QuantumOpticswithMesoscopicSystemsII

Outline

1) Cavity-QED with a single quantum dot2) Optical pumping of quantum dot spins3) Conditional dynamics of two quantum dots

A. ImamogluQuantum Photonics Group, Department of Physics

ETH-Zürich

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 isdescribed by the Jaynes-Cummings (JC) Hamiltonian

|e>

|g>ωeg

An exactlysolvable model

• ωeg : emitter frequency

• ωc : cavity frequency

• gc : cavity-emitter coupling strength

In all optical realizations:

gc << ωc ≈ ωeg

1010 1015 1015

ħgc = e <e| ε.r |g>ω

For electric-dipole coupling:

Eigenstates of the JC Hamiltonian

|0> = |g; nc = 0>

|+;1> = |g; nc=1> + |e; nc=0>

|−;1> = |g; nc=1> − |e; nc=0>

|−;2> = |g; nc=2> + |e; nc=1>

|−;2> = |g; nc=2> − |e; nc=1>

2gc

2√2 gc

ωc = ωeg

• The eigenstates of the coupledsystem are entangled emitter-cavity states

• The spectrum is anharmonic: thenonlinearity of the two-levelemitter ensures that the coupledsystem is also anharmonic.

• The eigenstates of the coupledsystem are entangled emitter-cavity states

• The spectrum is anharmonic: thenonlinearity of the two-levelemitter ensures that the coupledsystem is also anharmonic.

|0> = |g; nc = 0>

|+;1> = |g; nc=1> + |e; nc=0>

|−;1> = |g; nc=1> − |e; nc=0>

|−;2> = |g; nc=2> + |e; nc=1>

|−;2> = |g; nc=2> − |e; nc=1>

2gc

2√2 gc

ωc = ωeg

Eigenstates of the JC Hamiltonian

Ex: A laser (ωL) that is resonant withthe |0>-|+;1> transition will be off-resonant with all other transitions;the emitter-cavity molecule mayact as a two-level system.

⇒ Single photon blockade!

ωL

ωL

• Single two-level (anharmonic) emitter coupled to a single cavity mode isdescribed by the non-Hermitian JC Hamiltonian

+ noise terms describing quantum jumps associated with spontaneous emissionand cavity decay processes.

≡ description using a markovian master equation

Dissipative cavity-emitter coupling: an open quantum system

|e>

|g>Γsp

κc

Purcell regime:

Strong coupling:

spccg !>> "2

4,

4csp

cgκΓ

>

• Quantum dots do not have random thermal motion and they can easily beintegrated in nano-scale cavities.

• Just like atoms, quantum dots have nearly spontaneous emissionbroadened emission lines.

• Exciton transition oscillator (dipole) strength ranges from 10 to 100(depending on the QD type) due to a collective enhancement effect.

Quantum dots as two-level emitters in cavity-QED

• QDs nucleate in random locations during molecular beam epitaxy growth;

• The size and hence the emission energy of each QD is different; thestandard deviation in emission energy is ~20 meV.

⇒ Obtaining spatial and spectral overlap between the QD and the cavityelectric field is a major challenge.

Light confinement in photonic crystals

• A photonic crystal (PC) has a periodic modulation in dielectric constantand modifies the dispersion relation of electromagnetic fields to yieldallowed energy bands for light propagation. For certain crystal structuresthere are band-gaps: light with frequency falling within these band-gapscannot propagate in PCs.

Light confinement in photonic crystals

• A photonic crystal (PC) has a periodic modulation in dielectric constantand modifies the dispersion relation of electromagnetic fields to yieldallowed energy bands for light propagation. For certain crystal structuresthere are band-gaps: light with frequency falling within these band-gapscannot propagate in PCs.

• A defect that disrupts the periodicity in a PC confines light fields (at certainfrequencies) around that defect, allowing for optical confinement on lengthscales ~ wavelength.

Light confinement in photonic crystals

• A photonic crystal (PC) has a periodic modulation in dielectric constantand modifies the dispersion relation of electromagnetic fields to yieldallowed energy bands for light propagation. For certain crystal structuresthere are band-gaps: light with frequency falling within these band-gapscannot propagate in PCs.

