Quantum geometric phase

Quantum geometric phase WANG ZISHENG (M.Sc, China Institute of Atomic Energy, Beijing) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgements I am deeply indebted to my supervisors, Professor C. H. Lai and Professor C. H. Oh, for his guidance, advice and kindness throughout my thesis work. I am grateful to Dr. Feng Xunli and Associate Professor L. C. Kwek for his valuable helps, discussions and suggestions. I would like to thank the National University of Singapore for the research scholarship, which enables me to complete the research project and thesis. Finally, I sincerely acknowledge my family because of supporting and encouraging during my studies in NUS. i Contents Acknowledgements i Table of Contents iii Summary iv List of Figures vii Introduction 1.1 Quantum geometric phase . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Geometric quantum computation . . . . . . . . . . . . . . . . . . . . 1.3 Quantum tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Motivations and goals 1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Dynamical symmetry and geometric phase 21 2.1 Dynamical symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Geometric phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Invariant operator and geometric phase . . . . . . . . . . . . . . . . . 28 2.4 Non-Abelian geometric phase . . . . . . . . . . . . . . . . . . . . . . 32 2.5 An example for a two-level atomic system . . . . . . . . . . . . . . . 34 Geometric phase for open system 40 3.1 Open system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Geometric phase for an open two-level system . . . . . . . . . . . . . 42 3.3 Interpretation of differential geometry . . . . . . . . . . . . . . . . . . 48 3.4 Extension to a three-level system . . . . . . . . . . . . . . . . . . . . 51 3.5 Geometric phase expressed by a density matrix . . . . . . . . . . . . 56 ii CONTENTS iii 3.6 Examples for possible decay sources . . . . . . . . . . . . . . . . . . . 61 3.7 Effects of squeezed vacuum reservoir on geometric phase . . . . . . . 68 Geometric quantum computation 81 4.1 Nonadiabatic geometric quantum computation . . . . . . . . . . . . . 82 4.2 Unconventional geometric quantum computation in cavity QED . . . 94 4.3 Expanding to two-mode cavity . . . . . . . . . . . . . . . . . . . . . . 117 Quantum tunneling via geometric phase 131 5.1 Transformational symmetry . . . . . . . . . . . . . . . . . . . . . . . 131 5.2 Quantum tunneling time . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.3 Geometric phase for quantum tunneling . . . . . . . . . . . . . . . . . 136 5.4 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Conclusions 144 6.1 Fundament of geometric phase . . . . . . . . . . . . . . . . . . . . . . 145 6.2 Applications of geometric phase . . . . . . . . . . . . . . . . . . . . . 148 Summary The wave function of a quantum system retains a memory of its motion in the form of a geometric phase factor. This phase factor can be measured by interfering the wave function with another coherent wave function enabling one to discern whether or not the system has undergone an evolution. By elucidating the intimating connection between dynamical symmetry and geometric phase, we investigate the relation between the geometric phase and dynamical invariants, where the Liouville-von-Neumann equation is directly deduced. Furthermore, we show that an arbitrary shift of the Hamiltonian leaves the geometric phase invariant. The study is expanded for geometric phase in open system, where it is formulated entirely in terms of geometric structures on a complex projective Hilbert space because a general belief is that Berry’s phases are geometric in their nature, i.e., proportional to the area spanned in parameter space. The geometric phase in the open system is given in terms of both the wave function and the density matrix of open system. Our results have in an agreement with the one directly from the nonunitary evolution. The results are applied to the spin-1/2 system with all possible decay. It is known that the geometric quantum computation is largely insensitive to local inaccuracies and fluctuations, and thus provides us a possible way to achieve fault-tolerant quantum gates. A new scheme to realize nonadiabatic geometric quantum computation is proposed by varying parameters in the Hamiltonian for nuclear-magnetic-resonance systems, where the dynamical and geometric phases are implemented separately without the usual operational process. Therefore, the phase accumulated in the geometric gate is a pure geometric phase for any input state. The results are expanded iv Summary v to the unconventional quantum computation. At last, we define a new