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arXiv:2409.18215 · Physical Review Research

Engineering quantum droplet formation by cavity-induced long-range interactions

Leon Mixa, Milan Radonjić, Axel Pelster, Michael Thorwart

Published May 29, 2025• 1 citation
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Abstract

We investigate a dilute Bose gas with both a short-range contact and an effective long-range interaction between the atoms. The latter is induced by the strong coupling to a cavity light mode and is spatially characterized by a periodic signature and a tunable envelope rooted in the pumping of the cavity. We formulate a Bogoliubov theory based on a homogeneous mean-field description and quantum fluctuations around it. The competition between the repulsive contact interaction and the long-range interaction allows the formation of self-bound quantum droplets. This generic approach is applied to two cavity setups, one without and one with a momentum-conserving effective long-range interaction between the atoms in the form of a driven dispersive cavity mode and a multimode cavity, respectively. For both cases we show analytically how the size and the central density of the cavity-induced quantum droplets depend on the contact interaction strength and on the shape of the spatial envelope of the long-range interaction.

DOI
10.1103/physrevresearch.7.023204

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We investigate a dilute Bose gas with both a short-range contact and an effective long-range interaction between the atoms. The latter is induced by the strong coupling to a cavity light mode and is spatially characterized by a periodic signature and a tunable envelope rooted in the pumping of the cavity. We formulate a Bogoliubov theory based on a homogeneous mean-field description and quantum fluctuations around it. The competition between the repulsive contact interaction and the long-range interaction allows the formation of self-bound quantum droplets. This generic approach is applied to two cavity setups, one without and one with a momentum-conserving effective long-range interaction between the atoms in the form of a driven dispersive cavity mode and a multimode cavity, respectively. For both cases we show analytically how the size and the central density of the cavity-induced quantum droplets depend on the contact interaction strength and on the shape of the spatial envelope of the long-range interaction.
Quantum droplets are formed in quantum many-body systems when the competition of quantum corrections with the mean-field interaction yields a stable self-bound quantum liquid. We predict the emergence of a quantum droplet when a Bose-Einstein condensate is placed in a transverse-pumped optical cavity. The strong coupling between the atoms and a cavity mode induces long-range interactions in the atoms, and a roton mode for negative cavity detuning emerges. Using Bogoliubov theory, we show that the roton mode competes with the repulsive atomic s -wave scattering. Due to the favorable scaling of the quantum fluctuations with respect to the volume, a self-bound stable quantum liquid emerges.
Multimode optical cavities can be used to implement interatomic interactions which are highly tunable in strength and range. For bosonic atoms trapped in an optical lattice we show that, for any finite range of the cavity-mediated interaction, quantum self-bound droplets dominate the ground state phase diagram. Their size and in turn density is not externally fixed but rather emerges from the competition between local repulsion and finite-range cavity-mediated attraction. We identify two different regimes of the phase diagram. In the strongly glued regime, the interaction range exceeds the droplet size and the physics resembles the one of the standard Bose-Hubbard model in a (self-consistent) external potential, where in the phase diagram two incompressible droplet phases with different filling are separated by one with a superfluid core. In the opposite weakly glued regime, we find instead direct first order transitions between the two incompressible phases, as well as pronounced metastability. The cavity field leaking out of the mirrors can be measured to distinguish between the various types of droplets.
We study one-dimensional quantum gases in continuous space with cavity-mediated infinite-range interactions using the variational and the diffusion Monte Carlo methods. Starting from the exact two-body solution, we construct a non-translationally invariant Jastrow wave function that accurately captures the spatial structure induced by the cavity field and provides an efficient many-body ansatz for both bosonic and fermionic systems. We analyze properties of three characteristic quantum systems, subject to long-range interactions: (i) ideal Bose gas, (ii) interacting Bose gas, (iii) ideal Fermi gas. In the absence of short-range interactions, we identify a crossover from a stable, weakly modulated phase realized at finite particle number for repulsive interactions to a self-organized state for attractive interactions, marked by clustering, loss of superfluidity, and, at fixed interaction strength, the absence of a thermodynamic limit. Introducing short-range repulsion, either through contact interactions or fermionic statistics, leads to the formation of a mesoscopic gas-like regime that disappears in the thermodynamic limit. A qualitative phase diagram is proposed to illustrate the combined effects of short- and long-range interactions, highlighting the emergence of distinct regimes with characteristic structural properties. Applying the Kac prescription, we obtain a well-defined thermodynamic limit and show that the periodic modulation found for a finite number of particles in the repulsive regime is a finite-size artifact, while the self-organized state survives with its onset shifted to substantially stronger coupling.
We investigate beyond-mean-field corrections to the energy of an elongated homogeneous Bose gas strongly confined in two directions, with dipoles aligned along the long axis of the system. When the dipolar interaction reaches its critical strength, the mean-field approach predicts instability. However, similar to the free-space case, beyond-mean-field effects significantly alter the ground state of the system, leading to the formation of a self-bound atomic cloud known as a quantum droplet. Our analysis demonstrates that the beyond-mean-field contribution to the energy in the quasi-one-dimensional region, in addition to the confinement-induced shift of the mean-field energy, is proportional to the third power of the density approximately equal to n3{n}^{3}n3. Therefore, it can be interpreted as an effective three-body repulsion that stabilizes the gas, preventing collapse and leading to a finite-density solution. We also show that the same effect plays a crucial role in the binding of strongly elongated dipolar droplets under harmonic confinement.
We investigate beyond-mean-field corrections to the energy of an elongated homogeneous Bose gas strongly confined in two directions, with dipoles aligned along the long axis of the system. When the dipolar interaction reaches its critical strength, the mean-field approach predicts instability. However, similar to the free-space case, beyond-mean-field effects significantly alter the ground state of the system, leading to the formation of a self-bound atomic cloud known as a quantum droplet. Our analysis demonstrates that the beyond-mean-field contribution to the energy in the quasi-1D region, in addition to the confinement induced shift of the mean field energy, is proportional to the third power of the density ∼n3\sim n^3∼n3. Therefore, it can be interpreted as an effective three-body repulsion that stabilizes the gas, preventing collapse and leading to a finite-density solution. We also show that the same effect plays a crucial role in the binding of strongly elongated dipolar droplets under harmonic confinement.
We analytically calculate the leading quantum corrections of the ground-state energy of two- and one-dimensional weakly interacting Rabi-coupled Bose-Bose mixtures in the frame of the Bogoliubov approximation. We show that to repulsive intraspecies and attractive interspecies interactions, the effect of quantum fluctuations favors the formation of self-bound droplets. These liquidlike states are crucially affected by the Rabi coupling, leading thus to the appearance of a quantum instability. We derive meaningful formulas to describe the droplet phase in the one-dimensional case.