Quasiparticle energy in a strongly interacting homogeneous Bose-Einstein condensate

TL;DR

This study uses two-photon Bragg spectroscopy to measure quasiparticle energies in a strongly interacting homogeneous Bose-Einstein condensate, revealing significant deviations from existing theories and highlighting the need for further theoretical development.

Contribution

It provides experimental data on quasiparticle energies in strongly interacting BECs, showing deviations from Bogoliubov theory and comparing results with Feynman and Wilson operator product expansion models.

Findings

Sign change in excitation resonance shift at a specific scattering length.

Breakdown of Bogoliubov theory at high interaction strengths.

Better agreement with Wilson operator product expansion at strong interactions.

Abstract

Using two-photon Bragg spectroscopy, we study the energy of particle-like excitations in a strongly interacting homogeneous Bose-Einstein condensate, and observe dramatic deviations from Bogoliubov theory. In particular, at large scattering length $a$ the shift of the excitation resonance from the free-particle energy changes sign from positive to negative. For an excitation with wavenumber $q$, this sign change occurs at $a \approx 4/(\pi q)$, in agreement with the Feynman energy relation and the static structure factor expressed in terms of the two-body contact. For $a \gtrsim 3/q$ we also see a breakdown of this theory, and better agreement with calculations based on the Wilson operator product expansion. Neither theory explains our observations across all interaction regimes, inviting further theoretical efforts.

Equations

$$\omega ={\omega }_0\sqrt{1+\frac{2}{q^2{\xi }^2}},$$

$$S(q)=1+\frac{C}{8n}\left(\frac{1}{q}-\frac{4}{\pi a{q}^2}\right),$$

$$\Delta {\omega }_{\mathrm{FT}}$$

$$\alpha \equiv \frac{mq}{4\pi \hslash n}\Delta \omega ,$$

$${\alpha }_{\mathrm{FT}}=qa\left(1-\frac{\pi }{4}qa\right),$$

Full text

Spectroscopy of elementary excitations in a many-body system is one of the primary methods for probing the effects of interactions and correlations in the ground state of the system, which are at the heart of macroscopic phenomena such as superfluidity Kapitza (1938); Allen and Jones (1938). In ultracold atomic gases, two-photon Bragg spectroscopy provides a measurement of the excitation energy $\hbar\omega$ at a well defined wavenumber $q$ Kozuma et al. (1999); Stamper-Kurn et al. (1999); Steinhauer et al. (2002); Papp et al. (2008); Kuhnle et al. (2010); Hoinka et al. (2013); Gotlibovych et al. (2014). For a weakly interacting homogeneous Bose-Einstein condensate (BEC), the excitation spectrum is given by the Bogoliubov dispersion relation Bogoliubov (1947), with low-$q$ phonon excitations and high-$q$ particle-like excitations. Predictions of the Bogoliubov theory have been experimentally verified both in harmonically trapped gases, invoking the local density approximation Stamper-Kurn et al. (1999); Steinhauer et al. (2002), and in homogeneous atomic BECs Gotlibovych et al. (2014).

Much richer physics, including phenomena traditionally associated with superfluid liquid helium, such as the roton minimum in the excitation spectrum Yarnell et al. (1959), is expected in strongly interacting atomic BECs (for a recent review see Chevy and Salomon (2016)). The strength of two-body interactions, characterised by the s-wave scattering length $a$, can be enhanced by exploiting magnetic Feshbach resonances Chin et al. (2010). However, this also enhances three-body inelastic collisions, making the experiments on strongly interacting bulk BECs Papp et al. (2008); Navon et al. (2011); Wild et al. (2012); Makotyn et al. (2014) challenging and still scarce 111This is in contrast to Fermi gases near Feshbach resonances, and Bose and Fermi gases in optical lattices, where the regime of strong correlations is readily reached experimentally.. A deviation from the Bogoliubov spectrum was observed in Bragg spectroscopy of large-$q$ excitations in a harmonically trapped 85Rb BEC Papp et al. (2008), and has inspired various theoretical interpretations Papp et al. (2008); Ronen (2009); Kinnunen and Holland (2009); Sarjonen et al. (2012); Sahlberg et al. (2013); Hofmann and Zwerger (2016); Chevy and Salomon (2016), with no consensus or complete quantitative agreement with the experiments being reached so far.

In this Letter, we use Bragg spectroscopy to study the large-$q$, particle-like excitations in a strongly interacting homogeneous 39K BEC, produced in an optical box trap Gaunt et al. (2013). Our homogeneous system allows more direct comparisons with theory, and we also explore stronger interactions than in previous experiments. We show that at large $a$ the excitation-energy shift from the free-particle dispersion relation strongly deviates from the Bogoliubov theory and even changes sign from positive to negative. For $a\lesssim 3/q$ our measurements are in excellent agreement with the calculation based on the Feynman energy relation, with a static structure factor that accounts for short-range two-particle correlations. However, for even stronger interactions we also observe a breakdown of this approximation, and find better agreement with a recent prediction Hofmann and Zwerger (2016) based on the Wilson operator product expansion.

