We solve the static isoperimetric problem underlying the Mandelstam-Tamm bound. Among one-dimensional confining potentials with a fixed spectral gap, we prove that the harmonic trap is the unique maximizer of the ground-state position variance. As a consequence, we obtain a sharp geometric quantum speed-limit bound on the position-position component of the quantum metric, and we give a necessary-and-sufficient condition for when the bound is saturated. Beyond the exact extremum, we establish quantitative rigidity. We control the Thomas-Reiche-Kuhn spectral tail and provide square-integrable structural stability for potentials that nearly saturate the bound. We further extend the analysis to magnetic settings, deriving a longitudinal necessary-and-sufficient characterization and transverse bounds expressed in terms of guiding-center structure. Finally, we outline applications to bounds on static polarizability, limits on the quantum metric, and benchmarking of trapping potentials.
We study the low-energy spectral properties of positive center-of-mass conserving two-body Hamiltonians as they arise in models of fractional quantum Hall states. Starting from the observation that positive many-body Hamiltonians must have ground-state energies that increase monotonously in particle number, we explore what general additional constraints can be obtained for two-body interactions with ``center-of-mass conservation'' symmetry, both in the presence and absence of particle-hole symmetry. We find general bounds that constrain the evolution of the ground-state energy with particle number, and in particular, constrain the chemical potential at
T=0. Special attention is given to Hamiltonians with zero modes, in which case similar bounds on the first excited state are also obtained, using a duality property. In this case, in particular, an upper bound on the charge gap is also obtained. We further comment on center of mass and relative decomposition in disk geometry within the framework of second quantization.
Abstract We resolve the static isoperimetric problem underlying the Mandelstam–Tamm limit: among one-dimensional confining potentials at fixed gap Δ, the harmonic trap uniquely maximizes Var 0 ( x ) , yielding the exact geometric quantum speed limit g x x ⩽ ( 2 m Δ ) − 1 with an iff criterion. Beyond the extremum we prove quantitative rigidity (Thomas–Reiche–Kuhn-tail and structural L 2 control), extend to magnetic settings (longitudinal iff; transverse guiding-center bounds), and note applications to static polarizability, quantum-metric limits, and trap benchmarking.
The center-of-mass (c.m.) oscillation of a many-body system in a harmonic trap is known to be independent of the interparticle interaction. However, this is not necessarily the case if the interactions are treated approximately. Here, we prove a simple general criterion for preservation of the c.m. mode: the approximation has to preserve density and momentum. The result equally applies to zero and finite temperatures, as well as to nonequilibrium situations, and to the linear and nonlinear response regimes.
We investigate the physical requirements to reach the ultimate limits of ground-state metrology -- i.e., the estimation of an unknown parameter
θ of an
N-body Hamiltonian via measurements on its ground state -- using many-body probes featuring either short-range or long-range interactions. We characterize the conditions to saturate two fundamental limits of the quantum Fisher information
Fθ: the static ground-state bound
Fθ≲N2/Δ2, where
Δ is the spectral gap, and the dynamical Heisenberg limit
Fθ≲N2τ2, where
τ is the total protocol duration. To saturate the static limit, we argue that short-range interacting systems require a spectral gap that closes with
N. At second-order quantum critical points this leads to a universal condition on the critical exponents,
ν(d+z)=1, which can be approached arbitrarily closely, as we show for the XXZ chain; cat-like ground states at first-order transitions provide an alternative route to static optimality. In contrast, we find that all-to-all interacting models can achieve optimality even in gapped regimes. Furthermore, reaching the dynamical bound is tightly related to the ground-state preparation time; we demonstrate that its optimal, inverse-gap profile can be obtained via local adiabatic protocols. By analyzing paradigmatic systems -- the transverse-field Ising chain, the XXZ chain, a squeezing Hamiltonian, and the two-mode Bose-Hubbard model -- we identify specific many-body probes capable of approaching both limits. Overall, our results provide a theoretical basis for the design of (local) many-body ground-state sensors operating at the ultimate limits of quantum precision.
A system of N two-dimensional weakly interacting bosons in a harmonic trap is considered. When the two-particle potential is a \ensuremath{\delta} function, Smith and Wilkin have proved analytically that the elementary symmetric polynomials of particle coordinates measured from the center of mass are exact eigenstates. In this study, we point out that their proof works equally well for an arbitrary two-particle potential that possesses translational and rotational symmetries. We find that the interaction energy associated with the eigenstate with angular momentum L is equal to $aN(N\ensuremath{-}1)/2+(b\ensuremath{-}a)NL/2,$ where a and b are the interaction energies of two bosons in the lowest-energy one-particle state with zero and one unit of angular momentum, respectively. Additionally, we study briefly the case of attractive quartic interactions. We prove rigorously that the lowest-energy state is the one in which all angular momentum is carried by the center of mass motion.