1. Introduction
In recent decades, ultracold atom gases are used as a wonderful platform for quantum simulations of few-body and many-body physics [1–4]. Thanks to their high degree of tunability, ultracold gases can be used not only to realize the important models of quantum physics, such as the Bose and Fermi Hubbard models [5–9], but also to prepare interesting novel quantum systems which have not been observed or studied in other natural systems, such as momentum space lattices [10] and the four-dimensional Hall system [11–13]. The realization of these system extended the research area of quantum physics.
In 2017, J Jie and one of the authors (PZ) demonstrated that by using the technique of laser modulated magnetic Feshbach resonance (MFR) [14–16], one can realize a new type of pairwise interaction, which depends on the two-body center-of-mass (CoM) momentum P, between ultracold atoms [17]. Explicitly, the two-atom scattering length a(P) is a function of P, the dispersion relation of each single atom is still p2/(2m), with p and m being the one atom momentum and mass, respectively. This CoM momentum dependent interaction (CoMMDI) is novel because for almost all natural systems the two-body interactions are independent of the CoM momentum. Further research shows various phenomena, such as Fulde–Ferrell superfluids and synthesis of majorana mass terms, may be realized via ultracold gases with this interaction [18–23]. Nevertheless, so far the studies are mainly focused on the many-body effects induced by atomic scattering, and the bound states of this system have not been studied systemically.
In this paper we investigate bound states of two ultracold atoms with a CoMMDI for various parameter regions. We find that B < B0, with B and B0 being the magnetic field and the MFR point, respectively, two shallow bound states can appear in our system. That is due to the fact that multi closed-channel bound states are involved in the laser modulated MFR. In other words, for B < B0 the CoMMDI can effectively support a ‘two-component’ shallow dimer or a dimer with pseudo-spin 1/2. Moreover, the dimers in different pseudo-spin states are different functions of the CoM momentum, i.e., there is an effective pseudo-spin-orbital coupling (SOC), and the dimer dispersion curve has two minimum points, or a ‘double-well’ shape in the momentum space. These results show that the atoms with CoMMDI can also be used for the quantum simulations with two-component ultracold molecules, such as the ultracold molecules with SOC.
The remainder of this paper is organized as follows. In section 2 , we introduce our system in detail and show the approach to calculate the energy and lifetime of the two-body bound states. Our results are shown and discussed in section 3 . In section 4 , there is a brief summary.
2. Calculation of the bound state energy
As in [17], we consider two ultracold Fermi atoms with interaction in electronic ground states (S states). As shown in figure 1, we assume that the system is near an MFR. Furthermore, the closed-channel bound state ∣φα⟩ of this MFR is effectively coupled to another bound state ∣φβ⟩ of the same channel. This coupling is induced by two Raman beams α and β propagating in different directions, which directly couples ∣φα⟩ and ∣φβ⟩ to a bound state ∣φe⟩ in an excited channel, i.e. the ‘S + P’ channel with one atom being in the electronic excited state (the P-state), respectively.
Figure 1. A schematic picture of the Raman-laser modulated MFR. |
The Hamiltonian of our system is given by1 ) is defined as
