Study of the Aharonov-Anandan quantum phase by NMR interferometry

Geometric phases play a central role in a variety of quantum phenomena, especially in condensed matter physics. Recently, it was shown that this fundamental concept exhibits a connection to quantum phase transitions where the system undergoes a qualitative change in the ground state when a control parameter in its Hamiltonian is varied. Here we report the first experimental study using the geometric phase as a topological test of quantum transitions of the ground state in a Heisenberg XY spin model. Using NMR interferometry, we measure the geometric phase for different adiabatic circuits that do not pass through points of degeneracy.

The discovery of the geometric phase is one of the most interesting and intriguing findings of the last few decades. It led to a deeper understanding of the concept of phase in quantum mechanics and motivated a surge of interest in fundamental quantum mechanical issues, disclosing unexpected applications in very diverse fields of physics. Although the key ideas underlying the existence of a purely geometrical phase had already been proposed in 1956 by Pancharatnam, it was Michael Berry who revived this issue 30 years later. The clarity of Berry's seminal paper, in 1984, was extraordinary. Research on the topic flourished at such a pace that it became difficult for non-experts to follow the many different theoretical ideas and experimental proposals which ensued. Diverse concepts in independent areas of mathematics, physics and chemistry were being applied, for what was (and can still be considered) a nascent arena for theory, experiments and technology. Although collections of papers by different authors appeared in the literature, sometimes with ample introductions, surprisingly, to the best of my knowledge, no specific and exhaustive book has ever been written on this subject. The Geometric Phase in Quantum Systems is the first thorough book on geometric phases and fills an important gap in the physical literature. Other books on the subject will undoubtedly follow. But it will take a fairly long time before other authors can cover that same variety of concepts in such a comprehensive manner. The book is enjoyable. The choice of topics presented is well balanced and appropriate. The appendices are well written, understandable and exhaustive - three rare qualities. I also find it praiseworthy that the authors decided to explicitly carry out most of the calculations, avoiding, as much as possible, the use of the joke `after a straightforward calculation, one finds...' This was one of the sentences I used to dislike most during my undergraduate studies. A student is inexperienced in such matters and needs to look at details. This book is addressed to graduate physics and chemistry students and was written thinking of students. However, I would recommend it also to young and mature physicists, even those who are already `into' the subject. It is a comprehensive work, jointly written by five researchers. After a simple introduction to the subject, the book gradually provides deeper concepts, more advanced theory and finally an interesting introduction and explanation of recent experiments. For its multidisciplinary features, this work could not have been written by one single author. The collaborative effort is undoubtedly one of its most interesting qualities. I would definitely recommend it to anyone who wants to learn more on the geometric phase, a topic that is both beautiful and intruiguing.

We show how geometric phases may be used to fully describe quantum systems, with or without gravity, by providing knowledge about the geometry and topology of its Hilbert space. We find a direct relation between geometric phases and von Neumann algebras. In particular, we show that a vanishing geometric phase implies the existence of a well-defined trace functional on the algebra. We discuss how this is realised within the AdS/CFT correspondence for the eternal black hole. On the other hand, a non-vanishing geometric phase indicates missing information for a local observer, associated to reference frames covering only parts of the quantum system considered. We illustrate this with several examples, ranging from a single spin in a magnetic field to Virasoro Berry phases and the geometric phase associated to the eternal black hole in AdS spacetime. For the latter, a non-vanishing geometric phase is tied to the presence of a centre in the associated von Neumann algebra.

A geometric approach to quantum control in projective hilbert spaces

The geometric (Berry-Pancharatnam) phase originates from the intrinsic geometry of the space of quantum states and can be observed in different situations, such as a cyclic evolution of a quantum system. Here, we utilize the geometric phase to obtain a surprising insight: It is possible to create nonlocal correlations in a fixed interferometer with independent photon inputs by varying the photons' internal states. In particular, we consider a cyclic interferometer that is fixed, i.e., that has no variable internal phase shifts or subsequent measurement settings. Instead, the measurement choices of the different parties correspond to the internal states of the input photons which influence the observed correlations via a collective $N$-photon geometric phase, constituting a different approach for the generation of nonlocality with respect to the usual paradigm. We observe a trade-off between the geometric phases and the visibility of the many-photon interference, impeding the generation of nonlocality. However, by making use of the dynamical quantum Zeno effect, we show that nonlocality can be created in the fixed cyclic interferometer using 12 (or more) independent photons.

