Transcript Barnett, Burnett, and Vaccaro

Bose-Einstein Condensation of
magnons in nanoparticles.
Lawrence H. Bennett
NSF Cyberinfrastructure for Materials Science
August 3-5, 2006
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”
Outline
●Bose-Einstein condensation
•Atoms
•Magnons in nanoparticles
●Aftereffect measurements
•Decay rates
•Fluctuation fields
●Quantum entanglement
Bose-Einstein Condensation
The 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.
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.
Metastablity
In contrast to atomic BEC, magnon number may not
be conserved. Nevertheless, when magnon decay
mechanisms are significantly slower than numberconserving 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.
Magnons
• Magnons are bosons
• They obey the Bose-Einstein distribution
n
1
e
( E  ) kT
1
n is boson distribution
k is Boltzmann’s constant
E is energy
T is temperature
ζ is chemical potential
Ni/Cu Compositionally-Modulated Alloys
A=-dM/d(ln t)
Atzmony et al., JMMM 69, 237 (1987)
Quantum Magazine
July/August 1997
Temperature variation of aftereffect
in nanograin iron powders
R=Maximum Decay Rate
U. Atzmony, Z. Livne, R.D. McMichael, and L.H. Bennett, J. Appl. Phys., 79, 5456 (1996).
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).
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).
Fluctuation Field
The 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
Hf0
kT

0 M sV
(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).
Chemical Potential, ζ = 0
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
EB  0 M S  H k  V .
(2)
Equation (1) can then be rewritten as
kT  H k 
Hf 
EB
(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.
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
Hf0
kTH k
Hf 

EB   1   / EB
(4)
where Hfo is the fluctuation field when ζ =0, and
Hf0
kT

 0 M SV
Calculated fluctuation field vs. temperature, assuming Hf 0 is
linear in temperature and EB is temperature independent.
The Chemical Potential
• The chemical potential obeys
d  SdT  vdp  H  dM
• With constant pressure and magnetization
d
S 
dT
|
p ,M
• If the entropy is a constant, then
  S (T  T )
BE
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).
Experimental chemical potential for Co/Pt
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).
Summary
Magnetic aftereffect measurements in
nanostructural
materials
show
nonArrhenius 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.