A) - correlations in solid-state systems

A. Imamoglu Quantum Photonics Group, Department of Physics ETH-Zürich Solid-state spins & emitters

• Solid-state emitters (artificial ) can be used to realize high brightness long-lived single-photon sources: - no need for trapping - easy integration into a directional (fiber-coupled) cavity - up to 109 /sec with >70% efficiency

• Three different classes of emitters: - rare-earth atoms embedded in a solid matrix (Er in glass) - Deep defects in insulators (NV centers in diamond) - Shallow defects in semiconductors (quantum dots)

Note: While the concepts & techniques apply to a wide range of solid-state emitters, we focus on quantum dots Quantum dots & single photons

A quantum dot (QD), is a mesoscopic semiconductor structure (~10nm confinement length-scale) consisting of 10,000 atoms and still having a discrete (anharmonic) optical excitation spectrum.

-MBE grown InGaAs quantum dot

-QDs in monolayer WSe2 Light generation by a quantum dot

Resonant laser excitation

-

Laser excitation

- - - - - Light generation by a quantum dot

Resonant laser excitation

- -

Laser excitation  - + - - - -

-hole pair = Light generation by a quantum dot

Resonant laser excitation Resonance fluorescence

12000 9000 photon 6000 1X

emission 3000 RF (counts/s) 0 - - -20 -10 0 10 20 - - Energy detuning (µeV) - - Quantum dot Spectroscopy

From laser Polarization filter

To detector

Polarization optics

NA=0.65 Spot size ≈1µm

Magnet

Piezo positioner GaAs InGaAs Liquid He GaAs Photon correlations from a single QD

(2) : I(t)I(t  ) : • Intensity (photon) correlation function: g ( )  2 I(t)

stop pulse • To measure g(2)(), photons from a quantum emitter are single photon detectors sent to a Hanbury-Brown Time-to- Twiss setup amplitude start (voltage) pulse converter • Single quantum emitter driven by a pulsed laser: absence of a center peak indicates that none of the pulses have > 1 photon (Robert, LPN).  Signature of a single-photon source Photon correlations from a single QD

(2) : I(t)I(t  ) : • Intensity (photon) correlation function: g ( )  2 I(t)

stop pulse • To measure g(2)(), photons from a quantum emitter are single photon detectors sent to a Hanbury-Brown Time-to- Twiss setup amplitude start (voltage) pulse converter • Photon correlations from a weak pulsed laser ( ~ 1); detection of a photon does not change the likelihood of detecting a second.  = 0 Photon correlations from a single QD

(2) : I(t)I(t  ) : • Intensity (photon) correlation function: g ( )  2 I(t)

stop pulse • To measure g(2)(), photons from a quantum emitter are single photon detectors sent to a Hanbury-Brown Time-to- Twiss setup amplitude start (voltage) pulse converter • Single quantum emitter driven by a pulsed laser: absence of a center peak indicates that none of the pulses have > 1 photon.  Signature of a single-photon source  = 0 How do we make sure that single-photons are not quantum correlated with any other system: Two-photon (HOM) interference • Two completely indistinguishable single-photon pulses incident on a beam-splitter never lead to coincidences at the output due to a quantum interference effect.

• The single photon pulses have to have the same spatio-temporal profile, center frequecy, polarization. • Indistinguishability ensures the absence of entanglement of single photons with uncontrolled degrees of freedom. How do we make sure that single-photons are not quantum correlated with any other system: Two-photon (HOM) interference • Two completely indistinguishable single-photon pulses incident on a beam-splitter never lead to coincidences at the output due to a quantum interference effect.

• The single photon pulses have to have the same spatio-temporal profile, center frequecy, polarization. • Indistinguishability ensures the absence of entanglement of single photons with uncontrolled degrees of freedom. A single-photon frequency-qubit from a QD: |> = α|blue> + β|red>

In a neutral QD, the elementary optical excitations are (X0); the two linearly polarized exciton X0 lines are split due to electron-hole exchange by ~ 5 GHz

Contrast limited by detector jitter

Single-photon wave-packet

By controlling the pulse-shape, detuning and polarization of the resonant laser, we could generate a single-color photon or a two-color photonic qubit Interference of photonic qubits (superposition of blue and red photons) coming from two quantum dots

