Action principle for continuous quantum measurement!

Post date: May 23, 2013 2:15:54 AM

Our next posted paper I want to discuss concerns a topic I am really excited about. Indeed, I have been working on this idea on and off for the past five years, and am delighted that it has turned out even better than I had hoped for. The paper combines two topics I have great interest in: the stochastic path integral approach to random processes and continuous quantum measurement. The paper is written with Areeya Chantasri as first author (it is her first paper) and Justin Dressel.

Throughout the development of physics, it has been found that a tremendous variety of physical phenomena can be explained via simple, unifying principles. The most elegant of these principles are those that pick out a class of phenomena that extremize some quantity. Famously, Fermat's principle states that of all possible paths a beam of light can take through an arbitrary physical media, it takes the path of shortest time. Similarly, in classical mechanics, the equations of Newton, Lagrange, or Hamilton can all be derived from a unifying principle: physical trajectories with fixed initial and final boundary conditions are those that extremize the action between the two points. This principle can in turn be understood from an underlying theory (wave mechanics or quantum mechanics) as emerging from a small \hbar limit of a path-integral formulation of the theory.

In quantum mechanics, the type of dynamics is qualitatively changed once measurement is brought to bear. In the case of continuous quantum measurement, the state undergoes a combination of Hamiltonian and measurement back-action dynamics that has been analyzed via stochastic differential equation methods up until now. The breakthrough made in the current paper is to reformulate this process in terms of a stochastic path integral. This approach allows the calculation of any average or correlation function of the continuous measurement results and/or the quantum state. This reformulation brings great physical insight into the conditional dynamics of the quantum measurement. From this approach, we derive an action principle for the most-likely path taken in quantum state space, given initial and final boundary conditions on the state. The initial condition corresponds to a preparation, and the final condition corresponds to a post-selection.

We can do really neat things with this formalism. My favorite is what we call the anatomy of a quantum jump. If a quantum system is quickly measured repeatedly, it tends to freeze in one state, even if its quantum dynamics is trying to make it move between states. This is called the quantum Zeno effect. However, the system does not say frozen forever. If we keep measuring for a long enough time, the state will suddenly (an unpredictably) jump from one state to another state. These jumps can be seen in the lab and have been modeled theoretically as a random process. We have been able to give a picture of the most likely path the state will take - provided we fix the initial state (the frozen one), the final state (the opposite one), for a fixed time duration - as a portrait in a quantum phase space. These videos show the most likely paths for a quantum jump for different time intervals:

This movie shows the quantum jump dynamics of a continuously measured system in a quantum phase space, where we plotted the canonical momentum versus the quantum state angle. The lines represent the most-likely paths the quantum state takes through its space. The dots shows the real time motion given initial and final boundary conditions. They all take different times.

This movie shows the quantum jump dynamics of a continuously measured system. The measurement record r is plotted versus the quantum state angle theta as time develops (left), and versus time (right). The lines represent the most-likely paths the measurement record takes as time develops. The dots shows the real time motion given initial and final boundary conditions on the quantum state.