$\begin{eqnarray}\begin{array}{rcl}H & = & \displaystyle \frac{{{\boldsymbol{P}}}^{2}}{2M}+{H}_{\mathrm{MFR}}+{E}_{\beta }| {\phi }_{\beta }\rangle \langle {\phi }_{\beta }| +{E}_{e}| {\phi }_{e}\rangle \langle {\phi }_{e}| \\ & & +\displaystyle \sum _{l=\alpha ,\beta }{{\rm{\Omega }}}_{l}{{\rm{e}}}^{{\rm{i}}({{\boldsymbol{k}}}_{l}\cdot {\boldsymbol{R}}-{\omega }_{l}t)}| {\phi }_{e}\rangle \langle {\phi }_{l}| +{\rm{h}}.\,{\rm{c}}.,\end{array}\end{eqnarray}$
where P (p) is the CoM (relative) momentum of the two atoms, R is the CoM coordinate, M (μ) is the total (reduced) mass, Ej (j = α, β, e) is the energy of the bound state ∣φj⟩, and Ωl (l = α, β) is the Rabi frequency of the Raman-laser l, with corresponding angular frequency ωl and the wave vector is kl. In addition, the term HMFR in equation ( $\begin{eqnarray}\begin{array}{rcl}{H}_{\mathrm{MFR}} & = & \left[\displaystyle \frac{{{\boldsymbol{p}}}^{2}}{2\mu }+{V}_{\mathrm{bg}}(r)\right]| O{\rangle }_{I}\langle O| +{E}_{\alpha }| {\phi }_{\alpha }\rangle \langle {\phi }_{\alpha }| \\ & & +{V}_{\mathrm{hf}}(r)| O{\rangle }_{I}\langle C| +{\rm{h}}.\,{\rm{c}}.,\end{array}\end{eqnarray}$
where r is the relative coordinate of the two atoms, and Vbg is the background potential energy of the open channel of the MFR.In this work we calculate the bound state of these two atoms. As shown in [17], it is convenient to do the calculation in the rotated frame induced by a unitary transformation ${ \mathcal U }$:
$\begin{eqnarray}{ \mathcal U }={{\rm{e}}}^{{\rm{i}}({\omega }_{\alpha }t-{{\boldsymbol{k}}}_{\alpha }\cdot {\boldsymbol{R}})| {\phi }_{e}\rangle \langle {\phi }_{e}| }{{\rm{e}}}^{{\rm{i}}[({\omega }_{\alpha }-{\omega }_{\beta })t-({{\boldsymbol{k}}}_{\alpha }-{{\boldsymbol{k}}}_{\beta })\cdot {\boldsymbol{R}}]| {\phi }_{\beta }\rangle \langle {\phi }_{\beta }| }.\end{eqnarray}$
The transformed Hamiltonian is ${H}_{\mathrm{rot}}={ \mathcal U }H{{ \mathcal U }}^{\dagger }$, which can be written as: $\begin{eqnarray}\begin{array}{rcl}{H}_{\mathrm{rot}} & = & \displaystyle \frac{{{\boldsymbol{P}}}^{2}}{2M}+{H}_{\mathrm{MFR}}+\displaystyle \sum _{l=\alpha ,\beta }{{\rm{\Omega }}}_{l}| {\phi }_{e}\rangle \langle {\phi }_{l}| +{\rm{h}}.\,{\rm{c}}.\\ & & +{{\rm{\Delta }}}_{1{\rm{p}}}({\boldsymbol{P}})| {\phi }_{e}\rangle \langle {\phi }_{e}| +{{\rm{\Delta }}}_{2{\rm{p}}}({\boldsymbol{P}})| {\phi }_{\beta }\rangle \langle {\phi }_{\beta }| ,\end{array}\end{eqnarray}$
where the P-dependent one-photon and two-photon detunings Δ1p(P) and Δ2p(P) are defined as $\begin{eqnarray}{{\rm{\Delta }}}_{1{\rm{p}}}({\boldsymbol{P}})={{\rm{\Delta }}}_{1{\rm{p}}}^{(0)}+\displaystyle \frac{| {{\boldsymbol{k}}}_{\alpha }{| }^{2}}{2M}+\displaystyle \frac{{{\boldsymbol{k}}}_{\alpha }\cdot {\boldsymbol{P}}}{M},\end{eqnarray}$
and $\begin{eqnarray}{{\rm{\Delta }}}_{2{\rm{p}}}({\boldsymbol{P}})={{\rm{\Delta }}}_{2{\rm{p}}}^{(0)}+\displaystyle \frac{| {{\boldsymbol{k}}}_{\alpha }-{{\boldsymbol{k}}}_{\beta }{| }^{2}}{2M}+\displaystyle \frac{({{\boldsymbol{k}}}_{\alpha }-{{\boldsymbol{k}}}_{\beta })\cdot {\boldsymbol{P}}}{M},\end{eqnarray}$
respectively, with ${{\rm{\Delta }}}_{1{\rm{p}}}^{(0)}={E}_{e}-{\omega }_{\alpha }$ and ${{\rm{\Delta }}}_{2{\rm{p}}}^{(0)}={E}_{\beta }\,-({\omega }_{\alpha }-{\omega }_{\beta })$ being the bare one-photon and two-photon detunings, respectively.As proved in our previous work [17], for our system, the energy of a two-atom bound state with CoM momentum P can be expressed as9 ) is defined as
$\begin{eqnarray}{{ \mathcal E }}_{b}=\displaystyle \frac{{{\boldsymbol{P}}}^{2}}{2M}+{E}_{b}({\boldsymbol{P}}),\end{eqnarray}$
where Eb(P) is the ‘CoM-momentum-dependent dimer energy’. We can derive Eb(P) by solving the equation: $\begin{eqnarray}\det [{E}_{b}I-{\rm{\Sigma }}({E}_{b},{\boldsymbol{P}})]=0.\end{eqnarray}$
Here Σ(Eb, P) is the self-energy matrix, and is given by $\begin{eqnarray}\begin{array}{l}{\rm{\Sigma }}(E,{\boldsymbol{P}})\approx \left[\begin{array}{ccc}{E}_{\alpha }+E{{\prime} }_{\alpha }+\chi (E) & {{\rm{\Omega }}}_{\alpha }^{* } & 0\\ {{\rm{\Omega }}}_{\alpha } & {{\rm{\Delta }}}_{1{\rm{p}}}({\boldsymbol{P}})-{\rm{i}}\displaystyle \frac{\gamma }{2} & {{\rm{\Omega }}}_{\beta }^{* }\\ 0 & {{\rm{\Omega }}}_{\beta } & {{\rm{\Delta }}}_{2{\rm{p}}}({\boldsymbol{P}})\end{array}\right],\end{array}\end{eqnarray}$
where $E{{\prime} }_{\alpha }$ and γ are the Lamb shift and the spontaneous decay rate of ∣φe⟩, and satisfy ${E}_{\alpha }+E{{\prime} }_{\alpha }=\delta {\mu }_{B}(B-{B}_{0})$, with δμB being the magnetic moment difference between the channels O and C and B0 being the resonance position. In addition, the function χ(E) in equation ( $\begin{eqnarray}\begin{array}{l}\chi (E)=\sqrt{(-E)}(\delta {\mu }_{B}){a}_{\mathrm{bg}}{{\rm{\Delta }}}_{B}\sqrt{2\mu },\end{array}\end{eqnarray}$
where abg and ΔB are the background scattering length and width of the MFR, respectively.In this work, we derive Eb(P) for ultracold 40K or 6Li atoms, by numerically solving equation (8 ) under the conditions $\mathrm{Re}[{E}_{b}]\lt 0$ and $\mathrm{Im}[{E}_{b}]\lt 0$. The binding energy of the bound state is $| \mathrm{Re}[{E}_{b}({\boldsymbol{P}})]| $, and the lifetime τ of the bound state is given by9 ) of the self-energy matrix Σ(E, P) is not applicable when $\mathrm{Re}[| {E}_{b}| ]$ is comparable or larger than the van der Waals energy of the system (of the order of (2π)10 MHz and (2π)100 MHz for K and Li atoms, respectively). Thus, in the following calculations we ignore the ‘unphysical’ solutions of equation (8 ) with $\mathrm{Re}[| {E}_{b}| ]\gt (2\pi )10$ MHz. In our calculations for the 40K or 6Li atoms, for each group of parameters we always find that there are two and one shallow dimers for B < B0 and B > B0, respectively. In the following section we introduce the properties of these dimers in detail.
$\begin{eqnarray}\begin{array}{l}\tau =-\displaystyle \frac{1}{\mathrm{Im}[{E}_{b}({\boldsymbol{P}})]}.\end{array}\end{eqnarray}$
Notice that the expression (3. Results and analysis
In this section, we show the energy and lifetime of dimers given by our calculations, for ultracold 40K or 6Li atoms, and investigate the dispersion relation of these dimers for various laser beams and magnetic fields.
3.1. Ultracold 40K atoms
We first study the bound state of two ultracold 40K atoms in the lowest two hyperfine states ∣F = 9/2, mF = − 9/2⟩ and ∣F = 9/2, mF = − 7/2⟩, respectively. As in [17], we consider the cases of the Raman-laser modulated MFR with B0 = 202.2 Gauss (G), ΔB = 8G, abg = 174a0, and δμB = 1.68μB, with a0 and μB being the Bohr’s radius and Bohr’s magneton, respectively.
In figure 2, we show the energy $\mathrm{Re}[{E}_{b}]$ and lifetime of the shallow bound states, as functions of the magnetic field, for the cases with typical parameters of Raman lasers and fixed CoM momentum P = 0. It is shown that, as mentioned above, for B < B0 there are two shallow bound sates b1 and b2, while for B > B0 there is only one bound state b2. The state b1 with the lowest binding energy and long lifetime is mainly the Feshbach bound of the MFR, and can appear in the absence of the Raman-laser beams. The state b2 with large binding energy is induced by the Raman lasers. The lifetime of b2 is significantly shorter, because b2 is a dressed state of the bound states ∣φα,β⟩ in the electronic ground manifold and the bound state ∣φe⟩ in the excited channel, which has a very short lifetime.
