1. Introduction
In recent years, the ultracold gases of alkaline-earth (like) atoms attracted much interest from both theorists and experimentalists [1, 2]. These atoms can be prepared in not only the electronic-orbit ground state (the 1S0 state) but also some long-lived electronic-orbit excited states (the 3P0 and 3P2 states). In addition, there is a spin-exchange interaction between two fermionic alkaline-earth (like) atoms in 1S0 and 3P0 states, respectively. Explicitly, the nuclear-spin states of these two atoms can be exchanged during collision (figure 1(a)) [3–18]. Therefore, the ultracold alkaline-earth (like) atoms can be used in quantum simulations for many-body physics related to the spin-exchange processes [19–28], e.g. the Kondo effects. To implement these quantum simulations, it is important to control the spin-exchange interaction between two atoms in 1S0 and 3P0 states, respectively. People have proposed various approaches to control this interaction with confinement potential or laser beams and experimentally realized the former in a quasi-(1+0) dimensional system of ultracold 173Yb atoms [13–21].
Figure 1. (a) A schematic of spin-exchange scattering process of two alkaline-earth (like) atoms in 1S0 and 3P0 states, respectively, with different nuclear-spin states. (b) A schematic of the laser-induced nuclear-spin dependent AC-Stark shifts (i.e. the EZEs) of 3P0 states of an 173Yb atom. A detailed discussion is given in appendix C of [19]. |
In 2018, an optical-control approach for the above spin-exchange interaction is proposed by Kan$\acute{{\rm{a}}}$sz-Nagy et al for 173Yb atoms [19], which has relatively large natural scattering length. In this scheme, a single laser beam is applied to far-off-resonantly couple the 3P0 states to the 3D1 states. This beam induces a nuclear-spin-dependent AC-Stark shifts for the 3P0 states (figure 1(b)), which are proportional to the laser intensity and can be regarded as the effective Zeeman energies (EZEs) of a synthetic magnetic field. These EZEs can couple the scattering channels with respect to nuclear-spin singlet and triplet states. In this method, the laser intensity is periodically modulated, and thus the laser-induced EZEs for 3P0 are ‘shaking'. By changing the shaking amplitude and frequency, one can tune the effective inter-atomic interaction, i.e. realize a Floquet engineering.
Explicitly, in the presence of the shaking EZEs, there is a rotated frame where the free Hamiltonian of each atom is time-independent and the two-body interaction potential is a periodical function of time, which depends on the shaking of the EZEs. Moreover, in [19] the effective inter-atomic interaction is approximated to be the time-averaged value of this explicit periodical interaction potential.
In this work we investigate the system of this scheme, and try to answer the following two questions:
To this end, we calculate the inter-atomic zero-energy scattering amplitudes corresponding to the explicit time-dependent interaction potential for the 173Yb atoms, via the Floquet scattering approach. We answer question (i) by comparing the results with the scattering amplitudes given by the time-averaged interaction potential and answer question (ii) by deriving the two-body loss rate from the imaginary part of the zero-energy scattering amplitudes.
| i | (i)How good is the above ‘time-averaging approximation'for the effective inter-atomic interaction? |
| ii | (ii)How serious is the two-body loss (i.e. the heating effect) induced by the shaking of the laser intensity? |
We study the typical cases with the shaking angular frequency λ ∼ kHz. We find that for these cases the time-averaging approximation is applicable only when the shaking amplitude of the EZE is small enough. In addition, the two-body loss rate K2 increases with δ0. when the time-averaging approximation is not applicable we have K2 ≳ 10−10 cm3 · s−1, which yields that for ultracold gases with typical density (1013–1014)/cm3 the lifetime can be decreased to the order of (0.1–1) ms.
Our results yield that, in the quantum simulations for a closed system (e.g. the quantum simulations for Kondo physics), the shaking amplitude δ0 should be smaller enough so that the two-body loss rate is low enough. In this parameter region, the time-averaging approximation is usually applicable. On the other hand, since the two-body loss rate can be controlled by the amplitude and frequency of the shaking field, this system may be used for the studies of open quantum many-body systems with atom-number dissipation.
The remainder of this paper is organized as follows. In section 2 , we introduce the principle of the control for the interaction between alkaline-earth (like) atoms in 1S0 and 3P0, respectively, with shaking EZEs. In section 3 we show the Floquet scattering approach for the calculation of inter-atomic zero-energy scattering amplitude. Our results are discussed in section 4 and a brief summary is given in section 5 .
2. Control of inter-atomic interaction with a shaking laser beam
In this section, we introduce the interaction-manipulation scheme proposed in [19].
