Rabi oscillations and Ramsey-type pulses in ultracold bosons: Role of interactions

Our theoretical analysis of the Raman-coupled system is based on the standard mean-field formulation zhai2015 , which writes the mean-field spinor in terms of the components $\psi_{a}({\bf{r}},t)$ and $\psi_{b}({\bf{r}},t)$. Here and in what follows, $\psi_{a}$ and $\psi_{b}$ are time-dependent; note that the discussion in Sec. I adopted a stationary framework for simplicity. The unrotated $2\times 2$ mean-field Hamiltonian $\hat{H}$ reads

	$\displaystyle\hat{H}=$		$\displaystyle\left(\frac{\hat{\bf{p}}^{2}}{2m}+V_{\text{trap}}({\bf{r}},t)\right)\otimes I_{2}+\begin{pmatrix}&g_{aa}|\psi_{a}({\bf{r}},t)|^{2}+g_{ab}|\psi_{b}({\bf{r}},t)|^{2}&0\\ &0&g_{ba}|\psi_{a}({\bf{r}},t)|^{2}+g_{bb}|\psi_{b}({\bf{r}},t)|^{2}\\ \end{pmatrix}+$		(17)
			$\displaystyle\begin{pmatrix}&0&\frac{\Omega_{R}(t)}{2}\exp(-2\imath k_{R}z+\imath\omega_{R}t)\\ &\frac{\Omega_{R}(t)}{2}\exp(2\imath k_{R}z-\imath\omega_{R}t)&E_{\text{Zeeman}}\end{pmatrix},$

where $I_{2}$ is the $2\times 2$ identity matrix and the normalization, expressed in terms of the fractional populations $N_{a}$ and $N_{b}$, is

$\displaystyle N_{a}+N_{b}=1$

(18)

with ($j=a$ or $b$)

$\displaystyle N_{j}=\int|\psi_{j}({\bf{r}},t)|^{2}d{\bf{r}}.$

(19)

The interaction strengths $g_{ij}$ between atoms in hyperfine states $i$ and $j$ are given by

$\displaystyle g_{ij}=\frac{4\pi\hbar^{2}(N-1)a_{ij}}{m}.$

(20)

For ⁸⁷Rb, we have $a_{aa}=100.4$ $a_{\text{B}}$, $a_{ab}=a_{ba}=100.4$ $a_{\text{B}}$, and $a_{bb}=100.9$ $a_{\text{B}}$ scatteringlength_rb , where $a_{\text{B}}$ denotes the Bohr radius. In the arguments presented in Sec. I, the four $g_{ij}$ were assumed to be the same; this simplifying assumption is again made in Sec. III.5. The time dynamics of the system is governed by

$\displaystyle\imath\hbar\frac{\partial}{\partial t}\begin{pmatrix}\psi_{a}({\bf{r}},t)\\ \psi_{b}({\bf{r}},t)\end{pmatrix}=\hat{H}\begin{pmatrix}\psi_{a}({\bf{r}},t)\\ \psi_{b}({\bf{r}},t)\end{pmatrix}.$

(21)

Defining the rotated states $\tilde{\psi}_{a}({\bf{r}},t)$ and $\tilde{\psi}_{b}({\bf{r}},t)$,

$\displaystyle\begin{pmatrix}\tilde{\psi}_{a}({\bf{r}},t)\\ \tilde{\psi}_{b}({\bf{r}},t)\end{pmatrix}=\hat{U}(z,t)\begin{pmatrix}\psi_{a}({\bf{r}},t)\\ \psi_{b}({\bf{r}},t)\end{pmatrix},$

(22)

in terms of the rotation operator $\hat{U}(z,t)$,

$\displaystyle\hat{U}(z,t)=\begin{pmatrix}&1&0\\ &0&\exp(-2\imath k_{R}z+\imath\omega_{R}t)\end{pmatrix},$

(23)

we obtain Eq. (21) with ${\psi}_{a}({\bf{r}},t)$, ${\psi}_{b}({\bf{r}},t)$, and $\hat{H}$ replaced by $\tilde{\psi}_{a}({\bf{r}},t)$, $\tilde{\psi}_{b}({\bf{r}},t)$, and $\hat{\tilde{H}}$, respectively, where the rotated Hamiltonian $\hat{\tilde{H}}$ is given by

