Bosonic Field Operators Research Articles

Fermionic Fock Spaces and Quantum States for Causal Fermion Systems

It is shown for causal fermion systems describing Minkowski-type spacetimes that an interacting causal fermion system at time t gives rise to a distinguished state on the algebra generated by fermionic and bosonic field operators. The proof of positivity of the state is given, and representations are constructed.

An Engineer's Brief Introduction to Microwave Quantum Optics and a Single-Port State-Space Representation

Classical microwave circuit theory is incapable of representing some phenomena at the quantum level. To include quantum statistical effects, various theoretical treatments can be employed. Quantum input-output network (QION) theory is one such treatment. Another formalism, called SLH theory, incorporates scattering matrices ( S ), coupling vectors ( L ), and system Hamiltonians ( H ). These theoretical treatments require a reformulation of classical microwave theory. To make these topics comprehensible to an electrical engineer, we demonstrate some underpinnings of microwave quantum optics in terms of microwave engineering. For instance, we equate traveling-wave phasors in a transmission line ( $V_0^+$ ) directly to bosonic field operators. Furthermore, we extend QION to include a state-space representation and a transfer function for a single port quantum network. This serves as a case study to highlight how microwave methodologies can be applied in open quantum systems. Although the same conclusion could be found from a full SLH theory treatment, our method was derived directly from first principles of QION.

Quantum Information Hidden in Quantum Fields

We investigate a possible reduction mechanism from (bosonic) Quantum Field Theory (QFT) to Quantum Mechanics (QM), in a manner that could explain the apparent loss of degrees of freedom of the original theory in terms of quantum information in the reduced one. This reduction mechanism consists mainly of performing an ansatz on the boson field operator, which takes into account quantum foam and non-commutative geometry. Through the reduction mechanism, QFT reveals its hidden internal structure, which is a quantum network of maximally entangled multipartite states. In the end, a new approach to the quantum simulation of QFT is proposed through the use of QFT’s internal quantum network. Finally, the entropic equilibrium of fully mixed and maximally entangled states in the quantum network seems to suggest that the black hole paradox of information loss might be solved under suitable conditions.

Higher‐Order Nonclassicality in Photon Added and Subtracted Qudit States

AbstractHigher‐order nonclassical properties of r photon added and t photon subtracted qudit states (referred to as rPAQS and tPSQS, respectively) are investigated here to answer: How addition and subtraction of photon can be used to engineer higher‐order nonclassical properties of qudit states? To obtain the answer, higher‐order moment of relevant bosonic field operators is first obtained and subsequently used to study the higher‐order nonclassical properties (e.g., higher‐order antibunching, higher‐order squeezing, and higher‐order sub‐Poissonian photon statistics) of the corresponding states. These witnesses establish that rPAQS and tPSQS are highly nonclassical. To quantitatively establish this observation and to make a comparison between rPAQS and tPSQS, volumes of the negative part of Wigner function are computed. Finally, for the sake of verifiability of the obtained results, optical tomograms are also reported. Throughout the study, a particular type of qudit state named as a new generalized binomial state is used as an example.

Dc to ac Josephson transition in a dc atom superconducting quantum interference device

We analyze the effect of the barrier motion on the Bose-Hubbard Hamiltonian of a ring-shaped Bose-Einstein condensate interrupted by a pair of Josephson junctions, a configuration which is the cold atom analog of the well-known dc superconducting quantum interference device (SQUID). Such an effect is also shown to modify the Heisenberg equation of motion of the boson field operator in the two-mode approximation, where a hysteretic contribution that could affect the dynamics for accelerated or overlapping barriers is identified. By studying the energy landscape as a function of order and control parameters, we determine the diagram with the location of the dc and ac Josephson regimes, along with the critical points that are shown to depend on the junctions position. We analyze the dc to ac Josephson transition for adiabatic barrier trajectories that lead to a final uniform velocity, or which perform symmetric velocity paths. We show that such symmetric trajectories may induce, when reaching the critical point, highly hysteretic oscillating return paths within the dc regime, similar to the underdamped hysteresis loops arising from the action of a resistive flow in the ac regime. We also consider nonequilibrium initial conditions resulting from a finite phase difference on either side of the junctions, along with the critical features of such a parameter. An excellent agreement between the Gross-Pitaevskii simulations and the two-mode results is found in all cases.

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.

