Quantization via Fractional Revivals

1. 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

2. 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.

3. 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.

4. Rydberg wave packet dynamics Decays and revivals involve non-integer orbits

5. 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.

6. 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

7. 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!

8. 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”.

9. 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

10. 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

11. 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 ]

12. 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

13. [multiple of ] [multiple of ] Necessary and sufficient condition for revivals Towards a more general theory of wave packet revivals Useful ancillary conditions

14. 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

15. 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

16. 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

17. 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

18. 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.

19. 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)

20. 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?

21. 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.

22. N harmonic oscillators with nearest neighbor coupling introduce reciprocal-space variables which diagonalize the Hamiltonian

23. 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.

24. 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.

25. 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.

26. Prepare initial oscillator in a kitten state Entangled coherent states of N harmonic oscillators

27. 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.

28. Rydberg Wave Packet Kitten States • For high enough excitation the kittens are orthogonal

29. Rydberg Wave Packet Kitten States • For high enough excitation the kittens are orthogonal • Multi-level logic possible with higher-order kitten states.

30. 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).

31. 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.