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Journal of Low Temperature Physics

Zero-Point Energy of a Trapped Ultracold Fermi Gas at Unitarity: Squeezing the Heisenberg Uncertainty Principle and Suppressing the Pauli Principle to Produce a Superfluid State

D. K. Watson

Published Aug 6, 2026• 0 citations
Publisher
DOI
10.1007/s10909-026-03456-x

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Zero-point energy of a trapped ultracold Fermi gas at unitarity: squeezing the Heisenberg uncertainty principle and suppressing the Pauli principle to produce a superfluid state

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The zero-point energy of a trapped ultracold Fermi gas at unitarity is investigated in relation to the combined effects of the Heisenberg uncertainty principle and the Pauli principle. This lowest allowed quantum state is a superfluid state which has been studied extensively both experimentally and theoretically. The method used for the current investigation is based on a recent series of papers that proposed microscopic dynamics based on normal modes to describe superfluidity instead of real-space Cooper pairs. This approach yielded excellent agreement with experimental data for multiple properties and allowed the microscopic behavior underlying these results as well as the basis of universal behavior to be analyzed in detail using the group theoretic basis of this general N-body approach. This microscopic picture is now used to illucidate the roles played by the uncertainty principle and the Pauli principle in determining the energy and character of the lowest allowed quantum state including the squeezed character of this superfluid state and the suppression of the Pauli principle.

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Abstract Understanding the universal behavior of strongly-interacting systems of particles has been a major goal in multiple fields of physics from atomic physics to cosmology. Using a recently introduced manybody perturbation formalism for fermions which uses group theoretic and graphical techniques to solve the N -body problem, I study the thermodynamic behavior of the unitary regime for a trapped Fermi gas and investigate the role of the Pauli principle in the emergence of collective behavior, specifically superfluidity at ultralow temperatures. The method, which is called symmetry invariant perturbation theory, has no adjustable parameters and currently offers a solution to the manybody Schrodinger equation through first order in inverse dimensionality exactly. The solution at first order defines collective coordinates in terms of five types of N -body normal modes, identified as symmetric stretch, symmetric bend, single particle angular excitation, single particle radial excitation and phonon. A correspondence is established ‘on paper’ that enforces the Pauli principle through the assignment of specific normal mode quantum numbers. Applied at ultracold temperatures, this normal mode assignment yields occupation only in an extremely low frequency N -body phonon mode. A single particle radial excitation mode at a much higher frequency creates a gap that stabilizes the superfluidity at low temperatures. This radial excitation involves excitation of a single particle out of the synced motion of the phonon mode, rather than the breaking of a fermion pair. Coupled with the corresponding values for the energies at unitarity obtained by this manybody calculation, I obtain good agreement with experimental thermodynamic results. In particular, the lambda transition in the specific heat is clearly seen. Based on these findings, I propose a mechanism for the emergence of superfluidity at ultralow temperatures in this unitary regime that is driven by quantum statistics and the enforcement of the Pauli principle through the selection of normal modes. My results for the calculated thermodynamic quantities suggest that the emergence of some collective behaviors in macroscopic systems is due to the Pauli principle producing manybody pairing and the resulting macroscopic occupation of an N -body phonon mode.

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Non-Equilibrium Enhancement of Superfluidity in a Trapped Fermi Gas

Andrew J. Robertson, Victor M. Galitski

Bulletin of the American Physical Society•Mar 19, 2009•0 citations
The zero-point energy of a trapped ultracold Fermi gas at unitarity is investigated in relation to the combined effects of the Heisenberg uncertainty principle and the Pauli principle. This lowest allowed quantum state is a superfluid state which has been studied extensively both experimentally and theoretically. The method used for the current investigation is based on a recent series of papers that proposed microscopic dynamics based on normal modes to describe superfluidity instead of real-space Cooper pairs. This approach yielded excellent agreement with experimental data for multiple properties and allowed the microscopic behavior underlying these results as well as the basis of universal behavior to be analyzed in detail using the group theoretic basis of this general N-body approach. This microscopic picture is now used to illucidate the roles played by the uncertainty principle and the Pauli principle in determining the energy and character of the lowest allowed quantum state including the squeezed character of this superfluid state and the suppression of the Pauli principle.
The transition from the weakly interacting BCS regime to the strongly interacting unitary regime is explored for ultracold trapped Fermi gases assuming a normal mode description of the gas instead of the conventional Cooper pairing. The Pauli principle is applied ``on paper'' by using specific normal mode assignments. Energies, entropies, critical temperatures, and an excitation frequency are studied and compared to existing results in the literature. These normal modes have been derived analytically for N identical, confined particles from a first-order L=0 group theoretic solution of a three-dimensional Hamiltonian with a general two-body interaction. In previous studies, normal modes were able to describe the unitary regime obtaining ground state energies comparable to benchmark results and thermodynamics quantities in excellent agreement with experiment. In a recent study, the behavior of the normal mode frequencies was investigated for Hamiltonians with a range of interparticle interaction strengths from BCS to unitarity in the first test of this approach beyond the unitary regime, and a microscopic basis of the large excitation gaps and universal behavior at unitarity was proposed. Based on the success of these earlier studies, the current paper continues to explore the ability of normal modes to describe superfluidity along the BCS to unitarity transition. The results confirm earlier conclusions that the physics of superfluidity can be described using normal modes across a wide range of interparticle interaction strengths and offer an alternative to the two-body pairing models commonly used to describe superfluidity along this transition.
Abstract Understanding the universal behavior of strongly-interacting systems of particles has been a major goal in multiple fields of physics from atomic physics to cosmology. Using a recently introduced manybody perturbation formalism for fermions which uses group theoretic and graphical techniques to solve the N -body problem, I study the thermodynamic behavior of the unitary regime for a trapped Fermi gas and investigate the role of the Pauli principle in the emergence of collective behavior, specifically superfluidity at ultralow temperatures. The method, which is called symmetry invariant perturbation theory, has no adjustable parameters and currently offers a solution to the manybody Schrodinger equation through first order in inverse dimensionality exactly. The solution at first order defines collective coordinates in terms of five types of N -body normal modes, identified as symmetric stretch, symmetric bend, single particle angular excitation, single particle radial excitation and phonon. A correspondence is established ‘on paper’ that enforces the Pauli principle through the assignment of specific normal mode quantum numbers. Applied at ultracold temperatures, this normal mode assignment yields occupation only in an extremely low frequency N -body phonon mode. A single particle radial excitation mode at a much higher frequency creates a gap that stabilizes the superfluidity at low temperatures. This radial excitation involves excitation of a single particle out of the synced motion of the phonon mode, rather than the breaking of a fermion pair. Coupled with the corresponding values for the energies at unitarity obtained by this manybody calculation, I obtain good agreement with experimental thermodynamic results. In particular, the lambda transition in the specific heat is clearly seen. Based on these findings, I propose a mechanism for the emergence of superfluidity at ultralow temperatures in this unitary regime that is driven by quantum statistics and the enforcement of the Pauli principle through the selection of normal modes. My results for the calculated thermodynamic quantities suggest that the emergence of some collective behaviors in macroscopic systems is due to the Pauli principle producing manybody pairing and the resulting macroscopic occupation of an N -body phonon mode.