	$\displaystyle\hat{\tilde{H}}=$		$\displaystyle\left(\frac{\hat{\bf{p}}^{2}}{2m}+V_{\text{trap}}({\bf{r}},t)\right)\otimes I_{2}+\begin{pmatrix}&g_{aa}|\tilde{\psi}_{a}({\bf{r}},t)|^{2}+g_{ab}|\tilde{\psi}_{b}({\bf{r}},t)|^{2}&0\\ &0&g_{ba}|\tilde{\psi}_{a}({\bf{r}},t)|^{2}+g_{bb}|\tilde{\psi}_{b}({\bf{r}},t)|^{2}\\ \end{pmatrix}+$
			$\displaystyle\begin{pmatrix}&0&\frac{\Omega_{R}(t)}{2}\\ &\frac{\Omega_{R}(t)}{2}&\frac{2\hbar k_{R}\hat{p}_{z}}{m}+\delta_{R}\end{pmatrix}.$

To obtain Eq. (III.1), we used the relation $|\tilde{\psi}_{j}({\bf{r}},t)|^{2}=|\psi_{j}({\bf{r}},t)|^{2}$, where $j=a$ or $b$. Importantly, the position- and time-dependent phase $\tilde{\gamma}_{b}({\bf{r}},t)$ of the rotated component $\tilde{\psi}_{b}({\bf{r}},t)$ differs from the phase ${\gamma}_{b}({\bf{r}},t)$ of the unrotated component ${\psi}_{b}({\bf{r}},t)$. Since the change of the phases of the unrotated spinor components is dominated by the laser coupling term, thereby masking the change due to the internal dynamics, it is more convenient to analyze the phases of the spinor-components in the rotated basis, whose phase dynamics is governed by “internal effects” as opposed to the laser coupling.

For the Rabi oscillation measurements and the Ramsey-type pulse sequence, the BEC is initially (i.e., at $t=0$) prepared in the state $\psi_{a}({\bf{r}},t)\lvert\uparrow\rangle=\tilde{\psi}_{a}({\bf{r}},t)\lvert\uparrow\rangle$, which is characterized by a vanishing average mechanical momentum along the $z$-direction, i.e., $\langle\hat{p}_{z}\rangle_{\text{initial}}=0$. Our calculations assume an axially symmetric harmonic trap with $\omega_{x}=\omega_{y}=\omega_{\rho}$. The trapping potential defines the harmonic oscillator lengths $a_{\text{ho},z}$ and $a_{\text{ho},\rho}$,

$\displaystyle a_{\text{ho},z/\rho}=\sqrt{\frac{\hbar}{m\omega_{z/\rho}}}.$

(25)

The coupled mean-field equations are solved using standard techniques. The initial state is obtained by imaginary time propagation. The real time dynamics is implemented by expanding the time evolution operator in terms of Chebychev polynomials chebychev1 ; chebychev2 . We use equally spaced grid points in $z$ and $\rho$. The convergence of the results presented has been tested with respect to the size of the simulation box, the number of grid points, and the time step.

This section discusses Rabi oscillation results for the Raman coupling case with $\delta_{R}=0$. The numerical solutions are obtained by solving the time-dependent mean-field equation for $\hat{\tilde{H}}$ [see Eq. (III.1)] with $E_{R}/h=1960$ Hz. The Raman coupling is turned on at $t=0$, i.e., we have $t_{\text{start}}=0$. Figs. 2(a)-2(c) show the difference $N_{a}-N_{b}$ between the fractional populations as a function of the dimensionless time $t_{\text{seq}}\Omega_{0,R}/h$ for different $N$, $\omega_{z}$, and $\Omega_{0,R}$, respectively.

Figure 2(a) shows numerical results for $\Omega_{0,R}=E_{R}$ and three different $N$, namely $N=1$, $3\times 10^{5}$, and $N=10^{6}$. Even though the Rabi coupling lasers are turned on, at time $t=0$, after the trapping potential has been switched off, the figure caption quotes the trapping frequencies since they determine the initial state and thus the distribution of the kinetic and potential energy, including the mean-field energy, in the system. For the non-interacting single-atom system [black solid line in Fig. 2(a)], the Rabi oscillation period is nearly constant for the times considered; the amplitude, however, is visibly damped. While this “non-perfect” sinusoidal behavior might be surprising at first sight, it can be explained as follows: The center of mass of the component $\tilde{\psi}_{b}({\bf{r}},t)$ moves relative to the center of mass of the component $\tilde{\psi}_{a}({\bf{r}},t)$ during the Rabi oscillations. Thus, the two components are not perfectly overlapping spatially. As a consequence, the relative phase of the spinor components at fixed ${\bf{r}}$ is changing slightly due to the relative motion of the two components with respect to each other. This phase difference is responsible for the non-perfect population transfer (“damping”). An alternative but equivalent picture is that the finite momentum width of the initial state corresponds to a small effective momentum-dependent detuning. This effective detuning decreases with increasing mean-field interactions due to the decrease of the width of the initial state in momentum space.

