We design a quasi-one-dimensional spin chain with engineered coupling strengths such that the natural dynamics of the spin chain evolves a single excitation localized at the left-hand site to any specified single-particle state on the whole chain. Our treatment is an exact solution to a problem which has already been addressed in approximate ways. As two important examples, we study the $W$ states and Gaussian states of arbitrary width.
We design a quasi-one-dimensional spin chain with engineered coupling strengths such that the natural dynamics of the spin chain evolves a single excitation localized at the left-hand site to any specified single-particle state on the whole chain. Our treatment is an exact solution to a problem which has already been addressed in approximate ways. As two important examples, we study the W states and Gaussian states of arbitrary width.
The production of quantum states required for use in quantum protocols & technologies is studied by developing the tools to re-engineer a perfect state transfer spin chain so that a separable input excitation is output over multiple sites. We concentrate in particular on cases where the excitation is superposed over a small subset of the qubits on the spin chain, known as fractional revivals, demonstrating that spin chains are capable of producing a far greater range of fractional revivals than previously known, at high speed. We also provide a numerical technique for generating chains that produce arbitrary single-excitation states, such as the W state.
The realization of a quantum computer, after being considered only a wishful thinking for many years, \nlooks now more solid, with the major issues about its realization looking like “engineering problems”, whose \nsolution is not far to be found. \nThe most considerable open problems concerning the realization of a quantum computer concern: the \nsensitivity to external disturbances, the addressability of any single qubit, and the scalability of the proposed \narchitectures. These problems may have opposite solutions, since a good scalability would require high \ndensity of qubits embedded in solid state matrices, making it more difficult to address and protect each of \nthem. Conversely, well separated qubits realized by atoms or ions are easily addressed and protected, but \nmake it more difficult to achieve scalability. \nAnother aspect related to the realization of generic quantum devices (i.e. devices which exploit the \nquantum features of their constituents to realize some operations) is connected to the way one interfaces \nwith them: we live in a “classical world”, and any quantum device will have to interface with it at some \nstage, e.g. to be controlled or measured. The problem, in this case, is keeping our classical disturbances as \nfar as possible from the action taking place at the heart of the quantum device. \nHowever, the problem of being able to find the proper “interface” can be turned into an advantage, \nsince macroscopic systems may posses robustness features which can be exploited for the realization of novel \nquantum devices where the sensitivity problem is much alleviated. A good interface can be thus provided by \nsemi-classical systems, i.e. objects whose nature is intimately quantum, but whose features and behaviour is \nwell framed within a classical description. Any scheme, application or device, where a semi-classical system \ninteracts with some pure quantum object (e.g. a qubit) for realizing a quantum operation, will be referred \nto as hybrid. \nIn this thesis, we have developed the idea of exploiting the robustness features of some non-linear classical \nsystems in order to realize quantum devices where the sensitivity problem is alleviated. Hence, we have \nproposed a hybrid scheme for accomplishing the most basic actions related with the realization of a magnetic- \nbased quantum device, i.e. the control of a qubit state and entanglement generation. In particular, we have \ndeveloped it focusing on magnetic one-dimensional systems, i.e. spin chains, as channels connecting one or \nmore qubits to some external control apparatus. \nThis choice follows from the observation that classical spin chains are non-linear systems which enumerate \nsolitons among their dynamical configurations which are solutions of their equations of motion. In fact, \nsolitons are those particular solutions of non-linear systems which show space-time localization and shape- \ninvariant evolution and are celebrated for their impressive properties of robustness against scattering and \nexternal disturbances. These properties make them suitable candidates for practical purposes. \nWe have considered systems made by one-dimensional discrete lattices hosting, at each lattice site, one \nclassical spin (classical spin chain) or a large-S spin, i.e. a quantum spin characterized by a S-value large \nenough for the spin to be well described by a semi-classical behaviour (large-S spin chain). \nA classical, or a large-S, spin chain (with its solitons) can play the role of the robust partner while \nthe role of the fragile quantum system is played by