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Register a new userIdentical Wells, Symmetry Breaking, and the Near-Unitary Limit
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N.L. Harshman
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Energy level splitting from the unitary limit of contact interactions to the near unitary limit for a few identical atoms in an effectively one-dimensional well can be understood as an example of symmetry breaking. At the unitary limit in addition to particle permutation symmetry there is a larger symmetry corresponding to exchanging the $N!$ possible orderings of $N$ particles. In the near unitary limit, this larger symmetry is broken, and different shapes of traps break the symmetry to different degrees. This brief note exploits these symmetries to present a useful, geometric analogy with graph theory and build an algebraic framework for calculating energy splitting in the near unitary limit.
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Quantum adiabatic evolution, an important fundamental concept inphysics, describes the dynamical evolution arbitrarily close to the instantaneous eigenstate of a slowly driven Hamiltonian. In most systems undergoing spontaneous symmetry-breaking transitions, their two lowest eigenstates change from non-degenerate to degenerate. Therefore, due to the corresponding energy-gap vanishes, the conventional adiabatic condition becomes invalid. Here we explore the existence of quantum adiabatic evolutions in spontaneous symmetry-breaking transitions and derive a symmetry-dependent adiabatic condition. Because the driven Hamiltonian conserves the symmetry in the whole process, the transition between different instantaneous eigenstates with different symmetries is forbidden. Therefore, even if the minimum energy-gap vanishes, symmetry-protected quantum adiabatic evolutioncan still appear when the driven system varies according to the symmetry-dependent adiabatic condition. This study not only advances our understandings of quantum adiabatic evolution and spontaneous symmetry-breaking transitions, but also provides extensive applications ranging from quantum state engineering, topological Thouless pumping to quantum computing.
The attractive inverse square potential arises in a number of physical problems such as a dipole interacting with a charged wire, the Efimov effect, the Calgero-Sutherland model, near-horizon black hole physics and the optics of Maxwell fisheye lenses. Proper formulation of the inverse-square problem requires specification of a boundary condition (regulator) at the origin representing short-range physics not included in the inverse square potential and this generically breaks the Hamiltonians continuous scale invariance in an elementary example of a quantum anomaly. The systems spectrum qualitatively changes at a critical value of the inverse-square coupling, and we here point out that the transition at this critical potential strength can be regarded as an example of a $mathcal{PT}$ symmetry breaking transition. In particular, we use point particle effective field theory (PPEFT), as developed by Burgess et al [J. High Energy Phys., 2017(4):106, 2017], to characterize the renormalization group (RG) evolution of the boundary coupling under rescalings. While many studies choose boundary conditions to ensure the system is unitary, these RG methods allow us to systematically handle the richer case of nonunitary physics describing a source or sink at the origin (such as is appropriate for the charged wire or black hole applications). From this point of view the RG flow changes character at the critical inverse-square coupling, transitioning from a sub-critical regime with evolution between two real, unitary fixed points ($mathcal{PT}$ symmetric phase) to a super-critical regime with imaginary, dissipative fixed points ($mathcal{PT}$ symmetry broken phase) that represent perfect-sink and perfect-source boundary conditions, around which the flow executes limit-cycle evolution.
We review the properties of neutron matter in the low-density regime. In particular, we revise its ground state energy and the superfluid neutron pairing gap, and analyze their evolution from the weak to the strong coupling regime. The calculations of the energy and the pairing gap are performed, respectively, within the Brueckner--Hartree--Fock approach of nuclear matter and the BCS theory using the chiral nucleon-nucleon interaction of Entem and Machleidt at N$^3$LO and the Argonne V18 phenomenological potential. Results for the energy are also shown for a simple Gaussian potential with a strength and range adjusted to reproduce the $^1S_0$ neutron-neutron scattering length and effective range. Our results are compared with those of quantum Monte Carlo calculations for neutron matter and cold atoms. The Tan contact parameter in neutron matter is also calculated finding a reasonable agreement with experimental data with ultra-cold atoms only at very low densities. We find that low-density neutron matter exhibits a behavior close to that of a Fermi gas at the unitary limit, although, this limit is actually never reached. We also review the properties (energy, effective mass and quasiparticle residue) of a spin-down neutron impurity immersed in a low-density free Fermi gas of spin-up neutrons already studied by the author in a recent work where it was shown that these properties are very close to those of an attractive Fermi polaron in the unitary limit.
The controlled generation and the protection of entanglement is key to quantum simulation and quantum computation. At the single-mode level, protocols based on photonic cat states hold strong promise as they present unprecedentedly long-lived coherence and may be combined with powerful error correction schemes. Here, we demonstrate that robust ensembles of many-body photonic cat states can be generated in a Bose-Hubbard model with pair hopping via a spontaneous U(1) symmetry breaking mechanism. We identify a parameter region where the ground state is a massively degenerate manifold consisting of local cat states which are factorized throughout the lattice and whose conserved individual parities can be used to make a register of qubits. This phenomenology occurs for arbitrary system sizes or geometries, as soon as long-range order is established, and it extends to driven-dissipative conditions. In the thermodynamic limit, it is related to a Mott insulator to pair-superfluid phase transition.
A numerical study of the Faddeev equation for bosons is made with two-body interactions at or close to the Unitary limit. Separable interactions are obtained from phase-shifts defined by scattering length and effective range. In EFT-language this would correspond to NLO. Both ground and Efimov state energies are calculated. For effective ranges $r_0 > 0$ and rank-1 potentials the total energy $E_T$ is found to converge with momentum cut-off $Lambda$ for $Lambda > sim 10/r_0$ . In the Unitary limit ($1/a=r_0= 0$) the energy does however diverge. It is shown (analytically) that in this case $E_T=E_uLambda^2$. Calculations give $E_u=-0.108$ for the ground state and $E_u=-1.times10^{-4}$ for the single Efimov state found. The cut-off divergence is remedied by modifying the off-shell t-matrix by replacing the rank-1 by a rank-2 phase-shift equivalent potential. This is somewhat similar to the counterterm method suggested by Bedaque et al. This investigation is exploratory and does not refer to any specific physical system.
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