When mean-field interactions are present [the red dashed and blue dotted lines in Fig. 2(a) are for $N=3\times 10^{5}$ and $N=10^{6}$, respectively], the amplitude of the Rabi oscillation data changes somewhat while the oscillation period is essentially unaffected by the interactions. For these two $N$-values, the chemical potential (in units of $h$) of the initial state is $\approx 3303$ Hz and $5347$ Hz (corresponding to $1.685E_{R}$ and $2.728E_{R}$), i.e., the chemical potential at $t=0$ is larger than $\Omega_{0,R}$.

To highlight the effect of the interactions, Fig. 2(b) shows the Raman-induced Rabi oscillations for $N=3\times 10^{5}$ [same atom number as used for the red dashed line in Fig. 2(a)] for weaker and stronger confinement along the $z$-direction than used in Fig. 2(a). Stronger confinement leads to higher density and thus to enhanced interaction effects. For the largest $\omega_{z}$ considered [blue dotted line in Fig. 2(b)], the fractional population difference $N_{a}-N_{b}$ deviates appreciably from a simple sinusoidal curve after a few oscillations. This indicates that care needs to be taken when calibrating the effective Raman coupling strength $\Omega_{0,R}$; in particular, a fit to a simple sinusoidal function, applicable to the ideal two-level model, might not yield the correct effective coupling strength.

The results presented in Figs. 2(a)-2(b) are for $\Omega_{0,R}=E_{R}$. Figure 2(c) shows the dynamics for smaller and larger coupling strengths, namely $\Omega_{0,R}=E_{R}/2$, $\Omega_{0,R}=3E_{R}/2$, $\Omega_{0,R}=5E_{R}/2$, and the same trap confinement as in Fig. 2(a). Even though the particle number is quite moderate (namely, $N=3\times 10^{5}$), the oscillation amplitude and period for $\Omega_{0,R}=E_{R}/2$ [solid line in Fig. 2(c)] deviate strongly from perfect sinusoidal behavior due to the enhanced effects of the mean-field energy with decreasing $\Omega_{0,R}$. This implies that care needs to be exercised if the calibration of the effective Raman coupling strength is done for low coupling strengths. In this regime, one has to make sure that the particle number is sufficiently low or that one allows for sufficient time-of-flight expansion prior to turning on the Raman Rabi coupling (Fig. 2 is for $t_{\text{start}}=0$). If this is not done, the value of the effective Rabi coupling strength, which enters into the underlying system Hamiltonian, may be impacted by interaction effects, potentially leading to errors in experiments that require reliable precision, such as quantum analog simulations and many-body studies. Alternatively, explicit comparisons with Gross-Pitaevskii equation results, as done in this work, would be very useful when interactions are present. Last, one may consider performing the calibration in the “large power” regime and extrapolating the resulting calibration curve instead of performing the calibration in the “low power” regime.

Figure 3 shows a comparison between theory and experiment for the Raman-induced Rabi oscillations for an initial state with chemical potential $\mu=0.8436$ $E_{R}$. This chemical potential corresponds to a mean-field energy per particle at $t=0$ of $0.5206$ $E_{R}$. To reduce the mean-field energy in the system, a $0.5$ ms free expansion step was inserted after tuning off the trap and prior to turning on the Raman coupling lasers. At the end of the free expansion, the mean-field energy per particle is about $20$ % smaller than at $t=0$. The resulting Rabi oscillations are slightly damped. Although the experimental data (red dots) are obtained for a small negative detuning $\delta_{R}$, the detuning is not the only cause for the damping. Extrapolating the mean-field Gross-Pitaevskii results for finite detuning to zero detuning, we conclude that even the zero detuning case displays damping [explicit calculations for $\delta_{R}=0$ (not shown) confirm this]. Combining the good agreement between the experimental data and theoretical curves with the discussion of the previous section, we conclude that the damping can be partially attributed to the mean-field interactions. Indeed, if we let the BEC expand longer prior to turning on the Raman coupling, the damping or dephasing, for the same detuning $\delta_{R}$, is reduced.

