Recent progress in the simula2on of 2me- resolved quantum nanoelectronics

Transcription

1 Recent progress in the simula2on of 2me- resolved quantum nanoelectronics 1. Why quantum nanoelectronics at GHz and THz frequencies. 2. A Jme dependent concept: dynamical control of an interference palern. 3. SimulaJng d.c. quantum transport: Kwant WRITE YOUR HAMILTONIAN AS YOU DO ON THE BLACKBOARD 4. SimulaJng a.c. quantum transport: T- Kwant 5. Time- resolved quantum transport: mathemajcal aspects 6. More fancy applicajons. Xavier Waintal, SPSMS, INAC CEA Grenoble With Benoit Gaury, Joseph Weston, Christoph Groth.

2 Quantum nanoelectronics in the Jme domain: why? 6 THz = 300K: should room temperature THz electronics ever exist, it must be a fully quantum electronics 20 GHz = 1K: Jme resolved quantum electronics is now possible in the lab. First coherent single electron sources. Emergence of new concepts specific to Jme- dependent physics.

3 were encrypted in the current noise. Here we demonstrate the + + are treated on the same foot in the SAW quantum channel, one expects experimental realization of high-efficiency single-electron source that the spin coherence during the transport conserved. Naturally, of a quantum-coherent resistor-capacitor (RC) and detector for a single electron propagating isolated from the new possibilities will emerge to address the question of scalability in other electrons through a one-dimensional channel. The moving circuit, spinwhere qubit systems coherence is seen to strongly affect. potential is excited by a surface acoustic wave, which carries the To transport a single electron from one quantum dot to the other single electron along the one-dimensional channel at a speed of separated by a 3-mm 1D channel (see Fig. 1 and Methods), the following 3 mmns 21. When this quantum channel is placed between two procedure is applied. First, the region between the two electrodes, quantum dots several micrometres apart, a single electron can be which define the 1D channel, is fully depleted. As a consequence, direct transported «Electronic from one quantum quantum dot to the other op2c with quantum» is now linear electron coming transportto fromthe one endlab of the channel to the other is efficiencies of emission and detection of 96% and 92%, respectively. blocked because the Fermi energy lies below the potential induced by Furthermore, the transfer of the electron can be triggered on a the gates. Second, by applying microwave excitation to the interdigitated transducer (IDT), SAW-induced moving quantum dots are timescale shorter than the coherence time T 2 * of GaAs spin qubits 6. Our work opens new avenues with which to study the teleportation generated 12 as a result of the piezoelectric properties of GaAs (see also of a single electron spin and the distant interaction between spatially separated qubits in a condensed-matter system. the 1D channel and tuning both quantum dots into the single electron Supplementary Information). By adding a quantum dot to each side of regime, it is then possible to transport a single electron from one photon experiments with flying electrons in nanostructures at the quantum dot across the 1D channel and catch it inside the second single-electron level. Important tools with which to infer complex quantum dot. Stability diagrams for both quantum dots as a function photon correlations inaccessible from ensemble measurements are single-photon sources and single-photon detectors. In contrast Cambridge) with V photons, electrons are strongly interacting particles and they usually c AWG V bias propagate in a Fermi sea filled with other electrons. Each electron Vp 100 MΩ therefore inevitably mixes with the others of the Fermi sea, which implies that the quantum information stored within the charge or V b the spin of the single electron will be lost over short lengths. To perform quantum electron-optical experiments at the single-electron RF level, one therefore needs a source of single electrons, a controlled propagating medium and a single-electron detector. It has been proposed that edge states in the quantum Hall effect can serve as a one- IDT V b 1 μm dimensional (1D) propagating channel for flying electrons. As a result 100 MΩ V bias of Coulomb blockade, quantum dots have been demonstrated to be a V c V p good source of single electrons 7,8 and can also serve as a single-electron detector. Indeed, once an electron has been stored in a quantum dot, its Figure 1 Experimental device and measurement setup. Scanning electron Nature 447, 435 (2011) presence can be inferred routinely by charge detection 9. Nevertheless, microscope image of the single-electron transfer device, and diagram of the re-trapping the electron in another quantum dot after propagation in an experimental setup. Two quantum dots, which can be brought into the singleelectron regime, are separated by a 1D channel 3 mm long, as shown. Each edge state turns out to be extremely difficult, and currently all the quantum dot is capacitively coupled to a QPC close by that is used as an information extracted from such experiments is coming from ensemble electrometer 9. By applying a microwave burst 65 ns long on the IDT (see measurements 10,11. Here we show that a single flying electron an electron surfing on a sound wave can be sent on demand from a quantum 1D channel. Gate 6-GHz V c islorentzian connected to a home-made bias tee to allow nanosecond Methods b 0.3for details), a train of about 150 moving quantum dots is created in the dot by b means of a 1D quantum channel and re-trapped in a second manipulation of 2W the = 30 dotps potential. RF, radio frequency. V(t) 16-GHz sine 1 Institut Néel, CNRS, and Université Joseph Fourier, Grenoble, France. 2 Department of Applied Physics, University24-GHz of Tokyo, sine Tokyo, , Japan. 3 ERATO-JST, Kawaguchi-shi, Saitama , Japan. 4 ICORP (International V G Cooperative Research Project) Quantum Spin Information Project, Atsugi-shi, Kanagawa, 4-GHz , square Japan. 5 Lehrstuhl für Angewandte Festkörperphysik, Ruhr- 0.2 Universität Bochum,Universitätsstrasse 150, Bochum, Germany. 6 Institut Universitaire de France, 103 boulevard Saint-Michel, Paris, France. D of nominal density n s = m 2 and mobility m =260V 1 m 2 s 1.Thedotiselectro- Mesoscopic physics in the Jme domain. Fig. 1. Single-charge injection. (A) Schematicof single-charge injection. Starting from an antiresonant situation where the Fermi energy lies between two energy levels of the dot (step 1), the dot potential is increased by D moving one occupied level above the Fermi energy (step 2). One electron then escapes the dot on the mean time t = h/dd. Thedotpotential is then brought back to its initial value (step 3), where one electron can enter it, leaving a hole in the Fermi sea. (Inset at right) The quantum RC circuit: one edge channel is transmitted inside the submicrometer dot, with transmission D tuned by the QPC gate voltage V G.Thedotpotentialis varied by a radio-frequency Single electron source (GlaPli et al. LPA, ENS Paris) Single electron source with SAW (Bauerle, Meunier et al Neel, Grenoble, Ritchie et et al, Science 316, 1169 (2007) The Leviton: (GlaPli et al. SPEC, CEA Saclay) V(t) 1D 22 SEPTEMBER 2011 VOL 477 NATURE 435 V(t) 2011 Macmillan V G Publishers Limited. All rights reserved excitation V exc applied on a macroscopic gate located on top of the dot. The electrostatic potential can 0.0 also be tuned by V G because of the electrostatic coupling between 0 the dot1and the QPC. 2 (B) Timedomain measurement of the average current (black curves) on one period of Nature 502, 659 (2013) Charge theper excitation pulse signal (red curves) at 2eV exc = D for three values of the transmission D. The relaxation time t is deduced from an exponential fit (blue curve). 0.1 V(t) amplitude excitation the linear regime. W brought above the F expected to escape rate t 1 = DD/h, frequency and D is This gives nanosec single-charge detec perimentally. To i ratio, a statistical a ual events is used quences of singleby single-electron a as shown in Fig. 1 ing a periodic squa D/e to the top ga temporal traces of few seconds for a re The single-electron nential current dec transmission D is 0.002, the relaxatio exponential decay, For the two highes t << T/2, the curr mean transferred cha For the smallest tran emitted charge decre reduced probability time-domain measu 1-GHz bandwidth give access to the few corresponding to sm In order to get a above results, we ex