• A defect that disrupts the periodicity in a PC confines light fields (at certainfrequencies) around that defect, allowing for optical confinement on lengthscales ~ wavelength.

• A two dimensional (2D) PC membrane/slab defect can confine light in allthree dimensions: the confinement in the third dimension is achievedthanks to total internal reflection between the high index membrane and thesurrounding low-index (air) region.

Note: Total internal reflection and mirrors based on periodic modulation ofindex-of-refraction are key ingredients in all solid-state nano-cavities.

1 µm

Q ≈ 2×104, Veff ≈ 0.7 (λ/n)3

z

Photonic crystal fabrication

L3 defect cavity

Best Q ~ Finesse value = 1,200,000 with Veff ≈ (λ/n)3 (Noda)

A single quantum dot in an L3 cavity

QD PL before PC fabrication QD PL after PC fabrication

• We observe emission at cavity resonance (following QD excitation) even when thecavity is detuned by 4 nm (1.5 THz) !

• Control experiments with cavities containing no QDs (confirmed by AFM) or a QDpositioned at the intra-cavity field-node yield no cavity emission.

Strategy: first identify the QD location and emission energy; then fabricate the cavity

A B A B

Nonresonant coupling of the nano-cavity mode to the QD

Strong photon antibunching in cross-correlation proves that the QD andcavity emission are quantum correlated.

Off-resonance QDlifetime is prolonged bya factor of 10 due tophotonic band-gap

Cavity emissionlifetime is determinedby the pump duration(cavity lifetime ~10 ps)

Photon (auto)correlation of (detuned) cavity emission

⇒ Poissonian photon statistics despite the fact that cross-correlation withthe 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 anygiven detuning.

• The center-peak on resonance is induced by cavity emission occuring when theQD 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 afactor of 120 on resonance

The combined PL from the 3emission peaks on resonanceexhibit 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 defectsin the PC (Vuckovic)

• High cavity collection efficiency to a high NA objective can be attainedusing proper cavity design or by using tapered-fiber coupling.

• It is possible to fine-tune cavity resonance using nano-technology (AFMoxidation, digital etching, N2 or Xe condensation).

• How does the cavity mode, detuned from the QD resonances by as much as15 nm, fluoresce upon QD excitation?

• Why does the detuned cavity emission have Poissonian statistics whileexhibiting 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: principleCoulomb blockade ensures that electrons

are injected into the QD one at a time

(a) V = V1

EFermi---

QD

- EFermi

(b) V = V2

n-GaAs

- - -

Quantum dot embeddedbetween n-GaAs and a top gate.

i-GaAssubstrate

35-nm i-GaAstunnel barrier

40-nm n-GaAs (Si ~1018)

12-nm i-GaAs

50-nm Al0.4Ga0.6As

tunnel barrier

88-nm i-GaAs

capping layer

V GSchottky Gate Single electron charging energy:

e2/C = 20 meV

Voltage-controlled Photoluminescence Absorption

Quantum dot emission energy depends on the charge state due toCoulomb effects.X0 and X1- lines shift with applied voltage due to DC-Stark effect.

X0

X1- X2-X3-

-

+

-

Gate Voltage (mV)

Vertical cut at afixed gate voltage

Charged QD X1- (trion) absorption/emission

⇒ 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

|↑>|↓>

Excitation

-|mz = -3/2> -

|mz = 3/2>

- -- -

|mz = -1/2> |mz = 1/2>

|↑>|↓>

Emission

|mz = -3/2>-

|mz = 3/2>

- -- -

|mz = -1/2> |mz = 1/2>

- - -

laser excitationσ− photon

Strong spin-polarization correlations

• 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~1Tesla

• A strong detuned σ+ laser field generates an ac-Stark field onlyfor the spin-up state – an effective magnetic field.

Ω

Γ: spontaneous emission rate

Ω: laser coupling (Rabi) frequency

γ: spin-flip spontaneous emission

ξ: spin-flip due to spin-reservoir coupling

B0e-

Trion transitions in a charged QD

Ω

105 nuclear spins

Spin pumping in a single-electron charged QD with99.8% efficiency

0.2 Tesla0 Tesla

⇒ For B > 15 mT, the applied resonant laser leads to very efficent spin pumping,which is evidenced by disappearing absorption

0.2 Tesla0 Tesla

0 Tesla 0.2 Tesla 0.2 Tesla

Spin pumping in a single-electron charged QD with99.8% efficiency

⇒ 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).

