Optomechanics with atoms - PowerPoint Presentation

Slide1

Optomechanics with atoms

Darrick

Chang

ICFO – The Institute of Photonic

Sciences

Barcelona, Spain

School on Quantum Nano- and Opto-Mechanics

July 8, 2016Slide2

MotivationOptomechanics: unprecedented levels of control over interactions between motion and light

Ground-state cooling

Chan et al, Nature 478, 89 (2011)

Generation of squeezed light

Safavi

-

Naeini

et al, Nature 500, 185 (2013)

Entanglement of light and motion

Palomaki

et al, Science 342, 710 (2013)

Future: exploring the boundaries of quantum physics with

optomechanical systems?

Optomechanical

arrays

Walter and Marquardt, arXiv:1510.06754v1 (2015)

???

Ultracold

atoms

High-Tc superconductorsSlide3

MotivationThe difficulties with conventional systems:Large motional massWeak optomechanical interactions (linearized equations)Short lifetimes/coherence times of phonons and photons

Levitated optomechanicsSlide4

MotivationWhat about atoms?Rich history of optical cooling/trapping (no back-action)Pristine control over long-lived internal (“spin”) states and their interactions with photons

Ion traps

Atomic ensembles

Cavity QED

Question:

Can we actively manipulate atomic quantum motion, and interact strongly with atomic spins and photons?Slide5

Goals of lecturesIntroduction to quantum atom-light interactionsJaynes-Cummings model (cavity QED)

Nanofibers

Photonic crystals

How to implement “conventional” optomechanics with atoms

Creating progressively richer behavior with atoms?

Self-organization

Tailoring

optomechanical

interactions with new platforms

Quantum many-body physics with atomic spin and motionSlide6

Introduction to atom-light interactionsSlide7

The HamiltonianA “believable” proofWhen you shine light (optical frequencies) on an atom, the response is essentially electric

Field induces a dipole moment, so…

How do we quantize the dipole moment and the field?Slide8

Quantization of dipole operatorConsider a hydrogen-like atomEigenstates of the Coulomb potential 

“1s”

“2p”

Energy

Transition energy from n to n+1:

Take matrix elements of

with eigenstates

 Slide9

Quantization of dipole operatorConsider symmetries 

“1s”

“2p”

Final form:

Contains details of atomic

wavefunction

, can relate to more observable quantities

Induces transitions between ground and excited statesSlide10

Quantization of dipole operatorConsider symmetries 

“1s”

“2p”

Easier notation:

Definition:

 Slide11

Field quantizationNow quantize field operator  

Let’s draw an analogy: a (

unitless

) harmonic oscillator mass

Hamiltonian

Ladder operator representation

Number of quantized excitations (phonons)

Physical interpretation:

Dynamics (Heisenberg picture)

 Slide12

Field quantizationNow quantize field operator  

Compare to free-space electromagnetic field (single mode

)

Hamiltonian

Dynamics (Heisenberg picture)

Ladder operator representation

Physical interpretation:

creates a photon of

wavevector

k, and energy

Spatial profile of photon is given by

Normalization – deal with this later…Slide13

Field normalizationWhat is the normalization ?i.e., what is the characteristic “electric field” of a single photon?Semi-classical argument: energy of photon in a box V 

Field strength:

Physically: energy of single photon is fixed, but its intensity grows if you pack it in a small box

 Slide14

To summarize:Interaction Hamiltonian 

Interaction g is small compared to bare frequencies of photon and atomic transition

The energy “non-conserving” terms have negligible impact

 Slide15

Jaynes-Cummings modelSlide16

Cavity QEDJaynes-Cummings model: interaction of atom with single mode of a cavity

(defining energy relative to atomic transition)

So far, ideal (no losses)

Conserves total number of excitations (

atomic+photonic

)

Can solve each number manifold separatelySlide17

Cavity QEDExample

When n=0, reversible “vacuum Rabi oscillations” between photon and excited atomSlide18

Cavity QEDMore generally, can diagonalize each number manifold

Limit where

Eigenstates are almost purely photonic or atomic

 Slide19

Eliminating degrees of freedomA priori, we have a complicated system with many degrees of freedom (motion, spin, photon)!In the far-detuned regime, we can get rid of one of them (spin or photon) in perturbation theoryPhoton branch (eliminating spin)

Interpretation: refractive index of atom shifts resonance frequency of cavitySlide20

Conventional optomechanics with atomsSlide21

Effective HamiltonianSimplified effective Hamiltonian in the photon branch:

Recovering “normal” optomechanics:

Take a

Fabry

-Perot cavity

Add a tight (harmonic) external trapping potential for atoms

Can linearize in the displacement if the trap confines atoms to distances

 Slide22

Effective HamiltonianStandard optomechanical interaction with atomic CM mode

(+ decoupled relative degrees of freedom)

“Typical” numbers

Experimental setup (Stamper-

Kurn

, UC Berkeley)

PRL 105, 133602 (2010)

Nature Phys. 12, 27 (2016)

Atom number

Total mass

kgTrap frequency

kHzCavity linewidth MHzOptomechanical coupling MHz 

Not sideband resolved

Can use other atomic physics tricks to reach motional ground stateSlide23

New physics?We worked rather hard to get to the conventional regime!

