Effects of Efimov states on quench dynamics in a three-boson trapped system

The Ramsey signal Cetina et al. (2016) is defined as the wavefunction overlap of the initial and final states,

where $\Psi_{\rm i}$ is the initial state with energy $E_{\rm i}$, $\Psi^{\prime}$ is the post-quench state and the $\Psi^{\prime}_{j}$s are the eigenstates of the post-quench system with energy $E^{\prime}_{j}$, $j$ is summing over all post-quench eigenstates.

As mentioned above the COM wavefunction is unaffected by the quench and integrates to one. Hence the Ramsey signal depends only on the relative wavefunction,

To evaluate the Ramsey signal we need to evaluate the hyperradial integral $\bra{F_{qs}}\ket{F_{q_{\rm i}s_{\rm i}}}$ and the hyperangular integral $\bra{\phi_{s}}\ket{\phi_{s_{\rm i}}}$. The appendix contains the details of the evaluation of these integrals. With the integrals known we can then calculate the Ramsey signal for the forwards and backwards quenches. There is a degree of freedom in the choice of the Efimov energy spectrum, determined by the value of $R_{t}$. Whatever the value of $R_{t}$ the normalisation is preserved but the post-quench behaviour is nonetheless affected by the choice of $R_{t}$.

The Ramsey signal is the weighted sum of oscillators, $S(t)=Ae^{-iat}+Be^{-ibt}+Ce^{-ict}+\dots$, where the weights are the square overlaps between initial state and post-quench eigenstates and the angular frequencies are the differences between the initial energy and post-quench eigenenergies. The magnitude is similarly a weighted sum of oscillators but the angular frequencies of the oscillatory terms are the differences between post-quench eigenenergies, $(a-b),(b-c),(a-c),\dots$. The phase of the Ramsey signal is dominated by the phase of the most heavily weighted terms.

In Fig. 2 we plot the Ramsey signal of the forwards quench for a number of initial states and values of $R_{t}$. The calculations of the Ramsey signal for the forwards quench are performed including only the $q\geq-1$ Efimov energies except for the $R_{t}=a_{\mu}$ calculation which includes only the $q\geq 0$ Efimov energies. The neglected energies are significantly lower (e.g. for $R_{t}=a_{\mu}$ the $q=-1$ Efimov energy is $\approx-566\hbar\omega$) and do not contribute meaningfully. Unlike with the two-body case Kerin and Martin (2020) the magnitude of the Ramsey signal of the forwards quench is aperiodic. This is because the post-quench eigenenergies are irrational because the unitary $s$-eigenvalues are irrational as are the Efimov energies in general. This means the angular frequencies in Eq. (20) ($a$, $b$, $c\dots$ from the previous paragraph) are irrational as are the differences between them, hence the magnitude and phase of the Ramsey signal are aperiodic.

In the $R_{t}=a_{\mu}$, $(q_{\rm i},s_{\rm i})=(0,2)$ case (solid red line in the upper panel of Fig. 2) the post quench states with the largest overlaps are $(E_{q=1},s)\approx(2.27\hbar\omega,i\cdot 1.006)$, with square overlap of $\approx 0.666$, $(E_{q=0},s)\approx(-0.85\hbar\omega,i\cdot 1.006)$, with square overlap of $\approx 0.14$ and $(q,s)=(0,4.465\dots)$, with square overlap $\approx 0.105$. The two largest modes in the magnitude have periods of $\approx 2\pi/3\omega$. The phase is dominated by a period of $\approx 2.7\pi/\omega$.

In the $R_{t}=5a_{\mu}$, $(q_{\rm i},s_{\rm i})=(0,2)$ case (solid red line in the middle panel of Fig. 2) the most significant terms are $(E_{q=0},s)\approx(1.077\hbar\omega,i\cdot 1.006)$, with a square overlap of $\approx 0.583$, $(E_{q=1},s)\approx(3.37\hbar\omega,i\cdot 1.006)$ with a square overlap of $\approx 0.24$ and $(q,s)=(0,4.465\dots)$ with a square overlap of $\approx 0.105$. The two largest modes in the magnitude have periods of $\approx 0.87\pi/\omega$ and $\approx 0.45\pi/\omega$. The phase is dominated by the period $\approx\pi/\omega$.

