Two-state Quantum System Research Articles

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.

Dephasing Dynamics in a Non-Equilibrium Fluctuating Environment.

We performed a theoretical study of the dephasing dynamics of a quantum two-state system under the influences of a non-equilibrium fluctuating environment. The effect of the environmental non-equilibrium fluctuations on the quantum system is described by a generalized random telegraph noise (RTN) process, of which the statistical properties are both non-stationary and non-Markovian. Due to the time-homogeneous property in the master equations for the multi-time probability distribution, the decoherence factor induced by the generalized RTN with a modulatable-type memory kernel can be exactly derived by means of a closed fourth-order differential equation with respect to time. In some special limit cases, the decoherence factor recovers to the expression of the previous ones. We analyzed in detail the environmental effect of memory modulation in the dynamical dephasing in four types of dynamics regimes. The results showed that the dynamical dephasing of the quantum system and the conversion between the Markovian and non-Markovian characters in the dephasing dynamics under the influence of the generalized RTN can be effectively modulated via the environmental memory kernel.

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.

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.

Hidden Phase of the Spin-Boson Model

A quantum two-level system immersed in a sub-Ohmic bath experiences enhanced low-frequency quantum statistical fluctuations which render the nonequilibrium quantum dynamics highly non-Markovian. Upon using the numerically exact time-evolving matrix product operator approach, we investigate the phase diagram of the polarization dynamics. In addition to the known phases of damped coherent oscillatory dynamics and overdamped decay, we identify a new third region in the phase diagram for strong coupling showing an aperiodic behavior. We determine the corresponding phase boundaries. The dynamics of the quantum two-state system herein is not coherent by itself but slaved to the oscillatory bath dynamics.

Crossing time in the dissipative Landau\u2013Zener quantum dynamics

We study the dynamics of a quantum two-state system driven through an avoided crossing under the influence of a super-Ohmic environment. We determine the Landau–Zener probability employing the numerical exact quasi-adiabatic path integral and a Markovian weak coupling approach. Increasing the driving time in the numerical protocol, we find converged results which shows that super-Ohmic environments only influence the Landau Zener probability within a finite crossing time window. This crossing time is qualitatively determined by the environmental cut-off energy. At weak coupling, we show that the Markovian weak coupling approach provides an accurate description. Since pure dephasing of a super-Ohmic bath is non-Markovian, this highlights that pure dephasing hardly influences the Landau–Zener probability. The finite crossing time window, thus, results from the suppression of relaxation once the energy splitting exceeds the environmental cut-off energy.Graphical abstract

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.

Observing a Quantum Measurement

With the example of a Stern-Gerlach measurement on a spin-1/2 atom, we show that a superposition of both paths may be observed compatibly with properties attributed to state collapse - for example, the singleness (or mutual exclusivity) of outcomes. This is done by inserting a quantum two-state system (an ancilla) in each path, capable of responding to the passage of the atom, and thus acting as a virtual detector. We then consider real measurements on the compound system of atomic spin and two ancillae. Nondestructive measurements of a set of compatible joint observables can be performed, one for a superposition and others for collapse properties. A novel perspective is given as to why, within unitary quantum theory, ordinary measurements are blind to such superpositions. Implications for the theory of measurement are discussed.

A Physically Meaningful Relativistic Description of the Spin State of an Electron

We introduced a new model to present the states of a two-state quantum system. The space is the complexified Minkowski space. The Lorentz group acts by the linear extension of its action on the four-vectors. We applied this model to represent the spin state of an electron or any relativistic spin 1/2 particle. The spin state of such particle is of the form U+iS, where U is the four-velocity of the particle in the lab frame, and S is the 4D spin in this frame. Under this description, the transition probability between two pure spin states ϱ1 and ϱ2 of particles moving with the same velocity are defined by use of Minkowski dot product as 12<ϱ2|ϱ1>. This transition probability is Lorentz invariant, coincide with the quantum mechanics prediction and thus agree with the experimental results testing quantum mechanics predictions based on Bell’s inequality. For a a particle of mass m and charge q with the spin state ϱ, the total momentum is mcϱ and the electromagnetic momentum is qϱ. This imply that the Landé g factor for such particles must be g=2. We obtain an evolution equation of the spin state in an electromagnetic field which defines correctly the anomalous Zeeman effect and the fine structure splitting.

Rational Integrability of the Maxwell–Bloch System

The Maxwell–Bloch system arises naturally from laser models, which describes the dynamics of a two-state quantum system coupled to the basic electromagnetic resonator model. In this paper, we study its integrability and give a complete classification of the irreducible Darboux polynomials, which can help us find all polynomial and rational first integrals of the Maxwell–Bloch system. To this end, in the case that the loss rate of field is zero, we use the characteristic curve method and weight homogeneous polynomials to investigate the integrability of the Maxwell–Bloch system, and in the case that the loss rate of field is nonzero, we provide a linear scaling of variables to transform the Maxwell–Bloch system into the Lorenz system and obtain its integrability from some corresponding results of the Lorenz system. Our results show that although the Maxwell–Bloch system has chaotic behaviors for a large range of its parameters, it is nonchaotic when its parameters satisfy one of seven conditions, which may help us understand the complex and rich dynamics of nonlinear laser models.

