Anne E. B. Nielsen

Quantum spin Hamiltonians for the SU(2)k WZW model

We propose to use null vectors in conformal field theories to derive model Hamiltonians of quantum spin chains and the corresponding ground state wavefunction(s). The approach is quite general, and we illustrate it by constructing a family of Hamiltonians whose ground states are the chiral correlators of the SU(2)k WZW model for integer values of the level k. The simplest example corresponds to k = 1 and is essentially a nonuniform generalization of the Haldane–Shastry model with long-range exchange couplings. At level k = 2, we analyse the model for N spin 1 fields. We find that the Renyi entropy and the two-point spin correlator show, respectively, logarithmic growth and algebraic decay. Furthermore, we use the null vectors to derive a set of algebraic, linear equations relating spin correlators within each model. At level k = 1, these equations allow us to compute the two-point spin correlators analytically for the finite chain uniform Haldane–Shastry model and to obtain numerical results for the nonuniform case and for higher-point spin correlators in a very simple way and without resorting to Monte Carlo techniques.

Local models of fractional quantum Hall states in lattices and physical implementation

The fractional quantum Hall (FQH) effect is one of the most striking phenomena in condensed matter physics. It is described by a simple Laughlin wavefunction and has been thoroughly studied both theoretically and experimentally. In lattice systems, however, such an effect has not been observed, there are few simple models displaying it, and only few mechanisms leading to it are known. Here we propose a new way of constructing lattice Hamiltonians with local interactions and FQH like ground states. In particular, we obtain a spin 1/2 model with a bosonic Laughlin like ground state, displaying a variety of topological features. We also demonstrate how such a model naturally emerges out of a Fermi-Hubbard like model at half filling, in which the kinetic energy part possesses bands with nonzero Chern number, and we show how this model can be implemented in an optical lattice setup with present technology.The fractional quantum Hall effect is one of the most striking phenomena in condensed matter physics. It is described by a simple Laughlin wavefunction and has been thoroughly studied both theoretically and experimentally. In lattice systems, however, much less is currently known, and only few models and mechanisms leading to it have been identified. Here we propose a new way of constructing lattice Hamiltonians with local interactions and fractional quantum Hall like ground states. In particular, we obtain a spin 1/2 model with a bosonic Laughlin-like ground state, displaying a variety of topological features. We also demonstrate how such a model naturally emerges out of a Fermi-Hubbard-like model at half filling, in which the kinetic energy part possesses bands with non-zero Chern number, and we show how this model can be implemented in an optical lattice setup with present or planned technologies.

Laughlin Spin-Liquid States on Lattices Obtained from Conformal Field Theory

We propose a set of spin system wave functions that is very similar to lattice versions of the Laughlin states. The wave functions are conformal blocks of conformal field theories and for a filling factor of ν = 1/2 we provide a parent Hamiltonian, which is valid for any even number of spins and is at the same time a 2D generalization of the Haldane-Shastry model. We also demonstrate that the Kalmeyer-Laughlin state is reproduced as a particular case of this model. Finally, we discuss various properties of the spin states and point out several analogies to known results for the Laughlin states.

Lattice Laughlin states of bosons and fermions at filling fractions 1/ q

We introduce a two-parameter family of strongly-correlated wave functions for bosons and fermions in lattices. One parameter, q, is connected to the filling fraction. The other one, η, allows us to interpolate between the lattice limit (η = 1) and the continuum limit (η → + 0 ) of families of states appearing in the context of the fractional quantum Hall effect or the Calogero–Sutherland model. We give evidence that the main physical properties along the interpolation remain the same. Finally, in the lattice limit, we derive parent Hamiltonians for those wave functions and in 1D, we determine part of the low-energy spectrum.

Quantum filter reduction for measurement-feedback control via unsupervised manifold learning

We derive simple models for the dynamics of a single atom coupled to a cavity field mode in the absorptive bistable parameter regime by projecting the time evolution of the state of the system onto a suitably chosen nonlinear low-dimensional manifold, which is found by use of local tangent space alignment. The output field from the cavity is detected with a homodyne detector allowing observation of quantum jumps of the system between states with different average numbers of photons in the cavity. We find that the models, which are significantly faster to integrate numerically than the full stochastic master equation, largely reproduce the dynamics of the system, and we demonstrate that they are sufficiently accurate to facilitate feedback control of the state of the system based on the predictions of the models alone.

Conditional generation of path-entangled optical |N,0>+|0,N> states

We propose a measurement protocol to generate path-entangled vertical bar N,0>+ vertical bar 0,N> states conditionally from two pulsed type II optical parametric oscillators. We calculate the fidelity of the produced states and the success probability of the protocol. The trigger detectors are assumed to have finite dead time, and for short pulse trigger fields they are modeled as on-off detectors with finite efficiency. Continuous-wave operation of the parametric oscillators is also considered.

Quantum spin models for the SU(n)1 Wess-Zumino-Witten model

We propose 1D and 2D lattice wave functions constructed from the SU(n)1 Wess–Zumino–Witten (WZW) model and derive their parent Hamiltonians. When all spins in the lattice transform under SU(n) fundamental representations, we obtain a two-body Hamiltonian in 1D, including the SU(n) Haldane– Shastry model as a special case. In 2D, we show that the wave function converges to a class of Halperin’s multilayer fractional quantum Hall states and belongs to chiral spin liquids. Our result reveals a hidden SU(n) symmetry for this class of Halperin states. When the spins sit on bipartite lattices with alternating fundamental and conjugate representations, we provide numerical evidence that the state in 1D exhibits quantum criticality deviating from the expected behaviors of the SU(n)1 WZW model, while in 2D they are chiral spin liquids being consistent with the prediction of the SU(n)1 WZW model.

Atomic spin squeezing in an optical cavity

We consider squeezing of one component of the collective spin vector of an atomic ensemble inside an optical cavity. The atoms interact with a cavity mode, and the squeezing is obtained by probing the state of the light field that is transmitted through the cavity. Starting from the stochastic master equation, we derive the time evolution of the state of the atoms and the cavity field, and we compute expectation values and variances of the atomic spin components and the quadratures of the cavity mode. The performance of the setup is compared to spin squeezing of atoms by probing of a light field transmitted only once through the sample.

Single-photon-state generation from a continuous-wave nondegenerate optical parametric oscillator

We present a theoretical treatment of conditional preparation of one-photon states from a continuous-wave nondegenerate optical parametric oscillator. We obtain an analytical expression for the output state Wigner function, and we maximize the one-photon state fidelity by varying the temporal mode function of the output state. We show that a higher production rate of high fidelity Fock states is obtained if we condition the outcome on dark intervals around trigger photo detection events.

Exact parent Hamiltonians of bosonic and fermionic Moore–Read states on lattices and local models

We introduce a family of strongly-correlated spin wave functions on arbitrary spin-1/2 and spin-1 lattices in one and two dimensions. These states are lattice analogues of Moore-Read states of particles at filling fraction 1/q, which are non-Abelian Fractional Quantum Hall states in 2D. One parameter enables us to perform an interpolation between the continuum limit, where the states become continuum Moore-Read states of bosons (odd q) and fermions (even q), and the lattice limit. We show numerical evidence that the topological entanglement entropy stays the same along the interpolation for some of the states we introduce in 2D, which suggests that the topological properties of the lattice states are the same as in the continuum, while the 1D states are critical states. We then derive exact parent Hamiltonians for these states on lattices of arbitrary size. By deforming these parent Hamiltonians, we construct local Hamiltonians that stabilize some of the states we introduce in 1D and in 2D.