Optomechanics with atoms

Optomechanics with atoms
Darrick Chang ICFO – The Institute of Photonic Sciences Barcelona, Spain School on Quantum Nano- and Opto-Mechanics July 8, 2016

Motivation Optomechanics: unprecedented levels of control over interactions between motion and light Generation of squeezed light Safavi-Naeini et al, Nature 500, 185 (2013) Entanglement of light and motion Palomaki et al, Science 342, 710 (2013) Ground-state cooling Chan et al, Nature 478, 89 (2011) Future: exploring the boundaries of quantum physics with optomechanical systems? ??? Ultracold atoms High-Tc superconductors Optomechanical arrays Walter and Marquardt, arXiv: v1 (2015)

Motivation The difficulties with conventional systems:
Large motional mass Weak optomechanical interactions (linearized equations) Short lifetimes/coherence times of phonons and photons Levitated optomechanics

Motivation What 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?

Goals of lectures Introduction to quantum atom-light interactions
Jaynes-Cummings model (cavity QED) How to implement “conventional” optomechanics with atoms Creating progressively richer behavior with atoms? Self-organization Nanofibers Photonic crystals Tailoring optomechanical interactions with new platforms Quantum many-body physics with atomic spin and motion

Introduction to atom-light interactions

The Hamiltonian A “believable” proof
When you shine light (optical frequencies) on an atom, the response is essentially electric Field induces a dipole moment, so… 𝐻=− 𝑑 ⋅ 𝐸 ( 𝑟 atom ) How do we quantize the dipole moment and the field?

Quantization of dipole operator
𝑑 =−𝑒 𝑟 elec Consider a hydrogen-like atom Eigenstates of the Coulomb potential Transition energy from n to n+1: ∝ 1 𝑛 2 “1s” “2p” Energy |𝒈〉 |𝒆〉 𝑛=2,𝑙=𝑝 𝑛=1,𝑙=𝑠 Take matrix elements of 𝑑 with eigenstates 𝑑 =−𝑒 𝑗, 𝑗 ′ =𝑒,𝑔 𝑗 〈𝑗| 𝑟 elec 𝑗′ 〈𝑗′|

|𝒆〉 “2p” 𝑛=2,𝑙=𝑝 |𝒈〉 𝑛=1,𝑙=𝑠 “1s” 𝑑 =−𝑒 𝑗, 𝑗 ′ =𝑒,𝑔 𝑗 〈𝑗| 𝑟 elec 𝑗′ 〈𝑗′| Consider symmetries 𝑔 𝑟 elec 𝑔 =∫𝑑𝑟 even fn. ×odd×even=0 𝑒 𝑟 elec 𝑒 =0 Easier notation: 𝑑 =− 𝑑 0 ( 𝜎 𝑒𝑔 + 𝜎 𝑔𝑒 ) Definition: 𝜎 𝑖𝑗 = 𝑖 〈𝑗|

Field quantization Now quantize field operator 𝐸 (𝑟)
Let’s draw an analogy: a (unitless) harmonic oscillator mass 𝐻=( 𝑥 2 + 𝑝 2 )/2 Hamiltonian Dynamics (Heisenberg picture) 𝑑𝑥/𝑑𝑡=𝑝 𝑑𝑝/𝑑𝑡=−𝑥 Ladder operator representation 𝑥= 𝑎  +𝑎 𝑝= 𝑖 2 ( 𝑎  −𝑎) 𝐻= 𝑎  𝑎 Number of quantized excitations (phonons) Physical interpretation:

Field quantization Now quantize field operator 𝐸 (𝑟)
Compare to free-space electromagnetic field (single mode 𝜔) Hamiltonian 𝐻=( 𝑥 2 + 𝑝 2 )/2 𝐻∼∫𝑑𝑟( 𝐸 2 + 𝐵 2 )/2 Dynamics (Heisenberg picture) 𝑑𝑥/𝑑𝑡=𝑝 𝑑𝑝/𝑑𝑡=−𝑥 𝑑𝐸/𝑑𝑡= c 2 𝛻×𝐵 𝑑𝐵/𝑑𝑡=−𝛻×𝐸 𝑥= 𝑎  +𝑎 𝑝= 𝑖 2 ( 𝑎  −𝑎) Ladder operator representation 𝐸(𝑧)= 𝐸 0 𝜖 𝑘 ( 𝑎 𝑘 𝑒 𝑖𝑘𝑧 + 𝑎 𝑘  𝑒 −𝑖𝑘𝑧 ) Normalization – deal with this later… Physical interpretation: 𝑎 𝑘  creates a photon of wavevector k, and energy 𝜔=𝑐𝑘 Spatial profile of photon is given by 𝑒 𝑖𝑘𝑧

