Static Effective Hamiltonian of a Rapidly Driven Nonlinear System.
Jayameenakshi Venkatraman1, Xu Xiao1, Rodrigo G Cortiñas1
1Department of Applied Physics and Physics, Yale University, New Haven, Connecticut 06520, USA.
Physical Review Letters
|September 16, 2022
Summary
We developed a new recursive formula to calculate the static effective Hamiltonian for systems with fast-oscillating drives. This method enhances computational capabilities for both quantum and classical systems in quantum engineering.
Area of Science:
- Physics
- Quantum Mechanics
- Computational Physics
Background:
- Calculating the static effective Hamiltonian for driven systems is crucial in various physics domains.
- Existing time-dependent perturbation methods face limitations in order and computational feasibility.
- Understanding driven quantum and classical systems requires robust analytical tools.
Purpose of the Study:
- To present a novel recursive formula for computing the static effective Hamiltonian of systems subjected to fast-oscillating drives.
- To overcome the limitations of existing perturbation methods in terms of order and computational power.
- To provide a unified framework applicable to both quantum and classical systems.
Main Methods:
- Development of a recursive formula for the static effective Hamiltonian.
- Implementation of the formula for symbolic computation to arbitrary order.
- Introduction of a diagrammatic tool to aid calculations.
- Testing the method with illustrative examples.
Main Results:
- An analytical recursive formula for the static effective Hamiltonian was derived.
- The formula is suitable for arbitrary-order symbolic computation, surpassing previous methods.
- A diagrammatic tool simplifies the calculation process.
- The method demonstrated applicability to both quantum and classical systems.
Conclusions:
- The new recursive formula offers a powerful and versatile tool for analyzing driven systems.
- This approach enables previously intractable computations in quantum and classical physics.
- The method unifies disparate techniques and has direct implications for quantum engineering.
Related Concept Videos
01:21Linear Approximation in Time Domain
120
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
120
01:26Linear Approximation in Frequency Domain
128
Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....
128
01:22Mechanical Systems
265
Mechanical systems are analogous to to electrical networks where springs and masses play similar roles to inductors and capacitors, respectively. A viscous damper in mechanical systems functions similarly to a resistor in electrical networks, dissipating energy. The forces acting on a mass in such systems include an applied force in the direction of motion, counteracted by forces from the spring, a viscous damper, and the mass's acceleration. This interplay of forces is mathematically...
265
01:18Second Order systems II
154
In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
154
01:06Forced Oscillations
6.7K
When an oscillator is forced with a periodic driving force, the motion may seem chaotic. The motions of such oscillators are known as transients. After the transients die out, the oscillator reaches a steady state, where the motion is periodic, and the displacement is determined.
6.7K
01:07Damped Oscillations
5.9K
In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...
Although friction and other non-conservative...
5.9K