geometric phase by considering both the energy and momentum conservation, where the corresponding dynamical phases have two parts differently from the conventional calculations for the geometric phase. The results are applied to quantum tunneling process, which is helpful to distinguish the concept about the tunneling time. List of Figures 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 4.1 4.2 4.3 4.4 5.1 5.2 Geometric phases(ξ, in radian) in a maximally squeezed state for Ω2r = Ω2 − γ |M |2 ≥ as functions of the atomic decay rate γ ≤ Ω/|M |(in unit of 1/second) and evolving time t(in unit of second) with the parameters Ω = 1/second, N = 2, |M | = N (N + 1), φ = π/4 and θ = π/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Corresponding radiuses of Bloch sphere for Fig. 3.1. . . . . . . . . . Geometric phases(ξ, in radian) induced by a squeezed vacuum reservoir for Ω2r = Ω2 − γ |M |2 ≥ as functions of the squeezed parameter |M | ≤ Ω/γ and evolving time t(in unit of second) with the parameters Ω = 1/second, N = 2, γ = 0.2/second, φ = π/4 and θ = π/4. . . . Corresponding radiuses of Bloch sphere for Fig. 3.3. . . . . . . . . . Same as Fig. 3.1 for Ω2r = Ω2 − γ |M |2 < with the parameters Ω = 1/second, N = 2, φ = π/4 and θ = π/4. . . . . . . . . . . . . . Corresponding radiuses of Bloch sphere for Fig. 3.5. . . . . . . . . . Same as Fig. 3.3 for Ω2r = Ω2 − γ |M |2 < with the parameters Ω = 1/second, N = 2, γ = 0.4/second, φ = π/4 and θ = π/4. . . . Corresponding radiuses of Bloch sphere for Fig. 3.7. . . . . . . . . . The two-channel Raman transition diagram. The detunings δ1 and δ2 are assumed to be sufficiently large, so that the excited state |c can be eliminated. . . . . . . . . . . . . . . . . . . . . . . . . . . . (Color online) The fidelity F of the computational basis state | + + (or | − − ) after gate operating time T = 2mπ/δ versus k/|g| with m = 1, 2, 3, 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Each atom in Λ configuration with levels |g , |e and |c interacts with the exciting fields. Here δ1 , δ2 , δ3 , ∆1 , ∆2 are frequency detunings, and g1 , g2 , Ω1 , Ω2 , Ω3 and Ω4 the respective coupling strengths. . . The fidelity F++/−− of the phase gate after operating time T = 2mπ/∆1 = 2nπ/∆2 versus x = κ1 /h1 = κ2 /h2 when ∆1 /h1 = ∆2 /h2 . . 73 . 75 . 75 . 76 . 76 . 77 . 77 . 78 . 96 . 113 . 119 . 127 3-D plots of geometric phase as a function of the barrier width d and the incoming energy E, where the barrier height V0 is set to unity. . . 139 Cross-sectional plot of geometric phase as a function of d for various incoming energy E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 vi LIST OF FIGURES 5.3 5.4 5.5 5.6 Graphs of phase, dwell and B¨ uttiker times with geometric phase for incoming energy E = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . Three dimensional parametric plots of phase time with geometric phase for various incoming energy E. . . . . . . . . . . . . . . . . . Three dimensional parametric plots of dwell time with geometric phase for various incoming energy E. . . . . . . . . . . . . . . . . . Three dimensional parametric plots of B¨ uttiker time with geometric phase for various incoming energy E. . . . . . . . . . . . . . . . . . vii . 142 . 142 . 142 . 143 Chapter Introduction 1.1 Quantum geometric phase Quantum theory is one of the most important physical theories of the 20th century. Quantum mechanics has not only advanced our understanding of nature in a profound way but it has also provided the basis on an undoubtedly and indispensably guided principle lying behind contemporary science and technology. Quantum mechanics has, more significantly, changed our view of microscopic world in a way to a completely surprising and unprecedented depth. An important feature of quantum theory is entanglement, which gives rise to correlations that cannot be explained by any local realistic description of quantum mechanics. Entanglement is a quintessential property of quantum mechanics that sets it apart from any classical physical theory. The idea of non-local correlation among remote particles was originally exploited in a classic paper on the incompleteness of quantum mechanics by Einstein, Podolsky and Rosen [1]. In this paper, Einstein and his co-workers [1] proposed a Gedenken experiment involving in two 1.1. Quantum geometric phase entangled particles, which showed that quantum mechanics cannot in all situations be a complete description of physical reality. This idea subsequently conceptualized in a seminal paper by Schr¨odinger [2], and a subsequent work by Bell [3]. Incidentally, these particles (EPR pairs) have now found wide applications in the area of quantum information theory [4, 5, 6, 7, 8]. Besides