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Quantum Tomography of an Electron T Jullien, P Roulleau, B Roche, a Cavanna, Y Jin, Christian Glattli

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Quantum Tomography of an Electron T Jullien, P Roulleau, B Roche, a Cavanna, Y Jin, Christian Glattli Quantum tomography of an electron T Jullien, P Roulleau, B Roche, A Cavanna, Y Jin, Christian Glattli To cite this version: T Jullien, P Roulleau, B Roche, A Cavanna, Y Jin, et al.. Quantum tomography of an electron. Nature, Nature Publishing Group, 2014, 514, pp.603 - 607. 10.1038/nature13821. cea-01409215 HAL Id: cea-01409215 https://hal-cea.archives-ouvertes.fr/cea-01409215 Submitted on 5 Dec 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LETTER doi:10.1038/nature13821 Quantum tomography of an electron T. Jullien1*, P. Roulleau1*, B. Roche1, A. Cavanna2, Y. Jin2 & D. C. Glattli1 The complete knowledge of a quantum state allows the prediction of pbyffiffiffiffiffiffiffiffi mixing with a coherent field (local oscillator) with amplitude the probability of all possible measurement outcomes, a crucial step NLOE0utðÞ{x=c , where NLO is the mean photon number. Then a 1 in quantum mechanics. It can be provided by tomographic methods classical measurement of u can be made.p becauseffiffiffiffiffiffiffiffi the fundamentalp quan-ffiffiffiffiffiffiffiffi 2,3 4 5,6 which havebeenappliedtoatomic ,molecular , spin and photonic tum measurement uncertainty , 1 NLO vanishes for large NLO. 7–9 10–13 states. For optical or microwave photons, standard tomogra- Only the determination of the bosonic state jbæ requires purely quan- phy is obtained by mixing the unknown state with a large-amplitude tum tomography. For fermions, however, the state occupation being coherent photon field. However, for fermions such as electrons in limited to 1, no classical amplitude level can be reached and a quantum condensed matter, this approach is not applicable because fermionic tomography is necessary for both the wavefunction Q and the fermion fields are limited to small amplitudes (at most one particle per state), state jf æ. In the present work, we prepare many electrons, each in the known and so far no determination of an electron wavefunction has been fermionicstatejf æ 5 j1æ. We show that, althoughextremelydemanding, made. Recent proposals involving quantum conductors suggest that a quantum tomography of the wavefunction of a time-resolved single the wavefunction can be obtained by measuring the time-dependent electron is possible using shot noise, and we determine the first-order 14 current of electronic wave interferometers or the current noise of coherence Q~ÃðÞe0 Qe~ðÞin the energy representation (where Qe~ðÞis the 15–17 electronic Hanbury-Brown/Twiss interferometers . Here weshow Fourier transform of Q(t) and the asterisk denotes complex conjugation). that such measurements are possible despite the extreme noise sensi- We first explain how we prepare the single-electron state in a quan- tivity required, and present the reconstructed wavefunction quasi- tum conductor and then how the quantum tomography is performed. 17 probability, or Wigner distribution function , of single electrons The quantum conductor is a quantum point contact (QPC) placed at injected into a ballistic conductor. Many identical electrons are pre- low temperature (electron temperature, Te < 35 mK). It is formed in a 18 pared in well-controlled quantum states called levitons by repeatedly high-mobility two-dimensional electron gas using split gates deposited 19–21 applying Lorentzian voltage pulses to a contact on the conductor . on top of a GaAs/GaAlAs heterojunction (Fig. 1a and Supplementary After passing through an electron beam splitter, the levitons are Information). Applying negative voltage VG to the gates creates a con- mixed with a weak-amplitude fermionic field formed by a coherent striction transmitting a limited number of electronic modes. Here we superposition of electron–hole pairs generated by a small alternat- select a single mode whose transmission D(VG) is obtained by measur- ing current with a frequency that is a multiple of the voltage pulse ing the conductance G~(2e2=h)D, where e is the electron charge and h frequency16. Antibunching of the electrons and holes with the levi- is Planck’s constant. The on-demand injection of charges into the con- tons at the beam splitter changes the leviton partition statistics, and ductor is obtained by applying a voltage pulse VL(t) on one of the ohmic the noise variations provide the energy density matrix elements of contactsÐ adjacent to the QPC, say the left. If the voltage flux satisfies the levitons. This demonstration of quantum tomography makes z? ~ e {? VLðÞt dt h then the charge is unity. Moreover, a Lorentzian pulse the developing