Feedback Control of Quantum Systems to Stabilize Superposition States

We derive stochastic master equations for a quantum system interacting with a Bose field prepared in a superposition of continuous-mode coherent states. To determine a conditional evolution of the quantum system, we use a collision model with an environment given as an infinite chain of not interacting between themselves qubits prepared initially in an entangled state being a discrete analogue of a superposition of coherent states of the Bose field. The elements of the environment chain interact with the quantum system in turn one by one, and they are subsequently measured. We determine a conditional evolution of the quantum system for continuous in time observations of the output field as a limit of discrete recurrence equations. We consider the stochastic master equations for a counting as well as for a diffusive stochastic process.

In this paper, a new state called superposition of three-mode photon-added trio coherent state is introduced and its nonclassical properties are studied. Based on the negative values of the Wigner function, we find that the nonclassical as well as non-Gaussian characteristics of the superposition of three-mode photon-added trio coherent state are more obvious than that of the original trio coherent state. The superposition of three-mode photon-added trio coherent state always appears three-mode sum squeezing while the trio coherent state does not. It also shows that the superposition of three-mode photon-added trio coherent state is genuinely entangled. The entanglement degree of this state is more improved compared with the trio coherent state. It becomes bigger and bigger through nonlocally increasing the number of added photons.

We calculate the second-order moments, the Robertson-Schrödinger uncertainty product, and the Mandel factor for various superpositions of coherent phase states with opposite arguments, comparing the results with similar superpositions of the usual (Klauder-Glauber-Sudarshan) coherent states. We discover that the coordinate variance in the analog of even coherent states can show the most strong squeezing effect, close to the maximal possible squeezing for the given mean photon number. On the other hand, the Robertson-Schrödinger (RS) uncertainty product in superpositions of coherent phase states increases much slower (as function of the mean photon number) than in superpositions of the usual coherent states. A nontrivial behavior of the Mandel factor for small mean photon numbers is discovered in superpositions with unequal weights of two components. An exceptional nature of the even and odd superpositions is demonstrated.

Quantum information and quantum computing offer a new range of possibilities in terms ofnew computing algorithms, the ability to perform new types of operations and make statementsabout information in terms of what it is physically possible to know about a system.Experiments in quantum information and quantum computing have been advancing significantly,especially over the last decade. In optical quantum computing there are two maintypes of experiments that are performed today. They are experiments involving single photonswith information encoded on the polarisation of the photon and experiments involvingstates with Gaussian noise profiles in their quadratures. There appears to be no way toperform the required operations for universal quantum computing using operations whichtake Gaussian states to Gaussian states. Universal quantum computing using single photonsis possible but uses either enormously large non-linearities or a large number of resources.Also, operations which take Gaussian states to non-Gaussian states are not well understoodin general and can often be very difficult to implement.This thesis presents three sources of optical non-Gaussian states which demonstrate someof the interesting properties of these states. First, a scheme for universal quantum computingusing superpositions of coherent states is presented. Error correction and the fault toleranceproperties of this scheme is also explored and compared with current single photon architectures.These gates require superpositions of coherent states as resources and a relativelysimple scheme is shown for how these states may be generated. Next a scheme for distillingnear ideal Gaussian entanglement using non-Gaussian operations is presented. The loss andnoise conditions under which ideal distillation can be achieved are presented. Finally, a verysimple way of generating mixed non-Gaussian states from two Gaussian states is presented.These states are shown to give rise to counter-intuitive results when measuring their entanglementand when analysing their operational ability in terms of teleportation and quantumkey distribution.

Quantum optics has proved a fertile field for experimental tests of quantum information science, from experimental verification of the violation of the Bell inequalities to quantum teleportation. However it was long believed that quantum optics would not provide a practical path to efficient and scaleable quantum computation, and most current efforts to achieve a scaleable quantum computer have focussed on solid state implementations. This orthodoxy was challenged recently when Knill et al. showed that given single photon sources and single photon detectors, linear optics alone would suffice to implement efficient quantum computation. While this result is surprising, the complexity of the optical networks required is daunting. In this talk we propose an efficient scheme which is elegant in its simplicity. We indicate how fundamental single and two qubit gates can be achieved. By encoding the quantum information in multi-photon coherent states, rather than single photon states, simple optical manipulations acquire unexpected power. As an application of this new information processing ability we investigate a class of high precision measurements. We show how superpositions of coherent states allow displacement measurements at the Heisenberg limit. Entangling many superpositions of coherent states offers a significant advantage over a single mode superposition states with the same mean photon number.

Role of quantum interference in the origin of higher-order nonclassical characteristics of radiation field has been probed vis-a-vis a discrete and a continuous superposition of coherent states. Specifically, the possibilities of observing higher-order nonclassical properties (e.g., higher-order antibunching (HOA), higher-order sub-Poissonian photon statistics (HOSPS), higher-order squeezing (HOS) of Hong-Mandel type and Hillery type) have been investigated using a shifted symmetric cat state that reduces to Yurke-Stoler, even and odd coherent states at various limits. This shifted symmetric cat state which can be viewed as a discrete superposition of coherent states is found to show HOA and HOSPS. Similarly, higher-order nonclassical properties of a one-dimensional continuous superposition of coherent states is also studied here. The investigation has revealed the existence of HOS and HOSPS in the one-dimensional continuous superposition of coherent states studied here. Effect of non-Gaussianity inducing operations (e.g., photon addition and addition followed by subtraction) on these superposition states have also been investigated. Finally, some comparisons have been made between the higher-order nonclassical properties of discrete and continuous superposition of coherent states.