Coherent transfer by adiabatic passage in two-dimensional lattices

Coherent tunneling by adiabatic passage (CTAP) is a well-established technique for robust spatial transport of quantum particles in linear chains. Here we introduce two exactly-solvable models where the CTAP protocol can be extended to two-dimensional lattice geometries. Such bi-dimensional lattice models are synthesized from time-dependent second-quantization Hamiltonians, in which the bosonic field operators evolve adiabatically like in an ordinary three-level CTAP scheme thus ensuring adiabatic passage in Fock space.

Cavity quantum electrodynamics in the ultrastrong coupling regime

We revisit the mathematical formulation of the famous Jaynes–Cummings–Paul Hamiltonian, which describes the interaction of a two-level atom with a single mode of an electromagnetic cavity reservoir. We rigorously show that under the condition of ultrastrong coupling between the atom and cavity in which the transition frequency is comparable to the coupling frequency, the bosonic field operators undergo non-sinusoidal time variations. As a result, the well-known solutions to the Jaynes–Cummings–Paul model are no longer valid, even when the rotating wave approximation is not used. We show how a correct mathematical solution could be found instead.

Causal signal transmission by quantum fields. III: Coherent response of fermions

Structural response properties of fermionic fields are investigated. In the presence of fermions the key technical concept becomes response combination, or R-normal product, of field operators. It generalises the notion of time-normal operator product to response problems. Time-normal products are a special case of R-normal products without inputs; this paper thus also generalises the concept of time-normal ordering to fermions. Explicit causality of R-normal products of arbitrary (bosonic and/or fermionic) field operators is proven, and explicit relations expressing them by conventional Green’s functions of quantum field theory are derived.

Particle creation by a moving boundary with a Robin boundary condition

We consider a massless scalar field in 1+1 dimensions satisfying a Robin boundary condition (BC) at a non-relativistic moving boundary. We derive a Bogoliubov transformation between input and output bosonic field operators, which allows us to calculate the spectral distribution of created particles. The cases of Dirichlet and Neumann BC may be obtained from our result as limiting cases. These two limits yield the same spectrum, which turns out to be an upper bound for the spectra derived for Robin BC. We show that the particle emission effect can be considerably reduced (with respect to the Dirichlet/Neumann case) by selecting a particular value for the oscillation frequency of the boundary position.

Quantum electrodynamics of moving media

The macroscopic electromagnetic field is quantized in the presence of a uniformly moving permeable dielectric medium. We begin with the Maxwell-Minkowski equations, and the fields are quantized with the help of the Green function scheme. It is shown that for a homogeneous material, though the medium is assumed to be isotropic in its rest frame, it displays anisotropic character for the observer who detects the field observables in the laboratory frame. The field operators are expressed in terms of two independent infinite sets of bosonic field operators. Special attention is paid to the vacuum field fluctuation in moving media.

Mean field loops versus quantum anti-crossing nets in trapped Bose-Einstein condensates

We study a Bose-Einstein condensate trapped in an asymmetric double well potential. Solutions of the time-independent Gross-Pitaevskii equation reveal intrinsic loops in the energy (or chemical potential) level behavior when the shape of the potential is varied. We investigate the corresponding behavior of the quantum (many-body) energy levels. Applying the two-mode approximation to the bosonic field operators, we show that the quantum energy levels create an anti-crossing net inside the region bounded by the loop of the mean field solution.

Cumulant expansion for studying damped quantum solitons

The quantum statistics of damped optical solitons is studied using cumulant-expansion techniques. The effect of absorption is described in terms of ordinary Markovian relaxation theory, by coupling the optical field to a continuum of reservoir modes. After introduction of local bosonic field operators and spatial discretization, pseudo-Fokker-Planck equations for multidimensional s-parametrized phase-space functions are derived. These partial differential equations are equivalent to an infinite set of ordinary differential equations for the cumulants of the phase-space functions. Introducing an appropriate truncation condition, the resulting finite set of cumulant evolution equations can be solved numerically. Solutions are presented in a Gaussian approximation and the quantum noise is calculated, with special emphasis on squeezing and the recently measured spectral photon-number correlations [Sp\"alter et al., Phys. Rev. Lett. 81, 786 (1998)].

Quantum cohomology and free-field representation

In our previous article we have proposed that the Virasoro algebra controls the quantum cohomology of Fano varieties at all genera. In this paper we construct a free-field description of Virasoro operators and quantum cohomology. We shall show that to each even (odd) homology class of a Kähler manifold we have a free bosonic (fermionic) field and Visasoro operators are given by a simple bilinear form of these fields. We shall show that the Virasoro condition correctly reproduces the Gromov-Witten invariants also in the case of manifolds with non-vanishing non-analytic classes ( h p, q ≠ 0, p ≠ q) and suggest that the Virasoro condition holds universally for all compact smooth Kähler manifolds.