field of electron quantum optics with ballistic con- VtðÞ~2Bw=e(t2zw2),whereB~h=2p and 2wis the pulse width at mid ductors a new test-bed for quantum information with fermions20,22–24. height, creates a clean single-electron state19 called a leviton18 with an These results may find direct application in probing the entanglement energy just greater than the Fermi energy of the conductor. Remarkably, of electron flying quantum bits25, electron decoherence17 and electron although the leviton originates from a collective displacement of the interactions. They could also be applied to cold fermionic (or spin- Fermi sea towards positive energy, it can be considered a single-particle 1/2) atoms26. state20,21 disentangled from the Fermi sea21. A wavefunctionpffiffiffiffiffiffiffiffiffiffiffi representa- A quantum state y contains all the information about a particle or a tion of the leviton in the energy domain is Qe~ðÞ~ 2w=BheðÞe{ew=B, system. Disregarding spin for simplicity, a practical representation is where h is the Heaviside step function. Experimentally, periodic trains given by the product of the wavefunction Q which contains the spatial of levitons have been generated18. When partitioned by a QPC18 they information and an occupation state in a Fock space representation. generate a current shot noise corresponding to electrons prepared in Although the determination of the latter requires similar experimental the occupation state j1æ. Using shot-noise spectroscopy and electron resources for a fermion and for a single boson, the complete determi- Hong–Ou–Mandel noise correlation techniques, their wavefunction nation of the wavefunction by tomographic methods is fundamentally probability in the energy and time domain was found to be consistent more demanding for fermions than for bosons. To emphasize this let with expectations. Thus, levitons provide a clean way of testing wave- us consider a single particle propagating in a single spatial mode (an elec- function quantum tomography in a quantum conductor. tron in an effectively one-dimensional ballistic conductor or a photon We now discuss how to probe the first-order coherence from which in an optical medium). In the space and time domain representation, the full wavefunction can be extracted. A first approach could be wave the electron quantum state at position x and time t is Q(t 2 x/vF)jf æ, interferometry. In quantum optics, this gives G1(t 2 t9), which is the first- 1 0 where vF is the velocity, Q is the wavefunction and jf æ is the fermionic order coherence Æy (t9)y(t)æ averaged over the mean timet~ðÞtzt =2 occupation state, j1æ or j0æ. Similarly, a photonic state is characterized (here y1 is the Hermitian conjugate of y). The information is incom- 27 by E0u(t 2 x/c)jbæ, where E0 is the electric field amplitude of a single plete, however, and other approaches must be used . Wave interfero- photon, u is the spatial mode which plays the role of Q, jbæ is, for exam- metry is more promising with electrons: measuring the time-dependent ple, a number state or a Glauber coherent state, and c is the speed of current of an electronic Mach–Zehnder interferometer has been pro- light. In quantum optics, the determination of the mode u can be done posed14 as an elegant way to determine the full coherence Æy1(t9)y(t)æ, 1Service de Physique de l’Etat Condense´, IRAMIS/DSM (CNRS URA 2464), CEA Saclay, F-91191 Gif-sur-Yvette, France. 2CNRS, Laboratoire de Photonique et de Nanostructures, Route de Nozay, 91460 Marcoussis, France. *These authors contributed equally to this work. 30 OCTOBER 2014 | VOL 514 | NATURE | 603 ©2014 Macmillan Publishers Limited. All rights reserved RESEARCH LETTER a V t b d LO( ) V Correlation Cross-correlation noise detection G Detector h Detector ν0 4-harmonic 2ν0 V e R frequency ν0 V t V synthesizer LO( ) Sine G 0 ν0 = 6 GHz wave 6 GHz 300 K V(t) V(t) Lorentzian τ Phase pulses shifter Photon LO source –70dB –70dB c Energy Energy Leviton ~ ϕ(ε + hν ) 0 Cryo-amp. Cryo-amp. 4 K ~ μ ϕ(ε) R ε ~ h ϕ(ε – ν0) 0 ~ ϕ 13 mK V (t) ƒ (ε) V(t) LO R Bias tee QPC Bias tee V R Figure 1 | Schematics of quantum wave tomography. a, Experiment c, Experiment principle in energy representation. All quantities are step principle. Periodic Lorentzian voltage pulses V(t) applied to the left contact of a functions of the energy with energy scale hn0. Left: the energy dependence of the two-dimensional electron gas inject unit charge pulses called levitons which are wavefunction of periodic levitons emitted from the left contact. Right: the partitioned by an electronic beam splitter called a quantum point contact energy distribution of the right reservoir when the voltage VR 1 VLO(t)is (QPC). The split-gate voltage VG controls the transmission D of the one- applied.
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