Okay, hello. We are just discussing some ways of producing Schrodinger cats and we were

just discussing optomechanics that is light and acting with mechanical motion which could

give rise to a Schrodinger cat state in principle so I'll just draw the setup again. So we have

an optical cavity but one of the mirrors can actually vibrate for example because of the

tension to a sun cantilever and so if a laser shines into the cavity and light circulates

inside the cavity there will be a radiation pressure force proportional to the intensity,

proportional to the energy of light stored inside the cavity. And so when we discuss

measurement we already discussed how this could be used to measure the motion of the

mechanical oscillator via the phase shift of the right beam for example but now we are

also discussing the fact that since the force depends on the energy stored inside the cavity

we can actually entangle the optical field and the mechanical motion. And the simplest

version of that idea would be to imagine you have a coherent superposition of many different

photon numbers, each photon number gives rise to a different radiation pressure force

so after some while you will end up in an entangled state where different photon numbers

are associated with different motion of the cantilever. So now I wanted to talk a bit

about decoherence in such a setup. For example imagine you have created a state involving

a superposition of 1 and 0 phonons, this is the most modest kind of cat we can think of

of course if you are more ambitious you will involve more phonons. Now how to calculate

the decoherence rate in such a case where if this is the harmonic oscillator potential

for my mechanical motion then these are the energy levels of the harmonic oscillator,

if it is in its round state then there are zero phonons, if it is in the first excited

state we say there is exactly one phonon and so on. So now imagine we have placed it for

example in a superposition of state number 0 and state number 1 then obviously this superposition

can be destroyed for the simple fact that because of the damping of the mechanical motion

the system can relax down to the ground state. Also if the system is actually at some fire

temperature T so there are always these thermal vibrations in the substrate to which the cantilever

is coupled then once in a while a thermal phonon, a thermal vibration will go into the

cantilever and excite its motion. So in that case we also have a rate that takes us up

from the ground state to the first excited state but nevertheless since this is a thermal

phonon which is completely incoherent it would also destroy the superposition. So these are

the bad processes. Now what is the rate for example to go down from 1 to 0? Well it turns

out of course this rate is connected to the damping rate and at 0 temperature that would

be everything but if we are at a finer temperature then in addition to this spontaneous emission

given by the rate gamma we also could have some thermally induced emission because there

is a finite number of thermal phonons. So this rate actually gets multiplied with n

thermal plus 1 and n thermal is really the occupation of the environment at this particular

frequency, the frequency of the cantilever mode called omega and also given by the temperature

so that would be given by the Bose Einstein distribution. So this makes things worse at

higher temperatures and if you ask me okay what is the corresponding rate to go up from

0 to 1? This is also given by gamma as a pre-factor but then we can only absorb thermal phonons

so that would only be given by n thermal and if the environment were at 0 temperature which

usually it is not in these experiments then this rate would completely vanish and likewise

there are ways for example taking rules. So if I then want to have an estimate for my

decoherence rate for such a situation where I am in a superposition of maybe the first

few excited states then it is really given by this gamma times n thermal maybe plus 1

but now I would imagine a situation where the temperature is much higher than the h

bar omega so that n thermal really over worms this extra factor of 1. So then I would allow

myself to just define gamma by this gamma times n thermal and in any given situation

that you want to describe you better solve the master equation to really find out for

your particular state what is the effective rate at which it decays but this sets the