Two-state Quantum Research Articles

N-bein formalism for the parameter space of quantum geometry

Abstract This work introduces a geometrical object that generalizes the quantum geometric tensor; we call it N-bein. Analogous to the vielbein (orthonormal frame) used in the Cartan formalism, the N-bein behaves like a ‘square root’ of the quantum geometric tensor. Using it, we present a quantum geometric tensor of two states that measures the possibility of moving from one state to another after two consecutive parameter variations. This new tensor determines the commutativity of such variations through its anti-symmetric part. In addition, we define a connection different from the Berry connection, and combining it with the N-bein allows us to introduce a notion of torsion and curvature à la Cartan that satisfies the Bianchi identities. Moreover, the torsion coincides with the anti-symmetric part of the two-state quantum geometric tensor previously mentioned, and thus, it is related to the commutativity of the parameter variations. We also describe our formalism using differential forms and discuss the possible physical interpretations of the new geometrical objects. Furthermore, we define different gauge invariants constructed from the geometrical quantities introduced in this work, resulting in new physical observables. Finally, we present two examples to illustrate these concepts: a harmonic oscillator and a generalized oscillator, both immersed in an electric field. We found that the new tensors quantify correlations between quantum states that were unavailable by other methods.

Experimental Investigation on the Dynamics Characteristics of a Two-State Quantum Dot Laser under Optical Feedback

We experimentally investigate the dynamics characteristics of a two-state quantum dot laser (TSQDL) subject to optical feedback. Firstly, we inspect the impact of the temperature on the power-current characteristics of the ground state (GS) lasing and the excited state (ES) lasing in the TSQDL operating at free-running. The results demonstrate that with the decrease in the temperature, the threshold current for GS lasing (IthGS) and the threshold current for ES lasing (IthES) decrease very slowly. There exists a current for GS quenching (IQGS), which is gradually increased with the decrease in the temperature. After introducing optical feedback, the overall trend of change is similar to those obtained under free-running. Next, through inspecting the time series and power spectrum of the output from the TSQDL under optical feedback, the dynamical characteristics of the TSQDL are investigated under different feedback ratios, and diverse dynamical states including quasi-chaos pulse package, chaos state, regular pulse package, quasi-period two, quasi-regular pulsing, and chaos regular pulse package have been observed. Finally, for the TSQDL biased at three different cases: lower than IthES, slightly higher than IthES, and higher than IthES, nonlinear dynamic state evolutions with the increase in feedback ratio are inspected, respectively. The results show that, for the TSQDL biased at lower than IthES, it presents an evolution route of stable state—quasi-chaos pulse package—chaos state—regular pulse package. For the TSQDL biased at slightly larger than IthES, it presents an evolution route of stable state—quasi-regular pulsing—quasi-period two—chaos regular pulse package. For the TSQDL biased at higher than IthES, the TSQDL always behaves stable state within the range of feedback ratio that the experiment can achieve. However, with the increase in optical feedback ratio, the number of longitudinal modes for GS lasing and ES lasing are changed.

Numerical investigation on nonlinear dynamics of two-state quantum dot laser subject to optical injection

Based on the cascade relaxation model of the quantum dot (QD) lasers, the nonlinear dynamics of a two-state QD laser subject to optical injection is numerically studied. The simulation results show that, for a two-state QD laser operating at the ground-state (GS) and excited-state (ES) emission simultaneously, through adjusting the injection parameters, the two emission states can display the stable (S), injection locking (IL), suppression (SP), period-one (P1), period-two (P2), multi-period (MP), and chaotic pulse (CP) states when the frequency of the injection light is close to the GS lasing frequency. Further mapping of these dynamic states in the parameter space of frequency detuning and injection strength, it is found that the complex dynamic behaviors are mainly distributed in the negative frequency detuning region. In addition, the complexity of the chaotic signal is qualified by the permutation entropy (PE) calculating, and the high complexity signal with PE value over 0.95 is obtained. Moreover, within a certain frequency detuning range, the ES emission can be effectively suppressed by increasing the injection strength.

