Unconventional electron quantum optics in condensed matter systems

Dario Ferraro Centre de Physique Théorique, Marseille

In collaboration with: J. Rech, T. Jonckheere, T. Martin

nanoQT-2016, Kyiv, October 10, 2016

Outline

•  The rise of electron quantum optics

•  Hanbury-Brown-Twiss and Hong-Ou-Mandel

interferometry with individual electrons

•  Two-dimensional topological insulators: when Pauli

meets Topology

•  SC/Hall hybrid systems: creation and collision of

individual Bogoliubov excitations

Electron quantum optics Revising the tools of quantum optics with photons to describe individual

electronic wave-packets in mesoscopic systems E. Bocquillon et al., Ann. Phys. (Berlin) 526, 1 (2014)

One-dimensional electron channels as chiral wave-guides (several micrometers of elastic mean free path)

Quantum point contacts as beam splitters

Electron injection on-demand

Theory: Moskalets et al., Phys. Rev. Lett. 100, 086601 (2008); Experiments: G. Fève et al., Science 316, 1169 (2007)

Periodic injection of one electron and one hole with exponential wave-packets in time

�(t) / e��2 te�i!0t⇥(t)

Alternative approach based on Lorentzian voltage pulses in time Theory: L. S. Levitov et al., J. Math. Phys. 37, 4845 (1996), J. Keeling, et al., Phys. Rev. Lett. 97,116403 (2006);

Experiments: J. Dubois et al., Nature 502, 659 (2013), T. Jullien et al., Nature 514, 603 (2014)

Single electron sources based on driven mesoscopic capacitors

Individual electrons interferometry Hanbury-Brown-Twiss (HBT) and Hong-Ou-Mandel (HOM) ✬ ✩

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E. Bocquillon et al., Science 339, 1054 (2013)

Perfect Pauli dip only in the free fermion case, suppression of the contrast due to interaction

C. Wahl et al., Phys. Rev. Lett. 112, 046802 (2014); D. F. et al. Phys. Rev. Lett. 113, 166403 (2014).

100 50 0 50 1000.0

0.2

0.4

0.6

0.8

1.0

0 T00

1

SHOM t

2 SHBTt

�q̄

T. Jonckheere et al., Phys. Rev. B 86, 125425 (2012)

Quantum spin Hall effect (QSHE) CdTe/HgTe quantum wells

Trivial insulator

Topological insulator: counter-propagating edge channels carrying opposite spin at zero magnetic field, robust against backscattering

Theory: A. B. Bernevig et al., Science 314, 1757 (2006); Experiments: M. König et al., Science 318, 766 (2007)

Observed also in InAs/GaSb quantum wells: C. Lui et al., Phys. Rev. Lett. 100, 236601 (2008); I. Knez et al., Phys. Rev. Lett. 107, 136603 (2011)

Pair electron source Driven mesoscopic capacitor coupled to helical edge states

A. Inhofer and D. Bercioux, Phys. Rev. B 88, 235412 (2013); P. P. Hofer and M. Buttiker, Phys. Rev. B 88, 241308(R) (2013)

Spin-preserving and spin-flipping tunneling at the QPC

Injection of electrons pairs with opposite spin and propagating in opposite directions (spin-momentum locking)

Same exponential wave-packets as in the IQH case

HOM experiments: two-electrons injection (1) D. F., C. Wahl, J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 89, 075407 (2014)

Equal spin injection (analogous to IQH)

Loss of contrast without interaction, only due to additional channels Visibility of the dip depends on QPC properties

I

q(2)R",L"(�) = 1� Ie��|�|

HOM experiments: two-electrons injection (2) D. F., C. Wahl, J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 89, 075407 (2014)

Opposite spin injection (no equivalent in IQH)

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Dip related to the topological structure of the edges J. M. Edge et al., Phys. Rev. Lett. 110, 246601 (2013)

q(2)(�)R",L# = 1�Ke��|�|q(2)(�)R",R# = 1� J e��|�|

HOM experiments: three-electrons injection D. F., C. Wahl, J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 89, 075407 (2014)

Configuration possible only in the QSH case

q(3)(�1, �2) = 1� ↵e��|�1| � �e��|�2| � (1� ↵� �)e��|�1��2|

Synchronized case: zero noise due to Pauli principle and topology Not synchronized case: exploring different interference contributions

Analogous three-photons HOM experiments proposed in quantum optics: R. A. Campos, Phys. Rev. A 62, 013809 (2000)

