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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 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? Ultracold atoms ??? Optomechanical arrays Walter and Marquardt, arXiv:1510.06754v1 (2015) High-Tc superconductors 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(𝑟atom ) • How do we quantize the dipole moment and the field? Quantization of dipole operator • 𝑑 = −𝑒𝑟elec • Consider a hydrogen-like atom • Eigenstates of the Coulomb potential 1 Transition energy from n to n+1: ∝ 𝑛2 Energy “2p” 𝑛 = 2, 𝑙 = 𝑝 |𝒆〉 “1s” 𝑛 = 1, 𝑙 = 𝑠 |𝒈〉 • Take matrix elements of 𝑑 with eigenstates 𝑑 = −𝑒 𝑗 〈𝑗|𝑟elec 𝑗′ 〈𝑗′| 𝑗,𝑗 ′ =𝑒,𝑔 Quantization of dipole operator “2p” 𝑛 = 2, 𝑙 = 𝑝 |𝒆〉 “1s” 𝑛 = 1, 𝑙 = 𝑠 |𝒈〉 𝑑 = −𝑒 𝑗 〈𝑗|𝑟elec 𝑗′ 〈𝑗′| 𝑗,𝑗 ′ =𝑒,𝑔 • Consider symmetries 𝑔 𝑟elec 𝑔 = ∫ 𝑑𝑟 even fn. × odd × even = 0 𝑒 𝑟elec 𝑒 = 0 • Final form: 𝑑 = −𝑑0 (|e〉〈𝑔| + |g〉〈𝑒|) Contains details of atomic wavefunction, can relate to more observable quantities • Induces transitions between ground and excited states Quantization of dipole operator “2p” 𝑛 = 2, 𝑙 = 𝑝 |𝒆〉 “1s” 𝑛 = 1, 𝑙 = 𝑠 |𝒈〉 𝑑 = −𝑒 𝑗 〈𝑗|𝑟elec 𝑗′ 〈𝑗′| 𝑗,𝑗 ′ =𝑒,𝑔 • Consider symmetries 𝑔 𝑟elec 𝑔 = ∫ 𝑑𝑟 even fn. × odd × even = 0 𝑒 𝑟elec 𝑒 = 0 • Easier notation: 𝑑 = −𝑑0 (𝜎𝑒𝑔 + 𝜎𝑔𝑒 ) Definition: 𝜎𝑖𝑗 = 𝑖 〈𝑗| Field quantization • Now quantize field operator 𝐸(𝑟) • Let’s draw an analogy: a (unitless) harmonic oscillator mass Hamiltonian Dynamics (Heisenberg picture) Ladder operator representation 𝐻 = (𝑥 2 + 𝑝2 )/2 𝑑𝑥/𝑑𝑡 = 𝑝 𝑑𝑝/𝑑𝑡 = −𝑥 𝑥= 𝑝= 1 2 𝑖 2 𝑎 +𝑎 (𝑎 − 𝑎) • Physical interpretation: 𝐻=𝑎 𝑎 Number of quantized excitations (phonons) Field quantization • Now quantize field operator 𝐸(𝑟) • Compare to free-space electromagnetic field (single mode 𝜔) Hamiltonian Dynamics (Heisenberg picture) Ladder operator representation 𝐻 = (𝑥 2 + 𝑝2 )/2 𝑑𝑥/𝑑𝑡 = 𝑝 𝑑𝑝/𝑑𝑡 = −𝑥 𝑥= 𝑝= 1 2 𝑖 2 𝑎 +𝑎 (𝑎 − 𝑎) 𝐻 ∼ ∫ 𝑑𝑟(𝐸 2 + 𝐵2 )/2 𝑑𝐸/𝑑𝑡 = c 2 𝛻 × 𝐵 𝑑𝐵/𝑑𝑡 = −𝛻 × 𝐸 𝐸(𝑧) = 𝐸0 𝜖𝑘 (𝑎𝑘 𝑒 𝑖𝑘𝑧 + 𝑎𝑘 𝑒 −𝑖𝑘𝑧 ) Normalization – deal with this later… • Physical interpretation: • 𝑎𝑘 creates a photon of wavevector k, and energy 𝜔 = 𝑐𝑘 • Spatial profile of photon is given by 𝑒 𝑖𝑘𝑧 Field normalization 𝐸(𝑧) = 𝐸0 𝜖𝑘 (𝑎𝑘 𝑒 𝑖𝑘𝑧 + 𝑎𝑘 𝑒 −𝑖𝑘𝑧 ) • What is the normalization 𝐸0 ? • i.e., what is the characteristic “electric field” of a single photon? • Semi-classical argument: energy of photon in a box V • Field strength: 𝐸0 ∼ ℏ𝜔 𝜖0 𝑉 • Physically: energy of single photon is fixed, but its intensity grows if you pack it in a small box To summarize: • Interaction Hamiltonian 𝐻int = −𝑑 ⋅ 𝐸(𝑟atom ) • Interaction g is small compared to bare frequencies of photon and atomic transition • The energy “non-conserving” terms have negligible impact 𝐻int ≈ 𝑔(𝜎𝑒𝑔 𝑎𝑓 𝑟 + 𝜎𝑔𝑒 𝑎 𝑓 ∗ 𝑟 ) 𝐻0 = 𝜔𝑒𝑔 𝜎𝑒𝑒 + 𝜔𝑎 𝑎 Jaynes-Cummings model Cavity QED • Jaynes-Cummings model: interaction of atom with single mode of a cavity 𝐻 = 𝑔(𝑟) 𝜎𝑒𝑔 𝑎 + 𝜎𝑔𝑒 𝑎† + Δ𝑎† 𝑎 |𝑒, 𝑛〉 𝑔 𝑛+1 Δ = 𝜔cavity − 𝜔atom (defining energy relative to atomic transition) |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) 𝜎𝑒𝑔 𝑎 + 𝜎𝑔𝑒 𝑎† + Δ𝑎† 𝑎 |𝑒, 𝑛〉 Δ = 𝜔cavity − 𝜔atom 𝑔 𝑛+1 |𝑔, 𝑛 + 1〉 • Example • When n=0, reversible “vacuum Rabi oscillations” between photon and excited atom Cavity QED 𝐻 = 𝑔(𝑟) 𝜎𝑒𝑔 𝑎 + 𝜎𝑔𝑒 𝑎† + Δ𝑎† 𝑎 Δ = 𝜔cavity − 𝜔atom |𝑒, 𝑛〉 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 𝑝𝑖2 𝑔2 (𝑥𝑖 ) † + 𝜔cav + 𝑎 𝑎 2𝑚 Δ • Recovering “normal” optomechanics: • Take a Fabry-Perot cavity 𝑔 𝑥 = 𝑔0 cos 𝑘𝑥 • Add a tight (harmonic) external trapping potential for atoms • Can linearize in the displacement if the trap confines atoms to distances 𝑥𝑖 ≪ 𝜆 𝐻𝑂𝑀 = 𝑖∈atoms 𝑔2 (𝑥𝑖 ) † 𝑎 𝑎≈ Δ 𝑖∈atoms 2𝑔 𝑥𝑒𝑞 𝑔′ 𝑥𝑒𝑞 𝑥𝑖 𝑎† 𝑎 ≡ 𝐺𝑥𝐶𝑀 𝑎† 𝑎 Δ Effective Hamiltonian • Standard optomechanical interaction with atomic CM mode 2 𝑝𝐶𝑀 1 2 𝐻= + 𝑀𝜔2𝑇 𝑥𝐶𝑀 + 𝐺𝑥𝐶𝑀 𝑎 † 𝑎 2𝑀 2 • “Typical” numbers Experimental setup (StamperKurn, UC Berkeley) PRL 105, 133602 (2010) Nature Phys. 12, 27 (2016) (+ decoupled relative degrees of freedom) Atom number 𝑁𝑎𝑡𝑜𝑚 ∼ 103 Total mass 𝑀 ∼ 10−22 kg Trap frequency 𝜔 𝑇 ∼ 2𝜋 × 100 kHz Cavity linewidth 𝜅 ∼ 2𝜋 × 1 MHz Optomechanical coupling 𝐺𝑥𝑧𝑝 ∼ 2𝜋 × 1 MHz • 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 wellknown concepts from cavity QED • Goal: start from full system dynamics (including losses) and eliminate the photon Free-space emission Γ′ Cavity decay |𝑒, 𝑛〉 𝜅 𝑔 𝑛+1 |𝑔, 𝑛 + 1〉 𝐻 = 𝑔 𝑟 𝜎𝑒𝑔 𝑎 + 𝜎𝑔𝑒 𝑎† ′ 𝑖Γ 𝑖𝜅 † † − 𝜎𝑒𝑒 − 𝑎 𝑎 + Δ𝑎 𝑎 2 2 • 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 secondorder perturbation theory 𝛿𝜔𝑒 = 𝑚 𝛿𝜔𝑒 = 𝑒, 0 𝐻int 𝑚 〈𝑚|𝐻int |𝑒, 0〉 0 − 𝜔𝑚 𝑒, 0 𝐻int 𝑔, 1 〈𝑔, 1|𝐻int |𝑒, 0〉 −(Δ − 𝑖𝜅 2) 𝑔2 𝑟 𝑔2 𝑟 Δ 𝛿𝜔𝑒 = − =− 2 Δ − 𝑖𝜅 2 Δ + 𝜅 2 𝜅 𝑔2 𝑟 2 − 𝑖 2 Δ2 + 𝜅 2 2 Cavity-induced decay and shifts • Cavity-enhanced decay rate • On resonance: 𝑔2 𝜅 Γtotal = Γ′ + 2 Δ + 𝜅/2 4𝑔2 Γtotal = Γ′ + 𝜅 • “Cooperativity” factor 𝐶 = • Far off resonance: Γtotal 𝑔2 𝜅Γ′ 2 gives the branching ratio 𝑔2 ≈ Γ′ + 𝜅 2 Δ • Cavity-induced shift of excited state Δ𝑔2 𝛿𝜔𝑒 = − 2 Δ + 𝜅/2 2 • Far off resonance: 𝛿𝜔𝑒 = −𝑔2 /Δ agrees with previous eigenvalue calculation Two atoms in cavity • Goal: coherent excitation exchange between two atoms Γ′ |𝑒2 , 0〉 |𝑒1 , 0〉 𝑔 𝑔 𝜅 |𝑔1 𝑔2 , 1〉 • Apply similar perturbation theory on atomic excited state manifold • Find an equivalent Hamiltonian to describe the coherent dynamics (energy shifts and exchange rate) 𝐻eff = 𝑖,𝑗=1,2 𝑔 𝑟𝑖 𝑔 𝑟𝑗 (𝑖) (𝑗) − 𝜎𝑒𝑔 𝜎𝑔𝑒 Δ Two atoms in cavity • Equivalent non-Hermitian Hamiltonian to describe dissipation 𝐻d = −𝑖 𝑖,𝑗=1,2 Γ′ 𝑖 𝜎𝑒𝑒 − 2 𝑖,𝑗=1,2 𝑔 𝑟𝑖 𝑔 𝑟𝑗 𝜅 (𝑖) (𝑗) 2𝑖 𝜎𝑒𝑔 𝜎𝑔𝑒 2 Δ Optimizing an exchange interaction • An effective spin exchange interaction |𝑒2 , 0〉 |𝑒1 , 0〉 𝑔 𝑔 • Effective Hamiltonian: 𝐻eff 𝜅 |𝑔1 𝑔2 , 1〉 𝑔2 1 2 ≈ − 𝜎𝑒𝑔 𝜎𝑔𝑒 + ℎ. 