Transcript Document

SPINS, CHARGES, LATTICES, & TOPOLOGY IN LOW d
Meeting of “QUANTUM CONDENSED MATTER” network of PITP
(Fri., Jan 30- Sunday, Feb 1, 2004; Vancouver, Canada)
For all current information on this workshop go to
http://pitp.physics.ubc.ca/Conferences/20030131/index.html
All presentations will go online in the next week on PITP archive page:
http://pitp.physics.ubc.ca/CWSSArchives/CWSSArchives.html
DECOHERENCE in SPIN NETS &
RELATED LATTICE MODELS
PCE Stamp (UBC)
+
YC Chen (Australia?)
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S2
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PROBLEM #1
The theoretical problem is to calculate the dynamics of the
“M-qubit” reduced density matrix for the following Hamiltonian,
describing a set of N interacting qubits (with N > M typically):
H = Sj ( Dj tjx + ej tjz ) + Sij Vij tiz tjz
+ Hspin({sk}) + Hosc({xq}) + int.
The problem is to integrate out the 2 different environments
coupling to the qubit system- this gives the N-qubit reduced density
matrix. We may then average over other qubits if necessary to get
the M-qubit density matrix operator:
rNM({tj}; t)
The N-qubit density matrix contains all information about the
dynamics of this QUIP (QUantum Information Processing) system& all the quantum information is encoded in it.
A question of some theoretical interest is- how do decoherence
rates in this quantity vary with N and M ?
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)), where
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-interacting
Oscillators, with canonical coordinates {xq,pq} and frequencies {wq}.
It then follows that we can write the effective Hamiltonian for 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
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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,
Rep Prog Phys 63, 669 (2000)
A qubit coupled to a bath of
localised excitations: the
CENTRAL SPIN Model
Now consider the coupling of our 2-level system to LOCALIZED modes.
These have a Hilbert space of finite dimension, in the energy range of
interest- in fact, often each localised excitation has a Hilbert space dimension 2. From this we
see that our central Qubit is coupling to a set of effective spins; ie., to a “SPIN BATH”. Unlike the
case of the oscillators, we cannot assume these couplings are weak.
For simplicity assume here that the bath spins are a set {sk} of 2-level systems. Now
actually these interact with each other very weakly (because they are localised), but we
cannot drop these interactions. What we then get is the following low-energy effective
Hamiltonian (recall previous slide):
H (Wo)
= { [Dt+ exp(-i Sk ak.sk) + H.c.] + eotz
+ tzwk.sk + hk.sk
+ inter-spin interactions
(qubit)
(bath spins)
The crucial thing here is that now the couplings
wk , hk to the bath spins- the first between bath
spin and qubit, the second to external fields- are
often very strong (much larger than either the
inter-spin interactions or even than D).
Dynamics of Spin-Boson System
The easiest way to solve for the dynamics of the spin-boson model is in a path integral
formulation. The qubit density matrix
propagator is written as an integral
over an “influence functional” :
The influence functional is defined as
For an oscillator bath:
with bath propagator:
For a qubit the path reduces to
Thence
Dynamics of Central Spin model
(Qubit coupled to spin bath)
Consider following averages
Topological
phase average
Orthogonality
average
Bias average
The reduced density matrix after a
spin bath is integrated out is quite
generally given by:
Eg., for a single qubit,
we get the return probability:
NB: can also deal with external noise
DYNAMICS of DECOHERENCE
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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
Decoherence
in
SQUIDs
A.J. Leggett et al., Rev.
Mod Phys. 59, 1 (1987)
AND
PCE Stamp, PRL 61, 2905
(1988)
Prokof’ev and Stamp
Rep Prog Phys 63, 669
(2000)
The oscillator bath decoherence rate goes like
tf-1 ~ Do g(D,T) coth (D/2kT)
with the spectral function g(w,T) shown below for an Al
SQUID (contribution from electrons & phonons). All of this is
well known and leads to a decoherence rate tf-1 ~ paDo once
kT < Do. By reducing the flux change df = (f+ - f- ) ~ 10-3 , it
has been possible to make a ~ 10-7 (Delft expts), ie., a
decoherence rate for electrons ~ O(100 Hz). This is v small!
