Predictions for cooling a solid to its ground state
Author:
William C. Troy
Journal:
Quart. Appl. Math. 71 (2013), 331338
MSC (2010):
Primary 82B10
Published electronically:
October 23, 2012
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Abstract: A major goal of quantum computing research is to drain all quanta of thermal energy from a solid at a positive temperature 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 However, current theory, which is based on the BoseEinstein equation, predicts that the temperature as We prove that this discrepancy between experiment and theory is due to previously unobserved errors in low temperature predictions of the BoseEinstein equation. We correct this error and derive a new formula for temperature which proves that as Simultaneously, the energy decreases to its `supersolid' ground state level as For experimental data our temperature formula predicts that in close agreement with the experimental result. Our results form a first step towards bridging the gap between existing theory and the construction of useful quantum computing devices.
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 P. Debye, Zur theorie der spezifischen warme Annalen der Physik (Leipzig) 39 (1912) 789.
 2.
 A. Einstein, Die plancksche theorie der strahlung und die theorie der spezifischen warme. Annalen der Physik 22 (1907) 180190.
 3.
 H. Eyring, D. Henderson, B. J. Stover and E. M. Eyring, Statistical Mechanics and Dynamics. Wiley, Second Edition, New York, 1982.
 4.
 S. Groblacher, B. Hertzberg, M. Vanner, D. Cole, G. Gigan, S. K. Schwab and M. Aspelmeyer, Demonstration of an ultracold microoptomechanical oscillator in a cryogenic cavity. Nature Physics 5 (2009) 485488.
 5.
 E. Kim and M. H. W. Chan, Probable observation of a supersolid helium phase. Nature 427 (2004) 225  227
 6.
 A. D. O'Connell, M. Hofheinz, M. Ansmann, C. Bialczak, M. Lenander, E. Lucero, E. M. Neeley, D. Sank, D. H. Wang, M. Weides, J. Wenner, J. M. Martinis and A. N. Cleland, Quantum ground state and singlephoton control of a mechanical resonator. Nature 464 (2010) 697703.
 7.
 Y. Park and H. Wang, Resolvedsideband and cryogenic cooling of an optomechanical resonator. Nature Physics 5 (2009) 489493.
 8.
 D. Powell, Moved by Light. Science News 179 (2011) 2425.
 9.
 D. K. Prathia, Statistical Mechanics. AddisonWesley, 1999.
 10.
 T. Rocheleau, T. Ndukum, C. Macklin, J. B. Hertzberg, A. A. Clerk and K. C. Schwab, Preparation and detection of a mechanical resonator near the ground state of motion. Nature 463 (2010) 7275.
 11.
 O. RomeroIsart, A. C. Pflanzer, F. Blaser, FR. Kaltenbaek, N. Kissel, M. Aspelmeyer and J. I. Cirac, Large quantum superpositions and interference of massive nanometersized objects. Phys. Rev. Lett. 107 (2011) 0240502408.
 12.
 A. Schliesser, O. Arcizet, R. Rivière, G. Anetsberger and T. J. Kippenberg, Resolvedsideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit. Nature Physics 5 (2009) 509514.
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 D. V. Schroeder, An Introduction to Thermal Physics. AddisonWesley, 1999.
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 J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker and R. W. Simmonds, Circuit electromechanics cavity in the strong coupling regime. Nature 471 (2011) 204208.
 15.
 J. D. Teufel, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert and R. W. Simmonds, Sideband cooling of micromechanical motion to the quantum ground state. Nature 475 (2011) 359363.
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 P. A. Tipler and R. A. Llewellyn, Modern Physics. W. H. Freeman and Co., New York, Fifth Edition, 2008
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Additional Information
William C. Troy
Affiliation:
Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email:
troy@math.pitt.edu
DOI:
http://dx.doi.org/10.1090/S0033569X2012012943
PII:
S 0033569X(2012)012943
Keywords:
BoseEinstein equation,
temperature
Received by editor(s):
July 21, 2011
Published electronically:
October 23, 2012
Article copyright:
© Copyright 2012 Brown University
