Source code for qutip.correlation

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__all__ = ['correlation_2op_1t', 'correlation_2op_2t', 'correlation_3op_1t',
           'correlation_3op_2t', 'coherence_function_g1',
           'coherence_function_g2', 'spectrum', 'spectrum_correlation_fft',
           'correlation_ss', 'correlation', 'correlation_4op_1t',
           'correlation_4op_2t', 'spectrum_ss', 'spectrum_pi']

from re import sub
from warnings import warn
import types

import numpy as np
import scipy.fftpack

from qutip.eseries import esval, esspec
from qutip.essolve import ode2es
from qutip.expect import expect
from qutip.mesolve import mesolve
from qutip.mcsolve import mcsolve
from qutip.operators import qeye
from qutip.qobj import Qobj, isket, issuper
from qutip.rhs_generate import rhs_clear, _td_wrap_array_str
from qutip.cy.utilities import _cython_build_cleanup
from qutip.settings import debug
from qutip.solver import Options, config
from qutip.steadystate import steadystate
from qutip.states import ket2dm
from qutip.superoperator import liouvillian, spre, mat2vec
from qutip.tensor import tensor

if debug:
    import inspect


# -----------------------------------------------------------------------------
# PUBLIC API
# -----------------------------------------------------------------------------

# low level correlation

[docs]def correlation_2op_1t(H, state0, taulist, c_ops, a_op, b_op, solver="me", reverse=False, args={}, options=Options(ntraj=[20, 100])): """ Calculate the two-operator two-time correlation function: :math:`\left<A(t+\\tau)B(t)\\right>` along one time axis using the quantum regression theorem and the evolution solver indicated by the `solver` parameter. Parameters ---------- H : Qobj system Hamiltonian, may be time-dependent for solver choice of `me` or `mc`. state0 : Qobj Initial state density matrix :math:`\\rho(t_0)` or state vector :math:`\\psi(t_0)`. If 'state0' is 'None', then the steady state will be used as the initial state. The 'steady-state' is only implemented for the `me` and `es` solvers. taulist : array_like list of times for :math:`\\tau`. taulist must be positive and contain the element `0`. c_ops : list list of collapse operators, may be time-dependent for solver choice of `me` or `mc`. a_op : Qobj operator A. b_op : Qobj operator B. reverse : bool {False, True} If `True`, calculate :math:`\left<A(t)B(t+\\tau)\\right>` instead of :math:`\left<A(t+\\tau)B(t)\\right>`. solver : str {'me', 'mc', 'es'} choice of solver (`me` for master-equation, `mc` for Monte Carlo, and `es` for exponential series). options : Options Solver options class. `ntraj` is taken as a two-element list because the `mc` correlator calls `mcsolve()` recursively; by default, `ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in the `mc` correlator; by default, `mc_corr_eps=1e-10`. Returns ------- corr_vec : ndarray An array of correlation values for the times specified by `tlist`. References ---------- See, Gardiner, Quantum Noise, Section 5.2. """ if debug: print(inspect.stack()[0][3]) if reverse: A_op = a_op B_op = b_op C_op = 1 else: A_op = 1 B_op = a_op C_op = b_op return _correlation_2t(H, state0, [0], taulist, c_ops, A_op, B_op, C_op, solver=solver, args=args, options=options)[0]
[docs]def correlation_2op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, solver="me", reverse=False, args={}, options=Options(ntraj=[20, 100])): """ Calculate the two-operator two-time correlation function: :math:`\left<A(t+\\tau)B(t)\\right>` along two time axes using the quantum regression theorem and the evolution solver indicated by the `solver` parameter. Parameters ---------- H : Qobj system Hamiltonian, may be time-dependent for solver choice of `me` or `mc`. state0 : Qobj Initial state density matrix :math:`\\rho_0` or state vector :math:`\\psi_0`. If 'state0' is 'None', then the steady state will be used as the initial state. The 'steady-state' is only implemented for the `me` and `es` solvers. tlist : array_like list of times for :math:`t`. tlist must be positive and contain the element `0`. When taking steady-steady correlations only one tlist value is necessary, i.e. when :math:`t \\rightarrow \\infty`; here tlist is automatically set, ignoring user input. taulist : array_like list of times for :math:`\\tau`. taulist must be positive and contain the element `0`. c_ops : list list of collapse operators, may be time-dependent for solver choice of `me` or `mc`. a_op : Qobj operator A. b_op : Qobj operator B. reverse : bool {False, True} If `True`, calculate :math:`\left<A(t)B(t+\\tau)\\right>` instead of :math:`\left<A(t+\\tau)B(t)\\right>`. solver : str choice of solver (`me` for master-equation, `mc` for Monte Carlo, and `es` for exponential series). options : Options solver options class. `ntraj` is taken as a two-element list because