Abstract This chapter presents three examples of applying of phase space methods in systems with only a few important modes. The boson application is to interferometry in two-mode Bose–Einstein condensates, as in a double-well potential. The fermion application is to Cooper pairing in a four-mode system involving two spin-1/2 fermions, initially occupying states with opposite spins and momenta. The one-atom two-level Jaynes–Cummings model provides an example combining a one-mode bosonic system (the cavity mode) with a two-mode fermionic system (the two atomic states). Fokker–Planck equations are derived in each case. In the first two cases, Ito stochastic differential equations are determined. These describe dephasing effects in the boson interferometry case, and are solved to demonstrate Cooper pairing in the fermion case. For the combined case, the Fokker–Planck equations are solved directly—the distribution function shown to give the well-known Jaynes–Cummings model results.
Abstract This chapter presents three examples of applying of phase space methods in systems with only a few important modes. The boson application is to interferometry in two-mode Bose–Einstein condensates, as in a double-well potential. The fermion application is to Cooper pairing in a four-mode system involving two spin-1/2 fermions, initially occupying states with opposite spins and momenta. The one-atom two-level Jaynes–Cummings model provides an example combining a one-mode bosonic system (the cavity mode) with a two-mode fermionic system (the two atomic states). Fokker–Planck equations are derived in each case. In the first two cases, Ito stochastic differential equations are determined. These describe dephasing effects in the boson interferometry case, and are solved to demonstrate Cooper pairing in the fermion case. For the combined case, the Fokker–Planck equations are solved directly—the distribution function shown to give the well-known Jaynes–Cummings model results.
Abstract This chapter presents two examples of applying phase space methods in systems where large numbers of modes may be involved, such as in fermion cases with large particle numbers. Functional Fokker–Planck equations (FFPEs) are derived directly using the field operator correspondence rules. The boson application is to colliding bosons in a trap potential, such as for Bose–Einstein condensates. FFPEs and Ito stochastic field equations (SFEs) are derived for both positive P and Wigner distribution functional cases, and expressions obtained for quantum correlation functions. The fermion application is to colliding spin-1/2 fermions in a trap potential, such as an optical lattice. FFPEs and Ito SFEs are derived for the unnormalised B distribution functional. Solutions to the Ito SFE are presented for the untrapped and optical-lattice cases. For the untrapped case with two fermions, evaluation of short-time Fock state coherences and populations gives the same result as obtained in a previous chapter.
Phase space distribution functional methods utilising c-number and Grassmann fields for bosons and fermions respectively are being developed for theoretical treatments of degenerate quantum gases with large fermionic and bosonic atom numbers. The Jaynes-Cummings model where a single mode bosonic system and a two mode fermionic system are coupled provides a simple testbed for phase space methods involving Grassmann variables. The Fokker-Planck equation for the distribution function is shown to be equivalent to six coupled equations for c-number bosonic functions. Numerical solutions enable well-known effects such as collapses/revivals of the atomic population difference to be demonstrated.
This topical review provides an overview of the key theoretical features of Bose–Einstein condensates (BECs) in cold atomic gases at near zero temperature in the situation where all the bosons occupy at most two single particle states or modes. This situation applies to single-component BECs in double well trap potentials and to two-component BEC in single well trap potentials, such as occur when BEC are used in interferometry experiments. The Hamiltonian is introduced in terms of field operators and mode expansions are restricted to a total of two modes. Spin operators and their eigenstates are introduced as the fundamental basis states for describing the two-mode N boson quantum system. The spin states have a macroscopic angular momentum quantum number of N/2 and the magnetic quantum number k specifies the relative number of bosons in the two modes. The treatment presented involves an extensive use of angular momentum theory, including unitary rotation operators. Important states of the two-mode system such as binomial or coherent states, relative phase eigenstates are discussed. Boson position measurements are specified via quantum correlation functions, and the use of these functions in describing coherence properties, interference patterns and fragmentation effects in BECs is presented. The Bloch vector is defined and related to the quantum correlation functions, with quantum fluctuations of the Bloch vector being treated in terms of the covariance matrix. Applications to important two-mode states are made. Spin squeezing is discussed. Based on applying variational principles, the general dynamical behaviour of the two-mode BEC is determined via generalised Gross–Pitaevskii equations for the modes and matrix mechanics equations for the probability amplitudes of the relative number basis states, the mode and amplitude equations being coupled and self-consistent. The single mode equations are also presented. The Hamiltonian is written in terms of the spin operators and the Josephson Hamiltonian obtained