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

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

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

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.

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

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

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

Excitation and Timing 53p 53s microwaves 50s field ramp laser 480 nm 0 1 2 5p t (μs) 53s/53p 50s 780 nm 5s

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

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

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

43d Forster resonances observed by microwave AC Stark shifts

Population transfer using chirped microwave fields

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

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!

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

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.

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.

Lasers mw Field ramp Time n=79 n=70 19 GHz 13 GHz 500 ns long pulse np Li n ~ 70 3s 2p 2s Lasers mw Field ramp Time n=79 n=70 19 GHz 13 GHz 500 ns long pulse n=70 n=79 -12 GHz/μs Maeda et al, Science

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

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

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

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).

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.

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

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

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

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

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

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

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.

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.

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

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

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

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

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

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

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.