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Towards Electron-Electron Entanglement in Penning Traps

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Towards Electron-Electron Entanglement in Penning Traps PHYSICAL REVIEW A 81, 022301 (2010) Towards electron-electron entanglement in Penning traps L. Lamata, D. Porras, and J. I. Cirac Max-Planck-Institut fur¨ Quantenoptik, Hans-Kopfermann-Strasse 1, D-85748 Garching, Germany J. Goldman and G. Gabrielse Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA (Received 5 May 2009; revised manuscript received 11 December 2009; published 4 February 2010) Entanglement of isolated elementary particles other than photons has not yet been achieved. We show how building blocks demonstrated with one trapped electron might be used to make a model system and method for entangling two electrons. Applications are then considered, including two-qubit gates and more precise quantum metrology protocols. DOI: 10.1103/PhysRevA.81.022301 PACS number(s): 03.67.Bg, 06.20.Jr, 13.40.Em, 14.60.Cd I. INTRODUCTION two-qubit gate, spin-cyclotron entanglement generation, and a quantum metrology protocol with two entangled electrons. Entanglement is one of the most remarkable features In Sec. V, we examine the challenges faced when applying of quantum mechanics [1]. Two entangled systems share this protocol to existing experimental conditions. In Sec. VI, the holistic property of nonseparability—their joint state we point out possible extensions to many electrons, and we cannot be expressed as a tensor product of individual states. present our conclusions in Sec. VII. Entanglement is also at the center of the rapidly developing field of quantum information science. A variety of systems have been entangled [2], including photons, ions, atoms, II. TWO ELECTRONS IN A PENNING TRAP and superconducting qubits. However, no isolated elementary A Penning trap [5,6] for an electron (charge −e and mass particles other than photons have been entangled. m) consists of a homogeneous magnetic field, B = Bzˆ, and an It is possible to perform quantum information protocols electrostatic quadrupole potential energy, with electrons in Penning traps, as opposed to ions, even 2 though the former cannot be laser cooled. This is possible 1 2 2 ρ Vq (x) = mω z − , (1) because at low temperature in a large magnetic field (100 mK 2 z 2 and 6 T in Harvard experiments [3]) the cyclotron motion with ρ = xxˆ + yyˆ. The magnetic field provides dynamic radial radiatively cools to the ground state. We describe how one confinement, while the quadrupole potential gives the axial could use this mode for quantum information applications confinement. The single-particle oscillation frequencies are since there is sufficiently small coupling to other modes. the trap-modified cyclotron frequency ωc (slightly smaller than In this paper, we describe a possible method for entangling the cyclotron frequency ω = eB/m), the axial frequency ω , two electrons. The model system and method we investigate c z the magnetron frequency ωm, and the spin precession are largely based on building blocks already demonstrated frequency ω . with one trapped electron. On the way to measuring the s 13 Quantum control of cyclotron and spin motions was electron’s magnetic moment to 3 parts in 10 [3], quantum recently achieved with a single electron (but not yet with nondemolition (QND) methods were used to reveal one- more). Harvard experiments [3,7] cool an electron’s cyclotron quantum cyclotron and spin transitions between the lowest motion to its quantum ground state, using methods that should energy levels of a single electron suspended for months also work for more trapped electrons. Quantum nondemolition in a Penning trap. We demonstrate how the two-electron detection and quantum jump spectroscopy of the lowest entanglement could make a universal two-qubit gate. We cyclotron and spin levels (with quantum numbers n and ms )[8] show how this gate could enable a metrology protocol that cleanly resolve one-quantum transitions to determine ωc and surpasses the shot-noise limit, and as an example we consider the anomaly frequency ω = ω − ω . These frequencies, with in detail the requirements for implementing this protocol in a s c the measured axial frequency ωz, determine the dimension- a measurement of the electron magnetic moment using two less magnetic moment g/2—the magnetic moment in Bohr trapped entangled electrons. The payoffs and requirements magnetons—to an unprecedented level of accuracy. for moving from two-electron to N-electron entanglement are Crucial to the one-electron measurements, and the multipar- listed. Possible applications include quantum simulators [4], ticle approach suggested here, are anomaly transitions between analysis of decoherence, and more precise electron magnetic | | 1 | | − 1 | | the states 0 2 and 1 2 , where