QuTiP lecture: Correlation functions

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 *

First-order coherence function

Consider an oscillator that is interacting with a thermal environment. If the oscillator initially is in a coherent state, it will gradually decay to a thermal (incoherent) state. The amount of coherence can be quantified using the first-order optical coherence function

$\displaystyle g^{(1)}(\tau) = \frac{\langle a^\dagger(\tau)a(0)\rangle}{\sqrt{\langle a^\dagger(\tau)a(\tau)\rangle\langle a^\dagger(0)a(0)\rangle}}$

For a coherent state $|g^{(1)}(\tau)| = 1$, and for a completely incoherent (thermal) state $g^{(1)}(\tau) = 0$.

The following code calculates and plots $g^{(1)}(\tau)$ as a function of $\tau$.

Example: Decay of a coherent state to an incoherent (thermal) state

In [3]:
N = 20
taulist = np.linspace(0,10.0,200)
a = destroy(N)
H = 2*pi*a.dag()*a

# collapse operator
G1 = 0.75
n_th = 2.00  # bath temperature in terms of excitation number
c_ops = [sqrt(G1*(1+n_th)) * a, sqrt(G1*n_th) * a.dag()]

# start with a coherent state
rho0 = coherent_dm(N, 2.0)

# first calculate the occupation number as a function of time
n = mesolve(H, rho0, taulist, c_ops, [a.dag() * a]).expect[0]

# calculate the correlation function G1 and normalize with n to obtain g1
G1 = correlation(H, rho0, None, taulist, c_ops, a.dag(), a)
g1 = G1 / sqrt(n[0] * n)
In [4]:
fig, axes = plt.subplots(2, 1, sharex=True, figsize=(12,6))

axes[0].plot(taulist, real(g1), 'b', label=r'First-order coherence function $g^{(1)}(\tau)$')
axes[1].plot(taulist, real(n),  'r', label=r'occupation number $n(\tau)$')
axes[0].legend()
axes[1].legend()
axes[1].set_xlabel(r'$\tau$');

Second-order coherence function

$\displaystyle g^{(2)}(\tau) = \frac{\langle a^\dagger(0)a^\dagger(\tau)a(\tau)a(0)\rangle}{\langle a^\dagger(0)a(0)\rangle^2}$

For a coherent state $|g^{(2)}(\tau)| = 1$, and for a thermal state $g^{(2)}(\tau) = 2$ (bunched photons, tend to appear together).

The following code calculates and plots $g^{(2)}(\tau)$ as a function of $\tau$.

In [5]:
def correlation_ss_gtt(H, tlist, c_ops, a_op, b_op, c_op, d_op, rho0=None):
    """
    Calculate the correlation function <A(0)B(tau)C(tau)D(0)>

    (ss_gtt = steadystate general two-time)
    
    See, Gardiner, Quantum Noise, Section 5.2.1

    .. note::
        Experimental. 
    """
    if rho0 == None:
        rho0 = steadystate(H, c_ops)

    return mesolve(H, d_op * rho0 * a_op, tlist, c_ops, [b_op * c_op]).expect[0]
In [6]:
# calculate the correlation function G2 and normalize with n to obtain g2
G2 = correlation_ss_gtt(H, taulist, c_ops, a.dag(), a.dag(), a, a, rho0=rho0)
g2 = G2 / n**2
In [7]:
fig, axes = plt.subplots(2, 1, sharex=True, figsize=(12,6))

axes[0].plot(taulist, real(g2), 'b', label=r'Second-order coherence function $g^{(2)}(\tau)$')
axes[1].plot(taulist, real(n),  'r', label=r'occupation number $n(\tau)$')
axes[0].legend(loc=0)
axes[1].legend()
axes[1].set_xlabel(r'$\tau$');

Leggett-Garg inequality

Definition: Given an observable $Q(t)$ that is bound below and above by $|Q(t)| \leq 1$, the assumptions of

  • macroscopic realism
  • noninvasive measurements

implies that

$L(t_1, t_2) = \langle Q(t_1)Q(0)\rangle + \langle Q(t_1+t_2)Q(t_1)\rangle - \langle Q(t_1+t_2)Q(0)\rangle \leq 1$

If $Q$ is at a steady state at the initial time of measurement, we can set $\tau = t_1 = t_2$ and the Leggett-Garg inequality then reads

$L(\tau) = 2\langle Q(\tau)Q(0)\rangle - \langle Q(2\tau)Q(0)\rangle \leq 1$

References

In [17]:
def leggett_garg(c_mat):
    """
    For a given correlation matrix c_mat = <Q(t1+t2)Q(t1)>, calculate the Leggett-Garg correlation.
    """
    
