1. Manipulating Rydberg Atoms with Microwaves
P. Pillet
H. Maeda
D. Norum
J. Nunkaew
J. Han

6. Why is this interesting?
Energy level tuning
Population transport
Making new states with controllable properties

7. Floquet description applied to two level system
AC Stark shifts for resonant energy transfer
Population transport by chirped pulses
New states in combined dc and microwave fields

8. Floquet eigenstates in a periodically varying field are periodic (2π/ω) .
Consider an s and a p state. p
s+1φ
p-1φ s p
Ecosωt ω μ s
We add or subtract integral multiples of the frequency ω to construct
the Floquet states.
All pairs of states are equivalent.

9. s and p Floquet (time average) energies vs ω
Ws+ω p Wp
Energy
s+1φ ω
The degeneracy at resonance
is removed by the dipole coupling
μE, producing an avoided crossing

10. The same levels plotted with a non zero field,
creating an avoided crossing Wp
Ws+ω Ω ω

11. Exploration of this idea using a second microwave field
as a probe. p’
EcosωAt s’ μE s s

12. Excitation and Timing 53p 53s microwaves 50s field ramp laser 480 nm 5p
t (μs)
53s/53p 50s
780 nm 5s

13. Observing the dressed states of Rb in a MOT.
53p
53s
Scan 50s-53s frequency at different fixed 53s-53p transition frequencies.
50s
resonance
laser

14. With the dressing field below resonance the 53s state shifts down;
With the dressing field above resonance the 53s state shifts up.
53s
53p
53s
Scan 50s-53s frequency at different fixed 53s-53p transition frequencies.
50s
resonance

15. The AC Stark shift has been used by Martin et al to observe otherwise
inaccessible Forster energy transfer resonances.
Bohloulli-Zanjani et al PRL 2006

16. 43d Forster resonances observed by microwave AC Stark shifts

17. Population transfer using chirped microwave fields

18. Adiabatic Rapid Passage
(through a one photon avoided level crossing)
Chirp ω from below to above ω0 to
transfer population from s to p. p
Ecosωt ω0 μ ω s Wp Ω
Floquet energies
Ws+ω ω
ARP condition: dω/dt > Ω2 Ω=μE

19. The Floquet energies are time average energies.
Floquet eigenstates are periodic (2π/ω). p
s+1φ
p-1φ s p
Ecosωt ω s
(s+1φ) + p μ aligned opposite E W ΩR
(s+1φ) – p μ aligned with E E
In both states the dipoles are phase locked to the field!

20. In adiabaic rapid passage the dipole is phaselocked.
Chirp ω from below to above ω0 to
transfer population from s to p. p
Ecosωt ω0 μ ω s μ↓ Wp Ω
Floquet energies μ↑
Ws+ω ω
ARP condition: dω/dt > Ω2 Ω=μE

21. Suggested by Kalinski and Eberly
If we apply a microwave field at the Kepler frequency the dipole moment, and the orbital motion of the electron becomes phase locked to the microwave field. E E
Can we speed up or slow the electron’s orbital motion by chirping the microwave frequency ω?
Suggested by Kalinski and Eberly
Since ω =1/n 3, this amounts to changing n or the binding energy.

22. Moving n=70 atoms to n=80 with a chirped microwave pulse+ +
EMW
time + + x
The orbital frequency decreases, n increases,
and the orbit becomes larger and more
weakly bound.
The electron’s motion remains phase locked.

23. Lasers mw Field ramp Time n=79 n=70 19 GHz 13 GHz 500 ns long pulsenp Li
n ~ 70 3s 2p 2s
Lasers mw
Field ramp
Time n=79 n=70
19 GHz GHz ns long pulse
n= n= GHz/μs
Maeda et al, Science

24. Time resolved field ionization signals
field ramp
time
excite n=70
low mw power
high power

26. n=75, 76, and 77 Floquet levels in no microwave field
W75, W76-ω, W77-2ω 1ω
Energy (GHz) 2ω
n (GHz)
Chirping ω from 17 to 14 GHz transfers population from
n = 75 to n =77 by two one photon resonances.
Chirping from 14 to 17GHz does it by a two photon resonance
Oreg, Hioe, and Eberly; Noordam et al, Bergmann

27. N=68 to 84 levels no microwave field
Energy (GHz)
Microwave frequency (GHz)

28. With a 0.1V/cm microwave field the single photon avoided
crossings overlap and become a smooth curve. A B
Energy (GHz)
ν (GHz)
Chirping the frequency from 19 to 13 GHz moves
population from n = 71(A) to n=82 (B).

29. Population transfer from n=72 to 82 by chirping the frequency from 18 to 12 GHz
The Floquet energies suggest a different approach. Maeda et al.

30. 0 V/cm
4 V/cm
Starting in n=72 at 13 GHz and
Chirping to 19 GHz leads to n=82
Only a chirp of 600 MHz is required.
finish
start

31. N=72 to n=82
0 V/cm
2 V/cm
3 V/cm

32. Energy transfer from dressed states in combined static and microwave fields
Pillet et al
For simplicity we consider one of the energy transfer resonances.

33. In the treatment of resonant collisions we introduced the
direct product states ss and pp’
At R=∞ the levels cross at the
resonance field E0.
At finite R there is an avoided
crossing due to the dipole-dipole
coupling
Wpp’
Wss E W
The energy transfer cross section is given by

34. Adding a microwave field Emwcosωt to the static field Es modulates
the energy of the pp’ state
Wpp’
Energy modulation
of the pp’ state
Wss W E Es
Field modulation

35. Just as modulating the frequency of laser beam breaks it into a carrier
and sidebands, the addition of the microwave field does the same to the pp’ state.
The pp’ wavefunction without microwaves
becomes
The original pp’ state has been replaced by a carrier and sideband states
The sidebands have appreciable amplitude for

36. In terms of the energy level picture
Microwave field
Wpp’
Wss E W
Wpp’
Wss E W
Each of the pp’ sideband states crosses the ss state,
and has an avoided crossing at finite R.

37. The dipole dipole coupling between
the mth sideband state and the ss
state is
Wpp’
Wss E W
Since the cross section is proportional to
the avoided crossing (at any R), the cross
section for the m photon assisted collision
Is given by
At finite R there are
avoided crossings
where σ is the cross section in the absence of a microwave field.

38. Plot of the expected cross sections for 0, 1, 2, 3 photon resonances

39. Resonant Dipole-dipole Collisions of two Na atoms
Populate 17s in an atomic
beam
Collisions (fast atoms hit slow
ones)
Field ramp to ionize 17p
Sweep field over many laser
shots

41. There should be sets of four transitions,
corresponding to the emission of 0, 1, 2, and 3 microwave photons

42. 18s+18s→18p+17p
collisions in the presence
of a 15.4 GHz microwave field.
0 V/cm
13.5 V/cm
50 V/cm
105 V/cm
165 V/cm

43. 0 photons ○ 1 photon ●
2 photons ▲ 3 photons ■

44. Adding the microwave field creates new states, the sideband states.
The strength of the dipole-dipole interaction can be tuned at will.
The strength of the coupling between the ss state and the mth sideband state is

45. Microwaves make it possible to make and manipulate Rydberg states.
The frequency characteristics of microwave sources are MUCH better
than those of lasers.
Due to the large, ~n2, dipole matrix elements very low powers are required.