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Quantum revivals and generation of non-classical states in an N spin system

Shane Dooley

Published Sep 30, 2014• 0 citations
OpenAlex

Abstract

The generation of non-classical states of large quantum systems is an important topic of study. It is of fundamental interest because the generation of larger and larger non-classical states extends quantum theory further and further into the classical domain, and it is also of practical interest because such states are an important resource for quantum technologies. \nThe focus of this thesis is the “spin star model” for the interaction of a single spin-1/2 particle with N other spin-1/2 particles. Although this is a simple model, we show that its dynamics include many interesting quantum phenomena, including fractional revival, and Jaynes- Cummings-like collapse and revival. Starting with a spin coherent state of the N spin system – an easily prepared state, in principle – we show that these dynamics can be used to generate a wide variety of non-classical states of the N spin system, including Schrodinger cat states, GHZ states, multiple-Schrodinger cat states and spin squeezed states.

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Fractional revivals, multiple-Schrödinger-cat states, and quantum carpets in the interaction of a qubit with N qubits

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Non-classical states in light-matter interactions.

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In this thesis we investigate the possibilities to create non-classical states by using light-matter interactions. Our basic approach to describe these interactions is the Jaynes-Cummings model. It describes interactions between a two-level atom and a quantized field in a lossless cavity (ultra high Q quality factor). The model can be easily generalized to describe multi-level atoms, Stark shifts, Kerr interactions, micromasers, etc. We use some of these generalizations in our study. Our main interest is in the production of Schrodinger cat states, that is, the coherent superposition of two or more classical macroscopic states; and the production of sub-Poissonian photostatistics. First of all, we use the simple Jaynes-Cummings model to investigate the interaction of coherent states with two-level atoms, and compare it to the case of interaction with a statistical mixture to show that we can discern between the two. We investigate as well the interaction of two level atoms and squeezed light, where we show that these states have a strong coherent character. However, the model itself is very simple, and as we have said before, some generalizations are necessary if we are to describe more realistically the atom field interactions. In particular, we use a phenomenological Hamiltonian, which incorporates Stark shifts in the atomic levels due to the electric field in the cavity, and show that some experimental results concerning the lineshapes in micromasers may be explained by such effects. Another step is to allow the atoms to enter and leave the cavity, which leads us to investigate a micromaser system. Within this structure, we allow for one- and two-photon transitions. In the latter case, it is shown that sub-Poissonian fields (by using special atom-field interaction times) and multi Schrodinger cat states (by measuring the state of the atoms as they leave the cavity) can be generated.

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Collapse and revival and cat states with an N -spin system

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Collapse, Revival and Decoherence of Entanglement in Two Qubits Systems

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We extend study of the Jaynes-Cummings model involving a pair of identical two-level atoms (or qubits) interacting with a single mode quantised field. We investigate the effects of replacing the radiation field mode with a ‘big spin’, comprising a collection of N qubits, or spin-1/2 particles. We demonstrate the similarities of this set-up to the qubits-field model in terms of the qubits state probability, occurrence of attractor states, generation of Schr¨odinger cat state, and in particular the collapse and revival of the entanglement between the two qubits in the qubit subsystem. We extend our analysis by taking into account a decoherence effect due to qubit imperfections. We study two cases of ‘error’ in the system for both the field mode and ‘big spin’ cases. In the first part, we consider the case of systems with non-resonance frequencies, and secondly we let the systems evolve with a difference in the dipole interaction strengths of the two qubits. We average over the errors in both of these parameters with distributions of varying width. We demonstrate the effects of such error modeling in both the field mode and the ‘big spin’ scenarios. We discover that increasing the width \nof the ‘error’ distribution increases suppression of the coherent dynamics of the coupled system, including the collapse and revival of the entanglement between the qubits. We also find out that the decoherence effects are more significant in the system with difference in the coupling strength as opposed to the nonresonance case that has higher robustness against errors. At the end of the study, we investigate the qubit-big spin system with a modest value of N to identify the smallest size of the big spin that exhibits the important events in such interacting \nmodel.

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Tailoring spin chain dynamics for fractional revivals

Alastair Kay

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.