On the other hand paramagnetic spin impurities
(particularly in the junctions), & nuclear spins
have a Zeeman coupling to the SQUID flux
peaking at low energies- at energies below Eo, this
will cause complete incoherence. Coupling to
charge fluctuations (also a spin bath of 2-level
systems) is not shown here, but also peaks at very
low frequencies.
However when Do >> Eo, the spin bath
decoherence rate is:
1/tf = Do (Eo/8D0)2
Pey 1.34
as before
WRITE on PAPER SHEETS
PROBLEM #2: The DISSIPATIVE HOFSTADTER Model
This problem describes a set of fermions on a periodic potential, with uniform
flux threading the plaquettes. The fermions are then coupled to a background
oscillator bath:
We will assume a square lattice, and
a simple
cosine potential:
There are TWO dimensionless
couplings in
the problemto the external
field, and
to the bath:
The coupling to the oscillator bath is
assumed ‘Ohmic’:
where
The W.A.H. MODEL
This famous model was first investigated in preliminary way by Peierls,
Harper,, Kohn, and Wannier in the 1950’s. The fractal structure was shown by
Azbel in 1964. This structure was first displayed on a computer by Hofstadter
in 1976, working with Wannier.
The Hamiltonian involves a set of
charged fermions moving on a
periodic lattice- interactions between
the fermions are ignored. The
charges couple to a uniform flux
through the lattice plaquettes.
Often one looks at a square
lattice, although it turns out
much depends on the lattice
symmetry.
One key dimensionless parameter in the problem is
the FLUX per plaquette, in units of the flux quantum
The HOFSTADTER BUTTERFLY
The graph shows the ‘support’ of the density of states- provided a is rational
The effective Hamiltonian is also written as:
H = - t Sij [ ci cj exp {iAij} + H.c. ]
……. “WAH” lattice
+ SnSq lq Rn . xq
+ Hosc ({xq}) …… coupled to
oscillators
(i) the the WAH (Wannier-Azbel-Hofstadter) Hamiltonian describes the motion of
spinless fermions on a 2-d square lattice, with a flux
from the gauge term
f per plaquette (coming
Aij).
(ii) The particles at positions Rn couple to a set of oscillators.
This can be related to many systems- from 2-d J. Junction arrays in
an external field to flux phases in HTc systems, to one kind of open
string theory. It is also a model for the dynamics of information
propagation in a QUIP array, with simple flux carrying the info.
There are also many connections with other models of interest in
mathematical physics and statistical physics.
EXAMPLE: S/cond arrays
The bare action is:
Plus coupling to Qparticles,
photons, etc:
Interaction kernel (shunt resistance is RN):
Expt (Kravchenko,
Coleridge,..)
PHASE DIAGRAM
Callan & Freed
result (1992)
Mapping of the line a=1
under z  1/(1 + inz)
Proposed phase diagram
(Callan & Freed, 1992)
Arguments leading to this phase diagram
based mainly on duality, and assumption of
localisation for strong coupling to bosonic bath. The duality is now that
of the generalised vector Coulomb gas, in the complex z- plane.
DIRECT CALCULATION of m (Chen & Stamp)
We wish to calculate directly the time evolution of the reduced density matrix
of the particle. It satisfies
the eqtn of motion:
The propagator on the
Keldysh contour g is:
The influence functional
is written in the form:
Influence of the periodic potential
We do a weak potential expansion, using the standard trick
Without the lattice potential, the path integral contains paths obeying the
simple Q Langevin eqtn:
The potential then adds a set of ‘delta-fn. kicks’:
One can calculate the dynamics now in a quite direct way, not by calculating
an autocorrelation function but rather by evaluating the long-time behaviour of
the density matrix.
