Lecture 0 - Introduction to QuTiP¶

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.pyplot as plt
import numpy as np
from IPython.display import Image
from qutip import (Qobj, about, basis, coherent, coherent_dm, create, destroy,
                   expect, fock, fock_dm, mesolve, qeye, sigmax, sigmay,
                   sigmaz, tensor, thermal_dm, anim_matrix_histogram,
                   anim_fock_distribution)
# set a parameter to see animations in line
from matplotlib import rc
rc('animation', html='jshtml')

%matplotlib inline

Introduction¶

QuTiP is a python package for calculations and numerical simulations of quantum systems.

It includes facilities for representing and doing calculations with quantum objects such state vectors (wavefunctions), as bras/kets/density matrices, quantum operators of single and composite systems, and superoperators (useful for defining master equations).

It also includes solvers for a time-evolution of quantum systems, according to: Schrodinger equation, von Neuman equation, master equations, Floquet formalism, Monte-Carlo quantum trajectors, experimental implementations of the stochastic Schrodinger/master equations.

For more information see the project web site at qutip.org, and the QuTiP documentation.

Installation¶

You can install QuTiP directly from pip by running:

pip install qutip

For further installation details, refer to the GitHub repository.

Quantum object class: `qobj`¶

At the heart of the QuTiP package is the Qobj class, which is used for representing quantum object such as states and operator.

The Qobj class contains all the information required to describe a quantum system, such as its matrix representation, composite structure and dimensionality.

In [2]:

Image(filename="images/qobj.png")

Out[2]:

Creating and inspecting quantum objects¶

We can create a new quantum object using the Qobj class constructor, like this:

In [3]:

q = Qobj([[1], [0]])

q

Out[3]:

Quantum object: dims=[[2], [1]], shape=(2, 1), type='ket', dtype=Dense$$\left(\begin{array}{cc}1\\0\end{array}\right)$$

Here we passed python list as an argument to the class constructor. The data in this list is used to construct the matrix representation of the quantum objects, and the other properties of the quantum object is by default computed from the same data.

We can inspect the properties of a Qobj instance using the following class method:

In [4]:

# the dimension, or composite Hilbert state space structure
q.dims

Out[4]:

[[2], [1]]

In [5]:

# the shape of the matrix data representation
q.shape

Out[5]:

(2, 1)

In [6]:

# the matrix data itself. in sparse matrix format.
q.data

Out[6]:

Dense(shape=(2, 1), fortran=True)

In [7]:

# get the dense matrix representation
q.full()

Out[7]:

array([[1.+0.j],
       [0.+0.j]])

In [8]:

# some additional properties
q.isherm, q.type

Out[8]:

(False, 'ket')

Using `Qobj` instances for calculations¶

With Qobj instances we can do arithmetic and apply a number of different operations using class methods:

In [9]:

sy = Qobj([[0, -1j], [1j, 0]])  # the sigma-y Pauli operator

sy

Out[9]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', dtype=Dense, isherm=True$$\left(\begin{array}{cc}0 & -1j\\1j & 0\end{array}\right)$$

In [10]:

sz = Qobj([[1, 0], [0, -1]])  # the sigma-z Pauli operator

sz

Out[10]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', dtype=Dense, isherm=True$$\left(\begin{array}{cc}1 & 0\\0 & -1\end{array}\right)$$

In [11]:

# some arithmetic with quantum objects

H = 1.0 * sz + 0.1 * sy

print("Qubit Hamiltonian = \n")
H

Qubit Hamiltonian =

Out[11]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', dtype=Dense, isherm=True$$\left(\begin{array}{cc}1 & -0.100j\\0.100j & -1\end{array}\right)$$

Example of modifying quantum objects using the Qobj methods:

In [12]:

# The hermitian conjugate
sy.dag()

Out[12]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', dtype=Dense, isherm=True$$\left(\begin{array}{cc}0 & -1j\\1j & 0\end{array}\right)$$

In [13]:

# The trace
H.tr()

Out[13]:

0.0

In [14]:

# Eigen energies
H.eigenenergies()

Out[14]:

array([-1.00498756,  1.00498756])

For a complete list of methods and properties of the Qobj class, see the QuTiP documentation or try help(Qobj) or dir(Qobj).