Field normalization What is the normalization 𝐸 0 ?
𝐸(𝑧)= 𝐸 0 𝜖 𝑘 ( 𝑎 𝑘 𝑒 𝑖𝑘𝑧 + 𝑎 𝑘  𝑒 −𝑖𝑘𝑧 ) What is the normalization 𝐸 0 ? i.e., what is the characteristic “electric field” of a single photon? Semi-classical argument: energy of photon in a box V Field strength: 𝐸 0 ∼ ℏ𝜔 𝜖 0 𝑉 Physically: energy of single photon is fixed, but its intensity grows if you pack it in a small box

To summarize: Interaction Hamiltonian
𝐻 int =− 𝑑 ⋅ 𝐸 ( 𝑟 atom ) Interaction g is small compared to bare frequencies of photon and atomic transition The energy “non-conserving” terms have negligible impact 𝐻 int ≈𝑔( 𝜎 𝑒𝑔 𝑎𝑓 𝑟 + 𝜎 𝑔𝑒 𝑎  𝑓 ∗ 𝑟 ) 𝐻 0 = 𝜔 𝑒𝑔 𝜎 𝑒𝑒 +𝜔 𝑎  𝑎

Jaynes-Cummings model

(defining energy relative to atomic transition)
Cavity QED Jaynes-Cummings model: interaction of atom with single mode of a cavity 𝐻=𝑔(𝑟) 𝜎 𝑒𝑔 𝑎+ 𝜎 𝑔𝑒 𝑎 † +Δ 𝑎 † 𝑎 |𝑒,𝑛〉 Δ= 𝜔 cavity − 𝜔 atom (defining energy relative to atomic transition) 𝑔 𝑛+1 |𝑔,𝑛+1〉 So far, ideal (no losses) Conserves total number of excitations (atomic+photonic) Can solve each number manifold separately

Cavity QED 𝐻=𝑔(𝑟) 𝜎 𝑒𝑔 𝑎+ 𝜎 𝑔𝑒 𝑎 † +Δ 𝑎 † 𝑎 |𝑒,𝑛〉 Δ= 𝜔 cavity − 𝜔 atom 𝑔 𝑛+1 |𝑔,𝑛+1〉 Example When n=0, reversible “vacuum Rabi oscillations” between photon and excited atom

Cavity QED More generally, can diagonalize each number manifold
𝐻=𝑔(𝑟) 𝜎 𝑒𝑔 𝑎+ 𝜎 𝑔𝑒 𝑎 † +Δ 𝑎 † 𝑎 |𝑒,𝑛〉 Δ= 𝜔 cavity − 𝜔 atom 𝑔 𝑛+1 |𝑔,𝑛+1〉 More generally, can diagonalize each number manifold Limit where Δ≫𝑔 𝑛 Eigenstates are almost purely photonic or atomic

Eliminating degrees of freedom
A 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 theory Photon branch (eliminating spin) Interpretation: refractive index of atom shifts resonance frequency of cavity

Conventional optomechanics with atoms

Effective Hamiltonian
Simplified effective Hamiltonian in the photon branch: 𝐻= 𝑖∈atoms 𝑝 𝑖 2 2𝑚 + 𝜔 cav + 𝑔 2 ( 𝑥 𝑖 ) Δ 𝑎 † 𝑎 Recovering “normal” optomechanics: Take a Fabry-Perot cavity 𝑔 𝑥 = 𝑔 0 cos 𝑘𝑥 Add a tight (harmonic) external trapping potential for atoms Can linearize in the displacement if the trap confines atoms to distances 𝑥 𝑖 ≪𝜆 𝐻 𝑂𝑀 = 𝑖∈atoms 𝑔 2 ( 𝑥 𝑖 ) Δ 𝑎 † 𝑎 ≈ 𝑖∈atoms 2𝑔 𝑥 𝑒𝑞 𝑔 ′ 𝑥 𝑒𝑞 Δ 𝑥 𝑖 𝑎 † 𝑎≡𝐺 𝑥 𝐶𝑀 𝑎 † 𝑎

