Nonlinearity and coherence

Date: Sat, 5 Oct 1996 00:18:34 -0700 (PDT)
From: A Matacz <andrewm@maths.su.oz.au> and L Crowell <lcrowell@unm.edu> 
To: quantum-d@teleport.com
Subject: Nonlinearity and coherence

A discussion between Andrew Matacz <andrewm@maths.su.oz.au>
and Lawrence Crowell <lcrowell@unm.edu> about nonlinear environmental
coupling, colored noise, "supra ohmic environments," mechanisms
of robust coherence, and modelling strategy.

   "Yes, if the system is submarkovian then the fluctuations at one time
    can direct those at a later time. This would be a mechanism that
    blunts decoherent tendencies..."
                                    - Lawrence Crowell

   "...[though] if the non-local damping was ignored you would still
    get decoherence."
                      - A. Matacz

LC: If quantum mechanics plays a role in biology then it is likely to
involve some sort of large scale coherence in analogy with superfluidity
or stimulated emission. The model for microtubules I advanced posits a
sort of boson "gas," where each boson is in the same state. If we have a
single boson in a pure state then its density matrix is,

   rho_{ij} = (a_i^*)a_j|j><i|.

Now if there is a thermal environment that the boson is coupled to
then the probabilities becomes disturbed. The off diagonal terms are
supressed and the density matrix becomes

   rho_{ij} = e^{E_i/kT}|i><j|delta(ij).

There are a lot of details I am blasting through to get to this. Once
this has occured the boson gas has assumed a Boltzmann statistical
distribution for the states that the "particles" of the gas assume.

On Fri, 13 Sep 1996, Andrew Matacz posted:

>     ...The critical issue then becomes how do biological systems avoid
> environmental quantum decoherence? It will be impossible for quantum
> biology to be a serious subject unless this is addressed theoretically
> or observed experimentally. The vast majority of the work done to date
> on environmental decoherence is based on the Quantum Brownian Motion
> paradigm of non-equilibrium statistical physics. This paradigm successfully
> describes quantum dissipation in a very large range of problems. Most work
> is based on the simplest models in which the environment is described as
> having an ohmic spectral density (which generates white noise at high
> temperatures). These models shows clearly that decoherence occurs very
> quickly at room temperatures. A very small amount of work has studied the
> more complex type of environments characterized as having what are known
> as "supra-ohmic" (colored noise) environments. Interestingly this class
> of environment induces very little decoherence.

LC: This is similar to what I am advocating.  A white noise system is one
where there is no memory to stochastic fluctuations. The fluctuations at
one time are independent of fluctuations at any previous or later time.
Yet a colored noise system is one that has nonMarkovian statistics.  If
the system is submarkovian then the fluctuations at one time can direct
those at a later time. This would be one mechanism that blunts decoherent
tendencies.

AM replies: I mentioned in my posting that "supra ohmic" environments of
quantum brownian motion greatly suppress decoherence. In this type of
environment the system couples relatively more strongly to high frequency
bath modes. An example is electrons coupled to phonons in a crystal.
This type of environment does indeed generate colored noise. However its
not correct to conclude that any type of non-markov noise will reduce
decoherence. This only happens for "supra ohmic" environments. Sub-ohmic
environments are also non-markov but are very good decoherers. In the
supra-ohmic case the damping (which is now non-local) plays just as
critical a role as the noise in supressing decoherence. They act together
to supress decoherence, and if the non-local damping was ignored you would
still get decoherence.

> Andrew Matacz:
> Penrose and Hameroff have suggested some novel mechanisms that may shield
> microtubules from the noisy environment. This may be partially true but I
> doubt if its possible to completely shield the microtubules from noise and
> we know that decoherence requires very little noise to occur. If mesoscopic
> quantum coherence is to occur in biology I think it will be necessary for
> the system (eg microtubules) to operate in a highly non-linear regime where
> quantum coherence is continously generated, hopefully at a rate strong
> enough to not be entirely lost to quantum decoherence effects. This may
> well require some form of strong pumping. In these regimes it would be
> necessary to consider system-environment couplings, used in the quantum
> brownian motion models, that are non-linear in the system variable. It
> is interesting to note that as far as I know there have been no detailed
> studies of quantum decoherence outside of the linear coupling models.
> Relaxing this assumption may give some unexpected and interesting results.

LC: Yes, here is the problem!  Most studies of decoherence are done for
linear systems with linear couplings to the environment.

Andrew Matacz:
I don't think its necessary initially to model the environment in a serious
way. As I mentioned above this is very complicated. Initially I think it
would be fine to consider white noise and local damping but coupled to the
system in a non-linear way. The important thing is to have an efficient
decoherence mechanism. Finding any quantum system which could remain
coherent despite such an environment would be highly significant. I have
never seen such a thing in the literature. As I said before I think such
a system wouldhave to be highly non-linear with some pumping. I think this
is more likely to be a useful approach that trying to find novel mechanisms
to shield the system from the environment. The problem with this approach
is that its very  hard to do any calculations when you consider the
environment from first principles. I would like to do a calculation as I
outline above but I have never seen a model system that I could use.
Penrose and Hameroff clearly identify the tubulin dimers as quantum sites.
Specifically the dimers existing in a superposition over conformations.
It should be possible to write down an action for the dimer which would be
like a double well potential, but with some driving term due to Frohlich
pumping. But I haven't seen such a thing.

Andrew