Several recent experiments have demonstrated the promise of atomic ensembles for quantum teleportation and quantum memory. In these cases, the collective internal state of the atoms is well described by continuous variables corresponding to the operators ${X}_{1}{,P}_{1}$ and the interaction with the optical field ${(X}_{2}{,P}_{2})$ by a quadratic Hamiltonian ${X}_{1}{X}_{2}.$ We show how this interaction can be used optimally to create entanglement and squeezing. We derive conditions for the efficient simulation of quadratic Hamiltonians and the engineering of all Gaussian operations and states.
Several recent experiments have demonstrated the promise of atomic ensembles for quantum teleportation and quantum memory. In these cases, the collective internal state of the atoms is well described by continuous variables corresponding to the operators X1,P1 and the interaction with the optical field (X2,P2) by a quadratic Hamiltonian X1X2. We show how this interaction can be used optimally to create entanglement and squeezing. We derive conditions for the efficient simulation of quadratic Hamiltonians and the engineering of all Gaussian operations and states.
Quantum continuous variable systems are quantum systems obeying the canonical commutation relations. As we have seen in the previous chapter, such systems always require the adoption of infinite-dimensional Hilbert spaces, even when a finite number of degrees of freedom is considered. Besides making their dynamics potentially intractable, this fact also implies that the study of properties related to the information and correlation content of continuous variable systems turns out to be particularly difficult. A widely known way to make continuous variable systems more tractable is to consider the restriction to Gaussian states. Already well trodden in older research on quantum optics, the study of Gaussian systems has more recently become a central theme of continuous variable quantum information. If open quantum systems, coupled to an external environment, are considered, the Gaussian regime applies when the system couples linearly to its environment and when system and environment are governed by Hamiltonians which are at most quadratic in the canonical operators (or, equivalently, when the overall Hamiltonian of system and environment is at most quadratic). These conditions may seem rather restrictive but, as we shall point out in the course of the book, are actually ordinarily met by a vast number of existing experimental set-ups in the areas of quantum optics, trapped ions, optomechanics, atomic ensembles and certain superconducting degrees of freedom (all of these platforms are reviewed in Chapter 9 ). What we will refer to as Gaussian measurements – POVMs that preserve the Gaussian character of initial states – are also customarily carried out with comparatively high efficiency, while the description of quantum noise can be incorporated seamlessly into the Gaussian picture. What is perhaps even more important to note is that the restriction to the Gaussian realm, while not exhaustive of the full wealth of dynamics and situations allowed in the whole Hilbert space, still lets one include most of the processes relevant to quantum technologies. Squeezing, whereby certain canonical quadratures have uncertainties below the vacuum state noise, and can hence be used in precision measurements, quantum entanglement, and cooling, where the entropy is drained out of a system in order to reach a pure quantum state and thus initialise information protocols or low noise experiments, may all be described within the Gaussian regime. Within the quantum information discourse, the Gaussian restriction has been often criticised on the grounds that no real genuinely quantum effects can ever be observed without leaving it at some point, since the statistics obtained from Gaussian states through Gaussian measurements can be reproduced by classical probability distributions. While technically correct, the reader should be reminded that such critiques often disregard the fact that the object described by a Gaussian state does entail a genuinely quantum description in a Hilbert space. Hence, for instance, the preparation of a pure Gaussian quantum state is the preparation of a pure vector of the Hilbert space. Likewise, a highly squeezed Gaussian state prepared for a metrological protocol is a state with very low noise in a certain physical observable, regardless of whether it can be mimicked by a classical distribution or not. The testing and exploitation of quantum non-locality (stronger than classical correlations that imply the presence of quantum entanglement) with Gaussian states does instead require a departure from Gaussian measurements. However, as we will see, certain non-Gaussian measurements are customarily implemented with current technology, and it is hence still relevant to study in detail the creation of such a resource for Gaussian states. This chapter will introduce Gaussian states from a somewhat unusual angle: as ground and thermal states of at most second-order Hamiltonians. Our treatment will highlight the privileged role played by quadratic canonical ( i.e. , ‘symplectic’) transformations in this context, which we will take care to describe in full generality, and will hinge on the reduction of Gaussian states into their normal modes, which allows for their systematic diagonalisation. We will then proceed to explore an equivalent characterisation in terms of statistical moments of the canonical operators, and to introduce the implications of non-commutativity on such statistics in the form of the Robertson–Schrödinger uncertainty relation.
