Predictions for cooling a solid to its ground state
A major goal of quantum computing research is to drain all quanta (q) of thermal energy from a solid at a positive temperature T0 > 0, leaving the object in its ground state. In 2010 the first complete success was reported when a quantum drum was cooled to its ground state at T0 = 20mK. However, current theory, which is based on the Bose-Einstein equation, predicts that temperature T → 0 as q → 0. We prove that this discrepancy between experiment and theory is due to previously unobserved errors in low temperature predictions of the Bose-Einstein equation. We correct this error and derive a new formula for temperature which proves that T → T0 > 0 as q → 0. Simultaneously, the energy decreases to its ‘supersolid’ ground state level as q → 0. For experimental data our temperature formula predicts that T0 = 9.8mK, in close agreement with the 20mK experimental result. Our results form a first step towards bridging the gap between existing theory and the construction of useful quantum computing devices. 1. Background. In 2009-2010 Park and Wang [7], Schliesser et al [12], Groblacher et al [4], and Rocheleau et al [10] used laser cooling techniques to cool solids to temperatures where the number of remaining quanta (q) are 63, 37, 30 and 4, respectively. In 2010 O’Connell et al [6] achieved complete success in reducing the number of quanta in a quantum drum to zero at T0 = 20mK. The key to their breakthrough was to use a ceramic wafer whose natural frequency, 6GHz., is much higher than frequencies in previous studies. In 2011 Teufel et al [14, 15] reduced the number of quanta in a drumlike membrane to ‘less than one.’ They claim that their device remains in the ground state for 100 microseconds, significantly longer than the 6 nano second result in [6]. Theoretical investigations [14, 15] have focused on the assumption that the Bose-Einstein equation q = 1 exp ( hν kT ) − 1 (1.1) for a single atom represents the behavior of the entire solid. Here k is Boltzman’s constant, h is Planck’s constant, ν is frequency and T > 0 is temperature. In particular, 2000 Mathematics Subject Classification. Primary 82B10.