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CritPt / Challenge_20_main / Two identical dielectric ellipsoids are confined in two Gaussian optical…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
code template
Code
import numpy as np
from scipy.constants import c
def answer(a, b, rho, k, epsilon_r, P_0, w_0, R):
r"""
Return the values of $\omega_t$ and $g$.
Inputs
----------
a, b: float, The semi-major and semi-minor axes of the ellipsoids, in nm.
rho: float, The mass density of the ellipsoids, in kg/m^3.
k: float, The wave vector of the laser, in 1/m.
epsilon_r: float, The relative permittivity of the ellipsoids, dimensionless.
P_0: float, The power of the laser, in mW.
w_0: float, The beam waist of the laser, in nm.
R: float, The distance between the ellipsoids, in nm.
Outputs
----------
omega_t : float
The torsional frequency $\omega_t$, in 1/s.
g : float
The coupling rate $g$, in 1/s.
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
omega_t = ...
g = ...
# ---------------------------------------------------------------
return omega_t, gproblem description
# Problem setup:
Two identical dielectric ellipsoids are confined in two Gaussian optical tweezers that propagate along the -axis, are separated by a distance along the -axis, and are also linearly polarized along the -axis. The trapping lasers have wave vector , beam waist and power . Each ellipsoid possesses a semi-major axis of length , two equal semi-minor axes of length , a relative permittivity , and a mass density . Assume that the long axes execute small torsional oscillations about the -direction and that the second-quantized Hamiltonian governing this system is
# Main problem:
Derive and with the given parameters.
Plain-text mathematical notation (without MathML)
# Problem setup: Two identical dielectric ellipsoids are confined in two Gaussian optical tweezers that propagate along the z-axis, are separated by a distance R along the x-axis, and are also linearly polarized along the x-axis. The trapping lasers have wave vector k, beam waist w₀ and power P₀. Each ellipsoid possesses a semi-major axis of length a, two equal semi-minor axes of length b, a relative permittivity ϵ_(r), and a mass density ρ. Assume that the long axes execute small torsional oscillations about the x-direction and that the second-quantized Hamiltonian governing this system is H=ℏω_(t)(a₁^(†)a₁+a₂^(†)a₂)+ℏg(a₁^(†)a₂+a₁a₂^(†)). # Main problem: Derive ω_(t) and g with the given parameters.
Original LaTeX notation
# Problem setup:
Two identical dielectric ellipsoids are confined in two Gaussian optical tweezers that propagate along the $z$-axis, are separated by a distance $R$ along the $x$-axis, and are also linearly polarized along the $x$-axis. The trapping lasers have wave vector $k$, beam waist $w_0$ and power $P_0$. Each ellipsoid possesses a semi-major axis of length $a$, two equal semi-minor axes of length $b$, a relative permittivity $\epsilon_r$, and a mass density $\rho$. Assume that the long axes execute small torsional oscillations about the $x$-direction and that the second-quantized Hamiltonian governing this system is
$$H = \hbar {\omega _t}\left( {a_1^\dagger {a_1} + a_2^\dagger {a_2}} \right) + \hbar g\left( {a_1^\dagger {a_2} + {a_1}a_2^\dagger } \right).$$
# Main problem:
Derive $\omega_t$ and $g$ with the given parameters.Discussion
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