the `mc` correlator calls `mcsolve()` recursively; by default, `ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in the `mc` correlator; by default, `mc_corr_eps=1e-10`. Returns ------- corr_mat : ndarray An 2-dimensional array (matrix) of correlation values for the times specified by `tlist` (first index) and `taulist` (second index). If `tlist` is `None`, then a 1-dimensional array of correlation values is returned instead. References ---------- See, Gardiner, Quantum Noise, Section 5.2. """ if debug: print(inspect.stack()[0][3]) if tlist is None: return correlation_2op_1t(H, state0, taulist, c_ops, a_op, b_op, solver=solver, reverse=reverse, args=args, options=options) else: if reverse: A_op = a_op B_op = b_op C_op = 1 else: A_op = 1 B_op = a_op C_op = b_op return _correlation_2t(H, state0, tlist, taulist, c_ops, A_op, B_op, C_op, solver=solver, args=args, options=options)
[docs]def correlation_3op_1t(H, state0, taulist, c_ops, a_op, b_op, c_op, solver="me", args={}, options=Options(ntraj=[20, 100])): """ Calculate the three-operator two-time correlation function: :math:`\left<A(t)B(t+\\tau)C(t)\\right>` along one time axis using the quantum regression theorem and the evolution solver indicated by the `solver` parameter. Note: it is not possibly to calculate a physically meaningful correlation of this form where :math:`\\tau<0`. Parameters ---------- H : Qobj system Hamiltonian, may be time-dependent for solver choice of `me` or `mc`. rho0 : Qobj Initial state density matrix :math:`\\rho(t_0)` or state vector :math:`\\psi(t_0)`. If 'state0' is 'None', then the steady state will be used as the initial state. The 'steady-state' is only implemented for the `me` and `es` solvers. taulist : array_like list of times for :math:`\\tau`. taulist must be positive and contain the element `0`. c_ops : list list of collapse operators, may be time-dependent for solver choice of `me` or `mc`. a_op : Qobj operator A. b_op : Qobj operator B. c_op : Qobj operator C. solver : str choice of solver (`me` for master-equation, `mc` for Monte Carlo, and `es` for exponential series). options : Options solver options class. `ntraj` is taken as a two-element list because the `mc` correlator calls `mcsolve()` recursively; by default, `ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in the `mc` correlator; by default, `mc_corr_eps=1e-10`. Returns ------- corr_vec : array An array of correlation values for the times specified by `taulist` References ---------- See, Gardiner, Quantum Noise, Section 5.2. """ if debug: print(inspect.stack()[0][3]) return _correlation_2t(H, state0, [0], taulist, c_ops, a_op, b_op, c_op, solver=solver, args=args, options=options)[0]
[docs]def correlation_3op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, solver="me", args={}, options=Options(ntraj=[20, 100])): """ Calculate the three-operator two-time correlation function: :math:`\left<A(t)B(t+\\tau)C(t)\\right>` along two time axes using the quantum regression theorem and the evolution solver indicated by the `solver` parameter. Note: it is not possibly to calculate a physically meaningful correlation of this form where :math:`\\tau<0`. Parameters ---------- H : Qobj system Hamiltonian, may be time-dependent for solver choice of `me` or `mc`. rho0 : Qobj Initial state density matrix :math:`\\rho_0` or state vector :math:`\\psi_0`. If 'state0' is 'None', then the steady state will be used as the initial state. The 'steady-state' is only implemented for the `me` and `es` solvers. tlist : array_like list of times for :math:`t`. tlist must be positive and contain the element `0`. When taking steady-steady correlations only one tlist value is necessary, i.e. when :math:`t \\rightarrow \\infty`; here tlist is automatically set, ignoring user input. taulist : array_like list of times for :math:`\\tau`. taulist must be positive and contain the element `0`. c_ops : list list of collapse operators, may be time-dependent for solver choice of `me` or `mc`. a_op : Qobj operator A. b_op : Qobj operator B. c_op : Qobj operator C. solver : str choice of solver (`me` for master-equation, `mc` for Monte Carlo, and `es` for exponential series). options : Options solver options class. `ntraj` is taken as a two-element list because the `mc` correlator calls `mcsolve()` recursively; by default, `ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in the `mc` correlator; by default, `mc_corr_eps=1e-10`. Returns ------- corr_mat : array An 2-dimensional array (matrix) of correlation values for the times specified by `tlist` (first index) and `taulist` (second index). If `tlist` is `None`, then a 1-dimensional array of correlation values is returned instead. References ---------- See, Gardiner, Quantum Noise, Section 5.2. """ if debug: print(inspect.stack()[0][3]) if tlist is None: return correlation_3op_1t(H, state0, taulist, c_ops, a_op, b_op, c_op, solver=solver, args=args, options=options) else: return _correlation_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, solver=solver, args=args, options=options)