    N, M = shape(c_mat)

    lg_mat = np.zeros([N//2,M//2], dtype=complex)
    lg_vec = np.zeros(N//2, dtype=complex)

    # c_mat(i, j) = <Q(dt i+dt j)Q(dt i)>
    # LG = <Q(t_1)Q(0)> + <Q(t_1+t_2)Q(t_1)> - <Q(t_1+t_2)Q(0)>

    for i in range(N//2):
        lg_vec[i] = 2*c_mat[0, i] - c_mat[0, 2*i];
    
        for j in range(M//2):
            lg_mat[i, j] = c_mat[0, i] + c_mat[i, j] - c_mat[0, i+j];


    return lg_mat, lg_vec

Example: Leggett-Garg inequality for two coupled resonators (optomechanical system)

References:

In [18]:
wc = 1.0  * 2 * pi  # cavity frequency
wa = 1.0  * 2 * pi  # resonator frequency
g  = 0.3  * 2 * pi  # coupling strength
kappa = 0.075       # cavity dissipation rate
gamma = 0.005       # resonator dissipation rate
Na = Nc = 3         # number of cavity fock states
n_th = 0.0          # avg number of thermal bath excitation

tlist = np.linspace(0, 7.5, 251)
tlist_sub = tlist[0:(len(tlist)/2)]
In [19]:
# start with an excited resonator
rho0 = tensor(fock_dm(Na,0), fock_dm(Nc,1))    

a = tensor(qeye(Nc), destroy(Na))
c = tensor(destroy(Nc), qeye(Na))

na = a.dag() * a
nc = c.dag() * c

H = wa * na + wc * nc - g * (a + a.dag()) * (c + c.dag())
In [20]:
# measurement operator on resonator
Q = na                                              # photon number resolving detector
#Q = tensor(qeye(Nc), 2 * fock_dm(Na, 1) - qeye(Na)) # fock-state |1> detector
#Q = tensor(qeye(Nc), qeye(Na) - 2 * fock_dm(Na, 0)) # click or no-click detector
In [21]:
c_op_list = []

rate = kappa * (1 + n_th)
if rate > 0.0:
    c_op_list.append(sqrt(rate) * c)

rate = kappa * n_th
if rate > 0.0:
    c_op_list.append(sqrt(rate) * c.dag())

rate = gamma * (1 + n_th)
if rate > 0.0:
    c_op_list.append(sqrt(rate) * a)
    
rate = gamma * n_th
if rate > 0.0:
    c_op_list.append(sqrt(rate) * a.dag())

Calculate the correlation function $\langle Q(t_1+t_2)Q(t_1)\rangle$

Using the regression theorem, and QuTiP function correlation.

In [22]:
corr_mat = correlation(H, rho0, tlist, tlist, c_op_list, Q, Q)

Calculate the Leggett-Garg correlation

In [23]:
LG_tt, LG_t = leggett_garg(corr_mat)

Plot results

In [27]:
fig, axes = plt.subplots(1, 2, figsize=(12,4))

axes[0].pcolor(tlist,tlist,abs(corr_mat),edgecolors='none')
axes[0].set_xlabel(r'$t_1 + t_2$')
axes[0].set_ylabel(r'$t_1$')
axes[0].autoscale(tight=True)

axes[1].pcolor(tlist_sub,tlist_sub,abs(LG_tt),edgecolors='none')
axes[1].set_xlabel(r'$t_1$')
axes[1].set_ylabel(r'$t_2$')
axes[1].autoscale(tight=True)

fig, axes = plt.subplots(1, 1, figsize=(12,4))
axes.plot(tlist_sub, np.diag(real(LG_tt)), label=r'$\tau = t_1 = t_2$')
axes.plot(tlist_sub, np.ones(shape(tlist_sub)), 'k', label=r'quantum boundary')
axes.fill_between(tlist_sub, np.diag(real(LG_tt)), 1, where=(np.diag(real(LG_tt))>1), color="green", alpha=0.5)
axes.set_xlim([0, max(tlist_sub)])
axes.legend(loc=0)
axes.set_xlabel(r'$\tau$', fontsize=18)
axes.set_ylabel(r'LG($\tau$)', fontsize=18);

Software versions

In [28]:
from qutip.ipynbtools import version_table

version_table()
Out[28]:
SoftwareVersion
matplotlib1.3.1
QuTiP3.0.0.dev-5a88aa8
Python3.4.1 (default, Jun 9 2014, 17:34:49) [GCC 4.8.3]
Numpy1.8.1
OSposix [linux]
IPython2.0.0
Cython0.20.1post0
SciPy0.13.3
Thu Jun 26 14:59:46 2014 JST