Interestingly, fitting of the mean-field Gross-Pitaevskii results for $\delta_{R}/h=-200$ Hz (this detuning gives the best agreement with the experimental data) to a damped periodic function of the form

$\displaystyle N_{a}-N_{b}=\cos(2\pi ft)\exp(-t/\tau)$

(26)

yields a frequency $f$ of $2578$ Hz. This “fitted Rabi coupling strength” is about 1.2 % larger than the Rabi coupling strength $\Omega_{0,R}$ used in the simulations. This indicates that the interactions impact, for the parameter combinations considered, the oscillation frequency much less than the amplitude. More specifically, for the parameter combination considered in Fig. 3, the effect of the interactions on the Rabi oscillations can be described, to a good approximation, phenomenologically by the time constant $\tau$.

Throughout this section, the initial BEC ($N=3\times 10^{5}$) is prepared for a confinement with $\omega_{\rho}=2\pi\times 200$ Hz and $\omega_{z}=2\pi\times 40$ Hz. The dynamics are analyzed for the Ramsey-type pulse sequence with Raman coupling strength $\Omega_{0,R}=E_{R}$, detuning $\delta_{R}=0$, and—as in Sec. III.2—$E_{R}/h=1960$ Hz and $t_{\text{start}}=0$. The main emphasis lies on developing, motivated by numerical simulations of the time-dependent mean-field equation for the Hamiltonian given in Eq. (III.1), a benchmark and physical picture that provides the motivation for the analytical treatment presented in Sec. III.5.

When the Raman coupling is turned on, population is transferred from the component $\tilde{\psi}_{a}({\bf{r}},t)$ to the component $\tilde{\psi}_{b}({\bf{r}},t)$. As discussed in the previous sections, the interactions can notably impact the Rabi oscillations, in particular for longer times. Despite of this, we measure the lengths of our pulses in terms of the characteristic time scale of the non-interacting system, i.e., we refer to a $\pi/2$-pulse as a pulse that transfers, in the absence of interactions and assuming an infinitely narrow momentum space wave packet, half of the atoms from the state $\tilde{\psi}_{a}({\bf{r}},t)$ to the state $\tilde{\psi}_{b}({\bf{r}},t)$. For the parameters employed in Fig. 4, a $\pi/2$-pulse corresponds to $\pi h/(2\Omega_{0,R})\approx 0.1276$ ms. Figure 4(ai) shows that the population of state $\tilde{\psi}_{b}({\bf{r}},t)$ after the first $\pi/2$-pulse is close to $50$ % (it is $49.95$ %). In addition, it can be seen that the population in the component $\tilde{\psi}_{b}({\bf{r}},t)$ moves a tiny bit relative to the population in the component $\tilde{\psi}_{a}({\bf{r}},t)$ during the first $\pi/2$-pulse. The reason is that the population in state $\tilde{\psi}_{b}({\bf{r}},t)$ has an average mechanical momentum of about $2\hbar k_{R}$ along the $z$-direction while the population in state $\tilde{\psi}_{a}({\bf{r}},t)$ has an average mechanical momentum very close to zero along the $z$-direction.

During the variable hold time, no population transfer occurs since the Raman coupling lasers are turned off. The two key characteristics during the hold time are: First, the population in state $\tilde{\psi}_{b}({\bf{r}},t)$ continues to move relative to that in component $\tilde{\psi}_{a}({\bf{r}},t)$. Second, the interacting BEC expands a tiny bit. Figures 4(aii) and 4(aiii) show $\rho=0$ cuts and are for $t_{\text{hold}}=1$ ms and $t_{\text{hold}}=2$ ms, respectively.