4 Dynamical control of an interference pattern Machzender Interferometer 1.1 m B=1.8T 110nm 2.1 m V(t) Real sample SchemaJc sample Simulated sample

5 Dynamical control of an interference pattern Mach-Zehnderinterferometer Time-dependentcurent V b Phys.Rev.Let.100, (2008) Oscilationfrequency:eV b /h A B

6 Dynamical control of an interference pattern Fabry-Perotinterferometer

7 Kwant Kwant is our new so`ware for simulajng quantum transport. (26 lines) Kwant paradigms: Low entrance cost for physicists. Spend your Jme on physics not wrijng code (let the computer do the book keeping) Make the computer aware of physics concepts (symmetries, laeces, ) Simultaneously Fast AND Flexible (faster than recursive Green s funcjon: nested dissecjon A collaborajon with Christoph Groth, Anton Akhmerov (Del`) and Michael Wimmer (Del`) An open source so`ware available on Linux, Mac and even Windows. hlp://kwant- project.org

8 Kwant WRITE YOUR HAMILTONIAN AS YOU DO ON THE BLACKBOARD

9 Kwant KWANT SUPPORT MANY PHYSICAL CONCEPTS SUCH AS BRAVAIS LATTICE

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12 Kwant Arbitrary geometry Arbitrary internal degrees of freedom (spin, nambu, orbitals ) FuncJon as values MulJ- terminal Arbitrary dimension T(E) ZT max E n ad λ so ZT θ (b) Superconducting center contact ( = 0)