States and operators¶

Normally we do not need to create Qobj instances from stratch, using its constructor and passing its matrix represantation as argument. Instead we can use functions in QuTiP that generates common states and operators for us. Here are some examples of built-in state functions:

State vectors¶

In [15]:

# Fundamental basis states (Fock states of oscillator modes)

N = 2  # number of states in the Hilbert space
n = 1  # the state that will be occupied

basis(N, n)  # equivalent to fock(N, n)

Out[15]:

Quantum object: dims=[[2], [1]], shape=(2, 1), type='ket', dtype=Dense$$\left(\begin{array}{cc}0\\1\end{array}\right)$$

In [16]:

fock(4, 2)  # another example

Out[16]:

Quantum object: dims=[[4], [1]], shape=(4, 1), type='ket', dtype=Dense$$\left(\begin{array}{cc}0\\0\\1\\0\end{array}\right)$$

In [17]:

# a coherent state
coherent(N=10, alpha=1.0)

Out[17]:

Quantum object: dims=[[10], [1]], shape=(10, 1), type='ket', dtype=Dense$$\left(\begin{array}{cc}0.607\\0.607\\0.429\\0.248\\0.124\\0.055\\0.023\\0.009\\0.003\\0.001\end{array}\right)$$

Density matrices¶

In [18]:

# a fock state as density matrix
fock_dm(5, 2)  # 5 = hilbert space size, 2 = state that is occupied

Out[18]:

Quantum object: dims=[[5], [5]], shape=(5, 5), type='oper', dtype=Dia, isherm=True$$\left(\begin{array}{cc}0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0\\0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0\end{array}\right)$$

In [19]:

# coherent state as density matrix
coherent_dm(N=8, alpha=1.0)

Out[19]:

Quantum object: dims=[[8], [8]], shape=(8, 8), type='oper', dtype=Dense, isherm=True$$\left(\begin{array}{cc}0.368 & 0.368 & 0.260 & 0.150 & 0.075 & 0.034 & 0.014 & 0.006\\0.368 & 0.368 & 0.260 & 0.150 & 0.075 & 0.034 & 0.014 & 0.006\\0.260 & 0.260 & 0.184 & 0.106 & 0.053 & 0.024 & 0.010 & 0.004\\0.150 & 0.150 & 0.106 & 0.061 & 0.031 & 0.014 & 0.006 & 0.002\\0.075 & 0.075 & 0.053 & 0.031 & 0.015 & 0.007 & 0.003 & 0.001\\0.034 & 0.034 & 0.024 & 0.014 & 0.007 & 0.003 & 0.001 & 5.276\times10^{ -4 }\\0.014 & 0.014 & 0.010 & 0.006 & 0.003 & 0.001 & 4.990\times10^{ -4 } & 2.126\times10^{ -4 }\\0.006 & 0.006 & 0.004 & 0.002 & 0.001 & 5.276\times10^{ -4 } & 2.126\times10^{ -4 } & 9.058\times10^{ -5 }\end{array}\right)$$

In [20]:

# thermal state
n = 1  # average number of thermal photons
thermal_dm(8, n)

Out[20]:

Quantum object: dims=[[8], [8]], shape=(8, 8), type='oper', dtype=Dia, isherm=True$$\left(\begin{array}{cc}0.502 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0.251 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0.125 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0.063 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0.031 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0.016 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0.008 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.004\end{array}\right)$$

Operators¶

Qubit (two-level system) operators¶

In [21]:

# Pauli sigma x
sigmax()

Out[21]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', dtype=CSR, isherm=True$$\left(\begin{array}{cc}0 & 1\\1 & 0\end{array}\right)$$

In [22]:

# Pauli sigma y
sigmay()

Out[22]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', dtype=CSR, isherm=True$$\left(\begin{array}{cc}0 & -1j\\1j & 0\end{array}\right)$$

In [23]:

# Pauli sigma z
sigmaz()

Out[23]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', dtype=CSR, isherm=True$$\left(\begin{array}{cc}1 & 0\\0 & -1\end{array}\right)$$

Harmonic oscillator operators¶

In [24]:

#  annihilation operator

destroy(N=8)  # N = number of fock states included in the Hilbert space

Out[24]:

Quantum object: dims=[[8], [8]], shape=(8, 8), type='oper', dtype=Dia, isherm=False$$\left(\begin{array}{cc}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1.414 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1.732 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 2.236 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 2.449 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 2.646\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\end{array}\right)$$

In [25]:

# creation operator

create(N=8)  # equivalent to destroy(5).dag()

Out[25]:

Quantum object: dims=[[8], [8]], shape=(8, 8), type='oper', dtype=Dia, isherm=False$$\left(\begin{array}{cc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 1.414 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1.732 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 2.236 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 2.449 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 2.646 & 0\end{array}\right)$$

In [26]:

# the position operator is easily constructed from the annihilation operator
a = destroy(8)

x = a + a.dag()

x

Out[26]:

Quantum object: dims=[[8], [8]], shape=(8, 8), type='oper', dtype=Dia, isherm=True$$\left(\begin{array}{cc}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\1 & 0 & 1.414 & 0 & 0 & 0 & 0 & 0\\0 & 1.414 & 0 & 1.732 & 0 & 0 & 0 & 0\\0 & 0 & 1.732 & 0 & 2 & 0 & 0 & 0\\0 & 0 & 0 & 2 & 0 & 2.236 & 0 & 0\\0 & 0 & 0 & 0 & 2.236 & 0 & 2.449 & 0\\0 & 0 & 0 & 0 & 0 & 2.449 & 0 & 2.646\\0 & 0 & 0 & 0 & 0 & 0 & 2.646 & 0\end{array}\right)$$

Using `Qobj` instances we can check some well known commutation relations:¶

In [27]:

def commutator(op1, op2):
    return op1 * op2 - op2 * op1

$[a, a^1] = 1$

In [28]:

a = destroy(5)

commutator(a, a.dag())

Out[28]:

Quantum object: dims=[[5], [5]], shape=(5, 5), type='oper', dtype=Dia, isherm=True$$\left(\begin{array}{cc}1 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0\\0 & 0 & 1.000 & 0 & 0\\0 & 0 & 0 & 1 & 0\\0 & 0 & 0 & 0 & -4\end{array}\right)$$

Ops... The result is not identity! Why? Because we have truncated the Hilbert space. But that's OK as long as the highest Fock state isn't involved in the dynamics in our truncated Hilbert space. If it is, the approximation that the truncation introduces might be a problem.

$[x,p] = i$

In [29]:

x = (a + a.dag()) / np.sqrt(2)
p = -1j * (a - a.dag()) / np.sqrt(2)

In [30]:

commutator(x, p)

Out[30]:

Quantum object: dims=[[5], [5]], shape=(5, 5), type='oper', dtype=Dia, isherm=False$$\left(\begin{array}{cc}1.000j & 0 & 0 & 0 & 0\\0 & 1j & 0 & 0 & 0\\0 & 0 & 1.000j & 0 & 0\\0 & 0 & 0 & 1.000j & 0\\0 & 0 & 0 & 0 & -4.000j\end{array}\right)$$

Same issue with the truncated Hilbert space, but otherwise OK.

Let's try some Pauli spin inequalities

$[\sigma_x, \sigma_y] = 2i \sigma_z$

In [31]:

commutator(sigmax(), sigmay()) - 2j * sigmaz()

Out[31]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', dtype=CSR, isherm=True$$\left(\begin{array}{cc}0 & 0\\0 & 0\end{array}\right)$$

$-i \sigma_x \sigma_y \sigma_z = \mathbf{1}$

In [32]:

-1j * sigmax() * sigmay() * sigmaz()

Out[32]:

Quantum object: dims=[[2], [2]], shape=(2, 2), type='oper', dtype=CSR, isherm=True$$\left(\begin{array}{cc}1 & 0\\0 & 1\end{array}\right)$$

$\sigma_x^2 = \sigma_y^2 = \sigma_z^2 = \mathbf{1}$

In [33]:

sigmax() ** 2 == sigmay() ** 2 == sigmaz() ** 2 == qeye(2)

Out[33]:

True

Composite systems¶

In most cases we are interested in coupled quantum systems, for example coupled qubits, a qubit coupled to a cavity (oscillator mode), etc.

To define states and operators for such systems in QuTiP, we use the tensor function to create Qobj instances for the composite system.