the qubit, which is the agent of the relevant quantum \noperations in our hybrid quantum device.We have first introduced a method for generating solitons on discrete, classical Heisenberg chains by \napplying a time-dependent magnetic field to one of the chain extremities. The method has been numerically \nchecked, revealing the actual possibility of producing soliton-like dynamical configurations running on the \nspin chain, which resemble the known analytical solitons of the continuous chain if their typical width is \nlarge with respect to the chain spacing. The robustness of the generated solitons has been also tested with \nrespect to thermal noise present in the system. \nWe have then proposed a set-up where the generated soliton acts as a magnetic signal that travels along \nthe chain and eventually reaches a qubit and changes its quantum state. Since any unitary action on a single \nqubit can be represented in terms of a Zeeman interaction lasting for a precise time interval, qubit state \ncontrol is usually assumed to be obtained by applying suitable sequences of external magnetic fields. Indeed, \nfor this particular application, the spin chain needs not to be quantum and the suitable magnetic field is \nprovided by the moving deformation of the uniform chain represented by the magnetic soliton travelling \nalong the chain. Numerical results confirm that solitons are suitable for this task, giving the possibility to \nremotely control the qubit state by an appropriate choice of soliton shape and qubit-chain coupling. \nWe have finally addressed the problem of generating entanglement between distant qubits by introducing \na model where two qubits, distant and non-interacting, are locally coupled with a large-S Heisenberg chain, \nwhose dynamics is assumed to be characterized by the presence of solitons. The aim of this study was to \nverify if, by properly choosing the state of the spin chain, the evolution of such a robust semi-classical system \ncan bring the two qubits from a separable initial state to a non-separable state after a certain amount of \ntime, i.e., if a semi-classical channel could generate entanglement between the two qubits. \nAt variance with the previous application, the spin chain must here be considered, and consequently \ntreated, as a quantum system in order to allow for entanglement generation/transfer, but a suitable approximation \nis needed to solve the chain dynamics, as accounting for the exact evolution of large-S spin chain \nis out of reach, even numerically. This leads us to introduce a particular set of chain states built as product \nof single-spin coherent states which are in one-to-one correspondence with the configurations of the classical \nspin chain and are thus referred to as the semi-classical states of the chain. Being able to solve the evolution \nof the coherent state products, allow us to complete the hybrid scheme for entanglement generation: in \nfact, it is shown that, choosing the chain initial state as a semi-classical state corresponding to a soliton \nconfiguration, the correlations, generated between one qubit and the corresponding spin-S, are efficiently \ntransferred along the chain up to the other qubit, finally leaving the two qubits in an entangled state. \nThe results about the proposed hybrid schemes, showing their effectiveness for remotely controlling qubit \nstates and generating entanglement between distant qubits, encourage further studies opening also new \nperspectives for the realization of novel quantum devices based on the exploitation of the robustness features \nof semi-classical systems.
The spin chain is a theoretical work-horse of the physicist, providing a convenient, tractable model that yields insight into a host of physical phenomena including conduction, frustration, superconductivity, topological phases, localisation, phase transitions, quantum chaos and even string theory. Our ultimate aim, however, is not just to understand the properties of a physical system, but to harness it for our own ends. We therefore study the possibilities for engineering a special class of spin chain, envisaging the potential for this to feedback into the original physical systems. We pay particular attention to the generation of multipartite entangled states such as the W (Dicke) state, superposed over multiple sites of the chain.
We solve the Schrödinger equation for an interacting spin chain locally coupled to a quantum environment with a specific degeneracy structure. The reduced dynamics of the whole spin chain as well as of single spins is analyzed. We show that the total spin chain relaxes to a thermal equilibrium state independently of the internal interaction strength. In contrast, the asymptotic states of each individual spin are thermal for weak but nonthermal for stronger spin-spin coupling. The transition between both scenarios is found for couplings of the order of 0.1 x deltaE , with deltaE denoting the Zeeman splitting. We compare these results with a master-equation treatment; when time averaged, both approaches lead to the same asymptotic state and finally with analytical results.
