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arXiv:1012.3477 · Journal of Physics B Atomic Molecular and Optical Physics

Quantum tomography of the full hyperfine manifold of atomic spins via continuous measurement on an ensemble

Carlos A. Riofrío, Poul Jessen, Ivan Deutsch

Published Jul 25, 2011• 28 citations
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Abstract

Quantum state reconstruction based on weak continuous measurement has the advantage of being fast, accurate and almost non-perturbative. In this work we present a pedagogical review of the protocol proposed by Silberfarb et al (2005 Phys. Rev. Lett. 95 030402), whereby an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner so as to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time through a maximum-likelihood estimate. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sublevels of a large-spin alkali atom, 133 Cs. We detail two different protocols for control. Using magnetic interactions and a quadratic ac Stark shift, we can reconstruct a chosen hyperfine manifold F , e.g. the seven-dimensional F = 3 manifold in the electronic ground state of Cs. We review the procedure as implemented in experiments (Smith et al 2006 Phys. Rev. Lett. 97 180403). We extend the protocol to the more ambitious case of reconstruction of states in the full 16-dimensional electronic ground subspace ( F = 3⊕ F = 4), controlled by microwaves and radio-frequency (RF) magnetic fields. We give detailed derivations of all physical interactions, approximations, numerical methods and fitting procedures, tailored to the realistic experimental setting. For the case of light-shift and magnetic control, reconstruction fidelities of ∼0.95 have been achieved, limited primarily by inhomogeneities in the light-shift. For the case of microwave/RF-control we simulate fidelity >0.97, limited primarily by signal-to-noise.

DOI
10.1088/0953-4075/44/15/154007

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Quantum state reconstruction based on weak continuous measurement has the advantage of being fast, accurate and almost non-perturbative. In this work we present a pedagogical review of the protocol proposed by Silberfarb et al (2005 Phys. Rev. Lett. 95 030402), whereby an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner so as to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time through a maximum-likelihood estimate. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sublevels of a large-spin alkali atom, 133 Cs. We detail two different protocols for control. Using magnetic interactions and a quadratic ac Stark shift, we can reconstruct a chosen hyperfine manifold F , e.g. the seven-dimensional F = 3 manifold in the electronic ground state of Cs. We review the procedure as implemented in experiments (Smith et al 2006 Phys. Rev. Lett. 97 180403). We extend the protocol to the more ambitious case of reconstruction of states in the full 16-dimensional electronic ground subspace ( F = 3⊕ F = 4), controlled by microwaves and radio-frequency (RF) magnetic fields. We give detailed derivations of all physical interactions, approximations, numerical methods and fitting procedures, tailored to the realistic experimental setting. For the case of light-shift and magnetic control, reconstruction fidelities of ∼0.95 have been achieved, limited primarily by inhomogeneities in the light-shift. For the case of microwave/RF-control we simulate fidelity >0.97, limited primarily by signal-to-noise.
Quantum state tomography is a fundamental tool in quantum information processing. It allows us to estimate the state of a quantum system by measuring different observables on many identically prepared copies of the system. This is, in general, a very time-consuming task that requires a large number of measurements. There are, however, systems in which the data acquisition can be done more efficiently. In fact, an ensemble of quantum systems can be prepared and manipulated by external fields while being continuously and collectively probed, producing enough information to estimate its state. This provides a basis for continuous measurement quantum tomography. In this protocol, an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time. We use two different estimation methods: maximum likelihood and compressed sensing. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sub-levels of a large-spin alkali atom, 133{}^{133}133Cs. We apply this protocol to the case of reconstruction of states in the full 16-dimensional electronic-ground subspace (F=3⊕F=4F=3 \oplus F=4F=3⊕F=4), controlled by microwaves and radio-frequency magnetic fields. We present an experimental demonstration of continuous measurement quantum tomography in an ensemble of cold cesium atoms with full control of its 16-dimensional Hilbert space. We show the exquisite level of control achieved in the lab and the excellent agreement between the theory discussed in this dissertation and the experimental results. This allows us to achieve fidelities >95% for low complexity quantum states, and >92% for arbitrary random states, which is a formidable accomplishment for a space of this size.
Quantum state tomography is a fundamental tool in quantum information processing. It allows us to estimate the state of a quantum system by measuring different observables on many identically prepared copies of the system. This is, in general, a very time-consuming task that requires a large number of measurements. There are, however, systems in which the data acquisition can be done more efficiently. In fact, an ensemble of quantum systems can be prepared and manipulated by external fields while being continuously and collectively probed, producing enough information to estimate its state. This provides a basis for continuous measurement quantum tomography. In this protocol, an ensemble of identically prepared systems is collectively probed and controlled in a time-dependent manner to create an informationally complete continuous measurement record. The measurement history is then inverted to determine the state at the initial time. We use two different estimation methods: maximum likelihood and compressed sensing. The general formalism is applied to the case of reconstruction of the quantum state encoded in the magnetic sub-levels of a large-spin alkali atom, 133{}^{133}133Cs. We apply this protocol to the case of reconstruction of states in the full 16-dimensional electronic-ground subspace (F=3⊕F=4F=3 \oplus F=4F=3⊕F=4), controlled by microwaves and radio-frequency magnetic fields. We present an experimental demonstration of continuous measurement quantum tomography in an ensemble of cold cesium atoms with full control of its 16-dimensional Hilbert space. We show the exquisite level of control achieved in the lab and the excellent agreement between the theory discussed in this dissertation and the experimental results. This allows us to achieve fidelities >95% for low complexity quantum states, and >92% for arbitrary random states, which is a formidable accomplishment for a space of this size.
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.
Laboratory techniques to manipulate and observe ultracold atoms make these an attractive platform for testing new ideas in quantum control and measurement. Over the last decade we have revisited the tensor interaction between light fields and multilevel atoms, and have developed a theoretical framework suited for applications in quantum control and measurement (see [1] for a review). One important finding is that the combined action of a light shift and magnetic field on an atomic ground state can be used to implement a non-linear Hamiltonian for a hyperfine spin, and that its action can lead to nonlinear spin dynamics such as wavepacket collapse and revival. Using concepts from classical control theory it is straightforward to show that this Hamiltonian is sufficiently general for full control of an arbitrarily large spin. On this foundation we have developed a new protocol for quantum state reconstruction, based on continuous weak measurement of a spin observable during carefully designed coherent evolution [2]. We have further used our tools for control and state reconstruction to implement and verify protocols for optimal control of Cs hyperfine spns (Fig. 1), showing that an initial fiducial state can be transformed into any desired target state with a fidelity in the 80-90% range [3].
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.
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.