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Davide Candoli edited this page Mar 22, 2021 · 30 revisions

File Hamiltonians.py includes the definition of all the functions which generate the terms of the Hamiltonian of the nuclear spin.

In quantum mechanics, the Hamiltonian of a physical system is the operator associated to its energy. It is a privileged observable, since it plays a central role in the problem of the dynamics of the system.

In the present simulation program, we study the dynamics of a nuclear spin under the effect of three main interactions:

  1. Zeeman interaction with an external stationary magnetic field;
  2. Quadrupolar interaction with the electric field gradient generated by surrounding electrons;
  3. Electromagnetic coupling with an oscillating magnetic field applied in the time-interval [0, t].

Each one of these contributions is represented by a distinct function, which takes as arguments the dynamical and fixed parameters which appear in the expression of the Hamiltonian (including time, in the third case) and returns an Observable object which embodies the corresponding Hamiltonian. In the general case, the third contribution is made up of the superposition of many independent single-mode electromagnetic waves. For this reason, two functions have been defined: H_Single_Mode_Pulse and H_Multiple_Mode_Pulse.

Functions

  • h_zeeman(spin, theta_z, phi_z, B_0)

    Computes the term of the Hamiltonian associated with the Zeeman interaction between the nuclear spin and the external static field. The mathematical formula for this quantity is the following:

    Parameters

    • spin: Nuclear_Spin

      Spin under study;

    • theta_z: float

      Polar angle of the magnetic field in the laboratory coordinate system (expressed in radians);

    • phi_z: float

      Azimuthal angle of the magnetic field in the laboratory coordinate system (expressed in radians);

    • B_0: non-negative float

      Magnitude of the external magnetic field (expressed in tesla).

    Returns

    An Observable object which represents the Zeeman Hamiltonian in the laboratory reference frame (expressed in MHz).

    Raises

    ValueError, when the passed B_0 is a negative number.

  • h_quadrupole(spin, e2qQ, eta, alpha_q, beta_q, gamma_q)

    Computes the term of the Hamiltonian associated with the quadrupolar interaction. The mathematical formula for this quantity is the following:

    where V0, V+/-1, V+/-2 are the electric field gradient (EFG) tensor's spherical components (divided by the parameter eq), represented by the functions v0_EFG, v1_EFG, v2_EFG described below.

    Parameters

    • spin: Nuclear_Spin

      Spin under study;

    • e2qQ: float

      Product of the quadrupole moment constant, eQ, and the eigenvalue of the EFG tensor which is greatest in absolute value, eq. e2qQ is measured in MHz;

    • eta: float in the interval [0, 1]

    Asymmetry parameter of the EFG;

    • alpha_q, beta_q, gamma_q: float

    Euler angles for the conversion from the lab coordinate system (LAB) to the system of the principal axes of the EFG tensor (PAS) (expressed in radians).

    Returns

    If the quantum number of the spin is 1/2, the whole calculation is skipped and a null Observable object is returned.

    Otherwise, the function returns the Observable object which represents the quadrupolar Hamiltonian in the laboratory reference frame (expressed in MHz).

  • v0_EFG(eta, alpha_q, beta_q, gamma_q)

    Returns the component V0 of the EFG tensor (divided by eq) as seen in the LAB system. This quantity is expressed in terms of the Euler angles which relate PAS and LAB systems and the parameter eta as shown in the following formula:

    Parameters

    • eta: float in the interval [0, 1]

      Asymmetry parameter of the EFG;

    • alpha_q, beta_q, gamma_q: float

      Euler angles for the conversion from the lab coordinate system (LAB) to the system of the principal axes of the EFG tensor (PAS) (expressed in radians).

    Returns

    A float representing the component V0 (divided by eq) of the EFG tensor evaluated in the LAB system.

    Raises

    ValueError, when the passed eta is not in the interval [0, 1].

  • v1_EFG(sign, eta, alpha_q, beta_q, gamma_q)

    Returns the components V+/-1 of the EFG tensor (divided by eq) as seen in the LAB system. These quantities are expressed in terms of the Euler angles which relate PAS and LAB systems and the parameter eta as shown in the following formula:

    Parameters

    • sign: float

      Specifies wether the V+1 or the V-1 component is to be computed;

    • eta: float in the interval [0, 1]

      Asymmetry parameter of the EFG;

    • alpha_q, beta_q, gamma_q: float

      Euler angles for the conversion from the lab coordinate system (LAB) to the system of the principal axes of the EFG tensor (PAS) (expressed in radians).

    Returns

    A complex number representing the component:

    • V+1, if sign is positive;

    • V-1, if sign is negative.

    of the EFG tensor (divided by eq).

    Raises

    ValueError, when the passed eta is not in the interval [0, 1].

