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Geometriska Faser I Rena Och Blandade Kvantmekaniska System

Haider, Miran

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

Note that equations and expressions has been omitted here and is instead presented in the work. This thesis focuses on the geometric phase in pure and mixed quantum states. For the case of a pure quantum state, Berry's adiabatic approach (4.1.10) and Aharonov & Anandan's non-adiabatic generalization of Berry's approach (4.2.8) are included in this work. Mixed quantum state involves Uhlmanns approach (5.1.42), which is used extensively in Section 7 and Sjöqvist's et al. approach (5.2.22), used extensively in Section 6. Sjöqvist's approach states that the Uhlmann phase is an observable and provides the experimental groundworkusing an interferometer.􀀀This was later proven, by Du et al.[45] to reproduce experimental data (Figure 19) on page 56. The Uhlmann phase can be used to observe the behaviour of topological kinks. This was tested on 3 models, the Creutz-ladder, the Majorana chain andthe SSU-model. It is found that the Uhlmann phase is split into two regimes with the dividing parameter being the temperature. This temperature is called the critical temperature, Tc. If the temperatureis is below the critical temperature, the Uhlmann phase yields Π and if thetemperature is above the critical temperature, the Uhlmann phase yields zero.

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Note that equations and expressions has been omitted here and is instead presented in the work. This thesis focuses on the geometric phase in pure and mixed quantum states. For the case of a pure quantum state, Berry's adiabatic approach (4.1.10) and Aharonov & Anandan's non-adiabatic generalization of Berry's approach (4.2.8) are included in this work. Mixed quantum state involves Uhlmanns approach (5.1.42), which is used extensively in Section 7 and Sjöqvist's et al. approach (5.2.22), used extensively in Section 6. Sjöqvist's approach states that the Uhlmann phase is an observable and provides the experimental groundworkusing an interferometer.􀀀This was later proven, by Du et al.[45] to reproduce experimental data (Figure 19) on page 56. The Uhlmann phase can be used to observe the behaviour of topological kinks. This was tested on 3 models, the Creutz-ladder, the Majorana chain andthe SSU-model. It is found that the Uhlmann phase is split into two regimes with the dividing parameter being the temperature. This temperature is called the critical temperature, Tc. If the temperatureis is below the critical temperature, the Uhlmann phase yields Π and if thetemperature is above the critical temperature, the Uhlmann phase yields zero.
Two geometric phases of mixed quantum states, known as the interferometric phase and Uhlmann phase, are generalizations of the Berry phase of pure states. After reviewing the two geometric phases and examining their parallel-transport conditions, we specify a class of cyclic processes that are compatible with both conditions and therefore accumulate both phases through their definitions, respectively. Those processes then facilitate a fair comparison between the two phases. We present exact solutions of two-level and three-level systems to contrast the two phases. While the interferometric phase exhibits finite-temperature transitions only in the three-level system but not the two-level system, the Uhlmann phase shows finite-temperature transitions in both cases. Thus, using the two geometric phases as finite-temperature topological indicators demonstrates the rich physics of topology of mixed states.
Geometrical phases have been applied in virtually every major branch of physics and they play an important role in topology and knot theory in mathematics and quantum computation. However, most of the early works focus on pure quantum states which are very unrealistic and almost never occur in practice. The two existed definitions -Uhlmann's and Sjöqvist's- would result in different values for the geometric phase in general. The definition of geometric phase in mixed state scenario is still an open question. Here we present a unified framework for both approaches within one and the same formalism based on simple interferometry. We then present experimental results which confirm both approaches to mixed state geometric phase and clearly demonstrate that their unification is possible. Our experiments are furthermore the first such to measure Uhlmann's mixed state geometric phase and show in addition that it is different to the Sjöqvist's phase.
The Berry phase arises naturally from the adiabatic dynamics of a pure quantum state, whereas the Uhlmann phase is usually formulated kinematically for a prescribed path of mixed states. For a qubit subject to a uniformly gapped conical drive, we obtain the Uhlmann connection and holonomy in closed form for the equilibrium Gibbs cycle. We then solve the corresponding Lindblad equation exactly in the rotating frame and show that its steady state forms a nonequilibrium limit cycle that lags the instantaneous Gibbs cycle. As the driving becomes slow, the Uhlmann phase approaches the equilibrium Uhlmann phase, which cooling then reduces to the Berry phase: in the joint adiabatic and low-temperature limit, the dynamical Uhlmann phase therefore converges to the Berry phase. This grounds the Uhlmann--Berry correspondence in the physical dynamics of an open system. We also characterize the finite-driving corrections, together with the discontinuous πππ jump that the Uhlmann phase undergoes on the equatorial cycle as the temperature is varied. Finally, we examine an isolated transversal gap closing, where this zero-temperature correspondence can break down: the pure-state path becomes open, its closure is ambiguous, and gap-opening regularizations close it along a geodesic set by the direction of the bias field, splitting the Berry phase into a one-parameter family of values. The Gibbs path instead closes smoothly through the maximally mixed state, so the Uhlmann holonomy requires no regularization and remains unique and continuous at every finite temperature. In the low-temperature limit the Uhlmann phase selects a single member of the Berry family---the geodesic closure in the osculating plane, fixed by the velocity and acceleration of the driving field. The Uhlmann--Berry correspondence of the gapped regime thus survives the gap closing as a geodesic selection rule.
The Berry phase is a geometric phase of a pure state when the system is adiabatically transported along a loop in its parameter space. The concept of geometric phase has been generalized to mixed states by the so-called Uhlmann phase. However, the Uhlmann phase is constructed from the Uhlmann connection that possesses a well-defined global section. This property implies that the Uhlmann connection is topologically trivial and, as a consequence, the corresponding Chern character vanishes. We propose a modified Chern character whose integral gives the thermal Uhlmann Chern number, which is related to the winding number of the mapping defined by the Hamiltonian. Therefore, the thermal Uhlmann Chern number reflects the topological properties of the underlying Hamiltonian of a mixed state. By including the temperature dependence in the volume integral, we also introduce the nontopological thermal Uhlmann Chern number, which varies with temperature but is not quantized at finite temperatures. We illustrate the applications to a two-band model and a degenerate four-band model.
Pure-state manifestations of geometric phase are well established and have found applications across essentially all branches of physics, yet their generalization to mixed-state regimes remains largely unexplored experimentally. The Uhlmann geometric phase offers a natural extension of pure-state paradigms and can exhibit a topological character. However, observation of this invariant is impeded by the incompatibility between Uhlmann parallel transport and Hamiltonian dynamics, as well as the difficulty of preparing topologically nontrivial mixed states. To address this challenge, we report an experimentally accessible protocol for directly measuring the mixed-state topological invariant. By engineering controlled nonunitary dynamics in a photonic quantum walk, we prepare topologically nontrivial mixed states from a trivial initial state. Furthermore, by machine-learning the full density matrix in momentum space, we directly extract the quantized geometric phase of the nontrivial mixed states. These results highlight a geometric phase framework that naturally extends to open quantum systems both in and out of thermal equilibrium.