.. _time:

*************************************************
Solving Problems with Time-dependent Hamiltonians
*************************************************


Methods for Writing Time-Dependent Operators
============================================

In the previous examples of quantum evolution,
we assumed that the systems under consideration were described by time-independent Hamiltonians.
However, many systems have explicit time dependence in either the Hamiltonian,
or the collapse operators describing coupling to the environment, and sometimes both components might depend on time.
The time-evolutions  solvers
:func:`qutip.mesolve`, :func:`qutip.mcsolve`, :func:`qutip.sesolve`, :func:`qutip.bloch_redfield.brmesolve`
:func:`qutip.stochastic.ssesolve`, :func:`qutip.stochastic.photocurrent_sesolve`, :func:`qutip.stochastic.smesolve`, and :func:`qutip.stochastic.photocurrent_mesolve`
are all capable of handling time-dependent Hamiltonians and collapse terms.
There are, in general, three different ways to implement time-dependent problems in QuTiP:


1. **Function based**: Hamiltonian / collapse operators expressed using [qobj, func] pairs, where the time-dependent coefficients of the Hamiltonian (or collapse operators) are expressed using Python functions.

2. **String (Cython) based**: The Hamiltonian and/or collapse operators are expressed as a list of [qobj, string] pairs, where the time-dependent coefficients are represented as strings.  The resulting Hamiltonian is then compiled into C code using Cython and executed.

3. **Array Based**: The Hamiltonian and/or collapse operators are expressed as a list of [qobj, np.array] pairs. The arrays are 1 dimensional and dtype are complex or float. They must contain one value for each time in the tlist given to the solver. Cubic spline interpolation will be used between the given times.

4. **Hamiltonian function (outdated)**: The Hamiltonian is itself a Python function with time-dependence.  Collapse operators must be time independent using this input format.


Given the multiple choices of input style, the first question that arrises is which option to choose?
In short, the function based method (option #1) is the most general,
allowing for essentially arbitrary coefficients expressed via user-defined functions.
However, by automatically compiling your system into C++ code,
the second option (string based) tends to be more efficient and will run faster
[This is also the only format that is supported in the :func:`qutip.bloch_redfield.brmesolve` solver].
Of course, for small system sizes and evolution times, the difference will be minor.
Although this method does not support all time-dependent coefficients that one can think of,
it does support essentially all problems that one would typically encounter.
Time-dependent coefficients using any of the following functions,
or combinations thereof (including constants) can be compiled directly into C++-code::

  'abs', 'acos', 'acosh', 'arg', 'asin', 'asinh', 'atan', 'atanh', 'conj',
   'cos', 'cosh','exp', 'erf', 'zerf', 'imag', 'log', 'log10', 'norm', 'pi',
   'proj', 'real', 'sin', 'sinh', 'sqrt', 'tan', 'tanh'

In addition, QuTiP supports cubic spline based interpolation functions [:ref:`time-interp`].

If you require mathematical functions other than those listed above,
it is possible to call any of the functions in the NumPy library using the prefix ``np.``
before the function name in the string, i.e. ``'np.sin(t)'`` and  ``scipy.special`` imported as ``spe``.
This includes a wide range of functionality, but comes with a small overhead created by going from C++->Python->C++.

Finally option #4, expressing the Hamiltonian as a Python function,
is the original method for time dependence in QuTiP 1.x.
This method is somewhat less efficient then the previously mentioned ones.
However, in contrast to the other options
this method can be used in implementing time-dependent Hamiltonians that cannot be
expressed as a function of constant operators with time-dependent coefficients.

A collection of examples demonstrating the simulation of time-dependent problems can be found on the `tutorials <https://qutip.org/tutorials.html>`_ web page.