arXiv:1609.01397•Quantum•Aug 10, 2017•9 citations
The generation of non-classical states of large quantum systems is an important topic of study. It is of fundamental interest because the generation of larger and larger non-classical states extends quantum theory further and further into the classical domain, and it is also of practical interest because such states are an important resource for quantum technologies. \nThe focus of this thesis is the “spin star model” for the interaction of a single spin-1/2 particle with N other spin-1/2 particles. Although this is a simple model, we show that its dynamics include many interesting quantum phenomena, including fractional revival, and Jaynes- Cummings-like collapse and revival. Starting with a spin coherent state of the N spin system – an easily prepared state, in principle – we show that these dynamics can be used to generate a wide variety of non-classical states of the N spin system, including Schrodinger cat states, GHZ states, multiple-Schrodinger cat states and spin squeezed states.
We study the dynamics of a system comprised of a single qubit interacting equally with NNN qubits (a ``spin star'' system). Although this model can be solved exactly, the exact solution does not give much intuition for the dynamics of the model. Here, we find an approximation that gives some insight into the dynamics for a particular class of initial spin-coherent states of the NNN qubits. We find an effective Hamiltonian for the system that is a finite Kerr (one-axis twisting) Hamiltonian for the N+1N+1N+1 qubits. The initial spin-coherent state evolves to spin-squeezed states on short time scales, and to ``multiple-Schr\"odinger-cat'' states (superpositions of many spin-coherent states) on longer time scales, a manifestation of the phenomenon of fractional revivals of the initial state. The evolution of the system is visualized with phase-space plots (QQQ functions) that, when plotted against time, reveal a ``quantum carpet'' pattern. Of particular interest is the fact that our approximation captures the qualitative features of the model even for small values of NNN. This suggests the possibility of observing the phenomenon of fractional revival in this model for systems of few qubits.
We analyze the quantum dynamics of the fractional-time Jaynes-Cummings model using a recent unitary framework for the fractional-time Schrödinger equation. We examine how the fractional derivative order ααα influences non-classical features under different initial conditions. For an initial Fock state, fractional evolution introduces transient dynamics and heightened sensitivity to coupling strength. Through an inverse problem approach, we interpret these effects as arising from an effective time-dependent coupling with a strong initial pulse. For an initial coherent state, the fractional order tunes the system between dynamical regimes, with a transition at $α= 0.50 $ where standard collapse-and-revival is replaced by stable, periodic evolution. This regime enhances non-classical field properties, including stronger sub-Poissonian statistics, periodic quadrature squeezing, and the formation of Schrödinger cat states.
In this thesis we investigate the possibilities to create non-classical states by using light-matter interactions. Our basic approach to describe these interactions is the Jaynes-Cummings model. It describes interactions between a two-level atom and a quantized field in a lossless cavity (ultra high Q quality factor). The model can be easily generalized to describe multi-level atoms, Stark shifts, Kerr interactions, micromasers, etc. We use some of these generalizations in our study. Our main interest is in the production of Schrodinger cat states, that is, the coherent superposition of two or more classical macroscopic states; and the production of sub-Poissonian photostatistics. First of all, we use the simple Jaynes-Cummings model to investigate the interaction of coherent states with two-level atoms, and compare it to the case of interaction with a statistical mixture to show that we can discern between the two. We investigate as well the interaction of two level atoms and squeezed light, where we show that these states have a strong coherent character. However, the model itself is very simple, and as we have said before, some generalizations are necessary if we are to describe more realistically the atom field interactions. In particular, we use a phenomenological Hamiltonian, which incorporates Stark shifts in the atomic levels due to the electric field in the cavity, and show that some experimental results concerning the lineshapes in micromasers may be explained by such effects. Another step is to allow the atoms to enter and leave the cavity, which leads us to investigate a micromaser system. Within this structure, we allow for one- and two-photon transitions. In the latter case, it is shown that sub-Poissonian fields (by using special atom-field interaction times) and multi Schrodinger cat states (by measuring the state of the atoms as they leave the cavity) can be generated.
We discuss collapse and revival of Rabi oscillations in a system comprising a qubit and a ``big spin'' (made of NNN qubits, or spin-1/21/21/2 particles). We demonstrate a regime of behavior analogous to conventional collapse and revival for a qubit-field system, employing spin coherent states for the initial state of the big spin. These dynamics can be used to create a cat state of the big spin. Even for relatively small values of NNN, states with significant potential for quantum metrology applications can result, giving sensitivity approaching the Heisenberg limit.
We extend study of the Jaynes-Cummings model involving a pair of identical two-level atoms (or qubits) interacting with a single mode quantised field. We investigate the effects of replacing the radiation field mode with a ‘big spin’, comprising a collection of N qubits, or spin-1/2 particles. We demonstrate the similarities of this set-up to the qubits-field model in terms of the qubits state probability, occurrence of attractor states, generation of Schr¨odinger cat state, and in particular the collapse and revival of the entanglement between the two qubits in the qubit subsystem. We extend our analysis by taking into account a decoherence effect due to qubit imperfections. We study two cases of ‘error’ in the system for both the field mode and ‘big spin’ cases. In the first part, we consider the case of systems with non-resonance frequencies, and secondly we let the systems evolve with a difference in the dipole interaction strengths of the two qubits. We average over the errors in both of these parameters with distributions of varying width. We demonstrate the effects of such error modeling in both the field mode and the ‘big spin’ scenarios. We discover that increasing the width \nof the ‘error’ distribution increases suppression of the coherent dynamics of the coupled system, including the collapse and revival of the entanglement between the qubits. We also find out that the decoherence effects are more significant in the system with difference in the coupling strength as opposed to the nonresonance case that has higher robustness against errors. At the end of the study, we investigate the qubit-big spin system with a modest value of N to identify the smallest size of the big spin that exhibits the important events in such interacting \nmodel.
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.