A Quantum physics phenomena that have spiral-phase portraits?

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1. Nov 26, 2017

SeM

Hi, I was looking for a quantum physics phenomenon including in quantum field theory where the solutions of a related phase-plane system (i.e. the harmonic oscillator) have a spiral sink in the phase portrait?
The harmonic oscillator has saddle points in the phase-portrait, given its eigenvalue signs, however is there a phenomenon one can confirm has a spiral sink (attractor) and no singularities for its general solution form?

2. Nov 26, 2017

thephystudent

Angular motion in the phase-plane corresponds to hamiltonian dynamics, radial motion corresponds with damping; ergo for a spiral you need both: a damped harmonic oscillator.

In quantum physics this means you 'll be looking for open quantum systems, like a leaking cavity.

3. Nov 26, 2017

SeM

Thanks!! That was an excellent outline.

4. Nov 26, 2017

hilbert2

Make a harmonic oscillator where the mass of the oscillator has a small imaginary part: Im$(m)\neq 0$, and you get a spiral phase portrait with the cost that the Hamiltonian operator is not hermitian (and time evolution not unitary).

5. Nov 26, 2017

A. Neumaier

Any quantum system described by Lindblad equations will be dissipative and has spiral trajectories.

6. Nov 26, 2017

A. Neumaier

and probability is not preserved. Thus this is not really good....

7. Nov 26, 2017

SeM

Does it mean it is physically impossible?

8. Nov 26, 2017

A. Neumaier

Not if correctly interpreted.

It means that there is a positive rate that the system decays to something for which the description stops being applicable. For example, one can use it to model radioactive decay of a substance without taking into account the decay products.

9. Nov 26, 2017

SeM

Is thati why physstudent put the imaginary part on the mass? Wouldn't it be better to put the imaginary part on the angular part to make some physical sense? I cant imagine what an imaginary mass means, unless we are talking about dark matter.

10. Nov 26, 2017

A. Neumaier

In general one can add to the Hermitian Hamiltonian $-i$ times another Hermitian and positive semidefinite term, called (for historical reasons) the optical potential. Of course, the concrete choice is dictated by the system intended to be modelled.
An imaginary mass means a decay rate, as one can see by looking at the solution formula for a damped classical harmonic oscillator.

11. Nov 26, 2017

SeM

Thanks! I will look into the damped classical harmonic oscillator.

12. Nov 26, 2017

A. Neumaier

13. Nov 27, 2017

SeM

Thanks

14. Nov 27, 2017

hilbert2

As far as I know, a quantum measurement is a similar process where an initial state changes to an eigenstate of the measured observable as a result of interaction with many degrees of freedom (of the measuring device), and information of the initial state is lost irreversibly. The difference to simple damping is that the final state after measurement does not have to be the ground state.

15. Nov 28, 2017

SeM

Thanks Hilbert2. That was also very useful.

16. Nov 28, 2017

thephystudent

Well, if I am not mistaken A. Neumaier was hinting at the quantum trajectory method, where there is a decaying norm complemented by discrete random 'jumps'. In effect, its common interpretation is a sampling of the Lindblad equation through continuous weak measurements.
Just to say that there is some deep relationship between damping and measurement.

17. Nov 28, 2017

Agreed.