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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 via fractional revivals

Quantum Optics II

Cozumel, December, 2004

Quantization via Fractional Revivals

Collaborators: David Aronstein

Ashok Muthukrishnan

Hideomi Nihira

Mayer Landau

Alberto Marino

Carlos Stroud, University of Rochester stroud@optics.rochester.edu

quantization via stationary states

Quantization via Stationary States

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.

Bohr Orbits

only orbits with integer n

are allowed.

Feynman path integral shows us that more

general orbits are included in the propagator.

feynman propagator

Feynman 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.

rydberg wave packet dynamics

Rydberg wave packet dynamics

Decays and revivals involve non-integer orbits

rydberg wave packet dynamics5

Schrödinger “Kitten” States

Rydberg wave packet dynamics

Such superpositions of classically distinguishable states of a single

degree of freedom are often termed “Schrödinger “Kitten” states.

schr dinger kitten states

two coherent states  radians apart

in their phase space trajectory.

two coherent states 2/N radians apart

in their phase space trajectory.

Schrödinger “Kitten” States

Analogous states of harmonic oscillators can be formed with coherent states

or more generally

bohr sommerfeld racetrack ensemble

Classical ensemble of runners with Bohr velocities

Bohr-Sommerfeld Racetrack Ensemble

Decay, revival, and fractional revival with classical ensemble,

but the revival is on the wrong side of the track!

bohr sommerfeld racetrack ensemble8

Bohr-Sommerfeld Racetrack Ensemble

Proper phase of the full revival if we choose Bohr velocities with n + ½

but, then phase is wrong at ½ fractional revival!

  • This can be understood via the semiclassical approximation to the quantum
  • propagator.
  • Propagation from the initial wave packet to the revival wave packets can
  • be described in terms of the integral of the action over classical orbits.
  • The classical orbits that contribute in general include all orbits, both those
  • of the integer and non-integer Bohr orbits.
  • At the fractional revivals only a discrete subset of the classical orbits contribute,
  • sometimes the Bohr orbits, and sometimes other orbits.
  • These discrete sets form other schemes for “quantization”.
quantization of wave packet revival intervals

The problem

Quantization of wave packet revival intervals

Describe the system in an energy basis

Given a wave packet

Find times t such that

so that

towards a more general theory of wave packet revivals

Towards a more general theory of wave packet revivals

requires

The are orthogonal thus

for some t for all n

[ multiple of ]

General solution not known, but often problem reduces to

towards a more general theory of wave packet revivals11

Towards a more general theory of wave packet revivals

[ multiple of ]

where

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 ]

towards a more general theory of wave packet revivals12

Finite difference equations for discrete polynomials

Towards a more general theory of wave packet revivals

Corresponding continuous variable problem

is a Nth order polynomial in x and t , then

Discrete version

is an Nth order polynomial in discrete variable n and continuous variable t

towards a more general theory of wave packet revivals13

[multiple of ]

[multiple of ]

Necessary and sufficient condition for revivals

Towards a more general theory of wave packet revivals

Useful ancillary conditions

towards a more general theory of wave packet revivals14

Example: Infinite square well

Towards a more general theory of wave packet revivals

problem has not been solved for general initial condition.

Special case: Ladder States

The only nonzero in the initial state are those satisfying

or

towards a more general theory of wave packet revivals15

Example: Infinite square well

Towards a more general theory of wave packet revivals

We also have the ancillary condition

which is easily evaluated as

or

this is a necessary, but not sufficient condition. Substitute it back into the

first difference equation

towards a more general theory of wave packet revivals16

The smallest integer R must contain all prime factors of not present

in

Example: Infinite square well

Towards a more general theory of wave packet revivals

For the first revival of our ladder state then

The spacing of the initially excited states determines time to first revival

towards a more general theory of wave packet revivals17

Example: Infinite square well

Towards a more general theory of wave packet revivals

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

towards a more general theory of wave packet revivals18

Example: Highly excited systems

Towards a more general theory of wave packet revivals

Autocorrrelation function

compared with predicted revival

times near second and third

superrevivals.

application of schr dinger kitten states

Application of Schrödinger Kitten States

  • Quantum discrete Fourier transform
  • Energy basis and time basis are related by a transform.
  • One can take a transform by preparing a state in one basis
  • and reading out in the complementary basis.
  • Ashok Muthukrishnan and CRS, J. Mod. Opt. 49, 2115-2127 (2002)
application of schr dinger kitten states20

Application of Schrödinger Kitten States

  • Quantum discrete Fourier transform
  • Energy basis and time basis are related by a transform.
  • One can take a transform by preparing a state in one basis
  • and reading out in the complementary basis.
  • Ashok Muthukrishnan and CRS, J. Mod. Opt. 49, 2115-2127 (2002)

Generally quantum algorithms require entanglement.

Can we entangle multi-particle systems in kitten states?

entanglement of schr dinger kitten states

Entanglement of Schrödinger Kitten States

N harmonic oscillators with nearest neighbor coupling

  • Model for lattice of interacting Rydberg atoms
  • Model for lattice of single-mode optical fibers.
n harmonic oscillators with nearest neighbor coupling

N harmonic oscillators with nearest neighbor coupling

introduce reciprocal-space variables

which diagonalize the Hamiltonian

n harmonic oscillators with nearest neighbor coupling23

Solve the Heisenberg equation of motion

N harmonic oscillators with nearest neighbor coupling

apply to initial state with only first oscillator in a coherent state.

n harmonic oscillators with nearest neighbor coupling24

N harmonic oscillators with nearest neighbor coupling

transform to the Schrödinger picture

The time dependent state is a product of coherent states for the

separate oscillators.

No entanglement here.

n harmonic oscillators with nearest neighbor coupling25

Investigate the nature of the coherent states

N harmonic oscillators with nearest neighbor coupling

Each oscillator is in a coherent state with an amplitude that varies

as a Bessel function.

entangled coherent states of n harmonic oscillators27

Prepare initial oscillator in a kitten state

Entangled coherent states of N harmonic oscillators

applying the time evolution operator to each term we find

An N -particle GHZ state if the kittens were orthogonal.

rydberg wave packet kitten states

Rydberg Wave Packet Kitten States

  • For high enough excitation the kittens are orthogonal
rydberg wave packet kitten states29

Rydberg Wave Packet Kitten States

  • For high enough excitation the kittens are orthogonal
  • Multi-level logic possible with higher-order kitten states.
making rydberg wave packet kitten states

Laboratory creation of arbitrary kitten state

Making Rydberg Wave Packet Kitten States

“Shaping an atomic electron wave packet,”

Michael W. Noel and CRS, Optics Express 1, 176 (1997).

quantization via fractional revivals31

Conclusions

Quantization via Fractional Revivals

  • For dynamics problems it may be useful to quantize via revivals rather
  • stationary states.
  • The resulting “kitten” states can be entangled.
  • Quantum logic and encryption may be carried out using these states.
  • Realizations of these states are possible with atoms and photons.

Support by ARO, NSF and ONR.