𝑐. Δ • Transfer of excitation from one atom to another in time 𝑇 ∼ Δ/𝑔2 • Total error (loss) during that time: 𝑇 Γ′ + 𝑔2 𝜅/Δ2 • Optimizing with respect to Δ: 𝑑 Error 1 = 0 → Error ∼ ∼ 𝑑Δ 𝐶 𝜅Γ′ 𝑔2 Optimizing an exchange interaction • An effective spin exchange interaction |𝑒2 , 0〉 |𝑒1 , 0〉 𝑔 𝑔 𝜅 |𝑔1 𝑔2 , 1〉 • Can re-write cooperativity in terms of more physical quantities 𝜆3 𝐶∼𝑄 𝑉 • Can achieve quantum coherent spin dynamics with high cooperativity Spin-motion coupling • Focus in cavity QED is usually on spin dynamics, or spin-photon coupling • Effective Hamiltonian 𝐻eff = 𝑖,𝑗=1,2 𝑔 𝑟𝑖 𝑔 𝑟𝑗 (𝑖) (𝑗) − 𝜎𝑒𝑔 𝜎𝑔𝑒 Δ • (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, 𝛿𝐿 Setup of self-organization • Schematic of idea: Buildup of standing wave intensity provides a force Pump Ω, 𝛿𝐿 Atomic position dictates coupling strength to cavity field 𝑔(𝑥) Back-action! Cavity intensity builds up and provides force on atoms • A priori, many degrees of freedom coupled together • Possibility for elegant “emergent” phenomena? • “Self-organization” Setup of self-organization • Schematic of idea: Buildup of standing wave intensity provides a force Pump Ω, 𝛿𝐿 • Effective Hamiltonian of system 𝐻eff = 𝑖 𝑝𝑖2 − 2𝑚 𝑔02 cos 𝑘𝑥𝑖 cos 𝑘𝑥𝑗 𝑖 𝑗 𝜎𝑒𝑔 𝜎𝑔𝑒 + Δ 𝑖,𝑗 cavity-mediated spin interaction −𝑖 𝑖 Γ′ 𝑖 𝜎𝑒𝑒 − 2 𝑖,𝑗 (𝑖) 𝑖 external pump 𝑔 𝑟𝑖 𝑔 𝑟𝑗 𝜅 (𝑖) (𝑗) −2𝑖 𝜎𝑒𝑔 𝜎𝑔𝑒 2 Δ dissipation (𝑗) Ω 𝜎𝑒𝑔 + 𝜎𝑔𝑒 (𝑖) − 𝛿𝐿 𝜎𝑒𝑒 Equations of motion • Let’s consider the Heisenberg equations of motion: 𝑑𝑥𝑖 𝑝𝑖 = 𝑑𝑡 𝑚 𝑑𝑝𝑖 𝜕𝐻 𝑔02 𝑘 =− =− 𝑑𝑡 𝜕𝑥𝑖 Δ (𝑖) (𝑗) sin 𝑘𝑥𝑖 cos 𝑘𝑥𝑗 𝜎𝑒𝑔 𝜎𝑔𝑒 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 𝜎𝑔𝑒 ≈ 𝑖Ω 𝑖𝛿𝐿 −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 𝑘𝑥𝑖 cos 𝑘𝑥𝑗 = 1 for all pairs • Atoms either all sit on “even” anti-nodes 𝑘𝑥𝑖 = 2𝜋𝑛 or “odd” anti-nodes 𝑘𝑥𝑖 = 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 antinode, 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 |𝑒2 , 0〉 |𝑒1 , 0〉 𝑔 𝑔 𝜅 |𝑔1 𝑔2 , 1〉 • Working in the limit when photons are negligible (spin branch): 𝐻eff 𝑔2 ≈− Δ 𝑗 𝑖 cos 𝑘𝑥𝑖 cos 𝑘𝑥𝑗 𝜎𝑒𝑔 𝜎𝑔𝑒 𝑖,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 𝑗 2 𝐻eff = −𝜇0 𝑑02 𝜔𝑒𝑔 𝑖 Re 𝐺(𝑟𝑗 , 𝑟𝑖 , 𝜔𝑒𝑔 ) 𝜎𝑒𝑔 𝜎𝑔𝑒 𝑖,𝑗 • Losses: 𝑗 2 𝜌 = 𝐿 𝜌 = −𝜇0 𝑑02 𝜔𝑒𝑔 𝑗 𝑗 𝑖 𝑖 𝑖 Im 𝐺(𝑟𝑗 , 𝑟𝑖 , 𝜔𝑒𝑔 ) 𝜎𝑒𝑔 𝜎𝑔𝑒 𝜌 + 𝜌𝜎𝑒𝑔 𝜎𝑔𝑒 − 2𝜎𝑔𝑒 𝜌𝜎𝑒𝑔 𝑖,𝑗 • In short (non-Hermitian Hamiltonian): 𝑗 2 𝐻eff = −𝜇0 𝑑02 𝜔𝑒𝑔 𝑖 𝜎 𝐺(𝑟𝑗 , 𝑟𝑖 , 𝜔𝑒𝑔 )𝜎𝑒𝑔 𝑔𝑒 𝑖,𝑗 A “trivial” example • Must work for a single atom in free-space too 𝑗 2 𝐻eff = −𝜇0 𝑑02 𝜔𝑒𝑔 𝑖 𝐺(𝑟𝑗 , 𝑟𝑖 , 𝜔𝑒𝑔 )𝜎𝑒𝑔 𝜎𝑔𝑒 𝑖,𝑗 2 𝐻eff = −𝜇0 𝑑02 𝜔𝑒𝑔 𝐺(𝑟𝑎𝑡𝑜𝑚 , 𝑟𝑎𝑡𝑜𝑚 , 𝜔𝑒𝑔 )𝜎𝑒𝑔 𝜎𝑔𝑒 𝐻eff = 2 −𝜇0 𝑑02 𝜔𝑒𝑔 𝑖𝜔𝑒𝑔 𝑖ℏΓ0 𝜎𝑒𝑒 ≡ − 𝜎𝑒𝑒 6𝜋𝑐 2 3 𝜇0 𝑑02 𝜔𝑒𝑔 Γ0 = 3𝜋ℏ𝑐 • Recover spontaneous emission rate of atom, usually derived by Fermi’s Golden Rule! Justification of Hamiltonian 𝑗 2 𝐻eff = −𝜇0 𝑑02 𝜔𝑒𝑔 𝑖 𝐺(𝑟𝑗 , 𝑟𝑖 , 𝜔𝑒𝑔 )𝜎𝑒𝑔 𝜎𝑔𝑒 𝑖,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〉 Γ ~10 meters Photon |𝑔〉 A universal model 𝑗 2 𝐻eff = −𝜇0 𝑑02 𝜔𝑒𝑔 𝑖 𝐺(𝑟𝑗 , 𝑟𝑖 , 𝜔𝑒𝑔 )𝜎𝑒𝑔 𝜎𝑔𝑒 𝑖,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 𝑛𝑐𝑜𝑟𝑒 > 1 𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔 = 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 𝑅fiber ≪ 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: |𝒆〉 |𝒆〉 |𝒈〉 |𝒈〉 𝝎𝑳 < 𝝎𝒆𝒈 𝝎𝑳 > 𝝎𝒆𝒈 𝜶 𝝎𝑳 > 𝟎 𝜶 𝝎𝑳 < 𝟎 Optical tweezer potential 1 𝑈 𝑟 = − Re 𝛼 𝜔𝐿 𝐸 𝑟 2 2 • 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 Trap potential 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 • Typical trap depth: 100’s 𝜇K • Lifetime (without cooling): 100 ms Trap minima 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 𝜆20 − 𝑁atom 𝐴 ~ exp(−OD) OD~0.08 for single atom 𝑁𝑎𝑡𝑜𝑚 ~103 Atom-nanofiber spin model Spin model revisited • Recall in general: 𝑗 2 𝐻eff = −𝜇0 𝑑02 𝜔𝑒𝑔 𝑖 𝜎 𝐺(𝑟𝑗 , 𝑟𝑖 , 𝜔𝑒𝑔 )𝜎𝑒𝑔 𝑔𝑒 𝑖,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 • Spin model Hamiltonian 𝐻eff 𝑖Γ1D =− 2 𝑗 𝑖 𝜎 exp(𝑖𝑘 𝑧𝑖 − 𝑧𝑗 ) 𝜎𝑒𝑔 𝑔𝑒 𝑖,𝑗 • For single atom, spontaneous emission into fiber 𝐻eff 𝑖Γ1D =− 