The Newton Propagator

The following overview has been adapted from Ref [1].

Quantum mechanical equations of motion

The behavior of a quantum system is usually described either by the Schrödinger equation

\[\frac{\partial}{\partial t} \ket{\Psi(t)} = -\frac{\ii}{\hbar} \Op{H} \ket{\Psi(t)},\]

or, for open quantum systems, by the Liouville-von-Neumann equation

\[\frac{\partial}{\partial t} \Op{\rho}(t) = \Liouville \Op{\rho}(t).\]

ODE solvers versus series expansion

There are two possible approaches to obtaining a solution. The first is to simply take the equation of motion and apply one of the generic numerical methods for solving ordinary differential equations (ODEs), like one of the Runge-Kutta (RK) methods [2][3].

The alternative approach is to solve the equation of motion analytically, and then to evaluate that solution numerically. In this way, results of arbitrary precision can be obtained, with the obvious caveat that the propagation scheme will be specific to a particular equation of motion and its solution.

The Schrödinger equation for a time-independent Hamiltonian has the solution

\[\Ket{\Psi(t)}= \Op{U}(t,0) \Ket{\Psi(0)} = \ee^{-\frac{\ii}{\hbar} \Op{H}\, t} \Ket{\Psi(0)}\,.\]

Similarly, the Liouville-von-Neumann equation has the solution

\[\Op{\rho}(t) = \DynMap(t,0) \ = \ee^{\Liouville\, t} \Op{\rho}(0)\,.\]

For time-dependent Hamiltonians, we approximate \(\Op{H}(t)\) as piecewise constant on a time grid with time step \(dt\). Then, the total time evolution operator is the product of the time evolution operators at each time step,

\[\Op{U}[T,0] = \prod_{i=1}^{nt-1} \underbrace{\Op{U}(t_i+dt, t_i)}_{\equiv \Op{U}_i} = \prod_{i=1}^{nt-1} \exp\bigg[-\frac{\ii}{\hbar} \underbrace{\Op{H}\left(t_i + \frac{dt}{2}\right)}_{\equiv \Op{H}_i} dt\bigg]\,.\]

The time step \(dt\) must be chosen sufficiently small that this is a good approximation; in practice, convergence is checked by verifying that the numerical results remain stable within a desired precision when \(dt\) is decreased. The same applies to the Liouvillian.

For a system of non-trivial size, \(\exp[-\frac{\ii}{\hbar} \Op{H} dt]\) is evaluated by expanding the exponential in a polynomial series,

\[\ee^{-\frac{\ii}{\hbar} \Op{H}\, dt} \Ket{\Psi} \approx \sum_{n=0}^{N-1} a_n P_n(\Op{H}) \Ket{\Psi}\,. \label{eq:poly_expansion}\]

where \(P_n(\Op{H})\) is a polynomial of degree \(n\) and \(\{a_n\}\) are the expansion coefficients. Applying \(P_n(\Op{H})\) to \(\ket{\Psi}\) then simply means repeated applications of \(\Op{H}\). For this reason, an efficient propagation relies on the proper use of sparsity in storing the Hamiltonian and spectral methods such as the Fourier grid.

The idea of evaluating the exponential as a polynomial series is already presupposed by the very definition of the exponential of an operator,

\[\exp[\Op{A}] \equiv \sum_{n=0}^{\infty} \frac{1}{n!} \Op{A}^n\,.\]

However, this expansion converges particularly slowly and is numerically unstable [4]. Thus, it is not suitable for time propagation. Instead, a polynomial basis must be chosen such that the series converges quickly and can be truncated as early as possible.

Chebychev expansion

For a function \(f(x)\) with \(x \in [-1, 1]\), it can be shown [5] that the fastest converging polynomial series is the expansion in Chebychev polynomials. When using the Chebychev expansion for propagation, the requirement that the argument of \(f(x)\) must be real translates into \(\Op{H}\) being Hermitian. Secondly, to account for the requirement that \(x \in [-1, 1]\), the Hamiltonian must be normalized as [6][7][8]

\[\Op{H}_{\text{norm}} = 2 \frac{\Op{H} - E_{\min}\,\identity}{\Delta} - \identity\,,\]

where \(\Delta = E_{\max} - E_{\min}\) is the spectral radius and \(E_{\max}\) and \(E_{\min}\) are the smallest and largest eigenvalue.

