# Bosonic Environments

In this section we consider a simple two-level system coupled to a Drude-Lorentz bosonic bath. The system Hamiltonian, \(H_{sys}\), and the bath spectral density, \(J_D\), are

\[ \begin{align}\begin{aligned}H_{sys} &= \frac{\epsilon \sigma_z}{2} + \frac{\Delta \sigma_x}{2}\\J_D(\omega) &= \frac{2\lambda \gamma \omega}{\gamma^2 + \omega^2},\end{aligned}\end{align} \]

We will demonstrate how to describe the bath using two different expansions of the spectral density correlation function (Matsubara’s expansion and a Padé expansion), how to evolve the system in time, and how to calculate the steady state.

First we will do this in the simplest way, using the built-in implementations of the two bath expansions, `DrudeLorentzBath` and `DrudeLorentzPadeBath`. We will do this both with a truncated expansion and show how to include an approximation to all of the remaining terms in the bath expansion.

Environment API

We will also explain how to achieve the same results using the `DrudeLorentzEnvironment` that was introduced in the section on environments. The “bath” classes are part of an older API that is less powerful than the “environment” API, but often more convenient to use when one only uses the HEOM solver and does not need any of the new features.

Afterwards, we will show how to calculate the correlation function expansion coefficients and to use those coefficients to construct your own bath description so that you can implement your own bosonic baths / environments.

Finally, we will demonstrate how to simulate a system coupled to multiple independent baths, as occurs, for example, in certain photosynthesis processes.

A tutorial notebook containing a complete example similar to this one is the HEOM example notebook 1a.

## Describing the system and bath

First, let us construct the system Hamiltonian `H_sys` and the initial system state `rho0`:

    from qutip import basis, sigmax, sigmaz

    # The system Hamiltonian:
    eps = 0.5  # energy of the 2-level system
    Del = 1.0  # tunnelling term
    H_sys = 0.5 * eps * sigmaz() + 0.5 * Del * sigmax()

    # Initial state of the system:
    rho0 = basis(2,0) * basis(2,0).dag()

Now let us describe the bath properties:

    # Bath properties:
    gamma = 0.5  # cut off frequency
    lam = 0.1  # coupling strength
    T = 0.5  # temperature

    # System-bath coupling operator:
    Q = sigmaz()

where \(\gamma\) (`gamma`), \(\lambda\) (`lam`) and the temperature \(T\) are the parameters of a Drude-Lorentz bath, and `Q` is the coupling operator between the system and the bath.

We may the pass these parameters to either `DrudeLorentzBath` or `DrudeLorentzPadeBath` to construct an expansion of the bath correlations:

    from qutip.solver.heom import DrudeLorentzBath
    from qutip.solver.heom import DrudeLorentzPadeBath

    # Number of expansion terms to retain:
    Nk = 2

    # Matsubara expansion:
    bath = DrudeLorentzBath(Q, lam, gamma, T, Nk)

    # Padé expansion:
    bath = DrudeLorentzPadeBath(Q, lam, gamma, T, Nk)

Here, `Nk` is the number of terms to retain within the expansion of the bath.

Environment API

Using the environment API, we first create an abstract `DrudeLorentzEnvironment` describing the bath, and then use its functions to create exponential expansions such as the Matsubara and Pade ones:

    from qutip.core.environment import DrudeLorentzEnvironment

    env = DrudeLorentzEnvironment(T, lam, gamma)

    # Matsubara expansion:
    approx = env.approximate("matsubara", Nk)

    # Padé expansion:
    approx = env.approximate("pade", Nk)

Note that the coupling operator `Q` is not part of the environment objects.

## System and bath dynamics

Now we are ready to construct a solver:

    from qutip.solver.heom import HEOMSolver

    max_depth = 5  # maximum hierarchy depth to retain
    options = {"nsteps": 15_000}

    solver = HEOMSolver(H_sys, bath, max_depth=max_depth, options=options)

and to calculate the system evolution as a function of time:

    tlist = [0, 10, 20]  # times to evaluate the system state at
    result = solver.run(rho0, tlist)

The `max_depth` parameter determines how many levels of the hierarchy to retain. As a first approximation, hierarchy depth may be thought of as similar to the order of Feynman Diagrams (both classify terms by increasing number of interactions).

The `result` is a standard QuTiP results object with the attributes:

- `times`: The times at which the state was evaluated (i.e. `tlist`).

- `states`: The system states at each time.

- `expect`: A list with the values of each expectation operator at each time.

- `e_data`: A dictionary with the values of each expectation operator at each time.

- `ado_states`: See below (a list of instances of `HierarchyADOsState`).