Landau–Zener–Stückelberg–Majorana-like dynamics induced by tunneling of spin qubit states in a multiband two-state magnetic quantum wire

We investigate the transfer between spin quantum bit (qubit) states and study the Landau–Zener–Stückelberg–Majorana (LZSM)-like dynamics of tunneling spin qubits in a multiband two-state magnetic quantum wire. Indeed, within the framework of an optical parabolic potential in a three-dimensional (3D) heterostructure quantum wire and under the influence of an external time-varying magnetic field, a model for a multiband two-state magnetic quantum wire is developed. Here, the external magnetic field is used to coherently manipulate and control spin qubit states. By driving the system through an avoided crossing, we consider the associated effective quantum two-level system (TLS) related to the spin qubit states in each band, driving by an external coherent magnetic field in which the related Hamiltonian is Hermitian, and which generally paves a way to LZSM interferometry. Thus, we establish the analytical expressions of the energy eigenvalues in each band and derive the analytical solution of the dynamical evolution of the tunneling probabilities of the associated TLS. Accordingly, nonadiabatic and adiabatic tunneling probabilities (survival and transition) are calculated for each band of the multiband TLS. In this respect, the nonadiabatic and adiabatic dynamical evolutions of the tunneling probabilities of spin qubit populations in the first four bands, with band quantum numbers n = 0, 1, 2 and 3 are analyzed. As a result, depending on the amplitude strength of the driven magnetic field and the magnitude of the driving frequency, we report two striking nonadiabatic and adiabatic scenarios in each band for both the diabatic and adiabatic states. In this context, driving the two states of each band of the multiband TLS related to the spin qubit states through an avoided level crossing can result in nontrivial and incoherent dynamics at certain phases, resulting to apparent inaccurate probabilities, especially in the case of strong driven magnetic field and high driving frequency.

Zombie cats on the quantum-classical frontier: Wigner-Moyal and semiclassical limit dynamics of quantum coherence in molecules.

In this paper, we investigate the time evolution of quantum coherence-the off-diagonal elements of the density matrix of a multistate quantum system-from the perspective of the Wigner-Moyal formalism. This approach provides an exact phase space representation of quantum mechanics. We consider the coherent evolution of nuclear wavepackets in a molecule with two electronic states. For harmonic potentials, the problem is analytically soluble for both a fully quantum mechanical description and a semiclassical description. We highlight the serious deficiencies of the semiclassical treatment of coherence for general systems and illustrate how even qualitative accuracy requires higher order terms in the Moyal expansion to be included. The model provides an experimentally relevant example of a molecular Schrödinger's cat state. The alive and dead cats of the exact two-state quantum evolution collapse into a "zombie" cat in the semiclassical limit-an averaged behavior, neither alive nor dead, leading to significant errors. The inclusion of the Moyal correction restores a faithful simultaneously alive and dead representation of the cat that is experimentally observable.

Noise-induced Parrondo's paradox in discrete-time quantum walks.

Parrondo's paradox refers to the apparently paradoxical effect whereby two or more dynamics in which a given quantity decreases are combined in such a way that the same quantity increases in the resulting dynamics. We show that noise can induce Parrondo's paradox in one-dimensional discrete-time quantum walks with deterministic periodic as well as aperiodic sequences of two-state quantum coins where this paradox does not occur in the absence of noise. Moreover, we show how the noise-induced Parrondo's paradox affects the time evolution of quantum entanglement for such quantum walks.

Challenges in addressing student difficulties with basics and change of basis for two-state quantum systems using a multiple-choice question sequence in online and in-person classes

Research-validated multiple-choice questions comprise an easy-to-implement instructional tool for scaffolding student learning and providing formative assessment of students’ knowledge. We present findings from the implementation of a research-validated multiple-choice question sequence on the basics of two-state quantum systems, including inner products, outer products, translation between Dirac notation and matrix representation in a particular basis, and change of basis. This study was conducted in an advanced undergraduate quantum mechanics course, in both online and in-person learning environments, across three years. For each cohort, students had their learning assessed after traditional lecture-based instruction in relevant concepts before engaging with the multiple-choice question sequence. Their performance was evaluated again afterward with a similar assessment and compared to their earlier performance. We analyze, compare, and discuss the trends observed in the three implementations.