Effective Hamiltonian
Standard optomechanical interaction with atomic CM mode 𝐻= 𝑝 𝐶𝑀 2 2𝑀 𝑀 𝜔 𝑇 2 𝑥 𝐶𝑀 2 +𝐺 𝑥 𝐶𝑀 𝑎 † 𝑎 (+ decoupled relative degrees of freedom) “Typical” numbers Atom number 𝑁 𝑎𝑡𝑜𝑚 ∼ 10 3 Total mass 𝑀∼ 10 −22 kg Trap frequency 𝜔 𝑇 ∼2𝜋×100 kHz Cavity linewidth 𝜅∼2𝜋×1 MHz Optomechanical coupling 𝐺 𝑥 𝑧𝑝 ∼2𝜋×1 MHz Experimental setup (Stamper-Kurn, UC Berkeley) PRL 105, (2010) Nature Phys. 12, 27 (2016) Not sideband resolved Can use other atomic physics tricks to reach motional ground state

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?

Jaynes-Cummings model: Spin branch

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

Being more careful… Let’s do the calculation more carefully, to relate to some well-known concepts from cavity QED Goal: start from full system dynamics (including losses) and eliminate the photon Free-space emission Γ′ |𝑒,𝑛〉 |𝑔,𝑛+1〉 𝑔 𝑛+1 𝜅 Cavity decay − 𝑖 Γ ′ 2 𝜎 𝑒𝑒 − 𝑖𝜅 2 𝑎 † 𝑎 𝐻=𝑔 𝑟 𝜎 𝑒𝑔 𝑎+ 𝜎 𝑔𝑒 𝑎 † +Δ 𝑎 † 𝑎 Rigorously, should go to density matrix formalism or add “quantum jumps,” but not necessary here

Perturbation theory Consider the effect of cavity coupling on state |𝑒,0〉 in second-order perturbation theory 𝛿 𝜔 𝑒 = 𝑚 𝑒,0 𝐻 int 𝑚 〈𝑚| 𝐻 int |𝑒,0〉 0− 𝜔 𝑚 𝛿 𝜔 𝑒 = 𝑒,0 𝐻 int 𝑔,1 〈𝑔,1| 𝐻 int |𝑒,0〉 −(Δ− 𝑖𝜅 2 ) 𝛿 𝜔 𝑒 =− 𝑔 2 𝑟 Δ− 𝑖𝜅 2 =− 𝑔 2 𝑟 Δ Δ 𝜅 −𝑖 𝜅 2 𝑔 2 𝑟 Δ 𝜅 2 2

Cavity-induced decay and shifts
Cavity-enhanced decay rate Γ total =Γ′+ 𝑔 2 𝜅 Δ 2 + 𝜅/2 2 On resonance: Γ total =Γ′+ 4𝑔 2 𝜅 “Cooperativity” factor 𝐶= 𝑔 2 𝜅Γ′ gives the branching ratio Far off resonance: Γ total ≈Γ′+𝜅 𝑔 2 Δ 2 Cavity-induced shift of excited state 𝛿 𝜔 𝑒 =− Δ 𝑔 2 Δ 2 + 𝜅/2 2 Far off resonance: 𝛿 𝜔 𝑒 =− 𝑔 2 /Δ agrees with previous eigenvalue calculation

Two atoms in cavity 𝐻 eff = 𝑖,𝑗=1,2 − 𝑔 𝑟 𝑖 𝑔 𝑟 𝑗 Δ 𝜎 𝑒𝑔 (𝑖) 𝜎 𝑔𝑒 (𝑗)
Goal: coherent excitation exchange between two atoms Γ′ | 𝑒 1 ,0〉 | 𝑒 2 ,0〉 𝜅 𝑔 𝑔 | 𝑔 1 𝑔 2 ,1〉 Apply similar perturbation theory on atomic excited state manifold Find an equivalent Hamiltonian to describe the coherent dynamics (energy shifts and exchange rate) 𝐻 eff = 𝑖,𝑗=1,2 − 𝑔 𝑟 𝑖 𝑔 𝑟 𝑗 Δ 𝜎 𝑒𝑔 (𝑖) 𝜎 𝑔𝑒 (𝑗)

Two atoms in cavity Equivalent non-Hermitian Hamiltonian to describe dissipation 𝐻 d =−𝑖 𝑖,𝑗=1,2 Γ′ 2 𝜎 𝑒𝑒 𝑖 − 𝑖,𝑗=1,2 2𝑖 𝑔 𝑟 𝑖 𝑔 𝑟 𝑗 𝜅 Δ 2 𝜎 𝑒𝑔 (𝑖) 𝜎 𝑔𝑒 (𝑗)

