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http://pitp.physics.ubc.ca/
MISSION of PITP
1. PITP is an international institute, funded internationally, with an international missionto bring together groups of high-quality researchers, from around the world, and foster
path-breaking new research in all branches of theoretical physics.
2. Recognizing that theoretical physics is central to the whole of science, PITP fosters
links to other subjects, including chemistry & biology, and tries to ‘seed’ new
developments in these areas.
3. Recognizing the decisive influence that physics and other sciences have in the modern
world, PITP will advise and assist in areas within its competence, and provide
information to the general public about theoretical physics and related topics.
Advisory Board
-Prof P.W. Anderson (Princeton)
<at large>
http://pitp.physics.ubc.ca/index.html
TOKYO
PiTP
KOREA
(APCTP,
SeaQUest, etc)
AUSTRALIA
(ARC, AAS, NSW,
Queensland)
STANFORD
RESEARCH NETWORKS
1.
2.
3.
4.
Quantum Condensed Matter
Complex Systems
Strings and Particles
Cosmology & Astrophysics
EUROPE
Alberta, Perimeter
CITA, CQIQC
Sherbrooke
+ other Canadian
centres
PITP: Main activities
1. Meetings (conferences, schools,
workshops); 30 since April 2003
2. Support of research networks
3. Support of PITP visitor centre
+ nodes
MEMBERS of QUANTUM CONDENSED MATTER NETWORK
British Columbia
IK Affleck
M Berciu
D Bonn
A Damascelli
J Folk
M Franz
WN Hardy
I Herbut
GA Sawatzky
PCE Stamp
A Zagoskin
F Zhou
Alberta
M Boninsegni
F Marsiglio
United States
PW Anderson
G Christou
S Hill
D Goldhaber-Gordon
M Jarrell
A Kitaev
S Kivelson
G Kotliar
RB Laughlin
AJ Leggett
H Manoharan
C Marcus
J Moore
DD Osheroff
BL Spivak
PB Wiegmann
SC Zhang
Ontario
P Brumer
T Devereaux
S John
C Kallin
HY Kee
YB Kim
E Sorenson
A Steinberg
Quebec
A Blais
C Bourbonnais
K LeHur
D Senechal
AM Tremblay
Newfoundland
S Curnoe
Europe
Australasia
G Aeppli
B Barbara
Y Imry
S Popescu
J van den Brink
W Wernsdorfer
RW Clark
R McKenzie
G Milburn
Y Nakamura
N Nagaosa
M Nielsen
M Oshikawa
G Vidal
see
http://pitp.physics.ubc.ca/
NOW for the TALK…….
PCE STAMP
Physics & Astronomy
UBC
Vancouver
Pacific Institute
for
Theoretical Physics
How do REAL Solids (%99.9999) behave at low Energy?
The very low-T properties of even simple SiO2
are very peculiar. One sees a large and
strongly T-dependent
extra contribution to the
low-E density of states.
At very low T this starts
to develop a ‘hole’ around
zero energy. The system
develops a hierachy of
Results for Capacitance (Above) &
relaxation times
Sound velocity and dielectric absorption
extending over many
(Below) for pure SiO2 , at very low T
rs in supersolid 4He
orders of magnitude.
The incredible
thing is that this
happens even in a
system like pure
crystalline 4He
solid- which at
the same
time becomes a
supersolid! This is
still not really
Relaxation times in supersolid 4He
understood
What happens to Heff at low Energy in Solids?
States in a ‘Quantum Glass’, pile up
at low energy. Their structure is Tdependent in any effective Hamiltonian
The main reasons for the peculiar
nature of the low-energy states in
most solids are (i) boundaries,
and (ii) interactions which are
long range and/or ‘frustrating’.
Both of these are ubiquitous, even
for
pure
systems
without
disorder! States pile up at low
energy,
but
they
can’t
communicate with each other.