Added an external potential

Tightly trap atoms to linearize the displacement

Numbers are not especially unique

Can we find physics more unique to atoms?Slide24

Jaynes-Cummings model:Spin branchSlide25

Eliminating the photonsA priori, we have a complicated system with many degrees of freedom (motion, spin, photon)!What if we eliminate the photons instead?Slide26

Being more careful…Let’s do the calculation more carefully, to relate to some well-known concepts from cavity QEDGoal: start from full system dynamics (including losses) and eliminate the photon

Free-space emission

Cavity decay

Rigorously, should go to density matrix formalism or add “quantum jumps,” but not necessary hereSlide27

Perturbation theoryConsider the effect of cavity coupling on state in second-order perturbation theory 

 Slide28

Cavity-induced decay and shiftsCavity-enhanced decay rate

On resonance:

“Cooperativity” factor

gives the branching ratio

Cavity-induced shift of excited state

Far off resonance:

agrees with previous eigenvalue calculation

Far off resonance:

 Slide29

Two atoms in cavityGoal: coherent excitation exchange between two atoms

Find an equivalent Hamiltonian to describe the coherent dynamics (energy shifts and exchange rate)

Apply similar perturbation theory on atomic excited state manifoldSlide30

Two atoms in cavityEquivalent non-Hermitian Hamiltonian to describe dissipation

 Slide31

Optimizing an exchange interactionAn effective spin exchange interaction

Effective Hamiltonian:

Transfer of excitation from one atom to another in time

Total error (loss) during that time:

Optimi

z

ing with respect to

:

 Slide32

Optimizing an exchange interactionAn effective spin exchange interaction

Can re-write cooperativity in terms of more physical quantities

Can achieve

quantum coherent spin dynamics

with high cooperativity

 Slide33

Spin-motion couplingFocus in cavity QED is usually on spin dynamics, or spin-photon couplingEffective Hamiltonian

(Mechanical potential) x (Spin

term)

spin-

dependent

force

What are the physical consequences and possibilities?Slide34

Self-organization of atoms in a cavitySlide35

Setup of self-organizationSchematic of idea:

Atoms excite and emit photons into cavity

Pump

 Slide36

Setup of self-organizationSchematic of idea:

Buildup of standing wave intensity provides a force

Pump

Atomic position dictates coupling strength to cavity field

Cavity intensity builds up and provides force on atoms

Back-action!

A priori, many degrees of freedom coupled together

Possibility for elegant “emergent” phenomena?

“Self-organization”Slide37

Setup of self-organizationSchematic of idea:

Buildup of standing wave intensity provides a force

Pump

Effective Hamiltonian of system

cavity-mediated spin interaction

external pump

dissipationSlide38

Equations of motionLet’s consider the Heisenberg equations of motion: 

In principle, quantum correlations could make the system very rich and challenging!

Would be interesting if correlations matter (seminar!)

Some reasons to think that correlations break down:

Motion should be initially cold (ground state, quantum degenerate)

Motional time scales are very slow (atoms scatter many photons)

Scattering leads to recoil heating and breaks spin correlationsSlide39

Equations of motionThus, we’ll assume that we can de-correlate all variablesSolve classical equations of motion

For simplicity, drop

symbols, with understanding that all operators are just expectation values now

In general, the forces are

not

derivable from a potential

Equations of motion for spinsSlide40

Weak scattering limitSolutions can be studied numerically, but the “weak scattering” limit is particularly simple

Each atom then has a

constant, identical

dipole moment

(ignoring atomic saturation)

Going back to forces:

Special case, derivable from a mechanical potential!Slide41

Self-organizationConsider positive detuning  

Energy would be lowest if

for all pairs

Atoms either all sit on “even” anti-nodes

or “odd” anti-nodes

“Even”

“Odd”

Atoms can

self-organize

starting from a random distribution, and

spontaneously break the symmetrySlide42

Physical origin of symmetry breakingConsider just two atoms

Pump

Positioned at different signs

The pump field drives both atoms equally, creating dipoles oscillating with same phase

Dipoles with same phase, but sitting in an even and odd anti-node, drive a cavity field with opposite phases

No cavity field due to interference!Slide43

Summary of self-organizationClassical behavior (at least in the limits of our solution)Spin nature is not importantMaybe not surprising? Cavity mode already has standing wave structure, so “of course” atoms should organize in that pattern

Phenomena related to back-action, going beyond conventional optomechanics

Nonlinear in the displacement of atomic positions

Emergence of phase transitions

Baumann et al, Nature 464, 1301 (2010)Slide44

Beyond cavity QEDHave seen the features and limitations of atom-optomechanics with cavity QEDNew possibilities with other platforms for atom-light coupling?

