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The angular frequency of the underdamped harmonic oscillator is given by =, the exponential decay of the underdamped harmonic oscillator is given by =. The Q factor of a damped oscillator is defined as Q = 2 π × energy stored energy lost per cycle . {\displaystyle Q=2\pi \times {\frac {\text{energy stored}}{\text{energy lost per cycle}}}.}
The Q factor is a parameter that describes the resonance behavior of an underdamped harmonic oscillator (resonator). Sinusoidally driven resonators having higher Q factors resonate with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around that frequency for which they resonate; the range of frequencies for which the oscillator resonates is called the ...
Here ¯ is the mean number of excitations in the reservoir damping the oscillator and γ is the decay rate. To model the quantum harmonic oscillator Hamiltonian with frequency ω c {\displaystyle \omega _{c}} of the photons, we can add a further unitary evolution:
The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient:
The simplest description of this decay process can be illustrated by oscillation decay of the harmonic oscillator. Damped oscillators are created when a resistive force is introduced, which is dependent on the first derivative of the position, or in this case velocity.
The circuit forms a harmonic oscillator for current, and resonates in a manner similar to an LC circuit. Introducing the resistor increases the decay of these oscillations, which is also known as damping. The resistor also reduces the peak resonant frequency.
The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems. [2] For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well (for an early reference, see e.g. Schiff's textbook [3]). The coherent state describes a state ...
This plot corresponds to solutions of the complete Langevin equation for a lightly damped harmonic oscillator, obtained using the Euler–Maruyama method. The left panel shows the time evolution of the phase portrait at different temperatures. The right panel captures the corresponding equilibrium probability distributions.