Boson Operators Research Articles

Continuous-time quantum Monte Carlo for fermion-boson lattice models: Improved bosonic estimators and application to the Holstein model

We extend the continuous-time interaction-expansion quantum Monte Carlo method with respect to measuring observables for fermion-boson lattice models. Using generating functionals, we express expectation values involving boson operators, which are not directly accessible because simulations are done in terms of a purely fermionic action, as integrals over fermionic correlation functions. We also demonstrate that certain observables can be inferred directly from the vertex distribution, and present efficient estimators for the total energy and the phonon propagator of the Holstein model. Furthermore, we generalize the covariance estimator of the fidelity susceptibility, an unbiased diagnostic for phase transitions, to the case of retarded interactions. The new estimators are applied to half-filled spinless and spinful Holstein models in one dimension. The observed renormalization of the phonon mode across the Peierls transition in the spinless model suggests a soft-mode transition in the adiabatic regime. The critical point is associated with a minimum in the phonon kinetic energy and a maximum in the fidelity susceptibility.

The linear–non-linear frontier for the Goldstone Higgs

The minimal $SO(5)/SO(4)$ sigma model is used as a template for the ultraviolet completion of scenarios in which the Higgs particle is a low-energy remnant of some high-energy dynamics, enjoying a (pseudo) Nambu-Goldstone boson ancestry. Varying the $\sigma$ mass allows to sweep from the perturbative regime to the customary non-linear implementations. The low-energy benchmark effective non-linear Lagrangian for bosons and fermions is obtained, determining as well the operator coefficients including linear corrections. At first order in the latter, three effective bosonic operators emerge which are independent of the explicit soft breaking assumed. The Higgs couplings to vector bosons and fermions turn out to be quite universal: the linear corrections are proportional to the explicit symmetry breaking parameters. Furthermore, we define an effective Yukawa operator which allows a simple parametrization and comparison of different heavy fermion ultraviolet completions. In addition, one particular fermionic completion is explored in detail, obtaining the corresponding leading low-energy fermionic operators.

A Relation of the Noncommutative Parameters in Generalized Noncommutative Phase Space**Supported by the National Natural Science Foundation of China under Grant Nos 11405060 and 11571119.

We introduce the deformed boson operators which satisfy a deformed boson algebra in some special types of generalized noncommutative phase space. Based on the deformed boson algebra, we construct coherent state representations. We calculate the variances of the coordinate operators on the coherent states and investigate the corresponding Heisenberg uncertainty relations. It is found that there are some restriction relations of the noncommutative parameters in these special types of noncommutative phase space.

Krawtchouk-Griffiths systems II: as Bernoulli systems

We call Krawtchouk-Griffiths systems, KG-systems, systems of multivariate polynomials orthogonal with respect to corresponding multinomial distributions. The original Krawtchouk polynomials are orthogonal with respect to a binomial distribution. Here we present a Fock space construction with raising and lowering operators. The operators of multiplication by X are found in terms of boson operators and corresponding recurrence relations presented. The Riccati partial differential equations for the differentiation operators, Berezin transform and associated partial differential equations are found. These features provide the specifications for a Bernoulli system as a quantization formulation of multivariate Krawtchouk polynomials.