Need to find a more general model for atom-light interactions, beyond Jaynes-Cummings

This new “spin model” almost

automatically

points us to photonic crystals as the route toward quantum behavior

Atom-nanofiber interfaces

Atoms coupled to photonic crystal waveguidesSlide45

Quantum spin model for atom-light interfacesSlide46

From Jaynes-Cummings to Green’s functionsWorking in the limit when photons are negligible (spin branch):

The spatial function looks like a

Green’s functionSlide47

Green’s functionIt is a tensor quantity (

because the source dipole can have three orientations, and the electric field at r is a vector

Can ignore tensor nature for our purposes

Physical interpretation

G describes the electric field at point r, due to a (normalized) oscillating dipole at r’

Simple case: free spaceSlide48

Green’s function form of spin modelClaim: coherent evolution given by

Losses:

In short (non-Hermitian Hamiltonian):

 Slide49

A “trivial” exampleMust work for a single atom in free-space too

Recover spontaneous emission rate

of atom, usually derived by Fermi’s Golden Rule!Slide50

Justification of HamiltonianAtoms produce non-classical states of light, but quantum and classical light propagate in the same wayCan use classical E&M Green’s functionRe and Im parts dictate coherent evolution and dissipationClassically: field in/out of phase with oscillating dipole stores time-averaged energy or does time-averaged work

Limits of validity

No strong coupling effects (e.g. vacuum Rabi oscillations)

Ignores time retardation

Photon

~10 metersSlide51

A universal modelThis Hamiltonian equally captures any system of atoms interacting with lightCavity QEDFree-space atomic ensemblesNanophotonic systemsEnables one to compare very different systems on an equal footing

 Slide52

Basics of atom-nanofiber experimentSlide53

Modes of nanofiberOptical fiber: guides light by total internal reflection

Method of solution: use separation of variables for fields in core and cladding, and apply E&M boundary conditions

Highlights of solution:

Field actually evanescently leaks into cladding (vacuum) region

The evanescent tail becomes very long for thin fibers

Solution respects diffraction limit

 Slide54

Optical trapping of atomsGuided mode intensity profile250 nm radius fibern = 1.45 (SiO2)937 nm (free-space) wavelength

How to trap atoms:

Optical tweezer potential

Atoms seek intensity maxima (minima) for red (blue) detuned beamsSlide55

Optical trapping of atomsGuided mode intensity profile250 nm radius fibern = 1.45 (SiO2)937 nm (free-space) wavelength

Use a combination of red and blue detuned beams to create a stable potential

Red-detuned (lower freq.) has a longer wavelength, so it attracts atoms toward fiber at large distances

Blue-detuned creates a short-range repulsion, preventing atom from crashing into fiber surface

Trap potential

Trap minima

Typical trap depth: 100’s

K

Lifetime (without cooling): 100

ms

 Slide56

Loading the trapExperimental setup:MOT

The red-detuned beam can be sent in from both sides to create a standing wave (1D optical lattice for atoms)

Many trapping minima, but need to fill them with atoms!

A magneto-optical trap probabilistically cools a cloud of cold atoms into the trap sites

Typically ~50% filling probabilitySlide57

Atom-light interactionsTransmission spectra reveal properties of atomic ensembleGood fit to broadened Lorentzian responseOn resonance:attenuation ~

for single atom

 Slide58

Atom-nanofiber spin modelSlide59

Spin model revisitedRecall in general:

In principle, we can solve for G exactly (cylindrical fiber)

Separation of variables, Bessel functions, …

 Slide60

A toy modelSuppose we have a perfect, translationally invariant 1D system

Physically, no diffraction, just propagation phase

Green’s function

Spin model Hamiltonian

For single atom, spontaneous emission into fiber

obtained from more exact calculations or fits to experiment

 Slide61

A toy modelSo far, not very physically realistic

An atom emits 100% of the time into the guided mode

Add phenomenological, independent emission rate

into free space

for nanofiber experiments

 Slide62

Self-organization of fibers in waveguide(recall the discussion session!)Slide63

Schematic of setupSimilar as in optical cavityInitially random atoms (transversely trapped, but free axially)Pump atoms from the sideAtoms scatter photons into the guided mode, which produces forces on other atoms