For example, consider a system composed of two qubits. If we want to create a Pauli $\sigma_z$ operator that acts on the first qubit and leaves the second qubit unaffected (i.e., the operator $\sigma_z \otimes \mathbf{1}$), we would do:

In [34]:

sz1 = tensor(sigmaz(), qeye(2))

sz1

Out[34]:

Quantum object: dims=[[2, 2], [2, 2]], shape=(4, 4), type='oper', dtype=CSR, isherm=True$$\left(\begin{array}{cc}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & -1 & 0\\0 & 0 & 0 & -1\end{array}\right)$$

We can easily verify that this two-qubit operator does indeed have the desired properties:

In [35]:

psi1 = tensor(basis(N, 1), basis(N, 0))  # excited first qubit
psi2 = tensor(basis(N, 0), basis(N, 1))  # excited second qubit

In [36]:

# this should not be true,
# because sz1 should flip the sign of the excited state of psi1
sz1 * psi1 == psi1

Out[36]:

False

In [37]:

# this should be true, because sz1 should leave psi2 unaffected
sz1 * psi2 == psi2

Out[37]:

True

Above we used the qeye(N) function, which generates the identity operator with N quantum states. If we want to do the same thing for the second qubit we can do:

In [38]:

sz2 = tensor(qeye(2), sigmaz())

sz2

Out[38]:

Quantum object: dims=[[2, 2], [2, 2]], shape=(4, 4), type='oper', dtype=CSR, isherm=True$$\left(\begin{array}{cc}1 & 0 & 0 & 0\\0 & -1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & -1\end{array}\right)$$

Note the order of the argument to the tensor function, and the correspondingly different matrix representation of the two operators sz1 and sz2.

Using the same method we can create coupling terms of the form $\sigma_x \otimes \sigma_x$:

In [39]:

tensor(sigmax(), sigmax())

Out[39]:

Quantum object: dims=[[2, 2], [2, 2]], shape=(4, 4), type='oper', dtype=CSR, isherm=True$$\left(\begin{array}{cc}0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{array}\right)$$

Now we are ready to create a Qobj representation of a coupled two-qubit Hamiltonian: $H = \epsilon_1 \sigma_z^{(1)} + \epsilon_2 \sigma_z^{(2)} + g \sigma_x^{(1)}\sigma_x^{(2)}$

In [40]:

epsilon = [1.0, 1.0]
g = 0.1

sz1 = tensor(sigmaz(), qeye(2))
sz2 = tensor(qeye(2), sigmaz())

H = epsilon[0] * sz1 + epsilon[1] * sz2 + g * tensor(sigmax(), sigmax())

H

Out[40]:

Quantum object: dims=[[2, 2], [2, 2]], shape=(4, 4), type='oper', dtype=CSR, isherm=True$$\left(\begin{array}{cc}2 & 0 & 0 & 0.100\\0 & 0 & 0.100 & 0\\0 & 0.100 & 0 & 0\\0.100 & 0 & 0 & -2\end{array}\right)$$

To create composite systems of different types, all we need to do is to change the operators that we pass to the tensor function (which can take an arbitrary number of operator for composite systems with many components).

For example, the Jaynes-Cumming Hamiltonian for a qubit-cavity system:

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

In [41]:

wc = 1.0  # cavity frequency
wa = 1.0  # qubit/atom frenqency
g = 0.1  # coupling strength

# cavity mode operator
a = tensor(destroy(5), qeye(2))

# qubit/atom operators
sz = tensor(qeye(5), sigmaz())  # sigma-z operator
sm = tensor(qeye(5), destroy(2))  # sigma-minus operator

# the Jaynes-Cumming Hamiltonian
H = wc * a.dag() * a - 0.5 * wa * sz + g * (a * sm.dag() + a.dag() * sm)

H

Out[41]:

Quantum object: dims=[[5, 2], [5, 2]], shape=(10, 10), type='oper', dtype=CSR, isherm=True$$\left(\begin{array}{cc}-0.500 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0.500 & 0.100 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0.100 & 0.500 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 1.500 & 0.141 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0.141 & 1.500 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 2.500 & 0.173 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0.173 & 2.500 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 3.500 & 0.200 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.200 & 3.500 & 0\\0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4.500\end{array}\right)$$

Note that

$a \sigma_+ = (a \otimes \mathbf{1}) (\mathbf{1} \otimes \sigma_+)$

so the following two are identical:

In [42]:

a = tensor(destroy(3), qeye(2))
sp = tensor(qeye(3), create(2))

a * sp

Out[42]:

Quantum object: dims=[[3, 2], [3, 2]], shape=(6, 6), type='oper', dtype=Dia, isherm=False$$\left(\begin{array}{cc}0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1.414 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\end{array}\right)$$

In [43]:

tensor(destroy(3), create(2))

Out[43]:

Quantum object: dims=[[3, 2], [3, 2]], shape=(6, 6), type='oper', dtype=Dia, isherm=False$$\left(\begin{array}{cc}0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 1.414 & 0\\0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 0 & 0 & 0 & 0\end{array}\right)$$

Unitary dynamics¶

Unitary evolution of a quantum system in QuTiP can be calculated with the mesolve function.

mesolve is short for Master-eqaution solve (for dissipative dynamics), but if no collapse operators (which describe the dissipation) are given to the solve it falls back on the unitary evolution of the Schrodinger (for initial states in state vector for) or the von Neuman equation (for initial states in density matrix form).

The evolution solvers in QuTiP returns a class of type Odedata, which contains the solution to the problem posed to the evolution solver.

For example, considor a qubit with Hamiltonian $H = \sigma_x$ and initial state $\left|1\right>$ (in the sigma-z basis): Its evolution can be calculated as follows:

In [44]:

# Hamiltonian
H = sigmax()

# initial state
psi0 = basis(2, 0)

# list of times for which the solver should store the state vector
tlist = np.linspace(0, 10, 100)

result = mesolve(H, psi0, tlist, [], [])

In [45]:

result

Out[45]:

<Result
  Solver: sesolve
  Solver stats:
    method: 'scipy zvode adams'
    init time: 0.00015783309936523438
    preparation time: 0.00011277198791503906
    run time: 0.005159139633178711
    solver: 'Schrodinger Evolution'
  Time interval: [0.0, 10.0] (100 steps)
  Number of e_ops: 0
  States saved.
>

The result object contains a list of the wavefunctions at the times requested with the tlist array.

In [46]:

len(result.states)

Out[46]:

In [47]:

result.states[-1]  # the finial state

Out[47]:

Quantum object: dims=[[2], [1]], shape=(2, 1), type='ket', dtype=Dense$$\left(\begin{array}{cc}-0.839\\0.544j\end{array}\right)$$

You can visualize the time evolution of the state. anim_matrix_histogram visualizes the elements of it. The animation below shows that the state changes periodically and that the coefficient of $\left|1\right>$ is real and that of $\left|0\right>$ is imaginary.

In [48]:

fig, ani = anim_matrix_histogram(result, limits=[0, 1],
                                 bar_style='abs', color_style='phase')
# close an auto-generated plot and animation
plt.close()
ani

Out[48]:

Expectation values¶

The expectation values of an operator given a state vector or density matrix (or list thereof) can be calculated using the expect function.

In [49]:

expect(sigmaz(), result.states[-1])

Out[49]:

0.40806479408005597

In [50]:

expect(sigmaz(), result.states)

Out[50]:

array([ 1.        ,  0.97966324,  0.9194801 ,  0.82189845,  0.69088728,
        0.53177528,  0.35103398,  0.15601481, -0.04535002, -0.24487027,
       -0.43443068, -0.60632131, -0.7535507 , -0.87013055, -0.951319  ,
       -0.99381383, -0.9958867 , -0.9574533 , -0.88007676, -0.76690429,
       -0.62253905, -0.45285285, -0.26474744, -0.06587387,  0.13567909,
        0.3317135 ,  0.51425587,  0.67588157,  0.81001677,  0.91120556,
        0.97533227,  0.99978876,  0.98358026,  0.92736593,  0.83343229,
        0.70559992,  0.54906818,  0.37020385,  0.17628203, -0.0248099 ,
       -0.22489269, -0.41582823, -0.58985057, -0.73988155, -0.85981879,
       -0.9447841 , -0.9913216 , -0.99753841, -0.96318177, -0.88964906,
       -0.77993099, -0.6384903 , -0.47107995, -0.284509  , -0.08636599,
        0.11528974,  0.31225626,  0.49652225,  0.66059275,  0.79779446,
        0.90254706,  0.97058988,  0.99915523,  0.98708135,  0.93485936,
        0.84461322,  0.7200135 ,  0.56612831,  0.38921658,  0.19647394,
       -0.00425997, -0.20482058, -0.39705049, -0.57313082, -0.72589982,
       -0.84914389, -0.93785015, -0.98841059, -0.99876884, -0.9685036 ,
       -0.89884578, -0.7926286 , -0.65417236, -0.48910854, -0.30415083,
       -0.10682227,  0.09485115,  0.29266668,  0.47857836,  0.64502446,
        0.78523515,  0.89350741,  0.96543742,  0.99809965,  0.9901656 ,
        0.9419579 ,  0.85543735,  0.73412306,  0.58294927,  0.40806479])

