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Quantum Electronics

Quantum geometric phase under pre- and post-selection

T. S. Yakovleva, A. M. Rostom, V. A. Tomilin, L. V. Il’ichev

Published May 8, 2019• 4 citations
Publisher

Abstract

Abstract We consider a quantum system subjected to a controlled phase transformation and interaction with the environment in between the acts of selection, which leads to the emergence of interference effects. It is shown that the shift of the dependence of the statistics of contacts’ information with the environement on the controlled phase shift can be interpreted as a geometric phase. This interpretation is consistent with the known operational approach to the geometric phase. As a result, we suggest generalising the operational approach to the realm of pre- and post-selected quantum states.

DOI
10.1070/qel17014

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Quantum Electronics•Jun 1, 2020•1 citation
Abstract We consider a quantum system subjected to a controlled phase transformation and interaction with the environment in between the acts of selection, which leads to the emergence of interference effects. It is shown that the shift of the dependence of the statistics of contacts’ information with the environement on the controlled phase shift can be interpreted as a geometric phase. This interpretation is consistent with the known operational approach to the geometric phase. As a result, we suggest generalising the operational approach to the realm of pre- and post-selected quantum states.
This thesis consists of several studies performed over different few-dof quantum systems exposed to the effect of an uncontrolled environment. The primary focus of the work is to explore the relation between decoherence and environmentally-induced dissipative effects, and the concept known as geometric phases. The first mention of such an object in the context of quantum mechanics goes back to the seminal work by Berry. He demonstrated that the phase acquired by an eigenstate of a time-dependent Hamiltonian in an adiabatic cycle consists of two distinct contributions: one termed 'geometric' and the other known as the dynamical phase. Since Berry's work, the notion of geometric phase has been extended far beyond the original context, encompassing definitions applicable to arbitrary unitary evolutions. These geometric phases naturally arise in the geometric description of Hilbert space, where they manifest as holonomies and possess significance in the fundamental understanding of quantum mechanics and its mathematical framework, and in explaining various physical phenomena, including the Fractional Hall Effect. Moreover, from a modern perspective, geometric phases hold promise for practical applications, such as constructing geometric gates for quantum information processing and storage. However, in practice, a pure state of a quantum system is an idealized concept, and every experimental or real-world implementation must account for the presence of an environment that interacts with the observed system. This interaction necessitates a description in terms of mixed states and non-unitary evolutions. The definition of a geometric phase applicable in such scenarios remains an open problem, giving rise to multiple proposed solutions. Consequently, characterizing these geometric phases encompase motivations from fundamental aspects of quantum mechanics to technological applications.
Abstract When subject to measurements, quantum systems evolve along stochastic quantum trajectories that can be naturally equipped with a geometric phase observable via a post-selection in a final projective measurement. When post-selecting the trajectories to form a close loop, the geometric phase undergoes a topological transition driven by the measurement strength. Here, we study the geometric phase of a subset of self-closing trajectories induced by a continuous Gaussian measurement of a single qubit system. We utilize a stochastic path integral that enables the analysis of rare self-closing events using action methods and develop the formalism to incorporate the measurement-induced geometric phase therein. We show that the geometric phase of the most likely trajectories undergoes a topological transition for self-closing trajectories as a function of the measurement strength parameter. Moreover, the inclusion of Gaussian corrections in the vicinity of the most probable self-closing trajectory quantitatively changes the transition point in agreement with results from numerical simulations of the full set of quantum trajectories.
We will consider graph states generated by the action of controlled phase shift operators on a separable state of a multiqubit system. The case where all the qubits are initially prepared in arbitrary states is investigated. We will obtain the geometric measure of entanglement of a qubit with the remaining system in graph states represented by arbitrary weighted graphs and will establish its relationship with state parameters. For two-qubit graph states, the geometric measure of entanglement is also quantified on IBM’s simulator Qiskit Aer and quantum processor ibmq lima based on auxiliary mean spin measurements. The results of quantum computations verify our analytic predictions.
We revisit the quantum weak measurement (QWM) by sketching the polarisation state dynamics on the Poincaré sphere and the associated geometric phase, which is considered the soul of QWM. Our experimental arrangement comprises a coherent laser beam as a source of pure state, two polarizers corresponding to pre- and post-selection and a tilted wave plate placed in-between them to introduce weak interaction. The pre-selected state is a linearly polarised beam that can be represented as a point on the Poincaré sphere equator and the weak interaction with the wave plate results in a small spread in S3 axis. Now, an orthogonal projection of this state leads two different geodesics on the surface of the sphere through its poles, starting from both sides of the spread to the orthogonal point. The post-selection is made by moving the projection point slightly away from the orthogonal position so that both geodesics shift to the equator which results in a rapid accumulation of geometric phase in the beam crosssection. A gradient in the linear momentum is developed as a consequence which gives amplified shift in the beam position.
Mixed states typically arise when quantum systems interact with the outside world. Evolution of open quantum systems in general are described by quantum operations which are represented by completely positive maps. We elucidate the notion of geometric phase for a quantum system described by a mixed state undergoing unitary evolution and non-unitary evolutions. We discuss parallel transport condition for mixed states both in the case of unitary maps and completely positive maps. We find that the relative phase shift of a system not only depends on the state of the system, but also depends on the initial state of the ancilla with which it might have interacted in the past. The geometric phase change during a sequence of quantum operations is shown to be non-additive in nature. This property can attribute a "memory" to a quantum channel. We explore these ideas and illustrate them with simple examples.
Abstract As an alternative approach for measuring the weak effects associated with the artificial preparation of rare events in quantum metrology, we propose the study of the interference pattern generated by acts of pre- and postselection of a quantum system. An example of two Mach – Zehnder interferometers connected by a cross-Kerr nonlinearity is considered. Postselection of photon states at the output of one of the interferometers and the application of a controlled phase shift in one of its arms induces interference phenomena in the photodetection statistics at the output of the second interferometer. The nonlinearity parameter determines the shift and width of the structures in the interference pattern. The main features of this pattern are studied depending on the magnitude of the Kerr nonlinearity and the number of photons at the input of the interferometers.