2.1. System and inter-atomic interaction
As shown in figure 1(a), we consider two ultracold 173Yb atoms in the 1S0- (g-) and 3P0 (e-) states, respectively. In the two-body problem, these two atoms can be considered as two distinguishable atoms, i.e. the g-atom and e-atom. We further assume that the nuclear-spin state of each atom can be either ↑ or ↓, with magnetic quantum numbers ${m}_{F}^{(\uparrow )}$ or ${m}_{F}^{(\downarrow )}$, respectively, while the nuclear-spin states of the g- and e-atoms are different. Therefore, for our system, the two-atom internal state could be either
$\begin{eqnarray}| \alpha \rangle \equiv | \downarrow {\rangle }_{g}| \uparrow {\rangle }_{e},\end{eqnarray}$
or $\begin{eqnarray}| \beta \rangle \equiv | \uparrow {\rangle }_{g}| \downarrow {\rangle }_{e}.\end{eqnarray}$
Furthermore, in the presence of a natural magnetic field B and a laser beam which can induce nuclear-spin dependent AC-Stark shifts (i.e. the EZEs) for the 3P0 states (figure 1(b)). Therefore, the total Zeeman energies Eα(β) of the state ∣α(β)⟩ is the summation of the one given by the natural magnetic field and the laser-induced EZE, and thus the Zeeman-energy difference between these two states is
$\begin{eqnarray}\begin{array}{rcl}\delta & \equiv & {E}_{\beta }-{E}_{\alpha }\\ & = & {\delta }_{B}+{\delta }_{L},\end{array}\end{eqnarray}$
where δB is the difference of the Zeeman energies induced by the real magnetic field for states ∣β⟩ and ∣α⟩, and δL is the EZE difference between these two states. Explicitly, we have $\begin{eqnarray}{\delta }_{B}=B\left[{m}_{I}^{(\downarrow )}-{m}_{I}^{(\uparrow )}\right]({\mu }_{e}-{\mu }_{g});\end{eqnarray}$
$\begin{eqnarray}{\delta }_{L}={{\rm{\Delta }}}_{\downarrow }^{\mathrm{AC}}-{{\rm{\Delta }}}_{\uparrow }^{\mathrm{AC}},\end{eqnarray}$
where μe(g) is the nuclear magnetic moment of the e- (g-) atom and ${{\rm{\Delta }}}_{\uparrow (\downarrow )}^{\mathrm{AC}}$ is the laser-induced AC-Stark shift of the 3P0 state with a magnetic quantum number ${m}_{I}^{(\uparrow )}({m}_{I}^{(\downarrow )})$.Now we consider the inter-atomic interaction of these two atoms. The bare interaction of these two atoms is diagonal on the basis of nuclear-spin singlet and triplet states, i.e.
$\begin{eqnarray}| \pm \rangle =\displaystyle \frac{1}{\sqrt{2}}\left(| \beta \rangle \mp | \alpha \rangle \right),\end{eqnarray}$
and thus can be expressed as $\begin{eqnarray}\hat{V}(r)={V}_{+}(r)| +\rangle \langle +| +{V}_{-}(r)| -\rangle \langle -| ,\end{eqnarray}$
where r ≡ ∣r∣ with r is the relative position of these two atoms, and V±(r) is the interaction potential corresponding to the states ∣±⟩. In this work, we consider the low-energy cases where both the relative kinetic energy of the two atoms and the amplitude and shaking frequency of the Zeeman energy gap δ are much smaller than the characteristic energy (the van der Waals energy) of the inter-atomic interaction potential5 (5 This low-energy condition is satisfied by the current experiments. For instance, for ultracold gases of 173Yb atoms EvdW is about ℏ(2π)3 MHz, while in the experiments the two-body relative kinetic energy is on the order of ℏ(2π)104 Hz, and the energy gap δ is also of this order when the magnetic field B is below 100 G). Therefore, we model the interaction potential $\hat{V}(r)$ with the energy-independent Huang-Yang pseudopotential (HYP), i.e. we have (ℏ = m = 1, with m being the single-atom mass) $\begin{eqnarray}{V}_{\pm }(r)=4\pi {a}_{\pm }\delta ({\boldsymbol{r}})\displaystyle \frac{\partial }{\partial r}\left(r\cdot \right),\end{eqnarray}$
where a±are the scattering lengths of the two atoms in states ∣±⟩. In our calculation we take a+ = 1900a0 and a− = 200a0, which are approximations of the theoretical computations or the experimentally measured values [4–6].According to the above discussions, the total Hamiltonian for the relative motion of these two atoms is given by
$\begin{eqnarray}\hat{H}=-{{\rm{\nabla }}}_{{\boldsymbol{r}}}^{2}+\delta | \beta \rangle \langle \beta | +\hat{V}(r).\end{eqnarray}$
2.2. Shaking of δ and rotated frame
In this scheme, the intensity of the laser beam is periodically modulated, and thus the EZE difference δL of equation (5 ) can be expressed as
$\begin{eqnarray}{\delta }_{L}(t)={\delta }_{0}\left[1+\cos \left(\lambda t\right)\right],\end{eqnarray}$
with δ0 > 0 and λ > 0 being the shaking amplitude and frequency, respectively. In addition, we further assume that the real magnetic field is tuned so that the difference δB of the real Zeeman energies of states ∣α⟩ and ∣β⟩ satisfies $\begin{eqnarray}{\delta }_{B}=-{\delta }_{0}.\end{eqnarray}$
Thus, the total energy gap δ between these two states is $\begin{eqnarray}\delta (t)={\delta }_{B}+{\delta }_{L}(t)={\delta }_{0}\cos (\lambda t).\end{eqnarray}$
Therefore, in the Schrödinger picture, the time-dependent Hamiltonian H of our two-body problem is given by equation (9 ), with δ and $\hat{V}(r)$ being given by equation (12 ) and equations (7 ), (8 ), respectively. Nevertheless, as [19], we do our calculation in the rotated frame induced by the unitary transformation
$\begin{eqnarray}{\hat{U}}_{R}(t)=\exp \left\{{\rm{i}}{\int }_{0}^{t}\delta (t^{\prime} )| \beta \rangle \langle \beta | {\rm{d}}t^{\prime} \right\}.\end{eqnarray}$
Explicitly, the state ∣ψ(t)⟩(R) in the rotated frame is related to the state ∣ψ⟩(S) in the Schr$\ddot{{\rm{o}}}$dinger picture via $\begin{eqnarray}| \psi (t){\rangle }^{(R)}={\hat{U}}_{R}(t)| \psi (t){\rangle }^{(S)}.\end{eqnarray}$
Direct calculation shows that in this frame the evolution of ∣ψ(t)⟩(R) satisfies $\begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}| \psi (t){\rangle }^{(R)}={\hat{H}}^{(R)}(t)| \psi (t){\rangle }^{(R)},\end{eqnarray}$
where the Hamiltonian ${\hat{H}}^{(R)}(t)$ in the rotated frame is given by $\begin{eqnarray}{\hat{H}}^{(R)}=-{{\rm{\nabla }}}_{{\boldsymbol{r}}}^{2}+{\hat{V}}_{R}(t),\end{eqnarray}$
with $\begin{eqnarray}{\hat{V}}_{R}(t)=4\pi {\hat{A}}_{R}(t)\delta ({\boldsymbol{r}})\displaystyle \frac{\partial }{\partial r}\left(r\cdot \right).\end{eqnarray}$
Here the operator ${\hat{A}}_{R}(t)$ is defined as $\begin{eqnarray}\begin{array}{rcl}{\hat{A}}_{R}(t) & = & {a}_{e}\left(| \alpha \rangle \langle \alpha | +| \beta \rangle \langle \beta | \right)\\ & & +{a}_{\mathrm{ie}}| \beta \rangle \langle \alpha | \left[\displaystyle \sum _{n\in {\mathbb{Z}}}{J}_{n}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right){{\rm{e}}}^{{\rm{i}}n\lambda t}\right]+{\rm{h}}.{\rm{c}}.,\end{array}\end{eqnarray}$
with $\begin{eqnarray}{a}_{e}=\displaystyle \frac{{a}_{-}+{a}_{+}}{2};\quad {a}_{\mathrm{ie}}=\displaystyle \frac{{a}_{-}-{a}_{+}}{2},\end{eqnarray}$
and Jn(z) being the nth order first-type Bessel function. Notice that in the rotated frame the two-body spin states before and after spin-exchanging (e.g. ∣α⟩ and ∣β⟩) are degenerate when the two atoms are far away with each other.In the rotated frame the inter-atomic interaction is described by the time-dependent two-component pseudopotential ${\hat{V}}_{R}(t)$, which depends on the amplitude δ0 and the shaking frequency λ of the laser beam. Therefore, one can control the inter-atomic interaction via this laser beam.