as a simplification in which the dynamical behaviour of the mode functions is ignored – for the one-component case the mode functions are also required to be localised and separate. Coefficients in the Josephson Hamiltonian describe tunneling/intercomponent coupling, asymmetry and collisions and these are defined via integrals involving the mode functions. The Josephson model involves using the Josephson Hamiltonian to give simple predictions of the energy states and dynamical behaviour of the two-mode system, dynamical effects on the mode functions being ignored. The three regimes – Rabi, Josephson and Fock are described, and the energy states obtained for the Fock and Rabi regimes. Dynamical behaviour treatments based on the Josephson model are outlined. In the situation where all bosons are in the same single particle state, semi-classical Bloch equations are derived and their solutions given in terms of elliptic functions. The quantum regime is treated using matrix mechanics equations for the probability amplitudes. Two representative applications of the Josephson model dynamics are treated, with graphs showing the results of numerical work being displayed. The first is in describing Heisenberg limited BEC interferometry for a single-component BEC in a double well, the treatment showing collapses and revivals in the probability distribution for the relative phase. The second treats Ramsey interferometry for a two-component BEC in a single well, the study revealing that oscillations of the Bloch vector collapse and revive, with the Bloch vector's departure from the Bloch sphere during the collapse period revealing that the BEC has fragmented. In both cases collisions cause the dephasing effects that result in the collapse, revival phenomena. The review ends with a brief outline of phase space and other approaches that extend the treatment beyond the two-mode theory, enabling decoherence effects associated with bosons in non-condensate modes to be studied. A summary of the review contents is included. Detailed mathematical derivations are included in several appendices, available as online supplementary material.
Abstract This chapter provides an overview of the phase space method for treating the physics of systems of bosons and fermions in regimes where quantum degeneracy occurs. Key concepts such as modes, creation and annihilation operators, distribution functions in phase space depending on c-number (bosons) or Grassmann (fermions) phase space variables, quantum correlation functions, and phase space integrals are introduced, together with how the dynamics of the system density operator as described via Liouville–von Neumann or master equations is translated into Fokker–Planck equations for the distribution function, and then how phase space integrals for quantum correlation functions can be determined via Langevin stochastic equations that are equivalent to the Fokker–Planck equations, and which involve stochastic phase space variables. The extension of the phase space approach to systems with large mode numbers, involving field annihilation and creation operators, distribution functionals, and functional Fokker–Planck and Langevin stochastic field equations is also reviewed.
Abstract Recent experimental progress has enabled cold atomic gases to be studied at nanokelvin temperatures, creating new states of matter where quantum degeneracy occurs—Bose–Einstein condensates and degenerate Fermi gases. Such quantum states are of macroscopic dimensions. As its title suggests, this book presents the phase space theory approach to treating the physics of degenerate quantum gases, an approach already widely used in quantum optics. However, degenerate quantum gases involve massive bosonic and fermionic atoms, not massless photons. The book begins with a review of Fock states for systems of identical atoms, where large numbers of atoms occupy the various single-particle states or modes. First, separate modes are considered, and here the quantum density operator is represented by a phase space distribution function of phase space variables which replace mode annihilation and creation operators, the dynamical equation for the density operator determines a Fokker–Planck equation for the distribution function, and measurable quantities such as quantum correlation functions are given as phase space integrals. Finally, the phase space variables are replaced by time-dependent stochastic variables satisfying Langevin stochastic equations obtained from the Fokker–Planck equation, with stochastic averages giving the measurable quantities. Second, a quantum field approach is then treated, the density operator being represented by a distribution functional of field functions which replace field annihilation and creation operators, the distribution functional satisfying a functional Fokker–Planck equation, etc. A novel feature of this book is that the phase space variables for fermions are Grassmann variables, not c-numbers. However, the book shows that Grassmann distribution functions and functionals still provide equations for obtaining both analytic and numerical solutions.
A full treatment of decoherence and dephasing effects in BEC interferometry has been developed based on using quantum correlation functions for treating interferometric effects. The BEC is described via a phase space distribution functional of the Wigner type for the condensate modes and the positive P type for the non-condensate modes. Ito equations for stochastic condensate and non-condensate field functions replace the functional Fokker-Planck equation for the distribution functional and stochastic averages of field function products determine the quantum correlation functions.