𝜎𝑒𝑒 2 • Γ1D obtained from more exact calculations or fits to experiment A toy model • So far, not very physically realistic 𝐻eff 𝑖Γ1D =− 2 exp(𝑖𝑘 𝑧𝑖 − 𝑖,𝑗 𝑖 𝜎𝑗 𝑧𝑗 ) 𝜎𝑒𝑔 𝑔𝑒 𝑖Γ′ − 2 𝑖 𝜎𝑒𝑒 𝑖 • An atom emits 100% of the time into the guided mode • Add phenomenological, independent emission rate Γ′ into free space • Γ ′ ∼ 10Γ1𝐷 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 𝑧𝑖 |𝑒〉 |𝑒〉 |𝑒〉 |𝑔〉 |𝑔〉 |𝑔〉 𝑧𝑖+1 • Stable self-organization configurations? 𝑧𝑖+2 Taking a closer look at the Hamiltonian 𝐻eff 𝑖Γ1D =− 2 exp 𝑖𝑘 𝑧𝑖 − 𝑧𝑗 𝑖 𝜎𝑗 𝜎𝑒𝑔 𝑔𝑒 𝑖,𝑗 𝑖Γ′ − 2 𝑖 𝜎𝑒𝑒 𝑖 • Let’s break up effective Hamiltonian into Hermitian and dissipative components • Hermitian part: 𝐻eff Γ1D = 2 𝑗 𝑖 𝜎 sin 𝑘 𝑧𝑖 − 𝑧𝑗 𝜎𝑒𝑔 𝑔𝑒 𝑖,𝑗 • Dissipative (anti-Hermitian) part: 𝐻eff iΓ1𝐷 =− 2 cos 𝑘 𝑧𝑖 − 𝑧𝑗 𝑖,𝑗 𝑖 𝜎𝑗 𝜎𝑒𝑔 𝑔𝑒 𝑖Γ′ − 2 𝑖 𝜎𝑒𝑒 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 𝐻eff = 𝑖 𝑝𝑖2 𝑖Γ ′ 𝑖 𝑖Γ1D 𝑖 𝑖 − 𝛿𝐿 + 𝜎𝑒𝑒 + Ω(𝜎𝑒𝑔 + 𝜎𝑔𝑒 ) − 2𝑚 2 2 exp 𝑖𝑘 𝑧𝑖 − 𝑧𝑗 𝑗 𝑖 𝜎 𝜎𝑒𝑔 𝑔𝑒 𝑖,𝑗 • Heisenberg equations of motion • De-correlate all operators (classical expectation values) • Ignore atomic saturation effects (𝜎𝑒𝑒 ≈ 0, 𝜎𝑔𝑔 ≈ 1) (Γ ≡ Γ1𝐷 + 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|𝑧𝑖 −𝑧𝑗 | ), so integers 𝑛𝑖 do not matter Weak-scattering limit 𝛿𝐿 = 𝜔laser − 𝜔0 𝑁Γ1𝐷 −Γ 0 Γ −𝑁Γ1𝐷 • External pump field is much larger than scattered field, atoms have same induced dipole moment • Minimization of mechanical potential energy 𝐻eff Γ1𝐷 = 2 𝑗 𝑖 𝜎𝑒𝑔 𝜎𝑔𝑒 all pairs sin 𝑘0 𝑧𝑖 − 𝑧𝑗 Γ1𝐷 𝜎𝑒𝑒 ≈ 2 sin 𝑘0 𝑧𝑖 − 𝑧𝑗 all pairs 2 atoms: 𝒅 = 𝟑𝝀 𝟒 Weak-scattering limit 𝛿𝐿 = 𝜔laser − 𝜔0 𝑁Γ1𝐷 −Γ 0 Γ −𝑁Γ1𝐷 • External pump field is much larger than scattered field, atoms have same induced dipole moment • Minimization of mechanical potential energy 𝐻eff Γ1𝐷 = 2 𝑗 𝑖 𝜎𝑒𝑔 𝜎𝑔𝑒 all pairs sin 𝑘0 𝑧𝑖 − 𝑧𝑗 Γ1𝐷 𝜎𝑒𝑒 ≈ 2 sin 𝑘0 𝑧𝑖 − 𝑧𝑗 all pairs N atoms: 𝒅 = 𝝀𝟎 (𝟏 − 𝟏 ) 𝟐𝑵 General numerical procedure • No analytical solution beyond weak scattering limit • Difficult to directly solve steady state for 3N highly nonlinear equations! −𝛾𝑝 /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 {𝑧𝑖,𝑠𝑠 } is reached • Decrease |𝛿| by small amount, take the previous steady state solution as the new initial condition Red detuning 𝛿𝐿 = 𝜔laser − 𝜔0 −𝑁Γ1𝐷 −Γ 0 Γ 𝑁Γ1𝐷 • Atoms have an effective refractive index 𝑛eff > 1 • Expect a contraction of lattice constant N atoms: 𝒅 ≈ 𝝀𝐞𝐟𝐟 (𝟏 − 𝟏 ), 𝟐𝑵 𝝀𝐞𝐟𝐟 < 𝝀𝟎 Simulation vs. effective index model, Γ N=150 atoms, 1𝐷 = 0.25 Γ Blue detuning 𝛿𝐿 = 𝜔laser − 𝜔0 −𝑁Γ1𝐷 −Γ 0 Γ 𝑁Γ1𝐷 • Naïvely: expansion of lattice constant • But if 𝑑 ≈ 𝜆0 , 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 𝑓𝑗 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 Γ N=150 atoms, 1𝐷 = 0.25 Γ 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: 𝐻eff Γ1D = 2 𝑗 𝑖 sin 𝑘 𝑧𝑖 − 𝑧𝑗 𝜎𝑒𝑔 𝜎𝑔𝑒 𝑖,𝑗 • Dissipative (anti-Hermitian) part: 𝐻eff iΓ1𝐷 =− 2 cos 𝑘 𝑧𝑖 − 𝑧𝑗 𝑖,𝑗 𝑖 𝜎𝑗 𝜎𝑒𝑔 𝑔𝑒 𝑖Γ′ − 2 𝑖 𝜎𝑒𝑒 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 𝑐 = 𝑘 𝑛 𝑛𝑐𝑜𝑟𝑒 𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔 < 𝑛𝑐𝑜𝑟𝑒 • Single defect: scattering 𝑘 • Periodic defects: band structure 𝑎 𝜔(𝑘) • Band gaps – forbidden propagation 𝑘 𝜋 𝑎 Atom interactions around a band edge • Consider atomic frequency near a band edge: • Single atom (spontaneous emission): • Enhanced near band edge due to high density of states (Γ1𝐷 ≫ Γ′) 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) 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, 12772 (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 𝑗 2 𝐻eff = −𝜇0 𝑑02 𝜔𝑒𝑔 𝑖 𝐺(𝑟𝑗 , 𝑟𝑖 , 𝜔𝑒𝑔 )𝜎𝑒𝑔 𝜎𝑔𝑒 𝑖,𝑗 Band gap 𝐻eff = 𝐴 𝑖,𝑗 𝑧𝑖 − 𝑧𝑗 exp − 𝐿 𝑗 𝑖 𝜎𝑒𝑔 𝜎𝑔𝑒 • 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) Naïve question 𝑈 𝑥𝑖 − 𝑥𝑗 𝜎 𝑖 𝜎 𝐻eff ≈ 𝑗 𝑖,𝑗 Mechanical potential Leads to crystallization, etc. Spin interactions Leads to entanglement, etc. • Can we create a crystal held together by spin entanglement?