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Quantum trajectory theory?

Before encountering this Quanta Magazine article today, I’d not heard of this aspect of quantum measurement theory: “The Quantum Theory That Peels Away the Mystery of Measurement” (July 3, 2019) by Philip Ball, Contributing Writer (author of Beyond Weird: Why everything you thought you knew about quantum physics is different).

Well, a quick Google search found some articles about quantum trajectory theory (QTT). (I’d noticed the Physics World article back in June.) > Quantum Trajectories and Quantum Measurement Theory (Submitted on 11 Feb 2003)

Beyond their use as numerical tools, quantum trajectories can be ascribed a degree of reality in terms of quantum measurement theory. In fact, they arise naturally from considering continuous observation of a damped quantum system. A particularly useful form of quantum trajectories is as linear (but non-unitary) stochastic Schrodinger equations. In the limit where a strong local oscillator is used in the detection, and where the system is not driven, these quantum trajectories can be solved. This gives an alternate derivation of the probability distributions for completed homodyne and heterodyne detection schemes. It also allows the previously intractable problem of real-time adaptive measurements to be treated. The results for an analytically soluble example of adaptive phase measurements are presented, and future developments discussed. > A simple model of quantum trajectories (Submitted on 29 Aug 2001)

Quantum trajectory theory, developed largely in the quantum optics community to describe open quantum systems subjected to continuous monitoring, has applications in many areas of quantum physics. In this paper I present a simple model, using two-level quantum systems (q-bits), to illustrate the essential physics of quantum trajectories and how different monitoring schemes correspond to different “unravelings” of a mixed state master equation. I also comment briefly on the relationship of the theory to the Consistent Histories formalism and to spontaneous collapse models.

Physics World > To catch a quantum jump (07 Jun 2019)

The researchers … managed to control the quantum jump once it had started by applying an electric pulse to the artificial atom. In this way, they intercepted it and sent it back to the ground state. They are only able to do this because the quantum jump is not truly instantaneous and random. Instead, quantum jumps take the same trajectory between the two energy levels every time, so it is possible to predict how to send them back.

According to the Yale team, this is an important point: “while quantum jumps appear discrete and random in the long run, reversing a quantum jump means the evolution of the quantum state possesses, in part, a deterministic and non-random character,” say Devoret and Minev. “The jump always occurs in the same, predictable manner from its random starting point.”

“The findings are in complete agreement with the predictions of modern quantum trajectory theory,” Devoret tells Physics World, “with essentially no adjustable parameters.

Ball introduces the topic by characterizing quantum mechanics as a type of statistical mechanics, a way of predicting probabilistic or average outcomes based on statistical ensembles. Sort of like local weather forecasting – likely scenarios and expectations. Or, only being able to talk about your average commute time to work (based on your daily diary) rather than predicting the current time (which relies on knowledge of recent and current road / traffic conditions).

In the early days of quantum mechanics, that seemed to be its inevitable limitation: It was a probabilistic theory, telling us only what we will observe on average if we collect records for many events or particles. To Erwin Schrödinger, whose eponymous equation prescribes how quantum objects behave, it was utterly meaningless to think about specific atoms or electrons doing things in real time. “It is fair to state,” he wrote in 1952, “that we are not experimenting with single particles. … We are scrutinizing records of events long after they have happened.” In other words, quantum mechanics seemed to work only for “ensembles” of many particles. “When the ensemble is large enough, it’s possible to acquire sufficient statistics to check if the predictions are correct or not,” said Michel Devoret, a physicist at Yale University.

Ball notes for QTT that: “The standard [quantum mechanical] description is recovered over long timescales after the average of many events is computed.”

The catch is that, to apply QTT, you need to have nearly complete knowledge about the behavior of the system you’re observing.

Achieving this degree of control and information capture is very challenging. That’s why, although QTT has been around for a couple decades, “it is only within the past five years that we can experimentally test it,” said William Oliver of the Massachusetts Institute of Technology.

You might loosely compare this to forecasting the trajectory of a single air molecule. The Schrödinger equation plays a role a bit like the classical diffusion equation, which predicts how far on average such a particle travels over time as it undergoes collisions. But QTT predicts where a specific particle will go, basing its forecast on detailed information about the collisions the particle has experienced already. Randomness is still at play: You can’t perfectly predict a trajectory in either case. But QTT will give you the story of an individual particle — and the ability to see where it might be headed next.