Quantum Optics II Cozumel, December, 2004. Quantization via Fractional Revivals. Collaborators: David Aronstein Ashok Muthukrishnan Hideomi Nihira Mayer Landau Alberto Marino.
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Quantization is normally described in terms of discrete transitions
between stationary states.
Stationary states are a complete basis so it cannot be wrong.
But, it leads to a particular way of looking at quantum
mechanics that is not the most general.
only orbits with integer n
Feynman path integral shows us that more
general orbits are included in the propagator.
Propagator for wave function from x,t to x’,t’ is sum of the exponential
of the classical action over all possible paths between the two points.
Stationarity limits us to integer-action orbits.
In dynamic problems other orbits may contribute.
Decays and revivals involve non-integer orbits
Decay, revival, and fractional revival with classical ensemble,
but the revival is on the wrong side of the track!
Proper phase of the full revival if we choose Bohr velocities with n + ½
but, then phase is wrong at ½ fractional revival!
The are orthogonal thus
for some t for all n
[ multiple of ]
General solution not known, but often problem reduces to
[ multiple of ]
is a polynomial of degree N in n for a given t
Eigenvalue problem with eigenvalues t and eigenfunctions
We want to find the eigenvalues.
Apply order N+1 difference operator to each side of the equation.
[ multiple of ]
Corresponding continuous variable problem
is a Nth order polynomial in x and t , then
is an Nth order polynomial in discrete variable n and continuous variable t
[multiple of ]
Necessary and sufficient condition for revivals
Useful ancillary conditions
problem has not been solved for general initial condition.
Special case: Ladder States
The only nonzero in the initial state are those satisfying
We also have the ancillary condition
which is easily evaluated as
this is a necessary, but not sufficient condition. Substitute it back into the
first difference equation
Example: Infinite square well
For the first revival of our ladder state then
The spacing of the initially excited states determines time to first revival
Even parity initial wave packets have only odd states in their expansion, b=1, d=2
Odd parity initial wave packets have only even states in their expansion, b=2, d=2
compared with predicted revival
times near second and third
Generally quantum algorithms require entanglement.
Can we entangle multi-particle systems in kitten states?
N harmonic oscillators with nearest neighbor coupling
introduce reciprocal-space variables
which diagonalize the Hamiltonian
apply to initial state with only first oscillator in a coherent state.
transform to the Schrödinger picture
The time dependent state is a product of coherent states for the
No entanglement here.
Each oscillator is in a coherent state with an amplitude that varies
as a Bessel function.
applying the time evolution operator to each term we find
An N -particle GHZ state if the kittens were orthogonal.
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