Hybridization of first and second sound in weakly interacting Bose gases

September, 3rd

Lucas Verney

Supervised by Pr. Sandro Stringari and
Pr. Lev P. Pitaevskii


Sounds in superfluid 4He

Sound velocities in liquid Helium Sound modes in liquid Helium
Donnelly, Physics Today, 2009

  • Two sound velocities in a superfluid system.
  • $u_2 < u_1$.
  • Pressure ($u_1$) / temperature ($u_2$) wave

What about BECs ?

Sound velocities

Pitaevskii & Stringari, BEC, OUP 2003

  • Two sound velocities as well
  • $u_2 < u_1$ ?
  • Hybridization at \[T \approx gn = \mu(T=0)\]

Nature of the sound waves ?

  • Second sound is a wave where the normal and superfluid components oscillate with opposite phases.”, Hou, PRA, 2013

An interesting quantity is the Landau-Placzek ratio $\frac{c_p}{c_v} - 1$ which is always small in the case of He.

Taylor, PRA, 2009 and Landau, Fluid Mechanics

For BECs,

  • Below hybridization, Landau-Placzek ratio is small.
    Same behavior as in He:
    Pressure ($u_1$) / Temperature ($u_2$) wave.
  • Above hybridization, LP ratio is not small ⇒ very different behavior is expected.

Landau's two fluid model

Description of the superfluid in terms of normal and superfluid component:

Sound modes in liquid Helium
Donnelly, Physics Today, 2009

\begin{equation} \left\lbrace \begin{array}{l} \vec{j} = \overbrace{\rho_s \vec{v}_s}^{\text{superfluid}} + \overbrace{\rho_n \vec{v}_n}^{\text{normal}}\\ \frac{\partial \rho}{\partial t} + \mathrm{div}(\vec{j}) = 0\\ \frac{\partial \vec{j}}{\partial t} + \vec{\nabla}{P} = 0\\ \\ \\ \frac{\partial \vec{v}_s}{\partial t} + \vec{\nabla}{\mu} = 0\\ \\ \frac{\partial (\rho \tilde{s})}{\partial t} + (\rho \tilde{s})\, \mathrm{div}(\vec{v}_n) = 0\\ \end{array} \right. \end{equation}

Linear equations ⇒ look for plane wave solutions.

Plane wave solutions must satisfy a fourth order equation on sound velocity $u$:

\begin{equation} u^4 - \left( \left.\frac{\partial P}{\partial \rho}\right|_{\tilde{s}} + \frac{\rho_s T \tilde{s}^2}{\rho_n \tilde{c}_v}\right) u^2 + \frac{\rho_s T \tilde{s}^2}{\rho_n \tilde{c}_v} \left.\frac{\partial P}{\partial \rho}\right|_{T} = 0 \end{equation}
  • Fourth order equation ⇒ 2 positive solutions.
  • Valid for any superfluid system: He + BEC.
  • Now we have to plug the right thermodynamic functions in this equation.
  • For BECs, we will use Bogoliubov thermodynamics at low T and Beliaev technique at high T.

Low temperatures

In this regime, we use Bogoliubov thermodynamics. The quasiparticles excitation spectrum is given by:

\[E(\vec{p}) = \sqrt{\frac{p^2}{2m}\left( \frac{p^2}{2m} + 2 g n\right)}\]

We express everything in terms of dimensionless units:

  • $\tilde{p} = \frac{p}{mgn}$
  • $\tilde{t} = \frac{k_B T}{gn}$

An Important parameter to have in mind is: \[\eta = \frac{gn}{k_B T_c^{(0)}} \approx 0.03\]

For example, for the free energy, we get:

\begin{equation} \frac{F(\tilde{t}\,)}{gn N}= E_0(na^3) + \eta^{3/2} \tilde{f}(\tilde{t}) \end{equation}

Express the normal density using Landau's formula:

\begin{equation} \rho_n = - \frac{1}{3} \int \frac{\mathrm{d} N_\mathbf{p}(\varepsilon)}{\mathrm{d} \varepsilon} p^2 \frac{\mathrm{d} \mathbf{p}}{(2\pi \hbar)^3} \end{equation}


  • Hybridization at \[k_BT = 0.6 gn.\]
  • Gap width $\propto \eta^{3/4}$.
  • If $\eta = 0$, crossing.

Below hybridization, same behaviour as in liquid helium:

  • First sound is an in-phase oscillation, a pressure wave.
  • Second sound is an out-of-phase oscillation, a temperature wave.

A theory for all $T < T_c$ ?

Yes ! Beliaev technique. (Giorgini, New Journal of Physics, 2010).

Self-consistent equations. For example:

\[n = n_0 + \sum_{\mathbf{k}} \left[ \frac{\varepsilon_k + gn_0}{2E_k} (1+2N_E) - \frac{1}{2}\right]\]


\[\left\{\begin{array}{l} \varepsilon_k = \hbar^2 k^2 / (2m)\\ E_k = \sqrt{\varepsilon_k (\varepsilon_k + 2gn_0)} \\ N_E = \left(e^{\beta E} - 1\right)^{-1}\\ \end{array}\right.\]
  • Valid on a wide range of temperatures,
    (except for $T \approx 0$ and $T \approx T_c$).
  • Good agreement with Quantum Monte Carlo results.
  • Recovers Bogoliubov at low T, and Hartree-Fock at high T.


Potentially some problems with the derivatives ($c_v$, $\chi_T$).

High T results

Sound speeds

Simple expression for second sound speed

A simple expression for second sound velocity is given by:

\[u_2^2 = \frac{\rho_s}{\rho}\left( \frac{\partial P}{\partial \rho} \right)_T\]

Analogous to fourth sound in liquid He: only the superfluid part is moving.

$(\delta n / n) / (\delta T / T)$

Isothermal compressibility


  • Started by studying the hybridization phenomenon in BECs.
  • Low temperature phenomenon
    Bogoliubov theory is simple and valid.
  • Inquiry about the behaviour at higher temperatures
    Looked for a valid theory for almost all $T$.
  • Obtained results over a wide range of temperatures:
    speeds, nature of the modes, simple expressions.

Thanks to S. Stringari, L. P. Pitaevskii, S. Giorgini.