Ω Ω

M. Atature et al,Science (2006)

B0e- 2e-

Trion transitions in the center of the absorption plateau:Hyperfine mixing of spin-states

B = 0 T: electron spin-mixing by B = 0.2 T: slow spin-flips the random hyperfine field

γ−1 ∼ 1 µs, Γ−1 = 1 ns

⇒ The electron is optically pumped into the |↑> state for B > 0.1 T

⇒ Spin-pumping only in the center of the absorption plateau?

Ω Ω

Exchange interactions with the Fermi-sea induce spin-flipco-tunneling

• Co-tunneling is enhanced at the edges ofthe absorption plateau where the virtaulstate energy ~ initial/final state energy

• Co-tunneling rate changes by 5-orders-of-magnitude from the plateau edge to thecenter

FINAL STATEVIRTUAL STATE

gEs + U

INITIAL STATE

εF

Es

Spins as qubits vs. Kondo effect

• To ensure well isolated QD spins, we need toincrease the length of the tunnel barrier andthereby suppress exchange interactions thatcause co-tunneling.

• In the opposite limit of a small tunnel barrier,the QD spin forms a singlet with one of theFermi sea electrons – the Kondo effect.

Controlled coupling of spins

• Goal: prepare two single-electron spins in two(initially) uncoupled QDsand use laser excitation toinduce spin-state dependentcoherent interaction betweenthe two QDs.

Red QD

25-nm i-GaAstunnel barrier

40-nm n-GaAs (Si ~1018)

i-GaAs

50-nm Al0.4Ga0.6As

tunnel barrier

X-STM by courtesy of P.M. Koenraad

10nm

Coupled quantum dots

Two stacked InGaAs: the ground state is such that the blue QD hasone excess electron and the red QD is neutral

The optically excited (exciton) state of the red QD couples to the blue QDground-state by resonant tunneling: both electron-hole exchange split states |3>and |2> couple to the indirect exciton state |4>.

Blue QDRed QD

Conditional resonant absorption of two coupled QDs

Resonantabsorption of an xor y polarizedlaser by the redQD, when theblue QD is in theground state

x-pol

y-poly-pol

x-pol

Trion resonance energyof the blue QD exhibitsa jump when the redQD is charged

⇒ charge sensingBlue QD

Conditional resonant absorption of two coupled QDs

Resonantabsorption of an xor y polarizedlaser by the redQD, when theblue QD is in theground state

x-pol

Resonantabsorption of an xor y polarizedlaser by the redQD, when theblue QD is alsoresonantly excited

y-poly-pol

x-pol

Trion resonance energyof the blue QD exhibitsa jump when the redQD is charged

⇒ charge sensing:optical analog of aquantum point contact

Blue QD

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 ofcoupled QDs, each with a single electron:optical selection rules ensure that upon excitation by right-hand circularly polarized lasers, the spins will interact only ifthe red QD is in │↑> and the blue-QD is in │↓> state.

• Optical excitations out of singlet/triplet statesprotection against hyperfine decoherence + favourable opticalselection rules

Effective magnetic fields:Overhauser field: Bnuc ~<Iz>Knight field: Bel ~<σz> Flip-flop terms:Transfer of spin polarization from theelectron to the QD nuclear spin systems.

ESS: Electron Spin SystemNSS: Nuclear Spin System

ESS

NSSSel (Bel)Inuc (Bnuc)

Excitation Detection

Hyperfine interactions in a single-electron charged QD

Signature of nuclear spin polarization

C. W. Lai et al., PRL 96, 167403 (2006)

High resolution spectroscopy on QD recombination line (X-1) at B = 0:circularly polarized excitation of p-shell resonances lead PLpolarization exceeding 80% → electron-spin polarization

■ : Cross-polarized▲ : Co-polarized

: Lorentzian fit

Circular excitation: Electron spin polarization + Hyperfine interaction

Dynamical nuclear spin polarization (DNSP)

Electron spin splits in Bnuc Overhauser shift (OS)

Linear excitation: No angular momentum transfer to nuclei, no DNSP.