quantum entanglement, quantum mechanics harbors another surprising elegant idea. The wave function of a quantum system retains a memory of its motion in the form of a geometric phase factor. This phase factor can be measured in principle by interfering the wave function which has undergone the above evolution with another coherent wave function that did not evolve, which enable one to discern whether or not the system has undergone an evolution [9, 10]by the geometric phase. This idea arose from the acquisition of a purely geometric phase when a state undergoes an adiabatic evolution. Historically, the concept of geometric phase was first introduced by Pancharatnam [11] in his study of interference between light waves in distinct states of polarization. It was subsequently rediscovered by Berry [12] for quantum systems undergoing a cyclic adiabatic evolution. The Berry’s phase has since been linked to the notion of parallel transport [13] and re-formulated elegantly and rigorously in differential geometric terms. It has also been found that the phase depends only on the area covered by the evolution of the system in 6.2. Applications of geometric phase 149 in the geometric gate is a pure geometric phase for any input state. It is noted that for the unconventional geometric gates the total and dynamical phases were calculated in the rotating frame at the cavity frequency, where the total phase was taken as the geometric phase. Thus the total, dynamical and geometric phases are all geometric. This implies that there exists a direct proportionality, which is independent of all parameters of the system, between the total, geometric and dynamical phases respectively. Therefore, it is obvious that, if the proportionality constant between the geometric and dynamical phases is or -1, the dynamical or total phases are zero respectively so that all phases disappear. In the laboratory frame, especially, the total phase is not geometric [146, 147]. In the case, the independent proportionality does not exist for both the cyclic and noncyclic motions. In contrast to the unconventional geometric gates by using global geometric features in the rotating frame, our approach distinguishes the total and geometric phases in the laboratory frame and offers a wide choice of the relations between the dynamical phase and geometric phase, where our proportionality constant between the geometric and dynamical phases depends on the parameters of the Hamiltonian. However, by choosing some parameters in the Hamiltonian, the proportionality is constant so that the errors of the proportionality constant disappears as the un- 6.2. Applications of geometric phase 150 conventional geometric gates. Moreover, the proportionality constant includes all possible values in the physical region. Therefore, our approach is more general compared to the approach based on the unconventional geometric gates and may be helpful for experimental implementation of geometric quantum computation. For two-qubit phase gate with nonzero dynamical phase based on two-channel Raman interaction of two atoms in a cavity, furthermore, we show that the dynamical phase and the total phase for a cyclic evolution are proportional to the geometric phase at the rotating frame for the same cyclic evolution, hence they possess the same geometric features as the geometric phase does. In our scheme the atomic excited state is adiabatically eliminated and the operation of the proposed logic gate involves only the metastable states of the atoms, thus the effect of the atomic spontaneous emission can be neglected. The influence of the cavity decay on our scheme is examined. It is found that the relations regarding the dynamical phase, the total phase and the geometric phase in the ideal situation are still valid in the case of weak cavity decay. However, the presence of the cavity decay results not only in amplitude damping of the wave function, but also stops us from performing the cyclic evolution for the cavity mode, thus it reduces the fidelity of the gate operation. The feasibility and the effect of the phase fluctuations of the driving laser fields are also discussed. the results are expanded to the two atoms interacting 6.2. Applications of geometric phase 151 with two quantized fields. It is known that one of the striking features in wave mechanics is quantum tunneling. The controversy surrounding the quantum tunneling time is evident in the historical development of the problem. The different arrangements for the tunneling process cound led to different relevant time scales. Therefore, it is important to apply the concept of geometric phase to quantum tunneling time. 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[...]