From Cavity QED to Waveguide QED · Cavity Quantum Electrodynamics (Cavity QED) It is quantum...
Transcript of From Cavity QED to Waveguide QED · Cavity Quantum Electrodynamics (Cavity QED) It is quantum...
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Nanjing University, July 2018
Luis A. Orozco www.jqi.umd.edu
From Cavity QED to Waveguide QED
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The slides of the course are available at:
http://www.physics.umd.edu/rgroups/amo/orozco/results/2018/Results18.htm
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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
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One atom interacting with light in free space.
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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.
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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.
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Rate of decay (Fermi’s golden rule)
rad
Phase space density Interaction
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Rate of decay free space (Fermi’s golden rule)
γ0 =ω03d 2
3πε0!c3
Where d is the dipole moment
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Cavity QED with atoms in the optical regime
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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
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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) • …
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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)
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g
Separation of 800 µm; Finesse 20,000 Electric field Eν~ 7 V/cm
Excitation of Rb D2 line at 780 nm
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SIGNAL
PD
EMPTY CAVITY
LIGHT
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Coupling
Spontaneous emission
Cavity decay
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C1=g2
κγC=C1N
g≈κ ≈γ
Coupling
Spontaneous emission
Cavity decay Cooperativity for
one atom: C1
Cooperativity for N atoms: C
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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
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y
x Excitation
-2Cx 1+x2 Atomic polarization:
Transmission x/y= 1/(1+2C)
Steady State weak excitation
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Jaynes Cummings Dynamics Rabi Oscillations
Exchange of excitation for N atoms:
Ng≈Ω
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2g Vacuum Rabi Splitting
Two normal modes
Entangled
Not coupled
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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
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Quantum Optics Results
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Vacuum Rabi Oscillation
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Fourier Transform
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For two photons
Oscillations Fourier Transform P(n)
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After 65,000 averages
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If the interest is just in g, the dipole-mode coupling constant then let us look at circuit QED
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The dipole d with characteristic length L is in a coaxial cavity of lengh λ/2 and radius r
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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
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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
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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).
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What changes to cavity QED if we do not have mirrors?
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• 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
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Optical Nanofibers
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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)
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The scale
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Optical Nanofibers
λ=780 nm
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Lowest order fiber modes Intensities
HE11 TM01 TE01 HE21
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Polarization of the lowest order modes
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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
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Polarization at the fiber waist
Rotates like a bicycle wheel
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Coupling atoms to cavities and waveguides
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PhD Thesis Jonathan Hood
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Density of modes in 1D
Decay into the nanofiber mode
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Proportional to the electric field of the guided mode.
Density of modes
Decay into the nanofiber mode
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Evanescent Coupling
Not to scale
: atom
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Evanescent Coupling γ rad
γ1D Not to scale
γTot = γ rad +γ1D
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Evanescent Coupling γ rad
γ1D Not to scale
γTot = γ rad +γ1D
γ0 γTot ≠ γ0
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Coupling Enhancement γ rad
γ1D
α =γ1Dγ0γ0
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Coupling Enhancement
Not to scale
γ rad
γ1D
α =γ1Dγ0
γ0
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Coupling Efficiency γ rad
γ1D
γ0β =
γ1DγTot
; γTot = γ1D +γ rad
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Coupling Efficiency
β =γ1DγTot
γTot = γ rad +γ1D
Not to scale
γ rad
γ1D
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Purcell Factor γ rad
γ1D
γ0γTot = γ1D +γ rad
FP =γ totγ0
=αβ
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Purcell Factor
FP =γ totγ0
=αβ
Not to scale
γ rad
γ1D
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Cooperativity γ rad
γ1D
γ0C1 =
β(1−β)
=γ1Dγ rad
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Cooperativity
C1 =β
(1−β)=γ1Dγ rad
Not to scale
γ rad
γ1D
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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
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Cooperativity γ rad
γ1D
γ0C1 =
σ 0Areamode
1T
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Cooperativity
Not to scale
γ rad
γ1D
C1 =σ 0
Areamode•Enhancement
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Cooperativity γ rad
γ1D
C1 =g2
κγ rad=
σ 0Amode
⎛
⎝⎜
⎞
⎠⎟cvg
⎛
⎝⎜⎜
⎞
⎠⎟⎟=
γ1Dγ rad
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Cooperativity
Not to scale
γ rad
γ1D
C1 =σ 0
Areamodeneff =
γ1Dγ rad
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What happens if now we have a photonic structure?
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PhD Thesis Jonathan Hood
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PhD Thesis Jonathan Hood
The alligator photonic crystal waveguide (Cal Tech)
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PhD Thesis Jonathan Hood
Scanning electron microscope
Cross section of the intensity
Mode area:
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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
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• 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).
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Atom trapping
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Trapping scheme
1064 nm 750 nm
Quasi-linear polarization Quasi-linear polarization
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Trapping scheme
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Optical Nanofiber Trapping
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JQI nanofiber
trap
α =γ1Dγ0
Coupling Enhancement
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~120 atoms
Trapping scheme
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~23 ms
Trapping scheme
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Reflection and Transmission from atoms trapped in the nanofiber. Periodic array
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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).
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Quantum Optics Experiments
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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( ){ }
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P = εΔt
= IA⇒Δt = εIA
Super- and Sub-radiance (a classical explanation)
For N dipoles ε⇒ Nε
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I = E02= I0
Δt = τ 0
Super- and Sub-radiance (a classical explanation)
Normal radiance
ℜe{E}
ℑm{E}
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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}
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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}
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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
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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!
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Preparing the atoms
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Measuring the Radiative Lifetime
1 mm
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F=1
F=2
F’=2F’=3
F’=1F’=0
De-pump Re-pump Probe
De-pump
Probe
Re-pump
Preparing the Atoms
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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
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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)
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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
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-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
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Polarization dependent signal
Vertically polarized probe Horizontally polarized probe
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Polarization dependent signal
Vertically polarized probe Horizontally polarized probe
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Fitting the Simulation
-4
-3
-2
-1
0
Log 1
0 nor
mal
ized
cou
nt ra
te
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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
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Fitting the Simulation
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Can we see a collective atomic effect of atoms around the
nanofiber?
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Long distance modification of the atomic radiation
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nanofiber
Left MOT
Splitting the MOT in two
Right MOT
≈ 400λ
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Right MOT
Left MOT
Both MOTs
Evidence of infinite-range interactions
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Thanks