Figure 2. The dimer energy $\mathrm{Re}[{E}_{b}]$ (a) and lifetime τ (b) of bound states of two 40K atoms with CoM momentum P = 0, Ωα = (2π)10 MHz, Ωβ = (2π)30 MHz, ${{\rm{\Delta }}}_{1{\rm{p}}}^{(0)}+\tfrac{| {{\boldsymbol{k}}}_{\alpha }{| }^{2}}{2M}=(2\pi )400\,\mathrm{MHz}$, ${{\rm{\Delta }}}_{2{\rm{p}}}^{(0)}+\tfrac{| {{\boldsymbol{k}}}_{\alpha }-{{\boldsymbol{k}}}_{\beta }{| }^{2}}{2M}=-(2\pi )2.1\,\times \,{10}^{4}\mathrm{Hz}$, γ = (2π)30 MHz, ωα ≈ ωβ = (2π)3.9 × 1014 Hz, and kα = − kβ. Other parameters are shown in the main text. |
In figure 3 we illustrate the dispersion relation and lifetime of the bound states of two 40K atoms, for three groups of typical parameters with B < B0. As shown above in equations (4 )–(6 ), the Raman beams influence the Hamiltonian via the vector kα − kβ. Without a loss of generality, here we assume kα − kβ is along the z-direction. As a result, the dimer depends on only the CoM momentum along the z-direction (i.e. Pz), and the dispersion relation is described by
$\begin{eqnarray}\begin{array}{l}{{ \mathcal E }}_{b}({P}_{z})\equiv \displaystyle \frac{{{P}_{z}}^{2}}{2M}+\mathrm{Re}[{E}_{b}({P}_{z}{{\boldsymbol{e}}}_{z})].\end{array}\end{eqnarray}$
Furthermore, the effect of the Raman beams is significant when ∣kα − kβ∣ is large. Due to this fact, we consider the case where the Raman beam α(β) is being propagating along the z- (−z-)direction, so that the value of ∣kα − kβ∣ is maximum for fixed frequencies of the Raman beam. When the Raman beams are not propagating along opposite directions, the results would be quantitatively different because the effect of the Raman beams would be weaker.
Figure 3. The dispersion relation ${{ \mathcal E }}_{b}({P}_{z})\equiv \tfrac{{{P}_{z}}^{2}}{2M}+\mathrm{Re}[{E}_{b}({P}_{z}{{\boldsymbol{e}}}_{z})]$ and lifetime of bound states of two ultracold 40K atoms with B < B0. We show the results for the case with Ωα = (2π)0.05 MHz, B − B0 = − 0.9935ΔB (a) and (b), the case with Ωα = (2π)1.3MHz, B − B0 = − 0.9935ΔB (c)and (d), and the case with Ωα = (2π)3MHz, B − B0 = − 0.98ΔB(e) and (f). The unit kF of Pz is the Fermi momentum of two-component 40K gases with density n = 2.545 × 1019m−3 i.e. kF = 9.1 × 106 m−1. Other parameters are the same as the ones of figure 2. |
As shown in figure 3, the dispersion curve of the two bound states depends on the parameters of the Raman lasers. Since the two bound states can be understood as two internal states of a single dimer (molecule), this result implies that an SOC of the molecule is induced by the Raman-laser beams applied to modulate the MFR. Explicitly, in the case of figure 3 (a) and (b), the dispersion curve of the two bound states are shifted parabolic curves with an avoided crossing. These dispersion curves are very similar to the ones of a single two-component ultracold atom with a synthetic one-dimensional SOC, which is induced by two Raman beams with relatively small Rabi frequency [24]. In addition, in the case of figure 3 (c-f), there is no avoided crossing between the dispersion curve of the two bound states, which are similar to the ones of ultracold atoms with SOC generated by strong Raman beams with large Rabi frequency. Therefore, our system may be used to realize ultracold gases of molecules with various types of SOC.
3.2. Ultracold 6Li atoms
Now we consider the bound states of two ultracold 6Li atoms in the lowest two hyperfine states ∣F = 1/2, mF = 1/2⟩ and ∣F = 1/2, mF = − 1/2⟩, respectively. We focus on the cases of the Raman-laser modulated MFR with B0 = 543.25 Gauss (G), ΔB = 0.1G, abg = 60a0, and δμB = 2μB, which is much narrower than the above one of 40K atoms. The Raman modulation of this MFR has been studied experimentally [25].
