Quantum Mechanical Two-level System Research Articles

Coherent charge oscillations in a bilayer graphene double quantum dot

The coherent dynamics of a quantum mechanical two-level system passing through an anti-crossing of two energy levels can give rise to Landau-Zener-Stückelberg-Majorana (LZSM) interference. LZSM interference spectroscopy has proven to be a fruitful tool to investigate charge noise and charge decoherence in semiconductor quantum dots (QDs). Recently, bilayer graphene has developed as a promising platform to host highly tunable QDs potentially useful for hosting spin and valley qubits. So far, in this system no coherent oscillations have been observed and little is known about charge noise in this material. Here, we report coherent charge oscillations and T2*\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${T}_{2}^{*}$$\\end{document} charge decoherence times in a bilayer graphene double QD. The charge decoherence times are measured independently using LZSM interference and photon assisted tunneling. Both techniques yield T2*\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${T}_{2}^{*}$$\\end{document} average values in the range of 400–500 ps. The observation of charge coherence allows to study the origin and spectral distribution of charge noise in future experiments.

Nested SU(2) symmetry of photonic orbital angular momentum

The polarization state is described by a quantum mechanical two-level system, which is known as special unitary group of degree 2 [SU(2)]. Polarization is attributed to an internal spin degree of freedom inherent to photons, while photons also possess an orbital degree of freedom. A fundamental understanding of the nature of spin and orbital angular momentum of photons is significant to utilize the degrees of freedom for various applications in optical communications, computations, sensing, and laser-patterning. Here, we show that the orbital angular momentum of coherent photons emitted from a laser diode can be incremented using a vortex lens, and the magnitude of orbital angular momentum increases with an increase in the topological charge inside the mode. The superposition state of the left and right vortices is described by the SU(2) state, similar to polarization; however, the radius of the corresponding Poincaré sphere depends on the topological charge. Consequently, we expect a nested SU(2) structure to describe various states with different magnitudes in orbital angular momentum. We have experimentally developed a simple system to realize an arbitrary SU(2) state of orbital angular momentum by controlling both amplitudes and phases of the left and right vortices using a spin degree of freedom, whose interplays were confirmed by expected far-field images of dipoles and quadruples.

Polariton condensate and Landau-Zener-Stückelberg interferometry transition in multilayer transition metal dichalcogenides

This paper gives a detailed description of a high-performance polariton condensate for a quantum mechanical two-level system (TLS). We propose a transition metal dichalcogenides (TMDs) setup and theoretically carry out the spectroscopy of these polariton condensates. Through theoretical and numerical analysis, we obtain many features in two dimensional (2D) multilayer TMDs. We compute the energy of the system and the Landau-Zener-Stückelberg (LZS) quantum tunneling probability under the effect of a sequence of laser light. At certain critical 2D TMDs parameters, the system exhibits a multi-crossing scenario in a privileged position of 2D multilayer TMDs. We predict the consecutive modulations and highlight the conservation of the LZS interference patterns mapped from the 2D TMDs system. At weak coupling regime, a successful conversion of interferometry signals is identified for some values of laser frequency. We explain such a result as a valley sensitive cavity rate model due to coherent exchange and incoherent scattering, meaning that polariton condensate is formed in the valley around the Brillouin zone. The latter is used quantitatively and qualitatively to achieve high-precision measurements beyond that of its elementary constituents. The obtained results confirm that has the highest sensitivity to radiation field as compared to other 2D multilayer TMDs materials. Therefore, stands as an appropriate candidate among other 2D TMDs to form polariton condensates.

Nonequilibrium Landau-Zener tunneling in exciton-polariton condensates

For a coherent quantum mechanical two-level system driven with a linearly time-dependent detuning, the Landau-Zener model has served over decades as a textbook model of quantum dynamics. A particularly intriguing question is whether that framework can be extended to capture an intrinsical nonequilibrium nature for a quantum system with coherent and dissipative dynamics occurring on an equal footing. In this work, we are motivated to investigate the Landau-Zenner problem of polariton condensates in a periodic potential under nonresonant pumping, considering driven-dissipative Gross-Pitaevskii equations coupled to the rate equation of a reservoir. Using a two-mode approach, we find fluctuation of the reservoir can be considered as a constant and the relative phase plays a very important role. The evolution of the dissipative Landau-Zener model we obtain presents its adiabatic process very different from the closed system because the fluctuation of the reservoir has a peak and leads to the damping of the condensates. We substitute the fluctuation of the reservoir to Hamiltonian and get an effective two-level model. The motion of Hamiltonian in phase space is also discussed and is directly corresponding to the pumping rate. The instability of the band structure can also be studied by the curvatures in phase space and there may be two loops in the middle of the Brillouin zone when the pumping rate is far beyond the threshold.

