Quantization via Fractional Revivals

Quantization via Fractional Revivals

Quantum Optics IICozumel, December, 2004

Carlos Stroud, University of Rochester [email protected]

Collaborators: David Aronstein Ashok Muthukrishnan Hideomi Nihira Mayer Landau Alberto Marino

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

npdq only orbits with integer nare allowed.

Feynman path integral shows us that more general orbits are included in the propagator.

Feynman Propagator

Propagator for wave function from x,t to x’,t’ is sum of the exponentialof 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

Decays and revivals involve non-integer orbits

Rydberg wave packet dynamics

Schrödinger “Kitten” States

Such superpositions of classically distinguishable states of a singledegree of freedom are often termed “Schrödinger “Kitten” states.

Schrödinger “Kitten” States

Analogous states of harmonic oscillators can be formed with coherent states

)()(2

1)(kitten2 ttt two coherent states radians apartin their phase space trajectory.

or more generally

1

0

/21)(kittenN

j

NjieN

tN two coherent states 2/N radians apartin their phase space trajectory.

Bohr-Sommerfeld Racetrack Ensemble

Classical ensemble of runners with Bohr velocities

Decay, revival, and fractional revival with classical ensemble,but the revival is on the wrong side of the track!

Bohr-Sommerfeld Racetrack Ensemble

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

Describe the system in an energy basis

n

nti

n xectx n )(),(

Given a wave packet

n

nn xcx )()0,(

Find times t such that

n

in

tin xexectx n )0,()(),(

22 )0,(),( xtx so that

The problem

The are orthogonal thus

Towards a more general theory of wave packet revivals

n

nnn

in

tin xcexec n )()(

requires

)(xn

tn

)0,(),( xetx i

iti ee n for some t for all n

[ multiple of ] 2

General solution not known, but often problem reduces to


where

)()( nPt Ntn [ multiple of ] 2

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.

)()( nP Nt is a polynomial of degree N in n for a given t

)()( nP Nt

][1

1

tn nN

N

[ multiple of ] 2


Finite difference equations for discrete polynomials

Corresponding continuous variable problem

),( tx is a Nth order polynomial in x and t , then 0),(1

1

txxN

N

Discrete version

)(tn is an N th order polynomial in discrete variable n and continuous variable t

0)(1

1

tn nN

N

1

nnnn

k

jkjn

kjnk

k

tjkj

kn 0

2/ )()!(!

!)1( 112

2

2

nnnnn


Necessary and sufficient condition for revivals

Useful ancillary conditions

][1

1

tn nN

N

[multiple of ] 2

][1

1

tn njN

jN

[multiple of ] 2


Example: Infinite square well

Lxx

LxxV

or 0 ,0 ,0

)( 2

rev2

22 22

nTmL

nn

1rev

2

T

n

nn xcnTtitx )()/(2exp),( 2rev

problem has not been solved for general initial condition.

Special case: Ladder States

The only nonzero in the initial state are those satisfyingnc djbn j

]2 of multiple [ )()/(2 2rev

djbj

Tt or ]integer [ )()/( 2rev

djbj

Tt



We also have the ancillary condition

]integer [ )()/( 22

2

rev djbj

Tt

which is easily evaluated as

]integer [ 2)/( 2rev dTt or Rinteger somefor

2 2rev Rd

Tt

this is a necessary, but not sufficient condition. Substitute it back into thefirst difference equation

)integer (22

or )integer (222

22

222 dbd

dRdjdbd

dR



)integer (22

22 dbd

dR

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

22d22 dbd

22rev

22

2

2,2gcdor

2,2gcd2

dbddRTt

dbdddR R

For the first revival of our ladder state then

rev12rev

2Tt

dT

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



22rev

2,2gcd

dbddRTtR

Even parity initial wave packets have only odd states in their expansion, b=1, d=2

8)8,8gcd(revrev

1TTt

Odd parity initial wave packets have only even states in their expansion, b=2, d=2

4)12,8gcd(revrev

1TTt


Example: Highly excited systems

3

2

32

2

21

)()()(2T

nnsT

nnsT

nnnn

Autocorrrelation function

compared with predicted revivaltimes near second and third superrevivals.

),()0,( txx

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 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 Schrödinger Kitten States

N harmonic oscillators with nearest neighbor coupling

1

1

0

1

012

1 ,..

aachaaaaH N

N

j

N

jjjjj

• Model for lattice of interacting Rydberg atoms

• Model for lattice of single-mode optical fibers.


1

1

0

1

012

1 ,..

aachaaaaH N

N

j

N

jjjjj

introduce reciprocal-space variables

1

0

/21

0

/2 1 ,1 N

j

NjiN

jjj

Nji beN

aaeN

b

which diagonalize the Hamiltonian

1

0

/2cos where,N

NbbH


tiebtb

)0()(

tiN

Njij ebe

Nta

)0(1)(1

0

/2

tim

N

m

Nmji eaeN

)0(1 1

0,

/)(2

Solve the Heisenberg equation of motion

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

11000)()0()(

njj tata

110

1

0,

/)(2 00)0(1

N

tim

N

m

Nmji eaeN

110,

1

0,

/)(2 001

N

tiom

N

m

Nmji eeN


transform to the Schrödinger picture

jtfta jj oscillatoreach for ,)0()()0()(

1

0

/2)( whereN

tiNjij ee

Ntf

jttfta jj oscillatoreach for ,)()()()0(

111100 )()()()(

NN tftftft

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

No entanglement here.


Investigate the nature of the coherent states

1

0

/2)( N

tiNjij ee

Ntf

1

0

)/2cos(/2)( N

NtitiNjij eee

Ntf

)()/2cos()( 1

0

2/

0

/2 tJNmeeNetf m

Nmi

m

Njiti

j

time tripround tocomparedshort for times )()( 2/ tJeetf jjiti

j

Each oscillator is in a coherent state with an amplitude that variesas a Bessel function.

Entangled coherent states of N harmonic oscillators

Prepare initial oscillator in a kitten state

1100

002

1)0(

N

110110

00002

1

NN

Entangled coherent states of N harmonic oscillators

Prepare initial oscillator in a kitten state

1100

002

1)0(

N

110110

00002

1

NN

applying the time evolution operator to each term we find

111100111100 )()()()()()(

21)(

NNNN tftftftftftft

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

Rydberg Wave Packet Kitten States

• For high enough excitation the kittens are orthogonal

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

“Shaping an atomic electron wave packet,” Michael W. Noel and CRS, Optics Express 1, 176 (1997).

Quantization via Fractional Revivals

Conclusions

• 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.

Documents

Quantization via Fractional Revivals