.. _time-function:

Function Based Time Dependence
==============================

A very general way to write a time-dependent Hamiltonian or collapse operator is by using Python functions as the time-dependent coefficients.  To accomplish this, we need to write a Python function that returns the time-dependent coefficient.  Additionally, we need to tell QuTiP that a given Hamiltonian or collapse operator should be associated with a given Python function.  To do this, one needs to specify operator-function pairs in list format: ``[Op, py_coeff]``, where ``Op`` is a given Hamiltonian or collapse operator and ``py_coeff`` is the name of the Python function representing the coefficient.  With this format, the form of the Hamiltonian for both ``mesolve`` and ``mcsolve`` is:

>>> H = [H0, [H1, py_coeff1], [H2, py_coeff2], ...] # doctest: +SKIP

where ``H0`` is a time-independent Hamiltonian, while ``H1`` and ``H2`` are time-dependent. The same format can be used for collapse operators:

>>> c_ops = [[C0, py_coeff0], C1, [C2, py_coeff2], ...] # doctest: +SKIP

Here we have demonstrated that the ordering of time-dependent and time-independent terms does not matter.  In addition, any or all of the collapse operators may be time-dependent.

.. note:: While, in general, you can arrange time-dependent and time-independent terms in any order you like, it is best to place all time-independent terms first.

As an example, we will look at an example that has a time-dependent Hamiltonian of the form :math:`H=H_{0}-f(t)H_{1}` where :math:`f(t)` is the time-dependent driving strength given as :math:`f(t)=A\exp\left[-\left( t/\sigma \right)^{2}\right]`.  The following code sets up the problem

.. plot::
    :context: reset
    :nofigs:

    ustate = basis(3, 0)
    excited = basis(3, 1)
    ground = basis(3, 2)

    N = 2 # Set where to truncate Fock state for cavity
    sigma_ge = tensor(qeye(N), ground * excited.dag())  # |g><e|
    sigma_ue = tensor(qeye(N), ustate * excited.dag())  # |u><e|
    a = tensor(destroy(N), qeye(3))
    ada = tensor(num(N), qeye(3))

    c_ops = []  # Build collapse operators
    kappa = 1.5 # Cavity decay rate
    c_ops.append(np.sqrt(kappa) * a)
    gamma = 6  # Atomic decay rate
    c_ops.append(np.sqrt(5*gamma/9) * sigma_ue) # Use Rb branching ratio of 5/9 e->u
    c_ops.append(np.sqrt(4*gamma/9) * sigma_ge) # 4/9 e->g

    t = np.linspace(-15, 15, 100) # Define time vector
    psi0 = tensor(basis(N, 0), ustate) # Define initial state

    state_GG = tensor(basis(N, 1), ground) # Define states onto which to project
    sigma_GG = state_GG * state_GG.dag()
    state_UU = tensor(basis(N, 0), ustate)
    sigma_UU = state_UU * state_UU.dag()

    g = 5  # coupling strength
    H0 = -g * (sigma_ge.dag() * a + a.dag() * sigma_ge)  # time-independent term
    H1 = (sigma_ue.dag() + sigma_ue)  # time-dependent term

Given that we have a single time-dependent Hamiltonian term, and constant collapse terms, we need to specify a single Python function for the coefficient :math:`f(t)`.  In this case, one can simply do

.. plot::
    :context:
    :nofigs:

    def H1_coeff(t, args):
        return 9 * np.exp(-(t / 5.) ** 2)

In this case, the return value depends only on time.  However, when specifying Python functions for coefficients, **the function must have (t,args) as the input variables, in that order**.  Having specified our coefficient function, we can now specify the Hamiltonian in list format and call the solver (in this case :func:`qutip.mesolve`)

.. plot::
    :context:
    :nofigs:

    H = [H0,[H1, H1_coeff]]
    output = mesolve(H, psi0, t, c_ops, [ada, sigma_UU, sigma_GG])

We can call the Monte Carlo solver in the exact same way (if using the default ``ntraj=500``):


..
  Hacky fix because plot has complicated conditional code execution

.. doctest::
    :skipif: True

    output = mcsolve(H, psi0, t, c_ops, [ada, sigma_UU, sigma_GG])