To “reunite” the populations of the states $\tilde{\psi}_{a}({\bf{r}},t)$ and $\tilde{\psi}_{b}({\bf{r}},t)$, a second $\pi/2$-pulse is applied. For the example shown in Fig. 4, the population of state $\tilde{\psi}_{a}({\bf{r}},t)$ after the second $\pi/2$-pulse is $51.74$ % for $t_{\text{hold}}=1$ ms and $50.15$ % for $t_{\text{hold}}=2$ ms. After the second $\pi/2$-pulse, the density profiles [see Figs. 4(aiv)-4(av) for $\rho=0$ cuts] show interference fringes in the “central” or “overlap” region, i.e., in the spatial region where the two clouds overlapped prior to the application of the second $\pi/2$-pulse. The interference fringes establish themselves through out-of-phase oscillations of $|\tilde{\psi}_{a}(z,0)|^{2}$ and $|\tilde{\psi}_{b}(z,0)|^{2}$. We observe analogous fringes for other $\rho$ values. The total density of the two components (not shown), in contrast, exhibits oscillations with comparatively small amplitude in the outer region and no oscillations in the central region.

Quite generally, the appearance of fringes such as those displayed in Figs. 4(aiv) and 4(av) suggests the existence of two interfering pathways, i.e., the existence of a spatially dependent phase difference. In the following, we introduce a theoretical framework that highlights how the fringe pattern develops for the Ramsey-type pulse sequence with Raman coupling. To this end, it is instructive to visualize the time-evolving rotated spinor on the Bloch sphere. Since the two components $\tilde{\psi}_{a}({\bf{r}},t)$ and $\tilde{\psi}_{b}({\bf{r}},t)$ can each be written in terms of one complex number for each $z$, $\rho$, and $t$ (the axial symmetry suggests the use of cylindrical coordinates), we define

	$\displaystyle\begin{pmatrix}\tilde{\psi}_{a}(z,\rho,t)\\ \tilde{\psi}_{b}(z,\rho,t)\end{pmatrix}=R(z,\rho,t)\exp\left[\imath\tilde{\gamma}_{a}(z,\rho,t)\right]\times$
	$\displaystyle\begin{pmatrix}\cos\big{(}\frac{\theta(z,\rho,t)}{2}\big{)}\\ \exp\left[\imath\phi(z,\rho,t)\right]\sin\big{(}\frac{\theta(z,\rho,t)}{2}\big{)}\end{pmatrix},$		(27)

where

$\displaystyle R(z,\rho,t)=\sqrt{|\tilde{\psi}_{a}(z,\rho,t)|^{2}+|\tilde{\psi}_{b}(z,\rho,t)|^{2}},$

(28)

$\displaystyle\theta(z,\rho,t)=2\;\text{arctan}\left(\frac{|\tilde{\psi}_{b}(z,\rho,t)|}{|\tilde{\psi}_{a}(z,\rho,t)|}\right),$

(29)

and

$\displaystyle\phi(z,\rho,t)=\tilde{\gamma}_{b}(z,\rho,t)-\tilde{\gamma}_{a}(z,\rho,t).$

(30)

Here, $\tilde{\gamma}_{a}(z,\rho,t)$ can be interpreted as an overall spatially dependent phase of the spinor wave function. This phase has no effect on the physical observables considered in this work. The quantity $R(z,\rho,t)$ corresponds to a “weight” at each spatial point. The spinor dynamics for a given $z$ and $\rho$ is thus conveniently visualized by a vector of length $R(z,\rho,t)$ on the Bloch sphere. The direction of the vector is given by $\theta(z,\rho,t)$ and the relative phase $\phi(z,\rho,t)$ between the components $\tilde{\psi}_{b}(z,\rho,t)$ and $\tilde{\psi}_{a}(z,\rho,t)$.

To visualize the motion of the spinor on the Bloch sphere, we define the local spin expectation values $\sigma_{j}(z,\rho,t)$, where $j=x$, $y$, or $z$, through

$\displaystyle\sigma_{j}(z,\rho,t)=\begin{pmatrix}[\tilde{\psi}_{a}(z,\rho,t)]^{*}\\ [\tilde{\psi}_{b}(z,\rho,t)]^{*}\end{pmatrix}^{T}\hat{\sigma}_{j}\begin{pmatrix}\tilde{\psi}_{a}(z,\rho,t)\\ \tilde{\psi}_{b}(z,\rho,t)\end{pmatrix},$

(31)

where $\hat{\sigma}_{x}$, $\hat{\sigma}_{y}$, and $\hat{\sigma}_{z}$ denote the “usual” Pauli matrices. Note that we use the term “spin expectation value” for convenience throughout this paper even though our definition in Eq. (31) excludes the conventional $\hbar/2$ factor. Physically, $\sigma_{z}(z,\rho,t)$ corresponds to the local $(z,\rho)$-specific population difference at time $t$. Mathematically, one finds

