Experimental heat-bath cooling of spins

Abstract

Abstract. Algorithmic cooling (AC) is a method to purify quantum systems, such as ensembles of nuclear spins, or cold atoms in an optical lattice. When applied to spins, AC produces ensembles of highly polarized spins, which enhance the signal strength in nuclear magnetic resonance (NMR). According to this cooling approach, spin-half nuclei in a constant magnetic field are considered as bits, or more precisely quantum bits, in a known probability distribution. Algorithmic steps on these bits are then translated into specially designed NMR pulse sequences using common NMR quantum computation tools. The algorithmic cooling of spins is achieved by alternately combining reversible, entropy-preserving manipulations (borrowed from data compression algorithms) with selective reset, the transfer of entropy from selected spins to the environment. In theory, applying algorithmic cooling to sufficiently large spin systems may produce polarizations far beyond the limits due to conservation of Shannon entropy. Here, only selective reset steps are performed, hence we prefer to call this process "heat-bath" cooling, rather than algorithmic cooling. We experimentally implemented two consecutive steps of selective reset, thus transferring entropy from two selected spins to the environment. We performed such cooling experiments, with commercially available labeled molecules, on standard liquid-state NMR spectrometers. We report in particular on our original experiment, unpublished until now except on the arXiv (quant-ph/0511156) in 2005, which was, to the best of our knowledge, the world's first experiment that yielded polarizations results that bypassed Shannon's entropy-conservation bound, so that the entire spin-system was cooled.

Achievable Polarization for Heat-Bath Algorithmic Cooling

“…Based on this work, many cooling algorithms have been designed [6][7][8][9][10][11]. HBAC is not only of theoretical interest, experiments have already demonstrated an improvement in polarization using this protocol with a few qubits [12][13][14][15][16][17][18], where a few rounds of HBAC were reached; and some studies have even included the impact of noise [19].…”

“…Based on this work, many cooling algorithms have been designed [6][7][8][9][10][11]. HBAC is not only of theoretical interest, experiments have already demonstrated an improvement in polarization using this protocol with a few qubits [12][13][14][15][16][17][18], where a few rounds of HBAC were reached; and some studies have even included the impact of noise [19].Through numerical simulations, Moussa [7] and Schulman et al [8] observed that if the polarization of the bath ( b ) is much smaller than 2 −n , where n is the number of qubits used, the asymptotic polarization reached will be ∼ 2 n−2 b ; but when b is greater than 2 −n , a polarization of order one can be reached. Inspired also by the work of Patange [20], who investigated the use of algorithmic cooling on spins bigger than 1 2 (using NV center where the defect has an effective spin 1), we investigate the case of cooling a qubit using a general spin l, and extra qubits which get contact with a bath.…”

Algorithmic cooling in liquid-state nuclear magnetic resonance

“…The current optimal control methods (GRAPE), and better ones such as a second order GRAPE [42] and Krotov based optimization [43] could enable various applications of AC in magnetic resonance spectroscopy [13,14,44] and maybe also other potential applications [45][46][47][48][49][50][51][52][53][54]. ln (4) (see [23]), where ε C,eq is the carbons' equilibrium polarization. On the right, the measured polarization and IC of C1 in each round.…”

“…on polarization enhancement) can be derived by interpreting the spin state in terms of information theory [22]. The information content (IC) of the spin was defined using the conventional notion of Shannon entropy H. The relation between a single spin's polarization and IC is given by the following equation [23,24] …”

“…ln(4) (see [23]), where ε C,eq is the equilibrium polarization of the carbons. The measured error of all polarizations is 0.02.…”