Multichannel Molecular State and Rectified Short-range Boundary Condition for Spin-orbit Coupled Ultracold Fermions Near p-wave Resonances

TL;DR

This paper investigates how spin-orbit coupling influences p-wave interactions in ultracold Fermi gases, revealing the breakdown of conventional boundary conditions and proposing a new boundary condition that accurately predicts molecular states.

Contribution

It introduces a new p-wave boundary condition accounting for spin-orbit coupling effects, improving predictions of molecular states near p-wave resonances.

Findings

SOC significantly alters p-wave boundary conditions

Conventional p-wave BC fails under SOC influence

New boundary condition accurately predicts molecules

Abstract

We study the interplay of spin-orbit coupling (SOC) and strong p-wave interaction to the scattering property of spin-1/2 ultracold Fermi gases. Based on a two-channel square-well potential generating p-wave resonance, we show that the presence of an isotropic SOC, even for its length much longer than the potential range, can greatly modify the p-wave short-range boundary condition(BC). As a result, the conventional p-wave BC cannot predict the induced molecules near p-wave resonance, which can be fully destroyed to vanish due to strong interference between s- and p-wave channels. By analyzing the intrinsic reasons for the breakdown of conventional BC, we propose a new p-wave BC that can excellently reproduce the exact molecule solutions and also equally apply for a wide class of single-particle potentials besides SOC. This work reveals the significant effect of SOC to both the short- and long-range properties of fermions near p-wave resonance, paving the way for future exploring interesting few- and many-body physics in such system.

Equations

$$V(\mathbf{r})=V_p{P}_{11;00}+V_s\theta (r_0-r)P_{00;00}.$$

$$V_{p}=\left(\begin{array}[]{cc}V_{p,o}&\Omega\\ \Omega&V_{p,c}\end{array}\right)\theta(r_{0}-r)+\left(\begin{array}[]{cc}0&0\\ 0&\infty\end{array}\right)\theta(r-r_{0}).$$

$${\psi }_p(r)\to \frac{1}{r^2}+\frac{k^2}{2}+\frac{r}{3}(-\frac{1}{v_p}+\frac{r_p{k}^2}{2})+o(r^3).$$

$$\frac{(r^2{\psi }_p{)}^{\prime \prime \prime }}{r^2{\psi }_p}=2\left(-\frac{1}{v_p}+\frac{r_p{k}^2}{2}\right)+o(r^2),$$

$${\Psi }^{sc}(\mathbf{r})=\int d{\mathbf{r}}^{\prime }⟨\mathbf{r}|G|{\mathbf{r}}^{\prime }⟩⟨{\mathbf{r}}^{\prime }|V|\Psi ⟩.$$

$$⟨\mathbf{r}|G|{\mathbf{r}}^{\prime }⟩=\frac{1}{2}\sum _{\sigma {\sigma }^{\prime }=\pm }\frac{⟨\mathbf{r}|{\mathbf{k}}^{(\sigma )},-{\mathbf{k}}^{({\sigma }^{\prime })}⟩⟨{\mathbf{k}}^{(\sigma )},-{\mathbf{k}}^{({\sigma }^{\prime })}|{\mathbf{r}}^{\prime }⟩}{E-{ϵ}_{\mathbf{k}}^{(\sigma )}-{ϵ}_{-\mathbf{k}}^{({\sigma }^{\prime })}+i{0}^{+}};$$

$$⟨\mathbf{r}|V|\Psi ⟩=F_p(r)Y_{10}({\Omega }_r)|10⟩+F_s(r)Y_{00}({\Omega }_r)|00⟩,(r<r_0)$$

$${\psi }_p=f_{p}A-f_{s}B;{\psi }_s=f_{s}C+f_{p}D.$$

$$A$$

$$B$$

$$C$$

$$D$$

$${\psi }_p$$

$${\eta }_p$$

$${\psi }_s$$

$${\eta }_s$$

$$\frac{(r^2{\psi }_p{)}^{\prime }}{r^2{\psi }_p}=\left(-\frac{1}{v_p}+\frac{r_p{k}^2}{2}\right)r^2+k^{2}r+o(r^4).$$