Features of the Electronic Structure of Excited Quadrupolar Molecules in Non-Polar Solvents

Interactions of the electronic subsystem of the quadrupolar fluorophore with intramolecular highfrequency antisymmetric vibrations and solvent polarization are responsible for charge transfer symmetry breaking (SB), which is observed after optical excitation of such molecules in polar solvents. It is known that although these two interactions are mathematically described in similar ways, only the interaction of the fluorophore with solvent orientational polarization can create a state with broken symmetry if this interaction is strong enough. Nevertheless, the interaction of a quadrupolar fluorophore with intramolecular highfrequency antisymmetric vibrations in nonpolar solvents leads to a considerable reconstruction of the electronic subsystem. The analysis of the excited state of quadrupolar molecules in nonpolar solvents performed in this study reveals that such molecules can behave like quantum twostate systems, that is, as a quasispin s = 1/2, having an electric dipole moment instead of a magnetic one. This feature of excited quadrupolar molecules may be of interest to emerging technologies of molecular electronics.

Coherent control techniques for two-state quantum systems: A comparative study

We evaluate various sources of errors that occur when attempting to produce a specified coherent change of a two-state quantum system using six popular coherent control techniques: resonant excitation, adiabatic following, composite adiabatic passage, universal composite pulses, shortcut to adiabaticity, and single-shot shaped pulses. As error sources, we consider spatial intensity distribution, transit time variation, inhomogeneous broadening, Doppler broadening, unwanted chirp, and shape errors, as well as errors generated by counter-rotating terms. For the various error types, different techniques emerge as the best performers, but overall we find that universal composite pulses perform most consistently and are most resilient to errors compared to all other procedures.

Two-State Quantum Systems Revisited: A Clifford Algebra Approach

We revisit the topic of two-state quantum systems using Geometric Algebra (GA) in three dimensions $\mathcal G_3$. In this description, both the quantum states and Hermitian operators are written as elements of $\mathcal G_3$. By writing the quantum states as elements of the minimal left ideals of this algebra, we compute the energy eigenvalues and eigenvectors for the Hamiltonian of an arbitrary two-state system. The geometric interpretation of the Hermitian operators enables us to introduce an algebraic method to diagonalize these operators in GA. We then use this approach to revisit the problem of a spin-$1/2$ particle interacting with an external arbitrary constant magnetic field, obtaining the same results as in the conventional theory. However, GA reveals the underlying geometry of these systems, which reduces to the Larmor precession in an arbitrary plane of $\mathcal G_3$.

Energy dissipation during two-state switching for quantum-dot cellular automata

We examine the energy dissipated by a two-state quantum system during a switching operation when interacting with a thermal environment. For an isolated system, the excess energy decreases exponentially with switching time. For classically damped systems, the energy dissipation decreases linearly with switching time. We model the quantum system coupled to a thermal environment using a Lindblad equation for the density matrix. For rapid switching, the exponential quantum adiabaticity holds. For slow enough switching, the damping from the bath yields linear dissipation, as in the classical limit. Between these two limits, when the switching time is comparable to the characteristic energy transfer time to the thermal bath, there is an inverted region when dissipation increases with longer switching times. Consequences for the design of molecular quantum-dot cellular automata are discussed.

Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems.

A novel uncertainty relation for errors of general quantum measurement is presented. The new relation, which is presented in geometric terms for maps representing measurement, is completely operational and can be related directly to tangible measurement outcomes. The relation violates the naïve bound for the position-momentum measurement, whilst nevertheless respecting Heisenberg’s philosophy of the uncertainty principle. The standard Kennard–Robertson uncertainty relation for state preparations expressed by standard deviations arises as a corollary to its special non-informative case. For the measurement on two-state quantum systems, the relation is found to offer virtually the tightest bound possible; the equality of the relation holds for the measurement performed over every pure state. The Ozawa relation for errors of quantum measurements will also be examined in this regard. In this paper, the Kolmogorovian measure-theoretic formalism of probability—which allows for the representation of quantum measurements by positive-operator valued measures (POVMs)—is given special attention, in regard to which some of the measure-theory specific facts are remarked along the exposition as appropriate.

Theory of emittance exchange through coupling resonance crossing

In circular accelerators, transverse beam emittances can be exchanged by means of a coupling resonance crossing. It was first demonstrated experimentally at CERN accelerators, before a theoretical study revealed its mechanism. The existing theory of emittance exchange seems, however, to be deficient. In this study, we establish a theory of emittance exchange using the matrix formalism. Furthermore, we have found that the mechanism of emittance exchange is analogous to the transition dynamics of two-state quantum systems, thereby allowing the Landau-Zener formula to be applied for the prediction of transverse emittance values after nonadiabatic crossing.