In Bogoliubov theory, the excitation energy $\hbar\omega$ is given by

[TABLE]

where $\omega_{0}=\hbar q^{2}/(2m)$ is the free-particle dispersion relation, $m$ the atom mass, $\xi=1/\sqrt{8\pi na}$ the healing length, and $n$ the BEC density. For particle-like excitations, with $q\gg 1/\xi$, the Bogoliubov prediction for the interaction shift $\Delta\omega=\omega-\omega_{0}$ is $\Delta\omega_{\rm B}=4\pi\hbar na/m$ [see Fig. 1(a)]. This theory assumes $\sqrt{na^{3}}\ll 1$. Moreover, it is valid only for $q\ll 1/a$, because it does not consider the short-range two-particle correlations, at distances $r\lesssim a$.

For $\sqrt{na^{3}}\ll 1$, the Feynman energy relation gives the excitation resonance at $\omega=\omega_{0}/S(q)$, where $S(q)$ is the static structure factor. Considering short-range correlations, for $q\xi\gg 1$:

[TABLE]

where $C(n,a)$ is the two-body contact density, and the expression in the brackets reflects the two-body correlations at short distances Pitaevskii and Stringari (2016); Hofmann and Zwerger (2016); this ‘factorisation’ of the effects of many-body correlations (captured by $C$) and the short-distance two-body physics was proposed by Tan Tan (2008). For $\sqrt{na^{3}}\ll 1$, the contact density is $C\approx(4\pi na)^{2}$, and for our experimental parameters $|S(q)-1|<0.03$, so $1/S(q)-1\approx 1-S(q)$. This ‘Feynman-Tan’ (FT) approach thus gives the interaction shift of the excitation resonance

[TABLE]

For $qa\rightarrow 0$, $\Delta\omega_{\rm FT}$ reduces to $\Delta\omega_{\rm B}$, but for increasing $a$ (at fixed $q$) it back-bends and changes sign at $a=4/(\pi q)$ [see Fig. 1(a)] 222In Papp et al. (2008) the largest value of $a$ reached was $0.8/q$ and back-bending was observed, but $\Delta\omega$ remained positive.. At the same time, for the low-$q$ phonons $\Delta\omega$ is positive at all $a$ Rota et al. (2013). As illustrated in Fig. 1(b), this implies an inflection point in the dispersion relation, $\omega(q)$ at fixed $a$, which is a precursor of the roton minimum that fully develops only for extremely strong interactions Rota et al. (2013); Hofmann and Zwerger (2016). In Eq. (2) the maximum in $S(q)$ for fixed $n$ and $a$, which is conceptually associated with the roton Steinhauer et al. (2005); Hofmann and Zwerger (2016), occurs at $q=8/(\pi a)$, independently of $n$, and only for $\sqrt{na^{3}}\sim 1$ does this coincide with the familiar result for liquid helium, $q_{\rm roton}\sim n^{1/3}$.

In our experiments the regime $\sqrt{na^{3}}\sim 1$ is not reachable due to significant losses on the timescale necessary to perform high-resolution Bragg spectroscopy. Nevertheless, we reach the regime where interactions are strong enough to observe a dramatic departure from Bogoliubov theory and the precursors of roton physics. Our setup is described in Ref. Eigen et al. (2016). We produce quasi-pure homogeneous 39K BECs of $N=(50-160)\times 10^{3}$ atoms in a cylindrical optical box trap of variable radius, $R=(15-30)~{}\mu$m, and length, $L=(30-50)~{}\mu$m. The BEC is produced in the lowest hyperfine state, which features a Feshbach resonance centred at $402.70(3)$ G Fletcher et al. (2017). By varying $N$, $L$, and $R$, we vary $n$ in the range $(0.2-2.0)\times 10^{12}$ cm*-3*. The three-body loss rate is $\propto n^{2}a^{4}$, so working at such low $n$ is favourable for increasing both $qa$ and $\sqrt{na^{3}}$. We prepare the BEC at $a=200~{}a_{0}$, where $a_{0}$ is the Bohr radius, and then ramp $a$ in 50 ms to the value at which we perform the Bragg spectroscopy. For each $n$ we limit $a$ to values for which the particle loss during the whole experiment is $<10\%$. By varying the angle between the Bragg laser beams we also explore three different $q$ values: 1.1, 1.7 and 2.0 $k_{\text{rec}}$, where $k_{\text{rec}}=2\pi/\lambda$ and $\lambda=767$ nm. For all our parameters we stay in the regime of particle-like excitations, with $q\xi$ values between 5 and 40.

In Fig. 2(a) we show an example of an absorption image taken after the Bragg diffraction, and in Fig. 2(b) an example of a Bragg spectrum used to determine the resonance shift $\Delta\omega$. The diffracted fraction of atoms is determined from the centre of mass of the atomic distribution Papp et al. (2008); Hoinka et al. (2013); in all our measurements we keep the maximal diffracted fraction to $\lesssim 10\%$.