Newton expansion

For a general function \(f(z)\) with \(z \in \Complex\), expansion into Chebychev polynomial is not possible. For a function defined on an operator, a complex \(z\) corresponds to an operator with complex eigenvalues. This is the case for a non-Hermitian Hamiltonian, and, more importantly, for a dissipative Liouvillian in the Liouville-von-Neumann equation. In this case, Newton polynomials must be used. The expansion in Newton polynomials \(R_n(z)\) reads

\[f(z) \approx \sum_{n=0}^{N-1} a_n R_n(z)\,, \quad R_n(z) = \prod_{j=0}^{n-1} \left( z-z_j \right)\,,\]

for a set of sampling points \(\{z_j\}\) at which the interpolation is exact. The coefficients are defined as the divided differences [9],

(1)\[\begin{split}\begin{aligned} a_0 &= f(z_0)\,, \\ a_1 &= f(z_1) - f(z_0)\,, \\ a_n &= \frac{f(z_n) - \sum_{j=0}^{n-1} a_j \prod_{k=0}^{j-1}\left(z_n - z_k\right)} {\prod_{j=0}^{n-1} \left(z_n - z_j\right)}\,.\end{aligned}\end{split}\]

For solving the Liouville-von Neumann equation, \(f(z)=\ee^{z \,dt}\), where the argument \(z\) is \(\Liouville\). Thus, the propagation is written as

\[\Op{\rho} = \ee^{\Liouville \,dt} \, \Op{\rho}_0 \approx \underbrace{% \sum_{n=0}^{N-1} a_n \left( \Liouville - z_n\identity \right) }_{% \equiv p_{N-1}(\Liouville)} \Op{\rho}_0\,,\]

where the polynomial is evaluated through repeated application the Liouvillian.

The central issue for obtaining a fast-converging series is a proper choice of the sampling points \(\{z_j\}\). The fastest convergence results from using the complex eigenvalues of \(\Liouville\) [10]. However, the exact eigenvalues of the Liouvillian are not readily available. More generally, arbitrary points from the spectral domain of \(\Liouville\) can be used as sampling points.

A widely used method is to estimate the spectral domain and to encircle it with a rectangle or ellipse [11][9][12]. Then, a large number of expansion coefficients are calculated from sampling points on that boundary. The same coefficients are used for the propagation of all Liouvillians on the time grid under the assumption that they all fit into the same encirclement. The series is truncated as soon as convergence is reached. This is similar to the method employed for the Chebychev propagator, where a set of coefficients is calculated once and then used for the propagation of any Hamiltonian that is within the same spectral range.

A middle path between the exact eigenvalues of \(\Liouville\) and the crude encirclement of the spectral domain is the use of the Krylov method to obtain approximate eigenvalues. The Arnoldi algorithm for \(\hat{A} = \Liouville\) and using \(\vec{v} = \Op{\rho}\) as a starting vector yields a set of approximate eigenvalues of \(\Liouville\), as well as a Hessenberg matrix \(\hat{H}\) that is the projection of \(\Liouville\) into the Krylov subspace, and the set of Arnoldi vectors that span that subspace. Instead of using \(\Liouville\) as the argument of the polynomial \(p_{N-1}\), the Hessenberg matrix may be used. If \(\hat{V}_{N}\) is the transformation matrix between the full Liouville space and the reduced Krylov space, consisting of the Arnoldi vectors as columns, the propagation is evaluated using

\[\Liouville \Op{\rho} \approx \hat{V}_{N}\, p_{m-1}(\hat{H}) \, \hat{V}_{N}^{\dagger} \Op{\rho}_0\,.\]

Assuming \(N\) is much smaller than the full dimension of the Liouville space, most of the numerical effort is in the Arnoldi algorithm, in constructing the Krylov space.