We introduce a self-consistent framework for the analysis of both Abelian and non-Abelian geometric phases associated with open quantum systems, undergoing cyclic adiabatic evolution. We derive a general expression for geometric phases, based on an adiabatic approximation developed within an inherently open-systems approach. This expression provides a natural generalization of the analogous one for closed quantum systems, and we prove that it satisfies all the properties one might expect of a good definition of a geometric phase, including gauge invariance. A striking consequence is the emergence of a finite time interval for the observation of geometric phases. The formalism is illustrated via the canonical example of a spin-$1∕2$ particle in a time-dependent magnetic field. Remarkably, the geometric phase in this case is immune to dephasing and spontaneous emission in the renormalized Hamiltonian eigenstate basis. This result positively impacts holonomic quantum computing.

A quantum system interacting with its environment is subject to dephasing, which ultimately destroys the information it holds. Here we use a superconducting qubit to experimentally show that this dephasing has both dynamic and geometric origins. It is found that geometric dephasing, which is present even in the adiabatic limit and when no geometric phase is acquired, can either reduce or restore coherence depending on the orientation of the path the qubit traces out in its projective Hilbert space. It accompanies the evolution of any system in Hilbert space subjected to noise.

For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.

Non-Markovian dynamics of a three-level system is studied with the help of stochastic Schrodinger equation (SSE). We derive a new form of SSE for a three-level system driven by four independent Ornstein-Uhlenbeck stochastic noises. The main advantage of the suggested SSE is the ensuring of the complete positivity of the reduced density operator. We demonstrate significant influence of the non-Markovian noises on the dynamics of the three-level quantum system.

By constructing a vector model we study the coherent dynamics of a special three-level quantum system--one with equal detunings and equal Rabi frequencies. This three-level problem can be fully related to a two-level problem. We find that complete population depletion can be achieved even in the case of constant nonzero detunings when the three-level system is prepared initially in the common level of the two transitions. Complete population depletion with constant detunings can also occur in other odd-number multilevel systems. An explanation in terms of an equivalent resonant excitation is presented.

This special issue celebrates the discovery of two of the most important aspects of quantum mechanics: the Aharonov–Bohm effect and the Berry phase. The issue includes work presented at two conferences, 50 Years of the Aharonov–Bohm Effect, 11–14 October 2009, Tel Aviv University, Israel, and the Aharonov–Bohm Effect and Berry Phase 50/25 Anniversary, 14–15 December 2009, Bristol University, UK.The Aharonov–Bohm (AB) effect and the Berry phase have many things in common, apart from the happy coincidence that they were discovered 25 years apart, making possible such a joint anniversary. First, they were both discovered in the same place, the H H Wills Physics Laboratory, University of Bristol, UK.Second, they both were there for everybody to see from day one, from the very beginning of quantum mechanics. They were, so to say, right on the surface; there was no need for sophisticated mathematics or involved constructions. Yet it took so long for somebody to see them. The explanation, we dare to speculate, is that they are so unusual, they challenge so much our everyday experience and ingrained intuitions about physics that it was difficult to recognise them.Third, both the AB effect and the Berry phase are quintessentially quantum; they both manifest themselves as topological/geometric quantum phases.The AB effect revolutionized our understanding of the role of potentials in physics. For the first time, it was shown that a particle moving in a field-free region could be affected by a field in a disjoint region. In doing so, it shattered one of the strongest held principles of physics. In the case of the Berry phase, the importance lies in its absolute generality—the topological/geometric phase named after Berry is acquired by all physical systems when the external parameters they depend upon undergo an adiabatic cyclic change. Remarkably this phase depends only on the geometry of the cyclic path in the abstract parameter space, a fact which deepens our appreciation of the underlying geometry of quantum mechanics.The AB effect and the Berry phase are ubiquitous in modern physics and have implications ranging from the most fundamental to the very applied. First and foremost, they give a deeper understanding of the nature of quantum mechanics itself. Furthermore, we find them in fields as varied as cosmology, particle physics, non-abelian gauge theories, condensed matter, chemical and molecular physics, and laser dynamics. Generalizations of the AB effect to non-abelian gauge theories, such as the Wilson and `t Hooft loops, are important tools for studying confinement and spontaneous symmetry breaking. They explain charge quantization, the quantum Hall effect and the Josephson junction and also play a crucial role in electron microscope holography. Without them an understanding of many effects in the new field of mesoscopic physics, where tiny electronic circuits exhibit quantum behaviour, would be impossible.Like good wine, the AB effect and the Berry phase become finer and more appreciated with time. Even though they appear in textbooks and encyclopaedias, the number of researchers thinking and publishing papers on these two effects is increasing annually. The AB effect and the Berry phase keep being observed in new systems and with every day that passes novel applications are routinely found. This special issue is just a small snapshot of the state of the art 50 and 25 years, respectively, from their discoveries.

