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Physical Review A

Space-time properties of a boson-dressed fermion for the Yukawa model

R. E. Wagner, Matthew R. Ware, Qichang C. Su, R. Grobe

Published Sep 13, 2010• 14 citations
Publisher

Abstract

We analyze the interaction of fermions and bosons through a one-dimensional Yukawa model. We numerically compute the energy eigenstates that represent a physical fermion, which is a superposition of bare fermionic and bosonic eigenstates of the uncoupled Hamiltonian. It turns out that even fast bare fermions require only low-momentum dressing bosons, which attach themselves to the fast fermion through quantum correlations. We compare the space-time evolution of a physical fermion with that of its bare counterpart and show the importance of using dressed observables. The time evolution of the center of mass as well as the wave packet's spatial width suggests that the physical particle has a lower mass than the sum of the masses of its bare constituents. The numerically predicted dressed mass agrees with that from lowest-order perturbation theory as well as with the renormalized mass obtained from the corresponding Feynman graphs. For a given momentum, this lower mass leads to a faster physical particle and a different relativistic spreading behavior of the wave packet.

DOI
10.1103/physreva.82.032108

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We analyze the interaction of fermions and bosons through a one-dimensional Yukawa model. We numerically compute the energy eigenstates that represent a physical fermion, which is a superposition of bare fermionic and bosonic eigenstates of the uncoupled Hamiltonian. It turns out that even fast bare fermions require only low-momentum dressing bosons, which attach themselves to the fast fermion through quantum correlations. We compare the space-time evolution of a physical fermion with that of its bare counterpart and show the importance of using dressed observables. The time evolution of the center of mass as well as the wave packet's spatial width suggests that the physical particle has a lower mass than the sum of the masses of its bare constituents. The numerically predicted dressed mass agrees with that from lowest-order perturbation theory as well as with the renormalized mass obtained from the corresponding Feynman graphs. For a given momentum, this lower mass leads to a faster physical particle and a different relativistic spreading behavior of the wave packet.
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We study the scattering of a boson with a fermion with full spatial and temporal resolution based on the one-dimensional Yukawa Hamiltonian. In quantum field theory this interaction is described by the annihilation and creation of bosons with intermediate virtual particle states. We show that this process can be modeled in the center-of-mass frame by a scattering potential, permitting us to interpret the absorption and re-emission processes in quantum mechanical terms of a characteristic force. This Compton force between the fermion and boson is repulsive for large distances and attractive for shorter spacings. We also examine the periodic dynamics of a fermion and a boson that are spatially confined to a ring cavity in which they counterpropagate, enabling us to study interactions independent of the transients that characterize the (one-time) scattering event of two wave packets.
Trapped-ion, neutral-atom, and circuit QED quantum computers naturally enable coupled qubit-boson simulations. Boson-fermion interactions arise in quantum field theory, where renormalization is required to obtain finite results. We study one of the simplest examples of a coupled fermion-boson system, the static Yukawa model, on the QSCOUT trapped-ion system. We use similarity renormalization to express the same physics in terms of effective particles described by different values of the renormalization parameter sss. This enables us to map data taken in the window of bosonic occupancy that is experimentally accessible to larger occupancies of distinct effective particles. We use this to study strongly coupled, high-occupancy regimes despite limited readout ranges. Using a trapped-ion qubit-boson device on Sandia National Laboratories' QSCOUT platform, we demonstrate the time evolution of the static Yukawa model.
The idea of the systematic Weisskopf-Wigner approximation as used sporadically in atomic physics and quantum optics is extended here to the interaction of a field of non-relativistic fermions with a field of relativistic bosons. It is shown that the usual (non-existing) interaction Hamiltonian of this system can be written as a sum of a countable number of self-adjoint and bounded partial Hamiltonians. The system of these Hamiltonians defines the order hierarchy of the approximation scheme. To demonstrate its physical utility the author shows that in a certain order it provides a satisfactory quantum theory of the 'self-energy' of the fermions. This is defined as the binding energy of bosons bound to the fermions and building up the latter's 'individual Coulomb or Yukawa fields' in the sense of expectation values of the corresponding field operator. In states of more than one fermion the bound photons act as a mediating agent between the fermions; this mechanism closely resembles the Coulomb or Yukawa 'forces' used in conventional non-relativistic quantum mechanics.
I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin, or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (nonrelativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the nonrelativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.
We present a comparison between the bosonization results for quantum quenches and exact diagonalizations in microscopic models of interacting spinless fermions in a one-dimensional lattice. We show that important features are missed by the bosonization technique, which predicts the persistence of long-wavelength critical properties in the long-time evolution. Instead, numerical analysis provides puzzling evidences: while the momentum distribution appears to be consistent with the presence of a singularity at kFk_FkF​, density-density correlations at small momenta clearly display a thermal-like behavior, namely Nˉq≃const{\bar{N}}_q \simeq {\rm const}Nˉq​≃const (where the overbar indicates the long-time average). This feature at small momenta is preserved in presence of an interaction term that breaks integrability, together with a rounding of the singularities at finite qqq's, showing that the bosonization approach is not able to represent the time evolution of generic one-dimensional models after a quantum quench.