... then the problem about quantum tunneling time is introduced In section 5.3, we calculate geometric phase for quantum tunneling In the section 5.4, we give some results and discussions In chapter 6, We give some discussions and conclusions for fundament of geometric phase in section 6.1 and applications of geometric phase in section 6.2 Chapter 2 Dynamical symmetry and geometric phase The purpose of... that for certain quantum evolution of a quantum system of interest one can implement fault-tolerant quantum computation by using the total phase accumulated in the evolution because it depends only on global geometric features of the evolution in the rotating frame at the cavity frequency However, this approach does not distinguish between the total phase and the geometric phase The total phase is, especially,... nonadiabatic geometric quantum computation is proposed by varying parameters in the Hamiltonian for nuclear-magnetic-resonance systems, 1.4 Motivations and goals 16 where the dynamical and geometric phases are implemented separately without the usual operational process Therefore, the phase accumulated in the geometric gate is a pure geometric phase for any input state In comparison with the conventional geometric. .. to the unconventional geometric gates, which are executed using global geometric features in the rotating frame, our approach distinguishes the total and geometric phases and offers a wide choice of the relations between the dynamical and geometric phases Furthermore, we present a scheme for implementing the unconventional geometric two-qubit phase gate with nonzero dynamical phase based on two-channel... with the dynamical gates, the geometric quantum computation possesses practical advantages It is well known that geometric phases depend only on some global geometric features, and do not depend on the details of the path, the time spent, the driving Hamiltonian, and the initial and final states 1.2 Geometric quantum computation 9 of the evolution [32] Therefore the geometric quantum computation is largely... of the geometric phase described by the wave functions and by the density matrix, we find that our results are in agreement with nonunitary evaluation Methods for the physical implementation of quantum computation via geometric phase have been proposed [18, 31] by the all -geometric approach called holonomic quantum computation, based on the feature that one can achieve the entangling universal quantum. .. Hamiltonian along several special closed loops, then the dynamical phases accumulated in different loops may be cancelled, with the geometric phases being added [62, 63, 64] This is the socalled multi-loop or single-loop scheme The geometric quantum computation based on the cancellation of dynamical phases is referred to as conventional geometric quantum computation, which was usually applied to nuclear-magnetic-resonance... cancel the dynamical phase More worryingly, the dynamical phase accumulated in the gate operation is possibly nonzero and can not be eliminated Correspondingly several schemes have been presented recently to realize the so- 1.2 Geometric quantum computation 10 called unconventional geometric quantum computation [65, 66, 67, 68, 69] The central idea of the unconventional geometric quantum computation... definition of the quantum tunneling time using the geometric phase for the tunneling process, which may be helpful to distinguish the concept of quantum tunneling time 1.5 Organization This thesis includes 6 chapters Chapter 1 is an introduction In chapter 2, we study the relation between dynamical symmetry and geometric phase In chapter 1.5 Organization 18 3, we give a definition of the geometric phase for... in studying about the geometric phase has been for open system [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54], where the 1.2 Geometric quantum computation 8 statistical mixtures was considered [55, 56], based on both the experimental context of quantum interferometry and the different generalizations of the parallel transport condition [57, 58] 1.2 Geometric quantum computation Quantum computation employs . reservoir on geometric phase . . . . . . . 68 4 Geometric quantum computation 81 4.1 Nonadiabatic geometric quantum computation . . . . . . . . . . . . . 82 4.2 Unconventional geometric quantum computation. Introduction 1 1.1 Quantum geometric phase . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Geometric quantum computation . . . . . . . . . . . . . . . . . . . . 8 1.3 Quantum tunneling [9, 10]by the geometric phase. This idea arose from the acquisition of a purely geometric phase when a state undergoes an adiabatic evolution. Historically, the concept of geometric phase was first