The Berry phase plays an important role in determining many physical properties of quantum systems. However, tuning the energy spectrum of a quantum system via Berry phase is comparatively rare because the Berry phase is usually a fixed constant. Here, we report the realization of an unusual valley-polarized energy spectra via continuously tunable Berry phases in Bernal-stacked bilayer graphene quantum dots. In our experiment, the Berry phase of electron orbital states is continuously tuned from about π to 2π by perpendicular magnetic fields. When the Berry phase equals π or 2π, the electron states in the two inequivalent valleys are energetically degenerate. By altering the Berry phase to noninteger multiples of π, large and continuously tunable valley-polarized energy spectra are realized. Our result reveals the Berry phase's essential role in valleytronics and the observed valley splitting, on the order of 10meV at a magnetic field of 1T, is about 100 times larger than Zeeman splitting for spin, shedding light on graphene-based valleytronics.

We use a combination of microwave fields and free precession to drive the spin of a nitrogen-vacancy (NV) center in diamond on different trajectories on the Bloch sphere and investigate the physical significance of the frame-dependent decomposition of the total phase into geometric and dynamic parts. The experiments are performed on a two-level subspace of the spin-1 ground state of the NV, where the Aharonov-Anandan geometric phase manifests itself as a global phase, and we use the third level of the NV ground-state triplet to detect it. We show that while the geometric Aharonov-Anandan phase retains its connection to the solid angle swept out by the evolving spin, it is generally accompanied by a dynamic phase that suppresses the geometric dependence of the system dynamics. These results offer insights into the physical significance of frame-dependent geometric phases.

Coherent population transfer in quantum systems is of fundamental importance in many fields spanning atomic and molecular collision dynamics, information processing for qubit systems. Stimulated Raman nonadiabatic passage technique, when implemented in an externally driven three-level system, provides an efficient approach for realizing accelerated population transfer while maintaining robust quantum coherence, without the rotating wave approximation. However, previous protocols employ multiple pulses and imply that Rabi frequencies have a few oscillations during dynamical evolution. In this paper, under two-photon resonance, we utilize the gauge transformation method to inversely design a Λ-configuration three-level system that can be solved exactly. By invoking a <inline-formula><tex-math id="M2">\begin{document}$SU(3)$\end{document}</tex-math></inline-formula> transformation, we establish the connection between Schrödinger representation and gauge representation with which the effective Hamiltonian is an Abelian operator. Subsequently, we construct the desired Hamiltonian and further investigate its dynamic behavior. The result demonstrate that, by imposing appropriate boundary conditions on the control parameters, high-fidelity population transfer can be achieved in ideal evolution. In addition, for the practical case with pulse truncation and intermediate state decay, the fidelities of specific models can reach about <inline-formula><tex-math id="M3">\begin{document}$99.996\%$\end{document}</tex-math></inline-formula> and for <inline-formula><tex-math id="M4">\begin{document}$99.983\%$\end{document}</tex-math></inline-formula>. Compared to other existing nonadiabatic quantum control schemes, we show that the present scheme has the distinctive advantages. Firstly, instead of introducing an additional microwave field, we achieve the desired quantum control by applying only a few sets of Stokes and pump pulses. Moreover this approach does not exhibit Rabi oscillations in the dynamic process, nor does it present singularities in the pulse itself.

A systematic derivative expansion of the adiabatic phase