Evolution Of Quantum System Research Articles

Effective Hamiltonian theory of the geometric evolution of quantum systems

In this work we present an effective Hamiltonian description of the quantum dynamics of a generalized Lambda system undergoing adiabatic evolution. We assume the system to be initialized in the dark subspace and show that its holonomic evolution can be viewed as a conventional Hamiltonian dynamics in an appropriately chosen extended Hilbert space. In contrast to the existing approaches, our method does not require the calculation of the non-Abelian Berry connection and can be applied without any parametrization of the dark subspace, which becomes a challenging problem with increasing system size.

Introduction to continuous photocounting effects on the quantized optical field

Abstract In this manuscript, we explore the effects of continuous measurements upon the quantized electromagnetic field through a series of simple examples. For this purpose, we consider the Srinivas-Davies model to describe the optical field dynamics probed continuously by a photodetector. Through the application of this continuous photodetection model to some specific situations, it is possible to cover some basic concepts of quantum mechanics such as the principle of superposition, the collapse of the wave function, the probabilistic character of the possible outcomes associated with projective measurements, as well as some advanced topics such as the description of open quantum systems, irreversible processes in quantum mechanics, decoherence and dissipation associated with non-unitary evolution of quantum systems. Besides, we also consider the important concept of entanglement between two electromagnetic fields and how it is affected by the photodetection process. This work aims to provide complementary material for undergraduate and graduate students interested in the effects of measuring devices acting on quantum systems.

Three-dimensional transmon coherence measurement method based on network analyser

<sec>Three-dimensional(3D) transmon is a kind of superconducting qubit with long decoherence time, which has important applications in superconducting quantum computation, quantum optics, cavity quantum electrodynamics, et al. Rabi oscillation is a vital method to characterize the decoherence time of quantum system, and it is also a basic experiment to demonstrate the energy level evolution of quantum system. In order to test the Rabi oscillation of 3D transmon, strict timing control is necessary, and the process of testing and debugging is complicated. In this paper, 3D transmon samples are fabricated and their basic parameters <i>E<sub>C</sub></i> = 348.74 MHz and <i>E<sub>J</sub></i> = 11.556 GHz are tested and characterized. The coupling coefficient <i>g<sup>2</sup></i>/Δ between qubit and the 3D cavity is 43 MHz, which is located in the dispersive regime. The qubit’s first transition frequency <i>f</i><sub>01</sub><italic/> = 9.2709 GHz, and the second transition frequency <i>f</i><sub>12</sub> = 9.0100 GHz. The 3D resonator is fabricated by the material 6061T6 aluminum, the loaded quality factor is 4.8 × 10<sup>5</sup>, and the bare frequency of the resonator is 8.108 GHz.</sec><sec>Through comparison, it is found that the Rabi oscillation time obtained by the proposed method is shorter than by the Jaynes-Cummings method. The main reasons are as follows. First, the measurement of network analyzer is a continuous measurement, and the test signal always affects the decoherence process of 3D transmon. Second, the quantum bit is in the ground state after decoherence, and the ground state measured by the network analyzer accounts for a relatively high proportion, which causes the curve measured by the network analyzer to be one-sided attenuation oscillation. Third, the dispersive readout method is related to the quality factor of the superconducting cavity. The storage time of microwave photons in the superconducting cavity is longer than the decoherence time of 3D transmon, so the quantum information is partially decohered before leaving the superconducting cavity, which will shorten the Rabi oscillation time.</sec><sec>An innovative Rabi oscillation test method based on network analyzer is presented. The test system based on this method is simple to build and can be used as a new way to quickly verify whether 3D transmon has quantum characteristics. This method can also be extended to other quantum systems for preliminarily verifying the time domain characteristics.</sec>

On estimation and feedback control of spin-1/2 systems with unknown initial states

In this paper, we consider stochastic master equations describing the evolutions of quantum systems interacting with electromagnetic fields undergoing continuous-time measurements. In particular, we study feedback control of quantum spin1/2 systems in the case of unawareness of initial states and in presence of measurement imperfections. We prove that the fidelity between the true quantum filter and its estimation (with arbitrary initial state) converges to one under appropriate assumption on the feedback controller. This shows the asymptotic convergence of such filters. In the more general case of spin-J systems, we discuss heuristically the asymptotic behavior of the true quantum filter and the associated estimation, and the possibility of exponentially stabilizing such systems towards an eigenvector of the measurement operator by an appropriate feedback.

Time evolution in quantum systems: a closer look at student understanding

Time evolution of quantum systems has been shown to be one of the most difficult components of a typical undergraduate quantum mechanics course. In this work, we examine the current literature, and then take a closer look at the process that students use to determine how the quantum state of a spin-1/2 particle evolves with time. We divide the process of writing a time-dependent state into five elements and use these to both directly probe student understanding and guide our coding of student responses. We focus on three elements of this process, including knowledge of the Hamiltonian, the energy eigenstates and eigenvalues, and what basis should be used when writing the state as a function of time using the phase . Analysis of four exam questions given at three institutions suggests that knowledge of the energy eigenbasis and its importance for time evolution may be a weak point in student understanding.

Explicit solutions of coupled-mode equations for multiple waveguide devices: The optical analogy of a three-state geometric phase

Coupled waveguides mediated by the engineered refractive indexes are the central devices in integrated optical system. Generically, the coupled-mode equations describing the optical transports in the multiple waveguide-coupled structure are relatively complexity and hard to be analytically solved. In this paper we first show that the waveguides with the nearest-neighbor interactions possess simply a SU(2) dynamical symmetry, and then the analytical solutions of the relevant coupled-mode equations can be easily obtained. As a consequence, by setting properly the coupling coefficients between the waveguides, the optical transports in these waveguides can be controlled and the relevant optics devices can be designed robustly. Specifically, we show how the famous geometric phase acquired by the cyclic evolution of a parameter-driven three-level quantum system can be analogously demonstrated with the classical optical pulses transporting in a three-waveguide structure with the transverse interactions. Hopefully, the optical analogue of geometric phase might be applied to design and fabricate various novel phase-modulated integrated devices.

Brief Review: Simulation of Novel Systems Using Duality Quantum Algorithm

In this article, we review the recent works of quantum simulation of novel systems briefly, the parity-time-reversal-symmetric (PT-symmetric) quantum system and the Yang-Baxter-equation (YBE) system, using duality quantum algorithm. Duality quantum algorithm studies the linear combinations of unitary operators, making it possible to simulate non-unitary evolutions of novel quantum systems. A PT-symmetric quantum system is a typical non-Hermitian system of which the evolution is not unitary and cannot be simulated directly by a conventional quantum computer. A recent work by C. Zheng has established a theory to simulate a general PT-symmetric two level system by duality quantum computing. The other typical example is the YBE quantum systems, of which the evolutions can be both unitary and non-unitary. C. Zheng and S. J. Wei described a theory that the two hand sides of the YBE can be simulated efficiently by the duality quantum algorithm in their recent research. Perspectives of future applications are expected at last. Copyright © VBRI Press.

All-optical generation of quantum entangled states with strictly constrained ultrafast laser pulses

We present a quantum optimal control theory study combined with theoretical analysis to determine a pulsed laser field, capable of generating a maximally entangled state in two trapped two-level atoms. By expanding the time-dependent unitary operator to the first leading term of Magnus expansion, we reexamine the pulse area theorem for the trapped atoms driven by an arbitrarily temporary field. Due to the dipole-dipole interaction blockade, we find that the two trapped atoms described by a three-level ladder system can be reduced into an equivalent two-level model by using narrow-bandwidth pulses, leading to an analytical solution for generating the maximally entangled state. We also solve a highly constrained optimization problem to search for optimal laser pulses with broad bandwidths. A zero pulse-area constraint is employed to remove the dc offset of the optimized laser pulses, and a fixed fluence limitation combined with a constant pulse-area constraint at the resonant frequency of the equivalent two-level system are utilized to restrict the unitary evolution of quantum systems by the first leading term of Magnus expansion. This work provides a potentially useful approach to find all-optical control schemes for generating the maximally entangled state by using ultrafast laser pulses while satisfying multiple strict limitations.

Shortcuts to adiabaticity in optical waveguides

In recent years, the concept of shortcuts to adiabaticity (STA), originally developed for speeding up slow adiabatic state evolution in quantum systems, has found numerous applications in guided-wave optics. Optical waveguides, enabled by the advanced fabrication technologies, provide an ideal platform to implement the STA protocols in terms of geometry variations; moreoever, STA has enabled the development of short and robust waveguide components, with applications in beam couplers, beam splitters, mode converters, mode (de)multiplexers, and polarization manipulation devices. Concepts such as counterdiabatic driving, invariant-based inverse engineering, fast-forward approach, or fast quasiadiabatic dynamics, have been shown to provide shortcuts to adiabatic mode evolution in optical waveguides, resulting in compact functional devices with large bandwidth and fabrication tolerance. Novel devices have recently been fabricated following years of theoretical efforts, showing that STA have emerged as a new paradigm in optical waveguides. In this work, we discuss the major STA protocols for applications in optical waveguides and illustrate the shortcuts with device examples.

Non-Abelian geometric phases in periodically driven systems

We consider a periodically driven quantum system described by a Hamiltonian which is the product of a slowly varying Hermitian operator $V\left(\boldsymbol{\lambda}\left(t\right)\right)$ and a dimensionless periodic function with zero average. We demonstrate that the adiabatic evolution of the system within a fully degenerate Floquet band is accompanied by non-Abelian (non-commuting) geometric phases appearing when the slowly varying parameter $\boldsymbol{\lambda}=\boldsymbol{\lambda}\left(t\right)$ completes a closed loop. The geometric phases can have significant values even after completing a single cycle of the slow variable. Furthermore, there are no dynamical phases masking the non-Abelian Floquet geometric phases, as the former average to zero over an oscillation period. This can be used to precisely control the evolution of quantum systems, in particular for performing qubit operations. The general formalism is illustrated by analyzing a spin in an oscillating magnetic field with arbitrary strength and a slowly changing direction.

Chirality from quantum walks without a quantum coin

Quantum walks (QWs) describe the evolution of quantum systems on graphs. An intrinsic degree of freedom---called the coin and represented by a finite-dimensional Hilbert space---is associated to each node. Scalar quantum walks are QWs with a one-dimensional coin. We propose a general strategy allowing one to construct scalar QWs on a broad variety of graphs, which admit embedding in Eulidean spaces, thus having a direct geometric interpretation. After reviewing the technique that allows one to regroup cells of nodes into new nodes, transforming finite spatial blocks into internal degrees of freedom, we prove that no QW with a two-dimensional coin can be derived from an isotropic scalar QW in this way. Finally we show that the Weyl and Dirac QWs can be derived from scalar QWs in spaces of dimension up to three, via our construction.

Chebyshev hierarchical equations of motion for systems with arbitrary spectral densities and temperatures.

The time evolution in open quantum systems, such as a molecular aggregate in contact with a thermal bath, still poses a complex and challenging problem. The influence of the thermal noise can be treated using a plethora of schemes, several of which decompose the corresponding correlation functions in terms of weighted sums of exponential functions. One such scheme is based on the hierarchical equations of motion (HEOM), which is built using only certain forms of bath correlation functions. In the case where the environment is described by a complex spectral density or is at a very low temperature, approaches utilizing the exponential decomposition become very inefficient. Here, we utilize an alternative decomposition scheme for the bath correlation function based on Chebyshev polynomials and Bessel functions to derive a HEOM approach up to an arbitrary order in the environmental coupling. These hierarchical equations are similar in structure to the popular exponential HEOM scheme, but are formulated using the derivatives of the Bessel functions. The proposed scheme is tested up to the fourth order in perturbation theory for a two-level system and compared to benchmark calculations for the case of zero-temperature quantum Ohmic and super-Ohmic noise. Furthermore, the benefits and shortcomings of the present Chebyshev-based hierarchical equations are discussed.

Preliminary study for developing instantaneous quantum computing algorithms (IQCA)

Since the mid-1990s theoretical quadratic exponential and polynomial Quantum Computing (QC) speedup algorithms have been discussed. Recently the advent of relativistic information processing (RIP) introducing a relativistic qubit (r-qubit) with additional degrees of freedom beyond the current Hilbert space Bloch 2-sphere qubit formalism extended theory has appeared. In this work a penultimate form of QC speedup – Instantaneous Quantum Computing Algorithms (IQCA) is proposed. Discussion exists on passing beyond the quantum limits of locality and unitarity heretofore restricting the evolution of quantum systems to the standard Copenhagen Interpretation. In that respect as introduced in prior work an ontological-phase topological QC avails itself of extended modeling. As well-known by EPR experiments instantaneous connectivity exists inherently in the nonlocal arena. As our starting point we utilize Bohm’s super-implicate order where inside a wave packet a super-quantum potential introduces nonlocal connectivity. Additionally EPR experiments entangle simultaneously emitted photon pairs by parametric down-conversion. Operating an IQCA requires a parametric up-conversion cycle an M-Theoretic Unified Field Mechanical (MUFM) set of topological transformations beyond the current Galilean Lorentz-Poincairé transforms of the standard model (SM). Yang-Mills Kaluza-Klein (YM-KK) correspondence is shown to provide a path beyond the semi-quantum limit to realize the local-nonlocal duality required to implement IQCA.

