def position(n=default_n):
r"""Position operator.
Returns the n-dimensional approximation of the
dimensionless position operator Q in the number basis.
.. math::
Q &= \sqrt{\frac{m \omega}{\hbar}} q = (a+a^\dagger) / \sqrt{2},\\
P &= \sqrt{\frac{1}{m \hbar \omega}} p = -i (a-a^\dagger) / \sqrt{2}.
(Equivalently, :math:`a = (Q + iP) / \sqrt{2}`).
These operators fulfill :math:`[q, p] = i \hbar, \quad [Q, P] = i`.
The Hamiltonian of the harmonic oscillator is
.. math::
H = \frac{p^2}{2m} +\frac{1}{2} m \omega^2 q^2
= \frac{1}{2} \hbar \omega (P^2 +Q^2)
= \hbar \omega (a^\dagger a +\frac{1}{2}).
"""
# Ville Bergholm 2010
a = mat(boson_ladder(n))
return array(a + a.H) / sqrt(2)
def test():
"""Testing script for the harmonic oscillator module."""
from numpy.random import randn
from utils import assert_o
def randc():
"""Random complex number."""
return randn() + 1j*randn()
a = mat(boson_ladder(default_n))
alpha = randc()
s = coherent_state(alpha)
s0 = state(0, default_n)
D = displace(alpha)
assert_o((s - s0.u_propagate(D)).norm(), 0, tol) # displacement
z = randc()
S = squeeze(z)
Q = position(); P = momentum()
q = randn(); p = randn()
sq = position_state(q)
sp = momentum_state(p)
temp = 1e-1 # the truncation accuracy is not amazing here
assert_o(sq.ev(Q), q, temp) # Q, P eigenstates
assert_o(sp.ev(P), p, temp)
temp = ones(default_n); temp[-1] = -default_n+1 # truncation...
assert_o(norm(comm(Q,P) - 1j * diag(temp)), 0, tol) # [Q, P] = i
assert_o(norm(mat(P)**2 +mat(Q)**2 - 2 * a.H * a -diag(temp)), 0, tol) # P^2 +Q^2 = 2a^\dagger * a + 1
def displace(alpha, n=default_n):
r"""Bosonic displacement operator.
Returns the n-dimensional approximation for the bosonic
displacement operator
.. math::
D(\alpha) := \exp\left(\alpha a^\dagger - \alpha^* a\right)
= \exp\left( i \sqrt{2} \left(Q \mathrm{Im}(\alpha) -P \mathrm{Re}(\alpha)\right)\right)
in the number basis. This yields
.. math::
D(\alpha) Q D^\dagger(\alpha) &= Q -\sqrt{2} \textrm{Re}(\alpha) \mathbb{I},\\
D(\alpha) P D^\dagger(\alpha) &= P -\sqrt{2} \textrm{Im}(\alpha) \mathbb{I},
and thus the displacement operator displaces the state of a harmonic oscillator in phase space.
"""
# Ville Bergholm 2010
if not isscalar(alpha):
raise TypeError('alpha must be a scalar.')
a = mat(boson_ladder(n))
return array(expm(alpha * a.H -alpha.conjugate() * a))
def momentum(n=default_n):
"""Momentum operator.
Returns the n-dimensional approximation of the
dimensionless momentum operator P in the number basis.
See :func:`position`.
"""
# Ville Bergholm 2010
a = mat(boson_ladder(n))
return -1j*array(a - a.H) / sqrt(2)
def squeeze(z, n=default_n):
r"""Bosonic squeezing operator.
Returns the n-dimensional approximation for the bosonic
squeezing operator
.. math::
S(z) := \exp\left(\frac{1}{2} (z^* a^2 - z a^{\dagger 2})\right)
= \exp\left(\frac{i}{2} \left((QP+PQ)\mathrm{Re}(z) +(P^2-Q^2)\mathrm{Im}(z)\right)\right)
in the number basis.
"""
# Ville Bergholm 2010
if not isscalar(z):
raise TypeError('z must be a scalar.')
a = mat(boson_ladder(n))
return array(expm(0.5 * (z.conjugate() * (a ** 2) - z * (a.H ** 2))))