Rydberg-Rydberg InteractionsF. Robicheaux

Auburn University

Rydberg gas goes to plasma

Dipole blockade

Coherent processes in frozen Rydberg gases (expts)

Theoretical investigation of an excitation hopping through a Rydberg gas

Speculations??????

Cool Rb in MOT Photo-excitee

ee

ee

e

e

e

BE = 100 – 140 KT = 300 µKn~1016 m−3

Number of ions vs t

Slow ionization then cascade

Threshold ionization to bind subsequent e-

Where does energy for ionization come from?

MechanismRydberg atoms interact with each other through electron collisions.

Electron collides with atom and gains energy. Higher energy electron excites/ionizes other Rydberg atoms. (Thermalization works against.)

Fraction of the atoms driven to very low-n.

Time scales determined by starting n, density, and (weakly) size of the Rydberg gas.

Important: electron-Rydberg collision cross section is largest for small energy changes.

Dotted: 36dDashed: ~35 high-lDot-dash: Other high-lDotDotDotDash: too deepSolid: Free electrons

Rydberg-Rydberg InteractionsCurrent interest in systems involving interaction between Rydberg atoms motivated us to study processes involving Rydberg atoms.

When is it OK to only use the (nearly) degenerate states? (essential states approximation)

Can we calculate the hopping of a Rydberg state of type A through a gas of type B? Regular lattice & amorphous placement

Propose an experiment that could image the hopping (Noordam).

Leading Order InteractionThe leading order term in the interaction potential is through the dipole-dipole term.

r2R

r132121

0

2

21 R)R̂r)(R̂r( 3 rr

4e )r,rV( ⋅⋅−⋅

−=rrrr

rr

επ

Two atoms in Rydberg state n have an energy difference proportional to n-4 with (n-1),(n+1). How close can atoms be before strong mixing?

Two atoms in state E-field, F, have stark states with maximum splitting of n-4. How close can atoms be before mixing to other stark states?

Simple ModelToo difficult to solve full Schrodinger equation so solve a model that does not have angular dependence: V(r1,r2) = -(e2/4 π ε0)(1/r1 + 1/r2 – r1 r2/R3)with R fixed.

Expand the wave function in a basis of Rydberg statesΨ(t) = S Cn,n’(t) ψn(r1) ψn’(r2)

Increase the range of n until convergence is achieved

An initial 60,60 state strongly mixes when R~ 5 n2 a0

The outer turning point ~2 n2 a0 so the atoms are almost touching.

For n < 80, essential states OK for R > 10 n2 a0

Numerical Example (60,60)

Continues to ~ 250 ns in similar fashion

red-R = 6 n2 a0, orange-R = 4.5 n2 a0225 basis: from n = 53 to 67

H-H Interaction Within n-manifoldH-atom has n2 degenerate states. The Runge-Lenz vector can be used to simplify the investigation.

[ ]

ijkba,ka,jb,ia,

2121

ijkkji

02

20

ε δ J i ]J,[J

J J L 2

A L J 2

A L J

ε A i ]A,[AVector Lenz-Runge Scaled K ) /4(e E2/ m- A

Vector Lenz-Runge r̂ Lp pL e m 2

4 K

h

rrrrr

rrr

r

h

rr

rrrrr

=

+=−

=+

=

==

+×−×=

επ

επ

Within an n-manifold the x operator has the same effect as the 3 n a0 Ax/2h operator, similar for y & z

H-H Interaction SummaryThe two independent angular momenta, J1 & J2, have magnitude (n – 1)/2. Within an n-manifold they can represent the position and angular momentum:

21210 J J L & )J J(

2an 3 r

rrrrr

h

r+=−=

For an H-atom in a static electric field in the z-direction, the interaction potential is:

)J (J 2an 3 F e z F e 2z1z

0 −=h

The eigenstates are superpositions of the l,m 0 field states using Clebsch-Gordon Coefficients < l,m|j,m1,j,m2> with eigenvalues e F 3 n a0 (m1 – m2)/2

H-H Interaction PotentialFor atom 1 use independent angular momenta, J1 & J2, with magnitude (n1 – 1)/2. For atom 2, use independent angular momenta, J3 & J4, with magnitude (n2 – 1)/2. Assume there is a static electric field, F, in the z-direction.

)]}J J(R̂)][J J(R̂[ 3 )J J()J J{( R 16a n n 9 e

)]J (J n )J (J [n 2

a F e 3 V

43214321230

2021

2

4z3z22z1z10

rrrrrrrr

h

h

−⋅−⋅−−⋅−−

−+−=

επ

There are (n1 n2)2 states. Can’t diagonalize full V matrix.

Essential states OK for R(µm) > 0.16 n2/3 F-1/3(V/cm)

Best case R(µm) > 10-4 n7/3 when F slightly less n-mixing

Regular Grid of Rydberg AtomsAssume you have a regular grid of Rydberg atoms. One of the atoms is a p-state and all others are s-states. The p-state can hop to other points in the lattice. With an amplitude that proportional to 1/R3.

This is equivalent to the tight-binding approximation in band structure calculations. Unlike typical band calculations, the “excitation” couples to more than the nearest neighbors.

Regular Grid of Rydberg Atoms (1-d)The two transverse orientations are degenerate & behave as holes. ε doesn’t depend very much on far atoms.

Similarity to metal nano-particles.

Longitudinal

Transverse

Regular Grid of Rydberg Atoms (2-d)Transverse waves have “photon” dispersion relation near k~0.Solid-(center to center of edge), dashed-(center to corner), dotted-(center of edge to corner)

Out of plane

In plane

Regular Grid of Rydberg Atoms (3-d)Slow convergence with number of atoms: R2/R3

Solid-(center to center of face), dashed-(center to center of edge), long dashed (center to corner), etc

Longitudinal

Transverse

Amorphous Placement (3-d)Density of states as a function of scaled energy.

Use periodic boundary conditions to speed convergence

Solid-63 atoms, dotted-23 atoms, dashed-63 atoms for simple model

Most of the energy distribution comes from randomness in R

Almost no change after ~33 atoms

Amorphous Placement (3-d)P = |<Ψ(0)| Ψ(t)>|2 and measures prob for hopping away

Solid-63 atoms, dotted-23 atoms, dashed-63 atoms for simple model

Most of the early t dependence comes from randomness in R

Almost no change t<1.5 after ~33

atoms

“Simple Experiment” (Noordam et al)

5 µm 20 µm

n = 61 n = 61n = 60

CCD Camera

In E-field, excite atoms to different n-state in different regions. Can watch the different state hop.

Coherent? (Previous theory---no)

Two boxes: 1 & 1

red: in left orange: in rightSinusoidal forever if box width = 0.

Two boxes: 1 & 2

red: in left orange: in right blue: other rightShift of energy in right box---can’t hop

Three boxes: 1 & 1 & 1

red: in left orange: middle blue: rightleft middle right, repeat?

Three boxes: 1 & 2 & 1

red: in left orange: middle blue: middle purple righthop over blocked box, but on much longer time scale

Five boxes: 1 & 1 & 0 & 1 & 1

red: 1 orange: 2 blue: 4 purple 5Fast oscillation 1-2 & 4-5, slow oscillation (1-2)-(4-5)