Source of individual Bogoliubov quasiparticles SES

SC

W

Hall edge channels at filling factor 2 degenerate in spin (neglect Zeeman and interaction) coupled to a SC contact

C. W. J. Beenakker, Phys. Rev. Lett. 112, 070604 (2014)

|e, "i ) We|e, "i+Wh|h, #i = cos

˜✓|e, "i+ sin

˜✓ei(��2�)|h, #iElectrons emerge as Bogoliubov quasiparticles

✓̃(W, ls, lm)induced coherence length ls

magnetic lengthlmlm � ls

Optimal condition: high critical field in the SC Experiments with NbN contacts on Graphene: P. Rickhaus et al., Nano Lett. 12, 1942 (2012);

G.-H. Lee et al., arXiv:1609.08104

|e, "i =Z +1

�1d⌧'e(⌧)

†"(⌧)|F i

Current and charge he, " |I(t)|e, "i = �evhe, " | : †

(t)⌧z (t) : |e, "i = �e cos(2˜✓)'e(t� �)'⇤e(t� �)

Q =

Zdthe, " |I(t)|e, "i = �e cos(2˜✓)

/ |We|2 � |Wh|2

Non conservation of the charge due to Andreev reflections A. F. Andreev, Sov. Phys. JETP 19, 1228 (1964)

✓̃ =⇡

4Particle and hole contributions compensate: zero outgoing current

Particle density and number he, " |⇢(t)|e, "i = vhe, " | : †(t) (t) : |e, "i = 'e(t� �)'⇤

e(t� �)

N =

Zdthe, " |⇢(t)|e, "i = 1

/ |We|2 + |Wh|2Conservation of the particle number

Characterization of the source D. F., J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 91, 075406 (2015)

HBT noise D. F., J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 91, 075406 (2015)

S =

Z +1

�1dtdt0he, " |I1(t)I2(t0)|e, "ic

Σ

SES1

SC1

c1 M1c1 a1

a2

I1(t)

I2(t)

SHBT1 = �e2R(1�R) cos

2(2

˜✓1)

/ Q2

Partition noise associated to a non-integer charged wave-packet ✓̃ = 0,

⇡

2

We recover the standard result for individual electrons or holes E. Bocquillon et al., Phys. Rev. Lett. 108, 196803 (2012)

✓̃ =⇡

4Zero partition noise associated to a zero emitted charge

HOM noise for two Bogoliubov quasiparticles D. F., J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 91, 075406 (2015)

Σ

SES1

SES2

SC1

SC2

c1 M1c1

c2M2c2

a1

a2

I1(t)

I2(t)

SHOM = �SHOM + SHBT1 + SHBT

2

�SHOM / |W1eW2

e⇤ �W1

hW2h⇤|2

Synchronized emission through SC differing only on the order parameter phase

SHOM2SC = e2R(1�R) sin

2(2

˜✓) [1� cos(�1 � �2)]

Non local dependence on the order parameter phase C. W. J. Beenakker, Phys. Rev. Lett. 112, 070604 (2014)

No dependence on SC phase in the current (first order coherence), oscillatory modulation in the noise (second order coherence)

Similar purely second order correlations discussed in: P. Samuelsson et al., Phys. Rev. Lett. 92, 026805 (2004); I. Neder et al., Nature (London) 448, 333 (2007); J. Splettstoesser et al., Phys. Rev. Lett. 103, 076804 (2009)...

HOM noise as a spectroscopic tool (1) D. F., J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 91, 075406 (2015)

Σ

SES1

SES2

SC1

c1 M1c1

c2

a1

a2

I1(t)

I2(t)

Interference between one Bogoliubov quasiparticle and one electron

Overlap of the wave-packets

Ratio usually discussed in experiments E. Bocquillon et al., Science 339, 1054 (2013)

HOM noise as a spectroscopic tool (2) D. F., J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 91, 075406 (2015)

Emission by a driven mesoscopic capacitor (exponential wave-packet)

Spectroscopy of Bogoliubov quasiparticles In the same spirit as: D. F. et al., Phys. Rev. B 88, 205303 (2013); D. F. et al. Phys. Rev. Lett. 113, 166403 (2014).

Conclusions

•  HBT and HOM interferometers as milestones in electron

quantum optics

•  Two and three electron interferometry in quantum spin Hall

systems

•  Interferometry and spectroscopy of individual Bogoliubov

quasiparticles

For a review: D. F., T. Jonckheere, J. Rech, T. Martin, arXiv:1610.01043