The output from the master equation solver is identical to that shown in the examples, the Monte Carlo however will be noticeably off, suggesting we should increase the number of trajectories for this example.  In addition, we can also consider the decay of a simple Harmonic oscillator with time-varying decay rate

.. plot::
    :context:
    :nofigs:

    kappa = 0.5

    def col_coeff(t, args):  # coefficient function
        return np.sqrt(kappa * np.exp(-t))

    N = 10  # number of basis states
    a = destroy(N)
    H = a.dag() * a  # simple HO
    psi0 = basis(N, 9)  # initial state
    c_ops = [[a, col_coeff]]  # time-dependent collapse term
    times = np.linspace(0, 10, 100)
    output = mesolve(H, psi0, times, c_ops, [a.dag() * a])


Using the args variable
------------------------
In the previous example we hardcoded all of the variables, driving amplitude :math:`A` and width :math:`\sigma`, with their numerical values.  This is fine for problems that are specialized, or that we only want to run once.  However, in many cases, we would like to change the parameters of the problem in only one location (usually at the top of the script), and not have to worry about manually changing the values on each run.  QuTiP allows you to accomplish this using the keyword ``args`` as an input to the solvers.  For instance, instead of explicitly writing 9 for the amplitude and 5 for the width of the gaussian driving term, we can make use of the ``args`` variable

.. plot::
    :context:
    :nofigs:

    def H1_coeff(t, args):
        return args['A'] * np.exp(-(t/args['sigma'])**2)

or equivalently,

.. plot::
    :context:
    :nofigs:

    def H1_coeff(t, args):
          A = args['A']
          sig = args['sigma']
          return A * np.exp(-(t / sig) ** 2)

where ``args`` is a Python dictionary of ``key: value`` pairs ``args = {'A': a, 'sigma': b}`` where ``a`` and ``b`` are the two parameters for the amplitude and width, respectively.  Of course, we can always hardcode the values in the dictionary as well ``args = {'A': 9, 'sigma': 5}``, but there is much more flexibility by using variables in ``args``.  To let the solvers know that we have a set of args to pass we append the ``args`` to the end of the solver input:

.. plot::
    :context:
    :nofigs:

    output = mesolve(H, psi0, times, c_ops, [a.dag() * a], args={'A': 9, 'sigma': 5})

or to keep things looking pretty

.. plot::
    :context:
    :nofigs:

    args = {'A': 9, 'sigma': 5}
    output = mesolve(H, psi0, times, c_ops, [a.dag() * a], args=args)

Once again, the Monte Carlo solver :func:`qutip.mcsolve` works in an identical manner.

.. _time-string:

String Format Method
=====================

.. note:: You must have Cython installed on your computer to use this format.  See :ref:`install` for instructions on installing Cython.

The string-based time-dependent format works in a similar manner as the previously discussed Python function method.  That being said, the underlying code does something completely different.  When using this format, the strings used to represent the time-dependent coefficients, as well as Hamiltonian and collapse operators, are rewritten as Cython code using a code generator class and then compiled into C code.  The details of this meta-programming will be published in due course.  However, in short, this can lead to a substantial reduction in time for complex time-dependent problems, or when simulating over long intervals.

Like the previous method, the string-based format uses a list pair format ``[Op, str]`` where ``str`` is now a string representing the time-dependent coefficient.  For our first example, this string would be ``'9 * exp(-(t / 5.) ** 2)'``.  The Hamiltonian in this format would take the form:

.. plot::
   :context:
   :nofigs:

   ustate = basis(3, 0)
   excited = basis(3, 1)
   ground = basis(3, 2)