# high level correlation
[docs]def coherence_function_g1(H, state0, taulist, c_ops, a_op, solver="me", args={}, options=Options(ntraj=[20, 100])): """ Calculate the normalized first-order quantum coherence function: .. math:: g^{(1)}(\\tau) = \\frac{\\langle A^\\dagger(\\tau)A(0)\\rangle} {\\sqrt{\\langle A^\\dagger(\\tau)A(\\tau)\\rangle \\langle A^\\dagger(0)A(0)\\rangle}} using the quantum regression theorem and the evolution solver indicated by the `solver` parameter. Parameters ---------- H : Qobj system Hamiltonian, may be time-dependent for solver choice of `me` or `mc`. state0 : Qobj Initial state density matrix :math:`\\rho(t_0)` or state vector :math:`\\psi(t_0)`. If 'state0' is 'None', then the steady state will be used as the initial state. The 'steady-state' is only implemented for the `me` and `es` solvers. taulist : array_like list of times for :math:`\\tau`. taulist must be positive and contain the element `0`. c_ops : list list of collapse operators, may be time-dependent for solver choice of `me` or `mc`. a_op : Qobj operator A. solver : str choice of solver (`me` for master-equation and `es` for exponential series). options : Options solver options class. `ntraj` is taken as a two-element list because the `mc` correlator calls `mcsolve()` recursively; by default, `ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in the `mc` correlator; by default, `mc_corr_eps=1e-10`. Returns ------- g1, G1 : tuple The normalized and unnormalized second-order coherence function. """ # first calculate the photon number if state0 is None: state0 = steadystate(H, c_ops) n = np.array([expect(state0, a_op.dag() * a_op)]) else: n = mesolve(H, state0, taulist, c_ops, [a_op.dag() * a_op], options=options).expect[0] # calculate the correlation function G1 and normalize with n to obtain g1 G1 = correlation_2op_1t(H, state0, taulist, c_ops, a_op.dag(), a_op, solver=solver, args=args, options=options) g1 = G1 / np.sqrt(n[0] * n) return g1, G1
[docs]def coherence_function_g2(H, state0, taulist, c_ops, a_op, solver="me", args={}, options=Options(ntraj=[20, 100])): """ Calculate the normalized second-order quantum coherence function: .. math:: g^{(2)}(\\tau) = \\frac{\\langle A^\\dagger(0)A^\\dagger(\\tau)A(\\tau)A(0)\\rangle} {\\langle A^\\dagger(\\tau)A(\\tau)\\rangle \\langle A^\\dagger(0)A(0)\\rangle} using the quantum regression theorem and the evolution solver indicated by the `solver` parameter. Parameters ---------- H : Qobj system Hamiltonian, may be time-dependent for solver choice of `me` or `mc`. state0 : Qobj Initial state density matrix :math:`\\rho(t_0)` or state vector :math:`\\psi(t_0)`. If 'state0' is 'None', then the steady state will be used as the initial state. The 'steady-state' is only implemented for the `me` and `es` solvers. taulist : array_like list of times for :math:`\\tau`. taulist must be positive and contain the element `0`. c_ops : list list of collapse operators, may be time-dependent for solver choice of `me` or `mc`. a_op : Qobj operator A. args : dict Dictionary of arguments to be passed to solver. solver : str choice of solver (`me` for master-equation and `es` for exponential series). options : Options solver options class. `ntraj` is taken as a two-element list because the `mc` correlator calls `mcsolve()` recursively; by default, `ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in the `mc` correlator; by default, `mc_corr_eps=1e-10`. Returns ------- g2, G2 : tuple The normalized and unnormalized second-order coherence function. """ # first calculate the photon number if state0 is None: state0 = steadystate(H, c_ops) n = np.array([expect(state0, a_op.dag() * a_op)]) else: n = mesolve(H, state0, taulist, c_ops, [a_op.dag() * a_op], args=args).expect[0] # calculate the correlation function G2 and normalize with n to obtain g2 G2 = correlation_3op_1t(H, state0, taulist, c_ops, a_op.dag(), a_op.dag()*a_op, a_op, solver=solver, args=args, options=options) g2 = G2 / (n[0] * n) return g2, G2