Optimizing an exchange interaction
An effective spin exchange interaction | 𝑒 1 ,0〉 | 𝑒 2 ,0〉 𝜅 𝑔 𝑔 | 𝑔 1 𝑔 2 ,1〉 Effective Hamiltonian: 𝐻 eff ≈− 𝑔 2 Δ 𝜎 𝑒𝑔 1 𝜎 𝑔𝑒 2 +ℎ.𝑐. Transfer of excitation from one atom to another in time 𝑇∼Δ/ 𝑔 2 Total error (loss) during that time: 𝑇 Γ′+ 𝑔 2 𝜅/ Δ 2 Optimizing with respect to Δ: 𝑑 Error 𝑑Δ =0 → Error∼ 1 𝐶 ∼ 𝜅Γ′ 𝑔 2

Optimizing an exchange interaction
An effective spin exchange interaction | 𝑒 1 ,0〉 | 𝑒 2 ,0〉 𝜅 𝑔 𝑔 | 𝑔 1 𝑔 2 ,1〉 Can re-write cooperativity in terms of more physical quantities 𝐶∼𝑄 𝜆 3 𝑉 Can achieve quantum coherent spin dynamics with high cooperativity

Spin-motion coupling 𝐻 eff = 𝑖,𝑗=1,2 − 𝑔 𝑟 𝑖 𝑔 𝑟 𝑗 Δ 𝜎 𝑒𝑔 (𝑖) 𝜎 𝑔𝑒 (𝑗)
Focus in cavity QED is usually on spin dynamics, or spin-photon coupling Effective Hamiltonian 𝐻 eff = 𝑖,𝑗=1,2 − 𝑔 𝑟 𝑖 𝑔 𝑟 𝑗 Δ 𝜎 𝑒𝑔 (𝑖) 𝜎 𝑔𝑒 (𝑗) (Mechanical potential) x (Spin term) → spin-dependent force What are the physical consequences and possibilities?

Self-organization of atoms in a cavity

Setup of self-organization
Schematic of idea: Atoms excite and emit photons into cavity Pump Ω, 𝛿 𝐿

Schematic of idea: Buildup of standing wave intensity provides a force Pump Ω, 𝛿 𝐿 Atomic position dictates coupling strength to cavity field 𝑔(𝑥) 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”

Schematic of idea: Buildup of standing wave intensity provides a force Pump Ω, 𝛿 𝐿 Effective Hamiltonian of system 𝐻 eff = 𝑖 𝑝 𝑖 2 2𝑚 − 𝑖,𝑗 𝑔 0 2 cos 𝑘 𝑥 𝑖 cos 𝑘 𝑥 𝑗 Δ 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 + 𝑖 Ω 𝜎 𝑒𝑔 (𝑖) + 𝜎 𝑔𝑒 (𝑗) − 𝛿 𝐿 𝜎 𝑒𝑒 (𝑖) cavity-mediated spin interaction external pump −𝑖 𝑖 Γ′ 2 𝜎 𝑒𝑒 𝑖 − 𝑖,𝑗 −2𝑖 𝑔 𝑟 𝑖 𝑔 𝑟 𝑗 𝜅 Δ 2 𝜎 𝑒𝑔 (𝑖) 𝜎 𝑔𝑒 (𝑗) dissipation

Equations of motion Let’s consider the Heisenberg equations of motion:
𝑑 𝑥 𝑖 𝑑𝑡 = 𝑝 𝑖 𝑚 𝑑 𝑝 𝑖 𝑑𝑡 =− 𝜕𝐻 𝜕 𝑥 𝑖 =− 𝑔 0 2 𝑘 Δ 𝑗 sin 𝑘 𝑥 𝑖 cos 𝑘 𝑥 𝑗 𝜎 𝑒𝑔 (𝑖) 𝜎 𝑔𝑒 (𝑗) 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 correlations

Equations of motion Thus, we’ll assume that we can de-correlate all variables Solve 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 spins

Weak scattering limit Solutions can be studied numerically, but the “weak scattering” limit is particularly simple Each atom then has a constant, identical dipole moment 𝜎 𝑔𝑒 ≈ 𝑖Ω 𝑖 𝛿 𝐿 − Γ ′ /2 (ignoring atomic saturation) Going back to forces: Special case, derivable from a mechanical potential!