In the $R_{t}=10a_{\mu}$, $(q_{\rm i},s_{\rm i})=(0,2)$ case (solid red line in the lower panel of Fig. 2) the terms with the largest overlaps are $(E_{q=0},s)\approx(1.603\hbar\omega,i\cdot 1.006)$ with a square overlap of $\approx 0.72$, $(q,s)=(0,4.465\dots)$ with a square overlap of $\approx 0.105$ and $(E_{q=1},s)\approx(3.875\hbar\omega,i\cdot 1.006)$ with a square overlap of $\approx 0.092$. This leads to two main modes in the magnitude with periods of $\approx 0.5\pi/\omega$ and $\approx 0.9\pi/\omega$ and the phase is dominated by the period of $\approx 1.4\pi/\omega$.

In Fig. 3 we plot the Ramsey signal of the backwards quench for a system initially in an Efimov state for a variety of Efimov energies. Unlike in the forwards quench the magnitude of the Ramsey signal of the backwards quench is periodic. This is because the non-interacting eigenenergies are all odd integer multiples of $\hbar\omega$. The difference between the post-quench eigenenergies are even integers leading to the magnitude having period $\pi/\omega$. However the phase is dominated by the largest term in Eq. (20) and the angular frequencies of each term are irrational because the initial state is an Efimov state which, in general, has an irrational energy. This leads to the irregularity in the phase.

For $R_{t}=a_{\mu}$, $(E_{q_{\rm i}=0},s_{\rm i})\approx(-0.850,i\cdot 1.006)$ (solid red line of the upper panel of Fig. 3) the largest terms are the overlaps with $(q,s)=(0,2)$ with square overlap $\approx 0.14$ and $(q,s)=(1,2)$ with square overlap $\approx 0.12$. These terms have periods of $\approx 0.52\pi/\omega$ and $\approx 0.34\pi/\omega$. For $R_{t}=5a_{\mu}$ $(E_{q_{\rm i}=0},s_{\rm i})\approx(1.077,i\cdot 1.006)$ (solid red line of the middle panel of Fig. 3) the largest term is the overlaps with $(q,s)=(0,2)$ with square overlap $\approx 0.58$ and period $\approx\pi/\omega$. For $R_{t}=10a_{\mu}$ $(E_{q_{\rm i}=0},s_{\rm i})\approx(1.602,i\cdot 1.006)$ (solid red line of the lower panel of Fig. 3) the largest term is the overlaps with $(q,s)=(0,2)$ with square overlap $\approx 0.72$ and period $\approx 1.43\pi/\omega$.

We are not limited to calculating only the Ramsey signal. It is also possible to calculate the particle separation, $\langle\tilde{R}(t)\rangle$.

The expectation value of $\tilde{R}(t)$ is given

where $\Psi_{\rm i}$ is the initial pre-quench state with energy $E_{\rm i}$ and $\Psi^{\prime}(t)$ is the post-quench state. $\Psi^{\prime}_{j}$ and $\Psi^{\prime}_{j^{\prime}}$ are eigenstates of the post-quench system with eigenenergy $E_{j}$ and $E_{j^{\prime}}$ respectively, with the sums over $j$ and $j^{\prime}$ taken over all post-quench eigenstates.

The COM wavefunction is independent of the interparticle interaction and does not impact the post-quench dynamics. Due to the hyperangular wavefunction’s orthogonality in $s$, two sums over $s$ and $s^{\prime}$ collapse into a single sum over $s$. Hence $\langle\tilde{R}(t)\rangle$ is given

As in Eq. (20) indices with subscript i refer to the initial state and indices with no subscript refer to the post-quench eigenstates. As with the Ramsey signal all relevant integrals are presented in the appendix.