# spectrum
[docs]def spectrum(H, wlist, c_ops, a_op, b_op, solver="es", use_pinv=False): """ Calculate the spectrum of the correlation function :math:`\lim_{t \\to \\infty} \left<A(t+\\tau)B(t)\\right>`, i.e., the Fourier transform of the correlation function: .. math:: S(\omega) = \int_{-\infty}^{\infty} \lim_{t \\to \\infty} \left<A(t+\\tau)B(t)\\right> e^{-i\omega\\tau} d\\tau. using the solver indicated by the `solver` parameter. Note: this spectrum is only defined for stationary statistics (uses steady state rho0) Parameters ---------- H : :class:`qutip.qobj` system Hamiltonian. wlist : array_like list of frequencies for :math:`\\omega`. c_ops : list list of collapse operators. a_op : Qobj operator A. b_op : Qobj operator B. solver : str choice of solver (`es` for exponential series and `pi` for psuedo-inverse). use_pinv : bool For use with the `pi` solver: if `True` use numpy's pinv method, otherwise use a generic solver. Returns ------- spectrum : array An array with spectrum :math:`S(\omega)` for the frequencies specified in `wlist`. """ if debug: print(inspect.stack()[0][3]) if solver == "es": return _spectrum_es(H, wlist, c_ops, a_op, b_op) elif solver == "pi": return _spectrum_pi(H, wlist, c_ops, a_op, b_op, use_pinv) else: raise ValueError("Unrecognized choice of solver" + "%s (use es or pi)." % solver)
[docs]def spectrum_correlation_fft(tlist, y, inverse=False): """ Calculate the power spectrum corresponding to a two-time correlation function using FFT. Parameters ---------- tlist : array_like list/array of times :math:`t` which the correlation function is given. y : array_like list/array of correlations corresponding to time delays :math:`t`. inverse: boolean boolean parameter for using a positive exponent in the Fourier Transform instead. Default is False. Returns ------- w, S : tuple Returns an array of angular frequencies 'w' and the corresponding two-sided power spectrum 'S(w)'. """ if debug: print(inspect.stack()[0][3]) tlist = np.asarray(tlist) N = tlist.shape[0] dt = tlist[1] - tlist[0] if not np.allclose(np.diff(tlist), dt*np.ones(N-1,dtype=float)): raise Exception('tlist must be equally spaced for FFT.') if inverse: F = N * scipy.fftpack.ifft(y) else: F = scipy.fftpack.fft(y) # calculate the frequencies for the components in F f = scipy.fftpack.fftfreq(N, dt) # re-order frequencies from most negative to most positive (centre on 0) idx = np.array([], dtype = 'int') idx = np.append(idx, np.where(f < 0.0)) idx = np.append(idx, np.where(f >= 0.0)) return 2 * np.pi * f[idx], 2 * dt * np.real(F[idx])
# ----------------------------------------------------------------------------- # LEGACY API # ----------------------------------------------------------------------------- # low level correlation
[docs]def correlation_ss(H, taulist, c_ops, a_op, b_op, solver="me", reverse=False, args={}, options=Options(ntraj=[20, 100])): """ Calculate the two-operator two-time correlation function: .. math:: \lim_{t \\to \\infty} \left<A(t+\\tau)B(t)\\right> along one time axis (given steady-state initial conditions) using the quantum regression theorem and the evolution solver indicated by the `solver` parameter. Parameters ---------- H : Qobj system Hamiltonian. taulist : array_like list of times for :math:`\\tau`. taulist must be positive and contain the element `0`. c_ops : list list of collapse operators. a_op : Qobj operator A. b_op : Qobj operator B. reverse : *bool* If `True`, calculate :math:`\lim_{t \\to \\infty} \left<A(t)B(t+\\tau)\\right>` instead of :math:`\lim_{t \\to \\infty} \left<A(t+\\tau)B(t)\\right>`. solver : str choice of solver (`me` for master-equation and `es` for exponential series). options : Options solver options class. `ntraj` is taken as a two-element list because the `mc` correlator calls `mcsolve()` recursively; by default, `ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in the `mc` correlator; by default, `mc_corr_eps=1e-10`. Returns ------- corr_vec : array An array of correlation values for the times specified by `tlist`. References ---------- See, Gardiner, Quantum Noise, Section 5.2. """ warn("correlation_ss() now legacy, please use correlation_2op_1t() with" + "initial state as None", FutureWarning) if debug: print(inspect.stack()[0][3]) return correlation_2op_1t(H, None, taulist, c_ops, a_op, b_op, solver=solver, reverse=reverse, args=args, options=options)
[docs]def correlation(H, state0, tlist, taulist, c_ops, a_op, b_op, solver="me", reverse=False, args={}, options=Options(ntraj=[20, 100])): """ Calculate the two-operator two-time correlation function: :math:`\left<A(t+\\tau)B(t)\\right>` along two time axes using the quantum regression theorem and the evolution solver indicated by the `solver` parameter. Parameters ---------- H : Qobj system Hamiltonian, may be time-dependent for solver choice of `me` or `mc`. state0 : Qobj Initial state density matrix :math:`\\rho(t_0)` or state vector :math:`\\psi(t_0)`. If 'state0' is 'None', then the steady state will be used as the initial state. The 'steady-state' is only implemented for the `me` and `es` solvers. tlist : array_like list of times for :math:`t`. tlist must be positive and contain the element `0`. When taking steady-steady correlations only one tlist value is necessary, i.e. when :math:`t \\rightarrow \\infty`; here tlist is automatically set, ignoring user input. taulist : array_like list of times for :math:`\\tau`. taulist must be positive and contain the element `0`. c_ops : list list of collapse