In [51]:

fig, axes = plt.subplots(1, 1)

axes.plot(tlist, expect(sigmaz(), result.states))

axes.set_xlabel(r"$t$", fontsize=20)
axes.set_ylabel(r"$\left<\sigma_z\right>$", fontsize=20);

If we are only interested in expectation values, we could pass a list of operators to the mesolve function that we want expectation values for, and have the solver compute then and store the results in the Odedata class instance that it returns.

For example, to request that the solver calculates the expectation values for the operators $\sigma_x, \sigma_y, \sigma_z$:

In [52]:

result = mesolve(H, psi0, tlist, [], [sigmax(), sigmay(), sigmaz()])

Now the expectation values are available in result.expect[0], result.expect[1], and result.expect[2]:

In [53]:

fig, axes = plt.subplots(1, 1)

axes.plot(tlist, result.expect[2], label=r"$\left<\sigma_z\right>$")
axes.plot(tlist, result.expect[1], label=r"$\left<\sigma_y\right>$")
axes.plot(tlist, result.expect[0], label=r"$\left<\sigma_x\right>$")

axes.set_xlabel(r"$t$", fontsize=20)
axes.legend(loc=2);

Dissipative dynamics¶

To add dissipation to a problem, all we need to do is to define a list of collapse operators to the call to the mesolve solver.

A collapse operator is an operator that describes how the system is interacting with its environment.

For example, consider a quantum harmonic oscillator with Hamiltonian

$H = \hbar\omega a^\dagger a$

and which loses photons to its environment with a relaxation rate $\kappa$. The collapse operator that describes this process is

$\sqrt{\kappa} a$

since $a$ is the photon annihilation operator of the oscillator.

To program this problem in QuTiP:

In [54]:

w = 1.0  # oscillator frequency
kappa = 0.1  # relaxation rate
a = destroy(10)  # oscillator annihilation operator
rho0 = fock_dm(10, 5)  # initial state, fock state with 5 photons
H = w * a.dag() * a  # Hamiltonian

# A list of collapse operators
c_ops = [np.sqrt(kappa) * a]

In [55]:

tlist = np.linspace(0, 50, 100)

# request that the solver return the expectation value
# of the photon number state operator a.dag() * a
result = mesolve(H, rho0, tlist, c_ops, [a.dag() * a],
                 options={"store_states": True})

In [56]:

fig, axes = plt.subplots(1, 1)
axes.plot(tlist, result.expect[0])
axes.set_xlabel(r"$t$", fontsize=20)
axes.set_ylabel(r"Photon number", fontsize=16);

anim_fock_distribution enables you to visualize the probability distribution at each time.

In [57]:

fig, ani = anim_fock_distribution(result)
# close an auto-generated plot and animation
plt.close()
ani

Out[57]:

Installation information¶

In [58]:

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, Asier Galicia, Paul Menczel, and Patrick Hopf.
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:      5.1.0.dev0+c874c4a
Numpy Version:      1.22.4
Scipy Version:      1.13.1
Cython Version:     3.0.10
Matplotlib Version: 3.5.2
Python Version:     3.10.4
Number of CPUs:     4
BLAS Info:          Generic
INTEL MKL Ext:      False
Platform Info:      Linux (x86_64)
Installation path:  /usr/share/miniconda3/envs/test-environment/lib/python3.10/site-packages/qutip
================================================================================
Please cite QuTiP in your publication.
================================================================================
For your convenience a bibtex reference can be easily generated using `qutip.cite()`