Bose-Einstein Condensation of magnons in nanoparticles.
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Bose-Einstein Condensation of magnons in nanoparticles.
Lawrence H. Bennett
NSF Cyberinfrastructure for Materials Science
August 3-5, 2006
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Condensed thinking“Clinging to tried and trusted methods, though, may not be the right approach. ... Developing existing technology for use in quantum computers might prove equally mistaken. In this context, a relatively newly discovered form of matter called a Bose-Einstein condensate may point the way ahead.”The Economist: 5-6-2006, Vol. 379 Issue 8476, p79-80 “One qubit at a time”
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Outline●Bose-Einstein condensation
•Atoms•Magnons in nanoparticles
●Aftereffect measurements•Decay rates•Fluctuation fields
●Quantum entanglement
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Bose-Einstein CondensationThe occupation of a single quantum state by a large fraction of bosons at low temperatures was predicted by Bose and Einstein in the 1920s. The quest for Bose-Einstein condensation (BEC) in a dilute atomic gas was achieved in 1995 using laser-cooling to reach ultra-cold temperatures of 10-7 K. BEC of dilute atomic gases, now regularly created in a number of laboratories around the world, have led to a wide range of unanticipated applications. Especially exciting is the effort to use BEC for the manipulation of quantum information, entanglement, and topological order.
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BEC of magnons in nanoparticles.The study of atomic BEC has yielded rich dividends. A promising extension is to magnons—spin-wave quanta that behave as bosonic quasiparticles—in magnetic nanoparticles. This system has unique characteristics differentiating it from atomic BEC, creating the potential for a whole new variety of interesting behaviors and applications that include high-temperature Bose condensation (at tens or possibly even hundreds of Kelvin) and novel nanomagnetic devices.
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MetastablityIn contrast to atomic BEC, magnon number may not be conserved. Nevertheless, when magnon decay mechanisms are significantly slower than number-conserving magnon-magnon and magnon-phonon interaction rates, a metastable population of magnons can quasi-thermalize and manifest BEC-like behavior, and the system’s quantum state can be probed and exploited for its unique properties. In atomic BEC, atom number, which is a critical parameter, is difficult to control and even more difficult to adjust after the BEC is created. In contrast, magnon number can be actively controlled via microwave pumping.
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Magnons
• Magnons are bosons• They obey the Bose-Einstein distribution
11)( −
=− kTEe
nς
is boson distribution k is Boltzmann’s constant
E is energy T is temperature
ζ is chemical potential
n
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Ni/Cu Compositionally-Modulated Alloys
Atzmony et al., JMMM 69, 237 (1987)
A=-dM/d(ln t)
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Quantum Magazine July/August 1997
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Temperature variation of aftereffect in nanograin iron powders
U. Atzmony, Z. Livne, R.D. McMichael, and L.H. Bennett, J. Appl. Phys., 79, 5456 (1996).
R=Maximum Decay Rate
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Fluctuation Field vs. Temperature (Co/Pt)
(0.3 nm Co/2 nm Pt)15
Circles = Measured
Line = Fit to Eq. 4
S. Rao, et al, J. Appl. Phys., 97, 10N113 (2005).
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Thermal Magnetic Aftereffect
We have measured the fluctuation field, Hf , as a function of temperature for a nanosize columnar (0.3 nm Co/2 nm Pt)15 multilayer sample 1.
The fluctuation field exhibits a peak at the temperature, TBE = 14 K, attributed to a magnon BEC. A requirement for a BEC is that, below TBE, the chemical potential is zero. Below 14 K, the fluctuation field varies linearly with temperature, implying such a zero value. 1 S. Rao, E. Della Torre, L.H. Bennett, H.M. Seyoum, and R.E. Watson, J. Appl. Phys. 97, 10N113 (2005).
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Fluctuation FieldThe fluctuation field 1,2 can be viewed as the driving force in the magnetic aftereffect. It is a random variable of time, a measure of which, Hf0, is given by
(1)
where Ms is the saturation magnetization, and the activation volume, V, is presumed 3 to be the average volume of individual single domain magnetic entities.
1 L. Néel, J. Phys. Radium, 12, 339 (1951). 2 R. Street and S.D. Brown, J. Appl. Phys., 76, 6386 (1994). 3 E.P. Wohlfarth, J. Phys. F: Met. Phys. 14, L155-L 159 (1984).
VMkTH
sf
00 μ=
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Chemical Potential, ζ = 0
.0 VHME kSB ><=μ
A quantity important to the aftereffect is the energy barrier to spin reversal, EB. For an assembly of single domain particles, with an average volume V and an average applied switching field <Hk> it is given by
(2)
Equation (1) can then be rewritten as
(3)
This equation assumes that the chemical potential is zero. When the temperature is below TBE, then Eq. 3 is applicable with the chemical potential being constant, (i.e., =0) with temperature.
B
kf E
HkTH ><=
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Adding the chemical potential to the fluctuation field
The effect of the chemical potential, ζ, is to reduce the energy barrier. Therefore, when ζ is not zero, Hf has to be modified to
B
f
B
kf E
HEkTH
H/10
ζζ −=
−=
where Hfo is the fluctuation field when ζ =0, and
VMkTHS
f0
0 μ=
(4)
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Calculated fluctuation field vs. temperature, assuming Hf 0 is linear in temperature and EB is temperature independent.
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The Chemical Potential
• The chemical potential obeys
• With constant pressure and magnetization
• If the entropy is a constant, then
dMH •−+−= vdpSdTdς
MpdTdS
,|ς
−=
)(BETTS −−=ς
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Fluctuation Field vs. Temperature (Co/Pt)
(0.3 nm Co/2 nm Pt)15
Circles = Measured
Line = Fit to Eq. 4
S. Rao, et al, J. Appl. Phys., 97, 10N113 (2005).
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Experimental chemical potential for Co/Pt
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Quantum entanglement of Magnons
The most important point is that a magnon propagates spatially all over the magnet. By the propagation, quantum coherence is established between spatially separated points. Therefore by exciting a macroscopic number of magnons, one can easily construct states with huge entanglement.T. Morimae, A. Sugita, and A. Shimizu, “Macroscopic entanglement of many-magnon states”, Phys. Rev. A 71, 032317 (2005).
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SummaryMagnetic aftereffect measurements in nanostructural materials show non-Arrhenius behavior, with a peak value of the decay at some temperature.
Replacing classical statistics with quantum statistics explains the experimental results, with the peak occurring at the Bose-Einstein condensation temperature.
Macroscopic entanglement of the magnons is a basis for quantum computation.
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