QuTiP lecture: Single-Atom-Lasing

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]:
# setup the matplotlib graphics library and configure it to show 
# figures inline in the notebook
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
In [2]:
# make qutip available in the rest of the notebook
from qutip import *

from IPython.display import Image

Introduction and model

Consider a single atom coupled to a single cavity mode, as illustrated in the figure below. If there atom excitation rate $\Gamma$ exceeds the relaxation rate, a population inversion can occur in the atom, and if coupled to the cavity the atom can then act as a photon pump on the cavity.

In [3]:
Image(filename='images/schematic-lasing-model.png')
Out[3]:

The coherent dynamics in this model is described by the Hamiltonian

$H = \hbar \omega_0 a^\dagger a + \frac{1}{2}\hbar\omega_a\sigma_z + \hbar g\sigma_x(a^\dagger + a)$

where $\omega_0$ is the cavity energy splitting, $\omega_a$ is the atom energy splitting and $g$ is the atom-cavity interaction strength.

In addition to the coherent dynamics the following incoherent processes are also present:

  1. $\kappa$ relaxation and thermal excitations of the cavity,
  2. $\Gamma$ atomic excitation rate (pumping process).

The Lindblad master equation for the model is:

$\frac{d}{dt}\rho = -i[H, \rho] + \Gamma\left(\sigma+\rho\sigma- - \frac{1}{2}\sigma-\sigma+\rho - \frac{1}{2}\rho\sigma-\sigma+\right)

  • \kappa (1 + n_{\rm th}) \left(a\rho a^\dagger - \frac{1}{2}a^\dagger a\rho - \frac{1}{2}\rho a^\dagger a\right)
  • \kappa n_{\rm th} \left(a^\dagger\rho a - \frac{1}{2}a a^\dagger \rho - \frac{1}{2}\rho a a^\dagger\right)$

in units where $\hbar = 1$.

References:

Problem parameters

In [5]:
w0 = 1.0  * 2 * pi  # cavity frequency
wa = 1.0  * 2 * pi  # atom frequency
g  = 0.05 * 2 * pi  # coupling strength

kappa = 0.04        # cavity dissipation rate
gamma = 0.00        # atom dissipation rate
Gamma = 0.35        # atom pump rate

N = 50              # number of cavity fock states
n_th_a = 0.0        # avg number of thermal bath excitation

tlist = np.linspace(0, 150, 101)

Setup the operators, the Hamiltonian and initial state

In [6]:
# intial state
psi0 = tensor(basis(N,0), basis(2,0)) # start without excitations

# operators
a  = tensor(destroy(N), qeye(2))
sm = tensor(qeye(N), destroy(2))
sx = tensor(qeye(N), sigmax())

# Hamiltonian
H = w0 * a.dag() * a + wa * sm.dag() * sm + g * (a.dag() + a) * sx
In [7]:
H
Out[7]:
Quantum object: dims = [[50, 2], [50, 2]], shape = [100, 100], type = oper, isherm = True\begin{equation*}\left(\begin{array}{*{11}c}0.0 & 0.0 & 0.0 & 0.314 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 6.283 & 0.314 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.314 & 6.283 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.314 & 0.0 & 0.0 & 12.566 & 0.444 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.444 & 12.566 & \cdots & 0.0 & 0.0 & 0.0 & 0.0 & 0.0\\\vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \vdots\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 301.593 & 2.177 & 0.0 & 0.0 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 2.177 & 301.593 & 0.0 & 0.0 & 2.199\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 307.876 & 2.199 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 0.0 & 2.199 & 307.876 & 0.0\\0.0 & 0.0 & 0.0 & 0.0 & 0.0 & \cdots & 0.0 & 2.199 & 0.0 & 0.0 & 314.159\\\end{array}\right)\end{equation*}

Create a list of collapse operators that describe the dissipation

In [8]:
# collapse operators
c_ops = []

rate = kappa * (1 + n_th_a)
if rate > 0.0:
    c_ops.append(sqrt(rate) * a)

rate = kappa * n_th_a
if rate > 0.0:
    c_ops.append(sqrt(rate) * a.dag())

rate = gamma
if rate > 0.0:
    c_ops.append(sqrt(rate) * sm)

rate = Gamma
if rate > 0.0:
    c_ops.append(sqrt(rate) * sm.dag())

Evolve the system

Here we evolve the system with the Lindblad master equation solver, and we request that the expectation values of the operators $a^\dagger a$ and $\sigma_+\sigma_-$ are returned by the solver by passing the list [a.dag()*a, sm.dag()*sm] as the fifth argument to the solver.