Universal energy scaling law for optimally excited nonlinear oscillators.

We compute the optimal temporal profile for an external driving force F(t) that can maximize the energy absorption of any driven nonlinear oscillator. The technique is based on constraining the maximum amplitude of the force field such that optimal control theory can provide quasianalytical solutions. We illustrate this computational technique for the undamped Duffing oscillator as well as for a driven quantum mechanical two-level system. We find that under optimal force conditions the asymptotic time-dependence of the maximum amplitude growth is given by a power law X(t)∼t^{2/α}, where the (possibly noninteger) exponent is determined by the highest degree of the oscillator's nonlinearity α. As a universal result, this predicts that the maximal energy absorption of any nonlinear oscillator grows (under an optimized force field) quadratically in time. We also find for the two-level system that-even under optimized excitation conditions-the maximally achievable inversion does not monotonically increase with the force amplitude. It is characterized by an interesting sequence of n-cycle thresholds as well as a self-termination of the growth.

Resonance Enhanced Multiphoton Ionization Detected Millimeter-Wave Absorption: The 115 GHz Line of CO.

We investigate the potential for sensitively detecting millimeter-wave absorption through multiphoton ionization using the lowest rotational transition in CO (X1Σ, J = 1 ← J = 0) at 115 GHz as a diagnostic tool. This transition represents an almost perfect realization of a quantum mechanical two-level system. In the experiment, the output of a powerful continuous millimeter-wave source with sub-kilohertz resolution propagates counter to a pulsed molecular beam, causing Doppler shifted resonances. Absorption is detected by monitoring the involved rotational levels through resonance enhanced multiphoton ionization (REMPI). Observed polarization, saturation and Doppler effects demonstrate that the interaction with the millimeter-wave field is coherent and can be simulated well using the results for the two-level system. The combination of these results indicates that the millimeter-wave field is approximated well by a Gaussian beam with beam waist around 15 mm and constant effective electric field amplitude on the molecular beam axis. Time-correlated frequency modulation in the form of a short rectangular pulse is used to follow the molecular trajectory from the source to the laser interaction region. The experiments clearly demonstrate a great potential of extending high-resolution molecular beam spectroscopy with mass specific REMPI detection into the sub-terahertz or terahertz regimes.

Landau-Zener tunneling of qubit states and Aharonov-Bohm interferometry in double quantum wires

This paper gives a detailed description of a high-performance tunneling wire “flying qubit” for a quantum mechanical two-level system. We propose a simple double-wire interferometer setup and theoretically carry out the spectroscopy of these flying qubits. As such, we present a tight binding theoretical model which effectively mimicks the experimental setup and then compute the energy diagram and the Landau-Zener quantum tunneling probability under the effect of a sequence of magnetic pulses and gate voltages. At certain critical electrostatic gate voltages, the system exhibits a step rise in the Landau-Zener tunneling transition by playing a multi-crossing scenario in a privileged position of two geometric wires. We predict the consecutive modulations and precise the conservation of the Landau-Zener-Stückelberg-Majorana interference patterns mapped from the two-terminal of the double-wire system. A successful conversion of interferometry signals is identified within the two-path interferometer with higher intrinsic quantum correction errors. The latter is used quantitatively and qualitatively to achieve high-precision measurements beyond the standard Landau-Zener limit. The obtained results confirm the possibility of the construction of Aharonov-Bohm interferometers following the proposed model.

CUSP-SINGULARITY OF THE TRAJECTORY OF THE QUBIT’S BLOCH VECTOR UNDER AN EXTERNAL PERIODIC LIGHT FIELD

The quantum dynamics of a two-level quantum-mechanical system subjected to the external monochromatic action beyond the rotating wave approximation was investigated. It was shown that under the condition of exact resonance on the trajectories of the Bloch vectors, special points are manifested under different initial conditions. These points are classified as cusps singularities. It is revealed that at such points, the instantaneous rotation axis, relative to which the Bloch vector rotates, reverses its direction. There is a movement stop. For a nonzero frequency detuning, the cusp singularities vanish. A numerical analysis of the singularities of the trajectories of the Bloch vector without rotating wave approximation was supplemented by a study based on the use of the Floquet methods. Within the framework of this approach, recurrence relations for the spectral components of the probability amplitudes were obtained and analyzed. An analytic expression was found for the two values of quasi-energies within a fourth order of magnitude in the interaction energy. It was shown that to obtain a singular behavior of the trajectories of the Bloch vector, it is sufficient to confine by four spectral harmonics in the Floquet expansion. The results obtained are important for achieving the accuracy when performing coherent transformations with two-level systems in the cases where the rotating wave approximation is inapplicable.

