107,697 research outputs found
Time-Symmetric ADI and Causal Reconnection: Stable Numerical Techniques for Hyperbolic Systems on Moving Grids
Moving grids are of interest in the numerical solution of hydrodynamical
problems and in numerical relativity. We show that conventional integration
methods for the simple wave equation in one and more than one dimension exhibit
a number of instabilities on moving grids. We introduce two techniques, which
we call causal reconnection and time-symmetric ADI, which together allow
integration of the wave equation with absolute local stability in any number of
dimensions on grids that may move very much faster than the wave speed and that
can even accelerate. These methods allow very long time-steps, are fully
second-order accurate, and offer the computational efficiency of
operator-splitting.Comment: 45 pages, 19 figures. Published in 1994 but not previously available
in the electronic archive
Reservoir engineering and dynamical phase transitions in optomechanical arrays
We study the driven-dissipative dynamics of photons interacting with an array
of micromechanical membranes in an optical cavity. Periodic membrane driving
and phonon creation result in an effective photon-number conserving non-unitary
dynamics, which features a steady state with long-range photonic coherence. If
the leakage of photons out of the cavity is counteracted by incoherent driving
of the photonic modes, we show that the system undergoes a dynamical phase
transition to the state with long-range coherence. A minimal system, composed
of two micromechanical membranes in a cavity, is studied in detail, and it is
shown to be a realistic setup where the key processes of the driven-dissipative
dynamics can be seen.Comment: 16 pages, 9 figure
Ancilla-based quantum simulation
We consider simulating the BCS Hamiltonian, a model of low temperature
superconductivity, on a quantum computer. In particular we consider conducting
the simulation on the qubus quantum computer, which uses a continuous variable
ancilla to generate interactions between qubits. We demonstrate an O(N^3)
improvement over previous work conducted on an NMR computer [PRL 89 057904
(2002) & PRL 97 050504 (2006)] for the nearest neighbour and completely general
cases. We then go on to show methods to minimise the number of operations
needed per time step using the qubus in three cases; a completely general case,
a case of exponentially decaying interactions and the case of fixed range
interactions. We make these results controlled on an ancilla qubit so that we
can apply the phase estimation algorithm, and hence show that when N \geq 5,
our qubus simulation requires significantly less operations that a similar
simulation conducted on an NMR computer.Comment: 20 pages, 10 figures: V2 added section on phase estimation and
performing controlled unitaries, V3 corrected minor typo
inflation to probe non-perturbative quantum gravity
It is natural to expect a consistent inflationary model of the very early
Universe to be an effective theory of quantum gravity, at least at energies
much less than the Planck one. For the moment, , or shortly ,
inflation is the most successful in accounting for the latest CMB data from the
PLANCK satellite and other experiments. Moreover, recently it was shown to be
ultra-violet (UV) complete via an embedding into an analytic infinite
derivative (AID) non-local gravity. In this paper, we derive a most general
theory of gravity that contributes to perturbed linear equations of motion
around maximally symmetric space-times. We show that such a theory is quadratic
in the Ricci scalar and the Weyl tensor with AID operators along with the
Einstein-Hilbert term and possibly a cosmological constant. We explicitly
demonstrate that introduction of the Ricci tensor squared term is redundant.
Working in this quadratic AID gravity framework without a cosmological term we
prove that for a specified class of space homogeneous space-times, a space of
solutions to the equations of motion is identical to the space of backgrounds
in a local model. We further compute the full second order perturbed
action around any background belonging to that class. We proceed by extracting
the key inflationary parameters of our model such as a spectral index (),
a tensor-to-scalar ratio () and a tensor tilt (). It appears that
remains the same as in the local inflation in the leading slow-roll
approximation, while and get modified due to modification of the
tensor power spectrum. This class of models allows for any value of
with a modified consistency relation which can be fixed by future observations
of primordial -modes of the CMB polarization. This makes the UV complete
gravity a natural target for future CMB probes.Comment: 37 page
Self-consistent theory of large amplitude collective motion: Applications to approximate quantization of non-separable systems and to nuclear physics
The goal of the present account is to review our efforts to obtain and apply
a ``collective'' Hamiltonian for a few, approximately decoupled, adiabatic
degrees of freedom, starting from a Hamiltonian system with more or many more
degrees of freedom. The approach is based on an analysis of the classical limit
of quantum-mechanical problems. Initially, we study the classical problem
within the framework of Hamiltonian dynamics and derive a fully self-consistent
theory of large amplitude collective motion with small velocities. We derive a
measure for the quality of decoupling of the collective degree of freedom. We
show for several simple examples, where the classical limit is obvious, that
when decoupling is good, a quantization of the collective Hamiltonian leads to
accurate descriptions of the low energy properties of the systems studied. In
nuclear physics problems we construct the classical Hamiltonian by means of
time-dependent mean-field theory, and we transcribe our formalism to this case.
