Lecture 3B - Jaynes-Cummings-like model in the ultrastrong coupling regime¶

Author: J. R. Johansson (robert@riken.jp), https://jrjohansson.github.io/

This lecture series was developed by J.R. Johannson. The original lecture notebooks are available here.

This is a slightly modified version of the lectures, to work with the current release of QuTiP. You can find these lectures as a part of the qutip-tutorials repository. This lecture and other tutorial notebooks are indexed at the QuTiP Tutorial webpage.

In [1]:

import matplotlib as mpl
import matplotlib.pyplot as plt
import numpy as np
from qutip import (about, basis, destroy, entropy_vn, expect, mesolve, ptrace,
                   qeye, tensor, wigner)

%matplotlib inline

Introduction¶

In the Jaynes-Cumming model, the dipole-interaction term between the atom and the cavity field is assumed to be weak, so that a rotating-wave approximation can be performed. For large coupling strengths between the atom and the cavity field the RWA is not justified, and for very large coupling strength interesting properties of the atom-cavity ground state is observed.

To explore this using QuTiP, consider the Hamiltonian

$H = \hbar \omega_c a^\dagger a + \frac{1}{2}\hbar\omega_a\sigma_z + \hbar g(a^\dagger + a)(\sigma_- + \sigma_+)$.¶

Note that here we have not transformed the interaction part of the Hamiltonian using the RWA, for which the Hamiltonian would have been

$H_{\rm RWA} = \hbar \omega_c a^\dagger a + \frac{1}{2}\hbar\omega_a\sigma_z + \hbar g(a^\dagger\sigma_- + a\sigma_+)$.¶

In this notebook we will calculate the ground state of the Hamiltonian $H$ as a function of the interaction strength $g$ (try to set use_rwa = True to use $H_{\rm RWA}$ instead).

The regime $g$ is large compared with all other energy scales in the Hamiltonian $H$ is called the ultrastrong coupling regime, and has been an active topic of research in recent years. See references below.

References:

Problem parameters¶

Here we use units where $\hbar = 1$:

In [2]:

wc = 1.0 * 2 * np.pi  # cavity frequency
wa = 1.0 * 2 * np.pi  # atom frequency

N = 15  # number of cavity fock states
use_rwa = False

Setup the operators and the Hamiltonian¶

In [3]:

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

na = sm.dag() * sm  # atom
nc = a.dag() * a  # cavity

# decoupled Hamiltonian
H0 = wc * a.dag() * a + wa * sm.dag() * sm

# interaction Hamiltonian
if use_rwa:
    H1 = a.dag() * sm + a * sm.dag()
else:
    H1 = (a.dag() + a) * (sm + sm.dag())

Find ground state as a function of coupling strength¶

In [4]:

g_vec = np.linspace(0, 2.0, 101) * 2 * np.pi  # coupling strength vector

psi_list = []

for g in g_vec:

    H = H0 + g * H1

    # find the groundstate and its energy
    gnd_energy, gnd_state = H.groundstate()

    # store the ground state
    psi_list.append(gnd_state)

Calculate the cavity and atom excitation probabilities as for the calculated ground states:

In [5]:

na_expt = expect(na, psi_list)  # qubit  occupation probability
nc_expt = expect(nc, psi_list)  # cavity occupation probability

Plot the ground state occupation probabilities of the cavity and the atom as a function of coupling strenght. Note that for large coupling strength (the ultrastrong coupling regime, where $g > \omega_a,\omega_c$), the ground state has both photonic and atomic excitations.

In [6]:

fig, axes = plt.subplots(1, 1, sharex=True, figsize=(8, 4))

axes.plot(g_vec / (2 * np.pi), nc_expt, "r", linewidth=2, label="cavity")
axes.plot(g_vec / (2 * np.pi), na_expt, "b", linewidth=2, label="atom")
axes.set_ylabel("Occupation probability", fontsize=16)
axes.set_xlabel("coupling strenght", fontsize=16)
axes.legend(loc=0)

fig.tight_layout()

Plot the wigner functions of the cavity as a function of coupling strength¶

In [7]:

g_idx = np.where([g_vec == 2 * np.pi * g for
                  g in [0.0, 0.5, 1.0, 1.5, 2.0]])[1]
psi_sublist = []
for idx in g_idx:
    psi_sublist.append(psi_list[idx])

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

fig_grid = (2, len(psi_sublist) * 2)
fig = plt.figure(figsize=(3 * len(psi_sublist), 6))

for idx, psi in enumerate(psi_sublist):
    rho_cavity = ptrace(psi, 0)
    W = wigner(rho_cavity, xvec, xvec)
    ax = plt.subplot2grid(fig_grid, (0, 2 * idx), colspan=2)
    ax.contourf(
        xvec,
        xvec,
        W,
        100,
        norm=mpl.colors.Normalize(-0.125, 0.125),
        cmap=plt.get_cmap("RdBu"),
    )
    ax.set_title(r"$g = %.1f$" % (g_vec[g_idx][idx] / (2 * np.pi)),
                 fontsize=16)