Full text

The interplay of spin-orbit coupling (SOC) and interaction has generated tremendous research interests in condensed matter physicsreview_1 ; review_2 , while ultracold atomic gases offer an ideal platform for its study giving successful realizations of synthetic SOC using Raman lasersgauge2exp ; fermisocexp1 ; fermisocexp2 ; 2dsoczhang1 ; Wu2015 and highly tunable interactions via Feshbach resonancesChin . Nevertheless, before studying the complex many-body physics the very first question to address is how to model the fundamental two-body interactions. A crucial factor here is the asymptotic behavior of two-body wave function in the short-range limit, called the short-range boundary condition(BC), which is the basis for constructing the Huang-Yang pseudo-potentialsLHY ; Stock05 and also equivalent to the use of renormalized contact modelsreview_Fermi_gas ; Gurarie . In the presence of SOC, studies have shown that the usual s-wave short-range BC, giving the conventional s-wave models, is hardly modified near s-wave resonances given the typical length of realistic SOC much longer than the potential rangeCui ; Yu ; Zhang . Despite the negligible short-range consequence, SOC can greatly change the long-range (low-energy) scattering properties from near the thresholdCui ; Yu ; Blume to intermediate energy regimeBlume ; You ; Greene . Moreover, with conventional s-wave models it has been found that SOC can induce shallow moleculesVijay and universal trimerstrimer_soc ; borromean_soc more easily, and lead to various fascinating many-body phenomena in both bosons and fermions atomic systemssocreview1 ; socreview2 ; socreview4 ; socreview5 ; socreview6 .

Besides s-wave, the p-wave interacting atomic gases have also attracted great attention in recent yearsChin ; thesis , in particular, in view of the very recent explorations of universal properties near p-wave resonanceToronto ; Ueda ; Yu_p ; Zhou . In this work, we study the interplay of SOC and strong p-wave interaction to the short-range and long-range two-body physics. Specifically, we ask the question how would SOC affect the p-wave short-range BC and induce shallow molecules? There have been a few related discussions in literature without full answersCui ; Yu . Addressing this problem will be fundamentally important for future exploring a new set of few- and many-body physics due to the interplay of SOC and high partial-wave scatterings.

We consider two spin-1/2 fermions subject to an isotropic SOC, which is hopefully realizable in future considering a number of proposalsproposal1 ; proposal2 . By adopting a two-channel square-well potential generating the p-wave Feshbach resonanceChin (see Fig.1a), we can exactly pin down the binding energy of induced molecule as well as its multi-channel structures and short-range behaviors. We find that even for the SOC length much longer than the potential range, it can still greatly modify the p-wave short-range BC near p-wave resonances, on contrary to previous findings in the s-wave channel. Consequently, the conventional p-wave BC cannot predict the induced molecules near p-wave resonance, which instead can be fully destroyed to vanish within a visible range of interaction strength. Its breakdown is partly due to the intrinsic sensitivity of p-wave scattering parameters to short-range details, and partly due to the strong interference between s- and p-wave channels via SOC. Based on these observations, we finally propose a new p-wave BC, which excellently reproduces the molecule solution in this case and also equally applies for a wide class of single-particle potentials besides SOC.

We start from a square-well model potential, as depicted in Fig.1a, for two spin-1/2 fermions with relative distance ${\bf r}$:

[TABLE]

Here $P_{L_{r}S;m_{l}m_{s}}$ is the projection operator to two-particle state with relative orbital angular momenta $\{L_{r}m_{l}\}$ and total spin angular momenta $\{Sm_{s}\}$. Explicitly, $|Sm_{s}\rangle$ can be expanded as: $|10\rangle=\frac{|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle}{\sqrt{2}},\ |00\rangle=\frac{|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle}{\sqrt{2}}$. For the p-wave interaction ($L_{r}=1$), we have selected out $m_{l}=0$ channel whose resonance can be well separated from the other channelsChin ; thesis ; to mimic the realistic p-wave Feshbach resonance, we adopt a two-channel potentialChin

[TABLE]

Here $V_{p,o}$ and $V_{p,c}$ are respectively the open- and closed-channel potentials within the range $r_{0}$. $V_{p,o}$ provides the background p-wave volume $v_{bg}=-j_{2}(q_{o}r_{0})/(3j_{0}(q_{o}r_{0}))$ ($q_{o}=\sqrt{-mV_{p,o}}$), and $V_{p,c}$ gives a closed-channel molecule at $\epsilon_{c}$ satisfying $j_{1}(\sqrt{m(\epsilon_{c}-V_{p,c})}r_{0})=0$. Given $\epsilon_{c}$ close to the scattering threshold and an inter-channel coupling $\Omega$, a p-wave resonance can be induced in the open-channel with scattering volume $v_{p}\rightarrow\infty$ and a finite range $r_{p}$. Explicitly, $v_{p}$ and $r_{p}$ are defined through the phase shift expansion in low-energy limit ($k\ll 1/r_{0}$) as $k^{3}\cot\delta_{p}=-1/v_{p}+r_{p}k^{2}/2$, where the phase shift $\delta_{p}$ appears in the open-channel wave function $\psi_{p,o}=j_{1}(kr)-\tan\delta_{p}n_{1}(kr)$ and can be determined by requiring $\psi_{p,c}(r_{0})=0$ and the continuity of $\psi^{\prime}_{p,o}/\psi_{p,o}$ at $r_{0}$ in the current model. In Fig.1b, we plot $v_{p}$ and $r_{p}$ as a function of $\epsilon_{c}$ for a weak $v_{bg}=1.2r_{0}^{3}$ and $\Omega\ll|V_{p,c}-V_{p,o}|$. A resonance of $v_{p}$ occurs at $\epsilon_{c}=-0.37$ with a finite range $r_{p}=-7/r_{o}$.