Newton propagation with an implicitly restarted Arnoldi method

However, even for moderate values of \(N\) (typically on the order of 100), the Arnoldi algorithm can require prohibitive amounts of memory. This is because a full set of \(N\) Arnoldi vectors, each of the dimension of the Liouville space, needs to be stored. To counter this problem, an iterative scheme has been developed [13]. Instead of performing the Arnoldi algorithm to a high order \(N\), until convergence is reached in the propagation, we stop at some small order \(m<10\). This gives a first approximation to the propagated density matrix,

(2)\[\Op{\rho}^{(1)} = p_{m-1}^{(0)}(\Liouville) \Op{\rho}_0 = \sum_{n=0}^{m-1} a_n R_n(\Liouville) \Op{\rho}_0\,. \label{eq:newton1stIter}\]

The idea is now to iteratively add remaining terms to the Newton series in chunks of size \(m\), retaining all coefficients and sampling points, but restarting the Arnoldi procedure in every iteration.

Adding the next \(m\) terms to Eq. (2) yields

\[\begin{split}\begin{split} \Op{\rho}^{(2)} &= \Op{\rho}^{(1)} + \sum_{n=m}^{2m-1} a_n R_n(\Liouville) \Op{\rho}_{0} \\ &= \Op{\rho}^{(1)} + \underbrace{\left(\sum_{n=0}^{m-1} a_{m+n} R_n^{(1)}(\Liouville)\right)}_{% \equiv p_{m-1}^{(1)} } \underbrace{\left(R_{m}^{(0)}(\Liouville) \Op{\rho}_{0} \right)}_{% \equiv \Op{\sigma}^{(1)} }\,, \end{split}\end{split}\]


\[R_n^{(0)}(\Liouville) = \prod_{j=0}^{n-1}(\Liouville - z_{j}\identity), \qquad R_n^{(1)}(\Liouville) = \prod_{j=0}^{n-1}(\Liouville - z_{n+j}\identity)\,.\]

That is, the terms in \(R_n(\Liouville)\) already known from the calculation of \(\Op{\rho}^{(1)}\) have been pulled out, and yield a new “starting vector” \(\Op{\sigma}^{(1)}\), which is the argument to a Newton series of only \(m\) new terms. The new sampling points on which the \(R_n^{(1)}\) are evaluated are obtained by applying the Arnoldi procedure to \(\Op{\sigma}^{(1)}\). The Newton coefficients continue recursively from the previous restart. The third iteration yields

\[\Op{\rho}^{(3)} = \Op{\rho}^{(2)} + \underbrace{\left(\sum_{n=0}^{m-1} a_{2m+n} R_n^{(2)}(\Liouville)\right)}_{% \equiv p_{m-1}^{(2)} } \underbrace{\left(R_{m}^{(1)}(\Liouville) \Op{\sigma}_{1} \right)}_{% \equiv \Op{\sigma}^{(2)} }\,.\]

The Newton propagator continues, adding the \(m\) terms evaluating

\[p_{m-1}^{(s)}(\Liouville) \Op{\sigma}^{(s)} = \sum_{n=0}^{m-1} a_{sm + n} \prod_{k=0}^{n-1} \left(\Liouville - z_{sm+k} \identity \right) \Op{\sigma}^{(s)}\]


\[\Op{\sigma}^{(s)} = p_{m-1}^{(s-1)} \Op{\sigma}^{(s-1)}\]

at every restart iteration.

In the implementation of the algorithm, there are two details that need to be taken into account for numerical stability. First, the denominator of the divided differences in Eq. (1) may become extremely small if consecutive sampling points are close to each other. This can be addressed by reordering the points such that the denominator in the divided differences is maximized. This process is called Leja ordering [14]. The reverse problem that the sampling points are too far apart, causing an underflow in the calculation of coefficients can be avoided by normalizing the Liouvillian as

\[\tilde{\Liouville} = \frac{1}{\rho} \left( \Liouville - c \right)\,,\]

where \(c\) is an estimate for the center of the spectrum of \(\Liouville\), and the eigenvalues are roughly contained in a radius \(\rho\) around \(c\). These values can be estimated from the sampling points obtained in the first iteration of the Newton propagator. The normalization of the Liouvillian is in some sense similar to the normalization of the Hamiltonian in the Chebychev propagator, but it is crucial there since the Chebychev polynomials are only defined in the domain \([-1, 1]\). For the Newton propagator, the normalization is only for numerical stability.


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