A FIELD THEORY FOR THE READ OPERATOR

We introduce a new field theory for studying quantum Hall systems. The quantum field is a modified version of the bosonic operator introduced by Read. In contrast to Read's original work we do not work in the lowest Landau level alone, and this leads to a much simpler formalism. We identify an appropriate canonical conjugate field, and write a Hamiltonian that governs the exact dynamics of our bosonic field operators. We describe a Lagrangian formalism, derive the equations of motion for the fields and present a family of mean-field solutions. Finally, we show that these mean field solutions are precisely the Laughlin states. We do not, in this work, address the treatment of fluctuations.

Global cocycle dynamics for infinite mean field quantum systems interacting with the boson gas

In the framework of operator algebraic quantum statistical mechanics, the global nonequilibrium dynamics for a general class of interactions of infinite mean field quantum lattice systems with the boson field is investigated. The associated interaction operators consist of arbitrary powers of the collective density operators of the mean field system and are linear with respect to the bosonic field operators. Instead of the usual perturbation expansions, here the interacting dynamics are studied by means of cocycle techniques. The cocycle methods are more appropriate to the considered class of interactions, and lead to explicit closed expressions for the dynamical automorphism groups. The cocycle equations connect the classical, collective dynamical behavior of the mean field system with the one-boson dynamics. In physical applications such systems are due to collectively ordered finite-level atoms or the Josephson junction in the thermodynamic limit weakly interacting with the electromagnetic field.

Nonlinear quantum fields in ≥4 dimensions and cohomology of the infinite Heisenberg group

Aspects of the cohomology of the infinite-dimensional Heisenberg group as represented on the free boson field over a given Hilbert space are treated. The 1-cohomology is shown to be trivial in certain spaces of generalized vectors. From this derives a canonical quantization mapping from classical (unquantized) forms to generalized operators on the boson field. An example, applied here to scalar relativistic fields, is the quantization of a given classical interaction Lagrangian or Hamiltonian, i.e., the establishment and characterization of corresponding boson field operators. For example, if ϕ \phi denotes the free massless scalar field in d-dimensional Minkowski space ( d ≥ 4 d \geq 4 , even) and if q is an even integer greater than or equal to 4, then ∫ M 0 : ϕ ( X ) q : d X {\smallint _{{{\mathbf {M}}_0}}}:\phi {(X)^q}:dX exists as a nonvanishing, Poincaré invariant, hermitian, selfadjointly extendable operator, where : ϕ ( X ) q \phi {(X)^q} : denotes the Wick power. Applications are also made to the rigorous establishment of basic symbolic operators in heuristic quantum field theory, including certain massive field theories; to a class of pseudo-interacting fields obtained by substituting the free field into desingularized expressions for the total Hamiltonian in the conformally invariant case d = q = 4 d = q = 4 and to corresponding scattering theory.

Singular operators on boson fields as forms on spaces of entire functions on Hilbert space

Invariant scales of entire analytic functions on Hilbert space are introduced and applied. Singular operators represented by sesquilinear forms on spaces of regular vectors are given explicit integral representations via kernels that are entire functions on the direct sum of the Hilbert space with its dual. The Weyl (or, exponentiated boson field) operators act smoothly and irreducibly on corresponding spaces of entire functions. Arbitrary symplectic operators on a single-particle Hilbert space are shown to be implementable on the corresponding boson field by appropriate generalized operators.

Quantization of bosonic string field and infinite-dimensional pseudodifferential operators. Fixed gauge

Quantization of bosonic string field and infinite-dimensional pseudodifferential operators. Fixed gauge

Alternate orderings: A new tool for the study of phase and photon statistics.

A new way to expand an arbitrary boson-field operator into ordered products of creation and annihilation operators is introduced: alternate ordering. A continuous order parameter s characterizes the order and takes the value s=+1 for normal-alternate order, s=0 for symmetric-alternate order, and s=-1 for antinormal-alternate order. All of the alternate moments of a number of well-known states of light are calculated. The expansion of density matrices and operators in terms of alternate products is briefly discussed as well as the relation with the concepts of phase and higher-order phase dependence. Finally, the relevance of the new ordering scheme in the study of photon statistics is outlined and illustrated in the derivation of moment equations for phase-sensitive light amplifiers and absorbers.