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Nanjing University, July 2018
Luis A. Orozco www.jqi.umd.edu
From Cavity QED to Waveguide QED
The slides of the course are available at:
http://www.physics.umd.edu/rgroups/amo/orozco/results/2018/Results18.htm
Review article: P. Solano, J. A. Grover, J. E. Hoffman, S. Ravets, F. K. Fatemi, L. A. Orozco, and S. L. Rolston “Optical Nanofibers: A New Platform for Quantum Optics.” Advances in Atomic Molecular and Optical Physics, Vol. 46, 355-403, Edited by E. Arimondo, C. C. Lin, and S. F. Yelin, Academic Press, Burlington 2017. Available at: ArXiv:1703.10533
One atom interacting with light in free space.
Dipole cross section (same result for a classical dipole or from a two level atom):
σ 0 =3λ 202π
This is the “shadow” caused by a dipole on a beam of light.
H =!d ⋅!E
Energy due to the interaction between a dipole and an electric field.
!d = e 5S1/2
!r 5P3/2
The dipole matrix element between two states is fixed by the properties of the states (radial part) and the Clebsh-Gordan coefficients from the angular part of the integral. It is a few times a0 (Bohr radius) times the electron charge e between the S ground and P first excited state in alkali atoms.
Rate of decay (Fermi’s golden rule)
rad
Phase space density Interaction
Rate of decay free space (Fermi’s golden rule)
γ0 =ω03d 2
3πε0!c3
Where d is the dipole moment
Cavity QED with atoms in the optical regime
Cavity Quantum Electrodynamics (Cavity QED)
It is quantum electrodynamics for pedestrians. It is not necessary to renormalize.
Interaction of one mode or a finite number of modes of the electromagnetic field in the cavity and a single or collective atomic dipole
DIPOLE + MODE
Cavity QED platforms
(A bosonic field and a two level system)
• Microwave cavity with Rydberg atoms. • Optical cavity with alkaline atoms in the D2 line. • Micro/nano cavity and NV centers in diamond. • Micro/nano cavity and quantum dots. • Mechanical modes (phonons) and trapped ions. • RF/µ-wave resonators and transmons (two level
systems on superconductors) • …
Dipolar coupling between an atom with resonant frequency ω and a cavity mode also at ω.
(Vacuum Rabi frequency)
The field associated with the energy of a single photon in a cavity with effective volume Veff:
�
!vEdg ⋅=
effv VE
02εω!
=
Disipation by the Q factor of the cavity (κ) y spontaneous emission (γ)
g
Separation of 800 µm; Finesse 20,000 Electric field Eν~ 7 V/cm
Excitation of Rb D2 line at 780 nm
SIGNAL
PD
EMPTY CAVITY
LIGHT
Coupling
Spontaneous emission
Cavity decay
C1=g2
κγC=C1N
g≈κ ≈γ
Coupling
Spontaneous emission
Cavity decay Cooperativity for
one atom: C1
Cooperativity for N atoms: C
g
cavity
Two level system
interaction
Jaynes Cummings model
Add the reservoirs for dissipation by the cavity and by spontaneous emission with
rates κ and γ
y
x Excitation
-2Cx 1+x2 Atomic polarization:
Transmission x/y= 1/(1+2C)
Steady State weak excitation
Jaynes Cummings Dynamics Rabi Oscillations
Exchange of excitation for N atoms:
Ng≈Ω
2g Vacuum Rabi Splitting
Two normal modes
Entangled
Not coupled
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-30 -20 -10 0 10 20 30Frequency [MHz]
Sca
led
Tran
smis
sion
Transmission doublet different from the Fabry Perot resonance
Quantum Optics Results
Vacuum Rabi Oscillation
Fourier Transform
For two photons
Oscillations Fourier Transform P(n)
After 65,000 averages
If the interest is just in g, the dipole-mode coupling constant then let us look at circuit QED
The dipole d with characteristic length L is in a coaxial cavity of lengh λ/2 and radius r
The coaxial mode volume is much more confined than λ3
g = dEυ!; d = eL
Veff = πr2λ / 2;
Eυ =1r!ω 2
2π 2ε0c
gω=
Lr
⎛
⎝⎜
⎞
⎠⎟
e2
2π 2ε0!c=
Lr
⎛
⎝⎜
⎞
⎠⎟2απ
Now the coupling constant can be a percentage of the frequency!
gω=
Lr
⎛
⎝⎜
⎞
⎠⎟2απ
= 0.068 Lr
⎛
⎝⎜
⎞
⎠⎟
Be careful as the Jaynes Cummings model may no longer be adequate
For multimode cavities there is very important work by Jonathan Simon (Chicago) and by Ben Lev (Stanford).
