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arXiv:1308.4731 · Physical Review Letters

Quantum Control in the Cs 6S1/2 Ground Manifold Using Radio-Frequency and Microwave Magnetic Fields

Aaron C.T. Smith, Brian E. Anderson, Hector Sosa Martinez, Carlos A. Riofrío, Ivan Deutsch, Poul Jessen

Published Oct 24, 2013• 62 citations
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Abstract

We implement arbitrary maps between pure states in the 16-dimensional Hilbert space associated with the ground electronic manifold of ^{133}Cs. This is accomplished by driving atoms with phase modulated radio-frequency and microwave fields, using modulation waveforms found via numerical optimization and designed to work robustly in the presence of imperfections. We evaluate the performance of a sample of randomly chosen state maps by randomized benchmarking, obtaining an average fidelity >99%. Our protocol advances state-of-the-art quantum control and has immediate applications in quantum metrology and tomography.

DOI
10.1103/physrevlett.111.170502

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We implement arbitrary maps between pure states in the 16-dimensional Hilbert space associated with the ground electronic manifold of ^{133}Cs. This is accomplished by driving atoms with phase modulated radio-frequency and microwave fields, using modulation waveforms found via numerical optimization and designed to work robustly in the presence of imperfections. We evaluate the performance of a sample of randomly chosen state maps by randomized benchmarking, obtaining an average fidelity >99%. Our protocol advances state-of-the-art quantum control and has immediate applications in quantum metrology and tomography.
Quantum systems with Hilbert space dimension greater than two (qudits) provide an alternative to qubits as carriers of quantum information, and may prove advantageous for quantum information tasks if good laboratory tools for qudit manipulation and readout can be developed. We have implemented a protocol for arbitrary unitary transformations in the 16 dimensional hyperfine ground manifold of Cesium 133 atoms, using phase modulated rf and microwave magnetic fields to drive the atomic evolution. Our phase modulation waveforms are designed numerically using a variant of the highly efficient GRAPE algorithm. The fidelity of the resulting transformations is verified experimentally through randomized benchmarking, which indicates an average fidelity better than 97% across a sample of random unitaries.
We study quantum control of the full hyperfine manifold in the ground-electronic state of alkali-metal atoms based on applied radio frequency and microwave fields. Such interactions should allow essentially decoherence-free dynamics and the application of techniques for robust control developed for NMR spectroscopy. We establish the conditions under which the system is controllable in the sense that one can generate an arbitrary unitary map on the system. We apply this to the case of 133Cs^{133}\mathrm{Cs}133Cs with its d=16d=16d=16 dimensional Hilbert space of magnetic sublevels in the 6S1∕26{S}_{1∕2}6S1∕2​ state, and design control wave forms that generate an arbitrary target state from an initial fiducial state. We develop a generalized Wigner function representation for this space consisting of the direct sum of two irreducible representations of SU(2), allowing us to visualize these states. The performance of different control scenarios is evaluated based on the ability to generate a high-fidelity operation in an allotted time with the available resources. We find good operating points commensurate with modest laboratory requirements.
Quantum control in large dimensional Hilbert spaces is essential for realizing the power of quantum information processing. For closed quantum systems the relevant input/output maps are unitary transformations, and the fundamental challenge becomes how to implement these with high fidelity in the presence of experimental imperfections and decoherence. For two-level systems (qubits) most aspects of unitary control are well understood, but for systems with Hilbert space dimension d>2 (qudits), many questions remain regarding the optimal design of control Hamiltonians and the feasibility of robust implementation. Here we show that arbitrary, randomly chosen unitary transformations can be efficiently designed and implemented in a large dimensional Hilbert space (d=16) associated with the electronic ground state of atomic 133Cs, achieving fidelities above 0.98 as measured by randomized benchmarking. Generalizing the concepts of inhomogeneous control and dynamical decoupling to d>2 systems, we further demonstrate that these qudit unitary maps can be made robust to both static and dynamic perturbations. Potential applications include improved fault-tolerance in universal quantum computation, nonclassical state preparation for high-precision metrology, implementation of quantum simulations, and the study of fundamental physics related to open quantum systems and quantum chaos.
Neutral atom systems are an appealing platform for the development and testing of quantum control and measurement techniques. This dissertation presents experimental investigations of control and measurement tools using as a testbed the 16-dimensional hyperfine manifold associated with the electronic ground state of 133^{133}133Cs atoms. On the control side, we present an experimental realization of a protocol to implement robust unitary transformations in the presence of static and dynamic perturbations. We also present an experimental realization of inhomogeneous quantum control. Specifically, we demonstrate our ability to perform two different unitary transformations on atoms that see different light shifts from an optical addressing field. On the measurement side, we present experimental realizations of quantum state and process tomography. The state tomography project encompasses a comprehensive evaluation of several measurement strategies and state estimation algorithms. Our experimental results show that in the presence of experimental imperfections, there is a clear tradeoff between accuracy, efficiency and robustness in the reconstruction. The process tomography project involves an experimental demonstration of efficient reconstruction by using a set of intelligent probe states. Experimental results show that we are able to reconstruct unitary maps in Hilbert spaces with dimension ranging from d=4d = 4d=4 to d=16d = 16d=16. To the best of our knowledge, this is the first time that a unitary process in d=16d = 16d=16 is successfully reconstructed in the laboratory.
Cold atomic spins are a great platform for developing and testing control and measurement techniques. This thesis presents experimental investigations into quantum control and measurement using laser cooled cesium atoms. On the control side, we present an experimental realization of a protocol to achieve full controllability of the entire hyperfine ground manifold of cesium. In particular, we demonstrate the ability to map between arbitrary states with fidelity greater than 0.99, using a combination of static, radio frequency, and microwave magnetic fields. On the measurement side, we present an experimental realization of quantum state tomography. The tomography protocol begins by measuring expectation values of an informationally complete set of observables using a weak optical probe in combination with dynamical control. The measurement record is processed using two different state estimation algorithms, allowing us to estimate a quantum state with fidelity greater than 0.9.
Neutral atom systems are an appealing platform for the development and testing of quantum control and measurement techniques. This dissertation presents experimental investigations of control and measurement tools using as a testbed the 16-dimensional hyperfine manifold associated with the electronic ground state of 133^{133}133Cs atoms. On the control side, we present an experimental realization of a protocol to implement robust unitary transformations in the presence of static and dynamic perturbations. We also present an experimental realization of inhomogeneous quantum control. Specifically, we demonstrate our ability to perform two different unitary transformations on atoms that see different light shifts from an optical addressing field. On the measurement side, we present experimental realizations of quantum state and process tomography. The state tomography project encompasses a comprehensive evaluation of several measurement strategies and state estimation algorithms. Our experimental results show that in the presence of experimental imperfections, there is a clear tradeoff between accuracy, efficiency and robustness in the reconstruction. The process tomography project involves an experimental demonstration of efficient reconstruction by using a set of intelligent probe states. Experimental results show that we are able to reconstruct unitary maps in Hilbert spaces with dimension ranging from d=4d = 4d=4 to d=16d = 16d=16. To the best of our knowledge, this is the first time that a unitary process in d=16d = 16d=16 is successfully reconstructed in the laboratory.