In Fig. 4 we plot $\langle\tilde{R}(t)\rangle$ for a system initially in the non-interacting ground state quenched to unitarity with the upper, middle and lower panels corresponding to $R_{t}=a_{\mu}$, $R_{t}=5a_{\mu}$, $R_{t}=10a_{\mu}$ respectively. For $R_{t}\geq 5a_{\mu}$ we include states with $q\geq-1$ in the calculations but for $R_{t}=a_{\mu}$ $E_{q=-1}\approx-566\hbar\omega$ and this state does not meaningfully contribute so we only include the $q\geq 0$ Efimov states in the calculation. In Eq. (22) terms with $q=q^{\prime}$ are constants and the $s$ contributions to the energies cancel out, the angular frequencies depend only on $q$ and $q^{\prime}$. The universal state terms oscillate with an angular frequency that is an even integer multiple of $\omega$ because $q-q^{\prime}$ is an integer but the Efimov state terms oscillate with irrational angular frequencies because the differences between the Efimov energies, $E_{q}$ and $E_{q^{\prime}}$, are irrational in general. For each plot we have calculated $\langle\tilde{R}(t)\rangle$ by summing up to $N_{\rm max}=3,6,12,24$ terms in each of the three sums in Eq. (22), and we find that the sum is convergent.

For $R_{t}=a_{\mu}$ the largest oscillating terms are $(q^{\prime},q,s)=(q,q^{\prime},s)=(0,1,i\cdot 1.006\dots)$ with total coefficient $\approx 0.17$ and $E_{q=1}-E_{q=0}\approx 3.12\hbar\omega$ and $(q,q^{\prime},s)=(1,2,i\cdot 1.006\dots)$ with total coefficient $\approx 0.09$ and $E_{q=2}-E_{q=1}\approx 2.12\hbar\omega$ . This implies characteristic periods of $\approx 2\pi/3\omega$ and $\approx\pi/\omega$. For $R_{t}=5a_{\mu}$ the largest oscillating terms are $(q,q^{\prime},s)=(0,1,i\cdot 1.006\dots)$ with total coefficient $\approx 0.3$ and $E_{q=1}-E_{q=0}\approx 2.3\hbar\omega$ and $(q,q^{\prime},s)=(0,1,4.465)$ with total coefficient $\approx 0.054$ and associated energy difference $2\hbar\omega$. This leads to characteristic periods of $\approx 0.9\pi/\omega$ and $\pi/\omega$. For $R_{t}=10a_{\mu}$ the largest oscillating terms are $(q,q^{\prime},s)=(0,1,i\cdot 1.006\dots)$ with total coefficient $\approx 0.22$ and $E_{q=1}-E_{q=0}\approx 2.3\hbar\omega$ and $(q,q^{\prime},s)=(0,1,4.465)$ with total coefficient $\approx 0.054$ and associated energy difference $2\hbar\omega$. This leads to characteristic periods of $\approx 0.9\pi/\omega$ and $\pi/\omega$.

In Fig. 5 we plot $\langle\tilde{R}(t)\rangle$ for the backwards quench where the system is initially in a variety of Efimov states. Unlike the forwards quench we find that $\langle\tilde{R}(t)\rangle$ is periodic for the backwards quench. This is because in the backwards quench the post-quench states are universal states where the differences between eigenenergies are always even multiples of $\hbar\omega$, leading to a period of $\pi/\omega$. However similar to how Ref. Kerin and Martin (2020) found a divergence in $r=|\vec{r}_{2}-\vec{r}_{1}|$ in the backwards quench we find that $\langle\tilde{R}(t)\rangle$ also diverges for the backwards quench. In particular we find that it is logarithmically divergent with the number of terms in the summation, i.e. $\langle\tilde{R}(t\neq n\pi/\omega)\rangle\propto\ln(N_{\rm max})$. This divergence is not exclusively due to the Efimov states as the divergence is present even when there are no Efimov states Kerin and Martin (2022b).

This divergence is quite unusual, it is not obvious why it occurs nor why it occurs only for the reverse quench. To investigate further we look at how the probability distribution of $R$, $P(R,t)$, evolves over time for both quenches. $P(R,t)$ is given

where $\Psi^{\prime}(t)$ is the post-quench state.

In Fig. 6 we plot the evolution of $P(R,t)$ for the forwards quench with $R_{t}/a_{\mu}=1,5$ and $10$ in the upper, middle and lower panels respectively. For all values of $R_{t}$ we see a qualitatively similar evolution, the system oscillates between a broad distribution and a tightly peaked one with a smaller mean value. This oscillation is only approximately periodic due to the influence of the irrational Efimov energies. The broad distribution corresponds to the initial universal state and the tightly peaked distribution is dominated by the Efimov states, the system oscillates between these two regimes.

