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Physical Review A

Geometric manipulation of the quantum states of two-level atoms

Ming Tian, Zeb W. Barber, Joe A. Fischer, Wm. Randall Babbitt

Published May 6, 2004• 45 citations
Publisher

Abstract

Manipulation of the quantum states of two-level atoms has been investigated using laser-controlled geometric phase change, which has the potential to build robust quantum logic gates for quantum computing. For a qubit based on two electronic transition levels of an atom, two basic quantum operations that can make any universal single qubit gate have been designed employing resonant laser pulses. An operation equivalent to a phase gate has been demonstrated using ${\mathrm{Tm}}^{3+}$ doped in a yttrium aluminum garnet crystal.

DOI
10.1103/physreva.69.050301

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Physical Review A•Jul 23, 2007•9 citations
Manipulation of the quantum states of two-level atoms has been investigated using laser-controlled geometric phase change, which has the potential to build robust quantum logic gates for quantum computing. For a qubit based on two electronic transition levels of an atom, two basic quantum operations that can make any universal single qubit gate have been designed employing resonant laser pulses. An operation equivalent to a phase gate has been demonstrated using Tm3+{\mathrm{Tm}}^{3+}Tm3+ doped in a yttrium aluminum garnet crystal.
With carefully prepared resonant laser-controlled pulses, the two quantum states of a cold ensemble of sodium atoms were manipulated by pure geometric rotations of the Bloch vector around axes 2 and 3 on the Bloch sphere. The fidelity for two geometric rotations was improved to be 90%. A universal single-qubit operation on the ground-state wave function was demonstrated by a pure geometric operation and the phase shift due to the rotation around axis 3 was measured using a geometric atom interferometer.
A fundamental challenge of quantum computation is being able to scale up a large number of high fidelity quantum gates while noise and error are affecting the gate’s physical control parameters. This dissertation focuses on the fidelity of single-qubit quantum gates constructed by a change in quantum geometric phase, while the control parameters are affected by random noise and systematic error. A unified model of geometric quantum computation is developed, in which a qubit state is controlled by a composite Hamiltonian, resulting in a multiple-segment rotation of the quantum state and allowing characterization of evolution paths depending on the associated geometric and dynamic phase. The fidelity of the quantum gates in the presence of different noise error is compared for pu rely geometric, hybrid (having both geometric and dynamic phase), and conventional dynamic quantum gates built on single Hamiltonians. Results showed hybrid quantum gates had the highest fidelities, followed by geometric gates, and conventional dynamic gates had the lowest fidelities. In addition, there was indication in some cases higher fidelities result from gates created from a larger number of segments in the quantum state rotation. These results can be understood by the relation of the control parameters with the evolution path geometry. By translating between control parameters, our model can be applied to different systems for quantum computation, including: the laser manipulation of a two-level atom, laser manipulation of trapped ions, nuclear magnetic resonance, polarization states of photons, superconducting qubits in cavity QED, and quantum dots.
We propose a scheme for the realization of unconventional geometric two-qubit phase gates with two identical two-level ions. In the present scheme, the two ions are simultaneously illuminated by a standing-wave laser pulse with its pulse frequency being tuned to the ionic transition. The gate operation time can be much shorter, making the system robust against decoherence. In addition, we choose the appropriate experimental parameters to construct the geometric phase gate in one step, and thus avoid implementing the pure geometric single qubit operation.
Multiple-qubit quantum logic gates are an important element in the implementation of quantum computers. The direct construction of multiple-qubit quantum logic gates in an efficient way has important values compared to the construction of multiple-qubit gates using a series of two-qubit and single-qubit gates. We propose a scheme to construct a multiple-qubit CkU{\mathrm{C}}_{k}\mathrm{U}Ck​U gate (kkk denotes the number of control qubits and U means the arbitrary universal operation performed on the target qubit) in a neutral atom platform through the Rydberg blockade effect by successively exciting them to Rydberg states. This scheme takes advantage of the shortcut to adiabaticity of inverse engineering, geometric quantum operations, as well as optimized control theory. The geometric quantum computation considered in this manuscript guarantees the robustness to operational errors. Meanwhile, inverse engineering-based shortcut to adiabaticity provides a further advantage in terms of the speed of the system evolution compared to adiabatic processes. An additional feature of our multiple-qubit quantum logic gate is that arbitrary operation on the target atom can be realized by adjusting the amplitude and phase of the laser fields. Numerical simulation of the master equation based on the full Hamiltonian demonstrates the high fidelity of the proposed scheme and its robustness to operational errors and spontaneous emission.
The geometric manipulation of the quantum states of two-level atoms with two laser-controlled resonant $\ensuremath{\pi}$ pulses with a relative phase difference was evaluated using a time-domain atom interferometer comprising a cold sodium atom ensemble and two copropagating two-photon Raman pulses. It was verified that the relative phase shift between two states is double the control phase difference and that the rotation around the axis 3 of the Bloch sphere can be controlled by geometric manipulation.