• Nathan Schine, Albert Ryou, Andrey Gromov, Ariel Sommer,
Jonathan Simon, Synthetic Landau levels for photons, Nature, 534, 671 (2016)
• V. Vaidya, Y. Guo, R. Kroeze, K. Ballantine, A. Kollár, J. Keeling, and B. L. Lev, Tunable-range, photon mediated atomic interactions in multimode cavity QED, Physical Review X 8, 011002 (2018). Featured in APS Physics Viewpoint: A Multimode Dial for Interatomic Interactions, Physics 11, 3, (2018).
What changes to cavity QED if we do not have mirrors?
• We will use a waveguide to confine the electromagnetic field; the mode area can be smaller than λ2
• Focus on how to use it with atoms, we
are not going to talk about other qubits (solid state, superconducting, etc.)
• Photonic Structures, for example optical nanofibers
Optical Nanofibers
Optical Nanofibers
Waist 480 nm, 7 mm long
Taper lenght 28 mm
Angle 2 mrad
Core diameter 5 μmCladding diameter 125 μm
Unmodified fiber
(not to scale)
(a)
The scale
Optical Nanofibers
λ=780 nm
Lowest order fiber modes Intensities
HE11 TM01 TE01 HE21
Polarization of the lowest order modes
Very large radial gradients of E • Div E=0 implies large
longitudinal components.
• The evanescent field can have a longitudinal component of the
polarization! €
∂Er∂r
+∂Ez∂z
= 0
Polarization at the fiber waist
Rotates like a bicycle wheel
Coupling atoms to cavities and waveguides
PhD Thesis Jonathan Hood
Density of modes in 1D
Decay into the nanofiber mode
Proportional to the electric field of the guided mode.
Density of modes
Decay into the nanofiber mode
Evanescent Coupling
Not to scale
: atom
Evanescent Coupling γ rad
γ1D Not to scale
γTot = γ rad +γ1D
Evanescent Coupling γ rad
γ1D Not to scale
γTot = γ rad +γ1D
γ0 γTot ≠ γ0
Coupling Enhancement γ rad
γ1D
α =γ1Dγ0γ0
Coupling Enhancement
Not to scale
γ rad
γ1D
α =γ1Dγ0
γ0
Coupling Efficiency γ rad
γ1D
γ0β =
γ1DγTot
; γTot = γ1D +γ rad
Coupling Efficiency
β =γ1DγTot
γTot = γ rad +γ1D
Not to scale
γ rad
γ1D
Purcell Factor γ rad
γ1D
γ0γTot = γ1D +γ rad
FP =γ totγ0
=αβ
Purcell Factor
FP =γ totγ0
=αβ
Not to scale
γ rad
γ1D
Cooperativity γ rad
γ1D
γ0C1 =
β(1−β)
=γ1Dγ rad
Cooperativity
C1 =β
(1−β)=γ1Dγ rad
Not to scale
γ rad
γ1D
Cooperativity
C1 =β
(1−β)=γ1Dγ rad
Not to scale
γ rad
γ1D
C1 is the ratio of what goes into the selected mode to what goes into all the rest
Cooperativity γ rad
γ1D
γ0C1 =
σ 0Areamode
1T
Cooperativity
Not to scale
γ rad
γ1D
C1 =σ 0
Areamode•Enhancement
Cooperativity γ rad
γ1D
C1 =g2
κγ rad=
σ 0Amode
⎛
⎝⎜
⎞
⎠⎟cvg
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
γ1Dγ rad
Cooperativity
Not to scale
γ rad
γ1D
C1 =σ 0
Areamodeneff =
γ1Dγ rad
What happens if now we have a photonic structure?
PhD Thesis Jonathan Hood
PhD Thesis Jonathan Hood
The alligator photonic crystal waveguide (Cal Tech)
PhD Thesis Jonathan Hood
Scanning electron microscope
Cross section of the intensity
Mode area:
PhD Thesis Jonathan Hood
Because there is a bandgap, the cooperativity grows with it. It can also create a “cavity mode”
that does not move attached to the atom
• Dipole-dipole interactions among atoms through the waveguide. Forces among them.
• Periodic structure of atoms in a
periodic nanostructure or periodic light trap.
• New platform for quantum information already coupled directly into waveguides (fibers).