Importantly, the scattering parameters $v_{p}$ and $r_{p}$ also appear in the asymptotic behavior of $\psi_{p,o}\equiv\psi_{p}$ in the short-range regime $r_{0}\ll r\ll 1/k$:

[TABLE]

Thus $v_{p},r_{p}$ determine the ratio between the coefficients of $r$ and most singular $1/r^{2}$ terms, which sets the conventional short-range BC in p-wave channel:

[TABLE]

here the superscript ′′′ denotes the third derivative in terms of $r$. In fact, the pseudo-potential methodLHY ; Stock05 and p-wave contact modelGurarie all correspond to guaranteeing above short-range BC, regardless of the presence of any external or internal single-particle potentials.

Note that Eq.1 also includes a weak potential $V_{s}$ in s-wave channel at $r<r_{0}$, giving the scattering length $a_{s}/r_{0}=1-\tan(q_{s}r_{0})/(q_{s}r_{0})$ ($q_{s}=\sqrt{-mV_{s}}$) and effective range $r_{s}$. The reason for its inclusion will be explained later.

Now we consider the SOC part. Different from previous studiesCui ; Yu , in this work we consider the single-particle SOC applied only to the region outside the interaction potential, as shown by shaded area in Fig.1a, since the laser-generated SOC in experimentsgauge2exp ; fermisocexp1 ; fermisocexp2 ; 2dsoczhang1 ; Wu2015 can hardly reach the very short-range regime given so deep interaction potentials therein. Thus SOC will not modify the scattering inside the potential ($r<r_{0}$). Nevertheless, we will show below that it does greatly modify the p-wave short-range BC in the regime $r_{0}\ll r\ll 1/k$.

Under the isotropic SOC ($(\lambda/m)\mathbf{k}\cdot\boldsymbol{\sigma}$ with $\boldsymbol{\sigma}$ the Pauli matrix), the single-particle eigen-state at momentum $\mathbf{k}$ has two orthogonal branches $|\mathbf{k}^{(\pm)}\rangle$ with eigen-energies $\epsilon^{(\pm)}_{\mathbf{k}}=(k\pm\lambda)^{2}/(2m)\ (k\equiv|\mathbf{k}|)$. In single-particle level, the isotropic SOC enables the conservation of total angular momentum $\mathbf{j}=\mathbf{l}+\mathbf{s}\ (\mathbf{s}=\frac{1}{2}\boldsymbol{\sigma})$. Consequently in two-body level, the total angular momentum $\mathbf{J}=\mathbf{L}+\mathbf{S}$ is also conserved, and the orbital $\mathbf{L}$ can be reduced to the relative component $\mathbf{L}_{r}$ when we consider the ground state scattering with zero center-of-mass momentumCui . Given the interaction potential (1), we expect the relevant $J$ can be 0 and 2, which can be composed by $S=0,\ L_{r}=0,2$ or $S=1,\ L_{r}=1,3$. Such multi-channel structure is revealed clearly in the scattered wave function studied below.

Based on the Lippmann-Schwinger equation, the scattered wave function ($r>r_{0}$) at energy $E$ reads:

[TABLE]

where the Green function can be expanded as

[TABLE]

and the interaction part can be parametrized as

[TABLE]

with $F_{s},\ F_{p}$ respectively the scattering amplitudes in s- and p-wave channel based on Eq.(1).

