Quantum Optics with Mesoscopic Systems II

Quantum Optics with Mesoscopic Systems II A. Imamoglu Quantum Photonics Group, Department of Physics ETH-Zürich Outline 1) Cavity-QED with a single quantum dot 2) Optical pumping of quantum dot spins 3) Conditional dynamics of two quantum dots

Why is it interesting to couple QDs to cavities? Near-unity collection efficiency of single photon sources Strong interactions between single photons mediated by a QD Cavity-mediated coupling between distant spins

Cavity Quantum Electrodynamics (cavity-qed) Single two-level (anharmonic) emitter coupled to a single cavity mode is described by the Jaynes-Cummings (JC) Hamiltonian An exactly solvable model ω eg : emitter frequency ω c : cavity frequency g c : cavity-emitter coupling strength In all optical realizations: g c << ω c ω eg 10 10 10 15 10 15 e> g> ω eg For electric-dipole coupling: ω ħg c = e <e ε.r g>

Eigenstates of the JC Hamiltonian 2 2 g c 2g c ;2> = g; n c =2> e; n c =1> ;2> = g; n c =2> + e; n c =1> ;1> = g; n c =1> e; n c =0> +;1> = g; n c =1> + e; n c =0> The eigenstates of the coupled system are entangled emittercavity states The spectrum is anharmonic: the nonlinearity of the two-level emitter ensures that the coupled system is also anharmonic. ω c = ω eg 0> = g; n c = 0>

Eigenstates of the JC Hamiltonian 2 2 g c 2g c ω c = ω eg ω L ω L ;2> = g; n c =2> e; n c =1> ;2> = g; n c =2> + e; n c =1> ;1> = g; n c =1> e; n c =0> +;1> = g; n c =1> + e; n c =0> 0> = g; n c = 0> The eigenstates of the coupled system are entangled emittercavity states The spectrum is anharmonic: the nonlinearity of the two-level emitter ensures that the coupled system is also anharmonic. Ex: A laser (ω L ) that is resonant with the 0>- +;1> transition will be offresonant with all other transitions; the emitter-cavity molecule may act as a two-level system. Single photon blockade!

Dissipative cavity-emitter coupling: an open quantum system Single two-level (anharmonic) emitter coupled to a single cavity mode is described by the non-hermitian JC Hamiltonian + noise terms describing quantum jumps associated with spontaneous emission and cavity decay processes. description using a markovian master equation Purcell regime: g >> "! 2 c c sp e> g> Γ sp κ c Strong coupling: g c > Γ sp 4, κ c 4

Quantum dots as two-level emitters in cavity-qed Quantum dots do not have random thermal motion and they can easily be integrated in nano-scale cavities. Just like atoms, quantum dots have nearly spontaneous emission broadened emission lines. Exciton transition oscillator (dipole) strength ranges from 10 to 100 (depending on the QD type) due to a collective enhancement effect. QDs nucleate in random locations during molecular beam epitaxy growth; The size and hence the emission energy of each QD is different; the standard deviation in emission energy is ~20 mev. Obtaining spatial and spectral overlap between the QD and the cavity electric field is a major challenge.

Light confinement in photonic crystals A photonic crystal (PC) has a periodic modulation in dielectric constant and modifies the dispersion relation of electromagnetic fields to yield allowed energy bands for light propagation. For certain crystal structures there are band-gaps: light with frequency falling within these band-gaps cannot propagate in PCs. A defect that disrupts the periodicity in a PC confines light fields (at certain frequencies) around that defect, allowing for optical confinement on length scales ~ wavelength. A two dimensional (2D) PC membrane/slab defect can confine light in all three dimensions: the confinement in the third dimension is achieved thanks to total internal reflection between the high index membrane and the surrounding low-index (air) region. Note: Total internal reflection and mirrors based on periodic modulation of index-of-refraction are key ingredients in all solid-state nano-cavities.

Photonic crystal fabrication L3 defect cavity z 1 µm Q 2 10 4, V eff 0.7 (λ/n) 3 Best Q ~ Finesse value = 1,200,000 with V eff (λ/n) 3 (Noda)

A single quantum dot in an L3 cavity Strategy: first identify the QD location and emission energy; then fabricate the cavity A B A B QD PL before PC fabrication QD PL after PC fabrication We observe emission at cavity resonance (following QD excitation) even when the cavity is detuned by 4 nm (1.5 THz)! Control experiments with cavities containing no QDs (confirmed by AFM) or a QD positioned at the intra-cavity field-node yield no cavity emission.