   N = 2 # Set where to truncate Fock state for cavity

   sigma_ge = tensor(qeye(N), ground * excited.dag())  # |g><e|
   sigma_ue = tensor(qeye(N), ustate * excited.dag())  # |u><e|
   a = tensor(destroy(N), qeye(3))
   ada = tensor(num(N), qeye(3))

   c_ops = []  # Build collapse operators
   kappa = 1.5 # Cavity decay rate
   c_ops.append(np.sqrt(kappa) * a)
   gamma = 6  # Atomic decay rate
   c_ops.append(np.sqrt(5*gamma/9) * sigma_ue) # Use Rb branching ratio of 5/9 e->u
   c_ops.append(np.sqrt(4*gamma/9) * sigma_ge) # 4/9 e->g

   t = np.linspace(-15, 15, 100) # Define time vector
   psi0 = tensor(basis(N, 0), ustate) # Define initial state
   state_GG = tensor(basis(N, 1), ground) # Define states onto which to project
   sigma_GG = state_GG * state_GG.dag()
   state_UU = tensor(basis(N, 0), ustate)
   sigma_UU = state_UU * state_UU.dag()

   g = 5  # coupling strength
   H0 = -g * (sigma_ge.dag() * a + a.dag() * sigma_ge)  # time-independent term
   H1 = (sigma_ue.dag() + sigma_ue)  # time-dependent term


.. plot::
    :context:
    :nofigs:

    H = [H0, [H1, '9 * exp(-(t / 5) ** 2)']]

Notice that this is a valid Hamiltonian for the string-based format as ``exp`` is included in the above list of suitable functions. Calling the solvers is the same as before:

.. plot::
   :context:
   :nofigs:

   output = mesolve(H, psi0, t, c_ops, [a.dag() * a])

We can also use the ``args`` variable in the same manner as before, however we must rewrite our string term to read: ``'A * exp(-(t / sig) ** 2)'``

.. plot::
    :context:
    :nofigs:

    H = [H0, [H1, 'A * exp(-(t / sig) ** 2)']]
    args = {'A': 9, 'sig': 5}
    output = mesolve(H, psi0, times, c_ops, [a.dag()*a], args=args)


.. important:: Naming your ``args`` variables ``exp``, ``sin``, ``pi`` etc. will cause errors when using the string-based format.

Collapse operators are handled in the exact same way.


.. _time-interp:

Modeling Non-Analytic and/or Experimental Time-Dependent Parameters using Interpolating Functions
=================================================================================================

Sometimes it is necessary to model a system where the time-dependent parameters are non-analytic functions, or are derived from experimental data (i.e. a collection of data points).  In these situations, one can use interpolating functions as an approximate functional form for input into a time-dependent solver.  QuTiP includes its own custom cubic spline interpolation class :class:`qutip.interpolate.Cubic_Spline` to provide this functionality.  To see how this works, lets first generate some noisy data:

.. plot::
    :context: close-figs

    t = np.linspace(-15, 15, 100)
    func = lambda t: 9*np.exp(-(t / 5)** 2)
    noisy_func = lambda t: func(t)+(0.05*func(t))*np.random.randn(t.shape[0])
    noisy_data = noisy_func(t)

    plt.figure()
    plt.plot(t, func(t))
    plt.plot(t, noisy_data, 'o')
    plt.show()


To turn these data points into a function we call the QuTiP :class:`qutip.interpolate.Cubic_Spline` class using the first and last domain time points, ``t[0]`` and ``t[-1]``, respectively, as well as the entire array of data points:


.. plot::
    :context: close-figs

    S = Cubic_Spline(t[0], t[-1], noisy_data)

    plt.figure()
    plt.plot(t, func(t))
    plt.plot(t, noisy_data, 'o')
    plt.plot(t, S(t), lw=2)
    plt.show()


Note that, at present, only equally spaced real or complex data sets can be accommodated.  This cubic spline class ``S`` can now be pasted to any of the ``mesolve``, ``mcsolve``, or ``sesolve`` functions where one would normally input a time-dependent function or string-representation.  Taking the problem from the previous section as an example.  We would make the replacement:

.. code-block:: python

    H = [H0, [H1, '9 * exp(-(t / 5) ** 2)']]

to

.. code-block:: python

    H = [H0, [H1, S]]


When combining interpolating functions with other Python functions or strings, the interpolating class will automatically pick the appropriate method for calling the class.  That is to say that, if for example, you have other time-dependent terms that are given in the string-format, then the cubic spline representation will also be passed in a string-compatible format.  In the string-format, the interpolation function is compiled into c-code, and thus is quite fast.  This is the default method if no other time-dependent terms are present.