In this thesis I explore the use of collective spin angular momentum as a platform for quantum information processing. Such systems have several nice features that make them excellent choices for such protocols, especially ones where information is also stored in photonic variables. The longer coherence times of atoms makes it possible to store information from light in atoms for future use, and it is generally easier to couple atomic variables than to create nonlinear interactions of light. Above and beyond the advantages of atomic systems, collective atomic systems have additional strengths. In the limit of a large number of atoms, the collective variables of atomic systems have a natural connection to the bosonic algebra of light (known as the Holstein-Primakoff or HP approximation) where components of the collective spin angular momentum effectively act as quadratures, making them natural systems for coupling to light. Also, collectively addressing the atoms allows one to access a large state space without individual addressing of the atoms. A natural first step towards control of these collective variables is generating large amounts of atomic spin squeezing. In the HP approximation going from rotations to squeezing corresponds to going from linear to quadratic interactions in the atomic quadratures; for extremely large squeezing one should see the breakdown of the HP approximation and the ability to generate arbitrary collective atomic states. I have sought to improve previous schemes for the spin squeezing of atomic ensembles, such as the proposal of Takeuchi et. al. based on coherent quantum feedback. In this scheme a beam of linearly polarized light passes through the atomic ensemble (prepared in a coherent state), coupling to the atoms through a state-dependent index of refraction (the Faraday effect). The light is then passed through a wave-plate and reflected back through the atoms for a second pass. This double-pass scheme leads to an effective nonlinearity as the atomic fluctuations are mapped onto the light on the first pass and then back on to the atoms in the second pass. The light acts as a bus coupling each atom to each of the others. This nonlinear interaction forms a shearing of the atomic coherent state that results in squeezing. The light is entangled to the atoms through these interactions, and remains entangled as it escapes the system. This leads to decoherence of the atoms as the light is lost to the environment, reducing the amount of spin squeezing achieved. The first step towards improving the double-pass scheme was to add a quantum eraser step in which the light is disentangled from the squeezed atoms. By first measuring one quadrature of the light, and then performing a measurement-dependent rotation on the atomic ensemble, it is possible to decouple the atoms and light so that the loss of the light does not reduce the atomic squeezing. This results in an improvement of the rate of atomic spin squeezing. The nonlinear shearing interaction that remains still falls short of the exponential squeezing seen in optical parametric amplification. The reason for this can be seen by decomposing the shearing interaction into squeezing along a given quadrature followed by a rotation. The rotation leads to a constantly changing axis along which the squeezing occurs, and thus the squeezing builds up more slowly. This can be corrected for by breaking up the light pulse into a sequence of small pulses, and performing a small phase matching rotation after each pulse. The squeezing then adds up along a constant direction, rather than along a constantly varying direction. This results in still further improvement of the rate of atomic spin squeezing. A complete model includes the effects of photon-atom scattering and other noise and loss effects on the overall rate of squeezing. Previous derivations of noise due to photon-atom scattering have started with the unjustified assumptions that the atomic and photonic decoherence channels were both Gaussian and independent of correlations between the two subsystems. They then proceed with very general statistical arguments that rely upon these simplifications. My work begins with the more fundamental master equation picture in which I find the Linblad jump operators. I find that in general the photonic and atomic loss channels are not independent, with the intensity of the light dictating the details of this dependence, and find the conditions under which the Gaussian approximation holds. Squeezing and loss is initially treated assuming ensembles of spin-1/2 atoms, but this work is further extended to higher dimensional subsystems. For a higher spin case, preparing the atoms in spin coherent states is not optimal. One can engineer a stronger interaction by preparing the atoms in an atomic "cat state", i.e., a superposition of the two stretch states along the direction of propagation of the light beam. The fluctuations of such a state are more strongly coupled to the light, resulting in a stronger nonlinearity. This leads to strong correlations between the atoms, but they are not immediately useful for squeezing; the cat state must be coherently mapped to a coherent state to achieve atomic spin squeezing. The state created in this manner is ultimately more squeezed than that achieved with the same interaction but prepared initially in a coherent state.