Classical Stückelberg interferometry of a nanomechanical two-mode system

The transition from classical to quantum mechanics has intrigued scientists in the past and remains one of the most fundamental conceptual challenges in state-of-the-art physics. Beyond the quantum mechanical correspondence principle, quantum-classical analogies have attracted considerable interest. In this work, we present classical two-mode interference for a nanomechanical two-mode system, realizing classical St\"uckelberg interferometry. In the past, St\"uckelberg interferometry has been investigated exclusively in quantum mechanical two-level systems. Here, we experimentally demonstrate a classical analog of St\"uckelberg interferometry taking advantage of coherent energy exchange between two-strongly coupled, high quality factor nanomechanical resonator modes. Furthermore, we provide an exact theoretical solution for the double passage St\"uckelberg problem which reveals the analogy of the return probabilities in the quantum mechanical and the classical version of the problem. This result qualifies classical two-mode systems at large as a testbed for quantum mechanical interferometry.

Geometry of adiabatic Hamiltonians for two-level quantum systems

We present the formulation of the problem of the coherent dynamics of quantum mechanical two-level systems in the adiabatic region in terms of the differential geometry of plane curves. We show that there is a natural plane curve corresponding to the Hamiltonian of the system for which the geometrical quantities have a simple physical interpretation. In particular, the curvature of the curve has the role of the nonadiabatic coupling.

Electric dipoles on the Bloch sphere

The time evolution of a two-level quantum mechanical system can be geometrically described using the Bloch sphere. By mapping the Bloch sphere evolution onto the dynamics of oscillating electric dipoles, we provide a physically intuitive link between classical electromagnetism and the electric dipole transitions of atomic and molecular physics.

Entanglement dynamics in a quantum–classical hybrid of two q-bits and one oscillator

We investigate new features, especially of entanglement dynamics, which arise in a quantum–classical hybrid. As a model, we study the coupling between two quantum mechanical two-level systems, i.e., two q-bits and a classical harmonic oscillator. Their interaction is described by a hybrid coupling, in accordance with a recently developed quantum–classical hybrid theory. We discuss various situations in which entanglement of the q-bits does (not) evolve. Furthermore, we point out an experimental application in a hybrid cooling scheme and indicate topics for future study.

Quantum phase transition and correlations in the multi-spin-boson model

We consider multiple non-interacting quantum mechanical two-level systems coupled to a common bosonic bath and study its quantum phase transition with Monte Carlo simulations using a continuous imaginary time cluster algorithm. The common bath induces an effective ferromagnetic interaction between the otherwise independent two-level systems, which can be quantified by an effective interaction strength. For degenerate energy levels above a critical value of the bath coupling strength $\alpha$ all two-level systems freeze into the same state and the critical value $\alpha_c$ decreases asymptotically as $1/N$ with increasing $N$. For a finite number, $N$, of two-level systems the quantum phase transition (at zero temperature) is in the same universality class as the single spin-boson model, in the limit $N\to\infty$ the system shows mean-field critical behavior independent of the power of the spectral function of the bosonic bath. We also study the influence of a spatial separation of the spins in a bath of bosonic modes with linear dispersion relation on the location and characteristics of the phase transition as well as on correlations between the two-level systems.

The classical Bloch equations

Coherent control of a quantum mechanical two-level system is at the heart of magnetic resonance imaging, quantum information processing, and quantum optics. Among the most prominent phenomena in quantum coherent control are Rabi oscillations, Ramsey fringes, and Hahn echoes. We demonstrate that these phenomena can be derived classically by use of a simple coupled-harmonic-oscillator model. The classical problem can be cast in a form that is formally equivalent to the quantum mechanical Bloch equations with the exception that the longitudinal and the transverse relaxation times (T1 and T2) are equal. The classical analysis is intuitive and well suited for familiarizing students with the basic concepts of quantum coherent control, while at the same time highlighting the fundamental differences between classical and quantum theories.

Temporal and spectral evolution of an interrupted virtual single-photon transition: creation of optical gain and loss

We examine the optical response of a virtual dipole transition of a quantum mechanical two-level system (TLS). In the case of off-resonant excitation the time-integrated dipole response (TIDR) is expected to be zero, which corresponds to transparency of the system with respect to the exciting pulse. Our new time-frequency representation reveals that even for a zero TIDR there are positive and negative contributions included in the response. Furthermore, we present a way to access these contributions by using a second electromagnetic field, which interrupts the temporal evolution of the dipole response. The theoretical results are confirmed by attosecond transient absorption spectroscopy in helium (He).

