J.R. Johansson and P.D. Nation
This ipython notebook demonstrates how to simulate the quantum vacuum rabi oscillations in the Jaynes-Cumming model, using QuTiP: The Quantum Toolbox in Python.
For more information about QuTiP see project web page: http://code.google.com/p/qutip/
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%matplotlib inline
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import matplotlib.pyplot as plt
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import numpy as np
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from qutip import *
The Jaynes-Cumming model is the simplest possible model of quantum mechanical light-matter interaction, describing a single two-level atom interacting with a single electromagnetic cavity mode. The Hamiltonian for this system is (in dipole interaction form)
$H = \hbar \omega_c a^\dagger a + \frac{1}{2}\hbar\omega_a\sigma_z + \hbar g(a^\dagger + a)(\sigma_- + \sigma_+)$
or with the rotating-wave approximation
$H_{\rm RWA} = \hbar \omega_c a^\dagger a + \frac{1}{2}\hbar\omega_a\sigma_z + \hbar g(a^\dagger\sigma_- + a\sigma_+)$
where $\omega_c$ and $\omega_a$ are the frequencies of the cavity and atom, respectively, and $g$ is the interaction strength.
In [5]:
wc = 1.0 * 2 * np.pi # cavity frequency
wa = 1.0 * 2 * np.pi # atom frequency
g = 0.05 * 2 * np.pi # coupling strength
kappa = 0.005 # cavity dissipation rate
gamma = 0.05 # atom dissipation rate
N = 15 # number of cavity fock states
n_th_a = 0.0 # temperature in frequency units
use_rwa = True
tlist = np.linspace(0,25,100)
In [6]:
# intial state
psi0 = tensor(basis(N,0), basis(2,1)) # start with an excited atom
# operators
a = tensor(destroy(N), qeye(2))
sm = tensor(qeye(N), destroy(2))
# Hamiltonian
if use_rwa:
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() * sm + a * sm.dag())
else:
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() + a) * (sm + sm.dag())
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c_op_list = []
rate = kappa * (1 + n_th_a)
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * a)
rate = kappa * n_th_a
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * a.dag())
rate = gamma
if rate > 0.0:
c_op_list.append(np.sqrt(rate) * sm)
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output = mesolve(H, psi0, tlist, c_op_list, [a.dag() * a, sm.dag() * sm])
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fig, ax = plt.subplots(figsize=(8,5))
ax.plot(tlist, output.expect[0], label="Cavity")
ax.plot(tlist, output.expect[1], label="Atom excited state")
ax.legend()
ax.set_xlabel('Time')
ax.set_ylabel('Occupation probability')
ax.set_title('Vacuum Rabi oscillations');
In [10]:
from qutip.ipynbtools import version_table
version_table()
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