The generation of entanglement in atomic systems plays a central topic in the fields of quantum information storage and processing. Moreover, a special category of entangled states of multi-atom ensembles, spin squeezed states, have been proven to lead to considerable improvement in the sensitivity of precision measurements compared to systems involving uncorrelated atoms. A treatment of entanglement in open systems is, however, incomplete without a precise description of the process of decoherence which necessarily accompanies it. The theory of entanglement and decoherence are the two main topics of this thesis. Methods are described for the generation of strong correlations in large atomic ensembles using either cavity quantum electrodynamics or measurement outcome conditioned quantum dynamics. Moreover, the description of loss of entanglement resulting from the coupling to a noise reservoir (electromagnetic vacuum) is explored. A spin squeezing parameter is used throughout this thesis as both a measure of entanglement strength and as an indication of the sensitivity improvement above the so-called standard quantum limit (sensitivity obtained with uncorrelated particles) in metrology. The first scheme considered consists of a single mode cavity field interacting with a collection of atoms for which spin squeezing is produced in both resonant and off-resonant regimes. In the resonant case, transfer of squeezing from a field state to the atoms is analyzed, while in the off-resonant regime squeezing is produced via an effective nonlinear interaction (one-axis twisting Hamiltonian). A second, more experimentally realistic case, is one involving the interaction of free space atoms with laser pulses; a projective measurement of a source field originating from atomic fluctuations provides a means of preparing atomic collective states such as spin squeezed and Schrodinger cat states. A new unravelling is proposed, that employs the detection of photon number in a single output channel and is capable of producing mesoscopic Schrodinger cat states in a single step. As a first step to understanding the role of cooperative decoherence, of importance in the case of dense pencil-shaped atomic ensembles, the collective spin decoherence of a two multilevel atom system is derived. This calculation is also relevant to entanglement loss for two qubits manipulated using with reading/writing pulses. Finally, a scheme is proposed in which arbitrarily strong entanglement can be produced in a four-wave mixing setup, where the preparation of the atoms in a dark state limits the decoherence to negligible values.
Results of experiments recently performed are reported, in which two optical parametric amplifiers were set up to generate two independently quadrature squeezed continuous wave laser beams. The transformation of quadrature squeezed states into polarization squeezed states and into states with spatial quantum correlations is demonstrated. By utilizing two squeezed laser beams, a polarization squeezed state exhibiting three simultaneously squeezed Stokes operator variances was generated. Continuous variable polarization entanglement was generated and the Einstein-Podolsky-Rosen paradox was observed. A pair of Stokes operators satisfied both the inseparability criterion and the conditional variance criterion. Values of 0.49 and 0.77, respectively, were observed, with entanglement requiring values below unity. The inseparability measure of the observed quadrature entanglement was 0.44. This value is sufficient for a demonstration of quantum teleportation, which is the next experimental goal of the authors.
This dissertation consists of three sections. In the first section, we discuss the generation \nof arbitrary two-qubit entangled states and present three generation methods. \nThe first method is based on the interaction of an atom with classical and quantized \ncavity fields. The second method is based on the interaction of two coupled two-level \natoms with a laser field. In the last method, we use two spin-1/2 systems which \ninteract with a tuned radio frequency pulse. Using those methods we have generated \ntwo qubit arbitrary entangled states which is widely used in quantum computing and \nquantum information. In the second section, we discuss a possible experimental implementation \nof quantum walk which is based on the passage of an atom through a \nhigh-Q cavity. The chirality is determined by the atomic states and the displacement \nis characterized by the photon number inside the cavity. Our scheme makes quantum \nwalk possible in a cavity QED system and the results could be widely used on \nquantum computer. In the last section, we investigate the properties of teleporting \nan arbitrary superposition of entangled Dicke states of any number of atoms (qubits) \nbetween two distant cavities. We also studied teleporting continuous variables of an \noptical field. Teleportation of Dicke states relies on adiabatic passage using multiatom \ndark states in each cavity and a conditional detection of photons leaking out of both cavities. In the continuous variables teleportation scheme we first reformulate \nthe protocol of quantum teleportation of arbitrary input optical field states in \nthe density matrix form, and established the relation between the P-function of the \ninput and output states. We then present a condition involving squeeze parameter \nand detection efficiency under which the P-function of the output state becomes the \nQ function of the input state such that any nonclassical features in the input state \nwill be eliminated in the teleported state. Based on the research in this section we \nhave made it possible of arbitrary atomic Dicke states teleportation from one cavity \nto another, and this teleortation will play an essential role in quantum communication. \nSince quantum properties is so important in quantum communication, the \ncondition we give in this section to distinguish classical and quantum teleportation is \nalso important.
Summary form only given. A new approach to the problem of the quantum interface between light and atoms has been recently developed. The method utilizes free space dispersive interaction between pulses of light and spin polarized atomic ensembles. Entanglement between the polarization state of light and the collective spin state of atoms is established by measurement, more precisely by detection of light in a certain polarization basis. In the first demonstration of this approach we have generated a long-lived entangled state of two separate macroscopic atomic samples by a polarization measurement on light transmitted through the samples. We then have shown that this approach also works for mapping of a quantum state of light onto long-lived atomic spin state paving the road towards realization of the quantum memory for light. Recently we have also extended the free space light-atoms interface beyond the continuous variables in the proposal, where a coherent superposition "cat" state of an atomic ensemble is generated by a measurement sequence. Progress with other communication protocols such as atomic state teleportation and multiparty networks will be presented.