Self-organization Consider positive detuning Δ>0
Energy would be lowest if cos 𝑘 𝑥 𝑖 cos 𝑘 𝑥 𝑗 =1 for all pairs Atoms either all sit on “even” anti-nodes 𝑘 𝑥 𝑖 =2𝜋𝑛 or “odd” anti-nodes 𝑘 𝑥 𝑖 =2𝜋(𝑛+ 1 2 ) “Even” “Odd” Atoms can self-organize starting from a random distribution, and spontaneously break the symmetry

Physical origin of symmetry breaking
Consider just two atoms Positioned at different signs Pump Ω 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!

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

Beyond cavity QED Have seen the features and limitations of atom-optomechanics with cavity QED New possibilities with other platforms for atom-light coupling? Atom-nanofiber interfaces Atoms coupled to photonic crystal waveguides 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

Quantum spin model for atom-light interfaces

From Jaynes-Cummings to Green’s functions
| 𝑒 1 ,0〉 | 𝑒 2 ,0〉 𝜅 𝑔 𝑔 | 𝑔 1 𝑔 2 ,1〉 Working in the limit when photons are negligible (spin branch): 𝐻 eff ≈− 𝑔 2 Δ 𝑖,𝑗 cos 𝑘 𝑥 𝑖 cos 𝑘 𝑥 𝑗 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 The spatial function looks like a Green’s function

Green’s function Physical interpretation
G describes the electric field at point r, due to a (normalized) oscillating dipole at r’ It is a tensor quantity (𝛼,𝛽=𝑥,𝑦,𝑧) because the source dipole can have three orientations, and the electric field at r is a vector Can ignore tensor nature for our purposes Simple case: free space

Green’s function form of spin model
Claim: coherent evolution given by 𝐻 eff =− 𝜇 0 𝑑 0 2 𝜔 𝑒𝑔 2 𝑖,𝑗 Re 𝐺( 𝑟 𝑗 , 𝑟 𝑖 , 𝜔 𝑒𝑔 ) 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 Losses: 𝜌 =𝐿 𝜌 =− 𝜇 0 𝑑 0 2 𝜔 𝑒𝑔 2 𝑖,𝑗 Im 𝐺( 𝑟 𝑗 , 𝑟 𝑖 , 𝜔 𝑒𝑔 ) 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 𝜌+𝜌 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 −2 𝜎 𝑔𝑒 𝑗 𝜌 𝜎 𝑒𝑔 𝑖 In short (non-Hermitian Hamiltonian): 𝐻 eff =− 𝜇 0 𝑑 0 2 𝜔 𝑒𝑔 2 𝑖,𝑗 𝐺( 𝑟 𝑗 , 𝑟 𝑖 , 𝜔 𝑒𝑔 ) 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗

A “trivial” example Must work for a single atom in free-space too
𝐻 eff =− 𝜇 0 𝑑 0 2 𝜔 𝑒𝑔 2 𝑖,𝑗 𝐺( 𝑟 𝑗 , 𝑟 𝑖 , 𝜔 𝑒𝑔 ) 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 𝐻 eff =− 𝜇 0 𝑑 0 2 𝜔 𝑒𝑔 2 𝐺( 𝑟 𝑎𝑡𝑜𝑚 , 𝑟 𝑎𝑡𝑜𝑚 , 𝜔 𝑒𝑔 ) 𝜎 𝑒𝑔 𝜎 𝑔𝑒 𝐻 eff =− 𝜇 0 𝑑 0 2 𝜔 𝑒𝑔 2 𝑖 𝜔 𝑒𝑔 6𝜋𝑐 𝜎 𝑒𝑒 ≡− 𝑖ℏ Γ 𝜎 𝑒𝑒 Γ 0 = 𝜇 0 𝑑 0 2 𝜔 𝑒𝑔 3 3𝜋ℏ𝑐 Recover spontaneous emission rate of atom, usually derived by Fermi’s Golden Rule!