σ- exc. σ+ exc. π exc.

The role of the Knight field

In low magnetic fields, nuclear spin polarization is suppressed due to nucleardipole-dipole interactions (characterized by Bloc):

bn* : “Maximal” nuclear field

Bloc : Mean field of dipolar interactionsξ : Numerical factor ~1-3S : Mean electron spin (here: S«½)

Incomplete cancellation of Bnuc: -Inhomogeneous coupling...

-Quadrupolar interaction......

Knight field measured by compensation with an external magnetic field!

σ+ excitation

σ- excitation

B-field dependence of DNSP

σ+ excitation: Bnuc

Bnuc

P.M. et al., PRB 75, 035409 (2007)

σ+ excitation

σ- excitation σ- excitation:

k,z

Bext

Sam

ple

Magnetic field in the“Faraday” geometry:

Dynamics of QD nuclear spins

P.M. et al., PRL 99, 056804 (2007)

•“Pump – probe” technique probes dynamics of nuclear spin polarization.

Timescale:Milliseconds

•“Pump – probe” technique probes dynamics of nuclear spin polarization.

•Timescales of buildup and decay: Milliseconds Relaxation mechanism?

Red: σ+ exc.Black:σ- exc.Blue: Exp. fit

Timescale:Seconds

Red: σ+ exc.Black:σ- exc.Blue: With el.

Nuclei depolarized by residual electron

Bext=0 !

P.M. et al., PRL 99, 056804 (2007)

Long lifetime of DNSP at Bext=0:Stabilization of nuclear spins through Knight field or quadrupolarinteraction!

Mechanisms:

•Co-tunelling to electron reservoir.

Ωe: Electron Larmor frequencyN : Number of QD nucleiτel : Electron spin correlation time

•Electron mediated nuclear spin-spin interactions.

valid for Ωe>>A. (D. Klauser et al., PRB, 73, 205302 (2006) )

Electron-mediated nuclear spin decay

Distinguish the two mechanisms by their different scaling with τel and Ωel.P.M. et al., PRL 99, 056804 (2007)

DNSP decay outside the 1e- plateau

Outside 1e- plateau, τdecay>>1 min:QD nuclei very well isolated frombulk. No spin diffusion detectable

Bext=1T, σ+ exc. Vg switching changesQD charging state:

DNSP decay due to co-tunneling

τe-1

Smith et al., PRL, 94, 197402 (2005)

Electron co-tunnelingrate depends stronglyon gate voltage!

Bext=1T, σ+ exc.

Bext+Bnuc=0

QND measurement of a single QD spin

• The spin-state selective absorption: right (left) hand circularlypolarized laser sees substantial absorption if the electron spinis 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 (dispersiveresponse) allows for shot-noise limited measurement, withoutinducing optical pumping.

• It is possible to obtain Faraday SNR>1 while keeping spin-flipRaman scattering events negligible (no need for a cavity):⇒ maximize σabs/Alaser

Assumptions: -Td:constant-T1e: Suppressed with increasing electron Zeeman-splitting

Observed behavior of DNSP well explained by a simple rate equation:

Nonlinear electron-nuclear spin coupling

Bnuc~<Iz>Nonlinearbehavior of DNSP

gel:Electron g-factorµB:Bohr magnetonτel:Spin correlation time

Relaxation to ‘thermalequilibrium’ with electron

Nuclear spin decay (dipole-dipoleinteractions, quadrupolar int.,…)

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

Cooling Heating

Relaxation to ‘thermalequilibrium’ with electron

Nuclear spin decay (dipole-dipoleinteractions, quadrupolar int.,…)

Observed behavior of DNSP well explained by a simple rate equation:

Absorptive vs. Dispersive response of a QD

Initialelectron

spin-statedetermineswhether thepolarizationrotation is+θ or –θ;

this rotationis measured

by thedifference

signal

Measurement of an optically prepared single spin-state using Faradayrotation of a far detuned (~50 GHz) linearly-polarized laser

electronprepared in

spin-upstate usinga resonant

laser

electronprepared inspin-downstate usinga resonant

laser