2.3. Time-averaging approximation
Nevertheless, it is not easy to directly use the pseudopotential ${\hat{V}}_{R}(t)$ in the many-body calculations. Therefore, it would be useful if ${\hat{V}}_{R}(t)$ can be approximated by an effective interaction in the rotated frame. In [19] the authors use the time average of ${\hat{V}}_{R}(t)$ as the effective potential, which can be denoted as ${\hat{V}}^{(\mathrm{ave})}$ and is given by
$\begin{eqnarray}\begin{array}{rcl}{\hat{V}}^{(\mathrm{ave})} & \equiv & \displaystyle \frac{\lambda }{2\pi }{\displaystyle \int }_{0}^{\tfrac{\lambda }{2\pi }}{\hat{V}}_{R}(t){\rm{d}}{t}\\ & = & 4\pi \displaystyle \sum _{l,j=\alpha ,\beta }{\bar{a}}_{{lj}}| l\rangle \langle j| \delta ({\boldsymbol{r}})\displaystyle \frac{\partial }{\partial r}\left(r\cdot \right),\end{array}\end{eqnarray}$
with $\begin{eqnarray}{\bar{a}}_{\alpha \alpha }={\bar{a}}_{\beta \beta }={a}_{e};\end{eqnarray}$
$\begin{eqnarray}{\bar{a}}_{\alpha \beta }={\bar{a}}_{\beta \alpha }={a}_{\mathrm{ie}}{J}_{0}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right).\end{eqnarray}$
3. Floquet scattering
In this section, we show our approach for the calculation of the inter-atomic zero-energy scattering amplitude and the two-body loss rate. To perform these calculations, we require to solve the scattering problem with respect to the Hamiltonian ${\hat{H}}^{(R)}$ of equation (16 ). Since the interaction potential ${\hat{V}}_{R}(t)$ of this Hamiltonian is a periodic function of time, we use Floquet scattering theory with the formalism of Sykes, Landa, and Petrov in [29].
3.1. Scattering with incident channel ∣α⟩
We first consider the scattering of the g-atom with nuclear spin ↓ and the e-atom with nuclear-spin ↑, i.e. the scattering process incident from spin channel ∣α⟩. This scattering is described by the Floquet scattering wave function ∣$\Psi$(α)(r, t)⟩ which satisfies
$\begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}| {{\rm{\Psi }}}^{(\alpha )}({\boldsymbol{r}},t)\rangle ={\hat{H}}^{(R)}| {{\rm{\Psi }}}^{(\alpha )}({\boldsymbol{r}},t)\rangle ,\end{eqnarray}$
and the out-going boundary condition in the limit r → ∞, and thus can be expressed as: $\begin{eqnarray}| {{\rm{\Psi }}}^{(\alpha )}({\boldsymbol{r}},t)\rangle =\displaystyle \frac{1}{{\left(2\pi \right)}^{\tfrac{3}{2}}}\left\{| \alpha \rangle +\displaystyle \frac{{f}_{\alpha \leftarrow \alpha }}{r}| \alpha \rangle +\displaystyle \frac{{f}_{\beta \leftarrow \alpha }}{r}| \beta \rangle +\sum _{n\ne 0}\displaystyle \frac{{{\rm{e}}}^{\left({\rm{i}}{k}_{n}r-{\rm{i}}{\omega }_{n}t\right)}}{r}\left[{g}_{\alpha \leftarrow \alpha }^{(n)}| \alpha \rangle +{g}_{\beta \leftarrow \alpha }^{(n)}| \beta \rangle \right]\right\},\end{eqnarray}$
where r = ∣r∣ and $\begin{eqnarray}{\omega }_{n}=n\lambda ;\qquad {k}_{n}=\left\{\begin{array}{l}\sqrt{| {\omega }_{n}| },\ (\mathrm{for}\ n\gt 0);\\ {\rm{i}}\sqrt{| {\omega }_{n}| }\ \ (\mathrm{for}\ n\lt 0).\end{array}\right.\end{eqnarray}$
Here ${\left(2\pi \right)}^{-\tfrac{3}{2}}| \alpha \rangle $ is the zero-momentum incident wave, fα←α is the elastic scattering amplitude of spin channel ∣α⟩, and fβ←α is the spin-exchange amplitude, i.e. the scattering amplitude for the process from the zero-momentum state ∣α⟩ to ∣β⟩. In addition, ${g}_{\alpha (\beta )\leftarrow \alpha }^{(n)}$ (n > 0) are the scattering amplitudes of the out-going states in other Floquet bands, which is related to the two-body loss effect induced by the shaking of δ, as discussed in the next subsection.Substituting the wave function ∣$\Psi$(α)(r, t)⟩ into the Schr$\ddot{{\rm{o}}}$dinger equation (15 ), and using the relation ${{\rm{\nabla }}}_{{\boldsymbol{r}}}^{2}(\tfrac{1}{r})=-4\pi \delta ({\boldsymbol{r}})$, we can find that $\Psi$(α)(r, t)⟩ should also satisfy the Bethe-Peierls boundary condition (BPC)18 ). Substituting equation (24 ) into (26 ), we derive the linear algebraic equations for the amplitudes fα(β)←α and ${g}_{\alpha (\beta )\leftarrow \alpha }^{(n)}$:
$\begin{eqnarray}\mathop{\mathrm{lim}}\limits_{r\to 0}\left\{\displaystyle \frac{\partial }{\partial r}\left[r| {{\rm{\Psi }}}^{(\alpha )}({\boldsymbol{r}},t)\rangle \right]\right\}=-{\hat{A}}_{R}{\left(t\right)}^{-1}\left\{\mathop{\mathrm{lim}}\limits_{r\to 0}\left[r| {{\rm{\Psi }}}^{(\alpha )}({\boldsymbol{r}},t)\rangle \right]\right\}.\end{eqnarray}$
Here ${\hat{A}}_{R}{\left(t\right)}^{-1}$ is the inverse operator of ‘scattering length matrix'${\hat{A}}_{R}(t)$ defined in equation ( $\begin{eqnarray}{f}_{\alpha \leftarrow \alpha }=-{a}_{e}-{\rm{i}}{a}_{\mathrm{ie}}\sum _{n\ne 0}\left[{k}_{n}{g}_{\beta \leftarrow \alpha }^{(n)}{J}_{-n}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)\right];\end{eqnarray}$
$\begin{eqnarray}{f}_{\beta \leftarrow \alpha }=-{a}_{\mathrm{ie}}{J}_{0}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)-{\rm{i}}{a}_{\mathrm{ie}}\sum _{n\ne 0}\left[{k}_{n}{g}_{\alpha \leftarrow \alpha }^{(n)}{J}_{n}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)\right];\end{eqnarray}$
$\begin{eqnarray}{g}_{\alpha \leftarrow \alpha }^{(n)}=-{\rm{i}}{a}_{e}{k}_{n}{g}_{\alpha \leftarrow \alpha }^{(n)}-{\rm{i}}{a}_{\mathrm{ie}}\sum _{m\ne n}\left[{k}_{n-m}{g}_{\beta \leftarrow \alpha }^{(n-m)}{J}_{m}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)\right];\end{eqnarray}$
$\begin{eqnarray}{g}_{\beta \leftarrow \alpha }^{(n)}=-{a}_{\mathrm{ie}}{J}_{-n}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)-{\rm{i}}{a}_{e}{k}_{n}{g}_{\beta \leftarrow \alpha }^{(n)}-{\rm{i}}{a}_{\mathrm{ie}}\sum _{m\ne -n}\left[{k}_{n+m}{g}_{\alpha \leftarrow \alpha }^{(n+m)}{J}_{m}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)\right].\end{eqnarray}$
Numerically solving equations (27 )–(29 ), we can derive elastic scattering amplitude fα←α and the spin-exchanging amplitude fβ←α.
3.2. Scattering with incident channel ∣β⟩
Similarly, we can also consider the zero-energy scattering of the g-atom with nuclear spin ↑ and the e-atom with nuclear-spin ↓, i.e. the scattering process incident from spin channel ∣β⟩. This process is described by the Floquet scattering wave function ∣$\Psi$(β)(r, t)⟩ which satisfies
$\begin{eqnarray}{\rm{i}}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}| {{\rm{\Psi }}}^{(\beta )}({\boldsymbol{r}},t)\rangle ={\hat{H}}^{(R)}| {{\rm{\Psi }}}^{(\beta )}({\boldsymbol{r}},t)\rangle ,\end{eqnarray}$
and the out-going boundary condition in the limit r → ∞. Thus, this wave function can be expressed as: $\begin{eqnarray}| {{\rm{\Psi }}}^{(\beta )}({\boldsymbol{r}},t)\rangle =\displaystyle \frac{1}{{\left(2\pi \right)}^{\tfrac{3}{2}}}\left\{| \beta \rangle +\displaystyle \frac{{f}_{\alpha \leftarrow \beta }}{r}| \alpha \rangle +\displaystyle \frac{{f}_{\beta \leftarrow \beta }}{r}| \beta \rangle +\sum _{n\ne 0}\displaystyle \frac{{{\rm{e}}}^{\left({\rm{i}}{k}_{n}r-{\rm{i}}{\omega }_{n}t\right)}}{r}\left[{g}_{\alpha \leftarrow \beta }^{(n)}| \alpha \rangle +{g}_{\beta \leftarrow \beta }^{(n)}| \beta \rangle \right]\right\}.\end{eqnarray}$
Similar to above, here ${\left(2\pi \right)}^{-\tfrac{3}{2}}| \beta \rangle $ is the incident wave, fβ←β and fα←β are the is the elastic and the spin-exchanging scattering amplitudes, respectively, while ${g}_{\beta (\alpha )\leftarrow \beta }^{(n)}$ are the inter-Floquet-band scattering amplitudes corresponding to the two-body loss. Using the BPC approach in the above subsection, we can also derive the linear algebraic equations for the coefficients fβ(α)←β and ${g}_{\beta (\alpha )\leftarrow \beta }^{(n)}$: $\begin{eqnarray}{f}_{\alpha \leftarrow \beta }=-{a}_{\mathrm{ie}}{J}_{0}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)-{\rm{i}}{a}_{\mathrm{ie}}\sum _{n\ne 0}\left[{k}_{n}{g}_{\beta \leftarrow \beta }^{(n)}{J}_{-n}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)\right];\end{eqnarray}$