the states n ms moment measurements using improved quantum metrology are labeled with their cyclotron and spin quantum numbers. protocols. An applied oscillating electric field drives harmonic axial In Sec. II, we introduce the formalism of two electrons = + motion, z z0 cos [(ωa )t], taking the electron through in a Penning trap. In Sec. III, we obtain a two-electron gate a “magnetic bottle” (MB) gradient, based on building blocks experimentally demonstrated for one electron. In Sec. IV, we identify some applications, including ρ2 B(x) = β z2 − zˆ − zρ . (2) protocols to entangle the spins of two electrons, a universal 2 2 1050-2947/2010/81(2)/022301(5) 022301-1 ©2010 The American Physical Society LAMATA, PORRAS, CIRAC, GOLDMAN, AND GABRIELSE PHYSICAL REVIEW A 81, 022301 (2010) The electron spin sees the transverse magnetic field oscillating cyclotron motion cools to its ground state by synchrotron near ωs as needed to flip the spin, at the same time as the radiation, and the thermal axial excursion is much less than cyclotron motion sees the azimuthal electric field oscillating the radial separation x0. near ωc, as needed to make a simultaneous cyclotron transition. The magnetic bottle also enables single-shot quantum nonde- III. TWO-ELECTRON GATE molition measurements of one-quantum changes in cyclotron and spin energies, by coupling these to small but detectable In this model system, the desired entanglement can be + shifts in ωz [9]. achieved by applying an axial drive at ωa in the presence We now consider as a model system two electrons in of a magnetic bottle, as in the one-electron experiments [3,7]. the same Penning trap. We describe an interaction that The electrons’ center of mass oscillates at this frequency, with = + will entangle them, we explore possible applications of the zCM z0 cos[(ωa )t]. The entanglement arises from terms ρ · σ + σ entanglement of these two electrons, and then we consider zCM CM ( 1 2) contained in HI in Eq. (4), the interaction the challenges faced when implementing these techniques in of the electrons and the magnetic bottle. The Hamiltonian the laboratory. The Hamiltonian starts with the sum of two can be written in the well-known Tavis-Cummings form [11] single-particle terms, after the following steps: First, the radial coordinates of the electrons are written as a sum of center-of-mass (CM) and + 2 hω¯ s z [pi eA(xi )] stretch (st) coordinates, as in x = x ± x /2, with the CM Hi = σ + + Vq (xi ). (3) 1,2 CM st 2 i 2m terms contributing to producing entanglement. Second, the CM Both the spin and orbital magnetic moments couple to the radial position is replaced by raising and lowering operators for magnetic bottle, adding the CM magnetron and cyclotron motions, as initially defined for the single-particle case in Ref. [5], but with the CM mass 2 M = 2m replacing m. g Li HI = µB σ i + · B(xi )(4) † † = 2 h¯ − + − i 1 aCM,c aCM,c aCM,m aCM,m xCM = i √ , (8) 2 − 2 1/4 to the Hamiltonian, along with 4m/h¯ ωc 2ωz † † 2 e 1 aCM,c + a − aCM,m − a = =− CM,c CM,m V12 , (5) yCM √ 1/4 . (9) 4π0 |x1 − x2| 2 − 2 4m/h¯ ωc 2ωz the Coulomb interaction of the two electrons. Third, the rotating wave approximation (RWA) retains only An equilibrium separation of the two electrons can be terms that can make anomaly transitions. Fourth, we switch to produced using a so-called rotating wall—a time-dependent the interaction picture with respect to the sum of single-particle oscillating potential in the x-y plane that rotates at angular Hamiltonians for δ = 0. frequency ωzˆ [10]. In the co-rotating reference frame, the The resulting Hamiltonian, with the appropriate simplifying effective potential energy is choice of the origin of time, is V˜ (x ) = 1 mω 2ρ2 + 1 mω2z2 − 1 mω2δ x2 − y2 , (6) q i 2 ρ i 2 z i 2 z i i gate = + −it + − † it HI h ¯ (σi aCM,c e σi aCM,c e ), (10) 2 = − − 1 2 i=1,2 with ωρ (ωc ω)ω 2 ωz . The first term on the right in Eq. (6) is from the fictitious forces and the transformed which couples the spins and the CM cyclotron motion. quadrupole, the second term is the unchanged axial potential The discarded terms in HI of Eq. (4) have been carefully energy, and the third term is the weak rotating wall potential. checked to make sure that they do not produce an unacceptable Due to the fictitious forces, there is radial as well as axial decoherence. A thorough analysis including an extension to 2 − confinement in the rotating frame, provided that (ωρ ) N electrons will be included in future work [12]. The Rabi 2 ωz δ>0. For small δ, this condition sets a lower limit frequency is on the rotating-wall frequency, ω>ωm. The total potential ˜ x + ˜ x + g µB β2z0 energy, Vq ( 1) Vq ( 2) V12, is a minimum when the two = √ / . (11) electrons are diametrically opposed along the direction of 2 4mh¯ ω2 − 2ω2 1 4 weakest confinement.
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