In Fig. 3(a) we plot $\Delta\omega$ versus $a$ for two different combinations of the BEC density $n$ and excitation wavenumber $q$. In both cases we observe good agreement with the prediction of Eq. (3), without any adjustable parameters; for the lower $n$ we reach higher $a$ and clearly observe that $\Delta\omega$ changes sign. Defining a dimensionless interaction frequency shift

[TABLE]

the FT prediction of Eq. (3) is recast as:

[TABLE]

which is a universal function of $qa$ only; with the same normalisation the Bogoliubov theory gives $\alpha_{\rm B}=qa$. In Fig 3(b) we show that all our measurements of $\alpha$ with three different combinations of $n$ and $q$ fall onto the same universal curve, in agreement with Eq. (5) 333Note that the normalisation in Eq. (4) also takes into account the small (10%) density variations between measurements taken with different values of $a$ and approximately the same $n$..

While in Fig. 3(b) all our data agree with Eq. (5), we note that for the points near $qa=2.5$ the validity of this theory is questionable and the agreement might be partly fortuitous. For these data $\sqrt{na^{3}}\approx 0.05$, which is already not negligible. At this point the Lee-Huang-Yang (LHY) prediction for the next-order correction to $C$ is of order 50% Lee and Yang (1957); Lee et al. (1957); Combescot et al. (2009) and even beyond-LHY corrections Braaten and Nieto (1999); Braaten et al. (2002) could be significant. Moreover, the Feynman relation is expected to be quantitatively reliable only for $\sqrt{na^{3}}\ll 0.1$ Rota et al. (2013, 2014). In the final part of the paper we explore even stronger interactions and the limits of validity of the FT prediction.

In Fig. 4(a) we show measurements of $\Delta\omega$ with $n\approx 0.2\times 10^{12}~{}\text{cm}^{-3}$ and $q=2\,k_{\text{rec}}$. Here we explore scattering lengths up to $\approx 8\times 10^{3}~{}a_{0}$, corresponding to $qa\approx 7$ and $\sqrt{na^{3}}\approx 0.1$, and observe a strong deviation from the FT prediction.

Tuning $a$ at fixed $n$ and $q$ simultaneously changes $qa$ and $\sqrt{na^{3}}$, making it non-obvious which of the two dimensionless interaction parameters is (primarily) responsible for the breakdown of the FT theory. In an attempt to disentangle the two effects, we collect data with many $\{n,q,a\}$ combinations, and group them into sets with (approximately) equal $\sqrt{na^{3}}$, but varying $qa$ values. In Fig. 4(b) we plot $\alpha-\alpha_{\rm FT}$ versus $qa$, with different symbols corresponding to different $\sqrt{na^{3}}$. These measurements strongly suggest that, at least for our range of parameters, the breakdown of the FT theory occurs for $qa\gtrsim 3$, independently of the value of $\sqrt{na^{3}}$.

At $qa\gtrsim 3$, the deviation of our data from the FT theory is captured well by a recent calculation based on the Wilson operator product expansion (OPE)  Hofmann and Zwerger (2016). Assuming $C=(4\pi na)^{2}$, and with the same normalisation as in Eq. (4), $\alpha_{\rm OPE}=qa[2/(1+(qa/2)^{2})-1]$ (see also Beliaev (1958)); in Fig. 4(b) the dashed black line shows $\alpha_{\rm OPE}-\alpha_{\rm FT}$. This theory also allows for self-consistent inclusion of the LHY corrections to $C$, in which case $\alpha_{\rm OPE}$ depends on both $qa$ and $\sqrt{na^{3}}$; we show the LHY-corrected $\alpha_{\rm OPE}$ (dot-dashed black line) only for our largest $\sqrt{na^{3}}$, where it appears to provide a slightly better agreement with the experiments, but this observation is not conclusive (see also Wild et al. (2012)).

Finally, we note that while the OPE theory successfully describes our large-$qa$ measurements, it does not agree with our low-$qa$ data, in particular because it predicts the zero-crossing of $\Delta\omega$ at $qa=2$ instead of $qa=4/\pi$; this is highlighted in the inset of Fig. 4(b). Providing a unified description of quasiparticle resonances in all interaction regimes thus remains a theoretical challenge.

In conclusion, we have probed the quasiparticle excitations in a strongly interacting homogeneous BEC, pushing the experiments far beyond the regime of validity of the Bogoliubov theory. For a range of interaction strengths ($qa\lesssim 3$), our data can still be quantitatively explained in the framework of the Feynman energy relation, by taking into account the short-range two-particle correlations in the spirit introduced by Tan. However, for our most strongly interacting samples this theory also fails, pointing to the need for more sophisticated theoretical approaches. One such approach, based on the Wilson operator product expansion, accounts well for some of our observations, but does not agree with the experiments in all the interaction regimes that we explored.

We thank Sandro Stringari, Wilhelm Zwerger, Johannes Hofmann, Eric Cornell, Nir Davidson and Stefano Giorgini for inspiring discussions. This work was supported by the Royal Society, EPSRC [Grant No. EP/N011759/1], ERC (QBox), AFOSR, and ARO. R.L acknowledges support from the E.U. Marie-Curie program [Grant No. MSCA-IF-2015 704832] and Churchill College, Cambridge. N.N. acknowledges support from Trinity College, Cambridge.