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arXiv:2503.08230 · Physical Review Research

Unitary transformations using robust optimal control on a cold atom qudit

E. Flament, Nicolas Ombredane, Floriane Arrouas, Dominique Ronco, Bruno Peaudecerf, Dominique Sugny, David Guéry-Odelin

Published May 12, 2025• 2 citations
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Abstract

In this paper, we design and experimentally implement various robust quantum unitary transformations (gates) acting on d -dimensional vectors (qudits) by tuning a single control parameter using optimal control theory. The quantum state is represented by the momentum components of a Bose-Einstein condensate placed in an optical lattice, with the lattice position varying over a fixed duration serving as the control parameter. To evaluate the quality of these transformations, we employ standard quantum process tomography. In addition, we show how controlled unitary transformations can be used to extend state stabilization to global stabilization within a controlled vector subspace. Finally, we apply them to state tomography, showing how the information about the relative phase between distant momentum components can be extracted by inducing an interference process.

DOI
10.1103/fvlj-yph6

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In this paper, we design and experimentally implement various robust quantum unitary transformations (gates) acting on d -dimensional vectors (qudits) by tuning a single control parameter using optimal control theory. The quantum state is represented by the momentum components of a Bose-Einstein condensate placed in an optical lattice, with the lattice position varying over a fixed duration serving as the control parameter. To evaluate the quality of these transformations, we employ standard quantum process tomography. In addition, we show how controlled unitary transformations can be used to extend state stabilization to global stabilization within a controlled vector subspace. Finally, we apply them to state tomography, showing how the information about the relative phase between distant momentum components can be extracted by inducing an interference process.
Quantum technologies will ultimately require manipulating many-body quantum systems with high precision. Cold atom experiments represent a stepping stone in that direction: a high degree of control has been achieved on systems of increasing complexity. However, this control is still sub-optimal. In many scenarios, achieving a fast transformation is crucial to fight against decoherence and imperfection effects. Optimal control theory is believed to be the ideal candidate to bridge the gap between early stage proof-of-principle demonstrations and experimental protocols suitable for practical applications. Indeed, it can engineer protocols at the quantum speed limit - the fastest achievable timescale of the transformation. Here, we demonstrate such potential by computing theoretically and verifying experimentally the optimal transformations in two very different interacting systems: the coherent manipulation of motional states of an atomic Bose-Einstein condensate and the crossing of a quantum phase transition in small systems of cold atoms in optical lattices. We also show that such processes are robust with respect to perturbations, including temperature and atom number fluctuations.
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.
We report on the efficient design of quantum optimal-control protocols to manipulate the motional states of an atomic Bose-Einstein condensate (BEC) in a one-dimensional optical lattice.Our protocols operate on the momentum comb associated with the lattice.In contrast to previous works also dealing with control in discrete and large Hilbert spaces, our control schemes allow us to reach a wide variety of targets by varying a single parameter, the lattice position.With this technique, we experimentally demonstrate a precise, robust, and versatile control: we optimize the transfer of the BEC to a single or multiple quantized momentum states with full control on the relative phase between the different momentum components.This also allows us to prepare the BEC in a given eigenstate of the lattice band structure, or superposition thereof.
Quantum technologies will ultimately require manipulating many-body quantum systems with high precision.Cold atom experiments represent a stepping stone in that direction: a high degree of control has been achieved on systems of increasing complexity.However, this control is still sub-optimal.In many scenarios, achieving a fast transformation is crucial to fight against decoherence and imperfection effects.Optimal control theory is believed to be the ideal candidate to bridge the gap between early stage proof-of-principle demonstrations and experimental protocols suitable for practical applications.Indeed, it can engineer protocols at the quantum speed limit -the fastest achievable timescale of the transformation.Here, we demonstrate such potential by computing theoretically and verifying experimentally the optimal transformations in two very different interacting systems: the coherent manipulation of motional states of an atomic Bose-Einstein condensate and the crossing of a quantum phase transition in small systems of cold atoms in optical lattices.We also show that such processes are robust with respect to perturbations, including temperature and atom number fluctuations.
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.
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.