Source code for qutip.floquet

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__all__ = ['floquet_modes', 'floquet_modes_t', 'floquet_modes_table',
           'floquet_modes_t_lookup', 'floquet_states', 'floquet_states_t',
           'floquet_wavefunction', 'floquet_wavefunction_t',
           'floquet_state_decomposition', 'fsesolve',
           'floquet_master_equation_rates', 'floquet_collapse_operators',
           'floquet_master_equation_tensor',
           'floquet_master_equation_steadystate', 'floquet_basis_transform',
           'floquet_markov_mesolve', 'fmmesolve']

import numpy as np
import scipy.linalg as la
import scipy
from numpy import angle, pi, exp, sqrt
from types import FunctionType
from qutip.qobj import Qobj, isket
from qutip.superoperator import vec2mat_index, mat2vec, vec2mat
#from qutip.mesolve import mesolve
from qutip.sesolve import sesolve
from qutip.rhs_generate import rhs_clear
from qutip.steadystate import steadystate
from qutip.states import ket2dm
from qutip.states import projection
from qutip.solver import Options
from qutip.propagator import propagator
from qutip.solver import Result, _solver_safety_check
from qutip.cy.spmatfuncs import cy_ode_rhs
from qutip.expect import expect
from qutip.utilities import n_thermal

[docs]def floquet_modes(H, T, args=None, sort=False, U=None): """ Calculate the initial Floquet modes Phi_alpha(0) for a driven system with period T. Returns a list of :class:`qutip.qobj` instances representing the Floquet modes and a list of corresponding quasienergies, sorted by increasing quasienergy in the interval [-pi/T, pi/T]. The optional parameter `sort` decides if the output is to be sorted in increasing quasienergies or not. Parameters ---------- H : :class:`qutip.qobj` system Hamiltonian, time-dependent with period `T` args : dictionary dictionary with variables required to evaluate H T : float The period of the time-dependence of the hamiltonian. The default value 'None' indicates that the 'tlist' spans a single period of the driving. U : :class:`qutip.qobj` The propagator for the time-dependent Hamiltonian with period `T`. If U is `None` (default), it will be calculated from the Hamiltonian `H` using :func:`qutip.propagator.propagator`. Returns ------- output : list of kets, list of quasi energies Two lists: the Floquet modes as kets and the quasi energies. """ if U is None: # get the unitary propagator U = propagator(H, T, [], args) # find the eigenstates for the propagator evals, evecs = la.eig(U.full()) eargs = angle(evals) # make sure that the phase is in the interval [-pi, pi], so that # the quasi energy is in the interval [-pi/T, pi/T] where T is the # period of the driving. eargs += (eargs <= -2*pi) * (2*pi) + # (eargs > 0) * (-2*pi) eargs += (eargs <= -pi) * (2 * pi) + (eargs > pi) * (-2 * pi) e_quasi = -eargs / T # sort by the quasi energy if sort: order = np.argsort(-e_quasi) else: order = list(range(len(evals))) # prepare a list of kets for the floquet states new_dims = [U.dims[0], [1] * len(U.dims[0])] new_shape = [U.shape[0], 1] kets_order = [Qobj(np.array(evecs[:, o]).T, dims=new_dims, shape=new_shape) for o in order] return kets_order, e_quasi[order]
[docs]def floquet_modes_t(f_modes_0, f_energies, t, H, T, args=None): """ Calculate the Floquet modes at times tlist Phi_alpha(tlist) propagting the initial Floquet modes Phi_alpha(0) Parameters ---------- f_modes_0 : list of :class:`qutip.qobj` (kets) Floquet modes at :math:`t` f_energies : list Floquet energies. t : float The time at which to evaluate the floquet modes. H : :class:`qutip.qobj` system Hamiltonian, time-dependent with period `T` args : dictionary dictionary with variables required to evaluate H T : float The period of the time-dependence of the hamiltonian. Returns ------- output : list of kets The Floquet modes as kets at time :math:`t` """ # find t in [0,T] such that t_orig = t + n * T for integer n t = t - int(t / T) * T f_modes_t = [] # get the unitary propagator from 0 to t if t > 0.0: U = propagator(H, t, [], args) for n in np.arange(len(f_modes_0)): f_modes_t.append(U * f_modes_0[n] * exp(1j * f_energies[n] * t)) else: f_modes_t = f_modes_0 return f_modes_t
