Transcript Document

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
ψ = 𝛼|𝑠〉 + 𝛽|𝑝〉
• 𝒔 𝑎𝑡𝑜𝑚𝑖𝑐 𝑠𝑡𝑎𝑡𝑒
• 𝟎 𝑞𝑢𝑏𝑖𝑡
• ↑ 𝑠𝑝𝑖𝑛
• 𝒑 𝑎𝑡𝑜𝑚𝑖𝑐 𝑠𝑡𝑎𝑡𝑒
• 𝟏 𝑞𝑢𝑏𝑖𝑡
• ↓ 𝑠𝑝𝑖𝑛
𝐻|𝑠〉 = 𝐸𝑠 |𝑠〉
𝐻|𝑠〉 = 𝐸𝑠 |𝑠〉
i or 𝑦𝑠
i
𝛼
𝛽
1 or 𝑥𝑠
𝛼 = 𝑥𝑠 + 𝑖𝑦𝑠
1
β = 𝑥𝑝 + 𝑖𝑦𝑝
The two state wave function (continued)
𝜓′ = 𝛼|𝑠〉 + 𝛽|𝑝〉
Identity
〈𝜓′ 𝜓′ = 𝛼
2
+ 𝛽
2
=1
𝛼 = 𝑥𝑠 + 𝑖𝑦𝑠 = 𝑟𝑠 𝑒 𝑖𝜙𝑠
𝛽 = 𝑥𝑝 + 𝑖𝑦𝑝 = 𝑟𝑝 𝑒
In general (just math):
𝑖𝜙𝑝
𝑦
𝑇𝑎𝑛 𝜑𝑎 = 𝑥𝑎 ⟹ 𝜑𝑎 = 𝐴𝑟𝑐𝑇𝑎𝑛
𝑎
𝑦𝑎
𝑥𝑎
= |𝑠〉
The Bloch Vector 𝝍
Get φ: Multiply times 𝑒 −𝑖𝜙𝑠
𝑒 −𝑖𝜑𝑠 𝜓′ = 𝜓 = 𝑟𝑠 |𝑠〉 + 𝑟𝑝 𝑒 𝑖𝜙 |𝑝〉
Identity
〈𝜓|𝜓〉 = 𝑟𝑠
2
+ 𝑟𝑝
2
=1
= |𝑝〉
Wikipedia: Bloch Vector
The Bloch Vector
𝜓 = 𝑟𝑠 |𝑠〉 + 𝑟𝑝 𝑒 𝑖𝜙 |𝑝〉
Get r: Identity on the surface
𝜓𝜓 = 𝑟
Two unknowns
𝑟𝑠 =?
𝑟𝑝 =?
2
= 𝑟𝑠
Get 𝜃
2
+ 𝑟𝑝
2
=1
Try
𝑟𝑠 = 𝑟 𝐶𝑜𝑠 𝜃/2
𝑟𝑝 = 𝑟 𝑆𝑖𝑛(𝜃/2)
= |𝑠〉
ψ = 𝑟𝐶𝑜𝑠(𝜃/2)|𝑠〉 + 𝑟𝑆𝑖𝑛(𝜃/2)𝑒 𝑖𝜙 |𝑝〉
𝑟 is a measure of the coherence of the two
states |𝑠〉 and |𝑝〉.
r = 1 ⟹ completely coherent
r = 0 ⟹ completely incoherent
= |𝑝〉
Wikipedia: Bloch Vector
Optical Interpretation of Bloch Sphere
Electric Dipole
𝑒 𝜓|𝑥 𝜓 = 𝑒𝑟𝑠 𝑟𝑝 𝑠 𝑥 𝑝 𝑒 𝑖𝜙 + 𝑝 𝑥 𝑠 𝑒 −𝑖𝜙
= 2𝜇𝑠𝑝 𝑟 2 𝑆𝑖𝑛(𝜃)𝐶𝑜𝑠(𝜙)
𝜇𝑠𝑝 = 𝑝 𝑥 𝑠
𝑆𝑖𝑛 𝜃 = 𝑆𝑖𝑛 𝜃/2 𝐶𝑜𝑠(𝜃/2)
• The atomic dipole is in the x-y plane.
• The electric field of the laser may also
be in the x-y plane.
3D Spatial Interpretation
NMR: In a semiclassical description |ψ(𝑟, 𝜃′, 𝜑′)〉 = 𝑒 −𝑖𝐻𝑡 ′ ψ(𝑟, 𝜃, 𝜑)
of spin, the magnetic dipole points
in the direction of the Bloch vector
Rotation operator
and precesses with it.
= |𝑠〉
ψ
= |𝑝〉
Wikipedia: Bloch Vector
Carrier Wave Rabi Flopping (CWRF)
“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.
