Pauli Spin Operators Research Articles

The double scaling limit of randomly coupled Pauli XY spins

We consider the double scaling limit of a model of Pauli spin operators recently studied in Hanada et al. [1] and evaluate the moments of the Hamiltonian by the chord diagrams. We find that they coincide with those of the double scaled SYK model, which makes it more likely that this model may play an important role in the study of holography. We compare the model with another previously studied model. We also speculate on the form of the Hamiltonian in the double scaling limit.

A model of randomly-coupled Pauli spins

We construct a model of Pauli spin operators with all-to-all 4-local interactions by replacing Majorana fermions in the SYK model with spin operators. Equivalently, we replace fermions with hard-core bosons. We study this model numerically and compare the properties with those of the SYK model. We observe a striking quantitative coincidence between the spin model and the SYK model, which suggests that this spin model is strongly chaotic and, perhaps, can play some role in holography. We also discuss the path-integral approach with multi-local fields and the possibility of quantum simulations. This model may be an interesting target for quantum simulations because Pauli spins are easier to implement than fermions on qubit-based quantum devices.

Direct tomography of quantum states and processes via weak measurements of Pauli spin operators on an NMR quantum processor

Direct tomography of quantum states and processes via weak measurements of Pauli spin operators on an NMR quantum processor

Local operator entanglement in spin chains

Understanding how and whether local perturbations can affect the entire quantum system is a fundamental step in understanding non-equilibrium phenomena such as thermalization. This knowledge of non-equilibrium phenomena is applicable for quantum computation, as many quantum computers employ non-equilibrium processes for computations. In this paper, we investigate the evolution of bi- and tripartite operator mutual information of the time-evolution operator and the Pauli spin operators in the one-dimensional Ising model with magnetic field and the disordered Heisenberg model to study the properties of quantum circuits. In the Ising model, the early-time evolution qualitatively follows an effective light cone picture, and the late-time value is well described by Page’s value for a random pure state. In the Heisenberg model with strong disorder, we find that many-body localization prevents the information from propagating and being delocalized. We also find an effective Ising Hamiltonian that describes the time evolution of bi- and tripartite operator mutual information for the Heisenberg model in the large disorder regime.

Spin 1/2 one‐ and two‐particle systems in physical space without eigen‐algebra or tensor product

Under the spin–position decoupling approximation, a vector with a phase in 3D orientation space endowed with geometric algebra substitutes the vector–matrix spin model built on the Pauli spin operator. The standard quantum operator‐state spin formalism is replaced with vectors transforming by proper and improper rotations in the same 3D space—isomorphic to the space of Pauli matrices. In the single‐spin case, the novel spin 1/2 representation (1) is Hermitian, (2) shows handedness, (3) yields all the standard results and its modulus equals the total spin angular momentum , (4) formalizes irreversibility in measurement, and (5) permits adaptive imbedding of the 2D spin space in 3D. Maximally entangled spin pairs (1) are in phase and have opposite handedness, (2) relate by one of the four basic improper rotations in 3D: plane reflections (triplets) and inversion (singlet), (3) yield the standard total angular momentum, and (4) all standard expectation values for bipartite and partial observations follow. Depending on whether proper and improper rotors act one—or two—sided, the formalism appears in two complementary forms, the “spinor” or the “vector” form, respectively. The proposed scheme provides a clear geometric picture of spin correlations and transformations entirely in the 3D physical orientation space.

Generalization of Spin Operators

The study shall look to the group of generators SU(4). From these generators, a new group spin operator will be constructed. We will classify these groups into right handed groups and left handed groups. These two groups will satisfy all the properties of Pauli spin operators Sx, Sy and Sz with respect to the frame xyz. The analysis shows that the number of groups spin operators depends on the order of the group. This leads us to construct the theorem which defines the number of the groups spin operators. The analysis also leads to two kinds of frames: left handed frame (LHF) and right handed frame (RHF). The right handed operators will act on the RHF, and left hand operators act on the LHF. The study shall discuss the notion of spin squeezing for pure spin 3/2 system by using our new frames and new spin operators. It will show that our calculation is equivalent to the calculation by using Pauli spin operators.