If one evaluates the long-time behaviour of the Wigner function one then
finds the following, after expanding in the potential:
We now go to some rather detailed exact results for this velocity, in the
next few slides ….
LONGITUDINAL COMPONENT:
TRANSVERSE
COMPONENT:
DIAGONAL & CROSS-CORRELATORS:
It turns out from these exact results that not all of the conclusions
which come from a simple analysis of the long-time scaling are
confirmed. In particular we do not get the same phase diagram,
as we now see …
We find that we can get some exact results on a
particular circle in the phase plane- the one for
which K = 1/2
The reason is that on this circle, one finds
that both the long- and short-range parts of the
interaction permit a ‘dipole’ phase, in which
the system form close dipoles, with the dipolar
widely separated. This happens nowhere else.
One then may immediately evaluate the
dynamics, which is well-defined. If we write
this in terms of a mobility we have the simple
results shown:
RESULTS on CIRCLE K = 1/2
The behaviour on
this circle should be
testable in
experiments.
The results can be summarized as shown
in the figure. For a set of points on the circle
the system is localised. At all other points
on the circle, it is delocalised.
Conclusions
(1) In the weak-coupling limit (with dimensionless couplings ~ l ), the
disentanglement rate for a set of N coupled qubits, is actually linear
in N provided Nl < 1
(2) In the coherence window, this is good for quite large N
(3) In the dissipative Hofstadter model duality apparently fails. There is
actually a whole set of ‘exact’ solutions possible on various circles.
It will be interesting to explore decoherence rates for topological
computation- note that the bath couplings are local but one still has
to determine the couplings to the non-local information
THE END
The dynamics of the density matrix is calculated using path integral
methods. We define the propagator for the density matrix as follows:
This propagator is written a a path integral along a Keldysh contour:
All effects of the bath are
contained in Feynman’s
influence functional, which
averages over the bath
dynamics, entangled with
that of the particle:
The ‘reactive’ part & the
‘decoherence’ part of the
influence functional depend
on the spectral function:
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DYNAMICS of the DIPOLAR SPIN NET
The dipolar spin net is of great interest to solid-state theorists
because it represents the behaviour of a large class of systems
with “frustrating” interactions (spin glasses, ordinary dipolar
glasses). It is also a fascinating toy model for quantum
computation:
H = Sj (Dj tjx + ej tjz) + Sij Vijdip tiz tjz
+ HNN(Ik) + Hf(xq)
+ interactions
For magnetic systems this leads to the
picture at right.
Almost all experiments so far are done in the
region where Do is small- whether the dynamics
is dipolar-dominated or single molecule, it is
incoherent. However one can give a theory of this
regime.
The next great challenge is to understand the
dynamics in the quantum coherence regime, with
or without important inter-molecule interactions
NV Prokof’ev, PCE Stamp,
PRL 80, 5794 (1998)
JLTP 113, 1147 (1998)
PCE Stamp, IS Tupitsyn
Rev Mod Phys (to be publ.).
Quantum Relaxation
of a single
NANOMAGNET
Structure of
Nuclear spin
Multiplet 
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Our Hamiltonian:
When D <<Eo (linewidth of the nuclear
multiplet states around each magbit
level), the magbit relaxes via
incoherent tunneling. The nuclear bias
acts like a rapidly varying noise field,
causing the magbit to move rapidly in
and out of resonance, PROVIDED
|gmBSHo| < Eo
Fluctuating noise field
Tunneling now proceeds over a range Eo of bias, governed
by the NUCLEAR SPINmultiplet. The relaxation rate is
G ~ D2/Eo
for a single qubit.
Nuclear spin diffusion paths
NV Prokof’ev, PCE Stamp, J
Low Temp Phys 104, 143 (1996)
The path integral splits
into contributions for
each M. They have the
effective action of a set
of interacting instantons
The effective interactions
can be mapped to a set
of fake charges to produce an action
having the structure of a “spherical
model” involving a spin S
The key step is to then reduce this
to a sum over Bessel functions
associated with each polarisation
group.
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