# Lecture 13: Resonance flourescence¶

Author: J. R. Johansson ([email protected]), http://dml.riken.jp/~rob/

The latest version of this IPython notebook lecture is available at http://github.com/jrjohansson/qutip-lectures.

The other notebooks in this lecture series are indexed at http://jrjohansson.github.com.

In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np

In [2]:
from qutip import *


## Introduction¶

$\displaystyle H_L = -\frac{\Omega}{2}(\sigma_+ + \sigma_-)$

$\displaystyle \frac{d}{dt}\rho = -i[H_L, \rho] + \gamma_0(N+1)\left(\sigma_-\rho(t)\sigma_+ - \frac{1}{2}\sigma_+\sigma_-\rho(t) - \frac{1}{2}\rho(t)\sigma_+\sigma_-\right) + \gamma_0 N \left(\sigma_+\rho(t)\sigma_- - \frac{1}{2}\sigma_-\sigma_+\rho(t) - \frac{1}{2}\rho(t)\sigma_-\sigma_+\right)$

### Problem definition in QuTiP¶

In [3]:
Omega = 1.0 * 2 * pi

In [4]:
gamma0 = 0.05
w_th = 0.0
N = n_thermal(Omega, w_th)

In [5]:
def system_spec(Omega, gamma0, N):
HL = -0.5 * Omega * (sigmap() + sigmam())
c_ops = [sqrt(gamma0 * (N + 1)) * sigmam(), sqrt(gamma0 * N) * sigmap()]
return HL, c_ops

In [6]:
HL, c_ops = system_spec(Omega, gamma0, N)

In [7]:
e_ops = [sigmax(), sigmay(), sigmaz(), sigmam(), sigmap(), num(2)]

In [8]:
psi0 = basis(2, 0)

In [9]:
tlist = np.linspace(0, 20/(2*pi), 200)
result = mesolve(HL, psi0, tlist, c_ops, e_ops)

In [10]:
fig, axes = plt.subplots(2, 1, figsize=(12, 6), sharex=True)

axes[0].plot(result.times, result.expect[0], 'r', label=r'$\langle\sigma_x\rangle$')
axes[0].plot(result.times, result.expect[1], 'g', label=r'$\langle\sigma_y\rangle$')
axes[0].plot(result.times, result.expect[2], 'b', label=r'$\langle\sigma_z\rangle$')
axes[0].legend()
axes[0].set_ylim(-1, 1);

axes[1].plot(result.times, result.expect[5], 'b', label=r'$P_e$')

#axes[1].set_ylabel(r'$\langle\sigma_z\rangle$', fontsize=16)
axes[1].set_xlabel("time", fontsize=16)
axes[1].legend()
axes[1].set_ylim(0, 1);

In [11]:
fig, ax = plt.subplots(1, 1, figsize=(12, 6), sharex=True)

for idx, gamma0 in enumerate([0.1 * Omega, 0.5 * Omega, 1.0 * Omega]):

HL, c_ops = system_spec(Omega, gamma0, N)
result = mesolve(HL, psi0, tlist, c_ops, e_ops)

ax.plot(result.times, result.expect[5], 'b', label=r'$\langle\sigma_z\rangle$')

ax.set_ylim(0, 1);

In [12]:
fig, ax = plt.subplots(1, 1, figsize=(12, 6), sharex=True)

for idx, gamma0 in enumerate([0.1 * Omega, 0.5 * Omega, 1.0 * Omega]):

HL, c_ops = system_spec(Omega, gamma0, N)
result = mesolve(HL, psi0, tlist, c_ops, e_ops)

ax.plot(result.times, imag(result.expect[4]), label=r'im $\langle\sigma_+\rangle$')

ax.set_ylim(-.5, 0.5);

In [13]:
fig, axes = plt.subplots(1, 2, figsize=(12, 6))

taulist = np.linspace(0, 100, 10000)

for idx, gamma0 in enumerate([2 * Omega, 0.5 * Omega, 0.25 * Omega]):

HL, c_ops = system_spec(Omega, gamma0, N)
corr_vec = correlation_2op_1t(HL, None, taulist, c_ops, sigmap(), sigmam())
w, S = spectrum_correlation_fft(taulist, corr_vec)

axes[0].plot(taulist, corr_vec, label=r'$<\sigma_+(\tau)\sigma_-(0)>$')
axes[1].plot(-w / (gamma0), S, 'b', label=r'$S(\omega)$')
axes[1].plot( w / (gamma0), S, 'b', label=r'$S(\omega)$')

axes[0].set_xlim(0, 10)
axes[1].set_xlim(-5, 5);

/usr/lib/python3/dist-packages/numpy/core/numeric.py:460: ComplexWarning: Casting complex values to real discards the imaginary part
return array(a, dtype, copy=False, order=order)


### Software versions¶

In [14]:
from qutip.ipynbtools import version_table; version_table()

Out[14]:
SoftwareVersion
Cython0.20.1post0
OSposix [linux]
Numpy1.8.1
QuTiP3.0.0.dev-5a88aa8
Python3.4.1 (default, Jun 9 2014, 17:34:49) [GCC 4.8.3]
matplotlib1.3.1
IPython2.0.0
SciPy0.13.3
Thu Jun 26 14:11:39 2014 JST