In [9]:
opt = Odeoptions(nsteps=2000) # allow extra time-steps 
output = mesolve(H, psi0, tlist, c_ops, [a.dag() * a, sm.dag() * sm], options=opt)

Visualize the results

Here we plot the excitation probabilities of the cavity and the atom (these expectation values were calculated by the mesolve above).

In [10]:
n_c = output.expect[0]
n_a = output.expect[1]

fig, axes = plt.subplots(1, 1, figsize=(8,6))

axes.plot(tlist, n_c, label="Cavity")
axes.plot(tlist, n_a, label="Atom excited state")
axes.set_xlim(0, 150)
axes.legend(loc=0)
axes.set_xlabel('Time')
axes.set_ylabel('Occupation probability');

Steady state: cavity fock-state distribution and wigner function

In [11]:
rho_ss = steadystate(H, c_ops)
In [13]:
fig, axes = plt.subplots(1, 2, figsize=(12,6))

xvec = np.linspace(-5,5,200)

rho_cavity = ptrace(rho_ss, 0)
W = wigner(rho_cavity, xvec, xvec)
wlim = abs(W).max()

axes[1].contourf(xvec, xvec, W, 100, norm=mpl.colors.Normalize(-wlim,wlim), cmap=plt.get_cmap('RdBu'))
axes[1].set_xlabel(r'Im $\alpha$', fontsize=18)
axes[1].set_ylabel(r'Re $\alpha$', fontsize=18)

axes[0].bar(arange(0, N), real(rho_cavity.diag()), color="blue", alpha=0.6)
axes[0].set_ylim(0, 1)
axes[0].set_xlim(0, N)
axes[0].set_xlabel('Fock number', fontsize=18)
axes[0].set_ylabel('Occupation probability', fontsize=18);

Cavity fock-state distribution and Wigner function as a function of time

In [14]:
tlist = np.linspace(0, 25, 5)
output = mesolve(H, psi0, tlist, c_ops, [], options=Odeoptions(nsteps=5000))
In [15]:
rho_ss_sublist = output.states

xvec = np.linspace(-5,5,200)

fig, axes = plt.subplots(2, len(rho_ss_sublist), figsize=(3*len(rho_ss_sublist), 6))

for idx, rho_ss in enumerate(rho_ss_sublist):

    # trace out the cavity density matrix
    rho_ss_cavity = ptrace(rho_ss, 0)
    
    # calculate its wigner function
    W = wigner(rho_ss_cavity, xvec, xvec)
    
    # plot its wigner function
    wlim = abs(W).max()
    axes[0,idx].contourf(xvec, xvec, W, 100, norm=mpl.colors.Normalize(-wlim,wlim), cmap=plt.get_cmap('RdBu'))
    axes[0,idx].set_title(r'$t = %.1f$' % tlist[idx])
    
    # plot its fock-state distribution
    axes[1,idx].bar(arange(0, N), real(rho_ss_cavity.diag()), color="blue", alpha=0.8)
    axes[1,idx].set_ylim(0, 1)
    axes[1,idx].set_xlim(0, 15)

Steady state average photon occupation in cavity as a function of pump rate

References:

In [16]:
def calulcate_avg_photons(N, Gamma):
       
    # collapse operators
    c_ops = []

    rate = kappa * (1 + n_th_a)
    if rate > 0.0:
        c_ops.append(sqrt(rate) * a)

    rate = kappa * n_th_a
    if rate > 0.0:
        c_ops.append(sqrt(rate) * a.dag())

    rate = gamma
    if rate > 0.0:
        c_ops.append(sqrt(rate) * sm)

    rate = Gamma
    if rate > 0.0:
        c_ops.append(sqrt(rate) * sm.dag())
      
    # Ground state and steady state for the Hamiltonian: H = H0 + g * H1
    rho_ss = steadystate(H, c_ops)
    
    # cavity photon number
    n_cavity = expect(a.dag() * a, rho_ss)
    
    # cavity second order coherence function
    g2_cavity = expect(a.dag() * a.dag() * a * a, rho_ss) / (n_cavity ** 2)

    return n_cavity, g2_cavity
In [17]:
Gamma_max = 2 * (4*g**2) / kappa
Gamma_vec = np.linspace(0.1, Gamma_max, 50)

n_avg_vec = []
g2_vec = []

for Gamma in Gamma_vec:
    n_avg, g2 = calulcate_avg_photons(N, Gamma)
    n_avg_vec.append(n_avg)
    g2_vec.append(g2)
In [18]:
fig, axes = plt.subplots(1, 1, figsize=(12,6))