Surface plasmon hybridization and exciton coupling

We derive a semianalytical model to describe the interaction of a single photon emitter and a collection of arbitrarily shaped metal nanoparticles. The theory treats the metal nanoparticles classically within the electrostatic eigenmode method, wherein the surface plasmon resonances of collections of nanoparticles are represented by the hybridization of the plasmon modes of the noninteracting particles. The single photon emitter is represented by a quantum mechanical two-level system that exhibits line broadening due to a finite spontaneous decay rate. Plasmon-emitter coupling is described by solving the resulting Bloch equations. We illustrate the theory by studying model systems consisting of a single emitter coupled to one, two, and three nanoparticles, and we also compare the predictions of our model to published experimental data.

How to best sample a periodic probability distribution, or on the accuracy of Hamiltonian finding strategies

Projective measurements of a single two-level quantum mechanical system (a qubit) evolving under a time-independent Hamiltonian produce a probability distribution that is periodic in the evolution time. The period of this distribution is an important parameter in the Hamiltonian. Here, we explore how to design experiments so as to minimize error in the estimation of this parameter. While it has been shown that useful results may be obtained by minimizing the risk incurred by each experiment, such an approach is computationally intractable in general. Here, we motivate and derive heuristic strategies for experiment design that enjoy the same exponential scaling as fully optimized strategies. We then discuss generalizations to the case of finite relaxation times, T_2 < \infty.

Many-body Landau–Zener dynamics in coupled one-dimensional Bose liquids

The Landau-Zener model of a quantum mechanical two-level system driven with a linearly time dependent detuning has served over decades as a textbook paradigm of quantum dynamics. In their seminal work [L. D. Landau, Physik. Z. Sowjet. 2, 46 (1932); C. Zener, Proc. Royal Soc. London 137, 696 (1932)], Landau and Zener derived a non-perturbative prediction for the transition probability between two states, which often serves as a reference point for the analysis of more complex systems. A particularly intriguing question is whether that framework can be extended to describe many-body quantum dynamics. Here we report an experimental and theoretical study of a system of ultracold atoms, offering a direct many-body generalization of the Landau-Zener problem. In a system of pairwise tunnel-coupled 1D Bose liquids we show how tuning the correlations of the 1D gases, the tunnel coupling between the tubes and the inter-tube interactions strongly modify the original Landau-Zener picture. The results are explained using a mean-field description of the inter-tube condensate wave-function, coupled to the low-energy phonons of the 1D Bose liquid.

Microscopic work distribution of small systems in quantum isothermal processes and the minimal work principle

For a two-level quantum mechanical system, we derive microscopically the exact expression for the fluctuation of microscopic work in a multistep nonequilibrium process, and we rigorously prove that in an isothermal process, the fluctuation is vanishingly small and the most probabilistic work is just equal to the difference of the free energy. Our study demonstrates that the convergence of the microscopic work in the isothermal process is due to the nature of the isothermal process rather than the usual thermodynamic limit condition. Our investigation justifies the validity of a "minimum work principle" formulation of the second law even for a small system far from the thermodynamic limit.

Spins in few-electron quantum dots

The canonical example of a quantum-mechanical two-level system is spin. The simplest picture of spin is a magnetic moment pointing up or down. The full quantum properties of spin become apparent in phenomena such as superpositions of spin states, entanglement among spins, and quantum measurements. Many of these phenomena have been observed in experiments performed on ensembles of particles with spin. Only in recent years have systems been realized in which individual electrons can be trapped and their quantum properties can be studied, thus avoiding unnecessary ensemble averaging. This review describes experiments performed with quantum dots, which are nanometer-scale boxes defined in a semiconductor host material. Quantum dots can hold a precise but tunable number of electron spins starting with 0, 1, 2, etc. Electrical contacts can be made for charge transport measurements and electrostatic gates can be used for controlling the dot potential. This system provides virtually full control over individual electrons. This new, enabling technology is stimulating research on individual spins. This review describes the physics of spins in quantum dots containing one or two electrons, from an experimentalist's viewpoint. Various methods for extracting spin properties from experiment are presented, restricted exclusively to electrical measurements. Furthermore, experimental techniques are discussed that allow for (1) the rotation of an electron spin into a superposition of up and down, (2) the measurement of the quantum state of an individual spin, and (3) the control of the interaction between two neighboring spins by the Heisenberg exchange interaction. Finally, the physics of the relevant relaxation and dephasing mechanisms is reviewed and experimental results are compared with theories for spin-orbit and hyperfine interactions. All these subjects are directly relevant for the fields of quantum information processing and spintronics with single spins (i.e., single spintronics).