13 T- Kwant Becomes: B. Gaury et al. / Physics Reports 534 (2014) 1 37 ARTICLE 29 NATURE COMMUNICATIONS DOI: /ncomms4844 Will be open source but not ready yet. 1.1 µm a B. Gaury et al. / Physics Reports 534 (2014) µm 5. Fe ve, G., Degiovanni, P. & Jolicoeur, Th. Quantum detection of electronic flying qubits in the integer quantum hall regime. Phys. Rev. B 77, (2008). 6. Schomerus, H. & Robinson, J. P. Entanglement between static and flying qubits in an aharonov-bohm double electrometer. New J. Phys. 9, 67 (2007). B =1.8 T 7. Ji, Y. et al. An electronic mach-zehnder interferometer. Nature 442, 415 (2003). 8. Roulleau, P. et al. Direct measurement of the coherence length of edge states in the integer quantum hall regime. Phys. Rev. Lett. 100, (2008). 110 nm 9. Haack, G., Moskalets, M., Splettstoesser, J. & Bu ttiker, M. Coherence of singleelectron sources from mach-zehnder interferometry. Phys. Rev. B 84, (2011). 10. Yamamoto, M. et al. Electrical control of a solid-state flying qubit. Nat. Nanotechnol. 7, (2012). 11. Fe ve, G. et al. An on-demand coherent single-electron source. Science 316, V(t ) Fig. 13. Current density as a function of space (in unit of v P ) and time (in unit of P ) for the (2007). Gaussian pulse of Fig. 11(c). Fermi level is set at EF = 1.8. Left panel: the color map goes from zero values (blue) to 0.6 (red). Right panel: cut of the left at three in spaces (a), (b) corresponding 12.panel Lesovik, G. B.positions & Levitov, L. S. Noise inand an (c) ac biased junction: nonstationary to the three dashed lines shown on the left panel. Orange:1x = 15v P, blue: x = 30v P, green: x = 45v P. Aharonov Bohm effect. Phys. Rev. Lett. 72, (1994) Levitov, L. S., Lee, H. & Lesovik, G. B. Electron counting statistics and coherent states of electric current. J. Math. Phys. 37, 4845 (1996). 14. Keeling, J., Klich, I. & Levitov, L. S. Minimal excitation states of electrons in Figure 9 Mach Zehnder interferometer. Snapshot of the local electronic one-dimensional wires. Phys. Rev. Lett. 97, (2006). 3 density at t ¼ 46 ps. The colour map indicates the deviation from 15. Ivanov, D. A., Lee, H. W. & Levitov, L. S. Coherent states of alternating current. equilibrium, which goes from 0 (salmon) to 0.22 & 1011 cm % 2 (black). Phys. Rev. B 56, (1997). 16. Gabelli, J. & Reulet, B. Shaping a time-dependent excitation to minimize the in a tunnel junction. Phys. Rev. B 87, (2013). shot noise with f the total magnetic flux through the central depleted region (in unit of /e) 17. Zhong, Z., Gabor, N. M., Sharping, J. E., Gaetal, A. & McEuen, P. L. Terahertz and tf the extra time needed for the upper paths with respect to the lower one. time-domain measurement of ballistic electron resonance in a single-walled After following the same steps as for the Fabry Perot geometry, one obtains (in the carbon nanotube. Nat. Nanotechnol. 3, (2008). limit of short pulses) the number of particles transmitted to contact 1 (2), 18. Dubois, J. et al. Minimal-excitation states for electron quantum optics using 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi levitons. Nature 502, (2013). n1 ¼ðDA DB þ RA RB Þ! nþ DA DB RA RB sinðp! nþcosðp! n þ fþ ð27þ 19. van Wees, B. J. et al. Observation of zero-dimensional states in a onep dimensional electron interferometer. Phys. Rev. Lett. 62, (1989). 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 20. Liang, W. et al. Fabry perot interference in a nanotube electron waveguide. n% nþcosðp! n þ fþ ð28þ n2 ¼ðDA RB þ RA DB Þ! DA DB RA RB sinðp! p Nature 411, (2001). 21. Kretinin, A. V., Popovitz-Biro, R., Mahalu, D. & Shtrikman, H. Multimode fabry-perot conductance oscillations in suspended stacking-faults-free inas Model for the Mach Zehnder geometry. We consider a two-dimensional nanowires. Nano. Lett. 10, 3439 (2010). electron gas made from a two-dimensional GaAs/AlGaAs heterostructures with S. Electronic Transport in Mesoscopic 18.Datta, Propagation of a voltage pulse within the Systems coupled (Cambridge wire system.university The figures show snapshots of the difference of the local charge density from high mobility m ¼ 2 & 106 cm2 V % 1 s % 1, an electronic density ns ¼ 1011 cm % 2 and Fig.22. Press,at1997). equilibrium different points in time. Figures (a) and (b) correspond to two different values of the tunneling gate voltage, 0.24 mv and 0.34 mv respectively. a perpendicular magnetic field B ¼ 1.8 T (corresponding to filling factor one, 23.results Gaury, B. et produced al. Numerical simulations of time-resolved quantum electronics. withcolor a Gaussian voltage described in protocol. the main text. theseon two runsside corresponds to a computing first Hall plateau). The three-terminal electronic Mach Zehnder interferometer is These FIG. 3. were Charge density map for the pulse stopasand release TheEach twoofgates each of the system (red/bluetime per Phys. Rep. 534, 1 37 energy and per channel of control 30(2014). min on one computing core (a the = 0direction.5, 7250 sites and time steps). sketched in Fig. 9. dashed rectangles) the edge states (hence of propagation of the pulse). The left gate is polarized for t t1 24. Kazymyrenko, K. & Waintal, X. Knitting algorithm for calculating green The system is modelled within the effective mass approximation in presence of and grounded for t t2. At t2, the pulse is frozen. At t = t3 one of the two gates is polarized again, which releases the pulse. functions in quantum systems. Phys. Rev. B 77, (2008). a small static disorder. The Schrodinger equation is discretized on a mesh with a Top: the left gate is polarized, the pulse follows its original edge state and is collected in the top left electrode. Bottom: the a b c d b

