P.D. Nation and J.R. Johansson
For more information about QuTiP see http://qutip.org
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%matplotlib inline
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import matplotlib.pyplot as plt
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import numpy as np
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from IPython.display import Image
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from qutip import *
The optomechanical Hamiltonian arrises from the radiation pressure interaction of light in an optical cavity where one of the cavity mirrors is mechanically compliant.
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Image(filename='images/optomechanical_setup.png', width=500, embed=True)
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Assuming that $a^{+}$, $a$ and $b^{+}$,$b$ are the raising and lowering operators for the cavity and mechanical oscillator, respectively, the Hamiltonian for an optomechanical system driven by a classical monochromatic pump term can be written as
where $\Delta=\omega_{p}-\omega_{c}$ is the detuning between the pump ($\omega_{p}$) and cavity ($\omega_{c}$) mode frequencies, $g_{0}$ is the single-photon coupling strength, and $E$ is the amplitude of the pump mode. It is known that in the single-photon strong-coupling regime, where the cavity frequency shift per phonon is larger than the cavity line width, $g_{0}/\kappa \gtrsim 1$ where $\kappa$ is the decay rate of the cavity, and a single single photon displaces the mechanical oscillator by more than its zero-point amplitude $g_{0}/\omega_{m} \gtrsim 1$, or equiviently, $g^{2}_{0}/\kappa\omega_{m} \gtrsim 1$, the mechanical oscillator can be driven into a nonclassical steady state of the system$+$environment dynamics. Here, we will use the steady state solvers in QuTiP to explore such a state and compare the various solvers.
The steady state density matrix of the optomechanical system plus the environment can be found from the Liouvillian superoperator $\mathcal{L}$ via
\begin{equation} \frac{d\rho}{dt}=\mathcal{L}\rho=0\rho, \end{equation}where $\mathcal{L}$ is typically given in Lindblad form \begin{align} \mathcal{L}[\hat{\rho}]=&-i[\hat{H},\hat{\rho}]+\kappa \mathcal{D}\left[\hat{a},\hat{\rho}\right]\\ &+\Gamma_{m}(1+n_{\rm{th}})\mathcal{D}[\hat{b},\hat{\rho}]+\Gamma_{m}n_{\rm th}\mathcal{D}[\hat{b}^{+},\hat{\rho}], \nonumber \end{align}
where $\Gamma_{m}$ is the coulping strength of the mechanical oscillator to its thermal environment with average occupation number $n_{th}$. As is customary, here we assume that the cavity mode is coupled to the vacuum.
Although, the steady state solution is nothing but an eigenvalue equation, the numerical solution to this equation is anything but trivial due to the non-Hermitian structure of $\mathcal{L}$ and its worsening condition number has the dimensionality of the truncated Hilbert space increases.
As of QuTiP version 3.0, the following steady state solvers are available:
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# System Parameters (in units of wm)
#-----------------------------------
Nc = 4 # Number of cavity states
Nm = 80 # Number of mech states
kappa = 0.3 # Cavity damping rate
E = 0.1 # Driving Amplitude
g0 = 2.4*kappa # Coupling strength
Qm = 1e4 # Mech quality factor
gamma = 1/Qm # Mech damping rate
n_th = 1 # Mech bath temperature
delta = -0.43 # Detuning
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# Operators
#----------
a = tensor(destroy(Nc), qeye(Nm))
b = tensor(qeye(Nc), destroy(Nm))
num_b = b.dag()*b
num_a = a.dag()*a
# Hamiltonian
#------------
H = -delta*(num_a)+num_b+g0*(b.dag()+b)*num_a+E*(a.dag()+a)
# Collapse operators
#-------------------
cc = np.sqrt(kappa)*a
cm = np.sqrt(gamma*(1.0 + n_th))*b
cp = np.sqrt(gamma*n_th)*b.dag()
c_ops = [cc,cm,cp]
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solvers = ['direct','eigen','power','iterative-gmres','iterative-bicgstab']
mech_dms = []
for ss in solvers:
if ss in ['iterative-gmres','iterative-bicgstab']:
use_rcm = True
else:
use_rcm = False
rho_ss, info = steadystate(H, c_ops, method=ss,use_precond=True,
use_rcm=use_rcm, tol=1e-15, return_info=True)
print(ss,'solution time =',info['solution_time'])
rho_mech = ptrace(rho_ss, 1)
mech_dms.append(rho_mech)
mech_dms = np.asarray(mech_dms)
Can check to see if the solutions are the same by looking at the number of nonzero elements (NNZ) in the difference between mechanical density matrices.
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for kk in range(len(mech_dms)):
print((mech_dms[kk]-mech_dms[0]).data.nnz)
It seems that the eigensolver solution is not exactly the same. Lets check the magnitude of the elements to see if they are small.
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for kk in range(len(mech_dms)):
print(any(abs((mech_dms[kk] - mech_dms[0]).data.data)>1e-11))
It is known that the density matrix for the mechanical oscillator is diagoinal in the Fock basis due to phase diffusion. However some small off-diagonal terms show up during the factorization process
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fig = plt.figure(figsize=(8,6))
plt.spy(rho_mech.data, ms=1);
Therefore, to remove this error, let use explicitly take the diagonal elements are form a new operator out of them
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diag = rho_mech.diag()
rho_mech2 = qdiags(diag, 0, dims=rho_mech.dims, shape=rho_mech.shape)
fig = plt.figure(figsize=(8,6))
plt.spy(rho_mech2.data, ms=1);
Now lets compute the oscillator Wigner function and plot it to see if there are any regions of negativity.
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xvec = np.linspace(-20, 20, 256)
W = wigner(rho_mech2, xvec, xvec)
wmap = wigner_cmap(W, shift=-1e-5)
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fig, ax = plt.subplots(figsize=(8,6))
c = ax.contourf(xvec, xvec, W, 256, cmap=wmap)
ax.set_xlim([-10, 10])
ax.set_ylim([-10, 10])
plt.colorbar(c, ax=ax);
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from qutip.ipynbtools import version_table
version_table()
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