Challenges in addressing student difficulties with quantum measurement of two-state quantum systems using a multiple-choice question sequence in online and in-person classes

Challenges in addressing student difficulties with quantum measurement of two-state quantum systems using a multiple-choice question sequence in online and in-person classes

Quantum approximate optimization algorithm for qudit systems

A frequent starting point of quantum computation platforms are two-state quantum systems, i.e., qubits. However, in the context of integer optimization problems, relevant to scheduling optimization and operations research, it is often more resource-efficient to employ quantum systems with more than two basis states, so-called qudits. Here, we discuss the quantum approximate optimization algorithm (QAOA) for qudit systems. We illustrate how the QAOA can be used to formulate a variety of integer optimization problems such as graph coloring problems or electric vehicle (EV) charging optimization. In addition, we comment on the implementation of constraints and describe three methods to include these into a quantum circuit of a QAOA by penalty contributions to the cost Hamiltonian, conditional gates using ancilla qubits, and a dynamical decoupling strategy. Finally, as a showcase of qudit-based QAOA, we present numerical results for a charging optimization problem mapped onto a max-$k$-graph coloring problem. Our work illustrates the flexibility of qudit systems to solve integer optimization problems.

A Novel Class of Phase Space Representations for the Exact Population Dynamics of Two-State Quantum Systems and the Relation to Triangle Window Functions

A Novel Class of Phase Space Representations for the Exact Population Dynamics of Two-State Quantum Systems and the Relation to Triangle Window Functions

Challenges in addressing student difficulties with measurement uncertainty of two-state quantum systems using a multiple-choice question sequence in online and in-person classes

Research-validated multiple-choice questions comprise an easy-to-implement instructional tool that serves to scaffold student learning and formatively assess students’ knowledge. We present findings from the implementation, in consecutive years, of a research-validated multiple-choice question sequence on measurement uncertainty as it applies to two-state quantum systems. This study was conducted in an advanced undergraduate quantum mechanics course, in online and in-person learning environments for consecutive years. Student learning was assessed after receiving traditional lecture-based instruction in relevant concepts, and their performance was compared with that of a similar assessment given after engaging with the multiple-choice question sequence. We analyze and discuss the similar and differing trends observed in the two modes of instruction.

Hybrid form of quantum theory with non-Hermitian Hamiltonians

In Schrödinger picture the unitarity of evolution is usually guaranteed by the Hermiticity of the Hamiltonian operator h=h† in a conventional Hilbert space Htextbook. After a Dyson-inspired operator-transformation (OT) non-unitary preconditioning Ω:h→H the simplified Hamiltonian H is, in its manifestly unphysical Hilbert space Hauxiliary, non-Hermitian. Besides its natural OT-based physical interpretation it can also be “Hermitized” (i.e., made compatible with the unitarity) via a metric-amendment (MA) change of the Hilbert space, Hauxiliary→Hphysical. In our present letter we propose another, third, hybrid form (HF) of the Hermitization of H in which the change involves, simultaneously, both the Hamiltonian and the metric. Formally this means that the original Dyson map is assumed factorizable, Ω=ΩMΩH. A key practical advantage of the new HF approach lies in the model-dependent adaptability of such a factorization. The flexibility and possible optimality of the balance between the MA-related (i.e., metric-amending) factor ΩM and the OT-related (i.e., Hamiltonian-changing) factor ΩH are explicitly illustrated via an elementary two-state quantum model.

Mixing times of three-state quantum walks on cycles

In this paper, we successfully obtain an explicit expression of the limit distribution [Formula: see text] of three-state quantum walks on cycles, the total variation distance between [Formula: see text] and the average probability [Formula: see text], and lower bound on the difference between two eigenvalues, among others. Based on the above conclusions, we finally get the mixing time [Formula: see text] of the quantum walk of Grover coin on the N-cycle. [Formula: see text] is the time required to characterize [Formula: see text] approaching [Formula: see text]. Our results show that the average probability of a three-state quantum walk on a cycle can approach its limit distribution faster than that of a two-state quantum walk, which might be of significance to quantum computation.

Dynamical charge susceptibility in nonequilibrium double quantum dots

Double quantum dots are one of the promising two-state quantum systems for realizing qubits. In the quest of successfully manipulating and reading information in qubit systems, it is of prime interest to control the charge response of the system to a gate voltage, as filled in by the dynamical charge susceptibility. We theoretically study this quantity for a nonequilibrium double quantum dot by using the functional integral approach and derive its general analytical expression. One highlights the existence of two lines of maxima as a function of the dot level energies, each of them being split under the action of a bias voltage. In the low frequency limit, we derive the capacitance and the charge relaxation resistance of the equivalent quantum RC-circuit with a notable difference in the range of variation for $R$ depending on whether the system is connected in series or in parallel. By incorporating an additional triplet state in order to describe the situation of a double quantum dot with spin, we obtain results for the resonator phase response which are in qualitative agreement with recent experimental observations in spin qubit systems. These results show the wealth of information brought by the knowledge of dynamical charge susceptibility in double quantum dots with potential applications for spin qubits.

