Lee-Huang-Yang Quantum Fluctuation Correction

• Lee-Huang-Yang quantum fluctuation correction is the primary beyond-mean-field term that quantifies quantum depletion in dilute Bose gases.
• It is derived via Bogoliubov theory and a modified Gross-Pitaevskii framework, introducing a supercubic nonlinear term to stabilize self-bound droplets and multi-component condensates.
• Its corrections to ground-state energy and chemical potential have been validated experimentally across Bose mixtures, dipolar systems, and disordered quantum fluids.

The Lee-Huang-Yang (LHY) quantum fluctuation correction provides the leading-order beyond-mean-field contribution to the energy, chemical potential, and effective dynamics of dilute Bose gases. Originating in the quantum depletion of the ground state due to atomic interactions, it appears as a supercubic nonlinear term in the generalized Gross-Pitaevskii framework and governs the equation of state, stabilization of self-bound droplets, and collective excitations for both single- and multi-component condensates. The LHY correction is universal in the zero-range limit but acquires nonuniversal modifications and sensitivity to interactions, dimensionality, disorder, and collective mode structure when generalized beyond these constraints.

1. Quantum-Field-Theoretic Derivation and Bogoliubov Theory

Starting from the Heisenberg equation for the bosonic field operator $\hat{\psi}(\mathbf{r}, t)$ under a contact pseudopotential $$V(\mathbf{r},\mathbf{r}') = g\delta^3(\mathbf{r}-\mathbf{r}')$$, the field is split into a condensate order parameter $\psi_0(\mathbf{r})$ and noncondensed fluctuations $\hat{\eta}(\mathbf{r},t)$ via the Bogoliubov prescription $\hat{\psi} = \psi_0 + \hat{\eta}$ (Salasnich, 2018). The zero-temperature stationary equation

$$\mu \psi_0 = \left[ -\frac{\hbar^2}{2m}\nabla^2 + U + g|\psi_0|^2 + 2g\tilde{n} \right]\psi_0 + g\tilde{m}\psi_0^*$$

includes corrections from noncondensed and anomalous densities, $\tilde{n}$ and $\tilde{m}$. Solving the Bogoliubov–de Gennes equations under the semiclassical and slowly-varying order parameter approximations yields explicit expressions:

$$\tilde{n} = \alpha |\psi_0|^3,\quad \tilde{m} = 3\alpha |\psi_0|^3,$$

where $\alpha = (\sqrt{2}/12\pi^2)(2mg/\hbar^2)^{3/2}$. The resulting modified Gross–Pitaevskii equation contains a quartic ($|\psi_0|^3\psi_0$) LHY term representing the effects of quantum depletion.

2. Ground-State Energy and Chemical Potential Correction

The ground-state energy density with the LHY correction is, for a uniform gas (Brietzke et al., 2019, Boudjemaa, 2017):

$$\frac{E}{V} = \frac{1}{2}g n^2 \left[1 + \frac{128}{15\sqrt{\pi}}(n a^3)^{1/2}\right],$$

with $g = 4\pi\hbar^2 a/m$. The chemical potential acquires the celebrated Lee-Huang-Yang shift:

$$\mu(n) = g n \left[1 + \frac{32}{3\sqrt{\pi}}(n a^3)^{1/2}\right] = g n + \Delta\mu_{\rm LHY}(n),$$

where $$\Delta\mu_{\rm LHY}(n) = (32/3\sqrt{\pi}) g n \sqrt{n a^3}$$.

This form is rigorously derived for repulsive pair interactions as a second-order correction in the dilute regime ($n a^3 \ll 1$), and its validity is confirmed by lower bound methods in soft potential models (Brietzke et al., 2019).

3. Modified Gross-Pitaevskii Energetics and Quantum Fluid Regimes

The generalized energy functional governing condensate dynamics is

$$E[\psi] = \int d^3 r \left[\frac{\hbar^2}{2m}|\nabla\psi|^2 + U|\psi|^2 + \frac{g}{2}|\psi|^4 + \frac{128}{15\sqrt{\pi}}g a_s^{3/2}|\psi|^5 \right]$$

or, in multi-component mixtures (Skov et al., 2020, Jørgensen et al., 2018),

$$E[\psi_1, \psi_2] = \int d^3r \bigl\{ \sum_{i=1,2} (\frac{\hbar^2}{2m}|\nabla\psi_i|^2 + V|\psi_i|^2) + \varepsilon_{\rm MF}(n_1,n_2) + \varepsilon_{\rm LHY}(n_1,n_2) \bigr\},$$

with the LHY energy density exhibiting explicit dependence on interspecies and intraspecies scattering lengths.

In mixtures where the mean-field interactions are canceled ($a_{12} = -\sqrt{a_{11} a_{22}}$), the condensate becomes a pure "Lee-Huang-Yang fluid" governed almost solely by the LHY correction. Here, the effective one-component equation exhibits dynamics and collective modes dominated by quartic nonlinearity (Jørgensen et al., 2018, Skov et al., 2020).