Nonresonant coupling of the nano-cavity mode to the QD Off-resonance QD lifetime is prolonged by a factor of 10 due to photonic band-gap Cavity emission lifetime is determined by the pump duration (cavity lifetime ~10 ps) Strong photon antibunching in cross-correlation proves that the QD and cavity emission are quantum correlated.

Photon (auto)correlation of (detuned) cavity emission Poissonian photon statistics despite the fact that cross-correlation with the exciton emission shows strong photon antibunching!

Observation of deterministic strong-coupling Cavity tuning by waiting: the spectrum depicted above is aquired in ~30 hours, allowing us to carry out photon correlation and lifetime measurements at any given detuning. The center-peak on resonance is induced by cavity emission occuring when the QD is detuned due to defect-charging (and is emitting at the B-resonance).

Quantum correlations in the strong coupling regime Emission lifetime is reduced by a factor of 120 on resonance The combined PL from the 3 emission peaks on resonance exhibit photon antibunching.

Photon correlation from a single cavity-polariton peak (data from another device exhibiting 25 GHz vacuum Rabi coupling) Proof of the quantum nature of strongly coupled system First step towards photon blockade

Tuning the nano-cavity through the QD exciton resonance Dark exciton is visible when it is degenrate with the lower polariton Biexciton resonance exhibits anti-crossing simulatenously with the exciton

Comments/open questions It is possible to couple nano-cavities via waveguides defined as 1D defects in the PC (Vuckovic) High cavity collection efficiency to a high NA objective can be attained using proper cavity design or by using tapered-fiber coupling. It is possible to fine-tune cavity resonance using nano-technology (AFM oxidation, digital etching, N 2 or Xe condensation). How does the cavity mode, detuned from the QD resonances by as much as 15 nm, fluoresce upon QD excitation? Why does the detuned cavity emission have Poissonian statistics while exhibiting strong quantum correlations with the exciton emission?

QD spin physics Spin states of an excess conduction-band electron: A two-level system with long coherence times (> 1µs) + fast optical manipulation + novel physics Need to control the charging state of the QD

Controlled charging of a single QD: principle Quantum dot embedded between n-gaas and a top gate. Coulomb blockade ensures that electrons are injected into the QD one at a time V G Schottky Gate 88-nm i-gaas capping layer Single electron charging energy: e 2 /C = 20 mev (a) V = V 1 50-nm Al 0.4 Ga 0.6 As tunnel barrier - - - E Fermi 12-nm i-gaas 35-nm i-gaas tunnel barrier 40-nm n-gaas (Si ~10 18 ) (b) V = V 2 - - - n-gaas QD - E Fermi i-gaas substrate

Voltage-controlled Photoluminescence Absorption X 1- X 2-X3- X 0 - + - Vertical cut at a fixed gate voltage Gate Voltage (mv) Quantum dot emission energy depends on the charge state due to Coulomb effects. X 0 and X 1- lines shift with applied voltage due to DC-Stark effect.

Charged QD X 1- (trion) absorption/emission > Excitation Emission > - > - - > laser excitation - m z = -3/2> - m z = 3/2> - - - - m z = -1/2> m z = 1/2> - m z = -3/2> σ photon m z = 3/2> - - - - m z = -1/2> m z = 1/2> σ+ resonant absorption is Pauli-blocked The polarization of emitted photons is determined by the hole spin Polarized PL implies that the resident electron spin is polarized

Strong spin-polarization correlations Ω Γ: spontaneous emission rate Ω: laser coupling (Rabi) frequency γ: spin-flip spontaneous emission ξ: spin-flip due to spin-reservoir coupling QD with a spin-up (down) electron only absorbs and emits σ+ (σ-) photons. The likelihood of a spin-flip Raman scattering is ~0.1% for B~1 Tesla A strong detuned σ+ laser field generates an ac-stark field only for the spin-up state an effective magnetic field.

