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 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.

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.

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.

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

Ni/Cu Compositionally-Modulated Alloys

Atzmony et al., JMMM 69, 237 (1987)

A=-dM/d(ln t)

Quantum Magazine July/August 1997

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

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 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=

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 ><=

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)

Calculated fluctuation field vs. temperature, assuming Hf 0 is linear in temperature and EB is temperature independent.

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

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).

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.