New Class of Quantum Error-Correcting Codes for a Bosonic Mode

We construct a new class of quantum error-correcting codes for a bosonic mode which are advantageous for applications in quantum memories, communication, and scalable computation. These 'binomial quantum codes' are formed from a finite superposition of Fock states weighted with binomial coefficients. The binomial codes can exactly correct errors that are polynomial up to a specific degree in bosonic creation and annihilation operators, including amplitude damping and displacement noise as well as boson addition and dephasing errors. For realistic continuous-time dissipative evolution, the codes can perform approximate quantum error correction to any given order in the timestep between error detection measurements. We present an explicit approximate quantum error recovery operation based on projective measurements and unitary operations. The binomial codes are tailored for detecting boson loss and gain errors by means of measurements of the generalized number parity. We discuss optimization of the binomial codes and demonstrate that by relaxing the parity structure, codes with even lower unrecoverable error rates can be achieved. The binomial codes are related to existing two-mode bosonic codes but offer the advantage of requiring only a single bosonic mode to correct amplitude damping as well as the ability to correct other errors. Our codes are similar in spirit to 'cat codes' based on superpositions of the coherent states, but offer several advantages such as smaller mean number, exact rather than approximate orthonormality of the code words, and an explicit unitary operation for repumping energy into the bosonic mode. The binomial quantum codes are realizable with current superconducting circuit technology and they should prove useful in other quantum technologies, including bosonic quantum memories, photonic quantum communication, and optical-to-microwave up- and down-conversion.

Quantum states of dark solitons in the 1D Bose gas

We present a series of quantum states that are characterized by dark solitons of the nonlinear Schrödinger equation (i.e. the Gross–Pitaevskii equation) for the one-dimensional Bose gas interacting through the repulsive delta-function potentials. The classical solutions satisfy the periodic boundary conditions and we simply call them classical dark solitons. Through exact solutions we show corresponding aspects between the states and the solitons in the weak coupling case: the quantum and classical density profiles completely overlap with each other not only at an initial time but also at later times over a long period of time, and they move together with the same speed in time; the matrix element of the bosonic field operator between the quantum states has exactly the same profiles of the square amplitude and the phase as the classical complex scalar field of a classical dark soliton not only at the initial time but also at later times, and the corresponding profiles move together for a long period of time. We suggest that the corresponding properties hold rigorously in the weak coupling limit. Furthermore, we argue that the lifetime of the dark soliton-like density profile in the quantum state becomes infinitely long as the coupling constant approaches zero, by comparing it with the quantum speed limit time. Thus, we call the quantum states quantum dark soliton states.

Diagonalizing quadratic bosonic operators by non-autonomous flow equation

We study a non-autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. We specify assumptions that ensure the global existence of its solutions and allow us to derive its asymptotics at temporal infinity. We demonstrate that these assumptions are optimal in a suitable sense and more general than those used before. The evolution equation derives from the Brocket-Wegner flow that was proposed to diagonalize matrices and operators by a strongly continuous unitary flow. In fact, the solution of the non-linear flow equation leads to a diagonalization of Hamiltonian operators in boson quantum field theory which are quadratic in the field.

Time-Dependent Correlation Functions for the Bimodal Bose–Hubbard Model

The bimodal Bose–Hubbard model is studied. The application of the Quantum Inverse Method allows us to calculate the time-dependent correlation functions of the model. Form-factors of the bosonic creation and annihilation operators in the wells are expressed in the determinantal form.

Nonclassical effects in the second harmonic generation

The higher-order nonclassical squeezing and quantum entanglement effects emerging from the second harmonic generation of the associated two-mode and two-photon Hamiltonian are investigated in the dispersive limit. The squeezed states of the field, including the normal and amplitude squared (higher-order) squeezing factors are generated in two ways, i.e., from the bosonic operators via amplitude powered quadrature variables, and through the SU(2) characterization of a passive and lossless device with two input and two output ports, which then allows one to visualize the operations of beam splitters and phase shifters as rotations of angular momentum operators in 3-space. Two criteria for intermodal higher-order quantum entanglement and different coherent states for the two modes in the initial state are used to compute these nonclassical effects. The unitary time evolution of the linear entropy, computed from the partial trace of the density matrix over the secondary mode, is also used as a criterion of quantum entanglement. These approaches show, in fact, that the present model exhibits a considerable amount of this nonclassical effect. The unitary time evolution of the linear entropy shows that the present nonlinear optical model does not preserve the modulus of the Bloch vector.

