Two-body quench dynamics of harmonically trapped interacting particles

After a quench the dynamics of the system can be probed through a Ramsey interferometric measurement Cetina et al. (2016); Goold et al. (2011); Knap et al. (2012); Parish and Levinsen (2016), which allows the contrast, to be measured. Previous work regarding the contrast (and other quantities) in one- and two-dimensional systems of two bosons has been undertaken Bougas et al. (2020, 2019); Budewig et al. (2019); Kehrberger et al. (2018).

For our three-dimension system, utilising Eq. (10), the Ramsey signal reduces to

where $\ket{\psi(r)}$ is the initial eigenstate, with eigenenergy $E_{{\rm Rel}}$, of $\hat{H}_{{\rm Rel}}$, with an s-wave scattering length $a_{\rm s}$ and the states $\ket{\psi_{j}^{\prime}(r)}$ are the eigenstates of $\hat{H}^{\prime}_{{\rm Rel}}$, with an s-wave-scattering length $a_{\rm s}^{\prime}$, with corresponding eigenenergies $E_{j}^{\prime}$. The general properties of Eq. (13) can be understood via the realization that $\ket{\psi_{j}^{\prime}(r)}$ rotates at an angular frequency $E_{j}^{\prime}$ and $\left|\bra{\psi(r)}\ket{\psi_{j}^{\prime}(r)}\right|^{2}$ determines the magnitude of the squared overlap of the pre-quench state and the $j^{th}$ post-quench eigenstate.

In general the coefficients, $\left|\bra{\psi(r)}\ket{\psi_{j}^{\prime}(r)}\right|^{2}$, of the sum in Eq. (13) can be evaluated. Specifically, defining $\nu$ to be the pseudo-quantum number for the initial state, $\ket{\psi(r)}$, and $\nu_{j}^{\prime}$ to be the energy pseudo-quantum numbers for the final states, $\ket{\psi_{j}^{\prime}(r)}$, then

where ${}_{3}F_{2}(a,b,c;d,e;f)$ is a generalized hypergeometric function. From Eqs. (7,13,LABEL:eq:coefs) it is possible to numerically evaluate $S(t)$ for any quench from $a_{\rm s}$ to $a_{\rm s}^{\prime}$.

Below we consider two scenarios where the s-wave scattering length is quenched from $a_{\rm s}/a_{{\rm ho}}=0$ to $a_{\rm s}^{\prime}/a_{{\rm ho}}=\pm\infty$ and from $a_{\rm s}/a_{{\rm ho}}=\pm\infty$ to $a_{\rm s}^{\prime}/a_{{\rm ho}}=0$. In these cases exact closed form expressions for $S(t)$ can be found. From Eq. (7), and as can be seen in Fig. 1, in the unitary limit $\nu_{j}\rightarrow j-1/2$. In the non-interacting to unitary case the initial state, $\ket{\psi(r)}$, is the simple harmonic wavefunction with $l=m=0$:

where $Y_{00}(\theta,\phi)$ is the lowest order spherical harmonic.

For the case where $a_{\rm s}/a_{{\rm ho}}=0$ and $a_{\rm s}^{\prime}/a_{{\rm ho}}=\pm\infty$ and the initial state has quantum number $n$,

In the case where the initial state is the ground state ($n=0$) this reduces to

The contrast is plotted in Fig. 2 (solid blue curve for Eq. (17) and dashed-dotted red curve for Eq. (LABEL:eq:S_full) with $n=1$), with the upper figure showing the magnitude, $|S(t)|$, and lower figure showing the phase, $\phi(t)$, where we have parameterized the Ramsey signal as $S(t)=|S(t)|e^{-i\phi(t)}$. In general the contrast oscillates, with period $\pi/\omega$, whilst the phase exhibits a period of $2\pi/\omega$ with a stepping behaviour ($\omega t\approx\pi/2$) that is particularly pronounced in the $n\geq 1$ case. A similar phase feature has been experimentally and theoretically observed for the case of a single impurity in a uniform Fermi sea Cetina et al. (2016).

It is also informative to examine the short time dynamics of the contrast in the limit $\omega t\ll 1$. Expanding Eq. (17) we find

which is also plotted in Fig. 2 (dashed green curve).

Previous calculations which perturbatively evaluate the Ramsey signal for a single Fermi impurity in a uniform sea of fermions provide a useful comparison for the exact two-body results presented above. For such a case, where the interaction is quenched from $a_{\rm s}/a_{{\rm ho}}=0$ to $a_{\rm s}^{\prime}/a_{{\rm ho}}=\pm\infty$ it has been shown that the short time dynamics follow Parish and Levinsen (2016)

where we have assumed that the Fermi energy is $3\hbar\omega/2$.

