Solving non-Markovian stochastic Schrödinger equations

I have just posted on the arXiv a new preprint (a short letter) about a recent work of mine on non-Markovian stochastic Schrödinger equations. It is an unexpected technical byproduct of a more fundamental and long term research effort to construct tractable relativistic collapse models (but it will likely not seem related at all). Before explaining what I did in this preprint, it is probably worthwhile to recall a few things about stochastic Schrödinger equations and their history. This will be a rather technical entry but as the preprint itself is about something technical, there is no point in making a summary for a broad audience.

The most well known objective of stochastic Schrödinger equation is to unravel an open-system evolution. One typically considers an open quantum system of density matrix $\rho_t$ with a Markovian evolution generated by a Linbladian $\mathcal{L}$:

$\partial_t \rho_t = \mathcal{L}(\rho_t)$

An unraveling of this open evolution is a random evolution for pure states $|\psi_t\rangle$ such that the open evolution is obtained as the probabilistic average of the pure state trajectories:

$\rho_t = \mathbf{E} [\, |\psi_t\rangle \langle \psi_t| \, ]$.

The interest of this rewriting becomes obvious for large systems. Indeed, one needs quadratically more memory to parametrize a density matrix than to parametrize a pure state and it is thus preferable to work with the latter in numerics. The price to pay is of course that an averaging is needed in the end and the stochastic pure states are thus often called “Monte Carlo wave functions” in this context.

Now, given a specific Lindbladian, I did not say what the evolution equation for the pure state was. Actually, there are infinitely many possible stochastic pure state trajectories, jumpy or diffusive, unraveling the same Lindblad equation. I can still show an example. For a simple Lindbladian $\mathcal{L}(\rho)=-i[H,\rho] + M\rho M^{\dagger} - \frac{1}{2} \{ M^{\dagger} M,\rho \}$ a possible diffusive (linear) unraveling of the open evolution is given by:

${\rm d}|\psi_t\rangle = \left(-i H{\rm d}t + M{\rm d}W_t + \frac{1}{2} M^\dagger M{\rm d}t \right) |\psi_t \rangle$,

where $W_t$ is a Wiener process, i.e. a normalized 1d Brownian motion. It is easy to check that with this evolution $|\psi_t\rangle$ verifies the unraveling condition $\rho_t = \mathbf{E} [\, |\psi_t\rangle \langle \psi_t| \, ]$ (if it does not seem to work, remember the “physicist” Itô rule ${\rm d} W_t ^2={\rm d} t$) . The pure state evolution at this stage is still linear but the downside is that it is not norm-preserving (the generator is clearly not Hermitian). This is an issue for numerical simulations because the norm of $|\psi_t\rangle$ almost surely converges to zero. The statistical average is then dominated by rare events that have a $|\psi_t\rangle$ with a huge norm; we hit the so called “importance sampling” problem: most of the states that are sampled are useless. One can be more clever and find, with a bit more work, a non-linear diffusive unraveling that is norm preserving:

${\rm d}|\psi_t\rangle = \left(-i H{\rm d}t + (M-\langle M \rangle_t){\rm d}W_t + \frac{1}{2} (M-\langle M \rangle_t)^2 {\rm d}t \right) |\psi_t \rangle$

where $\langle M \rangle_t = \langle \psi_t |M|\psi_t\rangle$ and I have chosen $M=M^\dagger$ to simplify. Knowing only the previous linear unraveling, it is not so trivial to find this non-linear version (the standard technique is to normalize the linear state and then transform the probability measure to preserve the unraveling condition). Nonetheless, all these (Markovian) unravelings, linear or non-linear, once they are known, share a nice feature: they are explicit and trivial to solve numerically (typically by discretization, naive or more elaborate). All is great in the Markovian realm.

Before I jump to the non-Markovian regime, it is important that I mention the other applications of stochastic Schrödinger equations, especially as my own result does not have much interest without this context. Although it is their use in numerics that made stochastic Schrödinger equations famous (especially the 1992 article by Dalibard, Castin and Molmer in the jump case), the equations were actually discovered earlier (in the mid eighties) by people trying to understand the foundations of quantum mechanics. The objective was to find modifications of the Schrödinger equations that would spontaneously project the quantum state in measurement situations without having to use the collapse postulate (this gave rise to the so called collapse models which I may discuss in more detail in the future). The Markovian equations that were found to be consistent are stochastic Schrödinger equations of the same form as those used for unravelings. Roughly at the same time, physicists and mathematicians studying continuous measurements also discovered the same equations. In that case, the unraveled process is “real” in the sense that it describes a quantum trajectory associated to an actual measurement readout. In that latter case, a nice thing is that one gets the non-linear equation immediately without having to guess or use subtle changes of probability measures. Hence finally, stochastic Schrödinger equations can be used as a numerical method for open quantum systems, as a hypothetical solution to the measurement problem, or as a way to understand actual continuous measurement situations in the lab.

This rather “multifaceted” success in the Markovian case immediately suggested to look for a generalization to the non-Markovian regime (i.e., with the unraveling picture in mind, for a bath with memory). This was done in the late nineties with only a small subtlety for the continuous measurement application. Non-Markovian stochastic Schrödinger equations were indeed found to not yield true observable real-time trajectories but only trajectories obtainable point by point a posteriori. This difficulty of interpretation (which gave rise to quite a few papers) is unfortunately not the biggest issue in the non Markovian realm. The biggest problem is that, except in some very specific cases, it is impossible to compute the solution of a non-Markovian stochastic Schrödinger equation, even numerically. This is because the equation possesses a memory term involving a functional derivative which is essentially impossible to evaluate. People have developed perturbation techniques to approximate the solutions in some limit but no general purpose method existed (as far as I am aware). This is were my preprint may come useful: I propose a non perturbative numerical method to compute the solutions of non-Markovian stochastic Schrödinger equations.

I find the method quite cute because it uses a sort of unraveling again. In a nutshell, the trick is to write the solution of the non-Markovian stochastic Schrödinger equation $| \psi_t \rangle$ as the statistical average of another state that obeys a simpler stochastic equation:

$|\psi_t\rangle = \mathbf{E} [ \, |\varphi_t\rangle \, ]$ ,

where $|\varphi_t\rangle$ is the simple auxiliary state. At that point, to give the details I would need to explain how one constructs general non-Markovian stochastic Schrödinger equations in the first place. Luckily I do this in the aforementioned preprint so at that point it is probably easier to simply just read it, especially as it is pretty short and self-contained.

Now I should be transparent about the limitations of my proposal. The method works well for linear stochastic Schrödinger equations but the extension to the non-linear ones is admittedly ugly and not of much help, as it relies on Metropolis sampling. Consequently, it suffers quite heavily from the importance sampling problem and becomes dramatically inefficient for large times (compared to the effective system-bath coupling timescale). It should also be added that if the method is used to solve an open-system evolution (by two successive unravelings) then it is not new and it coincides with the technique developed in 2002 by Stockburger and Grabert. Hence the real novelty comes only if one is interested, not in open-system evolutions, but in the two other applications of stochastic Schrödinger equations: foundations and measurement theory. That said, it does clearly solve a problem that was particularly annoying: the seeming impossibility to even plot a mathematical object central to three subfields of quantum theory. Short term, this is enough to make me happy.