Carrier Wave Rabi Flopping (CWRF)

Presentation by Nathan Hart

Conditions for CWRF:1. There must exist a one photon resonance with the ground state 2. The Rabi frequency between the ground state and the first excited

state must be on the order of the laser frequency Result of CWRF:1. Asymmetric Bloch sphere path for the block vector.2. Broad frequency generation resulting from beating between

the atomic dipole and the laser frequency.

The two state wave function

𝛼 𝛽

1 or

i or i

1

𝛼=𝑥𝑠+𝑖 𝑦 𝑠 β=𝑥𝑝+ 𝑖 𝑦𝑝

The two state wave function (continued)Identity

𝛼=𝑥𝑠+𝑖 𝑦 𝑠=𝑟𝑠𝑒𝑖 𝜙𝑠

𝛽=𝑥𝑝+ 𝑖 𝑦𝑝=𝑟𝑝 𝑒𝑖 𝜙𝑝

Get : Multiply times

In general (just math):

𝑒−𝑖𝜑 𝑠|𝜓 ′ ⟩=|𝜓 ⟩=𝑟 𝑠∨𝑠⟩+𝑟 𝑝𝑒𝑖𝜙∨𝑝⟩

Identity

¿∨𝑠⟩The Bloch Vector

¿∨𝑝 ⟩Wikipedia: Bloch Vector

The Bloch VectorGet r: Identity on the surface

¿∨𝑠⟩

¿∨𝑝 ⟩Wikipedia: Bloch Vector

Two unknowns Try

|ψ ⟩=𝑟 𝐶𝑜𝑠(𝜃 /2)∨𝑠 ⟩+𝑟 𝑆𝑖𝑛 (𝜃 /2)𝑒𝑖 𝜙∨𝑝 ⟩

is a measure of the coherence of the two states and .r = 1 ⟹ completely coherentr = 0 ⟹ completely incoherent

Get

Optical Interpretation of Bloch SphereElectric Dipole

• The atomic dipole is in the x-y plane.• The electric field of the laser may

also be in the x-y plane.

NMR: In a semiclassical description of spin, the magnetic dipole points in the direction of the Bloch vector and precesses with it.

|ψ ⟩

¿∨𝑠⟩

¿∨𝑝 ⟩Wikipedia: Bloch Vector

3D Spatial Interpretation

¿ψ (𝑟 ,𝜃 ′ ,𝜑 ′)⟩=𝑒− 𝑖 𝑡′

|ψ (𝑟 , 𝜃 ,𝜑 ) ⟩

Rotation operator

𝑠=(𝑒−𝑖 (𝜔 −𝜈 )𝑡 /2+𝑒𝑖 (𝜔+𝜈 )𝑡 /2)(𝐴𝑒−𝑖Ωt 𝑡+𝐵𝑒𝑖Ωt 𝑡)𝑝=(𝑒−𝑖 (𝜔+𝜈 ) 𝑡 /2+𝑒𝑖 (𝜔−𝜈) 𝑡 /2)(𝐶𝑒−𝑖Ωt 𝑡+𝐷𝑒𝑖Ωt 𝑡 )

“The source for the CWRF is due to fast oscillations in the polarization equations outside the RWA.”Hughes, S. "Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses." Physical review letters 81, no. 16 (1998): 3363.

Fast oscillationHigh frequency

Slow oscillationLow frequency

Carrier Wave Rabi Flopping (CWRF)

Dipole acceleration

Frequency spectrum

|s⟩|p⟩

𝜔 𝜈

Beating the Frequencies

𝜔−𝜈≈ 0 → 𝐴=𝐵=𝐶=𝐷

𝑠∝(1+𝑒𝑖𝜈𝑡)(𝑒−𝑖Ωt 𝑡+𝑒𝑖Ωt 𝑡 )𝑝∝(𝑒− 𝑖𝜈𝑡+1)(𝑒−𝑖Ω𝑡 𝑡+φ+𝑒𝑖Ω 𝑡𝑡+φ)

Electric Dipole

List of frequencies:

(𝜔+𝜈 ) 𝑡2

=𝜈

Approximations:

• 2

• 2

• 2

• 2

Probability amplitudes:

Pulse Area Theorem:

The laser’s electric field :The Rabi frequency :The pulse area Pulse Area Theorem: The laser pulse phase is not changed (only delayed in time) if the pulse area , where is an integer. Self-Induced Transparency

𝜇=𝑑𝑖𝑝𝑜𝑙𝑒𝑚𝑜𝑚𝑒𝑛𝑡

|𝑠 (𝑡 )|2=∑

𝑛=0

∞

𝐴𝑛𝐶𝑜𝑠 (𝜔𝑛𝑡 )+∑𝑛=0

∞

𝐵𝑛𝑆𝑖𝑛(𝜔𝑛𝑡)

Fourier Series Waveform Reconstruction

M. F. Ciappina, J. A. Pérez-Hernández, A. S. Landsman, T. Zimmermann, M. Lewenstein, L. Roso, and F. Krausz, Phys. Rev. Lett. 114, 143902

Wikipedia: Fourier Series, 2015

Pulse is delayed and distorted by Rabi Flopping Pulse is slightly delayed in medium

Hughes, S. "Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses." Physical review letters 81, no. 16 (1998): 3363.

Absorption

Absorption & Frequency generation

Hughes, S. "Breakdown of the area theorem: carrier-wave Rabi flopping of femtosecond optical pulses." Physical review letters 81, no. 16 (1998): 3363.

“For these pulses, peculiar behavior emerges when the driven light intensity is so high that the period of one Rabi oscillation is comparable with that of one cycle of light.” M. F. Ciappina, J. A. Pérez-Hernández, A. S. Landsman, T. Zimmermann, M. Lewenstein, L. Roso, and F. Krausz, Phys. Rev. Lett. 114, 143902

Ω𝑅 𝜈

Time [fs]

Prob

abili

ty

3s

4s

continuum

5p

Density Matrix Simulation of Sodium Atom Level Population

Linear polarization

Nathan Hart

• Transient population inversion of ground state 3s and the excited state 5p at sufficiently high intensities.

• Possible applications for new laser mediumsNathan Hart

Density Matrix Simulation of Sodium Atom Dipole Spectrum

1st

3rd

5p

energy [eV]

phot

on y

ield

[au]

Linear polarizationBroadened odd harmonic orders

Final Notes

• M. F. Ciappina et. al. showed that sodium does not satisfy the condition #1 (slide 1) for CWRF.

• However, sodium may have a CWRF-like 3-photon resonance with the 5p energy level, allowing for broad frequency generation at each odd harmonic.

M. F. Ciappina, J. A. Pérez-Hernández, A. S. Landsman, T. Zimmermann, M. Lewenstein, L. Roso, and F. Krausz, Phys. Rev. Lett. 114, 143902