$\begin{eqnarray}{f}_{\beta \leftarrow \beta }=-{a}_{e}-{\rm{i}}{a}_{\mathrm{ie}}\sum _{n\ne 0}\left[{k}_{n}{g}_{\alpha \leftarrow \beta }^{(n)}{J}_{n}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)\right];\end{eqnarray}$
$\begin{eqnarray}{g}_{\alpha \leftarrow \beta }^{(n)}=-{a}_{\mathrm{ie}}{J}_{n}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)-{\rm{i}}{a}_{e}{k}_{n}{g}_{\alpha \leftarrow \beta }^{(n)}-{\rm{i}}{a}_{\mathrm{ie}}\sum _{m\ne n}\left[{k}_{n-m}{g}_{\beta \leftarrow \beta }^{(n-m)}{J}_{m}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)\right];\end{eqnarray}$
$\begin{eqnarray}{g}_{\beta \leftarrow \beta }^{(n)}=-{\rm{i}}{a}_{e}{k}_{n}{g}_{\beta \leftarrow \beta }^{(n)}-{\rm{i}}{a}_{\mathrm{ie}}\sum _{m\ne -n}\left[{k}_{n+m}{g}_{\alpha \leftarrow \beta }^{(n+m)}{J}_{m}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)\right].\end{eqnarray}$
Numerically solving equations (33 )–(36 ), we can derive elastic scattering amplitude fβ←β and the spin-exchanging amplitude fβ←α.
3.3. Two-atom loss rate
Our above calculation shows that due to the shaking of the interaction ${\hat{V}}_{R}(t)$, the atoms have some probability to be scattered to the Floquet bands which differs from the incoming one. This is described by the terms with n > 0 in equations (24 ), (32 ). When the shaking frequency λ is high, the out-going momentum kn for these bands are also very large. As a result, the atoms scattered to these bands can escape from the trap. This is the two-body loss induced by the shaking of the control beam of our system.
According to the optical theorem, for our system, the two-body loss rate of atoms in state ∣α⟩ and in state ∣β⟩ are $8\pi \mathrm{Im}[{f}_{\alpha \leftarrow \alpha }]$ and $8\pi \mathrm{Im}[{f}_{\beta \leftarrow \beta }]$, respectively. Furthermore, the direct calculation for the above equations (36 ) and (29 ) show that fα←α = fβ←β. Thus, the two-body loss rate of our system can be expressed as
$\begin{eqnarray}{K}_{2}=8\pi \mathrm{Im}[{f}_{\alpha \leftarrow \alpha }]=8\pi \mathrm{Im}[{f}_{\beta \leftarrow \beta }].\end{eqnarray}$
4. Results
In this section, we use the results given by the calculations of section 3 to address the questions (i) and (ii) of section 1 .
We first notice that, according to direct calculations, the zero-energy scattering amplitudes ${f}_{l\leftarrow j}^{(\mathrm{ave})}$ from channel ∣l⟩ to ∣j⟩ (l, j = α, β) with respect to ${\hat{V}}^{(\mathrm{ave})}$ can be expressed as
$\begin{eqnarray}{f}_{\alpha \leftarrow \alpha }^{(\mathrm{ave})}={f}_{\beta \leftarrow \beta }^{(\mathrm{ave})}=-{a}_{e};\end{eqnarray}$
$\begin{eqnarray}{f}_{\beta \leftarrow \alpha }^{(\mathrm{ave})}={f}_{\alpha \leftarrow \beta }^{(\mathrm{ave})}=-{a}_{\mathrm{ie}}{J}_{0}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right).\end{eqnarray}$
On the other hand, the zero-energy scattering amplitudes fl←j (l, j = α, β) derived by the Floquet scattering approach satisfy fα←α = fβ←β and fβ←α = fα←β.The above facts yield that to answer the question (i), i.e. examine the applicability of the time-averaging approximation, we just require to compare fα←α and fβ←α with ${f}_{\alpha \leftarrow \alpha }^{(\mathrm{ave})}$ and ${f}_{\beta \leftarrow \alpha }^{(\mathrm{ave})}$, respectively. In figures 2 and 3 we perform this comparison for cases with typical shaking angular frequency λ = (2π) 3 kHz, λ = (2π) 5 kHz, and λ = (2π) 10 kHz. Our results show that the time-averaging approximation is applicable only when the shaking amplitude δ0 is small enough, or roughly speaking, under the condition δ0 ≲ λ. As shown in figure 2, when this condition is violated, either Re[fα←α] or Re[fβ←α] would be quite different from ${f}_{\alpha \leftarrow \alpha }^{(\mathrm{ave})}$ or ${f}_{\beta \leftarrow \alpha }^{(\mathrm{ave})}$, respectively. Furthermore, as shown in figure 3, when δ0 is too large $\mathrm{Im}[{f}_{\beta \leftarrow \alpha }]$ becomes comparable with the norm of fβ←α, while the time-averaging approximation yields $\mathrm{Im}[{f}_{\beta \leftarrow \alpha }^{(\mathrm{ave})}]=0$.
