Technical Report MSC-2012-28

Title: Algorithmic Cooling of Spins by Optimal Control
Authors: Yosi Atia
Supervisors: Tal Mor
Abstract: Nuclear magnetic resonance (NMR) has proven to be a leading implementation of quantum information processors where each molecule in the sample constitutes a register of quantum bits (qubits) realized by spin half nuclei (hereinafter spins). The quantum logic gates are implemented by radio frequency waves transmitted on an ensemble of indistinguishable molecules, and by using the scalar coupling between the spins. The large number of molecules of the sample generates a detectable signal corresponding to the average state of the spin system. At room temperature and under a constant and homogeneous magnetic field typical for NMR, the nuclear spins are in a highly mixed state. The probability of a spin to be aligned with the magnetic field is slightly higher than the probability of the spin to be aligned against the magnetic field - in terms of information theory, the Shannon entropy of the spin state is close to 1. Using data compression tools, the entropy of the spin system can be manipulated, effectively cooling some spins while heating others. However this closed-system technique is limited by Shannon’s entropy conservation on reversible operations. Algorithmic cooling (AC) of spins counter intuitively utilizes opening the system to the heat bath as a method for cooling spins beyond Shannon’s bound. AC requires a spin system where some spins, called reset spins, thermalize significantly faster than other spins, called computation spins. Polarization compression or polarization exchange is applied to the spin system, manipulating some of the computation spins’ entropy to the reset spins, which quickly lose most of it to the environment. The process is similar to a heat engine and can be repeated, converging the system to a stable trajectory (limit-cycle) in the density-matrix space. The efficiency of AC is limited by the thermalization time ratios between the cooled computation spins and the reset spins. In this work, multiple rounds of AC were applied on 13C2-trichloroethylene (TCE), a three-spin system, at the Technion NMR lab, following the heat-bath cooling experiments conducted on TCE by my collaborators Mor, Elias, and Weinstein. We applied the first AC in liquid state NMR, in continuation of the pioneering AC experiment conducted in solid state NMR in 2005 by Baugh et al. The parameters of the Hamiltonian were measured from the spectrum and fed to GRAPE, an optimal control algorithm utilized to design efficient and robust radio frequency (RF) pulses required for AC. Our main result is cooling a single spin by a factor of 4.61±0.02 to 65K, bypassing the Shannon bound for the system (4.22). The result matches the prediction of our simulation given the system parameters and the pulses’ efficiency. In a second experiment, focused on maximizing the information content (IC) of the two carbons, we measured polarizations of 3.78±0.02 and 3.40±0.02, with information content of 25.9±0.2. In comparison, polarizations of 2.04±0.01 and 2.93±0.01 with IC 12.75±0.01 were achieved in heat bath cooling applied in the previous work. Finally, in a third experiment aimed to maximize the IC of the entire molecule, we measured polarizations of 2.87±0.02, 2.64±0.02 for the carbons and 3.58±0.02 for the proton, to yield an IC of 28.0±0.2. In a heat bath cooling experiment with the same goal conducted in the previous work, the two carbons and the proton spins were cooled by factors of 1.74±0.01, 1.86±0.01 and 3.77±0.01 respectively, reaching IC of 20.7±0.1.
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