Justification of Hamiltonian
𝐻 eff =− 𝜇 0 𝑑 0 2 𝜔 𝑒𝑔 2 𝑖,𝑗 𝐺( 𝑟 𝑗 , 𝑟 𝑖 , 𝜔 𝑒𝑔 ) 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 Atoms produce non-classical states of light, but quantum and classical light propagate in the same way Can use classical E&M Green’s function Re and Im parts dictate coherent evolution and dissipation Classically: 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 |𝑒〉 |𝑔〉 Γ Photon ~10 meters

A universal model 𝐻 eff =− 𝜇 0 𝑑 0 2 𝜔 𝑒𝑔 2 𝑖,𝑗 𝐺( 𝑟 𝑗 , 𝑟 𝑖 , 𝜔 𝑒𝑔 ) 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 This Hamiltonian equally captures any system of atoms interacting with light Cavity QED Free-space atomic ensembles Nanophotonic systems Enables one to compare very different systems on an equal footing

Basics of atom-nanofiber experiment

Modes of nanofiber Optical fiber: guides light by total internal reflection 𝑛 𝑐𝑜𝑟𝑒 >1 𝑛 𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔 =1 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 𝑅 fiber ≪𝜆 Solution respects diffraction limit

Optical trapping of atoms
Guided mode intensity profile 250 nm radius fiber n = 1.45 (SiO2) 937 nm (free-space) wavelength How to trap atoms: |𝒈〉 |𝒆〉 𝝎 𝑳 > 𝝎 𝒆𝒈 𝜶 𝝎 𝑳 <𝟎 |𝒈〉 |𝒆〉 𝝎 𝑳 < 𝝎 𝒆𝒈 𝜶 𝝎 𝑳 >𝟎 𝑈 𝑟 =− 1 2 Re 𝛼 𝜔 𝐿 𝐸 𝑟 2 Optical tweezer potential Atoms seek intensity maxima (minima) for red (blue) detuned beams

Optical trapping of atoms
Guided mode intensity profile 250 nm radius fiber n = 1.45 (SiO2) 937 nm (free-space) wavelength Use a combination of red and blue detuned beams to create a stable potential Trap 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 minima Typical trap depth: 100’s 𝜇K Lifetime (without cooling): 100 ms

Loading the trap Experimental 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 probability

Atom-light interactions
Transmission spectra reveal properties of atomic ensemble Good fit to broadened Lorentzian response On resonance: attenuation ~ exp − 𝜆 0 2 𝐴 𝑁 atom ~ exp (−OD) OD~0.08 for single atom 𝑁 𝑎𝑡𝑜𝑚 ~ 10 3

Atom-nanofiber spin model

Spin model revisited Recall in general:
𝐻 eff =− 𝜇 0 𝑑 0 2 𝜔 𝑒𝑔 2 𝑖,𝑗 𝐺( 𝑟 𝑗 , 𝑟 𝑖 , 𝜔 𝑒𝑔 ) 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 In principle, we can solve for G exactly (cylindrical fiber) Separation of variables, Bessel functions, …

A toy model Suppose we have a perfect, translationally invariant 1D system Physically, no diffraction, just propagation phase Green’s function 𝐻 eff =− 𝑖 Γ 1D 2 𝑖,𝑗 exp (𝑖𝑘 𝑧 𝑖 − 𝑧 𝑗 ) 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 Spin model Hamiltonian For single atom, spontaneous emission into fiber 𝐻 eff =− 𝑖 Γ 1D 2 𝜎 𝑒𝑒 Γ 1D obtained from more exact calculations or fits to experiment

A toy model So far, not very physically realistic
𝐻 eff =− 𝑖 Γ 1D 2 𝑖,𝑗 exp (𝑖𝑘 𝑧 𝑖 − 𝑧 𝑗 ) 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 − 𝑖Γ′ 2 𝑖 𝜎 𝑒𝑒 𝑖 An atom emits 100% of the time into the guided mode Add phenomenological, independent emission rate Γ′ into free space Γ ′ ∼10 Γ 1𝐷 for nanofiber experiments

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

Schematic of setup Similar as in optical cavity
Initially random atoms (transversely trapped, but free axially) Pump atoms from the side Atoms scatter photons into the guided mode, which produces forces on other atoms |𝑔〉 |𝑒〉 𝑧 𝑖 𝑧 𝑖+1 𝑧 𝑖+2 Stable self-organization configurations?

Taking a closer look at the Hamiltonian
𝐻 eff =− 𝑖 Γ 1D 2 𝑖,𝑗 exp 𝑖𝑘 𝑧 𝑖 − 𝑧 𝑗 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 − 𝑖Γ′ 2 𝑖 𝜎 𝑒𝑒 𝑖 Let’s break up effective Hamiltonian into Hermitian and dissipative components Hermitian part: 𝐻 eff = Γ 1D 2 𝑖,𝑗 sin 𝑘 𝑧 𝑖 − 𝑧 𝑗 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 Dissipative (anti-Hermitian) part: 𝐻 eff =− iΓ 1𝐷 2 𝑖,𝑗 cos 𝑘 𝑧 𝑖 − 𝑧 𝑗 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 − 𝑖Γ′ 2 𝑖 𝜎 𝑒𝑒 𝑖 Γ′ is large in realistic systems Even if Γ ′ =0, coherent and dissipative strengths in waveguide have characteristically equal strengths Later… how to fix this!