From Ising Model to Kitaev Chain

In this general article, we map the one-dimensional transverse field quantum Ising model of ferromagnetism to Kitaev's one-dimensional p-wave superconductor, which has its application in fault-tolerant topological quantum computing. Mapping Pauli's spin operators of transverse Ising chain to spinless fermionic creation and annihilation operators by Inverse Jordan-Wigner transformation leads to a Hamiltonian form closely related to Kitaev Chain, which exhibits topological phase transition where phases are characterized by different topological invariant that changes discontinuously at the transition point. Kitaev Chain supports two Majorana zero modes (MZMs) in the non-trivial topological phase, while none is in the trivial phase. The doubly degenerate ground state of the transverse Ising in ferromagnetic phase corresponds to non-local free fermion degree made from MZMs. The quasi-particle excitations of Ising chain, viz., domain wall formation in the ferromagnetic phase and spin-flip in paramagnetic phase maps to Bogoliubon excitations. The mapping suggests that a non-local order parameter can be defined for Kitaev Chain to work with the usual paradigm of Landau's theory.

Spin-boson model of quantum dissipation in graphene: Nonlinear electrical response

Dissipation in the quantum solid of graphene is couched in the methodology of nonequilibrium statistical mechanics of an open quantum system. Of specific interest is the absorption accompanied by relaxation of energy because of an external frequency-dependent oscillatory electric field. At optical frequencies and for sizable amplitude of the electric field, the response is nonlinear, the treatment of which is simplified by means of a ``rotating wave approximation'' in which rapidly changing, off-resonance terms are omitted. The characteristic resonant frequency is what quantifies the tunneling between the valence and the conduction bands across the so-called Dirac point in graphene. The valence and the conduction states are mapped into the eigenstates of pseudo Pauli spin operators and the corresponding Hamiltonian, when embedded in a dissipative heat bath comprising the surrounding phonons and other electrons, makes possible a comprehensive analysis in terms of the much-studied spin-boson model of dissipative quantum statistical mechanics. A master equation for the density operator associated with this Hamiltonian yields rate equations for the mean population of the valence and the conduction states as well as the transition (matrix elements) between them. Further approximation of these rate equations allows contact with phenomenological treatments of the nonlinear optical conductivity. The present paper then provides a microscopic framework for investigating the response characteristics of a material of great topical interest using contemporary methods of quantum dissipation.

Non-Contextual and Local Hidden-Variable Model for the Peres–Mermin and Greenberger–Horne–Zeilinger Systems

A hidden-variable model for quantum-mechanical spin, as represented by the Pauli spin operators, is proposed for systems illustrating the well-known no-hidden-variables arguments by Peres and Mermin (1990) and by Greenberger, Horne, and Zeilinger (1989). Both arguments rely on an assumption of non-contextuality; the latter argument can also be phrased as a non-locality argument, using a locality assumption. The model suggested here is compatible with both assumptions. This is possible because the scalar values of spin observables are replaced by vectors that are components of orientations.

Relativistic electron-spin dynamics in a strong unipolar laser field

The behavior of an electron spin interacting with a linearly polarized laser field is analyzed. In contrast to previous considerations of the problem, the initial state of the electron represents a localized wave packet, and a spatial envelope is introduced for the laser pulse, which allows one to take into account the finite size of both objects. Special attention is paid to ultrashort pulses possessing a high degree of unipolarity. Within a classical treatment (both nonrelativistic and relativistic), proportionality between the change of the electron spin projections and the electric field area of the pulse is clearly demonstrated. We also perform calculations of the electron spin dynamics according to the Dirac equation. Evolving the electron wave function in time, we compute the mean values of the spin operator in various forms. It is shown that the classical relativistic predictions are accurately reproduced when using the Foldy-Wouthuysen operator. The same results are obtained when using the Lorentz transformation and the nonrelativistic (Pauli) spin operator in the particle's rest frame.