$\displaystyle\sigma_{x}(z,\rho,t)=|R(z,\rho,t)|^{2}\cos(\phi(z,\rho,t))\sin(\theta(z,\rho,t)),$

(32)

$\displaystyle\sigma_{y}(z,\rho,t)=|R(z,\rho,t)|^{2}\sin(\phi(z,\rho,t))\sin(\theta(z,\rho,t)),$

(33)

and

$\displaystyle\sigma_{z}(z,\rho,t)=|R(z,\rho,t)|^{2}\cos(\theta(z,\rho,t)).$

(34)

The initial state at $t=0$ corresponds to a vector pointing to the north pole on the Bloch sphere. From the Hamiltonian $\hat{\tilde{H}}$ [Eq. (III.1)], it can be seen that the non-vanishing Raman coupling term introduces a torque along the positive $x$-axis. Thus, neglecting interactions, the first $\pi/2$-pulse rotates the spinor wave function by $-\pi/2$ about the $x$-axis on the Bloch sphere. As a result, the spinor points along the negative $y$-axis on the Bloch sphere after the first $\pi/2$-pulse. Our numerical mean-field results, which show that $\theta$ and $\phi$ are approximately equal to $\pi/2$ and $-\pi/2$ across the entire BEC after the first $\pi/2$-pulse are consistent with this simple picture. Correspondingly, $\sigma_{z}(z,0,t)$ is approximately zero and $\sigma_{y}(z,0,t)$ has—except for a minus sign—the same $z$-dependence as the density [red dashed and black solid lines Fig. 4(avi)]. We conclude that mean-field effects can, in a first-order approximation, be neglected during the first $\pi/2$-pulse.

During the hold time, two effects need to be accounted for: First, as already pointed out earlier, the population in state $\tilde{\psi}_{b}({\bf{r}},t)$ moves relative to that in state $\tilde{\psi}_{a}({\bf{r}},t)$. Second, the phases of the spinor components $\tilde{\psi}_{a}(z,\rho,t)$ and $\tilde{\psi}_{b}(z,\rho,t)$ evolve independently. For each $(z,\rho)$, the combination of these two effects leads to a rotation of the two-component spinor on the Bloch sphere. As an example, Figs. 4(avii)-4(aviii) show $\sigma_{z}(z,0,t)$ and $\sigma_{y}(z,0,t)$ for two different hold times. For both hold times, $\sigma_{z}(z,0,t)$ changes approximately linearly with $z$ in the region where the two components overlap. This follows immediately from the approximately parabolic shapes of the two density components, which are offset from each other: the difference leads to a term that is, to leading order, linear in $z$. The local spin expectation value $\sigma_{y}(z,0,t)$ develops “wiggles” during the hold time in the region where the two components overlap. The number of wiggles increases with increasing hold time. The wiggles arise from the relative phase dynamics and indicate interference; importantly, the densities do not show any indication of interference prior to the application of the second $\pi/2$-pulse.

The second $\pi/2$-pulse “rotates” $\sigma_{y}(z,\rho,t)$ to $\sigma_{z}(z,\rho,t)$. This can be seen clearly by comparing Fig. 4(aix) with Fig. 4(avii) (both these figures are for a hold time of $1$ ms) and by comparing Fig. 4(ax) with Fig. 4(aviii) (both these figures are for a hold time of $2$ ms). Since the relative phase information has been “moved” from $\sigma_{y}(z,\rho,t)$ to $\sigma_{z}(z,\rho,t)$ and since $\sigma_{z}(z,\rho,t)$ is equal to $|\tilde{\psi}_{a}(z,\rho,t)|^{2}-|\tilde{\psi}_{b}(z,\rho,t)|^{2}$, the interference is—after the second $\pi/2$-pulse—visible in the densities of the components [Figs. 4(aiv)-4(av)].

The numerical results presented in the previous section are obtained using the scattering lengths $a_{ij}$ for ⁸⁷Rb. Repeating the numerical calculations for equal scattering lengths $a_{ij}$ reveals that the effects due to the difference in the scattering lengths are quite small for the time scales considered in this paper. Motivated by this observation, the analytical treatment presented in this section makes the simplifying assumption that the scattering lengths are all equal ($a_{aa}=a_{ab}=a_{ba}=a_{bb}$). Moreover, the treatment assumes that $\delta_{R}$ and $t_{\text{start}}$ are equal to zero. Our analytical model is motivated by Refs. castin1996 ; other_castin ; however, the application to the Ramsey-type pulse sequence discussed here has, to the best of our knowledge, not been discussed previously.