The New Multipartite Squeezing Operator and Some of its Properties

In this paper, we introduce a pair of mutually conjugate multipartite entangled state representations for defining the squeezing operator of entangled multipartite Sn(λ) which involves an n-mode bosonic operator realization of the SU(1,1) Lie algebra. This operator squeezes the multipartite entangled state in a natural way. We discuss the transform properties of aj and \(a_{j}^{\dagger }\) under the operation of Sn(λ) and derive the interaction Hamiltonian which can generate such an evolution. In addition, the corresponding multipartite squeezed vacuum state |λ〉 is obtained. Based on this, the variances of the n-mode quadratures in |λ〉 are evaluated and the violation of the Bell inequality for |λ〉 is examined by using the formalism of Wigner representation.

Quantum filtering for multiple diffusive and Poissonian measurements

We provide a rigorous derivation of a quantum filter for the case of multiple measurements being made on a quantum system. We consider a class of measurement processes which are functions of bosonic field operators, including combinations of diffusive and Poissonian processes. This covers the standard cases from quantum optics, where homodyne detection may be described as a diffusive process and photon counting may be described as a Poissonian process. We obtain a necessary and sufficient condition for any pair of such measurements taken at different output channels to satisfy a commutation relationship. Then, we derive a general, multiple-measurement quantum filter as an extension of a single-measurement quantum filter. As an application we explicitly obtain the quantum filter corresponding to homodyne detection and photon counting at the output ports of a beam splitter.

Matrix product formula for Macdonald polynomials

We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik–Zamolodchikov equations, which arise by considering representations of the Zamolodchikov–Faddeev and Yang–Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalized probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalizations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.

Dissipative dynamics of quantum fluctuations

One way to look for complex behaviours in many‐body quantum systems is to let the number N of degrees of freedom become large and focus upon collective observables. Mean‐field quantities scaling as tend to commute, whence complexity at the quantum level can only be inherited from complexity at the classical level. Instead, fluctuations of microscopic observables scale as and exhibit collective Bosonic features, typical of a mesoscopic regime half‐way between the quantum one at the microscopic level and the classical one at the level of macroscopic averages. Here, we consider the mesoscopic behaviour emerging from an infinite quantum spin chain undergoing a microscopic dissipative, irreversible dynamics and from global states without long‐range correlations and invariant under lattice translations and dynamics. We show that, from the fluctuations of one site spin observables whose linear span is mapped into itself by the dynamics, there emerge bosonic operators obeying a mesoscopic dissipative dynamics mapping Gaussian states into Gaussian states. Instead of just depleting quantum correlations because of decoherence effects, these maps can generate entanglement at the collective, mesoscopic level, a phenomenon with no classical analogue that embodies a peculiar complex behaviour at the interface between micro and macro regimes.

Hiking a generalized Dyck path: A tractable way of calculating multimode boson evolution operators

A time evolution operator in the interaction picture is given by exponentiating an interaction Hamiltonian H. Important examples of Hamiltonians, often encountered in quantum optics, condensed matter and high energy physics, are of a general form H=r(A†−A), where A is a multimode boson operator and r is the coupling constant. If no simple factorization formula for the evolution operator exists, the calculation of the evolution operator is a notoriously difficult problem. In this case the only available option may be to Taylor expand the operator in r and act on a state of interest ψ. But this brute-force method quickly hits the complexity barrier since the number of evaluated expressions increases exponentially. We relate a combinatorial structure called Dyck paths to the action of a boson word (monomial) and a large class of monomial sums on a quantum state ψ. This allows us to cross the exponential gap and make the problem of a boson unitary operator evaluation computationally tractable by achieving polynomial-time complexity for an extensive family of physically interesting multimode Hamiltonians. We further test our method on a cubic boson Hamiltonian whose Taylor series is known to diverge for all nonzero values of the coupling constant and an analytic continuation via a Padé approximant must be performed.