• Two distant QDs rendered «identical» using local electric and magnetic fields. • 80% visibility in intereference of two photonic (color) qubits Quantum dot charge control

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(a) Vg = V1 Exciton emission - - - EFermi - - - - - Coulomb blockade ensures charge control at a single electron level - - (b) Vg = V2 - - - E Trion - Fermi emission n-GaAs - - - - - Optical transition from a quantum dot spin qubit in Voigt geometry

( Bext = Bx) Excitation of a trion state results in either emission of a H polarized red photon to |↓> state or a V polarized blue photon to |↑> state, with equal probability. Optical transition from a quantum dot spin qubit in Voigt geometry

( Bext = Bx) Excitation of a trion state results in either emission of a H polarized red photon to |↓> state or a V polarized blue photon to |↑> state, with equal probability.

 Spin-photon entanglement: potentially near-deterministic entanglement generation at ~1 GHz rate

Similar results by Yamamoto, Steel groups; earlier work by Monroe,Lukin Measurement of classical correlations

An additional π-pulse (dashed curve) is applied to realize a heralded measurement in the spin-up state.

Identical (unconditional) counts for red and blue photons confirm the selection rules.

The g(2) measurement shows that for the [1.2ns, 1.64ns] time range, probability of two-photon emission is negligible.

A spin down (up) measurement event ensures that the detected photon is red (blue).

F1=0.87+/-0.09 in the computational basis measure.ment Measurement of classical correlations

An additional π-pulse (dashed curve) is applied to realize a heralded measurement in the spin-up state.

Identical (unconditional) counts for red and blue photons confirm the selection rules.

The g(2) measurement shows that for the [1.2ns, 1.64ns] time range, probability of two-photon emission is negligible.

A spin down (up) measurement event ensures that the detected photon is red (blue).

F1=0.87+/-0.09 in the computational basis measure.ment Measurement of classical correlations

An additional π-pulse (dashed curve) is applied to realize a heralded measurement in the spin-up state.

Identical (unconditional) counts for red and blue photons confirm the selection rules.

The g(2) measurement shows that for the [1.2ns, 1.64ns] time range, probability of two-photon emission is negligible.

A spin down (up) measurement event ensures that the detected photon is red (blue).

F1=0.87+/-0.09 in the computational basis measure.ment Measurement of classical correlations

An additional π-pulse (dashed curve) is applied to realize a heralded measurement in the spin-up state.

Identical (unconditional) counts for red and blue photons confirm the selection rules.

The g(2) measurement shows that for the [1.2ns, 1.64ns] time range, probability of two-photon emission is negligible.

A spin down (up) measurement event ensures that the detected photon is red (blue).

F1=0.87 ± 0.05 in the computational basis measure.ment Measurement of quantum correlations

- An additional π/2 or 3π/2- pulse (dashed curve) is applied to measure the spin in |↑> ± |↓>.

- The data in b & c shows the coincidence measurement when π/2-pulse is applied.

- The data in d & e shows the coincidence measurement when 3 π/2-pulse is applied.

- F2=0.46+/-0.04 in the rotated basis measurement Measurement of quantum correlations

- An additional π/2 or 3π/2- pulse (dashed curve) is applied to measure the spin in |↑> ± |↓>.

- The data in b & c shows the coincidence measurement when π/2-pulse is applied.

- The data in d & e shows the coincidence measurement when 3 π/2-pulse is applied.  Coherent oscillations in conditional detection- demonstrateF2=0.46+/-quantum0.04 in the correlations rotated between spinbasis and measurementphoton Measurement of quantum correlations

- An additional π/2 or 3π/2- pulse (dashed curve) is applied to measure the spin in |↑> ± |↓>.

- The data in b & c shows the coincidence measurement when π/2-pulse is applied.

- The data in d & e shows the coincidence measurement when 3 π/2-pulse is applied.

