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arXiv:1008.5197 · Physical Review A

Dynamics of the collective modes of an inhomogeneous spin ensemble in a cavity

Janus H. Wesenberg, Z. Kurucz, Klaus Mølmer

Published Feb 25, 2011• 15 citations
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Abstract

We study the excitation dynamics of an inhomogeneously broadened spin ensemble coupled to a single cavity mode. The collective excitations of the spin ensemble can be described in terms of generalized spin waves, and, in the absence of the cavity, the free evolution of the spin ensemble can be described as a drift in the wavenumber without dispersion. In this article we show that the dynamics in the presence of coupling to the cavity mode can be described solely by a modified time evolution of the wavenumbers. In particular, we show that collective excitations with a well-defined wavenumber pass without dispersion from negative to positive-valued wavenumbers without populating the zero wavenumber spin wave mode. The results are relevant for multimode collective quantum memories where qubits are encoded in different spin waves.

DOI
10.1103/physreva.83.023826

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We study the excitation dynamics of an inhomogeneously broadened spin ensemble coupled to a single cavity mode. The collective excitations of the spin ensemble can be described in terms of generalized spin waves, and, in the absence of the cavity, the free evolution of the spin ensemble can be described as a drift in the wavenumber without dispersion. In this article we show that the dynamics in the presence of coupling to the cavity mode can be described solely by a modified time evolution of the wavenumbers. In particular, we show that collective excitations with a well-defined wavenumber pass without dispersion from negative to positive-valued wavenumbers without populating the zero wavenumber spin wave mode. The results are relevant for multimode collective quantum memories where qubits are encoded in different spin waves.
We study the dynamics of a spin ensemble strongly coupled to a single-mode resonator driven by external pulses. When the mean frequency of the spin ensemble is in resonance with the cavity mode, damped Rabi oscillations are found between the spin ensemble and the cavity mode which we describe very accurately, including the dephasing effect of the inhomogeneous spin broadening. We demonstrate that a precise knowledge of this broadening is crucial both for a qualitative and a quantitative understanding of the temporal spin-cavity dynamics. On this basis we show that coherent oscillations between the spin ensemble and the cavity can be enhanced by a few orders of magnitude, when driving the system with pulses that match special resonance conditions. Our theoretical approach is tested successfully with an experiment based on an ensemble of negatively charged nitrogen-vacancy centers in diamond strongly coupled to a superconducting coplanar single-mode waveguide resonator.
The enhanced collective coupling to weak quantum fields may turn atomic or spin ensembles into an important component in quantum information processing architectures. Inhomogeneous broadening can, however, significantly reduce the coupling and the lifetime of the collective excitation that represent the quantum information. In this paper we show that the width and shape of the inhomogeneous broadening have a striking influence on the dynamics of the cavity-ensemble system and may lead to narrowing of the linewidth of the collective states. We underpin our findings with the examples of a Gaussian and a Lorentzian profile of the inhomogeneity.
In the last couple of years much effort has been dedicated to the development and the investigation of hybrid quantum devices, which combine the advantages of very dissimilar quantum systems for the realization of efficient quantum computation and communication technologies. Particularly interesting in this regard are solid-state quantum memories based on ensembles of spins (e.g. nitrogen vacancy defects in diamond or rare-earth doped crystals) coupled to superconducting microwave cavities. Here the spin-ensemble acts as a robust memory where the collective coupling to the cavity mode allows for the coherent transfer of quantum information. The major downside of ensembles inside solid-state systems is their natural tendency to exhibit inhomogeneous broadening of the transition frequencies, which makes experiments as well as their theoretical description very challenging. On the theoretical side it is primarily the computational complexity due to the exponential growth of the associated Hilbert space with the system size that inevitably calls for approximation schemes. To date, the theoretical description of large spin-cavity systems is mainly limited to weak driving fields, where the number of spin-excitations in the ensemble is negligible, corresponding to a Holstein-Primakoff approximation, or to mean-field approaches, where correlations within the system are neglected. In this thesis we aim to develop a model that accurately accounts for the dynamics of large inhomogeneously broadened spin ensembles, coupled to a single cavity mode, even in the case of strong driving fields. In particular we employ the generalized cumulant expansion method to account for correlations within the system, which allows us to go beyond the Holstein-Primakoff and mean-field approximations. As a first step, we demonstrate the applicability of the cumulant expansion technique to the well-studied Jaynes-Cummings model and investigate its main limitations. We then move on to the Tavis-Cummings model with multiple spins inside the cavity and generalize our approach to ensembles containing a very large number of spins (ca. 10 12). Our model aims not only to provide a proper description of ensemble-based quantum memories, but also to serve as a novel tool for the investigation of interesting new physics arising from cooperative effects in inhomogeneous ensembles.
Ensembles of quantum-mechanical spins offer a promising platform for quantum memories, but proper functionality requires accurate control of unavoidable system imperfections. We present an efficient control scheme for a spin ensemble strongly coupled to a single-mode cavity based on a set of Volterra equations relying solely on weak classical control pulses. The viability of our approach is demonstrated in terms of explicit storage and readout sequences that will serve as a starting point towards the realization of more demanding full quantum-mechanical optimal control schemes.
We consider the dynamics of a disordered ensemble of qubits interacting with a single-mode photon field, which is described by an exactly solvable inhomogeneous Dicke model. In particular, we concentrate on the crossover from few-qubit systems to the system of many qubits and analyze how the collective behavior of a coupled qubit-cavity system emerges despite the broadening. We show that quantum interference effects survive in the mesoscopic regime---the dynamics of an entangled Bell state encoded into the qubit subsystem remains highly sensitive to the symmetry of the total wave function. Moreover, relaxation of these states is slowed down due to the formation of collective dark states weakly coupled to light. Dark states also significantly influence the dynamics of the excitations of a photon subsystem by absorbing them into the qubit subsystem and releasing quasiperiodically in time. We argue that the predicted phenomena can be useful in quantum technologies based on superconducting qubits. For instance, they provide tools to deeply probe both collective and quantum properties of such artificial macroscopic systems.
We propose a multimode quantum memory protocol able to store the quantum state of the field in a microwave resonator into an ensemble of electronic spins. The stored information is protected against inhomogeneous broadening of the spin ensemble by spin-echo techniques resulting in memory times orders of magnitude longer than previously achieved. By calculating the evolution of the first and second moments of the spin-cavity system variables for current experimental parameters, we show that a memory based on nitrogen vacancy center spins in diamond can store a qubit encoded on the |0> and |1> Fock states of the field with 80% fidelity and outperform classical memory strategies for storage times ≤69 μs.