Simulating Quantum Trajectories

From Qwiki

There are several techniques for numerically generating quantum trajectories from a given physical scenario. In all of the below descriptions, imagine the following physical situation for simplicity. There exists a system of interest, e.g. an atom, which is interacting with the electromagnetic field around it. It is our task to describe the state of the atom, conditioned upon what we measure in the field. Imagine many detectors surrounding the atom. When one detector registers a photon click, that corresponds to a measurement result unique to that detector, and the observer must condition his density matrix (state of knowledge) for the atom in a correspondingly unique way. The model-dependent update rule describing this random evolution is known as the Stochastic Master Equation. If detection is perfectly efficient, and all radiant energy is collected, then the conditional state remains pure and a corresponding Stochastic Schrodinger Equation can be used. (Note that the Quantum Optics Toolbox is a convenient tool for quantum trajectory simulations.)

Method A

In every timestep dt, we calculate from our state the probabilities for every measurement outcome to occur (including no detection events). We flip a coin weighted with all of those probabilities. We then conditionally evolve the state according to the result.

The advantage of this method is that it produces very likely trajectories only a few of which have to be averaged to get the correct unconditional behavior. The disadvantage is that we have to calculate the probabilities at every step.

Method B

In every timestep dt, we flip an evenly weighted coin such that every possible measurement outcome occurs with the same probability.

The advantage of this method is that we don't have to calculate the new probabilities at every step. The disadvantage is that each trajectory will have little probability of actually happening, thus we need lots of trajectories to correctly approximate the unconditional behavior.

Method C

This is a compromise between methods A and B. Like method B, we don't calculate the probabilities at every step, but instead choose the fixed weightings of the coin more wisely.

Method D

Here we take an entirely different approach. Instead of deciding which measurement outcome happens at every infinitesimal time step, we choose the times that the non-trivial measurement outcomes will occur. We begin the simulation by choosing a random number between 0 and 1 with a flat distribution. Then we evolve the state according to the effective Hamiltonian. When the norm of our state reaches the random number we chose, we decide it's time to condition due to a measurement. If there is more than one outcome to choose from we flip an evenly weighted coin between them and apply the winner. Next, we renormalize the state, pick a new random number, and continue. Importantly, the system Hamiltonian doesn't influence the norm of the state. With this method we can evolve the system with a high-order integrator using standard methods between jumps. Higher order integration can be done with the other methods (A-C), but it gets ugly. This is the method that the Quantum Optics Toolbox employs in such discrete measurement situations.

Method E

Here the goal is to replace the act of flipping a weighted coin into turning the above conditional equation into a (nonlinear) stochastic equation. We imagine making many coin tosses per step of evolution. To do this effectively, we have to be measuring faster than the system can evolve appreciably. We then replace the result of many coin tosses with the expected mean of the state plus some Gaussian white noise term which captures the randomness of the result.

In this final case, the conditional evolution is often called Quantum State Diffusion as opposed to the Quantum Jump methods A-D. This is physically the case for homodyne detection of a decaying cavity mode.