[docs]def floquet_modes_table(f_modes_0, f_energies, tlist, H, T, args=None): """ Pre-calculate the Floquet modes for a range of times spanning the floquet period. Can later be used as a table to look up the floquet modes for any time. Parameters ---------- f_modes_0 : list of :class:`qutip.qobj` (kets) Floquet modes at :math:`t` f_energies : list Floquet energies. tlist : array The list of times at which to evaluate the floquet modes. H : :class:`qutip.qobj` system Hamiltonian, time-dependent with period `T` T : float The period of the time-dependence of the hamiltonian. args : dictionary dictionary with variables required to evaluate H Returns ------- output : nested list A nested list of Floquet modes as kets for each time in `tlist` """ # truncate tlist to the driving period tlist_period = tlist[np.where(tlist <= T)] f_modes_table_t = [[] for t in tlist_period] opt = Options() opt.rhs_reuse = True rhs_clear() for n, f_mode in enumerate(f_modes_0): output = sesolve(H, f_mode, tlist_period, [], args, opt) for t_idx, f_state_t in enumerate(output.states): f_modes_table_t[t_idx].append( f_state_t * exp(1j * f_energies[n] * tlist_period[t_idx])) return f_modes_table_t
[docs]def floquet_modes_t_lookup(f_modes_table_t, t, T): """ Lookup the floquet mode at time t in the pre-calculated table of floquet modes in the first period of the time-dependence. Parameters ---------- f_modes_table_t : nested list of :class:`qutip.qobj` (kets) A lookup-table of Floquet modes at times precalculated by :func:`qutip.floquet.floquet_modes_table`. t : float The time for which to evaluate the Floquet modes. T : float The period of the time-dependence of the hamiltonian. Returns ------- output : nested list A list of Floquet modes as kets for the time that most closely matching the time `t` in the supplied table of Floquet modes. """ # find t_wrap in [0,T] such that t = t_wrap + n * T for integer n t_wrap = t - int(t / T) * T # find the index in the table that corresponds to t_wrap (= tlist[t_idx]) t_idx = int(t_wrap / T * len(f_modes_table_t)) # XXX: might want to give a warning if the cast of t_idx to int discard # a significant fraction in t_idx, which would happen if the list of time # values isn't perfect matching the driving period # if debug: print "t = %f -> t_wrap = %f @ %d of %d" % (t, t_wrap, t_idx, # N) return f_modes_table_t[t_idx]
[docs]def floquet_states(f_modes_t, f_energies, t): """ Evaluate the floquet states at time t given the Floquet modes at that time. Parameters ---------- f_modes_t : list of :class:`qutip.qobj` (kets) A list of Floquet modes for time :math:`t`. f_energies : array The Floquet energies. t : float The time for which to evaluate the Floquet states. Returns ------- output : list A list of Floquet states for the time :math:`t`. """ return [(f_modes_t[i] * exp(-1j * f_energies[i] * t)) for i in np.arange(len(f_energies))]
[docs]def floquet_states_t(f_modes_0, f_energies, t, H, T, args=None): """ Evaluate the floquet states at time t given the initial Floquet modes. Parameters ---------- f_modes_t : list of :class:`qutip.qobj` (kets) A list of initial Floquet modes (for time :math:`t=0`). f_energies : array The Floquet energies. t : float The time for which to evaluate the Floquet states. H : :class:`qutip.qobj` System Hamiltonian, time-dependent with period `T`. T : float The period of the time-dependence of the hamiltonian. args : dictionary Dictionary with variables required to evaluate H. Returns ------- output : list A list of Floquet states for the time :math:`t`. """ f_modes_t = floquet_modes_t(f_modes_0, f_energies, t, H, T, args) return [(f_modes_t[i] * exp(-1j * f_energies[i] * t)) for i in np.arange(len(f_energies))]