14 MathemaJcal aspects: Non Equilibrium Green FuncJon versus ScaLering theory versus parjjonless approach Ĥ(t) = ij H ij (t)c i + c j Non Equilibrium Green s Func2on (NEGF) approach Write Dyson equajon in Keldysh space Integrate out the leads degrees of freedom One finds the standard framework (Jauho, Wingreen, Meir) i t G( r,t, r ',t') = H(t)G( r,t, r ',t')+ dudr ''Σ( r,t, r '',u) G( r '',u, r ',t') G < ( r,t, r ',t') = dudvdr ''dr '''G( r,t, r '',u)σ < ( r '',u, r ''', v) G + ( r ''', v, r ',t') Computationally prohibitive: CPU=t 2 L 7 (3D) One value of t per calculation Huge memory footprint Large time just to recover the initial stationary solution (before switching on any time-dependent field) Algorithms easily unstable. Fig. 4. Current as a function of time for a square voltage pulse w(t) = w 0 (t t 0 ) (t 1 t) with w 0 = 0.1, t 0 = 10 1, t 1 = 40 1 and E F = 0. The lines show w(t) (dashed), the GF-C result (red) and the WF-B result (black). Lower inset: current I(t = 5 1 ) as a function of t for the GF-B scheme (symbols) together with the fit 1/ t (line). Upper inset: zoom of the lower inset with the fit I = (0.1 + cos(4 t ))/ t. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

15 How topropagateanelectronicstate? MathemaJcal aspects: ParJJonless approach Propagatingawavepacketisawelknownproblem:simplyusetheSchrödingerequ System atequilibrium Perturbationswitchedon t=0 Problem: time Condensedmater notoptics,thegroundstatei notempty. : Pauli How doitakeintoaccounttheunderlyingfermiseaandthepauliprinciple? WavefunctionfulyequivalenttoNEGF Fermisea? Emergingconceptsintime-resolvedquantum nanoelectronics

16 MathemaJcal aspects: Non Equilibrium Green FuncJon versus ScaLering theory versus parjjonless approach Ĥ(t) = ij H ij (t)c i + c j H(t) = H 0 + Δ(t) Going to a pracjcal scheme: Ψ E ( r,t) = Ψ E ( r,t)+ Ψ E K ( r)e iet i t Ψ E ( r,t) = H(t)Ψ E ( r,t)+ Δ(t)Ψ K E ( r)e iet + iσ( r)ψ E ( r,t) I ( r,t) = de Im Ψ E ( r,t) Ψ E ( r,t) f (E) Almost computationally easy: CPU=t L 3 (3D) All values of t at once Small memory footprint L 3 Start from the exact stationary solution Stable differential equation Easily amenable to analytical solutions Currently we solve systems with more than 10 5 sites Algorithm very parallel. SCHRODINGER SOURCE SINK

17 Conclusion Quantum electronics in the 2me domain is only star2ng to emerge Time resolved simula2ons are now rela2vely easy Even the simplest cases provide nice new concepts