Equations (18) and (19) show that the short time dynamics are dominated by $1-\alpha t^{3/2}$ in each case. Perhaps what is more remarkable is that for $m_{1}=m_{2}$ we find that for perturbative calculation, Eq. (19), $\alpha=0.829\dots e^{-i\pi/4}\omega^{3/2}$ as compared to $\alpha=0.825\dots e^{-i\pi/4}\omega^{3/2}$ for the two-body calculation, Eq. (18). This supports the idea that the early time dynamics are dominated by two-body effects Sykes et al. (2014) and are independent of the trapping potential Corson and Bohn (2015).

We also consider the reverse case where the system is initially in the unitary limit and quenched to the non-interacting limit. In the unitary case ($a_{\rm s}/a_{{\rm ho}}=\pm\infty$) with pseudo-quantum number $\nu_{j}=j-1/2$:

where

The final states are in the non-interacting limit ($a_{\rm s}^{\prime}/a_{{\rm ho}}=0$), and so are simple harmonic wavefunctions as defined in Eq. (15). As a result the Ramsey signal is evaluated as

where $j$ denotes the excitation of the initial state. In the case where the initial state is the ground state ($j=0$) the Ramsey signal reduces to

For early times ($\omega t\ll 1$) this becomes

The exact ground state result, Eq. (22) (solid blue curve), the early time dynamics, Eq. (23) (dashed green curve), and the exact first excited state result, Eq. (LABEL:eq:S2_full) with $j=1$ (dashed-dotted red curve), are plotted in Fig. 3. In general the contrast oscillates, with period $\pi/\omega$, whilst the phase exhibits a period of $2\pi/\omega$ with a stepping behaviour ($\omega t\approx\pi/2$) that is again particularly pronounced in the $n\geq 1$ case. In contrast to the previous case, shown in Fig. 2, we find that when the system is quenched from unitarity to non-interacting $S(t)$ exhibits non-analytic cusps at $\omega t=0,\pi,2\pi\dots$.

The Ramsey signal has been experimentally measured using Ramsey interferometry techniques Cetina et al. (2016). As outlined by Knap et al. (2012); Parish and Levinsen (2016) the Ramsey signal can be related to the difference in probabilities of different states.

Consider the two particles $1$ and $2$. Suppose that particle $1$ has only one allowable state, $\ket{1}$, and particle $2$ has two: $\ket{2\uparrow}$ and $\ket{2\downarrow}$. We assume that $\ket{2\downarrow}$ does not interact with $\ket{1}$ but $\ket{2\uparrow}$ can, the interactions are parameterized by $a_{\rm s}$.
For a general initial state of the system

For such a state a Ramsey pulse sequence has the following form:

where

and $\phi$ is the phase of the second $\pi/2$ pulse relative to first. Evaluating the probability of particle $2$ being in the $\ket{2\uparrow}$ ($P_{\uparrow}$) and $\ket{2\downarrow}$ ($P_{\downarrow}$) states given that the quench is applied after the first $\pi/2$ pulse results in

and

Hence we can connect an experimentally measurable quantity, ${\rm P}_{\uparrow}-{\rm P}_{\downarrow}$, to the Ramsey signal. In all the observables calculated in this paper we have assumed $a=1$ and $b=0$.

Experiment has shown that it is possible to measure the separation of two trapped ⁶Li atoms following a quench in trap geometry Guan et al. (2019) we believe it is possible to extend this to a quench in interaction strength.

Following the same methodology as presented in the previous section it is possible to evaluate the expectation value of the inter-particle separation via,

We first consider the non-interacting $(a_{\rm s}/a_{\rm ho}=0)$ to interacting $(a^{\prime}_{\rm s}/a_{\rm ho}\neq 0)$ case, where $\psi_{n}(r)$ is the non-interacting simple harmonic oscillator wavefunction, Eq. (15), and $\psi^{\prime}_{j}(r)$ is the interacting wavefunction, Eq. (5), with pseudo-quantum number $\nu_{j}$. In this case

and

Note that by combining Eqs. (LABEL:eq:coefs, 29, LABEL:eq:roverlap) one can calculate the expectation value of the particle separation for the general quench case. In the case where the initial state is the ground state $(n=0)$ and the system is quenched to unitarity ($a^{\prime}_{\rm s}/a_{\rm ho}\rightarrow\infty$ and $\nu_{j}\rightarrow j-1/2$) we find

where $\beta(z,a,b)$ is the incomplete beta function:

Evaluating $\langle\tilde{r}(t)\rangle$ in this case results in an oscillating form, with period $\pi/\omega$, as shown in Fig. 4.