[docs]def floquet_wavefunction(f_modes_t, f_energies, f_coeff, t): """ Evaluate the wavefunction for a time t using the Floquet state decompositon, given the Floquet modes at time `t`. Parameters ---------- f_modes_t : list of :class:`qutip.qobj` (kets) A list of initial Floquet modes (for time :math:`t=0`). f_energies : array The Floquet energies. f_coeff : array The coefficients for Floquet decomposition of the initial wavefunction. t : float The time for which to evaluate the Floquet states. Returns ------- output : :class:`qutip.qobj` The wavefunction for the time :math:`t`. """ return sum([f_modes_t[i] * exp(-1j * f_energies[i] * t) * f_coeff[i] for i in np.arange(len(f_energies))])
[docs]def floquet_wavefunction_t(f_modes_0, f_energies, f_coeff, t, H, T, args=None): """ Evaluate the wavefunction for a time t using the Floquet state decompositon, given the initial Floquet modes. Parameters ---------- f_modes_t : list of :class:`qutip.qobj` (kets) A list of initial Floquet modes (for time :math:`t=0`). f_energies : array The Floquet energies. f_coeff : array The coefficients for Floquet decomposition of the initial wavefunction. t : float The time for which to evaluate the Floquet states. H : :class:`qutip.qobj` System Hamiltonian, time-dependent with period `T`. T : float The period of the time-dependence of the hamiltonian. args : dictionary Dictionary with variables required to evaluate H. Returns ------- output : :class:`qutip.qobj` The wavefunction for the time :math:`t`. """ f_states_t = floquet_states_t(f_modes_0, f_energies, t, H, T, args) return sum([f_states_t[i] * f_coeff[i] for i in np.arange(len(f_energies))])
[docs]def floquet_state_decomposition(f_states, f_energies, psi): r""" Decompose the wavefunction `psi` (typically an initial state) in terms of the Floquet states, :math:`\psi = \sum_\alpha c_\alpha \psi_\alpha(0)`. Parameters ---------- f_states : list of :class:`qutip.qobj` (kets) A list of Floquet modes. f_energies : array The Floquet energies. psi : :class:`qutip.qobj` The wavefunction to decompose in the Floquet state basis. Returns ------- output : array The coefficients :math:`c_\alpha` in the Floquet state decomposition. """ # [:1,:1][0, 0] patch around scipy 1.3.0 bug return [(f_states[i].dag() * psi).data[:1, :1][0, 0] for i in np.arange(len(f_energies))]
[docs]def fsesolve(H, psi0, tlist, e_ops=[], T=None, args={}, Tsteps=100): """ Solve the Schrodinger equation using the Floquet formalism. Parameters ---------- H : :class:`qutip.qobj.Qobj` System Hamiltonian, time-dependent with period `T`. psi0 : :class:`qutip.qobj` Initial state vector (ket). tlist : *list* / *array* list of times for :math:`t`. e_ops : list of :class:`qutip.qobj` / callback function list of operators for which to evaluate expectation values. If this list is empty, the state vectors for each time in `tlist` will be returned instead of expectation values. T : float The period of the time-dependence of the hamiltonian. args : dictionary Dictionary with variables required to evaluate H. Tsteps : integer The number of time steps in one driving period for which to precalculate the Floquet modes. `Tsteps` should be an even number. Returns ------- output : :class:`qutip.solver.Result` An instance of the class :class:`qutip.solver.Result`, which contains either an *array* of expectation values or an array of state vectors, for the times specified by `tlist`. """ if not T: # assume that tlist span exactly one period of the driving T = tlist[-1] # find the floquet modes for the time-dependent hamiltonian f_modes_0, f_energies = floquet_modes(H, T, args) # calculate the wavefunctions using the from the floquet modes f_modes_table_t = floquet_modes_table(f_modes_0, f_energies, np.linspace(0, T, Tsteps + 1), H, T, args) # setup Result for storing the results output = Result() output.times = tlist output.solver = "fsesolve" if isinstance(e_ops, FunctionType): output.num_expect = 0 expt_callback = True elif isinstance(e_ops, list): output.num_expect = len(e_ops) expt_callback = False if output.num_expect == 0: output.states = [] else: output.expect = [] for op in e_ops: if op.isherm: output.expect.append(np.zeros(len(tlist))) else: output.expect.append(np.zeros(len(tlist), dtype=complex)) else: raise TypeError("e_ops must be a list Qobj or a callback function") psi0_fb = psi0.transform(f_modes_0) for t_idx, t in enumerate(tlist): f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T) f_states_t = floquet_states(f_modes_t, f_energies, t) psi_t = psi0_fb.transform(f_states_t, True) if expt_callback: # use callback method e_ops(t, psi_t) else: # calculate all the expectation values, or output psi if # no expectation value operators where defined if output.num_expect == 0: output.states.append(Qobj(psi_t)) else: for e_idx, e in enumerate(e_ops): output.expect[e_idx][t_idx] = expect(e, psi_t) return output
[docs]def floquet_master_equation_rates(f_modes_0, f_energies, c_op, H, T, args, J_cb, w_th, kmax=5, f_modes_table_t=None): """ Calculate the rates and matrix elements for the Floquet-Markov master equation. Parameters ---------- f_modes_0 : list of :class:`qutip.qobj` (kets) A list of initial Floquet modes. f_energies : array The Floquet energies. c_op : :class:`qutip.qobj` The collapse operators describing the dissipation. H : :class:`qutip.qobj` System Hamiltonian, time-dependent with period `T`. T : float The period of the time-dependence of the hamiltonian. args : dictionary Dictionary with variables required to evaluate H. J_cb : callback functions A callback function that computes the noise power spectrum, as a function of frequency, associated with the collapse operator `c_op`. w_th : float The temperature in units of frequency. k_max : int The truncation of the number of sidebands (default 5). f_modes_table_t : nested list of :class:`qutip.qobj` (kets) A lookup-table of Floquet modes at times precalculated by :func:`qutip.floquet.floquet_modes_table` (optional). Returns ------- output : list A list (Delta, X, Gamma, A) containing the matrices Delta, X, Gamma and A used in the construction of the Floquet-Markov master equation. """ N = len(f_energies) M = 2 * kmax + 1 omega = (2 * pi) / T Delta = np.zeros((N, N, M)) X = np.zeros((N, N, M), dtype=complex) Gamma = np.zeros((N, N, M)) A = np.zeros((N, N)) nT = 100 dT = T / nT tlist = np.arange(dT, T + dT / 2, dT) if f_modes_table_t is None: f_modes_table_t = floquet_modes_table(f_modes_0, f_energies, np.linspace(0, T, nT + 1), H, T, args) for t in tlist: # TODO: repeated invocations of floquet_modes_t is # inefficient... make a and b outer loops and use the mesolve # instead of the propagator. # f_modes_t = floquet_modes_t(f_modes_0, f_energies, t, H, T, args) f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T) for a in range(N): for b in range(N): k_idx = 0 for k in range(-kmax, kmax + 1, 1): # [:1,:1][0, 0] patch around scipy 1.3.0 bug X[a, b, k_idx] += (dT / T) * exp(-1j * k * omega * t) * \ (f_modes_t[a].dag() * c_op * f_modes_t[b])[:1, :1][0, 0] k_idx += 1 Heaviside = lambda x: ((np.sign(x) + 1) / 2.0) for a in range(N): for b in range(N): k_idx = 0 for k in range(-kmax, kmax + 1, 1): Delta[a, b, k_idx] = f_energies[a] - f_energies[b] + k * omega Gamma[a, b, k_idx] = 2 * pi * Heaviside(Delta[a, b, k_idx]) * \ J_cb(Delta[a, b, k_idx]) * abs(X[a, b, k_idx]) ** 2 k_idx += 1 for a in range(N): for b in range(N): for k in range(-kmax, kmax + 1, 1): k1_idx = k + kmax k2_idx = -k + kmax A[a, b] += Gamma[a, b, k1_idx] + \ n_thermal(abs(Delta[a, b, k1_idx]), w_th) * \ (Gamma[a, b, k1_idx] + Gamma[b, a, k2_idx]) return Delta, X, Gamma, A