Observables and density matrices embedded in dual Hilbert spaces

The introduction of operator states and of observables in various fields of quantum physics has raised questions about the mathematical structures of the corresponding spaces. In the framework of third quantization it had been conjectured that we deal with Hilbert spaces although the mathematical background was not entirely clear, particularly, when dealing with bosonic operators. This in turn caused some doubts about the correct way to combine bosonic and fermionic operators or, in other words, regular and Grassmann variables. In this paper we present a formal answer to the problems on a simple and very general basis. We illustrate the resulting construction by revisiting the Bargmann transform and finding the known connection between and the Bargmann–Hilbert space. We pursue this line of thinking one step further and discuss the representations of complex extensions of linear canonical transformations as isometries between dual Hilbert spaces. We then use the formalism to give an explicit formulation for Fock spaces involving both fermions and bosons thus solving the problem at the origin of our considerations.

Beyond the Schwinger boson representation of the su(2)-algebra

With the use of two kinds of boson operators, a new boson representation of the su(2)-algebra is proposed. The basic idea comes from the pseudo su(1,1)-algebra recently given by the present authors. It forms a striking contrast to the Schwinger boson representation of the su(2)-algebra which is also based on two kinds of bosons. This representation may be suitable for describing time-dependence of the system interacting with the external environment in the framework of the thermo field dynamics formalism, i.e., the phase space doubling. Further, several deformations related to the su(2)-algebra in this boson representation are discussed. On the basis of these deformed algebra, various types of time-evolution of a simple boson system are investigated.

On the renormalization of the electroweak chiral Lagrangian with a Higgs

We consider the scalar sector of the effective non-linear electroweak Lagrangian with a light "Higgs" particle, up to four derivatives in the chiral expansion. The complete off-shell renormalization procedure is implemented, including one loop corrections stemming from the leading two-derivative terms, for finite Higgs mass. This determines the complete set of independent chiral invariant scalar counterterms required for consistency; these include bosonic operators often disregarded. Furthermore, new counterterms involving the Higgs particle which are apparently chiral non-invariant are identified in the perturbative analysis. A novel general parametrization of the pseudoescalar field redefinitions is proposed, which reduces to the various usual ones for specific values of its parameter; the non-local field redefinitions reabsorbing all chiral non-invariant counterterms are then explicitly determined. The physical results translate into renormalization group equations which may be useful when comparing future Higgs data at different energies.

Parametric Amplifier Studied as Time Evolution of Angular Momentum System in Entangled State Representation

We present a new approach for studying theoretical model of a parametric amplifier. We show that time-evolution of parametric amplifier can be treated as formalism of angular momentum system by using the newly found bosonic operator realization of three generators (J +, J −, J z ). The entangled state representation is essential in this approach. The transition probability from vacuum state to two-mode squeezed state in a parametric amplifier can be directly evaluated.

General three-point functions in 4D CFT

We classify and compute, by means of the six-dimensional embedding formalism in twistor space, all possible three-point functions in four dimensional conformal field theories involving bosonic or fermionic operators in irreducible representations of the Lorentz group. We show how to impose in this formalism constraints due to conservation of bosonic or fermionic currents. The number of independent tensor structures appearing in any three-point function is obtained by a simple counting. Using the Operator Product Expansion (OPE), we can then determine the number of structures appearing in 4-point functions with arbitrary operators. This procedure is independent of the way we take the OPE between pairs of operators, namely it is consistent with crossing symmetry, as it should be. An analytic formula for the number of tensor structures for three-point correlators with two symmetric and an arbitrary bosonic (non-conserved) operators is found, which in turn allows to analytically determine the number of structures in 4-point functions of symmetric traceless tensors.

New fractional entangling transform and its quantum mechanical correspondence

In this Letter, a new fractional entangling transformation (FrET) is proposed, which is generated in the entangled state representation by a unitary operator exp{iθ(ab†+a†b)} where a(b) is the Bosonic annihilate operator. The operator is actually an entangled one in quantum optics and differs evidently from the separable operator, exp{iθ(a†a+b†b)}, of complex fractional Fourier transformation. The additivity property is proved by employing the entangled state representation and quantum mechanical version of the FrET. As an application, the FrET of a two-mode number state is derived directly by using the quantum version of the FrET, which is related to Hermite polynomials.