4. Stabilization of Self-Bound Droplets, Phase Diagrams, and Anomalous Dynamics

In Bose mixtures with sufficiently strong interspecies attraction ($g_{12}^2 > g_{11} g_{22}$), the mean-field term becomes negative, driving collapse. The LHY correction stabilizes against collapse, yielding a finite equilibrium density and enabling the formation of self-bound quantum droplets (Guo et al., 2021):

$$\mathcal{E}_{\rm MF} + \mathcal{E}_{\rm LHY}:~ \partial_n [\cdots] = 0 \implies n_0 \sim [\delta g]^2/g^5$$

with $\delta g = g_{12} + \sqrt{g_{11}g_{22}}$. Droplets exist only above a critical atom number $N_c$, which depends on the residual coupling and experimental conditions.

In experiments (e.g., $^{23}$Na-$^{87}$Rb mixtures), the LHY-driven phase diagram is mapped experimentally via critical atom numbers and anomalous expansion ("release") energies upon transition from droplet to gas phases. The LHY energy per particle in these droplet phases is comparable to the residual mean-field energy and is essential for stabilization (Guo et al., 2021).

5. Extensions: Dipolar Gases, Infrared Cutoff Effects, and Nonuniversal LHY Terms

For dipolar Bose gases, the LHY term is generalized as (Boudjemaa, 2017, He et al., 2024, Triay, 2019):

$$\mathcal{E}_{\rm LHY}(n) = \frac{32}{15\sqrt{\pi}} \frac{\hbar^2}{m}(n a)^{5/2} \mathcal{Q}_5(\epsilon_{dd}),$$

with $\epsilon_{dd}$ the relative dipole strength, and $\mathcal{Q}_5$ an angular integral or cutoff-corrected function. Infrared cutoff schemes (geometric, spherical, healing-length cutoff) are indispensable for quantitative agreement with droplet experiments, as naïve local-density LHY (no cutoff) generally overestimates repulsive pressure. Healing-length-based cutoffs yield the most accurate reproduction of droplet stability boundaries across atom number and dipolar strength (He et al., 2024, Triay, 2019).

Finite-range corrections, encoded in higher-order pseudopotential coefficients and effective range $r_s$, induce "nonuniversal" LHY terms in the equation of state for quantum droplets. These corrections modify ground-state density, collective-mode frequencies (notably the fractional breathing-mode shift), and surface tension, and are experimentally accessible in regimes with large effective range or extremely weak diluteness (Zhang et al., 12 Dec 2025).

6. Implications for Nonlinear Waves, Disorder, and Collective Excitations

LHY corrections fundamentally alter nonlinear wave dynamics, as evident in rogue-wave, dispersive shock, and soliton structures in extended Gross-Pitaevskii models (Chandramouli et al., 3 Oct 2025, Gangwar et al., 2022). In 1D, for example, the LHY term assumes the form $-\frac{g^{3/2}}{\pi}\sqrt{n}$, leading to attractive quantum bright/stripe solitons entirely stabilized by quantum fluctuation effects even in absence of mean-field attraction (Gangwar et al., 2022).

In defected or disordered settings, LHY-induced modifications suppress glassy fractions and change Anderson-localization properties. Phonon-mode screening and component-selective localization in mixtures with LHY corrections are observable in disordered optical potentials (Mehri et al., 2024). In dipolar systems, anisotropic superfluid density and localized ground states arise due to LHY's competition with disorder and dipole-induced collapse (Boudjemaa, 2017, Shamriz et al., 2020).

Monopole and breathing-mode oscillations in LHY fluids exhibit robust frequency shifts, stable against tuning of underlying scattering lengths over realistic experimental ranges, providing a direct probe of quantum fluctuation energetics (Skov et al., 2020, Jørgensen et al., 2018).

7. Validity Regimes, Limitations, and Universality

The LHY correction is derived under strict assumptions: weak diluteness ($n a^3 \ll 1$), small quantum depletion (depletion fraction $ñ/n \ll 1$), and slowly-varying condensate backgrounds (neglect of gradients in kinetic terms) (Salasnich, 2018, Brietzke et al., 2019). In these regimes, it yields universal corrections, acts as the leading stabilizing mechanism in multi-component, dipolar, and self-bound systems, and enables direct experimental access to pure quantum fluctuation physics.

Extensions to finite-range interactions, strong coupling, or highly anisotropic/low-dimensional regimes must include nonuniversal corrections, explicit cutoff schemes, and refined BdG treatments to remain quantitatively accurate (Zhang et al., 12 Dec 2025, He et al., 2024). Experimental signatures—critical mass thresholds, altered phase boundaries, mode frequency shifts, and glassy disorder suppression—are all controlled by the specific LHY structure and its regime-dependent generalizations.

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References:

• (Salasnich, 2018, Brietzke et al., 2019, Guo et al., 2021, Boudjemaa, 2017, He et al., 2024, Triay, 2019, Skov et al., 2020, Minardi et al., 2019, Chandramouli et al., 3 Oct 2025, Zhang et al., 12 Dec 2025, Jørgensen et al., 2018, Shamriz et al., 2020, Mehri et al., 2024, Gangwar et al., 2022).