To understand this oscillation it is useful to consider $\langle\tilde{R}\rangle$ for the initial and post-quench states. For example for $R_{t}=a_{\mu}$ the states with the largest overlaps with the initial are $(E_{q=1},s)\approx(2.27\hbar\omega,i\cdot 1.006)$ with a square overlap of $\approx 0.666$ and $(E_{q=0},s)\approx(-0.85\hbar\omega,i\cdot 1.006)$ with square overlap $\approx 0.14$, these states have $\langle\tilde{R}\rangle\approx 1.5$ and $\approx 0.57$ respectively. The initial state is $(q,s)=(0,2)$ so we have $\langle\tilde{R}(t=0)\rangle\approx 1.66$, hence the position of the initial broad distribution is to the right of the tightly peaked Efimov distribution. As $R_{t}$ increases $\langle\tilde{R}\rangle$ of the strongly overlapping Efimov states increases but they are, on average, still less than $\langle\tilde{R}\rangle$ of the initial universal state, hence the peak of the narrow distribution moves rightward with increasing $R_{t}$. Additionally as $R_{t}$ increases the narrow Efimov distribution broadens because the higher energy Efimov states are simply broader. Note in the third panel of Fig. 6 one can see small local peaks in probability near $\tilde{R}=0$. These come from the $(E_{q=-1},s)\approx(-5.6,i\cdot 1.006)$ state and have a square overlap with the initial state of $\approx 0.01$ and $\langle\tilde{R}\rangle\approx 0.3$, the $(E_{q=-1},s)$ state is also accounted for in the $R_{t}=5a_{\mu}$ calculation but the overlap is approximately 50 times smaller.

In Fig, 7 we plot $P(R,t)$ for the backwards quench at $t=0,0.17\pi/\omega,0.34\pi/\omega$ and $\pi/2\omega$. Unlike the forwards quench the evolution of $P(R,t)$ here is periodic. The mean of $P(R,t)$ increases with time, reaching a maximum at $\pi/2\omega$ before returning to its initial shape at $t=\pi/\omega$, this then repeats with period $\pi/\omega$. Initially $P(R,t)$ is tightly peaked but develops a long tail as it evolves and in Fig. 8 we present the long tail in detail for various values of $N_{\rm max}$. The tail behaves approximately like $\tilde{R}^{-2}$ until ending in an exponential-like “cut-off”. This cut-off occurs at larger $R$ for larger $N_{\rm max}$ and in the limit of $N_{\rm max}\rightarrow\infty$ the behaviour of the tail of $P(R,t)$ approaches $\tilde{R}^{-2}$ with no cut-off. This means that the integral of $P(R,t)$ over $R$ from $R=0$ to $R\rightarrow\infty$ is finite and properly normalised in the $N_{\rm max}\rightarrow\infty$ limit, however $RP(R,t)$ has a $\tilde{R}^{-1}$ tail and so is the integral is not finite, hence the divergence in $\langle\tilde{R}(t)\rangle$ for the backwards quench.

Physically speaking there are two likely candidates for the source of the divergence; the zero-range contact interaction and the instantaneous nature of the quench, in reality atoms interact at some finite range and the quench in $a_{\rm s}$ occurs over some finite time. These are two non-physical inputs into this model and may be responsible for the non-physical outputs.

By considering the finite range of the interaction it is possible to estimate a maximum value of $\langle\tilde{R}(t)\rangle$. The lengthscale of the interaction provides a justification for a maximum energy and thus a cut-off in Eq. (22). Specifically, the range of interaction defines a minimum de Broglie wavelength which defines a maximum energy and thus the cut-off. For sodium in a 1kHz trap and assuming a van der Waals range of one nanometre we obtain an energy of $E_{\rm rel}\approx 8.7\times 10^{6}\hbar\omega$ and so we predict $\langle\tilde{R}(t)\rangle_{\rm max}\approx 21$ for an initial Efimov energy of $E_{q=0}\approx-0.85$. This is an order of magnitude larger than the amplitude of oscillations when the system is quenched from the non-interacting to the strongly interacting regime.

In contrast it is difficult to quantify the effects of a finite duration quench. In the formalism used here only quenches between the non-interacting and unitary regimes can be described, meaning a quench to or from the intermediate regime can’t be elucidated. However in the two-body case a quench between any two scattering lengths can be considered Kerin and Martin (2020) so the effects of a finite duration quench can be investigated.