We report studies of a model for monopole vibrations, of Si with a
realistic interaction, several qualitative models of heavier nuclei, and
preliminary results for a more realistic approach to heavy nuclei. Other topics
included are a nuclear Born-Oppenheimer approximation for an {\em ab initio}
quantum theory and a theory of the transfer of energy between collective and
non-collective degrees of freedom when the decoupling is not exact. The
explicit account is based on the work of the authors, but a thorough survey of
other work is included.Comment: 203 pages, many figure
The Dynamics of 1D Quantum Spin Systems Can Be Approximated Efficiently
In this Letter we show that an arbitrarily good approximation to the
propagator e^{itH} for a 1D lattice of n quantum spins with hamiltonian H may
be obtained with polynomial computational resources in n and the error
\epsilon, and exponential resources in |t|. Our proof makes use of the finitely
correlated state/matrix product state formalism exploited by numerical
renormalisation group algorithms like the density matrix renormalisation group.
There are two immediate consequences of this result. The first is that the
Vidal's time-dependent density matrix renormalisation group will require only
polynomial resources to simulate 1D quantum spin systems for logarithmic |t|.
The second consequence is that continuous-time 1D quantum circuits with
logarithmic |t| can be simulated efficiently on a classical computer, despite
the fact that, after discretisation, such circuits are of polynomial depth.Comment: 4 pages, 2 figures. Simplified argumen
Bose-Hubbard model with occupation dependent parameters
We study the ground-state properties of ultracold bosons in an optical
lattice in the regime of strong interactions. The system is described by a
non-standard Bose-Hubbard model with both occupation-dependent tunneling and
on-site interaction. We find that for sufficiently strong coupling the system
features a phase-transition from a Mott insulator with one particle per site to
a superfluid of spatially extended particle pairs living on top of the Mott
background -- instead of the usual transition to a superfluid of single
particles/holes. Increasing the interaction further, a superfluid of particle
pairs localized on a single site (rather than being extended) on top of the
Mott background appears. This happens at the same interaction strength where
the Mott-insulator phase with 2 particles per site is destroyed completely by
particle-hole fluctuations for arbitrarily small tunneling. In another regime,
characterized by weak interaction, but high occupation numbers, we observe a
dynamical instability in the superfluid excitation spectrum. The new ground
state is a superfluid, forming a 2D slab, localized along one spatial direction
that is spontaneously chosen.Comment: 16 pages, 4 figure
Observations Outside the Light-Cone: Algorithms for Non-Equilibrium and Thermal States
We apply algorithms based on Lieb-Robinson bounds to simulate time-dependent
and thermal quantities in quantum systems. For time-dependent systems, we
modify a previous mapping to quantum circuits to significantly reduce the
computer resources required. This modification is based on a principle of
"observing" the system outside the light-cone. We apply this method to study
spin relaxation in systems started out of equilibrium with initial conditions
that give rise to very rapid entanglement growth. We also show that it is
possible to approximate time evolution under a local Hamiltonian by a quantum
circuit whose light-cone naturally matches the Lieb-Robinson velocity.
Asymptotically, these modified methods allow a doubling of the system size that
one can obtain compared to direct simulation. We then consider a different
problem of thermal properties of disordered spin chains and use quantum belief
propagation to average over different configurations. We test this algorithm on
one dimensional systems with mixed ferromagnetic and anti-ferromagnetic bonds,
where we can compare to quantum Monte Carlo, and then we apply it to the study
of disordered, frustrated spin systems.Comment: 19 pages, 12 figure
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