At low T, the system splits into
subspaces that can never communicate
with each other- the effective vacuum &
its structure are physically quite
meaningless. A glass can only be
defined by its dynamic (non-equilibrium)
properties. A commonly used model
effective Hamiltonian is:
Frustrating interactions
‘Frustration’ means that at low energy, any
local change must re-organize simultaneously
a vast number of states. This forces the
Hilbert space of the effective Hamiltonian to
have an ‘ultrametric’ geometry.
where the ‘spins’ represent 2-level systems
‘Ultrametric geometry’ of a glass Hilbert space
SOLID-STATE QUBITS: Theoretical Designs & Experiments
Here are a few:
(1) Superconducting SQUID
qubits (where qubit states are flux
states); all parameters can be
controlled.
(2) Magnetic molecule qubits (where an
easy axis anisotropy gives 2 low
energy spin states, which communicate via
tunneling, and couple via exchange or
dipolar interactions. Control of
individual qubit fields is easy in
principle- interspin couplings
less so...
S1
MnIV
MnIII
S2
J'
S1
MnIV
MnIII
S2
S2
-J
S
-J
J'
S1
S
MnIV
MnIII
S
(3) Spins in semiconductors (or in Q Dots).
Local fields can be
partially controlled, &
the exchange coupling
is also controllable.
A qubit coupled to a bath of
delocalised excitations: the
SPIN-BOSON Model
Feynman & Vernon, Ann.
Phys. 24, 118 (1963)
PW Anderson et al, PR B1,
1522, 4464 (1970)
Caldeira & Leggett, Ann.
Phys. 149, 374 (1983)
AJ Leggett et al, Rev Mod
Phys 59, 1 (1987)
Suppose we have a system whose low-energy dynamics truncates to that
U. Weiss, “Quantum
of a 2-level system t. In general it will also couple to DELOCALISED modes
Dissipative Systems”
(World Scientific, 1999)
around (or even in) it. A central feature of many-body theory (and indeed
quantum field theory in general) is that
(i) under normal circumstances the coupling to each mode is WEAK (in fact ~ O (1/N1/2))
N is the number of relevant modes, just BECAUSE the modes are delocalised; and
(ii) that then we map these low energy “environmental modes” to a set of non-interactin
Oscillators, with canonical coordinates {xq,pq} and frequencies {wq}.
It then follows that we can write the effective Hamiltonian
this coupled system in the ‘SPIN-BOSON’ form:
H (Wo) = {[Dotx + eotz]
+ 1/2 Sq (pq2/mq + mqwq2xq2)
+ Sq [ cqtz + (lqt+ + H.c.)] xq }
Where Wo is a UV cutoff, and the {cq, lq} ~ N-1/2.
qubit
oscillator
interaction
P.C.E. Stamp, PRL 61, 2905
(1988)
AO Caldeira et al., PR B48,
13974 (1993)
NV Prokof’ev, PCE Stamp, J
Phys CM5, L663 (1993)
NV Prokof’ev, PCE Stamp,
Now consider the coupling of our 2-level system to LOCALIZED modes. Rep Prog Phys 63, 669 (2000)
These have a Hilbert space of finite dimension, in the energy range of interestin fact, often each localised excitation has a Hilbert space dimension 2. Our central Qubit is thus coupl
of effective spins; ie., to a “SPIN BATH”. Unlike for the oscillators, we cannot assume these couplings a
A qubit coupled to a bath of
localised excitations: the
CENTRAL SPIN Model
For simplicity assume here the bath spins are a set {sk} of 2-level systems, which interac
each other only very weakly (because they are localised). We then get the following low-en
effective Hamiltonian (compare previous slide):
H (Wo) = { [Dt+ exp(-i Sk ak.sk) + H.c.] + eotz
(qubit)
+ tz wk.sk + hk.sk
(bath spin
+ inter-spin interactions
Now the couplings
wk , hk to the bath spins (t
1st between bath spin & qubit, the 2nd to extern
fields) are often very STRONG (much larger tha
the inter-bath spin interactions or even than D)
DYNAMICS of DECOHERENCE
At first glance a solution of this seems very forbidding. However it turns out that
one can solve for the reduced density matrix of the central spin exactly, in the
interesting parameter regimes. From this soltn the decoherence mechanisms
are easy to identify:
(i) Noise decoherence: Random phases added to different Feynman paths by
the noise field.
