Unconventional electron quantum optics in condensed matter systems

Transcription

1 Unconventional electron quantum optics in condensed matter systems Dario Ferraro Centre de Physique Théorique, Marseille nanoqt-2016, Kyiv, October 10, 2016 In collaboration with: J. Rech, T. Jonckheere, T. Martin

2 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

3 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

4 Electron injection on-demand Single electron sources based on driven mesoscopic capacitors Theory: Moskalets et al., Phys. Rev. Lett. 100, (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 t e 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, (2006); Experiments: J. Dubois et al., Nature 502, 659 (2013), T. Jullien et al., Nature 514, 603 (2014)

5 Individual electrons interferometry Hanbury-Brown-Twiss (HBT) and Hong-Ou-Mandel (HOM) S HOM t q 2 S HBT T0 50 t 100 T. Jonckheere et al., Phys. Rev. B 86, (2012) 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, (2014); D. F. et al. Phys. Rev. Lett. 113, (2014).

6 Quantum spin Hall effect (QSHE) CdTe/HgTe quantum wells Theory: A. B. Bernevig et al., Science 314, 1757 (2006); Experiments: M. König et al., Science 318, 766 (2007) Trivial insulator Topological insulator: counter-propagating edge channels carrying opposite spin at zero magnetic field, robust against backscattering Observed also in InAs/GaSb quantum wells: C. Lui et al., Phys. Rev. Lett. 100, (2008); I. Knez et al., Phys. Rev. Lett. 107, (2011)

7 Pair electron source Driven mesoscopic capacitor coupled to helical edge states A. Inhofer and D. Bercioux, Phys. Rev. B 88, (2013); P. P. Hofer and M. Buttiker, Phys. Rev. B 88, (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

8 HOM experiments: two-electrons injection (1) D. F., C. Wahl, J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 89, (2014) Equal spin injection (analogous to IQH) I q (2) R",L" ( )=1 Ie Loss of contrast without interaction, only due to additional channels Visibility of the dip depends on QPC properties

9 HOM experiments: two-electrons injection (2) D. F., C. Wahl, J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 89, (2014) Opposite spin injection (no equivalent in IQH) q (2) ( ) R",L# =1 Ke q (2) ( ) R",R# =1 J e Dip related to the topological structure of the edges J. M. Edge et al., Phys. Rev. Lett. 110, (2013)

10 HOM experiments: three-electrons injection D. F., C. Wahl, J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 89, (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, (2000)

11 Source of individual Bogoliubov quasiparticles SES Hall edge channels at filling factor 2 degenerate in spin (neglect Zeeman and interaction) coupled to a SC contact W SC Electrons emerge as Bogoliubov quasiparticles e, "i ) W e e, "i + W h h, #i = cos e, "i +sin e i( 2 ) h, #i C. W. J. Beenakker, Phys. Rev. Lett. 112, (2014) l s (W, l s,l m ) induced coherence length l m l m l s magnetic length 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:

12 D. F., J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 91, (2015) Z +1 e, "i = he, " I(t) e, "i = evhe, " : Z Q = Characterization of the source 1 d ' e ( ) Current and charge " ( ) F i (t) z (t) : e, "i = e cos(2 )' e (t )' e(t ) dthe, " I(t) e, "i = / W e 2 W h 2 e cos(2 ) Non conservation of the charge due to Andreev reflections A. F. Andreev, Sov. Phys. JETP 19, 1228 (1964) = 4 Particle and hole contributions compensate: zero outgoing current he, " (t) e, "i = vhe, " : Z N = Particle density and number (t) (t) : e, "i = ' e (t )' e(t ) dthe, " (t) e, "i =1 / W e 2 + W h 2 Conservation of the particle number

13 HBT noise D. F., J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 91, (2015) Z +1 S = SES1 1 dtdt 0 he, " I 1 (t)i 2 (t 0 ) e, "i c c 1 M 1 c 1 a 1 I 1 (t) I 2 (t) SC1 a 2 Σ S1 HBT = e 2 R(1 R) cos 2 (2 1 ) / Q 2 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, (2012) = 4 Zero partition noise associated to a zero emitted charge

14 HOM noise for two Bogoliubov quasiparticles D. F., J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 91, (2015) SES1 c 1 M 1 c 1 a 1 I 1 (t) I 2 (t) SC1 a 2 Σ M 2 c 2 c 2 SC2 SES2 S HOM = S HOM + S1 HBT + S2 HBT S HOM / We 1 We 2 WhW 1 h 2 2 Synchronized emission through SC differing only on the order parameter phase S HOM 2SC = e 2 R(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, (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, (2004); I. Neder et al., Nature (London) 448, 333 (2007); J. Splettstoesser et al., Phys. Rev. Lett. 103, (2009)...

15 HOM noise as a spectroscopic tool (1) D. F., J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 91, (2015) SES1 c 1 M 1 c 1 a 1 I 1 (t) I 2 (t) SC1 a 2 Σ c 2 SES2 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)

16 HOM noise as a spectroscopic tool (2) D. F., J. Rech, T. Jonckheere, T. Martin, Phys. Rev. B 91, (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, (2013); D. F. et al. Phys. Rev. Lett. 113, (2014).

17 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: