tkwant.onebody.WaveFunction

class tkwant.onebody.WaveFunction(H0, W, psi_init, energy=None, params=None, time_start=0, time_name='time', kernel_type=<class 'tkwant.onebody.kernels.Simple'>, solver_type=<class 'tkwant.onebody.solvers.Scipy'>, work=None, inplace=False, tmax=None)

A class to solve the time-dependent single-particle wavefunction.

The time-dependent single-particle Schrödinger equation is

\(i \partial_t \psi = (H_0 + W(t)) \psi\),

where the total Hamiltonian \(H(t) = H_0 + W(t)\) has been splitted into a static part \(H_0\) and a time-dependent perturbation \(W(t)\). Moreover, the initial condition is \(\psi(t_0)\) and \(W(t)\) is expected to be absent before the initial time \(t_0\):

\(W(t) = 0 \,\, \text{for} \,\, t \leq t_0\).

If an energy \(E\) is provided, this routine expects that the initial condition represents the scattering state \(\psi_{st}\) that solves the time-independent Schrödinger equation

\(H_0 \psi_{st} = E \psi_{st}, \, \psi_{st} = \psi(t_0)\).

For numerical reasons, the evolution is then performed in the variable

\(i \partial_t \bar{\psi} = (H_0 + W(t) - E) \bar{\psi} + W(t) \psi_{st}\),

where

\(\psi = (\bar{\psi} - \psi_{st}) e^{-i E (t - t_0)}\).

See J. Weston and X. Waintal, Phys. Rev. B 93, 134506 (2016).

Parameters

• H0 (array-like) – The static part of the Hamiltonian matrix, \(H_0\).

• W (callable or None) – Time-dependent part of the Hamiltonian matrix, \(W(t)\). Typically the object returned by tkwant.system.extract_perturbation.

• psi_init (array of complex) – The state \(\psi(t_0)\) from which to start, defined over the central region.

• energy (float, optional) – If provided, then psi_init is assumed to be an eigenstate of energy \(E\). If the Hamiltonian represents an open quantum system with leads, then psi_init is assumed to be the projection of a scattering state at energy \(E\) on to the central part of the system.

• params (dict, optional) – Extra arguments to pass to the time-dependent Hamiltonian function \(W(t)\), excluding time.

• time_start (float, optional) – The initial time \(t_0\). Default value is zero.

• time_name (str, optional) – The name of the time argument \(t\). Default name: time.

• kernel_type (tkwant.onebody.solvers.default, optional) – The kernel to calculate the right-hand-site of the Schrödinger equation.

• solver_type (tkwant.onebody.solvers.default, optional) – The solver used to evolve the wavefunction forward in time.

• work (ndarray of complex, optional) – Workarray of size H0.shape[0] for performance and memory optimization.

• inplace (bool, optional) – If true and energy is not None, the term H0 - E is calculated inplace in order to save memory. If energy is None, inplace has no effect.

• tmax (float, optional) – Optional maximal time until when the boundary is valid.

Methods

add_perturbation(qt)

Add a time-dependent perturbation to the Hamiltonian

H(t) = H_0 + W(t)

it is modified to

H(t) = H_0 + W(t) + Q(t)

evaluate(observable)

Evaluate the expectation value of an operator at the current time t.

For an operator \(\hat{O}\) the expectation value is \(O(t) = \langle \psi(t) | \hat{O} |\psi(t) \rangle\).

Parameters

observable (callable or kwant.operator) – An operator \(\hat{O}\) to evaluate the expectation value. Must have the calling signature of kwant.operator.

Returns

result – The expectation value \(O(t)\) of observable.

Return type

numpy array

evolve(time)

Evolve the wavefunction \(\psi(t)\) foreward in time up to \(t =\) time.

Parameters

time (int or float) – time argument up to which the solver should be evolved

classmethod from_kwant(syst, psi_init, boundaries=None, energy=None, params=None, time_start=0, time_name='time', kernel_type=<class 'tkwant.onebody.kernels.Simple'>, solver_type=<class 'tkwant.onebody.solvers.Scipy'>, perturbation_type=<class 'tkwant.onebody.kernels.PerturbationInterpolator'>)

Set up a time-dependent onebody wavefunction from a kwant system.

Parameters

• syst (kwant.builder.FiniteSystem) – The low level system for which the wave functions are to be calculated.

• psi_init (array of complex) – The state from which to start, defined over the central region.

• boundaries (sequence of BoundaryBase, optional) – The boundary conditions for each lead attached to syst. Must be provided for a system with leads.

• energy (float, optional) – If provided, then psi_init is assumed to be an eigenstate of energy E. If syst has leads, then psi_init is assumed to be the projection of a scattering state at energy E on to the central part of the system.

• params (dict, optional) – Extra arguments to pass to the Hamiltonian of syst, excluding time.

• time_start (float, optional) – The initial time \(t_0\). Default value is zero.

• time_name (str, optional) – The name of the time argument \(t\). Default name: time.

• kernel_type (tkwant.onebody.solvers.default, optional) – The kernel to calculate the right-hand-site of the Schrödinger equation.

• solver_type (tkwant.onebody.solvers.default, optional) – The solver used to evolve the wavefunction forward in time.

• perturbation_type (tkwant.onebody.kernels.ExtractPerturbation, optional) – Class to extract the time dependent perturbation \(W(t)\) out of syst.

Returns

wave_function – A time-dependent onebody wavefunction at the initial time.

Return type

tkwant.onebody.WaveFunction

h0()

Return the static Hamiltonian matrix H0 in the central scattering region

hamiltonian()

Return the full Hamiltonian matrix H = H0 + W(time) in the central scattering region. If add_perturbation has been called previously, this routine returns H0 + W(t) + Q(t).

psi()

Return the wavefunction \(\psi(t)\) at the current time t.

psindarray

The wavefunction at time. If this wavefunction is for a system with leads, then the wavefunction projected onto the central region is returned.

wt()

Return the time dependent Hamiltonian matrix W(time) in the central scattering region. If add_perturbation has been called previously, this routine returns W(t) + Q(t).