We develop an all-optical scheme to generate superpositions of macroscopically distinguishable coherent states in traveling optical fields. It non-deterministically distills coherent state superpositions (CSSs) with large amplitudes out of CSSs with small amplitudes using inefficient photon detection. The small CSSs required to produce CSSs with larger amplitudes are extremely well approximated by squeezed single photons. We discuss some remarkable features of this scheme: it effectively purifies mixed initial states emitted from inefficient single photon sources and boosts negativity of Wigner functions of quantum states.

Chapter 3 - Superpositions of Degenerate Quantum States: Preparation and Detection in Atomic Beams

We calculate and plot 2D slices of the Wigner functions of several families of highly excited even and odd superpositions of nonlinear coherent states, looking for conditions under which such superpositions can be interpreted as models of the “Schrödinger cat” states. The decisive factor seems to be the form of the number distribution function over the Fock basis: it must have well localized peaks. Otherwise, no “cat-like” structures are observed.

The neutrino oscillation patterns can be modified by neutrino interactions with external environments including electromagnetic fields that can influence on neutrinos in the case neutrinos have nonzero electromagnetic properties [1]. The phenomenon of neutrino oscillations can proceed only in the case of the coherent superposition of neutrino mass states. An external environment can modify a neutrino evolution in a way that conditions for the coherent superposition of neutrino mass states are violated. Such a violation is called quantum decoherence of neutrino states and leads to the suppression of flavor neutrino oscillations [2,3]. Note that neutrino decoherence appeared due to the wave separation of different mass states is usually not related to quantum neutrino decoherence, the effect that is not considered below. We consider the neutrino quantum decoherence due to neutrino radiative decay in the presence of an electron medium and radiation field. The corresponding damping of neutrino oscillations is calculated.

In this paper we discuss the nonclassical properties of quantum superpositions of coherent states of light. Using general expressions for the Wigner functions of superposition states we analyze the consequences of quantum interference between coherent states. We describe in detail nonclassical properties of a superposition of two coherent states. In particular, we study the oscillatory behavior of the photon number distribution of the even and odd coherent states. We show under which conditions a superposition of two coherent states can exhibit second- and fourth-order squeezing or sub-Poissonian photon statistics. We examine the sensitivity of nonclassical effects such as oscillations in the photon number distribution or second-order squeezing to dissipation. We demonstrate that quantities such as the photon number distribution and interferences in phase space are highly sensitive to even a quite small dissipative coupling, because they depend on all moments of the field observables, and higher moments decay more rapidly than lower moments. Quantities such as quadrature squeezing, on the other hand, are more robust against dissipation because they involve only lower moments. Finally, we find a remarkable effect whereby fourth-order squeezing is generated by damping.

Evolutions of quantum noise, characterized by quadrature squeezing parameter and Fano factor, and of mixedness, quantified by quantum von Neumann and linear entropies, of a pumped dissipative non-linear oscillator are studied. The model can describe a signal mode interacting with a thermal reservoir in a parametrically pumped cavity with a Kerr non-linearity. It is discussed that the initial pure states, including coherent states, Fock states, and finite superpositions of coherent states evolve into the same steady mixed state as verified by the quantum relative entropy and the Bures metric. It is shown analytically and verified numerically that the steady state can be well approximated by a nonclassical Gaussian state exhibiting quadrature squeezing and sub-Poissonian statistics for the cold thermal reservoir. A rapid increase is found in the mixedness, especially for the initial Fock states and superpositions of coherent states, during a very short time interval, and then for longer evolution times a decrease in the mixedness to the same, for all the initial states, and relatively low value of the nonclassical Gaussian state.

As a generalization of the optical Schr\"odinger cats, discrete sets of coherent states are considered on a circle in the \ensuremath{\alpha} plane. It is shown that simple superpositions of Schr\"odinger cats exhibit amplitude squeezing, similarly to the case of a superposition of several coherent states along a straight line that shows quadrature squeezing. The interference fringes between the coherent states form the annuli of the Fock states in the Wigner-function picture. It is also shown that a continuous superposition of coherent states on a circle can serve as a basis for the representation of any state.

We propose a scheme to generate a superposition of coherent states with arbitrary coefficients on a line in phase space for the center-of-mass vibrational mode of ions by means of isolating all other spectator vibrational modes from the center-of-mass mode. It can be viewed as the generalization of previous methods for preparing motional states of one ion. For a large number of ions, only one cyclic operation enables one to generate such a superposition of many coherent states.

We examine various uncertainty relations: (electromagnetic-field phase (the so-called “Pegg–Barnett operator”))–(number of photons), (trigonometric functions of the phase)–(number of photons), and (trigonometric functions of the phase)–(field phase). An electromagnetic field in quantum states of phase superpositions of coherent states, which are general superpositions of two coherent states with identical moduli but arbitrary phases, is considered. The rigorous uncertainty relation (Cauchy inequality) and the soft uncertainty relation (Heisenberg inequality) are examined and compared.