  • v2_EFG(sign, eta, alpha_q, beta_q, gamma_q)

    Returns the components V+/-2 of the EFG tensor (divided by eq) as seen in the LAB system. These quantities are expressed in terms of the Euler angles which relate PAS and LAB systems and the parameter eta as shown in the following formula:

    Parameters

    • sign: float

      Specifies wether the V+2 or the V-2 component is to be returned;

    • eta: float in the interval [0, 1]

      Asymmetry parameter of the EFG tensor;

    • alpha_q, beta_q, gamma_q: float

      Euler angles for the conversion from the lab coordinate system (LAB) to the system of the principal axes of the EFG tensor (PAS) (expressed in radians).

    Returns

    A float representing the component:

    • V+2, if sign is positive;

    • V-2, if sign is negative.

    of the EFG tensor (divided by eq).

    Raises

    ValueError, when the passed eta is not in the interval [0, 1].

  • h_single_mode_pulse(spin, frequency, phase, B_1, theta_1, phi_1, t)

    Computes the term of the Hamiltonian describing the interaction with a monochromatic and linearly polarized electromagnetic pulse. The mathematical formula for this quantity is the following:

    where all quantities labelled with the subscript 1 belong to the oscillating magnetic component B1 of the electromagnetic wave.

    Parameters

    • spin: Nuclear_Spin

      Spin under study.

    • frequency: non-negative float

      Frequency of the monochromatic wave (expressed in MHz).

    • phase: float

      Inital phase of the wave (at t=0) (expressed in radians).

    • B_1: non-negative float

      Maximum amplitude of the oscillating magnetic field (expressed in tesla).

    • theta_1, phi_1: float

      Polar and azimuthal angles of the direction of polarization of the magnetic wave in the LAB frame (expressed in radians);

    • t: float

      Time of evaluation of the Hamiltonian (expressed in microseconds).

    Returns

    An Observable object which represents the Hamiltonian of the coupling with the electromagnetic pulse evaluated at time t (expressed in MHz).

    Raises

    ValueError, in two distinct cases:

    1. When the passed frequency parameter is a negative quantity;
    2. When the passed B_1 parameter is a negative quantity.

  • h_multiple_mode_pulse(spin, mode, t)

    Computes the term of the Hamiltonian describing the interaction with a superposition of single-mode electromagnetic pulses. If the passed argument spin is a Nuclear_Spin object, the returned Hamiltonian will describe the interaction between the pulse of radiation and the single spin; if it is a Many_Spins object, it will represent the interaction with the whole system of many spins. The mathematical formula of this term of the Hamiltonian for a single spin is the following:

    where all quantities labelled with the subscript i belong to the single mode of the magnetic field Bi.

    Parameters

    • spin: Nuclear_Spin or Many_Spins

      Spin or spin system under study;

    • mode: pandas.DataFrame

      Table of the parameters of each electromagnetic mode in the superposition. It is organised according to the following template:

      index 'frequency' 'amplitude' 'phase' 'theta_p' 'phi_p'
      (MHz) (T) (rad) (rad) (rad)
      0 omega_0 B_0 phase_0 theta_0 phi_0
      1 omega_1 B_1 phase_1 theta_1 phi_1
      ... ... ... ... ... ...
      N omega_N B_N phase_N theta_N phi_N

      where the meaning of each column is analogous to the corresponding parameters in h_single_mode_pulse.

    • t: float

      Time of evaluation of the Hamiltonian (expressed in microseconds).

    Returns

    An Observable object which represents the Hamiltonian of the coupling with the superposition of the given modes evaluated at time t (expressed in MHz).

  • h_changed_picture(spin, mode, h_unperturbed, h_change_of_picture, t)

    Returns the global Hamiltonian of the system, made up of the time-dependent term h_multiple_mode_pulse(spin, mode, t) and the stationary term h_unperturbed, cast in the picture generated by h_change_of_picture.

    Parameters

    • spin, mode, t: same meaning as the corresponding arguments of h_multiple_mode_pulse;

    • h_unperturbed: Operator

      Stationary term of the global Hamiltonian (in MHz);

    • h_change_of_picture: Operator

      Operator which generates the new picture (in MHz).

    Returns

    Observable object representing the Hamiltonian of the pulse evaluated at time t in the new picture (in MHz).

  • h_j_coupling(spins, j_matrix)

    Returns the term of the Hamiltonian describing the J-coupling between the spins of a system of many nuclei, computed through the following formula

    Parameters

    • spins: Many_Spins

      Spins' system under study;

    • j_matrix: np.ndarray

      Array storing the coefficients Jmn which enter the formula above.

      Remark: j_matrix doesn't have to be symmetric, since the function reads only those elements located in the upper half of the array with respect to its diagonal. This means that the elements j_matrix[m, n] which matter are those for which m<n.

    Returns

    Observable object acting on the full Hilbert space of the spins' system representing the Hamiltonian of the J-coupling between the spins.

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