Atom trapping
Trapping scheme
1064 nm 750 nm
Quasi-linear polarization Quasi-linear polarization
Trapping scheme
Optical Nanofiber Trapping
JQI nanofiber
trap
α =γ1Dγ0
Coupling Enhancement
~120 atoms
Trapping scheme
~23 ms
Trapping scheme
Reflection and Transmission from atoms trapped in the nanofiber. Periodic array
N. V. Corzo, B. Gouraud, A. Chandra, A. Goban, A. S. Sheremet, D. Kupriyanov, J. Laurat. “Large Bragg reflection from one-dimensional chains of trapped atoms near a nanoscale waveguide.” Phys. Rev. Lett. 117, 133603 (2016).
Quantum Optics Experiments
Super- and Sub-radiance (a classical explanation)
We need the response of one oscillator due to a nearby oscillator:
!!a1 +γ0 !a1 +ω02a1 =
32ω0γ0d̂1 ⋅
!E 2!r( )a1,
γ = γ0 +32γ0Im d̂1 ⋅
!E 2!r( ){ }, with d̂1 ⋅
!E 2 0( ) =
23
ω =ω0 −34γ0 Re d̂1 ⋅
!E 2!r( ){ }
P = εΔt
= IA⇒Δt = εIA
Super- and Sub-radiance (a classical explanation)
For N dipoles ε⇒ Nε
I = E02= I0
Δt = τ 0
Super- and Sub-radiance (a classical explanation)
Normal radiance
ℜe{E}
ℑm{E}
I = E02= I0 I = 4I0
Δt = τ 0 Δt =12τ 0
ℜe{E}
ℑm{E}
Super- and Sub-radiance (a classical explanation)
Normal radiance Super-radiance
ℜe{E}
ℑm{E}
I = E02= I0 I = 4I0 I = 0
Δt = τ 0 Δt =12τ 0 Δt =∞
ℜe{E}
ℑm{E}
Super- and Sub-radiance (a classical explanation)
Normal radiance Super-radiance Sub-radiance
ℑm{E}
ℜe{E}ℜe{E}
ℑm{E}
Δt = τ 0 Δt =12τ 0 Δt =∞
ℜe{E}
ℑm{E}
Super- and Sub-radiance (a classical explanation)
Normal radiance Super-radiance Sub-radiance
ℑm{E}
ℜe{E}ℜe{E}
ℑm{E}
ge + eg ge − eg
Δt = 12τ 0 Δt =∞
ℜe{E}
ℑm{E}
Super- and Sub-radiance (a classical explanation)
Super-radiance Sub-radiance
ℑm{E}
ℜe{E}
ge + eg ge − eg
Super- and sub-radiance are interference
effects!
93
Preparing the atoms
Measuring the Radiative Lifetime
1 mm
F=1
F=2
F’=2F’=3
F’=1F’=0
De-pump Re-pump Probe
De-pump
Probe
Re-pump
Preparing the Atoms
F=1
F=2
F’=2F’=3
F’=1F’=0
De-pump Re-pump Probe
Preparing the Atoms
20 30 40 50 60 70-30
-25
-20
-15
-10
-5
0
Atom-surface position [nm]
Frecuencyshift
[MHz] Δ
Δ
Atomic distribution with optical pumping
Distribution of atoms with optical pumping
0 50 100 150 200 2500.0
0.2
0.4
0.6
0.8
1.0
Atom-surface distance (nm)
Densitydistribution(arb.unit)
Experiment with nanofibers at JQI
P. Solano, P. Barberis-Blostein, F. K. Fatemi, L. A. Orozco, and S. L. Rolston, "Super-radiance
reveals infinite-range dipole interactions through a nanofiber," Nat. Commun. 8, 1857 (2017).
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0 5 10 15 20- 6
- 5
- 4
- 3
- 2
- 1
0
time (in units of τ0)
Log 1
0no
rmal
ized
coun
t ra
te
-4
-3
-2
-1
0
Log 1
0 cou
nt r
ate
0 5 10 15 20 25 30t/τ
0
Two distinct mean lifes
τ ≈ 7.7τ 0
τ ≈ 0.9τ 0
Polarization dependent signal
Vertically polarized probe Horizontally polarized probe
Polarization dependent signal
Vertically polarized probe Horizontally polarized probe
Fitting the Simulation
-4
-3
-2
-1
0
Log 1
0 nor
mal
ized
cou
nt ra
te
-4-2024
Nor
mal
ized
Res
idua
ls
(a)
(b)
0
-1
-2
-3
-4
t/τ0
0 5 10 15 20 25 30
0 5 10 15 20
t/τ0
Log 1
0 nor
mal
ized
cou
nt ra
te
Fitting the Simulation
Can we see a collective atomic effect of atoms around the
nanofiber?
Long distance modification of the atomic radiation
nanofiber
Left MOT
Splitting the MOT in two
Right MOT
≈ 400λ
Right MOT
Left MOT
Both MOTs
Evidence of infinite-range interactions
Thanks
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