- F2=0.46 ± 0.04 in the rotated basis measurement; overall fidelity F = 0.67 ± 0.05

W. Gao et al. Nature (2012) B) Heralded generation of distant spin entanglement Spin-photon quantum interface

• Goal: Creation of quantum entanglement between a single photon and a condensed quantum system (qubit) and to use the ensuing nonlocal correlations to entangle distant spins/qubits. • Application: for (unconditionally secure) quantum communication, entangled qubits separated by > 1000 km is required. Using controlled interactions to generate entanglement at such large distances is impossible. • Method: heralded probabilistic generation of entanglement without direct interaction. Required resource: single-photons quantum correlated with spins stationary qubits  single spin |> = α|↑> + β|↓> propagating qubits  single photon |> = α|blue> + β|red> Spin-photon quantum interface

• Goal: Creation of quantum entanglement between a single photon and a condensed matter quantum system (qubit) and to use the ensuing nonlocal correlations to entangle distant spins/qubits. • Application: for (unconditionally secure) quantum communication, entangled qubits separated by > 1000 km is required. Using controlled interactions to generate entanglement at such large distances is impossible. • Method: heralded probabilistic generation of entanglement without direct interaction. Required resource: single-photons quantum correlated with spins stationary qubits  single spin |> = α|↑> + β|↓> propagating qubits  single photon |> = α|blue> + β|red> Spin-photon quantum interface

• Goal: Creation of quantum entanglement between a single photon and a condensed matter quantum system (qubit) and to use the ensuing nonlocal correlations to entangle distant spins/qubits. • Application: for (unconditionally secure) quantum communication, entangled qubits separated by > 1000 km is required. Using controlled interactions to generate entanglement at such large distances is impossible. • Method: heralded probabilistic generation of entanglement without direct interaction. Required resource: single-photons quantum correlated with confined spins  ↑,blue ↓,red  Maximally entangled | > = (| > + | >)/ 2 spin-photon pair Spin-photon quantum interface

• Goal: Creation of quantum entanglement between a single photon and a condensed matter quantum system (qubit) and to use the ensuing nonlocal correlations to entangle distant spins/qubits. • Application: for (unconditionally secure) quantum communication, entangled qubits separated by > 1000 km is required. Using controlled interactions to generate entanglement at such large distances is impossible. • Method: heralded probabilistic generation of entanglement without direct interaction. Required resource: single-photons quantum correlated with confined spins  ↑,0 ↓,1 Vanishingly small or | > = | > + ε| > , ε<<1 entanglement! Key requirements for heralded generation of entanglement

• Single photon pulses: if spin is correlated with 2 or more photons, quantum information is lost • Indistinguishable photons: there should be no way to tell which spin has generated a given photon when which-path information is erased • Coherent spins: qubit coherence should last at least until entanglement is heralded • Efficient spin-photon interface: fast emission + high photon extraction leads to a high rate of spin entanglement → photonic structures. Heralded entanglement of two ions (Monroe group 2007)

| 1,-1 | 1,0 | 1,1 2P 2,1 GHz 1/2 | 0,0

λ = 369.5 nm 500 ns long laser pulse (red) initializes the ion in state |0,0>

2 S1/2 | 1,-1 | 1,0 | 1,1 12.6 GHz

| 0,0 Heralded entanglement of two ions (Monroe group 2007)

| 1,-1 | 1,0 | 1,1 2P 2,1 GHz 1/2 | 0,0 2 ps long laser pulse (red) excites the ion s- to state |1,-1>; p |w spontaneous s- emission then

|w populates the 3 of the 4 ground states

2 S1/2 | 1,-1 | 1,0 | 1,1 12.6 GHz Heralded entanglement of two ions (Monroe group 2007)

| 1,-1 | 1,0 | 1,1 2P 2,1 GHz 1/2 | 0,0 After polarization s- filtering, one obtains |w   | = ||w + ||w s-

|w

2 S1/2 | 1,-1 | 1,0 | 1,1 12.6 GHz

| 0,0 Heralded entanglement of two ions HOM interference

|in  (|1| w1  |1| w1)(|2|w2  |2 |w2)

upon detection of a coincidence

|out  |1|2 - |2|2 Verification of entanglement

Classical correlations p + p = 0.78 ± 0.02

Quantum V = 0.47 ± 0.05 correlations  F = 0.63 ± 0.03

coincidences as a function of the delay-dependent phase difference φ = φb - φa Entanglement of distant spins • Erasing which-path information in single-photon scattering from distant spins, leads to entanglement upon detection [Cabrillo et al. Phys. Rev. A 59, 1025 (1999)]

