Nonlocal field theory of quasiparticle scattering in dipolar Bose-Einstein condensates

{\it Strengths: The article attempts to address the problem of a domain wall of contact interaction in a one dimensional gas of dipolar atoms in the framework of the one-dimensional Gross Pitaevskii equation. It points out that the linearized Gross Pitaevskii equation is an integrodifferential equation with an integral kernel whose Fourier transform has a branch point for imaginary wavevector that gives rise to a continuum of evanescent waves near the discontinuity of contact interactions. It attempts to solve the linearized equation by singular integral equation techniques and discuss current conservation.}


$\Longrightarrow$ We thank the Referee for reading our manuscript and pointing out its strengths in an organized manner. One of the major issues of the Referee's report, is that they suggest the manuscript did not present a clear definition of the conservation law being discussed. However, apparently the Referee refers to particle number conservation implied by the theory's $U(1)$ symmetry. The conservation law discussed in the present paper is {\em not} related to particle number conservation, as we address in detail below.
We also discuss separately each of the weaknesses claimed by the Referee.


{\it Weakness (1): The presentation of the manuscript is very sketchy. The potential energy terms are described in Eq. (4) for the dipolar interaction and the first paragraph of Sec. III-A for the discontinuous contact interaction. The full Gross-Pitaevskii equation (GPE) is never given, only the assumed linearized equation (8). However, that form is incorrect.}

$\Longrightarrow$ The main objective of our work is to study the solutions for the Bogoliubov-de Gennes (BdG) equation, Eq.~(8) of the submitted manuscript, and the Referee asserts that our assumed form for Eq.~(8) is not correct, which would raise serious concerns regarding the manuscript's physical validity. However, Eq.~(8) as we presented is indeed correct as we clarify now. We assumed throughout the work a stationary condensate of homogeneous particle density. The system inhomogeneity at the level of the linearized perturbation is, in our work, modeled by a variable contact interaction which changes the local sound velocity, for a fixed and constant condensate particle density. In this way, $\psi_{0}$ written by the Referee in their BdG equation is a constant, and the resulting equation is precisely the same as Eq.~(8) of the manuscript.

The Referee also pointed out that the manuscript does not present a deduction for Eq.~(8) from the GPE, which we believe prevented the Referee's understanding of how the condensate background is determined in the analysis of our model. We have amended the manuscript with the details of obtaining Eq.~(8), by showing how the external potential should be chosen in the experiment such that the condensate background is homogeneous. Furthermore, we also stress that, as mentioned in the manuscript, the main result of our work, namely, the recipe for how to construct solutions for the perturbations in a 1D condensate with dipolar interactions with a single sound interface, does not depend on the homogeneous condensate density assumption, and the same technique can be applied to a condensate with a variable particle density at $x=0$. We call attention of the Referee to the fact that the solutions we built explore precisely the splitting of integral kernel into two parts $x'<0$ and $x>0$, as shown in detail in the Appendices A and B of the submitted ms. For a condensate with a variable density modeling the sound barrier, exactly the same steps that lead to the singular integral equation (SIE) (15) can be used, with the difference that the resulting SIE depends explicitly on the density jump at $x=0$, and the potential term $\partial_x^2\psi_{0}/2\psi_{0}$ pointed out by the Referee becomes a delta derivative potential, explored, for instance, in [Ann. of Phys. {\bf 407,} 148-165 (2019)].

{\it Weakness (2): A related issue is the discussion of the current conservation. The full GPE gives a conserved current that is exactly and the linearized conserved current is obtained by inserting the expansion $\psi_0+\delta\psi$. The conserved current derived from the linearized GPE should match that exact current. It is not clear that Eq. (25) satisfies that property.}

$\Longrightarrow$ We thank the Referee for raising this point, whose detailed resolution we think contributes for our manuscript's clarity. The Bogoliubov expansion we use is known to lead to a spontaneous breakdown of the theory's $U(1)$ symmetry [cf. Phys. Rev. A {\bf 57}, 3008 (1998)], and thus there is no particle number conservation at the level of linearized $\delta\psi$ alone.
This can be deduced directly from the BdG equation, Eq.~(8) from the ms:

\begin{equation*}
i\partial_t\delta\psi=-\frac{\partial_x^2}{2}\delta\psi+n(g_{\rm c}+g_{\rm d})(\delta\psi+\delta\psi^*)-3ng_{\rm d}[G*(\delta\psi+\delta\psi^*)].
\end{equation*}

Multiplying the equation above by $\delta\psi^*$, and subtracting the result from its complex conjugate results in

\begin{equation*}
\partial_t|\delta\psi|^2+\frac{1}{2i}\partial_x[\delta\psi^*\partial_x\delta\psi-(\partial_x\delta\psi^*)\delta\psi]=\frac{n(g_{\rm c}+g_{\rm d})}{i}(\delta\psi^2-\delta\psi^{*2})-3ng_{\rm d}\frac{\delta\psi^*-\delta\psi}{i}[G*(\delta\psi+\delta\psi^*)].
\end{equation*}

