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arXiv:quant-ph/0002024 · Physical Review A

Entanglement and quantum computation with ions in thermal motion

Anders Søndberg Sørensen, Klaus Mølmer

Published Jul 18, 2000• 787 citations
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Abstract

With bichromatic fields, it is possible to deterministically produce entangled states of trapped ions. In this paper we present a unified analysis of this process for both weak and strong fields, for slow and fast gates. Simple expressions for the fidelity of creating maximally entangled states of two or an arbitrary number of ions under nonideal conditions are derived and discussed.

DOI
10.1103/physreva.62.022311

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With bichromatic fields, it is possible to deterministically produce entangled states of trapped ions. In this paper we present a unified analysis of this process for both weak and strong fields, for slow and fast gates. Simple expressions for the fidelity of creating maximally entangled states of two or an arbitrary number of ions under nonideal conditions are derived and discussed.
In the ion trap quantum computer, quantum bits are represented by the internal state of the trapped ions, and the collective vibrations of the ions are exploited for the coherent manipulation of the state of several bits. We have developed a method to perform gates on different ions by illuminating the ions with bichromatic light. Our gates are independent of the motional state of the ions and they may be applied to ions in any state or, e.g., thermal mixture of vibrational states. Moreover, our proposal allows the production of multiparticle entangled states even in situations without experimental access to the individual ions in the trap. Maximally entangled states of any number of ions are produced by simply illuminating all ions with the same beams of bichromatic light.
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.
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.
A robust scheme is proposed for producing maximally entangled states for many trapped ions in thermal motion. In the scheme the ions are simultaneously illuminated by two standing-wave laser fields. During the operation the phases of the lasers are inverted, which not only cancels the vibration-dependent parts in the evolution operator, but also suppresses direct off-resonant coupling of the internal states. Thus, our scheme allows the production of entanglement for hot trapped ions with laser fields of high intensity, which makes the entanglement speed extremely high.
Trapped atomic ions are a promising medium for quantum computing, due to their long coherence times and potential for scalability. Current methods of entangling ions rely on addressing individual modes of motion within the trap and applying qubit state dependent forces with external fields. This approach can limit the speed of entangling gates and make them vulnerable to decoherence due to coupling to unwanted modes or ion heating. This thesis is directed towards demonstrating novel entanglement schemes which are not limited by the trap frequency, and can be made almost arbitrarily fast. Towards this goal, I report here on the first experiments using ultrafast laser pulses to control the internal and external states of a single trapped ion. I begin with experiments in ultrafast spin control, showing how a single laser pulse can be used to completely control both spin degrees of freedom of the ion qubit in tens of picoseconds. I also show how a train of weak pulses can be used to drive Raman transitions based on a frequency comb. I then discuss experiments using pulses to rapidly entangle the spin with the motion, and how careful spectral redistribution allows a single pulse to execute a spin-dependent momentum kick. Finally, I explain how these spin-dependent momentum kicks can be used in the future to create an ultrafast entangling gate. I go over how such a gate would work, and present experimentally realizable timing sequences which would create a maximally entangled state of two ions in a time faster than the period of motion in the trap.
Trapped atomic ions are a promising medium for quantum computing, due to their long coherence times and potential for scalability. Current methods of en-tangling ions rely on addressing individual modes of motion within the trap and applying qubit state dependent forces with external fields. This approach can limit the speed of entangling gates and make them vulnerable to decoherence due to coupling to unwanted modes or ion heating. This thesis is directed towards demon-strating novel entanglement schemes which are not limited by the trap frequency, and can be made almost arbitrarily fast. Towards this goal, I report here on the first experiments using ultrafast laser pulses to control the internal and external states of a single trapped ion. I begin with experiments in ultrafast spin control, showing how a single laser pulse can be used to completely control both spin degrees of freedom of the ion qubit in tens of picoseconds. I also show how a train of weak pulses can be used to drive Raman transitions based on a frequency comb. I then discuss exper-iments using pulses to rapidly entangle the spin with the motion, and how careful spectral redistribution allows a single pulse to execute a spin-dependent momentum kick. Finally, I explain how these spin-dependent momentum kicks can be used in the future to create an ultrafast entangling gate. I go over how such a gate would work, and present experimentally realizable timing sequences which would create a maximally entangled state of two ions in a time faster than the period of motion in the trap.