Figure 2. The scattering amplitudes of 173Yb atoms, as a function of δ0 for λ = (2π) 3 kHz, λ = (2π) 5 kHz, and λ = (2π) 10 kHz. (a1)–(a3): Re[fα←α] (black solid line) and ${f}_{\alpha \leftarrow \alpha }^{(\mathrm{ave})}$ (blue dotted line). (b1)–(b3): Re[fβ←α] (black solid line) and ${f}_{\beta \leftarrow \alpha }^{(\mathrm{ave})}$ (blue dotted line). The insets show the behavior of each quantity for δ0 ≤ 5 kHz. |
Figure 3. Im[fβ←α]/∣fβ←α∣ of 173Yb atoms as a function of δ0, for λ = (2π) 3 kHz (red solid line), λ = (2π) 5 kHz (blue dashed line), and λ = (2π) 10 kHz (green dotted line). Here we show the results for (a): δ0 ≤ (2π)100 kHz and (b): δ0 ≤ (2π)4 kHz. |
In figure 4 we show the two-body loss rate K2 for these three cases, which is calculated by the Floquet scattering approach. It is shown that K2 increases with the shaking amplitude δ0, and can be enhanced to the order of 10−10 cm3 · s−1 or even larger when δ0 is so large that the time-averaging approximation is not applicable.
Figure 4. The two-body loss rate K2 of 173Yb atoms as a function of δ0. Here we show the results for (a): δ0 ≤ (2π)100 kHz and (b): δ0 ≤ (2π)5 kHz. |
In addition, figure 2 also shows that a scattering resonance occurs for λ = (2π) 3 kHz and δ0 ≈ (2π) 6.58 kHz, where both the scattering amplitude and the two-body loss rate are significantly enhanced. To understand this resonance more clearly, in figure 5 we illustrate the real parts of the scattering amplitudes fα←α and fβ←α as a function of λ for fixed shaking amplitude δ0 = (2π) 6.58 kHz. Five significant resonances A, B, C, D and E are clearly shown in this figure, which occur at λ ≈ (2π)6.16 kHz, λ ≈ (2π)3.00 kHz, λ ≈ (2π)1.99 kHz, λ ≈ (2π)1.49 kHz and λ ≈ (2π)1.19 kHz, respectively. The resonance B is just the one in figure 2. These resonances are due to the coupling between the incident channel and other Floquet channels corresponding to the terms proportional to ${{\rm{e}}}^{-{\rm{i}}{\omega }_{n}t}$ (n ≠ 0) of equation (32 ).
Figure 5. The scattering amplitudes of 173Yb atoms, as a function of λ for δ0 = (2π) 6.58 kHz. (a): Re[fα←α] (black solid line) and ${f}_{\alpha \leftarrow \alpha }^{(\mathrm{ave})}$ (blue dotted line). (b): Re[fβ←α] (black solid line) and ${f}_{\beta \leftarrow \alpha }^{(\mathrm{ave})}$ (blue dotted line). The insets show the behavior of each quantity for λ ≤ 0.8 kHz. |
In the following, we provide more discussion on these resonances. We first notice that, for our Floquet scattering problem the total Hilbert space can be expressed as ${\mathscr{H}}={{\mathscr{H}}}_{{\rm{r}}}\otimes {{\mathscr{H}}}_{{\rm{S}}}\otimes {{\mathscr{H}}}_{{\rm{F}}}$, with ${{\mathscr{H}}}_{{\rm{r}}}$ and ${{\mathscr{H}}}_{{\rm{S}}}$ being the Hilbert space for the two-atom relative motion and internal states, respectively, and ${{\mathscr{H}}}_{{\rm{F}}}$ being the space of periodical functions of time, i.e. the space spanned by the functions ${{\rm{e}}}^{-{\rm{i}}{\omega }_{n}t}$ (n = 0, ±1, ±2, ...). Here we formally denote the states in ${{\mathscr{H}}}_{{\rm{S}}}$, ${{\mathscr{H}}}_{{\rm{F}}}$, and ${{\mathscr{H}}}_{{\rm{S}}}\otimes {{\mathscr{H}}}_{{\rm{F}}}$ with ∣⟩, ∣⟩F and ∣⟩SF, respectively, and denote the functions ${{\rm{e}}}^{-{\rm{i}}{\omega }_{n}t}$ (n = 0, ±1, ±2, ...) as ∣n). With these notations, the Schr$\ddot{{\rm{o}}}$dinger equation (23 ) can be re-expressed as46 ) by the inter-channel couplings, i.e. the inter-channel coupling can contribute to Lamb shifts for the self-energies of the closed-channel bound states. This physical picture is justified by our results of figure 5. Using direct calculations we find that the values of (δ0, λ) corresponding to the resonances A, B, C, D and E of this figure approximately satisfy equation (46 ) with ζ = + and n = 1.21, 5.29, 18.33, 24.12 and 15.82, respectively. In the parameter region of figure 5 there may be other resonances that are too narrow to be clearly resolved.
