Post on 04-Jan-2016
Quantization via Fractional Revivals
Quantum Optics IICozumel, December, 2004
Carlos Stroud, University of Rochester stroud@optics.rochester.edu
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)(kitten
N
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
Towards a more general theory of wave packet revivals
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
Towards a more general theory of wave packet revivals
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
k
n 02/ )(
)!(!
!)1( 112
2
2
nnnnn
Towards a more general theory of wave packet revivals
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
Towards a more general theory of wave packet revivals
Example: Infinite square well
Lxx
LxxV
or 0 ,
0 ,0)( 2
rev2
22 2
2n
TmL
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
Towards a more general theory of wave packet revivals
Example: Infinite square well
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
dbdd
Rdjdbd
d
R
Towards a more general theory of wave packet revivals
Example: Infinite square well
)integer (22
22
dbdd
R
The smallest integer R must contain all prime factors of not presentin
22d22 dbd
22rev
22
2
2,2gcdor
2,2gcd
2
dbdd
RTt
dbdd
dR R
For the first revival of our ladder state then
rev12rev
2Tt
d
T
The spacing of the initially excited states determines time to first revival
Towards a more general theory of wave packet revivals
Example: Infinite square well
22rev
2,2gcd
dbdd
RTtR
Even parity initial wave packets have only odd states in their expansion, b=1, d=2
8)8,8gcd(revrev
1
TTt
Odd parity initial wave packets have only even states in their expansion, b=2, d=2
4)12,8gcd(revrev
1
TTt
Towards a more general theory of wave packet revivals
Example: Highly excited systems
3
2
32
2
21
)()()(2
T
nns
T
nns
T
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.
N harmonic oscillators with nearest neighbor coupling
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
N harmonic oscillators with nearest neighbor coupling
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
N harmonic oscillators with nearest neighbor coupling
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.
N harmonic oscillators with nearest neighbor coupling
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 tJNmeeN
etf 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 )()()()()()(
2
1)(
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.