant. |
| Zero Point Fluctuation Amplitude (a0) | ~5 x 10-3 | m | Amplitude of zero point fluctuations. |
| Energy Scale (εV) | ~10-9 | J | Energy scale defined by nonlinear terms (ωr2ma03). |
| Time Scale (t) | ~1 | µs | Order of magnitude for system dynamics. |
| Spin-Oscillator Coupling (Q) | 0.5 to 25 | (Dimensionless) | Interaction strength used in simulations. |
| Damping Constant (γ) | 0.01 to 0.04 | (Unit of ω0-1) | Markovian dephasing parameter used in simulations. |
| Purity (P) | 0.5 to 1.0 | (Dimensionless) | Purity range during Markovian evolution. |
| Von Neumann Entropy (S) | 0 to 1.0 | (Dimensionless) | Entropy range during Markovian evolution (0 = pure, 1 = mixed). |
Key Methodologies
Section titled “Key Methodologies”The study employed a rigorous theoretical approach involving multiple transformations and quantum mechanical models to achieve analytical solutions for the hybrid system dynamics:
- Hamiltonian Definition: The total system Hamiltonian was defined, including the NV spin (Hs), the linear oscillator (H0), the nonlinear oscillator terms (HNL), external periodic driving (V(x, t)), and the spin-oscillator coupling term (g cos(ωt)xSz).
- Classical Transformation to Action-Angle Variables: The classical equation of motion for the nonlinear cantilever was analyzed using perturbation theory and transformed into canonical action-angle (I, φ) variables, simplifying the description of the nonlinear frequency Ω(I).
- Mathieu-Schrödinger Equation Mapping: The cantilever part of the Hamiltonian (Hm) was mapped onto the Hamiltonian of a mathematical pendulum, which corresponds directly to the analytically solvable Mathieu-Schrödinger equation. This provided the eigenfunctions (Mathieu functions c en, s en) and eigenvalues (characteristic values an, bn) for the mechanical subsystem.
- Unitary Spin Dynamics: The total Hamiltonian was expressed in the joint basis of the pendulum eigenfunctions and the Pauli spin matrix basis. The time evolution of the spin state was calculated by solving the Schrödinger equation, yielding the density matrix ρAB(t).
- Markovian Dissipation Model: The system’s evolution under environmental coupling was modeled using the Liouville-von Neumann-Lindblad master equation, allowing for analytical solutions for the density matrix elements, purity (P), and von Neumann entropy (S).
- Non-Markovian Noise Model: Decoherence due to the 13C nuclear spin bath was modeled by coupling the NV spin to N independent bosonic reservoirs, calculating the time-dependent coefficients using the correlation function Sn(t - t1).
- Coherence Quantification: Quantum coherence generation under unitary evolution was quantified using the relative entropy measure, C(ρ(t)|ρd), where ρd is the diagonal part of the propagated density matrix.
Commercial Applications
Section titled “Commercial Applications”The findings regarding robust coherence generation in hybrid NEMS-NV systems are highly relevant for several advanced technological sectors:
- Quantum Computing: NV centers are leading solid-state qubits. The ability to generate and sustain coherence, even in the presence of nonlinear mechanical coupling and noise, is crucial for building scalable quantum processors.
- Quantum Information Transfer: NEMS cantilevers can act as mechanical quantum buses, transferring information between distant NV qubits. Maximizing coherence during this transfer is essential for high-fidelity quantum networks.
- High-Sensitivity Quantum Sensing: Hybrid NV-NEMS systems are used for ultra-sensitive magnetic field, force, and acceleration sensing. Understanding and controlling the spin dynamics in the chaotic regime could lead to new sensing modalities or enhanced sensitivity limits.
- Hybrid Quantum Systems Development: This research supports the engineering of complex hybrid devices that integrate mechanical, optical, and spin degrees of freedom, which are foundational for next-generation quantum technologies.
- Chaotic System Engineering: The demonstration that controlled quantum chaos (near the homoclinic tangle) enhances coherence provides a novel design principle for quantum devices, moving beyond traditional linear system optimization.
View Original Abstract
This study is focused on the quantum dynamics of a nitrogen-vacancy (NV) center coupled to a nonlinear, periodically driven mechanical oscillator. For a continuous periodic driving that depends on the position of the oscillator, the mechanical motion is described by Mathieu elliptic functions. This solution is employed to study the dynamics of the quantum spin system including environmental effects and to evaluate the purity and the von Neumann entropy of the NV-spin. The unitary generation of coherence is addressed. We observe that the production of coherence through a unitary transformation depends on whether the system is prepared initially in mixed state. Production of coherence is efficient when the system initially is prepared in the region of the separatrix (i.e., the region where classical systems exhibit dynamical chaos). From the theory of dynamical chaos, we know that phase trajectories of the system passing through the homoclinic tangle have limited memory, and therefore the information about the initial conditions is lost. We proved that quantum chaos and diminishing of information about the mixed initial state favors the generation of quantum coherence through the unitary evolution. We introduced quantum distance from the homoclinic tangle and proved that for the initial states permitting efficient generation of coherence, this distance is minimal.