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Model Hamiltonians (qit.hamiltonian)

This module has methods that generate several common types of model Hamiltonians used in quantum mechanics.

Contents

heisenberg(dim[, C, J, B]) Heisenberg spin network model.
jaynes_cummings(om_a, om_c, omega[, m]) Jaynes-Cummings model, a two-level atom in a single-mode cavity.
hubbard(C[, U, mu]) Hubbard model, fermions on a lattice.
bose_hubbard(C[, U, mu, m]) Bose-Hubbard model, bosons on a lattice.
holstein(C[, omega, g, m]) Holstein model, electrons on a lattice coupled to phonons.
qit.hamiltonian.heisenberg(dim, C=None, J=(0, 0, 2), B=(0, 0, 1))

Heisenberg spin network model.

Returns the Hamiltonian H for the Heisenberg model, describing a network of n interacting spins in an external magnetic field.

dim is an n-tuple of the dimensions of the spins, i.e. dim == (2, 2, 2) would be a system of three spin-1/2’s.

C is the n \times n connection matrix of the spin network, where C[i,j] is the coupling strength between spins i and j. Only the upper triangle is used.

J defines the form of the spin-spin interaction. It is either a 3-tuple or a function J(i, j) returning a 3-tuple for site-dependent interactions. Element k of the tuple is the coefficient of the Hamiltonian term S_k^{(i)} S_k^{(j)}, where S_k^{(i)} is the k-component of the angular momentum of spin i.

B defines the effective magnetic field the spins locally couple to. It’s either a 3-tuple (homogeneous field) or a function B(a) that returns a 3-tuple for site-dependent field.

H = \sum_{\langle i,j \rangle} \sum_{k = x,y,z} J(i,j)[k] S_k^{(i)} S_k^{(j)}  +\sum_i \vec{B}(i) \cdot \vec{S}^{(i)})

Examples:

C = np.eye(n, n, 1)  linear n-spin chain
J = (2, 2, 2)        isotropic Heisenberg coupling
J = (2, 2, 0)        XX+YY coupling
J = (0, 0, 2)        Ising ZZ coupling
B = (0, 0, 1)        homogeneous Z-aligned field
qit.hamiltonian.jaynes_cummings(om_a, om_c, omega, m=10)

Jaynes-Cummings model, a two-level atom in a single-mode cavity.

Returns the Hamiltonian H and the dimension vector dim for an implementation of the Jaynes-Cummings model, describing a two-level atom coupled to a harmonic oscillator (e.g. a single EM field mode in an optical cavity).

H/\hbar = \frac{\omega_a}{2} \sigma_z +\omega_c a^\dagger a +\frac{\Omega}{2} \sigma_x (a+a^\dagger)

The dimension of the Hilbert space of the bosonic cavity mode (infinite in principle) is truncated to m.

qit.hamiltonian.hubbard(C, U=1, mu=0)

Hubbard model, fermions on a lattice.

Returns the Hamiltonian H and the dimension vector dim for an implementation of the Hubbard model.

The model consists of spin-1/2 fermions confined in a graph defined by the symmetric connection matrix C (only upper triangle is used). The fermions interact with other fermions at the same site with interaction strength U, as well as with an external chemical potential mu. The Hamiltonian has been normalized by the fermion hopping constant t.

H = -\sum_{\langle i,j \rangle, \sigma} c^\dagger_{i,\sigma} c_{j,\sigma}
  +\frac{U}{t} \sum_i n_{i,up} n_{i,down} -\frac{\mu}{t} \sum_i (n_{i,up}+n_{i,down})

qit.hamiltonian.bose_hubbard(C, U=1, mu=0, m=10)

Bose-Hubbard model, bosons on a lattice.

Returns the Hamiltonian H and the dimension vector dim for an implementation of the Bose-Hubbard model.

The model consists of spinless bosons confined in a graph defined by the symmetric connection matrix C (only upper triangle is used). The bosons interact with other bosons at the same site with interaction strength U, as well as with an external chemical potential mu. The Hamiltonian has been normalized by the boson hopping constant t.

H = -\sum_{\langle i,j \rangle} b^\dagger_i b_{j} +\frac{U}{2t} \sum_i n_i (n_i-1) -\frac{\mu}{t} \sum_i n_i

The dimensions of the boson Hilbert spaces (infinite in principle) are truncated to m.

qit.hamiltonian.holstein(C, omega=1, g=1, m=10)

Holstein model, electrons on a lattice coupled to phonons.

Returns the Hamiltonian H and the dimension vector dim for an implementation of the Holstein model.

The model consists of spinless electrons confined in a graph defined by the symmetric connection matrix C (only upper triangle is used), coupled to phonon modes represented by a harmonic oscillator at each site. The dimensions of phonon Hilbert spaces (infinite in principle) are truncated to m.

The order of the subsystems is [e1, ..., en, p1, ..., pn]. The Hamiltonian has been normalized by the electron hopping constant t.

H = -\sum_{\langle i,j \rangle} c_i^\dagger c_j  +\frac{\omega}{t} \sum_i b_i^\dagger b_i
  -\frac{g \omega}{t} \sum_i (b_i + b_i^\dagger) c_i^\dagger c_i