Outline of procedure to solve
Full effective Hamiltonian 𝐻 eff = 𝑖 𝑝 𝑖 2 2𝑚 − 𝛿 𝐿 + 𝑖 Γ ′ 2 𝜎 𝑒𝑒 𝑖 +Ω( 𝜎 𝑒𝑔 𝑖 + 𝜎 𝑔𝑒 𝑖 ) − 𝑖 Γ 1D 2 𝑖,𝑗 exp 𝑖𝑘 𝑧 𝑖 − 𝑧 𝑗 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 Heisenberg equations of motion De-correlate all operators (classical expectation values) Ignore atomic saturation effects ( 𝜎 𝑒𝑒 ≈0, 𝜎 𝑔𝑔 ≈1) (Γ≡ Γ 1𝐷 + Γ ′ )

A convenient parametrization
Describe spacing between atoms in terms of an integer + fractional number of wavelengths Spin model is periodic in distances ( 𝑒 𝑖𝑘| 𝑧 𝑖 − 𝑧 𝑗 | ), so integers 𝑛 𝑖 do not matter

Weak-scattering limit
𝛿 𝐿 = 𝜔 laser − 𝜔 0 −𝑁 Γ 1𝐷 −Γ Γ 𝑁 Γ 1𝐷 External pump field is much larger than scattered field, atoms have same induced dipole moment 𝐻 eff = Γ 1𝐷 2 all pairs 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 sin 𝑘 0 𝑧 𝑖 − 𝑧 𝑗 ≈ Γ 1𝐷 𝜎 𝑒𝑒 2 all pairs sin 𝑘 0 𝑧 𝑖 − 𝑧 𝑗 Minimization of mechanical potential energy 2 atoms: 𝒅= 𝟑𝝀 𝟒

Weak-scattering limit
𝛿 𝐿 = 𝜔 laser − 𝜔 0 −𝑁 Γ 1𝐷 −Γ Γ 𝑁 Γ 1𝐷 External pump field is much larger than scattered field, atoms have same induced dipole moment 𝐻 eff = Γ 1𝐷 2 all pairs 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 sin 𝑘 0 𝑧 𝑖 − 𝑧 𝑗 ≈ Γ 1𝐷 𝜎 𝑒𝑒 2 all pairs sin 𝑘 0 𝑧 𝑖 − 𝑧 𝑗 Minimization of mechanical potential energy N atoms: 𝒅= 𝝀 𝟎 (𝟏− 𝟏 𝟐𝑵 )

General numerical procedure
No analytical solution beyond weak scattering limit Difficult to directly solve steady state for 3N highly nonlinear equations! −𝛾 𝑝 /2 Approach Start at large laser detuning |𝛿|, use initial atomic positions corresponding to weak scattering solution Add an artificial momentum damping Integrate differential equations in time until steady state { 𝑧 𝑖,𝑠𝑠 } is reached Decrease |𝛿| by small amount, take the previous steady state solution as the new initial condition

Red detuning Atoms have an effective refractive index 𝑛 eff >1
𝛿 𝐿 = 𝜔 laser − 𝜔 0 −𝑁 Γ 1𝐷 −Γ Γ 𝑁 Γ 1𝐷 Atoms have an effective refractive index 𝑛 eff >1 Expect a contraction of lattice constant N atoms: 𝒅≈ 𝝀 𝐞𝐟𝐟 (𝟏− 𝟏 𝟐𝑵 ), 𝝀 𝐞𝐟𝐟 < 𝝀 𝟎 Simulation vs. effective index model, N=150 atoms, Γ 1𝐷 Γ =0.25

Blue detuning Actual: two “bound collective super-atoms”
𝛿 𝐿 = 𝜔 laser − 𝜔 0 −𝑁 Γ 1𝐷 −Γ Γ 𝑁 Γ 1𝐷 Naïvely: expansion of lattice constant But if 𝑑≈ 𝜆 0 , it is known that the atoms become a good Bragg reflector, and the “refractive index” argument is not consistent Actual: two “bound collective super-atoms” Minimize effective two- “super-atom” potential

Simulation Numerical simulation of N=150 atoms, random initial positions Fractional positions 𝑓 𝑗 versus time Possible because of infinite-range interactions!