As in the above subsection, in figure 4 we illustrate the dimer energy $\mathrm{Re}[{E}_{b}]$ and lifetime τ as functions of the magnetic field for a typical case. Similar to the systems of 40K atoms, there are two and one shallow bound states for B < B0 and B > B0, respectively. Moreover, in figure 5 we show the dispersion relations and lifetime of the 6Li two-body bound states for two typical cases with B < B0. Similar to the dimers of 40K atoms, the dispersion curves of these two bound states may either have or not have an avoided crossing.4(4 Notice that in figure 3 (b) the lifetimes shown exhibit a sudden increasing/decreasing around the avoided crossing point of the dispersion curves, while in figure 5(b) the lifetimes show continuous change. This can be explained as follows. The Rabi frequency Ωα of figure 3 (b) is small, and thus the expressions of the bound states (especially the probability of the excited state ∣φe⟩ with respect to each bound state) changes rapidly with Pz. Since the bound state lifetime is mainly determined by the corresponding probability of ∣φe⟩, the variation of the lifetimes with Pz of figure 3 (b) are also rapid. Our calculations show that for the system of figure 3 (b), when Ωα is increased, the lifetime can also show a continuous change with Pz, which is similar to the one of figure 5(b).) This yields that by tuning the Raman-laser and magnetic field one can manipulate the effective SOC experienced by the 6Li dimer.
Figure 4. The dimer energy $\mathrm{Re}[{E}_{b}]$ (a) and lifetime τ (b) of bound states of two 6Li atoms with CoM momentum P = 0, Ωα = (2π)10 MHz, Ωβ = (2π)30 MHz, ${{\rm{\Delta }}}_{1p}^{(0)}+\tfrac{| {{\boldsymbol{k}}}_{\alpha }{| }^{2}}{2M}=(2\pi )400\,\mathrm{MHz}$, ${{\rm{\Delta }}}_{2p}^{(0)}+\tfrac{| {{\boldsymbol{k}}}_{\alpha }-{{\boldsymbol{k}}}_{\beta }{| }^{2}}{2M}=0\,\mathrm{Hz}$, ωα ≈ ωβ = (2π) × 3.9 × 1014 Hz, γ = 2π × 30 MHz, and kα = − kβ. Other parameters are shown in the main text. |
Figure 5. The dispersion relation ${{ \mathcal E }}_{b}({P}_{z})\equiv \tfrac{{{P}_{z}}^{2}}{2M}+\mathrm{Re}[{E}_{b}({P}_{z}{{\boldsymbol{e}}}_{z})]$ and lifetime of bound states of two ultracold 6Li atoms with B < B0. In (a) and (b) we show the results for the case with Ωα = (2π)0.1 MHz, Ωβ = (2π)9.685 MHz, B − B0 = − 1.28ΔB, in (c) and (d) we show the ones of the case with Ωα = (2π)6.9 MHz, Ωβ = (2π)22.6 MHz, B − B0 = − 0.888ΔB. The unit kF of Pz is the Fermi momentum of two-component 6Li gases with density n = 2.545 × 1019m−3, i.e. kF = 3.5 × 106 m−1. Other parameters are the same as the ones of figure 4. |
4. Summary and discussion
In this work we calculate the energy and lifetime of the bound states of two ultracold atoms under a Raman-laser modulated MFR proposed in [17]. The results for ultracold 6Li and 40K atoms are investigated in detail. Our results show that for B < B0 two bound states can appear in our system. As a result, the two-atom dimer behaves as a two-component molecule with effective SOC which can be manipulated by the Raman-laser and magnetic fields. These results imply that one may use our system to realize the ultracold gases of molecules with SOC.
However, it should be noticed that for the systems of 6Li and 40K atoms studied above, the bound state lifetime is as short as μs in many cases. This fact, which is due to the large spontaneous decay rate γ, may strongly restrict the application of these bound states. One possible approach to solve this problem is to use MFR between an alkaline atom and an alkaline-earth (like) atom in the 1S0 state [26], and modulate this MFR with Raman beams coupling the atoms to an excited bound state with the alkaline-earth (like) atom being in the 3P1 state. Since the linewidth of the 3P1 state is only of the order of kHz, the spontaneous decay rate of this excited bound state may be much shorter than one of the two alkaline atoms.