We now turn our attention to the case where the initial state is in the unitary limit $(a_{\rm s}/a_{\rm ho}=\infty)$ and the quench is to the non-interacting limit $(a^{\prime}_{\rm s}/a_{\rm ho}=0)$ where we find

where $j$ denotes the initial state in the unitary limit. Eq. (LABEL:eq:divergent) is logarithmically divergent in the summation. This divergence is most clearly demonstrated in Figs. 5 and 6. Fig. 5 shows, for $j=0$ that although $\langle\tilde{r}(t)\rangle$ is periodic, with period $\pi/\omega$ and the sum converges for $t\omega=k\pi$, where $k$ is an integer, it appears to diverge away from these points. Examining $\langle\tilde{r}(t=\pi/2\omega)\rangle$ elucidates this more clearly as demonstrated in Fig. 6, which plots $\langle\tilde{r}(t=\pi/2\omega)\rangle$ as a function of the maximum number of terms ($N_{\rm max}$) in the sum in Eq. (LABEL:eq:divergent). As can be seen $\langle\tilde{r}(t=\pi/2\omega)\rangle$ diverges logarithmically.

The origins of this divergence can be understood by examining

where for given initial state ($n$)

In the ground state ($n=0$) this reduces to,

In Fig. 7 $I_{0}({\tilde{r}},t=\pi/2\omega,N_{\rm max})$ is plotted for increasing values of the cut-off ($N_{\rm max}$) in the sum over $m$ and $l$. From this it is clear that as $N_{\rm max}$ increases there is $1/\tilde{r}$ tail in $I_{0}({\tilde{r}},t=\pi/2\omega,N)$. A similar analysis for $t=k\pi/\omega$, where $k$ is an integer, reveals that at these specific times $I_{0}({\tilde{r}},t=k\pi/\omega,N_{\rm max})$ does not exhibit a $1/\tilde{r}$ tail and hence the sum is convergent. However, away from $t=k\pi/\omega$ it is this $1/\tilde{r}$ dependence which leads to the divergence in $\langle\tilde{r}(t)\rangle$.

This divergence naturally leads to three questions: where does it come from, what does it mean in the context of experiment and why does this not happen in the reverse case? Answering the second question first: in the context of an experiment this result shows that if the system started in the ground state, with $a_{\rm s}/a_{\rm ho}\rightarrow\pm\infty$, and is quenched to the non-interacting regime an ensemble measurement of the separation between the two particles would exhibit a periodic structure but would also, away from $t=k\pi/\omega$, diverge. This is concerning on many levels and as such it is worth considering the first question which can be addressed by evaluating the wavefunction of the initial state (Eq. (5)), which in the limit $\tilde{r}\rightarrow 0$ approaches $1/\tilde{r}$. A careful analysis of this reveals that completing the sum to $\infty$ in Eq. (5) is crucial in describing the properties of the interacting wavefunction as $\tilde{r}\rightarrow 0$. This means that after the quench has occurred (to the non-interacting regime) this cusp leads to a divergence in momenta which can only be transferred into high-energy non-interacting states. Since this work considers a zero-range interaction this means a divergence in momenta and hence a $1/\tilde{r}$ dependence in $I_{0}({\tilde{r}},t,N\rightarrow\infty)$. This then helps to understand what this means in the context of an actual experiment. In reality there is a cut-off, the interaction is not zero-range. However, this cut-off is short range and can be estimated to be of order $<10^{-9}$m and hence one expects that the oscillations in $\langle\tilde{r}(t=\pi/2\omega)\rangle$ should have an amplitude of order $>7a_{\rm ho}$. This is still a very large oscillation, an order of magnitude larger than the case where the interactions are quenched from $a_{\rm s}/a_{\rm ho}\rightarrow 0$ to $a^{\prime}_{\rm s}/a_{\rm ho}\rightarrow\pm\infty$, see Fig. 4. For the third question; in the non-interacting to unitary case we believe there is no divergence because the wavefunctions being projected onto (the non-interacting states) do not have a cusp at $\tilde{r}=0$ and so there is no corresponding divergence in momenta.