Robust Dynamic Hamiltonian Engineering of Many-Body Spin Systems

We introduce a new approach for the robust control of quantum dynamics of strongly interacting many-body systems. Our approach involves the design of periodic global control pulse sequences to engineer desired target Hamiltonians that are robust against disorder, unwanted interactions and pulse imperfections. It utilizes a matrix representation of the Hamiltonian engineering protocol based on time-domain transformations of the Pauli spin operator along the quantization axis. This representation allows us to derive a concise set of algebraic conditions on the sequence matrix to engineer robust target Hamiltonians, enabling the simple yet systematic design of pulse sequences. We show that this approach provides an efficient framework to (i) treat any secular many-body Hamiltonian and engineer it into a desired form, (ii) target dominant disorder and interaction characteristics of a given system, (iii) achieve robustness against imperfections, (iv) provide optimal sequence length within given constraints, and (v) substantially accelerate numerical searches of pulse sequences. Using this systematic approach, we develop novel sets of pulse sequences for the protection of quantum coherence, optimal quantum sensing and quantum simulation. Finally, we experimentally demonstrate the robust operation of these sequences in a system with the most general interaction form.

Generation of Spin 1 System

In this paper, we shall establish the connection between the group theory and quantum mechanics by showing how the group theory helps us to construct the spin operators. We look to the group generators SU (3). From these generators, new spin 1 operators will be constructed. These operators S-x, S-y and S-z satisfy all the properties of Pauli spin operators S-x, S-y and S-z. We shall discuss the notion of spin squeezing and correlations for pure spin 1 system using our spin operators S-x, S-y and S-z.

Incompatibility of observables as state-independent bound of uncertainty relations

For a pair of observables, they are called "incompatible", if and only if the commutator between them does not vanish, which represents one of the key features in quantum mechanics. The question is, how can we characterize the incompatibility among three or more observables? Here we explore one possible route towards this goal through Heisenberg's uncertainty relations, which impose fundamental constraints on the measurement precisions for incompatible observables. Specifically, we quantify the incompatibility by the optimal state-independent bounds of additive variance-based uncertainty relations. In this way, the degree of incompatibility becomes an intrinsic property among the operators, but not on the quantum state. To justify our case, we focus on the incompatibility of spin systems. For an arbitrary setting of two or three linearly-independent Pauli-spin operators, the incompatibility is analytically solved, the spins are maximally incompatible if and only if they are orthogonal to each other. On the other hand, the measure of incompatibility represents a versatile tool for applications such as testing entanglement of bipartite states, and EPR-steering criteria.

One and two spin-1/2 particle systems under the Lorentz transformations

Lorentz transformation (LT) was used to link two inertial frames, consisted of moving and lab frames. In addition, the effects of LT on the states of two and one spin-½ particle systems are addressed. Throughout the paper, we only consider two spin operators including Czachor's and the Pauli spin operators. It is shown that the system's state predictions made by Pauli spin operator for one spin-½ particle systems is better than that of made by Czachor's spin operator. Thereinafter, we focused on entangled systems consisted of two spin-½ particles moving away from each other and the treatment of system state under Lorentz transformation was studied .We also use both Pauli and Czachor’s operators to build the Bell’s operator. Additionally, we address the behavior of Bell’s inequality under LT and compare the results made by considering Pauli’s operator with that of from Czachor's spin operator. In the last part, some results of considering the Pauli-Lubanski spin operator are also addressed.

Bose description of the generalized Pauli spin operators and para super symmetry

In this paper, the generalized fermion algebra is introduced so as to have the [Formula: see text]-dimensional Fock space. The generalized Pauli spin matrices of order [Formula: see text] are introduced by identifying these matrices with the step operators of the generalized fermion algebra. From the [Formula: see text]-graded parity relation [Formula: see text] generalized Pauli matrices are obtained. Finally, the para super symmetry (SUSY) is realized in terms of these matrices and ordinary bosonic operators.

