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arXiv:0810.0670 · New Journal of Physics

Deterministic entanglement of ions in thermal states of motion

Gerhard Kirchmair, Jan Benhelm, F. Zähringer, R. Gerritsma, Christian F. Roos, R. Blatt

Published Feb 4, 2009• 120 citations
arXivHTMLPDF

Abstract

We give a detailed description of the implementation of a Mølmer–Sørensen gate entangling two 40 Ca + ions using a bichromatic laser beam near-resonant with a quadrupole transition. By amplitude pulse shaping and compensation of ac-Stark shifts we achieve a fast gate operation without compromising the error rate. Subjecting different input states to concatenations of up to 21 individual gate operations reveals Bell state fidelities above 0.80. In principle, the entangling gate does not require ground state cooling of the ions as long as the Lamb–Dicke criterion is fulfilled. We present the first experimental evidence for this claim and create Bell states with a fidelity of 0.974(1) for ions in a thermal state of motion with a mean phonon number of in the mode coupling to the ions' internal states.

DOI
10.1088/1367-2630/11/2/023002

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We give a detailed description of the implementation of a Mølmer–Sørensen gate entangling two 40 Ca + ions using a bichromatic laser beam near-resonant with a quadrupole transition. By amplitude pulse shaping and compensation of ac-Stark shifts we achieve a fast gate operation without compromising the error rate. Subjecting different input states to concatenations of up to 21 individual gate operations reveals Bell state fidelities above 0.80. In principle, the entangling gate does not require ground state cooling of the ions as long as the Lamb–Dicke criterion is fulfilled. We present the first experimental evidence for this claim and create Bell states with a fidelity of 0.974(1) for ions in a thermal state of motion with a mean phonon number of in the mode coupling to the ions' internal states.
We implement a 2-qubit entangling Mølmer-Sørensen interaction by transporting two cotrapped ^{40}Ca^{+} ions through a stationary, bichromatic optical beam within a surface-electrode Paul trap. We describe a procedure for achieving a constant Doppler shift during the transport, which uses fine temporal adjustment of the moving confinement potential. The fixed interaction duration of the ions transported through the laser beam as well as the dynamically changing ac Stark shift require alterations to the calibration procedures used for a stationary gate. We use the interaction to produce Bell states with fidelities commensurate to those of stationary gates performed in the same system. This result establishes the feasibility of actively incorporating ion transport into quantum information entangling operations.
The high-fidelity multi-ion entangled states and quantum gates are the basis for trapped-ion quantum computing. Among the developed quantum gate schemes, Mølmer-Sørensen gate is a relatively mature experimental technique to realize multi-ion entanglement and quantum logic gates. In recent years, there have also been schemes to realize ultrafast quantum entanglement and quantum logic gates that operate outside the Lamb-Dicke regime by designing ultrafast laser pulse sequences. In such a many-body quantum system, these entanglement gates couple the spin states between ions by driving either the phonon energy level or the motional state of the ion chain. To improve the fidelity of quantum gates, the modulated laser pulses or the appropriately designed pulse sequences are applied to decouple the multi-mode motional states. In this review, we summarize and analyze the essential aspects of realizing these entanglement gates from both theoretical and experimental points of view. We also reveal that the basic physical process of realizing quantum gates is to utilize nonlinear interactions in non-equilibrium processes through driving the motional states of an ion chain with laser fields.
This thesis describes schemes for both a fast two-qubit gate operation and the steady-state preparation of a Bell state with trapped ions. A critical figure of merit for quantum computing with trapped ions is the gate duration relative to the decoherence timescale. We propose a fast gate scheme that off ers improvements in time, fidelity and simplicity of implementation over existing fast gate proposals. Our scheme can operate on both neighbouring and non-neighbouring ions in a long ion crystal. This provides a simpler and faster mechanism than traditional gates for complex quantum computing operations on large numbers of ions. The scheme achieves fidelities well above quantum error correction thresholds around 0.0001, and operates arbitrarily fast given arbitrary laser repetition rates. The production of these ultra-fast pulses is an experimental challenge, and fast gates have not yet been implemented; we present an implementation scheme using pulse splitting to provide a higher repetition rate and the pulse timing freedoms required for the gate scheme. We also analyse the effects of errors in the pulses on the gate operation. We analyse another strategy to generate entanglement using a driven dissipative process. Typically, environmental couplings cause decoherence. However, by combining dissipative dynamics with suitably chosen Hamiltonian evolution, the system can be steered to the desired steady states. Our steady-state scheme prepares a maximally-entangled Bell state with fidelity above 0.99, much higher than for schemes implemented with trapped ions. The driven dissipation continuously pumps the system towards the antisymmetric Bell steady-state, which is dark to the system dynamics and robust to parameter variations. The dominant loss mechanism is anomalous heating of the motional modes, reducing our fidelity by less than 0.01 for current experimental rates. Our scheme jointly addresses the ions and does not use sympathetic cooling. We enhance our scheme by combining the dissipative state preparation with the detection of photons, and obtain a significant fidelity enhancement.
Entanglement generation in trapped-ion systems has relied thus far on two distinct but related geometric phase gate techniques: Mølmer-Sørensen and light-shift gates. We recently proposed a variant of the light-shift scheme where the qubit levels are separated by an optical frequency [B. C. Sawyer and K. R. Brown, Phys. Rev. A 103, 022427 (2021)PLRAAN2469-992610.1103/PhysRevA.103.022427]. Here we report an experimental demonstration of this entangling gate using a pair of ^{40}Ca^{+} ions in a cryogenic surface-electrode ion trap and a commercial, high-power, 532 nm Nd:YAG laser. Generating a Bell state in 35 μs, we directly measure an infidelity of 6(3)×10^{-4} without subtraction of experimental errors. The 532 nm gate laser wavelength suppresses intrinsic photon scattering error to ∼1×10^{-5}.
A single 40 Ca + -ion is trapped and laser-cooled to its motional ground state. Laser radiation which couples off-resonantly to a motional sideband of the ion's S 1/2 to D 5/2 transition causes a phase shift proportional to the ion's motional quantum state | n ⟩. As the phase shift is conditional upon the ion's motion, we are able to demonstrate a universal 2-qubit quantum gate operation where the electronic target state { S , D } is flipped depending on the motional qubit state | n ⟩ = {|0⟩,|1⟩}. Finally, we discuss scaling properties of this universal quantum gate for linear ion crystals and present numerical simulations for the generation of a maximally entangled state of five ions.
Abstract We propose and study ways speeding up of the entangling operations in the trapped ions system with high fidelity. First, we find a scheme to increase the speed of a two-qubit gate without the limitation of trap frequency, which was considered as the fundamental limit. Second, we study the fast gate scheme for entangling more than two qubits simultaneously. We apply the method of applying multiple frequency components on laser beams for the gate operations. In particular, in order to avoid infinite terms from the coupling to carrier transition, we focus on the phase-insensitive gate scheme here. We carefully study the effect of large excitation of motional mode beyond the limit of Lamb–Dicke approximation by including up to second order terms of the Lamb–Dicke parameter. We study the speed limit of multi-qubit global entangling gates without individual addressing requirements. Furthermore, our gates can be made insensitive to the fluctuation of initial motional phases which are difficult to stabilise in the phase-insensitive gate scheme.