Trimers in the resonant ($2+1$)-fermion problem on a narrow Feshbach resonance: Crossover from Efimovian to hydrogenoid spectrum

We study the quantum three-body free-space problem of two same-spin-state fermions of mass $m$ interacting with a different particle of mass $M$, on an infinitely narrow Feshbach resonance with infinite $s$-wave scattering length. This problem is made interesting by the existence of a tunable parameter, the mass ratio $\alpha =m/M$. By a combination of analytical and numerical techniques, we obtain a detailed picture of the spectrum of three-body bound states, within each sector of fixed total angular momentum $l$. For $\alpha$ increasing from 0, we find that the trimer states first appear at the $l$-dependent Efimovian threshold ${\alpha }_c^{\left(l\right)}$, where the Efimov exponent $s$ vanishes, and that the entire trimer spectrum (starting from the ground trimer state) is geometric for $\alpha$ tending to ${\alpha }_c^{\left(l\right)}$ from above, with a global energy scale that has a finite and nonzero limit. For further increasing values of $\alpha$, the least bound trimer states still form a geometric spectrum, with an energy ratio $\exp (2\pi /|s\left|\right)$ that becomes closer and closer to unity, but the most bound trimer states deviate more and more from that geometric spectrum and eventually form a hydrogenoid spectrum.

Figure 1

In the complex plane, illustration of the procedure used to determine the Fourier transform of the function $x\to e^x{F}^{\left(l\right)}\left(x\right)$: Under the a posteriori checkable hypothesis of Eq. (65) (no pole of ${\tilde{F}}^{\left(l\right)}\left(z\right)$ in the gray area), one can shift the integration contour in Eq. (64) upward by one unity along the vertical axis to obtain the Fourier representation [Eq. (66)] of $x\to e^x{F}^{\left(l\right)}\left(x\right)$. (Circles) Poles of ${\tilde{F}}^{\left(l\right)}\left(z\right)$ due to the function hyperbolic sine in the denominator of Eq. (68). (Pluses) Poles of ${\tilde{F}}^{\left(l\right)}\left(z\right)$ due to the factors $\Gamma (1+iz-i{v}_n)$, $n\ge 0$, in the numerators of the infinite product (71). (Black crosses) Poles of ${\tilde{F}}^{\left(l\right)}\left(z\right)$ due to the factor $\Gamma (i{S}_l-iz)$ in Eq. (71). Colored crosses on the $\mathrm{Im}z<0$ part of the imaginary axis: Poles of ${\tilde{F}}^{\left(l\right)}\left(z\right)$ due to the factors $\Gamma (-i{u}_n-iz)$ for $n>0$ in Eq. (71) [(red) $n=1$; (blue) $n=2$; (violet) $n=3$; (cyan) $n=4$]; from top to bottom, $n=1$ (three crosses), $n=2$ (two crosses), two triplets $n=1,2,3$, and a quadruplet $n=1,2,3,4$. The figure corresponds to $l=1$ and $\alpha =20$.

Figure 2

Values of the Efimov exponent $s$ as a function of the fermion-to-extra-particle mass ratio $\alpha =m/M$ for several values of the angular momentum $l$. The modulus of $s$ is shown only over the interval of mass ratio where $s$ is purely imaginary (that is, where the Efimov effect takes place). This was found to occur only for odd values of $l$, and for a single pair of $\pm s$ roots of ${\Lambda }_l\left(s\right)=0$. (Solid line) Numerical result from the exact function ${\Lambda }_l\left(s\right)$ as given by Eq. (42). (Dashed line) Born-Oppenheimer approximation [Eq. (76)].

Figure 3

Matching of two limiting solutions of the integral Eq. (32), the solution $E=-{\hslash }^2{q}^2/\left(2\mu \right)<0,R_{*}=0$ (see text in black, in the upper half of the figure) given by Eq. (52) and the solution $E=0,R_{*}>0$ (see the text in red in the lower half of the figure) deducible from Eqs. (27), (57), (60), (68), and (71), over a common interval of values of $k$ (the “matching interval,” blue segment on $k$ axis) where they are both in their asymptotic regimes, Eqs. (53) and (73). This matching procedure leads to the Efimovian spectrum formula (84) with a global scale given by Eq. (85). This procedure makes sense when $q{R}_{*}\ll 1$, and it is expected to be exact when $q{R}_{*}\to 0$, that is, for the quantum number $n$ tending to infinity for a fixed (purely imaginary) Efimov exponent $s_l$ (as in Efimov's historical solution) or (less usually) for $|s_l|$ tending to zero at fixed quantum number $n\ge 1$. For simplicity of the figure, we have dropped factors slowly depending on the mass ratio $\alpha$, such as $\cos \nu$; that is, we have assumed that the angular momentum $l$ and $|s_l|$ are not much larger than unity.

Figure 4

In (a) value of the global wave-number scale $q_{\mathrm{global}}^{\left(l\right)}$ of the Efimovian part of the spectrum [see Eq. (85)] as a function of the difference of the mass ratio $\alpha$ from its critical value, for all odd angular momenta up to $l=7$. (Thin solid lines) Exact result obtained from Eq. (86) and from the infinite product representation Eq. (71) [the more rapidly converging form, Eqs. (C3) and (C5), was used in the numerics]. (Thick dashed lines) Zeroth-order approximation omitting all the terms in the sum over $k$ in Eq. (87). In (b) the (small) difference between the exact ${\theta }_l$ and the zeroth-order approximation ${\theta }_l^{k=0}$ is plotted as thin solid lines, and the value ${\theta }_l^{k=1}$ of the $k=1$ term of Eq. (87) is plotted as thick dashed lines. In (c) the values of $q_{\mathrm{global}}^{\left(l\right)}$ resulting from the Born-Oppenheimer-plus-semiclassical approximation are shown as dashed lines [see Eq. (106)], as functions of $1/{(\alpha -{\alpha }_c^{\left(l\right)})}^{1/2}$, showing how the $l$-independent $\alpha \to +\infty$ limit is reached. The solid lines correspond to the exact numerical data of (a) and the dotted straight lines correspond to the asymptotic expansion [Eq. (108)].

Figure 5

Dependence, with the angular momentum $l$, of the global wave-number scale $q_{\mathrm{global}}^{\left(l\right)}$ in the Efimov trimer spectrum, as obtained from the exact result (85). The figure shows the quantity $q_{\mathrm{global}}^{\left(l\right)}{R}_{*}$ multiplied by ${(l+1/2)}^3$, as a function of $\alpha -{\alpha }_c^{\left(l\right)}$, for (odd) angular momenta equal to $l=1$, $l=3$, $l=5$, and $l=7$, from bottom to top. This quantity is observed to converge rapidly to a finite limit for increasing $l$. The dashed line represents the analytical prediction (90) for that limit. We recall that $E_{\mathrm{global}}=-{\hslash }^2{q}_{\mathrm{global}}^2/\left(2\mu \right)$.

Figure 6

Numerical study of the Efimovian part of the spectrum: For an angular momentum $l=1$, quantities $q_n$ (labeled in descending order) from $n=1$ (ground trimer state) to $n=8$, as functions of the mass ratio $\alpha =m/M$. We recall that the trimer energies are then $E_n=-{\hslash }^2{q}_n^2/2\mu$. In (a) the numerical values are shown in log scale, restricting the figure to experimentally accessible values $q{R}_{*}>{10}^{-3}$ (see text). In (b) the absolute value of the deviation from unity of the ratio of the numerical $q_n$ to the analytical approximation (84). The solid (respectively dashed) lines correspond to numerical values of $q_n$ larger (respectively smaller) than the analytical ones. For both (a) and (b), the vertical axis is in log scale, and the curves $n=1$ to $n=8$ are from top to bottom.

Figure 7

For $l=1$ and a mass ratio $\alpha =14$, the ground eigenvector ($n=1$) and first excited eigenvector ($n=2$) of the eigenvalue problem [Eq. (114)], corresponding to the ground and first excited trimer states. (Solid lines) Numerical results. (Dashed lines) Analytical form [Eq. (116)], meaningful over the matching interval $1/\cos \nu <k/q<1/\left(q{R}_{*}\cos \nu \right)$ whose meaning is explained in Fig. 3 and whose borders are indicated by the vertical dotted lines. The expression of $D\left(\mathbf{k}\right)$ in terms of ${\check{F}}^{\left(1\right)}\left(x\right)$ and $Y_1^0\left(\mathbf{k}\right)$ can be obtained from the text, with $x=\ln (k/q)$. An overall factor $\exp (-x/2)$ was applied to ${\check{F}}^{\left(1\right)}\left(x\right)$ for convenience, and the resulting functions are normalized to the maximal value of unity.

Figure 8

Numerical study of the hydrogenoid part of the spectrum: For an angular momentum $l=1$, wave numbers $q_n^{\left(l\right)}$ for $n=1$ (ground trimer state), $n=2$, $n=3$, and $n=4$ (from top to bottom), divided by the hydrogenoid asymptotic Born-Oppenheimer prediction [Eq. (102)], as functions of $1/{\alpha }^{1/2}$. (Solid circles) Numerical results. (Straight lines) Asymptotic result [Eq. (103)], including the first deviation of the Born-Oppenheimer potential from the Coulomb form at short range. The color code is as follows: black for $n=1$, red for $n=2$, green for $n=3$, and blue for $n=4$.