axes.plot(Gamma_vec * kappa / (4*g**2), n_avg_vec, color="blue", alpha=0.6, label="numerical")

axes.set_xlabel(r'$\Gamma\kappa/(4g^2)$', fontsize=18)
axes.set_ylabel(r'Occupation probability $\langle n \rangle$', fontsize=18)
axes.set_xlim(0, 2);
In [19]:
fig, axes = plt.subplots(1, 1, figsize=(12,6))

axes.plot(Gamma_vec * kappa / (4*g**2), g2_vec, color="blue", alpha=0.6, label="numerical")

axes.set_xlabel(r'$\Gamma\kappa/(4g^2)$', fontsize=18)
axes.set_ylabel(r'$g^{(2)}(0)$', fontsize=18)
axes.set_xlim(0, 2)
axes.text(0.1, 1.1, "Lasing regime", fontsize=16)
axes.text(1.5, 1.8, "Thermal regime", fontsize=16);

Here we see that lasing is suppressed for $\Gamma\kappa/(4g^2) > 1$.

Let's look at the fock-state distribution at $\Gamma\kappa/(4g^2) = 0.5$ (lasing regime) and $\Gamma\kappa/(4g^2) = 1.5$ (suppressed regime):

Case 1: $\Gamma\kappa/(4g^2) = 0.5$

In [20]:
Gamma = 0.5 * (4*g**2) / kappa
In [21]:
c_ops = [sqrt(kappa * (1 + n_th_a)) * a, sqrt(kappa * n_th_a) * a.dag(), sqrt(gamma) * sm, sqrt(Gamma) * sm.dag()]

rho_ss = steadystate(H, c_ops)
In [22]:
fig, axes = plt.subplots(1, 2, figsize=(16,6))

xvec = np.linspace(-10,10,200)

rho_cavity = ptrace(rho_ss, 0)
W = wigner(rho_cavity, xvec, xvec)
wlim = abs(W).max()
axes[1].contourf(xvec, xvec, W, 100, norm=mpl.colors.Normalize(-wlim,wlim), cmap=plt.get_cmap('RdBu'))
axes[1].set_xlabel(r'Im $\alpha$', fontsize=18)
axes[1].set_ylabel(r'Re $\alpha$', fontsize=18)

axes[0].bar(arange(0, N), real(rho_cavity.diag()), color="blue", alpha=0.6)
axes[0].set_xlabel(r'$n$', fontsize=18)
axes[0].set_ylabel(r'Occupation probability', fontsize=18)
axes[0].set_ylim(0, 1)
axes[0].set_xlim(0, N);

Case 2: $\Gamma\kappa/(4g^2) = 1.5$

In [23]:
Gamma = 1.5 * (4*g**2) / kappa
In [24]:
c_ops = [sqrt(kappa * (1 + n_th_a)) * a, sqrt(kappa * n_th_a) * a.dag(), sqrt(gamma) * sm, sqrt(Gamma) * sm.dag()]

rho_ss = steadystate(H, c_ops)
In [26]:
fig, axes = plt.subplots(1, 2, figsize=(16,6))

xvec = np.linspace(-10,10,200)

rho_cavity = ptrace(rho_ss, 0)
W = wigner(rho_cavity, xvec, xvec)
wlim = abs(W).max()
axes[1].contourf(xvec, xvec, W, 100, norm=mpl.colors.Normalize(-wlim,wlim), cmap=plt.get_cmap('RdBu'))
axes[1].set_xlabel(r'Im $\alpha$', fontsize=18)
axes[1].set_ylabel(r'Re $\alpha$', fontsize=18)

axes[0].bar(arange(0, N), real(rho_cavity.diag()), color="blue", alpha=0.6)
axes[0].set_xlabel(r'$n$', fontsize=18)
axes[0].set_ylabel(r'Occupation probability', fontsize=18)
axes[0].set_ylim(0, 1)
axes[0].set_xlim(0, N);

Too large pumping rate $\Gamma$ kills the lasing process: reversed threshold.

Software version

In [27]:
from qutip.ipynbtools import version_table

version_table()
Out[27]:
SoftwareVersion
IPython2.0.0
OSposix [linux]
Python3.4.1 (default, Jun 9 2014, 17:34:49) [GCC 4.8.3]
QuTiP3.0.0.dev-5a88aa8
Numpy1.8.1
matplotlib1.3.1
Cython0.20.1post0
SciPy0.13.3
Thu Jun 26 14:28:35 2014 JST