def floquet_collapse_operators(A): """ Construct collapse operators corresponding to the Floquet-Markov master-equation rate matrix `A`. .. note:: Experimental. """ c_ops = [] N, M = np.shape(A) # # Here we really need a master equation on Bloch-Redfield form, or perhaps # we can use the Lindblad form master equation with some rotating frame # approximations? ... # for a in range(N): for b in range(N): if a != b and abs(A[a, b]) > 0.0: # only relaxation terms included... c_ops.append(sqrt(A[a, b]) * projection(N, a, b)) return c_ops def floquet_master_equation_tensor(Alist, f_energies): """ Construct a tensor that represents the master equation in the floquet basis (with constant Hamiltonian and collapse operators). Simplest RWA approximation [Grifoni et al, Phys.Rep. 304 229 (1998)] Parameters ---------- Alist : list A list of Floquet-Markov master equation rate matrices. f_energies : array The Floquet energies. Returns ------- output : array The Floquet-Markov master equation tensor `R`. """ if isinstance(Alist, list): # Alist can be a list of rate matrices corresponding # to different operators that couple to the environment N, M = np.shape(Alist[0]) else: # or a simple rate matrix, in which case we put it in a list Alist = [Alist] N, M = np.shape(Alist[0]) Rdata_lil = scipy.sparse.lil_matrix((N * N, N * N), dtype=complex) for I in range(N * N): a, b = vec2mat_index(N, I) for J in range(N * N): c, d = vec2mat_index(N, J) R = -1.0j * (f_energies[a] - f_energies[b])*(a == c)*(b == d) Rdata_lil[I, J] = R for A in Alist: s1 = s2 = 0 for n in range(N): s1 += A[a, n] * (n == c) * (n == d) - A[n, a] * \ (a == c) * (a == d) s2 += (A[n, a] + A[n, b]) * (a == c) * (b == d) dR = (a == b) * s1 - 0.5 * (1 - (a == b)) * s2 if dR != 0.0: Rdata_lil[I, J] += dR return Qobj(Rdata_lil, [[N, N], [N, N]], [N*N, N*N])
[docs]def floquet_master_equation_steadystate(H, A): """ Returns the steadystate density matrix (in the floquet basis!) for the Floquet-Markov master equation. """ c_ops = floquet_collapse_operators(A) rho_ss = steadystate(H, c_ops) return rho_ss
[docs]def floquet_basis_transform(f_modes, f_energies, rho0): """ Make a basis transform that takes rho0 from the floquet basis to the computational basis. """ return rho0.transform(f_modes, True)
# ----------------------------------------------------------------------------- # Floquet-Markov master equation # #
[docs]def floquet_markov_mesolve(R, ekets, rho0, tlist, e_ops, f_modes_table=None, options=None, floquet_basis=True): """ Solve the dynamics for the system using the Floquet-Markov master equation. """ if options is None: opt = Options() else: opt = options if opt.tidy: R.tidyup() # # check initial state # if isket(rho0): # Got a wave function as initial state: convert to density matrix. rho0 = ket2dm(rho0) # # prepare output array # n_tsteps = len(tlist) dt = tlist[1] - tlist[0] output = Result() output.solver = "fmmesolve" output.times = tlist if isinstance(e_ops, FunctionType): n_expt_op = 0 expt_callback = True elif isinstance(e_ops, list): n_expt_op = len(e_ops) expt_callback = False if n_expt_op == 0: output.states = [] else: if not f_modes_table: raise TypeError("The Floquet mode table has to be provided " + "when requesting expectation values.") output.expect = [] output.num_expect = n_expt_op for op in e_ops: if op.isherm: output.expect.append(np.zeros(n_tsteps)) else: output.expect.append(np.zeros(n_tsteps, dtype=complex)) else: raise TypeError("Expectation parameter must be a list or a function") # # transform the initial density matrix to the eigenbasis: from # computational basis to the floquet basis # if ekets is not None: rho0 = rho0.transform(ekets) # # setup integrator # initial_vector = mat2vec(rho0.full()) r = scipy.integrate.ode(cy_ode_rhs) r.set_f_params(R.data.data, R.data.indices, R.data.indptr) r.set_integrator('zvode', method=opt.method, order=opt.order, atol=opt.atol, rtol=opt.rtol, max_step=opt.max_step) r.set_initial_value(initial_vector, tlist[0]) # # start evolution # rho = Qobj(rho0) t_idx = 0 for t in tlist: if not r.successful(): break rho = Qobj(vec2mat(r.y), rho0.dims, rho0.shape) if expt_callback: # use callback method if floquet_basis: e_ops(t, Qobj(rho)) else: f_modes_table_t, T = f_modes_table f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T) e_ops(t, Qobj(rho).transform(f_modes_t, True)) else: # calculate all the expectation values, or output rho if # no operators if n_expt_op == 0: if floquet_basis: output.states.append(Qobj(rho)) else: f_modes_table_t, T = f_modes_table f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T) output.states.append(Qobj(rho).transform(f_modes_t, True)) else: f_modes_table_t, T = f_modes_table f_modes_t = floquet_modes_t_lookup(f_modes_table_t, t, T) for m in range(0, n_expt_op): output.expect[m][t_idx] = \ expect(e_ops[m], rho.transform(f_modes_t, False)) r.integrate(r.t + dt) t_idx += 1 return output
# ----------------------------------------------------------------------------- # Solve the Floquet-Markov master equation # #
[docs]def fmmesolve(H, rho0, tlist, c_ops=[], e_ops=[], spectra_cb=[], T=None, args={}, options=Options(), floquet_basis=True, kmax=5, _safe_mode=True): """ Solve the dynamics for the system using the Floquet-Markov master equation. .. note:: This solver currently does not support multiple collapse operators. Parameters ---------- H : :class:`qutip.qobj` system Hamiltonian. rho0 / psi0 : :class:`qutip.qobj` initial density matrix or state vector (ket). tlist : *list* / *array* list of times for :math:`t`. c_ops : list of :class:`qutip.qobj` list of collapse operators. e_ops : list of :class:`qutip.qobj` / callback function list of operators for which to evaluate expectation values. spectra_cb : list callback functions List of callback functions that compute the noise power spectrum as a function of frequency for the collapse operators in `c_ops`. T : float The period of the time-dependence of the hamiltonian. The default value 'None' indicates that the 'tlist' spans a single period of the driving. args : *dictionary* dictionary of parameters for time-dependent Hamiltonians and collapse operators. This dictionary should also contain an entry 'w_th', which is the temperature of the environment (if finite) in the energy/frequency units of the Hamiltonian. For example, if the Hamiltonian written in units of 2pi GHz, and the temperature is given in K, use the following conversion >>> temperature = 25e-3 # unit K # doctest: +SKIP >>> h = 6.626e-34 # doctest: +SKIP >>> kB = 1.38e-23 # doctest: +SKIP >>> args['w_th'] = temperature * (kB / h) * 2 * pi * 1e-9 \ #doctest: +SKIP options : :class:`qutip.solver` options for the ODE solver. k_max : int The truncation of the number of sidebands (default 5). Returns ------- output : :class:`qutip.solver` An instance of the class :class:`qutip.solver`, which contains either an *array* of expectation values for the times specified by `tlist`. """ if _safe_mode: _solver_safety_check(H, rho0, c_ops, e_ops, args) if T is None: T = max(tlist) if len(spectra_cb) == 0: # add white noise callbacks if absent spectra_cb = [lambda w: 1.0] * len(c_ops) f_modes_0, f_energies = floquet_modes(H, T, args) f_modes_table_t = floquet_modes_table(f_modes_0, f_energies, np.linspace(0, T, 500 + 1), H, T, args) # get w_th from args if it exists if 'w_th' in args: w_th = args['w_th'] else: w_th = 0 # TODO: loop over input c_ops and spectra_cb, calculate one R for each set # calculate the rate-matrices for the floquet-markov master equation Delta, X, Gamma, Amat = floquet_master_equation_rates( f_modes_0, f_energies, c_ops[0], H, T, args, spectra_cb[0], w_th, kmax, f_modes_table_t) # the floquet-markov master equation tensor R = floquet_master_equation_tensor(Amat, f_energies) return floquet_markov_mesolve(R, f_modes_0, rho0, tlist, e_ops, f_modes_table=(f_modes_table_t, T), options=options, floquet_basis=floquet_basis)