operators, may be time-dependent for solver choice of `me` or `mc`. a_op : Qobj operator A. b_op : Qobj operator B. reverse : *bool* If `True`, calculate :math:`\left<A(t)B(t+\\tau)\\right>` instead of :math:`\left<A(t+\\tau)B(t)\\right>`. solver : str choice of solver (`me` for master-equation, `mc` for Monte Carlo, and `es` for exponential series). options : Options solver options class. `ntraj` is taken as a two-element list because the `mc` correlator calls `mcsolve()` recursively; by default, `ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in the `mc` correlator; by default, `mc_corr_eps=1e-10`. Returns ------- corr_mat : array An 2-dimensional array (matrix) of correlation values for the times specified by `tlist` (first index) and `taulist` (second index). If `tlist` is `None`, then a 1-dimensional array of correlation values is returned instead. References ---------- See, Gardiner, Quantum Noise, Section 5.2. """ warn("correlation() now legacy, please use correlation_2op_2t()", FutureWarning) if debug: print(inspect.stack()[0][3]) return correlation_2op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, solver=solver, reverse=reverse, args=args, options=options)
[docs]def correlation_4op_1t(H, state0, taulist, c_ops, a_op, b_op, c_op, d_op, solver="me", args={}, options=Options(ntraj=[20, 100])): """ Calculate the four-operator two-time correlation function: :math:`\left<A(t)B(t+\\tau)C(t+\\tau)D(t)\\right>` along one time axis using the quantum regression theorem and the evolution solver indicated by the `solver` parameter. Note: it is not possibly to calculate a physically meaningful correlation of this form where :math:`\\tau<0`. Parameters ---------- H : Qobj system Hamiltonian, may be time-dependent for solver choice of `me` or `mc`. rho0 : Qobj Initial state density matrix :math:`\\rho(t_0)` or state vector :math:`\\psi(t_0)`. If 'state0' is 'None', then the steady state will be used as the initial state. The 'steady-state' is only implemented for the `me` and `es` solvers. taulist : array_like list of times for :math:`\\tau`. taulist must be positive and contain the element `0`. c_ops : list list of collapse operators, may be time-dependent for solver choice of `me` or `mc`. a_op : Qobj operator A. b_op : Qobj operator B. c_op : Qobj operator C. d_op : Qobj operator D. solver : str choice of solver (`me` for master-equation, `mc` for Monte Carlo, and `es` for exponential series). options : Options solver options class. `ntraj` is taken as a two-element list because the `mc` correlator calls `mcsolve()` recursively; by default, `ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in the `mc` correlator; by default, `mc_corr_eps=1e-10`. Returns ------- corr_vec : array An array of correlation values for the times specified by `taulist`. References ---------- See, Gardiner, Quantum Noise, Section 5.2. .. note:: Deprecated in QuTiP 3.1 Use correlation_3op_1t() instead. """ warn("correlation_4op_1t() now legacy, please use correlation_3op_1t()", FutureWarning) warn("the reverse argument has been removed as it did not contain any" + "new physical information", DeprecationWarning) if debug: print(inspect.stack()[0][3]) return correlation_3op_1t(H, state0, taulist, c_ops, a_op, b_op * c_op, d_op, solver=solver, args=args, options=options)
[docs]def correlation_4op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, d_op, solver="me", args={}, options=Options(ntraj=[20, 100])): """ Calculate the four-operator two-time correlation function: :math:`\left<A(t)B(t+\\tau)C(t+\\tau)D(t)\\right>` along two time axes using the quantum regression theorem and the evolution solver indicated by the `solver` parameter. Note: it is not possibly to calculate a physically meaningful correlation of this form where :math:`\\tau<0`. Parameters ---------- H : Qobj system Hamiltonian, may be time-dependent for solver choice of `me` or `mc`. rho0 : Qobj Initial state density matrix :math:`\\rho_0` or state vector :math:`\\psi_0`. If 'state0' is 'None', then the steady state will be used as the initial state. The 'steady-state' is only implemented for the `me` and `es` solvers. tlist : array_like list of times for :math:`t`. tlist must be positive and contain the element `0`. When taking steady-steady correlations only one tlist value is necessary, i.e. when :math:`t \\rightarrow \\infty`; here tlist is automatically set, ignoring user input. taulist : array_like list of times for :math:`\\tau`. taulist must be positive and contain the element `0`. c_ops : list list of collapse operators, may be time-dependent for solver choice of `me` or `mc`. a_op : Qobj operator A. b_op : Qobj operator B. c_op : Qobj operator C. d_op : Qobj operator D. solver : str choice of solver (`me` for master-equation, `mc` for Monte Carlo, and `es` for exponential series). options : Options solver