From Cavity QED to Waveguide QED · Cavity Quantum Electrodynamics (Cavity QED) It is quantum electrodynamics for pedestrians. It is not necessary to renormalize. Interaction of one

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


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


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


γ1D

γ0C1 =

σ 0Areamode

1T

Cooperativity

Not to scale

γ rad

γ1D

C1 =σ 0

Areamode•Enhancement


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?


The alligator photonic crystal waveguide (Cal Tech)


Scanning electron microscope

Cross section of the intensity

Mode area:


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


For N dipoles ε⇒ Nε

I = E02= I0

Δt = τ 0


Normal radiance

ℜe{E}

ℑm{E}

I = E02= I0 I = 4I0

Δt = τ 0 Δt =12τ 0

ℜe{E}

ℑm{E}


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}


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}


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-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

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

From Cavity QED to Waveguide QED · Cavity Quantum Electrodynamics (Cavity QED) It is quantum electrodynamics for pedestrians. It is not necessary to renormalize. Interaction of one

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