We apply the Sklyanin method of separation of variables to the reflection algebra underlying the open spin-1 2 XXX chain with non-diagonal boundary fields. The spectral problem can be formulated in terms of a T Q-equation which leads to the known Bethe equations for boundary parameters satisfying a constraint. For generic boundary parameters we study the asymptotic behaviour of the solutions of the T Q-equation.

We study the spectrum of the integrable open XXX Heisenberg spin chain subject to non-diagonal boundary magnetic fields. The spectral problem for this model can be formulated in terms of functional equations obtained by separation of variables or, equivalently, from the fusion of transfer matrices. For generic boundary conditions the eigenvalues cannot be obtained from the solution of finitely many algebraic Bethe equations. Based on careful finite size studies of the analytic properties of the underlying hierarchy of transfer matrices we devise two approaches to analyze the functional equations. First we introduce a truncation method leading to Bethe-type equations determining the energy spectrum of the spin chain. In a second approach, the hierarchy of functional equations is mapped to an infinite system of nonlinear integral equations of TBA type. The two schemes have complementary ranges of applicability and facilitate an efficient numerical analysis for a wide range of boundary parameters. Some data are presented on the finite-size corrections to the energy of the state which evolves into the antiferromagnetic ground state in the limit of parallel boundary fields.

We derive functional equations for the eigenvalues of the XXZ model subject to anti-diagonal twisted boundary conditions by means of fusion of transfer matrices and by Sklyanin's method of separation of variables. Our findings coincide with those obtained using Baxter's method and are compared to the recent solution of Galleas. As an application we study the finite size scaling of the ground-state energy of the model in the critical regime.

We formulate the functional Bethe ansatz for bosonic (infinite dimensional) representations of the Yang-Baxter algebra. The main deviation from the standard approach consists in a half infinite Sklyanin lattice made of the eigenvalues of the operator zeros of the Bethe annihilation operator. By a separation of variables, functional T Q-equations are obtained for this half infinite lattice. They provide valuable information about the spectrum of a given Hamiltonian model. We apply this procedure to integrable spin-boson models subject to both twisted and open boundary conditions. In the case of general twisted and certain open boundary conditions polynomial solutions to these T Q-equations are found and we compute the spectrum of both the full transfer matrix and its quasi-classical limit. For generic open boundaries we present a two-parameter family of Bethe equations, derived from T Q-equations that are compatible with polynomial solutions for Q. A connection of these parameters to the boundary fields is still missing.

We present experimental results on a dispersive scheme for reading out a Josephson phase qubit. A capacitively shunted dc-SQUID is used as a nonlinear resonator which is inductively coupled to the qubit. We detect the flux state of the qubit by measuring the amplitude and phase of a microwave pulse reflected from the SQUID resonator. By this low-dissipative method, we reduce the qubit state measurement time down to 25 µs, which is much faster than using the conventional readout performed by switching the SQUID to its non-zero dc voltage state. The demonstrated readout scheme allows for reading out multiple qubits using a single microwave line by employing frequency-division multiplexing. , consist of superconducting loops interrupted by one or more Josephson junctions. Since their readable states can be discriminated by the magnetic flux passing through the qubit loop, it is common to use inductively coupled dc-SQUIDs as sensitive detectors.The standard method to read out a Josephson phase qubit is to record the dc bias current at which the SQUID switches to its non-superconducting state [3][4][5][6]. This process generates heat directly on the chip and quasiparticles in the circuitry. Both effects are responsible for a relatively long cool-down time of about 1-2 ms that is required after each switching event. This, together with the time needed to ramp up the bias current of the SQUID, limits the repetition rate of the experiment.For the flux qubit, non-destructive dispersive readout schemes have successfully been realized already some time ago, either by coupling to a high quality LC-tank circuit [7], or to a dc-SQUID [8]. So far, most measurements of phase qubits were typically done by the above mentioned switching current measurement of an inductively coupled dc-SQUID. Recently, first experiment overcoming the limitations of the switching readout was reported [9]. In this approach, the phase qubit was capacitively coupled to a transmission line which allows for direct probing its resonance frequency with a microwave pulse. This approach eliminates a readout dc-SQUID, but in turn introduce decoherence via the line coupled directly to the qubit.In this letter, we present experiments on dispersive * Electronic address: ustinov@kit.edu readout of a SQUID weakly coupled to a phase qubit. By using weak coupling between the SQUID and the qubit, this scheme protects the qubit from decoherence sources introduced by the readout circuitry. Moreover, while preserving the intrinsic coherence of the qubit, this method is suitable for reading out many qubits using a single microwave line and frequency-division multiplexing addressing individual readout SQUID resonators. We couple the qubit to a capacitively shunted dc-SQUID which forms a tank circuit having a resonance frequency around 2 GHz. It is connected to a microwave line by a coupling capacitor C 0 shown in Fig. 1. Our sample was fabricated in a standard niobium-aluminium trilayer process. Measurement of the amplitude and phase of a reflected microwave pulse allows ...