Trion transitions in a charged QD Ω 10 5 nuclear spins B0e -

Spin pumping in a single-electron charged QD with 99.8% efficiency 0 Tesla 0.2 Tesla For B > 15 mt, the applied resonant laser leads to very efficent spin pumping, which is evidenced by disappearing absorption

Spin pumping in a single-electron charged QD with 99.8% efficiency 0 Tesla 0.2 Tesla 0.2 Tesla M. Atature et al, Science (2006) Ω Ω For B > 15 mt, the applied resonant laser leads to very efficent spin pumping, which is evidenced by disappearing absorption Resonant absorption is fully recovered by applying a second laser (i.e. spin-state is randomized).

Trion transitions in the center of the absorption plateau: Hyperfine mixing of spin-states Ω Ω B = 0 T: electron spin-mixing by the random hyperfine field B0e - 2e - γ 1 1 µs, Γ 1 = 1 ns B = 0.2 T: slow spin-flips The electron is optically pumped into the > state for B > 0.1 T Spin-pumping only in the center of the absorption plateau?

Exchange interactions with the Fermi-sea induce spin-flip co-tunneling INITIAL STATE VIRTUAL STATE FINAL STATE ε F g E s E s + U Co-tunneling is enhanced at the edges of the absorption plateau where the virtaul state energy ~ initial/final state energy Co-tunneling rate changes by 5-orders-ofmagnitude from the plateau edge to the center

Spins as qubits vs. Kondo effect To ensure well isolated QD spins, we need to increase the length of the tunnel barrier and thereby suppress exchange interactions that cause co-tunneling. In the opposite limit of a small tunnel barrier, the QD spin forms a singlet with one of the Fermi sea electrons the Kondo effect.

Controlled coupling of spins Goal: prepare two singleelectron spins in two (initially) uncoupled QDs and use laser excitation to induce spin-state dependent coherent interaction between the two QDs. 50-nm Al 0.4 Ga 0.6 As tunnel barrier i-gaas 25-nm i-gaas tunnel barrier Red QD 40-nm n-gaas (Si ~10 18 )

Coupled quantum dots Two stacked InGaAs: the ground state is such that the blue QD has one excess electron and the red QD is neutral Red QD Blue QD 10nm X-STM by courtesy of P.M. Koenraad The optically excited (exciton) state of the red QD couples to the blue QD ground-state by resonant tunneling: both electron-hole exchange split states 3> and 2> couple to the indirect exciton state 4>.

Conditional resonant absorption of two coupled QDs Trion resonance energy of the blue QD exhibits a jump when the red QD is charged Blue QD charge sensing Resonant absorption of an x or y polarized laser by the red QD, when the blue QD is in the ground state x-pol x-pol y-pol y-pol

Conditional resonant absorption of two coupled QDs Trion resonance energy of the blue QD exhibits a jump when the red QD is charged Blue QD charge sensing: optical analog of a quantum point contact Resonant absorption of an x or y polarized laser by the red QD, when the blue QD is in the ground state x-pol x-pol Resonant absorption of an x or y polarized laser by the red QD, when the blue QD is also resonantly excited y-pol y-pol

Conditional dynamics of two QDs: dipole sensing or optical gating of tunnel coupling L. Robledo et al, Science (2006)

Future Directions Spin-state dependent conditional dynamics of coupled QDs, each with a single electron: optical selection rules ensure that upon excitation by righthand circularly polarized lasers, the spins will interact only if the red QD is in > and the blue-qd is in > state. Optical excitations out of singlet/triplet states protection against hyperfine decoherence + favourable optical selection rules

Hyperfine interactions in a single-electron charged QD Excitation I nuc (B nuc ) ESS NSS Detection S el (B el ) ESS: Electron Spin System NSS: Nuclear Spin System Effective magnetic fields: Overhauser field: B nuc ~<I z > Knight field: B el ~<σ z > Flip-flop terms: Transfer of spin polarization from the electron to the QD nuclear spin systems.