Signatures of self-organization
Distinct transmission and reflection spectra of probe beams Photonic band structure and band gaps Reflection versus pump and probe detunings N=150 atoms, Γ 1𝐷 Γ =0.25

Summary of self-organization
Phenomena related to back-action, going beyond conventional optomechanics Nonlinear in the displacement of atomic positions Surprising: order emerges from a truly translationally invariant system Classical behavior Spin nature is not important (spin-dependent forces)

Recall the problem Hermitian part of fiber Hamiltonian:
𝐻 eff = Γ 1D 2 𝑖,𝑗 sin 𝑘 𝑧 𝑖 − 𝑧 𝑗 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 Dissipative (anti-Hermitian) part: 𝐻 eff =− iΓ 1𝐷 2 𝑖,𝑗 cos 𝑘 𝑧 𝑖 − 𝑧 𝑗 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 − 𝑖Γ′ 2 𝑖 𝜎 𝑒𝑒 𝑖 Even if Γ ′ =0, 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!

The fix – photonic crystals

Photonic crystal waveguides
Normal fiber: light guided by total internal reflection 𝜔(𝑘) 𝜔 𝑘 = 𝑐 𝑛 𝑛 𝑐𝑜𝑟𝑒 𝑛 𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔 < 𝑛 𝑐𝑜𝑟𝑒 Single defect: scattering 𝑘 Periodic defects: band structure 𝑎 𝑘 𝜋 𝑎 𝜔(𝑘) Band gaps – forbidden propagation

Atom interactions around a band edge
Consider atomic frequency near a band edge: Wrong place to do physics! Single atom (spontaneous emission): Enhanced near band edge due to high density of states ( Γ 1𝐷 ≫Γ′) Theory: S. John and T. Quang, PRA 50, 1764 (1994) Expts with QD’s: M. Arcari et al, PRL 113, (2014) Expts with atoms: A. Goban et al, Nature Commun. 5, 3808 (2014)

Atom interactions around a band edge
Consider atomic frequency near a band edge: physics Spontaneous emission shuts off (ideally), Im 𝐺=0 Coherent interactions still remain! Re 𝐺≠0 S. John and J. Wang, PRB 43, (1991) J.S. Douglas et al, Nature Photonics 9, 326 (2015)

Green’s function in bandgap
What does the Green’s function 𝐺(𝑧, 𝑧 ′ ,𝜔) look like? From the outside, a photonic bandgap just looks like a distributed Bragg reflector A source inside also produces an exponentially localized field 𝐺 𝑧, 𝑧 ′ ,𝜔 ∼ exp (− 𝑧− 𝑧 ′ /𝐿) (inside bandgap)

Green’s function in bandgap
Attenuation length L 𝐿→∞ as one approaches the band edge 𝐿 decreases as one moves deeper into the gap (limited by diffraction to 𝐿∼𝜆) Near band edge, L is just determined by the band curvature

Spin model in a bandgap General spin model Band gap
𝐻 eff =− 𝜇 0 𝑑 0 2 𝜔 𝑒𝑔 2 𝑖,𝑗 𝐺( 𝑟 𝑗 , 𝑟 𝑖 , 𝜔 𝑒𝑔 ) 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 Band gap 𝐻 eff =𝐴 𝑖,𝑗 exp − 𝑧 𝑖 − 𝑧 𝑗 𝐿 𝜎 𝑒𝑔 𝑖 𝜎 𝑔𝑒 𝑗 Purely coherent interaction (no dissipation, at least ideally!) Tunable range of interaction L Now have all the ingredients to see coherent spin-motion coupling

A sneak preview of the seminar…

Magnetism vs. crystallization
Spin physics (encoded in electron spins) has been studied forever “Quantum magnetism” Curie Law 𝑴∝ 𝑩 𝑻 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)

Leads to crystallization, etc. Leads to entanglement, etc.
Naïve question 𝐻 eff ≈ 𝑖,𝑗 𝑈 𝑥 𝑖 − 𝑥 𝑗 𝜎 𝑖 𝜎 𝑗 Mechanical potential Leads to crystallization, etc. Spin interactions Leads to entanglement, etc. Can we create a crystal held together by spin entanglement?

Optomechanics with atoms

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