$\begin{eqnarray}\left[\sum _{n}{\hat{{ \mathcal H }}}_{n}| n{\rangle }_{{\rm{F}}}\langle n| +\sum _{m\ne n}{\hat{{ \mathcal V }}}_{{mn}}| m{\rangle }_{{\rm{F}}}\langle n| \right]| \psi (r){\rangle }_{\mathrm{SF}}=0,\end{eqnarray}$
where ∣ψ(r)⟩SF is the scattering wave function, $\begin{eqnarray}{\hat{{ \mathcal H }}}_{n}=-{{\rm{\nabla }}}_{{\boldsymbol{r}}}^{2}-{\omega }_{n}+4\pi {\hat{{ \mathcal B }}}_{n}\delta ({\boldsymbol{r}})\displaystyle \frac{\partial }{\partial r}\left(r\cdot \right),\end{eqnarray}$
and $\begin{eqnarray}{\hat{{ \mathcal V }}}_{{mn}}=4\pi {\hat{{ \mathcal C }}}_{{mn}}\delta ({\boldsymbol{r}})\displaystyle \frac{\partial }{\partial r}\left(r\cdot \right).\end{eqnarray}$
Here ${\hat{{ \mathcal B }}}_{n}$ and ${\hat{{ \mathcal C }}}_{{mn}}$ are defined as: $\begin{eqnarray}{\hat{{ \mathcal B }}}_{n}={a}_{e}\left(| \alpha \rangle \langle \alpha | +| \beta \rangle \langle \beta | \right)+{a}_{\mathrm{ie}}{J}_{0}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)| \beta \rangle \langle \alpha | +{\rm{h}}.{\rm{c}}.,\end{eqnarray}$
and $\begin{eqnarray}{\hat{{ \mathcal C }}}_{{mn}}={a}_{\mathrm{ie}}{J}_{m-n}\left(\displaystyle \frac{{\delta }_{0}}{\lambda }\right)| \beta \rangle \langle \alpha | +{\rm{h}}.{\rm{c}}.\end{eqnarray}$
In our problem, the atoms are incident from the channel ∣0⟩F with zero kinetic energy, and the channels ∣n⟩F with n ≥ 0 and n < 0 are open and closed channels, respectively. Furthermore, direct calculations also show that for each given n < 0, the self Hamiltonian ${\hat{{ \mathcal H }}}_{n}$ of the closed channel can support two bound states with energies $\begin{eqnarray}{E}_{b}^{(\pm )}(n,\lambda )=-\displaystyle \frac{1}{{\left[{a}_{e}\pm {a}_{\mathrm{ie}}{J}_{0}\left(\tfrac{{\delta }_{0}}{\lambda }\right)\right]}^{2}}-{\omega }_{n}.\end{eqnarray}$
In the weak-inter-channel-coupling cases with small shaking amplitude δ0, the scattering resonances can occur when these closed-channel bound states are resonant with the zero-energy incident state, i.e. when any one of the following equations $\begin{eqnarray}{E}_{b}^{(\zeta )}(n,\lambda )=0;\quad (\zeta =+,-;n=-1,-2,\ldots )\end{eqnarray}$
is approximately satisfied by δ0 and λ. Here we say ‘approximately'because the resonance points can actually be slightly shifted from the solutions of equation (5. Summary
We calculate the scattering amplitude and two-atom loss rate for the ultracold gases of 173Yb atoms with interaction being modulated with the Floquet-engineering approach proposed in [19]. We use the energy-independent two-channel Huang-Yang pseudopotential to model the bare interaction between these two atoms. A more quantitatively accurate may be derived via the calculations with quantum defect theory.
Our results show that for the cases with typical shaking angular frequency λ ∼ (2π) kHz, the time-averaging approximation is applicable and the two-body loss rate K2 is low-enough only when the shaking amplitude δ0 is low enough. When δ0 is too large, the inter-atomic scattering amplitude would be quite different from the one given by the simple time-averaged potential, and K2 can be enhanced to the order of 10−10 cm3 · s−1, which is quite large for typical ultracold gases.
According to these results, when our system is used for the simulations of closed quantum systems, the shaking amplitude δ0 should be small enough so that the two-body loss effect is weak enough. On the other hand, since one can increase the two-body loss rate by increasing δ0, this system may be used for the studies of open systems with particle-number dissipations.