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%matplotlib inline
import matplotlib.pyplot as plt
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import numpy as np
In [3]:
from qutip import *
Model and parameters¶
We use the Jaynes-Cumming model of a single two-level atom interacting with a single-mode cavity via a dipole interaction and under the rotating wave approximation.
In [4]:
kappa = 2
gamma = 0.2
g = 5
wc = 0
w0 = 0
wl = 0
N = 5
E = 0.5
tlist = np.linspace(0,10.0,500)
Correlation functions and spectrum using eseries¶
In [5]:
def probcorr(E,kappa,gamma,g,wc,w0,wl,N):
#
# returns the two-time correlation and covariance of the intracavity
# field as exponential series for the problem of a coherently driven
# cavity with a two-level atom
#
# E = amplitude of driving field, kappa = mirror coupling,
# gamma = spontaneous emission rate, g = atom-field coupling,
# wc = cavity frequency, w0 = atomic frequency, wl = driving field frequency,
# N = size of Hilbert space for intracavity field (zero to N-1 photons)
#
# Hamiltonian
a = tensor(destroy(N),qeye(2))
sm = tensor(qeye(N),destroy(2))
H = (w0-wl)*sm.dag()*sm + (wc-wl)*a.dag()*a + 1j*g*(a.dag()*sm - sm.dag()*a) \
+ E*(a.dag()+a)
# collapse operators
C1 = np.sqrt(2*kappa) * a
C2 = np.sqrt(gamma) * sm.dag()
c_op_list = [C1, C2]
# Calculate the Liouvillian
L = liouvillian(H, c_op_list)
# Find steady state density matrix and field
rhoss = steadystate(L);
ass = expect(a,rhoss);
# Initial condition for regression theorem
arho = a*rhoss;
# Solve differential equation with this initial condition
solES = ode2es(L,arho);
# Find trace(a' * solution)
corrES = expect(a.dag(),solES);
# Calculate the covariance by subtracting product of means
covES = corrES - np.real(np.conjugate(ass) * ass)
return corrES, covES
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corrES, covES = probcorr(E, kappa, gamma, g, wc, w0, wl, N)
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corr= esval(corrES, tlist)
cov = esval(covES, tlist)
Correlation function¶
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fig, ax = plt.subplots(figsize=(12, 6))
ax.plot(tlist, np.real(corr))
ax.plot(tlist, np.real(cov))
ax.set_xlabel('Time')
ax.set_ylabel('Correlation and covariance')
ax.legend(("corr", "cov"));
Spectrum¶
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wlist = np.linspace(-10,10,200)
Ps = esspec(covES, wlist)
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fig, ax = plt.subplots(figsize=(12, 6))
ax.plot(wlist, np.real(Ps))
ax.set_xlabel('Frequency')
ax.set_ylabel('Spectral amplitude');
Comparison: Correlation function using correlation_ss with mesolve and eseries¶
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def calc_correlation(E, kappa, gamma, g, wc, w0, wl, N, tlist):
#
# returns the two-time correlation of the intracavity field as exponential
# series for the problem of a coherently driven cavity with a two-level atom
#
# E = amplitude of driving field, kappa = mirror coupling,
# gamma = spontaneous emission rate, g = atom-field coupling,
# wc = cavity frequency, w0 = atomic frequency, wl = driving field frequency,
# N = size of Hilbert space for intracavity field (zero to N-1 photons)
#
# Define cavity field and atomic operators
a = tensor(destroy(N),qeye(2))
sm = tensor(qeye(N),sigmam())
# Hamiltonian
H = (w0-wl)*sm.dag()*sm + (wc-wl)*a.dag()*a + 1j*g*(a.dag()*sm - sm.dag()*a) \
+ E*(a.dag()+a)
# collapse operators
C1 = np.sqrt(2*kappa)*a
C2 = np.sqrt(gamma)*sm.dag()
A = a
corr_ode = correlation_2op_1t(H, None, tlist, [C1, C2], A.dag(), A, solver="me")
corr_es = correlation_2op_1t(H, None, tlist, [C1, C2], A.dag(), A, solver="es")
print("real corr at 0 [me]:", corr_ode[0])
print("real corr at 0 [es] :", corr_es[0])
return corr_ode, corr_es
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corr1, corr2 = calc_correlation(E, kappa, gamma, g, wc, w0, wl, N, tlist)
real corr at 0 [me]: (0.04995548405912743+0j) real corr at 0 [es] : (0.04995548405912682+0j)
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fig, ax = plt.subplots(figsize=(12, 6))
ax.plot(tlist, np.real(corr1),
tlist, np.real(corr2))
ax.set_xlabel('Time')
ax.set_ylabel(r'Correlation $\langle a^\dag(t)a(0)\rangle$')
ax.legend(("me", "es"));
Correlation functions and spectrum at two different coupling strengths¶
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def calc_spectrum(N, wc, wa, g, kappa, gamma, tlist, wlist):
# Hamiltonian
a = tensor(destroy(N), qeye(2))
sm = tensor(qeye(N), destroy(2))
H = wc * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() * sm + a * sm.dag())
# collapse operators
c_op_list = []
n_th_a = 0.5
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)
A = a.dag() + a
B = A
# calculate the power spectrum
corr = correlation_2op_1t(H, None, tlist, c_op_list, A, B, solver='es')
# calculate the power spectrum
spec = spectrum(H, wlist, c_op_list, A, B)
return corr, spec
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N = 4 # number of cavity fock states
wc = 1.00 * 2 * np.pi # cavity frequency
wa = 1.00 * 2 * np.pi # atom frequency
g = 0.10 * 2 * np.pi # coupling strength
kappa = 1.0 # cavity dissipation rate
gamma = 0.2 # atom dissipation rate
wlist = np.linspace(0, 4 * np.pi, 200)
tlist = np.linspace(0, 15, 500)
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corr1, spec1 = calc_spectrum(N, wc, wa, g, kappa, gamma, tlist, wlist)
corr2, spec2 = calc_spectrum(N, wc, wa, 0, kappa, gamma, tlist, wlist)
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fig, ax = plt.subplots(figsize=(12,6))
ax.plot(tlist, np.real(corr1),
tlist, np.real(corr2))
ax.set_xlabel('Time')
ax.set_ylabel('Correlation')
ax.legend(("g = 0.1", "g = 0.0"));
In [18]:
fig, ax = plt.subplots(figsize=(12,6))
ax.plot(wlist/(2*np.pi), abs(spec1),
wlist/(2*np.pi), abs(spec2))
ax.set_xlabel('Frequency')
ax.set_ylabel('Power spectrum')
ax.legend(("g = 0.1", "g = 0.0"));
Versions¶
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from qutip.ipynbtools import version_table
version_table()
Out[19]:
| Software | Version |
|---|---|
| QuTiP | 4.4.0.dev0+ce05217 |
| Numpy | 1.15.2 |
| SciPy | 1.1.0 |
| matplotlib | 3.0.0 |
| Cython | 0.28.5 |
| Number of CPUs | 2 |
| BLAS Info | INTEL MKL |
| IPython | 6.5.0 |
| Python | 3.5.4 |Anaconda, Inc.| (default, Oct 5 2017, 08:00:22) [GCC 7.2.0] |
| OS | posix [linux] |
| Sun Jul 21 08:58:20 2019 PDT | |
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