We present an introductory overview of the use of spin chains as quantum wires, which has recently developed into a topic of lively interest. The principal motivation is in connecting quantum registers without resorting to optics. A spin chain is a permanently coupled 1D system of spins. When one places a quantum state on one end of it, the state will be dynamically transmitted to the other end with some efficiency if the spins are coupled by an exchange interaction. No external modulations or measurements on the body of the chain, except perhaps at the very ends, is required for this purpose. For the simplest ( uniformly coupled) chain and the simplest encoding (single qubit encoding), however, dispersion reduces the quality of transfer. We present a variety of alternatives proposed by various groups to achieve perfect quantum state transfer through spin chains. We conclude with a brief discussion of the various directions in which the topic is developing.
An essential role for the description of correlated quantum many-particle systems is played by impurity problems. These consist of a small number of localized orbitals, where the electrons are subject to a Coulomb-interaction, coupled to noninteracting bath degrees of freedom. Impurity models are used for the description of magnetic impurities in metals, of quantum points in nanostructures, and in the context of the dynamical mean field theory (DMFT). For their solution a large number of numerical methods exists. We mention here quantum Monte Carlo methods, the numerical renormalization group, or the exact diagonalization method. In this thesis we introduce a new approach to impurity problems, which is based on the functional renormalization group (fRG). In contrast to the methods mentioned above this approach is approximative, but it can be applied with comparatively lower numerical effort. The aim of this thesis is to examine the advantages and limits of this method. An important role in this connection is played by the single impurity Anderson model (SIAM), which is already understood very well with other methods and is seen as a prototype for models with strong local correlations. The spin fluctuations of this model are governed by the so-called Kondo energy scale and a central question of this thesis is whether this scale can be reproduced with our approach. We begin with a short introduction to the Green's function formalism for correlated quantum many particle systems. Then the fRG flow equations for the one-particle irreducible vertex functions are deduced, and several approximation schemes are discussed. In chapter 4 we introduce the fRG scheme used in this thesis. At first the SIAM is mapped to a semi-infinite chain, in which the interacting orbital is given by the first site of this chain. The system is then subdivided into two parts. The first part, which is called 'core', consists of the interacting orbital and the first L bath sites. The remaining bath sites form the second part. At the beginning both parts are decoupled from each other such that the core can be solved exactly. Starting with this exact solution the coupling between the core and the remaining bath sites is switched on slowly in a renormalization group flow. This way it is possible to calculate the one-particle and two-particle correlation functions on the correlated site. We call this flow scheme 'hybridization flow'. The flow equations are formulated in an effective theory on the first bath site outside the core, which turns out to be advantageous compared to other implementations of the hybridization flow. In chapter 5 this hybridization flow method is applied to the SIAM with semi-elliptic bath density of states. We discuss the differences which arise for different core-sizes (L = 0,1,2,3) and for different truncations of the fRG flow equations. In the cases L=0 and L=2 the local Fermi liquid properties of the SIAM and the associated Kondo scale can not be reproduced in both approximation schemes. For the core sizes L=1 and L=3 the ground state of the core is already a spin singlet state in the beginning of the flow and the Fermi liquid properties are reproduced. The Kondo energy scale, which determines the width of the central quasi-particle peak and the size of the local spin-susceptibility, is however not accurately resolved. This is partly due to the finite temperature used in our implementation such that for larger interaction strength the temperature is above the Kondo energy scale. The application of the hybridization flow method to quantum impurity problems in the context of the DMFT is discussed in chapter 6. In this approach lattice models with a local interaction, like the Hubbard model, are mapped to an effective impurity problem for a single lattice site embedded in a dynamical mean field representing the influence of the other electrons. This scheme is exact in infinitely many dimensions and a nonperturbative approximation method for finite-dimensional systems. In particular the Mott metal-insulator transition in the Hubbard model is successfully described by this method. One class of quantities, which is easily accessible from the hybridization flow scheme, is given by the local two-particle correlation functions. These play an essential role in non-local extensions of the DMFT. Therefore we mainly focus on these quantities. At first we apply our flow scheme to the Hubbard model on the Bethe lattice in infinite dimensions and calculate the local one-particle irreducible vertex functions in the insulating and the metallic phase in good agreement with previous calculations, which use the exact diagonalization method as impurity solver. After this we extend our flow scheme to a cluster DMFT method, which includes short-range antiferromagnetic correlations and calculate again the local and non-local vertex functions for the two-dimensional Hubbard model. We compare these to the results of the single-site DMFT.