# plot the cavity occupation probability in the ground state
ax = plt.subplot2grid(fig_grid, (1, 1), colspan=(fig_grid[1] - 2))
ax.plot(g_vec / (2 * np.pi), nc_expt, label="Cavity")
ax.plot(g_vec / (2 * np.pi), na_expt, label="Atom excited state")
ax.legend(loc=0)
ax.set_xlabel("coupling strength")
ax.set_ylabel("Occupation probability");

Entropy of atom/cavity as a measure of entanglement¶

In [8]:

entropy_cavity = np.zeros(g_vec.shape)
entropy_atom = np.zeros(g_vec.shape)

for idx, psi in enumerate(psi_list):

    rho_cavity = ptrace(psi, 0)
    entropy_cavity[idx] = entropy_vn(rho_cavity, 2)

    rho_atom = ptrace(psi, 1)
    entropy_atom[idx] = entropy_vn(rho_atom, 2)

In [9]:

fig, axes = plt.subplots(1, 1, figsize=(12, 6))
axes.plot(g_vec / (2 * np.pi), entropy_cavity, "b", label="cavity",
          linewidth=2)
axes.plot(g_vec / (2 * np.pi), entropy_atom, "r--", label="atom", linewidth=2)
axes.set_ylim(0, 1)
axes.set_ylabel("entropy", fontsize=16)
axes.set_xlabel("coupling strength", fontsize=16)
axes.legend(loc=0);

Dynamics of an initially excited cavity¶

In [10]:

H = H0 + 1.0 * 2 * np.pi * H1

psi0 = tensor(basis(N, 1), basis(2, 0))

In [11]:

tlist = np.linspace(0, 20, 1000)
output = mesolve(H, psi0, tlist, [], [a.dag() * a, sm.dag() * sm])

In [12]:

fig, axes = plt.subplots(1, 1, sharex=True, figsize=(8, 4))

axes.plot(tlist, np.real(output.expect[0]), "r", linewidth=2, label="cavity")
axes.plot(tlist, np.real(output.expect[1]), "b", linewidth=2, label="atom")
axes.legend(loc=0)

fig.tight_layout()

Fock-state distribution and Wigner function for the cavity as a function of time¶

In [13]:

tlist = np.linspace(0, 0.35, 8)
output = mesolve(H, psi0, tlist, [], [])

In [14]:

rho_ss_sublist = output.states  # [::4]

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

fig, axes = plt.subplots(2, len(rho_ss_sublist),
                         figsize=(2 * len(rho_ss_sublist), 4))

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
    axes[0, idx].contourf(
        xvec,
        xvec,
        W,
        100,
        norm=mpl.colors.Normalize(-0.25, 0.25),
        cmap=plt.get_cmap("RdBu"),
    )

    # plot its fock-state distribution
    axes[1, idx].bar(
        np.arange(0, N), np.real(rho_ss_cavity.diag()), color="blue", alpha=0.6
    )
    axes[1, idx].set_ylim(0, 1)
    axes[1, idx].set_xlim(0, N)

Same thing with a little bit of dissipation¶

In [15]:

kappa = 0.25

In [16]:

tlist = np.linspace(0, 20, 1000)
output = mesolve(H, psi0, tlist, [np.sqrt(kappa) * a],
                 [a.dag() * a, sm.dag() * sm])

In [17]:

fig, axes = plt.subplots(1, 1, sharex=True, figsize=(8, 4))
axes.plot(tlist, output.expect[0], "r", linewidth=2, label="cavity")
axes.plot(tlist, output.expect[1], "b", linewidth=2, label="atom")
axes.legend(loc=0);

In [18]:

tlist = np.linspace(0, 10, 8)
output = mesolve(H, psi0, tlist, [np.sqrt(kappa) * a], [])

In [19]:

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

fig, axes = plt.subplots(2, len(output.states),
                         figsize=(2 * len(output.states), 4))

for idx, rho_ss in enumerate(output.states):

    # 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
    axes[0, idx].contourf(
        xvec,
        xvec,
        W,
        100,
        norm=mpl.colors.Normalize(-0.25, 0.25),
        cmap=plt.get_cmap("RdBu"),
    )

    # plot its fock-state distribution
    axes[1, idx].bar(
        np.arange(0, N), np.real(rho_ss_cavity.diag()), color="blue", alpha=0.6
    )
    axes[1, idx].set_ylim(0, 1)
    axes[1, idx].set_xlim(0, N)