Stable self-organization configurations?Slide64

Taking a closer look at the HamiltonianLet’s break up effective Hamiltonian into Hermitian and dissipative components

Hermitian part:

Dissipative (anti-Hermitian) part:

is large in realistic systems

Even if

, coherent and dissipative strengths in waveguide have characteristically equal strengths

Later… how to fix this!Slide65

Outline of procedure to solveFull effective Hamiltonian

Heisenberg equations of motion

De-correlate all operators (classical expectation values)

Ignore atomic saturation effects (

)

 Slide66

A convenient parametrizationDescribe spacing between atoms in terms of an integer + fractional number of wavelengths

Spin model is periodic in distances (

), so integers

do

not

matter

 Slide67

Weak-scattering limitExternal pump field is much larger than scattered field, atoms have same induced dipole moment

Minimization of mechanical potential energy

2 atoms:

 Slide68

Weak-scattering limit

Minimization of mechanical potential energy

N atoms:

External pump field is much larger than scattered field, atoms have same induced dipole moment

 Slide69

General numerical procedureNo analytical solution beyond weak scattering limitDifficult to directly solve steady state for 3N highly nonlinear equations!

Approach

Start at large laser detuning

, use initial atomic positions corresponding to weak scattering solution

Add an artificial momentum damping

Integrate differential equations in time until steady state

is reached

Decrease by small amount, take the previous steady state solution as the new initial condition

  Slide70

Red detuningAtoms have an effective refractive index Expect a contraction of lattice constant N atoms:

,

Simulation vs. effective index model,

N=150 atoms,

 Slide71

Blue detuningNaïvely: expansion of lattice constantBut if , it is known that the atoms become a good Bragg reflector, and the “refractive index” argument is not consistent 

Actual:

t

wo “bound collective super-atoms”

Minimize effective two- “super-atom” potential

 Slide72

SimulationNumerical simulation of N=150 atoms, random initial positionsFractional positions versus time Possible because of infinite-range interactions!Slide73

Signatures of self-organizationDistinct transmission and reflection spectra of probe beamsPhotonic band structure and band gapsReflection versus pump and probe detuningsN=150 atoms,

 Slide74

Summary of self-organizationPhenomena related to back-action, going beyond conventional optomechanicsNonlinear in the displacement of atomic positionsSurprising: order emerges from a truly translationally invariant systemClassical behaviorSpin nature is not important (spin-dependent forces)Slide75

Recall the problemHermitian part of fiber Hamiltonian:

Dissipative (anti-Hermitian) part:

Even if

, coherent and dissipative strengths in waveguide have characteristically equal strengths

Expect emission to break down correlations

Dissipation comes from having a set of optical modes at the atomic resonance frequency

Need to get rid of this!Slide76

The fix – photonic crystalsSlide77

Photonic crystal waveguidesNormal fiber: light guided by total internal reflection

Single defect: scattering

Periodic defects: band structure

Band gaps – forbidden propagationSlide78

Atom interactions around a band edgeConsider atomic frequency near a band edge:

Single atom (spontaneous emission):

Enhanced near band edge due to high density of states (

)

Theory:

S. John and T.

Quang

, PRA 50, 1764 (1994)

Expts

with QD’s:

M. Arcari et al, PRL 113, 093603 (2014)Expts with atoms: A. Goban et al, Nature Commun. 5, 3808 (2014)

Wrong place to do physics!Slide79

Atom interactions around a band edgeConsider atomic frequency near a band edge:physics

Spontaneous emission shuts off (ideally),

Coherent interactions still remain!

S. John and J. Wang, PRB 43, 12772 (1991)

J.S. Douglas et al, Nature Photonics 9, 326 (2015)Slide80

Green’s function in bandgapWhat does the Green’s function look like? 

From the outside, a photonic bandgap just looks like a distributed Bragg reflector

A source inside also produces an exponentially localized field

(inside bandgap)

 Slide81

Green’s function in bandgapAttenuation length L as one approaches the band edge decreases as one moves deeper into the gap (limited by diffraction to ) 

Near band edge, L is just determined by the band curvatureSlide82

Spin model in a bandgapGeneral spin model

Purely coherent interaction (no dissipation, at least ideally!)

Tunable range of interaction L

Band gap

Now have all the ingredients to see coherent spin-motion couplingSlide83

A sneak preview of the seminar…Slide84

Magnetism vs. crystallizationSpin physics (encoded in electron spins) has been studied forever“Quantum magnetism”Curie Law  

Can destroy

paramagnetism

at low temperatures, without melting the material

Physics of spin and crystallization have different origin and different strengths (Bohr magneton vs. Coulomb)Slide85

Naïve question

Can we create a crystal held together by

spin entanglement

?

Spin interactions

Leads to entanglement, etc.

Mechanical potentialLeads to crystallization, etc.