options class. `ntraj` is taken as a two-element list because the `mc` correlator calls `mcsolve()` recursively; by default, `ntraj=[20, 100]`. `mc_corr_eps` prevents divide-by-zero errors in the `mc` correlator; by default, `mc_corr_eps=1e-10`. Returns ------- corr_mat : array An 2-dimensional array (matrix) of correlation values for the times specified by `tlist` (first index) and `taulist` (second index). If `tlist` is `None`, then a 1-dimensional array of correlation values is returned instead. References ---------- See, Gardiner, Quantum Noise, Section 5.2. """ warn("correlation_4op_2t() now legacy, please use correlation_3op_2t()", FutureWarning) warn("the reverse argument has been removed as it did not contain any" + "new physical information", DeprecationWarning) if debug: print(inspect.stack()[0][3]) return correlation_3op_2t(H, state0, tlist, taulist, c_ops, a_op, b_op * c_op, d_op, solver=solver, args=args, options=options)
# spectrum
[docs]def spectrum_ss(H, wlist, c_ops, a_op, b_op): """ Calculate the spectrum of the correlation function :math:`\lim_{t \\to \\infty} \left<A(t+\\tau)B(t)\\right>`, i.e., the Fourier transform of the correlation function: .. math:: S(\omega) = \int_{-\infty}^{\infty} \lim_{t \\to \\infty} \left<A(t+\\tau)B(t)\\right> e^{-i\omega\\tau} d\\tau. using an eseries based solver Note: this spectrum is only defined for stationary statistics (uses steady state rho0). Parameters ---------- H : :class:`qutip.qobj` system Hamiltonian. wlist : array_like list of frequencies for :math:`\\omega`. c_ops : *list* of :class:`qutip.qobj` list of collapse operators. a_op : :class:`qutip.qobj` operator A. b_op : :class:`qutip.qobj` operator B. use_pinv : *bool* If `True` use numpy's `pinv` method, otherwise use a generic solver. Returns ------- spectrum : array An array with spectrum :math:`S(\omega)` for the frequencies specified in `wlist`. """ warn("spectrum_ss() now legacy, please use spectrum()", FutureWarning) return spectrum(H, wlist, c_ops, a_op, b_op, solver="es")
[docs]def spectrum_pi(H, wlist, c_ops, a_op, b_op, use_pinv=False): """ Calculate the spectrum of the correlation function :math:`\lim_{t \\to \\infty} \left<A(t+\\tau)B(t)\\right>`, i.e., the Fourier transform of the correlation function: .. math:: S(\omega) = \int_{-\infty}^{\infty} \lim_{t \\to \\infty} \left<A(t+\\tau)B(t)\\right> e^{-i\omega\\tau} d\\tau. using a psuedo-inverse method. Note: this spectrum is only defined for stationary statistics (uses steady state rho0) Parameters ---------- H : :class:`qutip.qobj` system Hamiltonian. wlist : array_like list of frequencies for :math:`\\omega`. c_ops : *list* of :class:`qutip.qobj` list of collapse operators. a_op : :class:`qutip.qobj` operator A. b_op : :class:`qutip.qobj` operator B. use_pinv : *bool* If `True` use numpy's pinv method, otherwise use a generic solver. Returns ------- spectrum : array An array with spectrum :math:`S(\omega)` for the frequencies specified in `wlist`. """ warn("spectrum_pi() now legacy, please use spectrum()", FutureWarning) return spectrum(H, wlist, c_ops, a_op, b_op, solver="pi", use_pinv=use_pinv)
# ----------------------------------------------------------------------------- # PRIVATE SOLVER METHODS # ----------------------------------------------------------------------------- # master 2t correlation solver def _correlation_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, solver="me", args={}, options=Options()): """ Internal function for calling solvers in order to calculate the three-operator two-time correlation function: <A(t)B(t+tau)C(t)> """ # Note: the current form of the correlator is sufficient for all possible # two-time correlations (incuding those with 2ops vs 3). Ex: to compute a # correlation of the form <A(t+tau)B(t)>: a_op = identity, b_op = A, # and c_op = B. if debug: print(inspect.stack()[0][3]) if min(tlist) != 0: raise TypeError("tlist must be positive and contain the element 0.") if min(taulist) != 0: raise TypeError("taulist must be positive and contain the element 0.") if config.tdname: _cython_build_cleanup(config.tdname) rhs_clear() H, c_ops, args = _td_wrap_array_str(H, c_ops, args, tlist) if solver == "me": return _correlation_me_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, args=args, options=options) elif solver == "mc": return _correlation_mc_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, args=args, options=options) elif solver == "es": return _correlation_es_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op) else: raise ValueError("Unrecognized choice of solver" + "%s (use me, mc, or es)." % solver) # master equation solvers def _correlation_me_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, args={}, options=Options()): """ Internal function for calculating the three-operator two-time correlation function: <A(t)B(t+tau)C(t)> using a master equation solver. """ # the solvers only work for positive time differences and the correlators # require positive tau if state0 is None: rho0 = steadystate(H, c_ops) tlist = [0] elif isket(state0): rho0 = ket2dm(state0) else: rho0 = state0 if debug: print(inspect.stack()[0][3]) rho_t = mesolve(H, rho0, tlist, c_ops, [], args=args, options=options).states corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex) H_shifted, c_ops_shifted, _args = _transform_L_t_shift(H, c_ops, args) if config.tdname: _cython_build_cleanup(config.tdname) rhs_clear() for t_idx, rho in enumerate(rho_t): if not isinstance(H, Qobj): _args["_t0"] = tlist[t_idx] corr_mat[t_idx, :] = mesolve( H_shifted, c_op * rho * a_op, taulist, c_ops_shifted, [b_op], args=_args, options=options ).expect[0] if t_idx == 1: options.rhs_reuse = True if config.tdname: _cython_build_cleanup(config.tdname) rhs_clear() return corr_mat # exponential series solvers def _correlation_es_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op): """ Internal function for calculating the three-operator two-time correlation function: <A(t)B(t+tau)C(t)> using an exponential series solver. """ # the solvers only work for positive time differences and the correlators # require positive tau if state0 is None: rho0 = steadystate(H, c_ops) tlist = [0] elif isket(state0): rho0 = ket2dm(state0) else: rho0 = state0 if debug: print(inspect.stack()[0][3]) # contruct the Liouvillian L = liouvillian(H, c_ops) corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex) solES_t = ode2es(L, rho0) # evaluate the correlation function for t_idx in range(len(tlist)): rho_t = esval(solES_t, [tlist[t_idx]]) solES_tau = ode2es(L, c_op * rho_t * a_op) corr_mat[t_idx, :] = esval(expect(b_op, solES_tau), taulist) return corr_mat def _spectrum_es(H, wlist, c_ops, a_op, b_op): """ Internal function for calculating the spectrum of the correlation function :math:`\left<A(\\tau)B(0)\\right>`. """ if debug: print(inspect.stack()[0][3]) # construct the Liouvillian L = liouvillian(H, c_ops) # find the steady state density matrix and a_op and b_op expecation values rho0 = steadystate(L) a_op_ss = expect(a_op, rho0) b_op_ss = expect(b_op, rho0) # eseries solution for (b * rho0)(t) es = ode2es(L, b_op * rho0) # correlation corr_es = expect(a_op, es) # covariance cov_es = corr_es - a_op_ss * b_op_ss # tidy up covariance (to combine, e.g., zero-frequency components that cancel) cov_es.tidyup() # spectrum spectrum = esspec(cov_es, wlist) return spectrum # Monte Carlo solvers def _correlation_mc_2t(H, state0, tlist, taulist, c_ops, a_op, b_op, c_op, args={}, options=Options()): """ Internal function for calculating the three-operator two-time correlation function: <A(t)B(t+tau)C(t)> using a Monte Carlo solver. """ if not c_ops: raise TypeError("If no collapse operators are required, use the `me`" + "or `es` solvers") # the solvers only work for positive time differences and the correlators # require positive tau if state0 is None: raise NotImplementedError("steady state not implemented for " + "mc solver, please use `es` or `me`") elif not isket(state0): raise TypeError("state0 must be a state vector.") psi0 = state0 if debug: print(inspect.stack()[0][3]) psi_t_mat = mcsolve( H, psi0, tlist, c_ops, [], args=args, ntraj=options.ntraj[0], options=options, progress_bar=None ).states corr_mat = np.zeros([np.size(tlist), np.size(taulist)], dtype=complex) H_shifted, c_ops_shifted, _args = _transform_L_t_shift(H, c_ops, args) if config.tdname: _cython_build_cleanup(config.tdname) rhs_clear() # calculation of <A(t)B(t+tau)C(t)> from only knowledge of psi0 requires # averaging over both t and tau for t_idx in range(np.size(tlist)): if not isinstance(H, Qobj): _args["_t0"] = tlist[t_idx] for trial_idx in range(options.ntraj[0]): if isinstance(a_op, Qobj) and isinstance(c_op, Qobj): if a_op.dag() == c_op: # A shortcut here, requires only 1/4 the trials chi_0 = (options.mc_corr_eps + c_op) * \ psi_t_mat[trial_idx, t_idx] # evolve these states and calculate expectation value of B c_tau = chi_0.norm()**2 * mcsolve( H_shifted, chi_0/chi_0.norm(), taulist, c_ops_shifted, [b_op], args=_args, ntraj=options.ntraj[1], options=options, progress_bar=None ).expect[0] # final correlation vector computed by combining the # averages corr_mat[t_idx, :] += c_tau/options.ntraj[1] else: # otherwise, need four trial wavefunctions # (Ad+C)*psi_t, (Ad+iC)*psi_t, (Ad-C)*psi_t, (Ad-iC)*psi_t if isinstance(a_op, Qobj): a_op_dag = a_op.dag() else: # assume this is a number, ex. i.e. a_op = 1 # if this is not correct, the over-loaded addition # operation will raise errors a_op_dag = a_op chi_0 = [(options.mc_corr_eps + a_op_dag + np.exp(1j*x*np.pi/2)*c_op) * psi_t_mat[trial_idx, t_idx] for x in range(4)] # evolve these states and calculate expectation value of B c_tau = [ chi.norm()**2 * mcsolve( H_shifted, chi/chi.norm(), taulist, c_ops_shifted, [b_op], args=_args, ntraj=options.ntraj[1], options=options, progress_bar=None ).expect[0] for chi in chi_0 ] # final correlation vector computed by combining the averages corr_mat_add = np.asarray( 1.0 / (4*options.ntraj[0]) * (c_tau[0] - c_tau[2] - 1j*c_tau[1] + 1j*c_tau[3]), dtype=corr_mat.dtype ) corr_mat[t_idx, :] += corr_mat_add if t_idx == 1: options.rhs_reuse = True if config.tdname: _cython_build_cleanup(config.tdname) rhs_clear() return corr_mat # pseudo-inverse solvers def _spectrum_pi(H, wlist, c_ops, a_op, b_op, use_pinv=False): """ Internal function for calculating the spectrum of the correlation function :math:`\left<A(\\tau)B(0)\\right>`. """ L = H if issuper(H) else liouvillian(H, c_ops) tr_mat = tensor([qeye(n) for n in L.dims[0][0]]) N = np.prod(L.dims[0][0]) A = L.full() b = spre(b_op).full() a = spre(a_op).full() tr_vec = np.transpose(mat2vec(tr_mat.full())) rho_ss = steadystate(L) rho = np.transpose(mat2vec(rho_ss.full())) I = np.identity(N * N) P = np.kron(np.transpose(rho), tr_vec) Q = I - P spectrum = np.zeros(len(wlist)) for idx, w in enumerate(wlist): if use_pinv: MMR = np.linalg.pinv(-1.0j * w * I + A) else: MMR = np.dot(Q, np.linalg.solve(-1.0j * w * I + A, Q)) s = np.dot(tr_vec, np.dot(a, np.dot(MMR, np.dot(b, np.transpose(rho))))) spectrum[idx] = -2 * np.real(s[0, 0]) return spectrum # auxiliary def _transform_L_t_shift(H, c_ops, args={}): """ Time shift the Hamiltonian with private time-shift variable _t0 """ # while this list doesn't seem exhaustive, mesolve has already been called # and has hence called _td_format_check from qutip.rhs_generate. important # to keep in mind is that mesolve already requires the types of # time-dependence to be the same for the hamiltonian as for the collapse # operators. if isinstance(H, Qobj): # constant hamiltonian H_shifted = H # not shifted! c_ops_is_td = False if isinstance(c_ops, list): for i in range(len(c_ops)): # test is collapse operators are time-dependent if isinstance(c_ops[i], list): c_ops_is_td = True if not c_ops_is_td: # constant collapse operators c_ops_shifted = c_ops # not shifted! if isinstance(H, Qobj) and not c_ops_is_td: # constant hamiltonian and collapse operators _args = args # not shifted! if isinstance(H, types.FunctionType): # function-callback based time-dependence if isinstance(args, dict) or args is None: if args is None: _args = {"_t0": 0} else: _args = args.copy() _args["_t0"] = 0 H_shifted = lambda t, args_i: H(t+args_i["_t0"], args_i) else: raise TypeError("If using function-callback based Hamiltonian" + "time-dependence, args must be a dictionary") if isinstance(H, list) or c_ops_is_td: # string/function-list based time-dependence if args is None: _args = {"_t0": 0} elif isinstance(args, dict): _args = args.copy() _args["_t0"] = 0 else: _args = {"_user_args": args, "_t0": 0} if isinstance(H, list): # hamiltonian is time-dependent H_shifted = [] for i in range(len(H)): if isinstance(H[i], list): # modify Hamiltonian time dependence in accordance with the # quantum regression theorem if isinstance(args, dict) or args is None: if isinstance(H[i][1], types.FunctionType): # function-list based time-dependence fn = lambda t, args_i: \ H[i][1](t + args_i["_t0"], args_i) else: # string-list based time-dependence # Again, note: _td_format_check already raises # errors formixed td formatting fn = sub("(?<=[^0-9a-zA-Z_])t(?=[^0-9a-zA-Z_])", "(t+_t0)", H[i][1]) else: if isinstance(H[i][1], types.FunctionType): # function-list based time-dependence fn = lambda t, args_i: \ H[i][1](t + args_i["_t0"], args_i["_user_args"]) else: raise TypeError("If using string-list based" + "Hamiltonian time-dependence, " + "args must be a dictionary") H_shifted.append([H[i][0], fn]) else: H_shifted.append(H[i]) if c_ops_is_td: # collapse operators are time-dependent c_ops_shifted = [] for i in range(len(c_ops)): if isinstance(c_ops[i], list): # modify collapse operators time dependence in accordance # with the quantum regression theorem if isinstance(args, dict) or args is None: if isinstance(c_ops[i][1], types.FunctionType): # function-list based time-dependence fn = lambda t, args_i: \ c_ops[i][1](t + args_i["_t0"], args_i) else: # string-list based time-dependence # Again, note: _td_format_check already raises # errors formixed td formatting fn = sub("(?<=[^0-9a-zA-Z_])t(?=[^0-9a-zA-Z_])", "(t+_t0)", c_ops[i][1]) else: if isinstance(H[i][1], types.FunctionType): # function-list based time-dependence fn = lambda t, args_i: \ c_ops[i][1](t + args_i["_t0"], args_i["_user_args"]) else: raise TypeError("If using string-list based" + "collapse operator" + "time-dependence, " + "args must be a dictionary") c_ops_shifted.append([c_ops[i][0], fn]) else: c_ops_shifted.append(c_ops[i]) return H_shifted, c_ops_shifted, _args