Optical pumping of the atom magnetic moment under spin polarization equal to zero

Possibility of preparing of an alkali-atom's state with non-zero magnetic moment and zero spin-polarization was demonstrated in this work. There are zero average values of Pauli spin operators for all coordinates in the case. We suggest using a biharmonic laser field for preparing such states. Moreover both components of the laser field should be set to resonance with two sublines in D1 line of the alkali and have circular polarization. The amplitudes of the components needed to prepare the observed states was found. It is found interesting that polarization of the components of the laser may be orthogonal or parallel for preparing the states.

Solvable Hamiltonians and Fermionization Transformations Obtained from Operators Satisfying Specific Commutation Relations

It is shown that a solvable Hamiltonian can be obtained from a series of operators satisfying specific commutation relations. A transformation that diagonalize the Hamiltonian is obtained simultaneously. The two-dimensional Ising model with periodic interactions, the one-dimensional XY model with period 2, the transverse Ising chain, the one-dimensional Kitaev model and the cluster model, and other composite quantum spin chains are diagonalized following this procedure. The Jordan–Wigner transformation, the transformation from the Pauli spin operators to the Majorana fermion used by Shankar and Murthy, and the transformation introduced by Nambu, are special cases of this treatment.

Three-particle Bell-like inequalities under Lorentz transformations

We study the effects of Lorentz transformations on three-particle non-local system states (GHZ and W) of spin 1/2 particles, using the Pauli spin operator and a three-particle generalization of Bell's inequality, introduced by Svetlichny. In our setup, the moving and laboratory frames used the (same) set of measurement directions that maximally violate Svetlichny's inequality in the laboratory frame. We also investigate the behavior of Mermin's and Collins' inequalities. We find that, regardless of the particles' type of entanglement, violation of Svetlichny's inequality in the moving frame is decreased by increasing the boost velocity and the energy of particles in the laboratory frame. In the relativistic regime Svetlichny's inequality is a good criterion to investigate the non-locality of the GHZ state. We also find that Mermin's and Collins' inequalities lead to reasonable predictions, in agreement with the behavior of the spin state, about non-locality of the W state in the relativistic regime. Then, comparing our results with those in which Czachor's relativistic spin is used instead of the Pauli operator, we find that the results obtained by considering the Pauli spin operator are in better agreement with the behavior of spin state of the system in the relativistic information theory.

Relativistic spin operators in various electromagnetic environments

Different operators have been suggested in the literature to describe the electron's spin degree of freedom within the relativistic Dirac theory. We compare concrete predictions of the various proposed relativistic spin operators in different physical situations. In particular, we investigate the so-called Pauli, Foldy-Wouthuysen, Czachor, Frenkel, Chakrabarti, Pryce, and Fradkin-Good spin operators. We demonstrate that when a quantum system interacts with electromagnetic potentials the various spin operators predict different expectation values. This is explicitly illustrated for the scattering dynamics at a potential step and in a standing laser field and also for energy eigenstates of hydrogenic ions. Therefore, one may distinguish between the proposed relativistic spin operators experimentally.

Scheme for directly observing the noncommutativity of the position and the momentum operators with interference

Although non-commutativity of a certain set of quantum operators (e.g., creation/annihilation operators and Pauli spin operators) has been shown experimentally in recent years, the commu- tation relation for the position and the momentum operators has not been directly demonstrated to date. In this paper, we propose and analyze an experimental scheme for directly observing the non-commutativity of the position and the momentum operators using single-photon quantum in- terference. While the scheme is studied for the single-photon state as the input quantum state, the analysis applies equally to matter-wave interference, allowing a direct test of the position-momentum commutation relation with a massive particle.