Non-Hermitian quantum systems and their geometric phases

We discuss the basic theoretical framework for non-Hermitian quantum systems with particular emphasis on the diagonalizability of non-Hermitian Hamiltonians and their $GL(1,\mathbb{C})$ gauge freedom, which are relevant to the adiabatic evolution of non-Hermitian quantum systems. We find that the adiabatic evolution is possible only when the eigen-energies are real. The accompanying geometric phase is found to be generally complex and associated with not only the phase of a wavefunction but also its amplitude. The condition for the real geometric phase is laid out. Our results are illustrated with two examples of non-Hermitian $\mathcal{PT}$ symmetric systems, the two-dimensional non-Hermitian Dirac fermion model and bosonic Bogoliubov quasi-particles.

All-optical implementation of collision-based evolutions of open quantum systems

We present a new optical scheme enabling the implementation of highly stable and configurable non-Markovian dynamics. Here one photon qubit can circulate in a multipass bulk geometry consisting of two concatenated Sagnac interferometers to simulate the so called collisional model, where the system interacts at discrete times with a vacuum environment. We show the optical features of our apparatus and three different implementations of it, replicating a pure Markovian scenario and two non-Markovian ones, where we quantify the information backflow by tracking the evolution of the initial entanglement between the system photon and an ancillary one.

Enhancing student visual understanding of the time evolution of quantum systems

A combined simulation tutorial helps students develop a correct visual understanding of the time evolution of the wave function and how this leads to the time evolution of the probability density.

Efficient machine-learning representations of a surface code with boundaries, defects, domain walls, and twists

Machine learning representations of many-body quantum states have recently been introduced as an ansatz to describe the ground states and unitary evolutions of many-body quantum systems. We explore one of the most important representations, restricted Boltzmann machine (RBM) representation, in stabilizer formalism. We give the general method of constructing RBM representation for stabilizer code states and find the exact RBM representation for several types of stabilizer groups with the number of hidden neurons equal or less than the number of visible neurons, which indicates that the representation is extremely efficient. Then we analyze the surface code with boundaries, defects, domain walls and twists in full detail and find that all the models can be efficiently represented via RBM ansatz states. Besides, the case for Kitaev's $D(\Zb_d)$ model, which is a generalized model of surface code, is also investigated.

Observer-dependent locality of quantum events

In general relativity, the causal structure between events is dynamical, but it is definite and observer-independent; events are point-like and the membership of an event A in the future or past light-cone of an event B is an observer-independent statement. When events are defined with respect to quantum systems however, nothing guarantees that the causal relationship between A and B is definite. We propose to associate a causal reference frame corresponding to each event, which can be interpreted as an observer-dependent time according to which an observer describes the evolution of quantum systems. In the causal reference frame of one event, this particular event is always localised, but other events can be ‘smeared out’ in the future and in the past. We do not impose a predefined causal order between the events, but only require that descriptions from different reference frames obey a global consistency condition. We show that our new formalism is equivalent to the pure process matrix formalism (Araújo et al 2017 Quantum 1 10). The latter is known to predict certain multipartite correlations, which are incompatible with the assumption of a causal ordering of the events—these correlations violate causal inequalities. We show how the causal reference frame description can be used to gain insight into the question of realisability of such strongly non-causal processes in laboratory experiments. As another application, we use causal reference frames to revisit a thought experiment Zych et al (arXiv:1708.00248) where the gravitational time dilation due to a massive object in a quantum superposition of positions leads to a superposition of the causal ordering of two events.

Application of Shemesh theorem to quantum channels

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators A(A1,…,AK). By applying the Shemesh and Amitsur-Levitzki theorems to analyse the structure of the algebra A(A1,…,AK), one can predict the asymptotic dynamics for a class of operations.

Speeding up thermalisation via open quantum system variational optimisation

Optimizing open quantum system evolution is an important step on the way to achieving quantum computing and quantum thermodynamic tasks. In this article, we approach optimisation via variational principles and derive an open quantum system variational algorithm explicitly for Lindblad evolution in Liouville space. As an example of such control over open system evolution, we control the thermalisation of a qubit attached to a thermal Lindbladian bath with a damping rate $\gamma$. Since thermalisation is an asymptotic process and the variational algorithm we consider is for fixed time, we present a way to discuss the potential speedup of thermalisation that can be expected from such variational algorithms.