Lengthy but straightforward calculation of Eq.5 indeed results in various $|J;L_{r}S\rangle$ states. These states include $|2;11\rangle$, $|2;31\rangle$, $|2;20\rangle$, $|0;11\rangle$, $|0;00\rangle$ generated by p-wave interaction ($F_{p}$ part in Eq.7), and $|0;11\rangle$, $|0;00\rangle$ generated by s-wave interaction ($F_{s}$ part in Eq.7). By analyzing their coefficients, it is found that the dominated $1/r^{L_{r}+1}$ singularity only occur in $L_{r}=0,1$ channelsnote_divergence , which reads $\psi_{p}Y_{10}(\Omega_{r})|10\rangle+\psi_{s}Y_{00}(\Omega_{r})|00\rangle$ with

[TABLE]

Here $f_{s}=\int_{0}^{r_{0}}drr^{2}F_{s}(r)$, $f_{p}=\int_{0}^{r_{0}}drr^{3}F_{s}(r)$, and $A,B,C,D$ are the functions of $r$ and $E^{+}\equiv E+i0^{+}$:

[TABLE]

Note that we have used $\epsilon_{\mathbf{k}}^{(\sigma)}=\epsilon_{-\mathbf{k}}^{(\sigma)}$ to simplify above equations. In small $r$ limit, $A\sim 1/r^{2}$, and $C,D\sim 1/r$. Here $D\sim 1/r$ shows an interesting fact that the p-wave interaction alone ($f_{p}\neq 0,\ f_{s}=0$) can cause short-range singularity in the s-wave channelCui . Thus to ensure a physical $\psi_{s}$ near $r=0$, an s-wave potential has to be activated at short-range, as shown in Fig.1a. In other words, one has to explicitly include s-wave interaction while considering the physics near p-wave resonance. Note that this is in sharp difference to the pure s-wave interaction ($f_{p}=0,\ f_{s}\neq 0$), which does not induce any singularity in p-wave channelCui ; Vijay and a single s-wave model is adequate to describe the physics near s-wave resonance.

Next we study the bound state solution with $E=-\kappa^{2}/m$, whose wave function is fully given by $\Psi^{sc}$ (Eq.5) for $r>r_{0}$. Specifically, $\kappa$ and the ratio $f_{sp}\equiv-if_{s}/f_{p}$ can be exactly determined by requiring the continuity of both $\psi_{s}^{\prime}/\psi_{s}$ and $\psi_{p}^{\prime}/\psi_{p}$ at the potential boundary $r=r_{0}$. These quantities also determine the asymptotic behaviors of $\psi_{s}$ and $\psi_{p}$ (Eq.8) in short-range regime ($r_{0}\ll r\ll 1/\kappa$):

[TABLE]

One can check that in $\lambda=0$ limit, these equations well reproduce the free-space results $\eta_{s}=-\kappa$ and $\eta_{p}=\kappa^{3}/3$. When turn on SOC ($\lambda\neq 0$), under certain limit of $f_{sp}$ they can reproduce the results from individual s- or p-wave BC. Namely, the individual s-wave BC corresponds to equating $\eta_{s}$ with $1/a_{s}-r_{s}\kappa^{2}$ in $f_{sp}\rightarrow\infty$ limitVijay ; Cui ; while the individual p-wave BC corresponds to equating $\eta_{p}$ with $-\frac{1}{3}(\frac{1}{v_{p}}+\frac{r_{p}}{2}\kappa^{2})$ in $f_{sp}\rightarrow 0$ limitCui . Of course one can further improve the theory by relaxing $f_{sp}$ and imposing the s- and p-wave BC simultaneously. We will show below that none of these theories predict correctly the bound states near p-wave resonance.

In Fig.2, we plot the exact solution of $\kappa/\lambda$ across the p-wave resonance (square dots), taking a small SOC strength $\lambda r_{0}=0.1$ and a weak s-wave interaction with $a_{s}=-4.5r_{0}$. It shows two branches of solutions. Far away from p-wave resonance, the two branches follow the predictions from individual s-wave and individual p-wave BC(short-dashed lines), respectively giving the s-wave and p-wave dominated molecules. However, when close to p-wave resonance with $1/(\lambda v_{p}^{1/3})\sim[-3,3]$, the exact solutions no longer follow the predictions from individual s-wave or individual p-wave BC, nor from the combined s-wave and conventional p-wave BC (Eq.4) (blue dashed). Instead, no molecule solution is found in this regime, which can be attributed to the enhanced interference between s- and p-wave channels due to the presence of SOC and strong p-wave interaction.

In the following, we show that the SOC-induced s-p interference (resulting in a finite $f_{sp}$) greatly modify the p-wave asymptotic parameter $\eta_{p}$, such that the usual p-wave BC breaks down even near the p-wave resonance. To see this clearly, in Fig.3 we plot $f_{sp}$ and its associated weight in $\eta_{p}$ ($W_{sp}$, i.e., the weight of the first term in $\eta_{p}$) for one branch of solution shown in Fig.2. We see that as $v_{p}$ is tuned to approach resonance, $|f_{sp}|$ gradually increases from zero to around $0.5$, and $W_{sp}$ increases from zero to nearly 1, suggesting the molecule evolves from the p-wave dominated to s-p strongly interfered regime. In the latter regime, the s-p interference play an essential role in determining the actual $\eta_{p}$, which can be very different from the predictions of usual p-wave BC (Eq.3).

