Section: New Results

System theory approach of some quantum systems

Participants : Hadis Amini, Zaki Leghtas, Mazyar Mirrahimi, Pierre Rouchon.

Most of this work is done in close collaboration with the Pierre Aigrain laboratory (LPA) at ENS Paris and the Quantronics Laboratory (Qlab) of Michel Devoret and the Rob Schoelkopf Lab at Yale University.

Modern scientific and technologic requirements have led the theoretical and experimental research toward an engineering of quantum systems. The technologies that are proposed or developed include nano-scale electromechanical devices, tools for implementing quantum computation and quantum communication, NMR applications, quantum chemistry synthesis, high-resolution sensors, etc. The recent theoretical and experimental researches have shown that the quantum dynamics can be studied in the framework of the theory of estimation and control of systems, but give place to models that are not completely explored yet.

Our activities lie in the theoretical and experimental interface of this progressing field of research with an accent on the applications in quantum information and computation as well as high-precision metrology. By focusing on two different but similar types of experimental setups, consisting of cavity quantum electro-dynamical systems and quantum Josephson circuits, we aim in preparing highly non-classical states of a micro-wave field and protect these states against decoherence. Two different approaches are considered: 1- real-time measurement, quantum filtering and feedback ; 2- dissipation engineering also called reservoir engineering. Through the first methodology, we try to propose new experimental feedback protocols based on a fast real-time processing of measurement signal, followed by a state estimation applying the filtered signal and finally designing simple feedback laws based on the estimated state. The second methodology consists of designing new quantum circuit schemes that allow to orient the system's coupling to its environment in such a way that evacuates the undesired entropy induced by un-controlled noise sources.

Measurement based feedback

We have developed the mathematical methods underlying a recent quantum feedback experiment stabilizing photon-number states [17] , [30] , [29] , [24] . We consider a controlled system whose quantum state, a finite dimensional density operator, is governed by a discrete-time nonlinear Markov process. In open-loop, the measurements are assumed to be quantum non-demolition (QND) measurements. This Markov process admits a set of stationary pure states associated to an orthonormal basis. These stationary states provide martingales crucial to prove the open-loop stability: under simple assumptions, almost all trajectories converge to one of these stationary states; the probability to converge to a stationary state is given by its overlap with the initial quantum state. From these open-loop martingales, we construct a supermartingale whose parameters are given by inverting a Metzler matrix characterizing the impact of the control input on the Kraus operators defining the Markov process. This supermartingale measures the "distance" between the current quantum state and the goal state chosen from one of the open-loop stationary pure states. At each step, the control input minimizes the conditional expectation of this distance. It is proven that the resulting feedback scheme stabilizes almost surely towards the goal state whatever the initial quantum state. This state feedback takes into account a known constant delay of arbitrary length in the control loop. This control strategy is proved to remain also convergent when the state is replaced by its estimate based on a quantum filter. It relies on measurements that can be corrupted by random errors with conditional probabilities described by a known left stochastic matrix. Closed-loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box [29] .

We have also investigated the stabilization of the dynamical state of a superconducting qubit. In a series of papers, A. Korotkov and his co-workers suggested that continuous weak measurement of the state of a qubit and applying an appropriate feedback on the amplitude of a Rabi drive, should maintain the coherence of the Rabi oscillations for arbitrary time. Here, in the aim of addressing a metrological application of these persistent Rabi oscillations, we explore a new variant of such strategies. This variant is based on performing strong measurements in a discrete manner and using the measurement record to correct the phase of the Rabi oscillations. Noting that such persistent Rabi oscillations can be viewed as an amplitude- to-frequency convertor (converting the amplitude of the Rabi microwave drive to a precise frequency), we propose another feedback layer consisting of a simple analog phase locked loop to compensate the low frequency deviations in the amplitude of the Rabi drive [60] .

Dissipation engineering

We have introduced a new quantum gate that transfers an arbitrary state of a qubit into a superposition of two quasi-orthogonal coherent states of a cavity mode, with opposite phases. This qcMAP gate is based on conditional qubit and cavity operations exploiting the energy level dispersive shifts, in the regime where they are much stronger than the cavity and qubit linewidths [77] , [26] . The generation of multi-component superpositions of quasi-orthogonal coherent states, non-local entangled states of two resonators and multi-qubit GHZ states can be efficiently achieved by this gate. We also propose a new method, based on the application of this gate, to autonomously correct for errors of a logical qubit induced by energy relaxation. This scheme encodes the logical qubit as a multi-component superposition of coherent states in a harmonic oscillator. The error correction is performed by transferring the entropy to an ancila qubit and reseting the qubit. We layout in detail how to implement these operations in a practical system [78] . This proposal directly addresses the task of building a hardware-efficient and technically realizable quantum memory [78] .

We have also studied the application of dissipation engineering techniques to perform a high-performance and fast qubit reset. Qubit rest is crucial at the start of and during quantum information algorithms. Our protocol, nicknamed DDROP (Double Drive Reset of Population) is experimentally tested on a superconducting transmon qubit and achieves a ground state preparation of at least 99.5$\%$ in times less than 3$\mu$s; faster and higher fidelity are predicted upon parameter optimization [74] . We are currently working on extending our protocol to prepare and protect two-qubit entangled states and to perform autonomous quantum error correction.