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).