In comparison to the robust s-wave BC near s-wave resonance under SOCCui ; Yu ; Zhang , here the fragile p-wave BC near the p-wave resonance has its intrinsic reasons. This is because the usual p-wave BC (Eq.4) is characterized by the ratio between very singular $1/r^{2}$ and very weak $r$ terms as shown in Eq.3. Such ratio can be very easily destroyed by perturbations outside the potential, since it is related to the third derivative of $\psi_{p}$ (see Eq.4) while any realistic BC can only guarantee the continuity of $\psi_{p}$ and its first derivative across the boundary. This is quite contrary to the s-wave case where the short-range behavior is characterized by the ratio between $1/r$ and constant terms. In this case the continuity of $\psi_{s}$ and its first derivative is adequate to guarantee the continuity of this ratio. Therefore the short-range behavior of $\psi_{s}$ outside the potential ($r>r_{0}$) is universally given by the physics inside the potential ($r<r_{0}$), which is hardly modified by small perturbations such as SOC.

Motivated by above observations, we can set up a new p-wave BC following the persistent continuity of $\psi_{p}^{\prime}/\psi_{p}$ at the short-range boundary. Given the wave function inside the potential is not modified by SOC, the new BC can be formulated using the original scattering parameters $v_{p}$ and $r_{p}$ in free space (see Eq.3). Namely, at the potential boundary where SOC just starts to take effect, we have ($r\ll k^{-1},\ |v_{p}|^{1/3}$)

[TABLE]

Note that this rectified BC reflects the whole structure of $\psi_{p}$ in short-range limit (Eq.3), including the constant term which has been omitted by usual p-wave BC (Eq.4). Because this constant is more dominated than $r$ term in short-range limit, and because the SOC greatly modifies such constant (Eq.9), it is important for its effect to be taken into account. In Fig.2, we plot the re-calculated $\kappa$ (red dashed-dot) using Eq.11 to replace the usual p-wave BC for $\psi_{p}$ at $r=r_{0}$. The obtained results are excellently consistent with the exact solutions across p-wave resonance. We also check its robustness by choosing other boundaries $r(>r_{0})$ in matching Eq.11, and find good consistence with exact solutions for $r$ up to a few times of $r_{0}$. This justifies Eq.11 as the correct p-wave BC in the presence of SOC. In fact, since Eq.11 only relies on the wave function continuity near the potential boundary, it will be generally applicable for any type of SOC or other single-particle potentials, as long as they do not modify the physics inside the short-range potential.

In summary, we have studied the interplay of an isotropic SOC and strong p-wave interaction to both the short-range physics and shallow molecules of two interacting fermions. We find that even for SOC length much longer than the range of interaction potential, it can still induce strong interference between s- and p-wave channels, which leads to the vanishing of molecules near p-wave resonance and the breakdown of usual p-wave short-range BC. The proposed new p-wave BC, which applies for a wide class of single-particle potentials including SOC, will hopefully play more roles in dealing with few- and many-body problems near p-wave resonance.

Finally, since our scheme to solve the two-body problem and the conclusion of s-p interference do not rely on the specific type of SOC, our results will shed light on the molecule formation in current experiments with one or two-dimensional SOCgauge2exp ; fermisocexp1 ; fermisocexp2 ; 2dsoczhang1 ; Wu2015 and near p-wave resonance. In particular, it would be practical to realize SOC in quite a number of two-species fermion systems with p-wave Feshbach resonance, such as the $F=9/2$ 40K Fermi gas with two hyperfine states $|m_{F1}=-9/2;m_{F2}=-5/2\rangle$ at $B_{0}=215$G, $|-3/2;-1/2\rangle$ at $338$G, and $|9/2;5/2\rangle$ at $44$Gthesis . It is interesting to explore in future the multichannel molecules in these systems.

Acknowledgment. I like to thank Doerte Blume, Qingze Guan and Peng Zhang for useful discussions on the few-body physics with spin-orbit coupling during the KITP program ”Universality in Few-Body Systems” in the winter of 2016. This project is supported by the National Natural Science Foundation of China (No.11374177, No.11626436, No. 11421092, No. 11534014), and the National Key Research and Development Program of China (2016YFA0300603).