If `ado_return=True` is passed to `.run(...)` the full set of auxilliary density operators (ADOs) that make up the hierarchy at each time will be returned as `result.ado_states`. We will describe how to use these to determine other properties, such as system-bath currents, later in the fermionic guide. If one has a full set of ADOs from a previous call of `.run(...)`, one may supply it as the initial state of the solver by calling `.run(result.ado_states[-1], tlist, ado_init=True)`.

As with other QuTiP solvers, if expectation operators or functions are supplied using `.run(..., e_ops=[...])` the expectation values are available in `result.expect` and `result.e_data`.

Environment API

When using the environment API, one needs to pass the coupling operator to the HEOM solver together with the approximated environment:

    solver = HEOMSolver(H_sys, (approx, Q), max_depth=max_depth, options=options)

Below we run the solver again, but use `e_ops` to store the expectation values of the population of the system states and the coherence:

    # Define the operators that measure the populations of the two
    # system states:
    P11p = basis(2,0) * basis(2,0).dag()
    P22p = basis(2,1) * basis(2,1).dag()

    # Define the operator that measures the 0, 1 element of density matrix
    # (corresonding to coherence):
    P12p = basis(2,0) * basis(2,1).dag()

    # Run the solver:
    tlist = np.linspace(0, 20, 101)
    result = solver.run(rho0, tlist, e_ops={"11": P11p, "22": P22p, "12": P12p})

    # Plot the results:
    fig, axes = plt.subplots(1, 1, sharex=True, figsize=(6, 6))
    axes.plot(result.times, np.real(result.e_data["11"]), 'b', linewidth=2, label="P11")
    axes.plot(result.times, np.real(result.e_data["12"]), 'r', linewidth=2, label="P12")
    axes.set_xlabel(r't', fontsize=16)
    axes.legend(loc=0, fontsize=16)

## Steady state

Using the same solver, we can also determine the steady state of the combined system and bath using:

    steady_state, steady_ados = solver.steady_state()

where `steady_state` is the steady state of the system and `steady_ados` is the steady state of the full hierarchy. The ADO states are described more fully in the section on determining currents and in the API documentation for `HierarchyADOsState`.

## Matsubara Terminator

When constructing the Drude-Lorentz bath we have truncated the expansion at `Nk = 2` terms and ignore the remaining terms.

However, since the coupling to these higher order terms is comparatively weak, we may consider the interaction with them to be Markovian, and construct an additional Lindbladian term that captures their interaction with the system and the lower order terms in the expansion.

This additional term is called the `terminator` because it terminates the expansion.

The `DrudeLorentzBath` and `DrudeLorentzPadeBath` both provide a means of calculating the terminator for a given expansion:

    # Matsubara expansion:
    bath = DrudeLorentzBath(Q, lam, gamma, T, Nk)

    # Padé expansion:
    bath = DrudeLorentzPadeBath(Q, lam, gamma, T, Nk)

    # Add terminator to the system Liouvillian:
    delta, terminator = bath.terminator()
    HL = liouvillian(H_sys) + terminator

    # Construct solver:
    solver = HEOMSolver(HL, bath, max_depth=max_depth, options=options)

This captures the Markovian effect of the remaining terms in the expansion without having to fully model many more terms.

The terminator amplitude `delta` is an approximation to the strength of the effect of the remaining terms in the expansion (i.e. how strongly the terminator is coupled to the rest of the system).

Environment API

Here, the terminator amplitude can be returned directly by the `approximate` method used earlier. Based on it, the special function `environment.system_terminator` can then be used to construct the terminator Liouvillian:

    from qutip.core.environment import system_terminator

    # Matsubara expansion:
    approx, delta = env.approximate("matsubara", Nk, compute_delta=True)

    # Padé expansion:
    approx, delta = env.approximate("pade", Nk, compute_delta=True)

    # Add terminator to the system Liouvillian:
    terminator = system_terminator(Q, delta)
    HL = liouvillian(H_sys) + terminator

    # Construct solver
    solver = HEOMSolver(HL, (approx, Q), max_depth=max_depth, options=options)

## Matsubara expansion coefficients

So far we have relied on the built-in `DrudeLorentzBath` to construct the Drude-Lorentz bath expansion for us. Now we will calculate the coefficients ourselves and construct a `BosonicBath` directly. A similar procedure can be used to apply `HEOMSolver` to any bosonic bath for which we can calculate the expansion coefficients.