(ii) Precessional decoherence: the
phase accumulated by environmental
spins between qubit flips.
(iii) Topological Decoherence: The
phase induced in the environmental
spin dynamics by the qubit flip itself
Noise decoherence source
USUALLY THE 2ND MECHANISM
(PRECESSIONAL DECOHERENCE)
is DOMINANT
Precessional
decoherence
The COHERENCE WINDOW
In solid-state qubit systems, the coherence window arises because of the large separation
of energy scales typically existing between spin and oscillator baths. This coherence window
exists in ALL solid-state systems- we look here at magnetic systems
ENERGY (K)
104
ELECTRONS
(in conductors)
102
PHONONS
M Dube, PCE Stamp, Chem Phys 268, 257 (2001)
PCE Stamp, J Q Comp & Computing 4, 20 (2003)
PCE Stamp, IS Tupitsyn, Phys Rev B69, 014401 (2004)
If we now fix the operating frequency D
of the qubits to lie well below the high
phonon frequencies, but well above the
characteristic nuclear spin frequencies
(given by hyperfine couplings, then the
phonons are too fast to cause decoherence,
& the nuclear spins too slow.
Log (td-1)
1
Nuclear spin
Decoherence
Phonon
Decoherence
Aij
10-2
NUCLEAR
SPINS
10-4
Vkk’
Log
D
NUCLEAR SPIN BATH in MAGNETIC
SYSTEMS: The LiHoxY1-xF4 system
This system is usually treated as the
archetypal Quantum Ising system:
However the Ho nuclear spin actually plays
a profound role in the physics:
(1) It blocks transitions until we get to very high fields (see left)
(2) The only way to understand the quantum spin glass phase is
by incorporating the nuclear spins (and also the transverse
dipolar terms); see below right
(3) The decoherence is completely governed by the
nuclear spins down to the lowest temperatures
(phonon effects disappear below roughly 250 mK
Stamp, P.C.E., Tupitsyn, I.S.,
Phys Rev B69, 014401 (2004)
M Schechter, PCE Stamp,
PRL 95, 267208 (2005)
DECOHERENCE
in the
Fe-8 Molecule
At low applied transverse
Fields, decoherence
switches on very fastexpect incoherent spin
relaxation:
Stamp, P.C.E., Tupitsyn, I.S.,
Phys Rev B69, 014401 (2004)
However, at high fields, system can be
in coherence window, in which qubit
dynamics is too fast for nuclear spins to
follow, but still much slower than phonons
This frequency window we call the
coherence window- note that typically
SOME
EXPTS
RIGHT: Expts on
Tunneling magn.
molecules & Ho
ions
Wernsdorfer et al, PRL 82, 3903
(1999); and
PRL 84, 2965 (2000); and
Science 284, 133 (1999)
R. Giraud et al., PRL 87, 057203 (2001
Expts on
the quantum
phase
transition
in LiHoF4
LEFT: ESR on Mn
dimer system
S Hill et al, Science 302,
1015 (2003)
RIGHT: NMR on
Mn-12 tunneling
molecules
H.M. Ronnow et al., Science 308, 389 (2005)
A. Morello et al., PRL 93, 197202 (2004)
DECOHERENCE in Superconducting Qubits
(1) The oscillator bath (electrons, photons, phonons) decoherence
rate:
tf-1 ~ Do g(D,T) coth (D/2kT)
(Caldeira-Leggett). This is often many orders of magnitude
smaller than the experimental decoherence rates.
