Bloch-Redfield Solver: Time dependent operators

Authors: C. Staufenbiel, 2022

following the instructions in the Bloch-Redfield documentation.

Introduction

This notebook introduces the usage of time-dependent operators in the Bloch-Redfield solver, which is also described in the corresponding documentation.

We will discuss time-dependent Hamiltonians and time-dependent dissipations. The Bloch-Redfield solver is especially efficient since it uses Cython internally. For correct functioning we have to pass the time dependence in a string-based format.

Imports

In [1]:
import numpy as np
from qutip import about, basis, brmesolve, destroy, plot_expectation_values

%matplotlib inline

For our small example, we setup a system with N states and the number operator as Hamiltonian. We can observe that for the constant Hamiltonian and no given a_ops the expectation value $\langle n \rangle $ is a constant.

In [2]:
# num modes
N = 2
# Hamiltonian
a = destroy(N)
H = a.dag() * a

# initial state
psi0 = basis(N, N - 1)

# times for simulation
times = np.linspace(0, 10, 100)

# solve using brmesolve
result_const = brmesolve(H, psi0, times, e_ops=[a.dag() * a])
In [3]:
plot_expectation_values(result_const, ylabels=["<n>"]);

Next we define a string, which describes some time-dependence. We can use functions that are supported by the Cython implementation. A list of all supported functions can be found in the docs. For example, supported functions are sin or exp. The time variable is denoted by t.

In [4]:
time_dependence = "sin(t)"

Time-dependent Hamiltonian

As a first example, we define a time-dependent Hamiltonian (as described here).

$$ H = \hat{n} + sin(t) \hat{x} $$

Again, we can solve the dynamics using brmesolve().

In [5]:
H_t = [H, [a + a.dag(), time_dependence]]
result_brme = brmesolve(H_t, psi0, times, e_ops=[a.dag() * a])
plot_expectation_values(result_brme, ylabels=["<n>"]);

Time-dependent dissipation

Above we did not use the noise-power-spectrum, which the Bloch-Redfield solver is mainly used for. This spectrum is passed in the argument a_ops. We can also add a string-based time dependence to a_ops and thereby make the dissipation itself time-dependent.

Here we will define a a noice power spectrum of the form:

$$ J(\omega, t) = \kappa * e^{-t} \quad \text{for} \; \omega \geq 0$$
In [6]:
# setup dissipation
kappa = 0.2
a_ops = [[a + a.dag(), "{kappa}*exp(-t)*(w>=0)".format(kappa=kappa)]]

# solve
result_brme_aops = brmesolve(H, psi0, times, a_ops, e_ops=[a.dag() * a])

plot_expectation_values([result_brme_aops], ylabels=["<n>"]);

The coupling to the bath is sometimes described by operators of the form

$$ A = f(t)a + f(t)^* a^\dagger $$

To add such a coupling to brmesolve we can pass tuple in the a_ops argument. For example if we have $f(t) = e^{i * t}$ we can define the coupling of operator $A$ with strength $\kappa$ by the following a_ops. Note that t

In [7]:
a_ops = [[(a, a.dag()),
          ('{kappa} * (w>=0)'.format(kappa=kappa),
           'exp(1j*t)', 'exp(-1j*t)')]]

# solve using brmesolve and plot expecation
result_brme_aops_sum = brmesolve(H, psi0, times, a_ops, e_ops=[a.dag() * a])
plot_expectation_values([result_brme_aops_sum], ylabels=["<n>"]);

About

In [8]:
about()
QuTiP: Quantum Toolbox in Python
================================
Copyright (c) QuTiP team 2011 and later.
Current admin team: Alexander Pitchford, Nathan Shammah, Shahnawaz Ahmed, Neill Lambert, Eric Giguère, Boxi Li, Jake Lishman, Simon Cross and Asier Galicia.
Board members: Daniel Burgarth, Robert Johansson, Anton F. Kockum, Franco Nori and Will Zeng.
Original developers: R. J. Johansson & P. D. Nation.
Previous lead developers: Chris Granade & A. Grimsmo.
Currently developed through wide collaboration. See https://github.com/qutip for details.

QuTiP Version:      4.7.1.dev0+9098716
Numpy Version:      1.22.4
Scipy Version:      1.8.1
Cython Version:     0.29.32
Matplotlib Version: 3.5.2
Python Version:     3.10.4
Number of CPUs:     2
BLAS Info:          Generic
OPENMP Installed:   False
INTEL MKL Ext:      False
Platform Info:      Linux (x86_64)
Installation path:  /home/runner/work/qutip-tutorials/qutip-tutorials/qutip/qutip
================================================================================
Please cite QuTiP in your publication.
================================================================================
For your convenience a bibtex reference can be easily generated using `qutip.cite()`

Testing

In [9]:
assert np.allclose(result_const.expect[0], 1.0)

# compare result from brme with a_ops to analytic solution
analytic_aops = (N - 1) * np.exp(-kappa * (1.0 - np.exp(-times)))
assert np.allclose(result_brme_aops.expect[0], analytic_aops)

assert np.all(np.diff(result_brme_aops_sum.expect[0]) <= 0.0)