The Matsubara expansion of the Drude-Lorentz correlation function is detailed in the section on the Drude-Lorentz Environment. Let us calculate the coefficients and exponents in Python:

    # Convenience functions and parameters:

    def cot(x):
        return 1. / np.tan(x)

    beta = 1. / T

    # Number of expansion terms to calculate:
    Nk = 2

    # C_real expansion terms:
    ck_real = [lam * gamma / np.tan(gamma / (2 * T))]
    ck_real.extend([
        (8 * lam * gamma * T * np.pi * k * T /
            ((2 * np.pi * k * T)**2 - gamma**2))
        for k in range(1, Nk + 1)
    ])
    vk_real = [gamma]
    vk_real.extend([2 * np.pi * k * T for k in range(1, Nk + 1)])

    # C_imag expansion terms (this is the full expansion):
    ck_imag = [lam * gamma * (-1.0)]
    vk_imag = [gamma]

After all that, constructing the bath is very straight forward:

    from qutip.solver.heom import BosonicBath

    bath = BosonicBath(Q, ck_real, vk_real, ck_imag, vk_imag)

And we’re done!

Environment API

The analogue of the `BosonicBath` is the `ExponentialBosonicEnvironment`. Its usage is very similar:

    from qutip.core.environment import ExponentialBosonicEnvironment

    env = ExponentialBosonicEnvironment(ck_real, vk_real, ck_imag, vk_imag)

The `BosonicBath` and the `ExponentialBosonicEnvironment` can be used with the `HEOMSolver` in exactly the same way as the baths and environments that we constructed previously using the built-in Drude-Lorentz bath expansions.

## Multiple baths

The `HEOMSolver` supports having a system interact with multiple reservoirs. All that is needed is to supply a list of baths or environments instead of a single bath or environment.

In the example below we calculate the evolution of a small system where each basis state of the system interacts with a separate bath. Such an arrangement can model, for example, the Fenna–Matthews–Olson (FMO) pigment-protein complex which plays an important role in photosynthesis (for a full FMO example see the HEOM example notebook 2).

For each bath expansion, we also include the terminator in the system Liouvillian.

At the end, we plot the populations of the system states as a function of time, and show the long-time beating of quantum state coherence that occurs:

    # The size of the system:
    N_sys = 3

    def proj(i, j):
        """ A helper function for creating an interaction operator. """
        return basis(N_sys, i) * basis(N_sys, j).dag()

    # Construct one bath for each system state:
    baths = []
    for i in range(N_sys):
        Q = proj(i, i)
        baths.append(DrudeLorentzBath(Q, lam, gamma, T, Nk))

    # Construct the system Liouvillian from the system Hamiltonian and
    # bath expansion terminators:
    H_sys = sum((i + 0.5) * eps * proj(i, i) for i in range(N_sys))
    H_sys += sum(
      (i + j + 0.5) * Del * proj(i, j)
      for i in range(N_sys) for j in range(N_sys)
      if i != j
    )
    HL = liouvillian(H_sys) + sum(bath.terminator()[1] for bath in baths)

    # Construct the solver (pass a list of baths):
    solver = HEOMSolver(HL, baths, max_depth=max_depth, options=options)

    # Run the solver:
    rho0 = basis(N_sys, 0) * basis(N_sys, 0).dag()
    tlist = np.linspace(0, 5, 200)
    e_ops = {
        f"P{i}": proj(i, i)
        for i in range(N_sys)
    }
    result = solver.run(rho0, tlist, e_ops=e_ops)

    # Plot populations:
    fig, axes = plt.subplots(1, 1, sharex=True, figsize=(8,8))
    for label, values in result.e_data.items():
        axes.plot(result.times, np.real(values), label=label)
    axes.set_xlabel(r't', fontsize=16)
    axes.set_ylabel(r"Population", fontsize=16)
    axes.legend(loc=0, fontsize=16)

Environment API

Instead of a list `[bath1, bath2, ...]`, one can of course also pass multiple environments with different coupling operators like

    HEOMSolver(Hsys, [(env1, Q1), (env2, Q2), ...], ...)

or even a mixed list of baths and environments.

## Shifted-Drude-Lorentz Bath

As discussed in Drude-Lorentz Environment, a Shifted-Drude-Lorentz bath with a shift of \(\Omega\) can be created by combining two regular Drude-Lorentz baths. Two Drude-Lorentz baths can be connected to the same site using a similar approach as in the previous section:

    Omega = 0.1 # Shift
    baths = []
    for i in range(N_sys):
        Q = proj(i, i)
        # Two baths are added with same coupling-operator
        baths.append(DrudeLorentzBath(Q, lam/2, gamma + 1j * Omega, T, Nk))
        baths.append(DrudeLorentzBath(Q, lam/2, gamma - 1j * Omega, T, Nk))

As a side note, it is easy to see that if \(\Omega=0\) then we get two Drude-Lorentz baths each with a coupling strength of \(\lambda/2\). Now, because \(\gamma\) and \(T\) are same for both of them, the two can be combined to form a Drude-Lorentz bath with coupling strength \(\lambda\).