𝑑
𝑠 𝒙 𝑝 + 𝑐𝑐
𝑑𝑡
∞
1
𝑎 𝜔 =
𝑎 𝑡 e−𝑖𝜔𝑡 𝑑𝑡
2𝜋 −∞
𝑒2
𝑊 𝜔 = 2 2 2 𝑎 𝜔 2
4𝜖0 𝑐 𝜔
𝑎 𝑡 =
Dipole acceleration
Frequency spectrum
Slow oscillation
Low frequency
𝑠=
(𝑒 −𝑖 𝜔−𝜈 𝑡/2
𝑝 = (𝑒 −𝑖
𝜔+𝜈 𝑡/2
|p⟩
Fast oscillation
High frequency
+ 𝑒 𝑖 𝜔+𝜈 𝑡/2 )(𝐴𝑒 −𝑖Ωt 𝑡
+
+ 𝑒𝑖
+ 𝐷𝑒 𝑖Ωt 𝑡 )
𝜔−𝜈 𝑡/2 )(𝐶𝑒 −𝑖Ωt 𝑡
𝐵𝑒 𝑖Ωt 𝑡
)
𝜔
𝜈
|s⟩
Beating the Frequencies
Approximations:
𝜔+𝜈 𝑡
=𝜈
2
𝜔−𝜈 ≈0 →𝐴 =𝐵 =𝐶 =𝐷
Probability amplitudes:
𝑠 ∝ (1 + 𝑒 𝑖𝜈𝑡 )(𝑒 −𝑖Ωt 𝑡 + 𝑒 𝑖Ωt 𝑡 )
𝑝 ∝ (𝑒 −𝑖𝜈𝑡 + 1)(𝑒 −𝑖Ω𝑡𝑡+φ + 𝑒 𝑖Ω𝑡 𝑡+φ )
Electric Dipole
𝑒 𝜓|𝑥 𝜓 = 𝜇𝑠𝑝 𝑝∗ 𝑠 + 𝑠 ∗ 𝑝
= 𝜇𝑠𝑝 (𝐸𝑥𝑝 𝑖 𝜈 − Ωt 𝑡 + φ + 𝐸𝑥𝑝 𝑖 𝜈 + Ωt 𝑡 + φ + ⋯
List of frequencies:
• 𝜈 + Ωt
• 𝜈
• 𝜈 − Ωt
• Ωt
•
•
•
•
𝜈 + 2Ωt
0
𝜈 − 2Ωt
2Ωt
•
•
•
•
2𝜈 + Ωt
2𝜈
2𝜈 − Ωt
2Ωt
• 2𝜈 + 2Ωt
• 2𝜈 − 2Ωt
Pulse Area Theorem:
The laser’s electric field ℰ 𝑥, 𝑡 :
ℰ 𝑥, 𝑡 = 𝐴 𝑥, 𝑡 𝑒 𝑖𝜈𝑡
𝜇 = 𝑑𝑖𝑝𝑜𝑙𝑒 𝑚𝑜𝑚𝑒𝑛𝑡
The Rabi frequency Ω(𝑥, 𝑡):
𝜇𝐴 𝑥, 𝑡
Ω 𝑥, 𝑡 =
ℏ
The pulse area θ:
∞
𝜃(𝑥) =
Ω 𝑥, 𝑡 𝑑𝑡
−∞
Pulse Area Theorem: The laser pulse phase is not changed (only
delayed in time) if the pulse area 𝜃(𝑥) = 2𝑛𝜋, where 𝑛 is an
integer. Self-Induced Transparency
Fourier Series Waveform Reconstruction
∞
𝑠 𝑡
2
=
∞
𝐴𝑛 𝐶𝑜𝑠(𝜔𝑛 𝑡) +
𝑛=0
𝐵𝑛 𝑆𝑖𝑛(𝜔𝑛 𝑡)
𝑛=0
Wikipedia: Fourier Series, 2015
M. F. Ciappina, J. A. Pérez-Hernández, A. S.
Landsman, T. Zimmermann, M. Lewenstein, L.
Roso, and F. Krausz, Phys. Rev. Lett. 114,
143902
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.
Ω𝑅 ~ 𝜈
“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
Density Matrix Simulation of
Sodium Atom Level Population
3s
continuum
Probability
5p
𝜏 = 56 𝑓𝑠
𝐼0 = 6 × 1012 𝑊/𝑐𝑚2
λ = 800 𝑛𝑚
Linear polarization
4s
Time [fs]
• Transient population inversion of ground state 3s and the excited
state 5p at sufficiently high intensities.
• Possible applications for new laser mediums
Nathan
NathanHart
Hart
Density Matrix Simulation of
Sodium Atom Dipole Spectrum
photon yield [au]
5p
1st
3rd
energy [eV]
𝜏 = 56 𝑓𝑠
𝐼0 = 6 × 1012 𝑊/𝑐𝑚2
λ = 800 𝑛𝑚
Linear polarization
Broadened 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