The L.H.S. of the equation above contains the contribution to the full particle density coming from the perturbations: $\delta n=|\delta\psi|^2$, and the corresponding contribution to the system full particle current: $\delta J=\mbox{Im}\ \delta\psi^*\partial_x\delta\psi$. Yet, even in the absence of dipolar interactions, $g_{\rm d}=0$, we find $\partial_t\delta n+\partial_x\delta J\neq0$, which is the mathematical expression for the no number conservation of the expansion. A thorough study of the physical meaning of this symmetry breakdown by the Bogoliubov expansion and how to correctly address the corrections to the condensate due to the depleted particles can be found, for instance, in Phys. Rev. D {\bf 72}, 105005 (2005).


However, and this seems to be the root of the Referee's stated problem, there exists a conserved quantity implied by Eq.~(8) which is linked to the exact {\em unitarity of the scattering process,} and not to the theory's number conservation. Specifically, the BdG equation in Nambu representation, $\Psi=(\delta\psi, \delta\psi^*)^{\rm t}$, reads

\begin{equation*}
i\sigma_3\partial_t\Psi=-\frac{\partial_x^2}{2}\delta\Psi+n(g_{\rm c}+g_{\rm d})(1+\sigma_1)\Psi-3ng_{\rm d}(1+\sigma_1)G*\Psi,
\end{equation*}

which is the equation solved in the ms. In order to deduce the conserved ``charge'' implied by the equation above, we let $\Psi$ be any solution, not necessarily satisfying $\sigma_1\Psi^*=\Psi$. By repeating the same steps as before, we multiply on the left by $\Psi^{\dagger}$ and subtract the complex conjugate to obtain

\begin{equation*}
\partial_t(\Psi^{\dagger}\sigma_3\Psi)+\frac{1}{2i}\partial_x[\Psi^{\dagger}\partial_x\Psi-(\partial_x\Psi^\dagger)\Psi]= 3ing_{\rm d}[\Psi^\dagger(1+\sigma_1)G*\Psi-(G*\Psi^\dagger)(1+\sigma_1)\Psi].
\end{equation*}

We observe that in the absence of dipolar interactions, a local conservation law exists: $\partial_t(\Psi^\dagger\sigma_3\Psi)+\partial_x\mbox{Im}\ \Psi^{\dagger}\partial_x\Psi=0$. This is the conservation law discussed in section IV of the ms, which assumes the form $\partial_xJ_{\omega}=0$ for the quasiparticle excitations in the absence of nonlocal interactions. For the dipolar system in the ms, however, the local conservation law no longer holds, but the full ``charge'' $\int dx \Psi^{\dagger}\sigma_3\Psi$ is stil conserved in time, as can be seen from the integration of the equation above. Furthermore, by following the same steps that lead to the equation above for any two solutions $\Psi$, $\Psi'$, we show that the BdG scalar product $$\langle\Psi,\Psi'\rangle:=\int dx\Psi^\dagger\sigma_3\Psi'$$ is conserved in time: $\partial_t\langle\Psi,\Psi'\rangle=0$. For the quasiparticle excitations, this condition is equivalent to the ``conservation law'' stated in the manuscript, from which we can assess the unitarity of the scattering process, i.e., the
balance between incident, reflected, and transmitted waves.
We cite in this regard two references
[Phys. Rev. D {\bf 93}, 124060 (2016), Phys. Rev. D {\bf 94}, 084027 (2016)] for a discussion of the scattering process in the context of analogue gravity.
By verifying the S-matrix unitarity for the built quasiparticles, we thus ensure that the field modes we constructed are indeed solutions to the BdG equation. We have amended the manuscript by stating explicitly the meaning of the S-matrix in terms of the Bogoliubov scalar product time-invariance.

{\it Weakness (3): Given that strong approximations have to be made to arrive at Eq. (8), it is not obvious that the sophisticated singular integral equation treatment of sec. III-C is physically relevant. Scattering processes taking place over a distance of the order of the healing length in the vicinity of the contact interaction discontinuity is neglected, yet they can contribute evanescent waves of larger range than the ones retained in Eq. (13).}

$\Longrightarrow$ As explained in the above, our development relies on no approximation besides the Bogoliubov expansion to linearize the GPE and study its perturbations, and thus Eq.~(8) is exact for the condensate with constant particle density we assumed. The implementation of our technique to the variable density case is a straightforward extension that leads to no qualitative differences to the scattering phenomena we have discovered, and whose addition to the manuscript would only further increase the already rather elevated mathematical complexity of the problem.