Signature of nuclear spin polarization High resolution spectroscopy on QD recombination line (X -1 ) at B = 0: circularly polarized excitation of p-shell resonances lead PL polarization exceeding 80% electron-spin polarization σ - exc. σ + exc. π exc. : Cross-polarized : Co-polarized : Lorentzian fit Circular excitation: Electron spin polarization + Hyperfine interaction Dynamical nuclear spin polarization (DNSP) Electron spin splits in B nuc Overhauser shift (OS) Linear excitation: No angular momentum transfer to nuclei, no DNSP. C. W. Lai et al., PRL 96, 167403 (2006)

The role of the Knight field In low magnetic fields, nuclear spin polarization is suppressed due to nuclear dipole-dipole interactions (characterized by Bloc): σ - excitation σ + excitation b * n : Maximal nuclear field B loc : Mean field of dipolar interactions ξ : Numerical factor ~1-3 S : Mean electron spin (here: S«½) Incomplete cancellation of B nuc : -Inhomogeneous coupling... -Quadrupolar interaction... Knight field measured by compensation with an external magnetic field!

B-field dependence of DNSP Magnetic field in the Faraday geometry: k,z Bext Sample σ + excitation σ + excitation: B nuc σ - excitation σ - excitation: B nuc P.M. et al., PRB 75, 035409 (2007)

Dynamics of QD nuclear spins Red: σ + exc. Black:σ - exc. Blue: Exp. fit Timescale: Milliseconds Pump probe technique probes dynamics of nuclear spin polarization. Timescales of buildup and decay: Milliseconds Relaxation mechanism? P.M. et al., PRL 99, 056804 (2007)

Nuclei depolarized by residual electron Red: σ + exc. Black:σ - exc. Blue: With el. B ext =0! Timescale: Seconds Long lifetime of DNSP at Bext=0: Stabilization of nuclear spins through Knight field or quadrupolar interaction! P.M. et al., PRL 99, 056804 (2007)

Electron-mediated nuclear spin decay Mechanisms: Co-tunelling to electron reservoir. Ω e : Electron Larmor frequency N : Number of QD nuclei τ el : Electron spin correlation time Electron mediated nuclear spin-spin interactions. valid for Ω e >>A. (D. Klauser et al., PRB, 73, 205302 (2006) ) Distinguish the two mechanisms by their different scaling with τ el and Ω el. P.M. et al., PRL 99, 056804 (2007)

DNSP decay outside the 1e - plateau B ext =1T, σ + exc. V g switching changes QD charging state: Outside 1e - plateau, τ decay >>1 min: QD nuclei very well isolated from bulk. No spin diffusion detectable

DNSP decay due to co-tunneling B ext =1T, σ + exc. τ e -1 Electron co-tunneling rate depends strongly on gate voltage! B ext +B nuc =0 Smith et al., PRL, 94, 197402 (2005)

QND measurement of a single QD spin The spin-state selective absorption: right (left) hand circularly polarized laser sees substantial absorption if the electron spin is in > ( >) and perfect transmission otherwise. Optical pumping of spin destroys the information about the initial spin before it can be measured. Faraday rotation of an off-resonant laser field (dispersive response) allows for shot-noise limited measurement, without inducing optical pumping. It is possible to obtain Faraday SNR>1 while keeping spin-flip Raman scattering events negligible (no need for a cavity): maximize σ abs /A laser

Nonlinear electron-nuclear spin coupling Observed behavior of DNSP well explained by a simple rate equation: Relaxation to thermal equilibrium with electron Nuclear spin decay (dipole-dipole interactions, quadrupolar int., ) Assumptions: -T d :constant -T 1e : Suppressed with increasing electron Zeeman-splitting g el :Electron g-factor µb:bohr magneton τ el :Spin correlation time B nuc ~<I z > Nonlinear behavior of DNSP M.I. D yakonov, JETP Lett. 16, 398 (1972) D. Gammon et al., PRL 86, 5176 (2001) P.-F. Braun et al., PRB 74, 245306 (2006) P.M. et al., PRB 75, 035409 (2007)

Hysteresis in DNSP Observed behavior of DNSP well explained by a simple rate equation: Relaxation to thermal equilibrium with electron Nuclear spin decay (dipole-dipole interactions, quadrupolar int., ) Cooling Heating

Initial electron spin-state determines whether the polarization rotation is +θ or θ; this rotation is measured by the difference signal Absorptive vs. Dispersive response of a QD

Measurement of an optically prepared single spin-state using Faraday rotation of a far detuned (~50 GHz) linearly-polarized laser electron prepared in spin-up state using a resonant laser electron prepared in spin-down state using a resonant laser