• Quantum interference erases which-path information, provided that photonic pulses are indistinguishable • Spin coherence should survive for 30 ns for 10 m separation. Outline of the experiment

• We need a solid-state spin with a long coherence time: heavy-hole spins. We find two QDs & prepare two hole spins in ⇓ state ⇓, 0 퐿 ⇓, 0 푅 Optically generated hole Outline of the experiment

• We need a solid-state spin with a long coherence time: heavy-hole spins. We find two QDs & prepare two hole spins in ⇓ state ⇓, 0 퐿 ⇓, 0 푅 Optically generated hole

Tb T • Weak laser excitation of the trion r leads to a small Raman photon Ω Raman photon generation amplitude ( ⇓, 0 + 휀 ⇑, 1 )( ⇓, 0 + 휀 ⇑, 1 ) ⇑ ⇓ Outline of the experiment

• We need a solid-state spin with a long coherence time: heavy-hole spins. We find two QDs & prepare two hole spins in ⇓ state ⇓, 0 퐿 ⇓, 0 푅 Optically generated hole

Tb T • Weak laser excitation of the trion r leads to a small Raman photon Ω Raman photon generation amplitude ( ⇓, 0 + 휀 ⇑, 1 )( ⇓, 0 + 휀 ⇑, 1 ) ⇑ ⇓

• A single Raman photon click after a beam splitter that erases which- path information and heralds successful spin entanglement: ⇑, ⇓ + ⇓, ⇑ Coherent heavy-hole spins

• We need spins with long coherence time: hole spins have 100 times weaker hyperfine interaction

Tb

Tr Ramsey measurements

res X 0

Bx   Optically generated hole spin

0 Spin coherence measured via Raman scattering

Detected Raman and Rayleigh scattering as a function of path length difference for an average delay time of 23 ns: • Rayleigh: coherence determined by laser • Raman: coherence determined by spin Indistinguishable Raman photons from two distant quantum dots

T b Tr

Ω

⇑ ⇓

100 900 orthogonal polarization orthogonal polarization parallel polarization 800 parallel polarization 80 700

600 60 500

400 40 300 Coincidence /h

Coincidence /h /bin 20 200 100

0 0 -1 0 1 -1 0 1 -1 0 1 -2 -1 0 1 2  - T0 (ns)  (ns)  + T0 (ns) Period Measurement of classical correlations between the two QD spins

Pulse 1: prepare spins in ⇓, 0 ⇓, 0 QD1 1 2 3 state 퐿 푅 Pulse 2: Generates a Raman photon, heralding QD2 1 2 4 spin entanglement: ⇑, ⇓ + ⇓, ⇑ Pulse 3&4: measures the

spin state in each QD 112 0.6 F = 80.6 ± 6.6 % coincidences Z ideal case 93 0.5

F = 100 % limit 3-fold correlation: 74 0.4 obtained by

56 0.3 averaging over 37 0.2 100 hours

coincidences / 106.5 h 19 0.1

0 0.0

measurement Quantum correlations

Pulse 1: prepare spins in state 1 2 3 4 ⇓, 0 퐿 ⇓, 0 푅 QD1 Pulse 2: Generates a Raman photon, heralding spin entanglement: ⇑, ⇓ + ⇓, ⇑ QD2 1 2 4 Pulse 3: rotates QD1 spin Pulse 4: generates a second Raman photon.

0.8 180 min • Coherent oscillations of two- 0.7 0.13 /s photon coincidence rate 0.6 (single photons from pulses 2 0.5 & 4) proves the existence of 0.4

ratio quantum correlations of spins. 0.3

0.2 data • Entanglement generation at a 0.1 F = 29.8 ± 2.6 % sine fit x rate of 2300 ebits/second – 3 0.0 -1 0 1 2 3 4 5 6 7 8 orders of magnitude faster pulse duration (arb. u.) than the previous results Future: Integrated spin photonics Outlook

• Spin-photon quantum interface with decoherence- free spin qubits (singlet-triplet states in QDs)

• Combine fast entanglement generation in QDs with long coherence time of qubits in other systems: interfacing QD spins with superconducting qubits, trapped ions or NV centers • Controlled introduction of QDs in monolayer TMDs: high photon extraction, long spin-coherence due to spin-valley locking, strong cavity coupling