(2) The spin bath decoherence will be caused by a combination
of charge & spin (nuclear & paramagnetic) defects- in
junction, SQUID, and substrate. 1/tf = Do (Eo/8D0)2
NV Prokof’ev, PCE Stamp, Rep Prog Phys 63, 669 (2000)
The basic problem with any theory-experiment comparison here is that most of the 2-level systems
are basically just junk (coming from impurities and defects), whose
characteristics are hard to
quantify. Currently ~10 groups
have seen coherent oscillations
in superconducting qubits, and
several have seen entanglement
between qubit pairs.
I Chiorescu et al., Science 299, 1869 (2003)
RW Simmonds et al., PRL 93, 077003 (2004)
New Kinds of Order: Topological Q Fluids
The archetype for all topological
Quantum fluids discussed so far is
the quantum Hall fluid. It has a
remarkable RG flow, predicted in
1984 by Laughlin, which leads to a
complex ‘nested’ phase diagram
of MI transitions (see Zhang et al).
The underlying symmetry
in the 2d parameter space
is SL(2,Z), the same as that
of an interacting set of
vortices and charges. This
is the same symmetry as
that possessed by a large
class of string theories.
There is a v interesting
model encapsulating all
these features- the ‘dissipative Hofstadter model’, (cf.
Callan et al). It describes open string theories, but also
flux phases, Josephson junction arrays, and even
interacting 3-wire Q wire junctions (Affleck et al).
Kitaev has shown that lattice anyon systems should
be able to do ‘topological quantum computing’, almost
immune from decoherence, & implementable on JJ
arrays (Doucot & Ioffe), or on ‘Kagome lattice’ systems
(Kitaev & Freedman).
SCHMID MODEL &
DISSIPATIVE HOFSTADTER MODEL:
SOME REVISIONISM
Two very well studied models in the
quantum dissipation community are
(1) Schmid model (particle in
periodic lattice potential
coupled to oscillator bath)
Proposed phase
diagram
(Callan & Freed)
Mapping of line a=1
under z  1/(1 + inz)
(2) Dissipative WAH model (now add
a uniform flux threading the 2-d
lattice plaquettes).
However, it looks as though some very interesting
features may have been missed. In the 1st place,
enforcing the natural constraint of lattice
periodicity on the oscillator bath changes thingsand produces some remarkable new solutions. In
the 2nd place, it seems as though duality actually
fails in the dissipative WAH model, again, some
exact results can be found.
(1) M Hasselfield, G Semenoff,
T Lee, PCE Stamp,
(2) PCE Stamp, YC Chen;
hep-th/0512219
preprint
Remarks on NETWORKS- the QUANTUM WALK
Computer scientists have been interested in RANDOM WALKS
on various mathematical GRAPHS, for many years. These
allow a general analysis of decision trees, search algorithms,
and indeed general computer programmes (a Turing machine
can be viewed as a walk). One of the most important
applications of this has been to error correction- which is
central to modern software.
Starting with papers by Aharonov et al (1994), & Farhi & Gutmann (1998), the same kind
of analysis has been applied to QUANTUM COMPUTATION. It is easy to show that ANY
quantum computation can be modeled as a QUANTUM WALK on some graph. The problem
then becomes one of QUANTUM DIFFUSION on this graph, and one easily finds either
power-law or exponential speed-up, depending on the graph. Great hopes have been
pinned on this new development- it allows very general analyses, and offers hope of new
kinds of algorithm, and new kinds of quantum error correction- and new ‘circuit designs’.
It also allows a very interesting general analysis of
decoherence in quantum computation (Prokof’ev &
Stamp: and Hines, Milburn & Stamp, 2005), with
extraordinary results. For example, for the Hamiltonian
we get ‘superdiffusion’ in the long time limit- part of the
density matrix still propagates diffusively (while another
part propagates SUB-diffusively). Note the general
implications of this result!