.. _time-dynargs:

Accesing the state from solver
==============================

New in QuTiP 4.4

The state of the system, the ket vector or the density matrix,
is available to time-dependent Hamiltonian and collapse operators in ``args``.
Some keys of the argument dictionary are understood by the solver to be values
to be updated with the evolution of the system.
The state can be obtained in 3 forms: ``Qobj``, vector (1d ``np.array``), matrix (2d ``np.array``),
expectation values and collapse can also be obtained.

+-------------------+-------------------------+----------------------+------------------------------------------------------------------+
|                   | Preparation             | usage                | Notes                                                            |
+-------------------+-------------------------+----------------------+------------------------------------------------------------------+
| state as Qobj     | ``name+"=Qobj":psi0``   | ``psi_t=args[name]`` | The ket or density matrix as a Qobj with ``psi0``'s dimensions   |
+-------------------+-------------------------+----------------------+------------------------------------------------------------------+
| state as matrix   | ``name+"=mat":psi0``    | ``mat_t=args[name]`` | The state as a matrix, equivalent to ``state.full()``            |
+-------------------+-------------------------+----------------------+------------------------------------------------------------------+
| state as vector   | ``name+"=vec":psi0``    | ``vec_t=args[name]`` | The state as a vector, equivalent to ``state.full().ravel('F')`` |
+-------------------+-------------------------+----------------------+------------------------------------------------------------------+
| expectation value | ``name+"=expect":O``    | ``e=args[name]``     | Expectation value of the operator ``O``, either                  |
|                   |                         |                      | :math:`\left<\psi(t)|O|\psi(t)\right>`                           |
|                   |                         |                      | or :math:`\rm{tr}\left(O \rho(t)\right)`                         |
+-------------------+-------------------------+----------------------+------------------------------------------------------------------+
| collpases         | ``name+"=collapse":[]`` | ``col=args[name]``   | List of collapse,                                                |
|                   |                         |                      | each collapse is a tuple of the pair ``(time, which)``           |
|                   |                         |                      | ``which`` being the indice of the collapse operator.             |
|                   |                         |                      | ``mcsolve`` only.                                                |
+-------------------+-------------------------+----------------------+------------------------------------------------------------------+

Here ``psi0`` is the initial value used for tests before the evolution begins.
:func:`qutip.bloch_redfield.brmesolve` does not support these arguments.

Reusing Time-Dependent Hamiltonian Data
=======================================

.. note:: This section covers a specialized topic and may be skipped if you are new to QuTiP.

When repeatedly simulating a system where only the time-dependent variables, or initial state change, it is possible to reuse the Hamiltonian data stored in QuTiP and there by avoid spending time needlessly preparing the Hamiltonian and collapse terms for simulation.  To turn on the the reuse features, we must pass a :class:`qutip.solver.Options` object with the ``rhs_reuse`` flag turned on.  Instructions on setting flags are found in :ref:`Options`.  For example, we can do

.. plot::
    :context: close-figs

    H = [H0, [H1, 'A * exp(-(t / sig) ** 2)']]
    args = {'A': 9, 'sig': 5}
    output = mcsolve(H, psi0, times, c_ops, [a.dag()*a], args=args)
    opts = Options(rhs_reuse=True)
    args = {'A': 10, 'sig': 3}
    output = mcsolve(H, psi0, times, c_ops, [a.dag()*a], args=args, options=opts)

The second call to :func:`qutip.mcsolve` does not reorganize the data, and in the case of the string format, does not recompile the Cython code.  For the small system here, the savings in computation time is quite small, however, if you need to call the solvers many times for different parameters, this savings will obviously start to add up.


.. plot::
    :context: reset
    :include-source: false
    :nofigs:

    # reset the context at the end