An eigenfunction expansion formula for one-dimensional two-state quantum walks

The purpose of this paper is to give a direct proof of an eigenfunction expansion formula for one-dimensional two-state quantum walks, which is an analog of that for Sturm–Liouville operators due to Weyl, Stone, Titchmarsh, and Kodaira. In the context of the theory of CMV matrices, it had been already established by Gesztesy–Zinchenko. Our approach is restricted to the class of quantum walks mentioned above, whereas it is direct and it gives some important properties of Green functions. The properties given here enable us to give a concrete formula for a positive-matrix-valued measure, which gives directly the spectral measure, in a simplest case of the so-called two-phase model.

Decoherence in the three-state quantum walk

Quantum walks are dynamic systems with a wide range of applications in quantum computation and quantum simulation of analog systems, therefore it is of common interest to understand what changes from an isolated process to one embedded in an environment. In the present work, we analyze the decoherence in a three-state uni-dimensional quantum walk. The approaches taken into consideration to account for the environment effects are phase and amplitude damping Kraus operators, unitary noise on the coin space, and broken links. The results are compared with the two-state quantum walk.

Challenges in addressing student difficulties with time-development of two-state quantum systems using a multiple-choice question sequence in virtual and in-person classes

Research-validated clicker questions as instructional tools for formative assessment are relatively easy to implement and can provide effective scaffolding when developed and implemented in a sequence. We present findings from the implementation of a research-validated clicker question sequence (CQS) on student understanding of the time-development of two-state quantum systems. This study was conducted in an advanced undergraduate quantum mechanics course for two consecutive years in virtual and in-person classes. The effectiveness of the CQS discussed here in both modes of instruction was determined by evaluating students’ performance after traditional lecture-based instruction and comparing it to their performance after engaging with the CQS.

Unitary two-state quantum operators realized by quadrupole fields in the electron microscope

In this work, a novel method for using a set of electromagnetic quadrupole fields is presented to implement arbitrary unitary operators on a two-state quantum system of electrons. In addition to analytical derivations of the required quadrupole and beam settings which allow an easy direct implementation, numerical simulations of realistic scenarios show the feasibility of the proposed setup. This is expected to pave the way not only for new measurement schemes in electron microscopy and related fields but even one day for the implementation of quantum computing in the electron microscope.

Parrondo's paradox in quantum walks with deterministic aperiodic sequence of coins.

Parrondo's effect is a well-known apparent paradox where a combination of biased random walks displays a counterintuitive reversal of the bias direction. We show that Parrondo's effect can occur not only in the case of one-dimensional discrete quantum walks with random or deterministic periodic sequence of two- or multistate quantum coins but also in the case of one-dimensional discrete quantum walks with deterministic aperiodic sequence of two-state quantum coins. Moreover, we show how Parrondo's effect affects the time evolution of the walker-coin quantum entanglement.

Unitary equivalence classes of split-step quantum walks

In this study, we investigate the unitary equivalence of split-step quantum walks (SSQWs). Introduced by Kitagawa and generalized by Suzuki, SSQWs are two-state discrete-time quantum walks on a one-dimensional lattice. In this work, we define a new class of quantum walks that includes all SSQWs by making use of the rank conditions, which state that (1) the rank of operators representing the walk from vertex x to vertex $$x\pm 1$$ is less than or equal to 1 and that (2) the rank from vertex x to x (i.e., to itself) is less than or equal to 2. Initially, we abstractly define quantum walks in this class, then clarify by presenting the explicit form of such quantum walks. We also calculate their unitary equivalence classes and show that four real parameters are required for each vertex to parameterize the unitary equivalence classes. Further, we consider the unitary equivalence of Suzuki’s SSQWs and prove that all parameters of Suzuki’s SSQWs can be real. Finally, we discuss the chiral symmetry of SSQWs.