We describe how sideband cooling techniques may be applied to large spin ensembles in magnetic resonance. Using the Tavis-Cummings model in the presence of a Rabi drive, we solve a Markovian master equation describing the joint spin-cavity dynamics to derive cooling rates as a function of ensemble size. Our calculations indicate that the coupled angular momentum subspaces of a spin ensemble containing roughly 10(11) electron spins may be polarized in a time many orders of magnitude shorter than the typical thermal relaxation time. The described techniques should permit efficient removal of entropy for spin-based quantum information processors and fast polarization of spin samples. The proposed application of a standard technique in quantum optics to magnetic resonance also serves to reinforce the connection between the two fields, which has recently begun to be explored in further detail due to the development of hybrid designs for manufacturing noise-resilient quantum devices.
We describe how sideband cooling techniques may be applied to large spin ensembles in magnetic resonance. Using the Tavis-Cummings model in the presence of a Rabi drive, we solve a Markovian master equation describing the joint spin-cavity dynamics to derive cooling rates as a function of ensemble size. Our calculations indicate that the coupled angular momentum subspaces of a spin ensemble containing roughly 10(11) electron spins may be polarized in a time many orders of magnitude shorter than the typical thermal relaxation time. The described techniques should permit efficient removal of entropy for spin-based quantum information processors and fast polarization of spin samples. The proposed application of a standard technique in quantum optics to magnetic resonance also serves to reinforce the connection between the two fields, which has recently begun to be explored in further detail due to the development of hybrid designs for manufacturing noise-resilient quantum devices.
Hybrid quantum systems consisting of a collection of N spin- 1 / 2 particles uniformly interacting with an electromagnetic field, such as one confined in a cavity, are important for the development of quantum information processors and will be useful for metrology, as well as tests of collective behavior. Such systems are often modeled by the Tavis-Cummings model, and having an accurate understanding of the thermal behaviors of this system is needed to understand their behavior in realistic environments. We quantitatively show in this work that the Dicke subspace approximation is at times invoked too readily. Specifically, we show that there is a temperature above which the degeneracies in the system become dominant and the Dicke subspace is minimally populated. This transition occurs at a lower temperature than previously considered. In such a temperature regime, the key constants of the motion are the total excitation count between the spin system and cavity and the collective angular momentum of the spin system. These enable perturbative expansions for thermal properties in terms of the energy shifts of dressed states, called Lamb shifts herein. They enable efficient numeric methods that scale in terms of the size of the spin system. Notably the runtime to obtain certain parameters of the system scales as O ( N ) , and is thus highly efficient. These provide methods for approximating, and bounding, properties of these systems as well as characterizing the dominant population regions, including under perturbative noise. In the regime of stronger spin-spin coupling, the perturbations outweigh the expansion series terms, and inefficient methods must likely be employed, removing the computational efficiency of simulating such systems. The results in this work can also be used for related systems such as coupled-cavity arrays, cavity-mediated coupling of collective spin ensembles, and collective spin systems.
We numerically solve the Liouville equation for the Tavis Cummings model of multiple spins coupled to a lossless single mode cavity, starting from an initial condition with small numbers of fully polarized spins tipped by a specified angle, and the cavity in its ground Fock state. Time evolution of the magnetizations and cavity states, following small to medium nutation by a classical field, yields a microscopic quantum mechanical picture of radiation damping in magnetic resonance, and the formation of the free induction signal, that is, the transfer of Zeeman energy, via spin coherence, to cavity coherence. Although the motion of the Bloch vector is nonclassical, our quantum description is related to the macroscopic picture of NMR reception, by showing the close relationship between the usual radiation damping constant, and the quantum mechanical Rabi nutation frequency (as enhanced by cavity coupling and stimulated emission.) That is, each is the product, of a nutation rate per oscillator current, and a current. Although the current in the damping constant is explicitly limited by cavity losses, which do not enter the formula for the Rabi frequency, we nonetheless show (in an appendix) how these losses can be introduced into our problem by means of a master equation. Numerical solution of the classical Bloch-Kirchhoff equations reinforces the conclusion that the strength of the free induction
We study the energy level structure of the Tavis-Cumming model applied to an ensemble of independent magnetic spins s=1/2 coupled to a variable number of photons. Rabi splittings are calculated and their distribution is analyzed as a function of photon number nmax and spin system size N. A sharp transition in the distribution of the Rabi frequency is found at ${n}_{\text{max}}\ensuremath{\approx}N$. The width of the Rabi frequency spectrum diverges as N at this point. For increased number of photons nmax>N, the Rabi frequencies converge to a value proportional to nmax. This behavior is interpreted as analogous to the classical spin-resonance mechanism where the photon is treated as a classical field and one resonance peak is expected. We also present experimental data demonstrating cooperative, magnetic strong coupling between a spin system and photons, measured at room temperature. This points toward quantum computing implementation with magnetic spins, using cavity quantum-electrodynamics techniques.
We apply the methods of cavity quantum electrodynamics (CQED), to obtain a microscopic and fully quantum-mechanical picture of radiation damping in magnetic resonance, and the nascent formation of the free induction signal. Numerical solution of the Tavis-Cummings model - i.e., multiple spins 1∕2 coupled to a lossless single-mode cavity - shows in fine detail the transfer of Zeeman energy, via spin coherence, to excite the cavity - represented here by a quantized LC resonator. The case of a single spin is also solved analytically. Although the motion of the Bloch vector is non-classical, we nonetheless show that the quantum mechanical Rabi nutation frequency (as enhanced by cavity coupling and stimulated emission) gives realistic estimates of macroscopic signal strength and the radiation damping constant in nuclear magnetic resonance. We also show how to introduce dissipation: cavity losses by means of a master equation, and relaxation by the phenomenological method of Bloch. The failure to obtain the full Bloch equations (unless semi-classical conditions are imposed on the cavity) is discussed in light of similar issues arising in CQED (and in earlier work in magnetic resonance as well), as are certain problems relative to quantization of the electromagnetic near-field.
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.
An efficient technique to generate ensembles of spins that are highly polarized by external magnetic fields is the Holy Grail in Nuclear Magnetic Resonance (NMR) spectroscopy. Since spin-half nuclei have steady-state polarization biases that increase inversely with temperature, spins exhibiting high polarization biases are considered cool, even when their environment is warm. Existing spin-cooling techniques are highly limited in their efficiency and usefulness. Algorithmic cooling is a promising new spin-cooling approach that employs data compression methods in open systems. It reduces the entropy of spins on long molecules to a point far beyond Shannon's bound on reversible entropy manipulations, thus increasing their polarization. Here we present an efficient and experimentally feasible algorithmic cooling technique that cools spins to very low temperatures even on short molecules. This practicable algorithmic cooling could lead to breakthroughs in high-sensitivity NMR spectroscopy in the near future, and to the development of scalable NMR quantum computers in the far future. Moreover, while the cooling algorithm itself is classical, it uses quantum gates in its implementation, thus representing the first short-term application of quantum computing devices.