We assume that the initial state $\tilde{\psi}_{a}({\bf{r}},t=0)$ is described well within the Thomas-Fermi approximation. We additionally assume that the population transfer process commutes with the relative moving and expansion processes during the first $\pi/2$-pulse. Specifically, our analytical model treats the population transfer associated with the first $\pi/2$-pulse as occuring instantaneously and then subsequently treats the relative moving and expansion of the two components for the duration $\tau_{1}$ of the actual $\pi/2$-pulse. For the second $\pi/2$-pulse, we reverse the order of the operations, i.e., we first treat the relative moving and expansion of the two components for the duration $\tau_{2}$ and then treat the population transfer associated with the second $\pi/2$-pulse as occuring instantaneously. In the following, we provide an analytical framework that yields approximate expressions for $\tilde{\psi}_{a}(z,\rho,t)$ and $\tilde{\psi}_{b}(z,\rho,t)$ right after the first instantaneous $\pi/2$-pulse ($t=0^{+}$), during the time $0^{+}<t<\tau_{1}+t_{\text{hold}}+\tau_{2}$, and right after the second instantaneous $\pi/2$-pulse ($t=t_{\text{end}}^{+}$).

Motivated by the discussion in Sec. III.2, we make the ansatz that $|\tilde{\psi}_{a}({\bf{r}},t)|^{2}$ has an inverted parabola-like form during the “effective” hold time, i.e., for $0^{+}<t<t_{\text{end}}$,

	$\displaystyle|\tilde{\psi}_{a}({\bf{r}},t)|^{2}=$
	$\displaystyle\frac{1}{\lambda_{z}(t)\lambda_{\rho}^{2}(t)}\left[-\alpha_{z}\left(\frac{z}{\lambda_{z}(t)}\right)^{2}-\alpha_{\rho}\left(\frac{\rho}{\lambda_{\rho}(t)}\right)^{2}+\frac{\mu}{2g}\right],$		(35)

where $\mu$ denotes the chemical potential of the initial state, i.e., of the system prior to the application of the first $\pi/2$-pulse stringariRMP ,

$\displaystyle\mu=\frac{1}{2}\left[\omega_{\rho}^{4}\omega_{z}^{2}m^{3}\left(\frac{15g}{4\pi}\right)^{2}\right]^{1/5},$

(36)

and

$\displaystyle\alpha_{z/\rho}=\frac{m\omega_{z/\rho}^{2}}{4g}.$

(37)

Compared to the “standard case” castin1996 , $\alpha_{\rho}$ and $\alpha_{z}$ are smaller by a factor of 2 since the population for $t=0^{+}$ is assumed to be equally distributed between the two components, i.e., $|\tilde{\psi}_{b}({\bf{r}},0^{+})|=|\tilde{\psi}_{a}({\bf{r}},0^{+})|$. In Eq. (III.5), it is understood that $|\tilde{\psi}_{a}({\bf{r}},t)|$ is zero when the right hand side of the equation takes negative values. The dimensionless scaling parameters $\lambda_{z}(t)$ and $\lambda_{\rho}(t)$ obey the initial conditions $\lambda_{z}(0^{+})=\lambda_{\rho}(0^{+})=1$. The differential equations that govern the time evolution of $\lambda_{z}(t)$ and $\lambda_{\rho}(t)$ are discussed below. We set $\tilde{\gamma}_{a}({\bf{r}},0^{+})=0$ and assume that the first $\pi/2$-pulse introduces a $-\pi/2$ phase shift onto the second component, i.e., $\tilde{\gamma}_{b}({\bf{r}},0^{+})=-\pi/2$.

For $0^{+}<t<t_{\text{end}}$, $\tilde{\psi}_{b}({\bf{r}},t)$ moves with approximately constant velocity $v_{z}$,

$\displaystyle v_{z}=\frac{2\hbar k_{R}}{m},$

(38)

along the $z$-direction while the center-of-mass of $|\tilde{\psi}_{a}({\bf{r}},t)|^{2}$ remains essentially unchanged. Due to the symmetry of the system, we enforce

$\displaystyle\tilde{\psi}_{b}({\bf{r}},t)=\tilde{\psi}_{a}(v_{z}t\hat{e}_{z}-{\bf{r}},t)\exp\left(-\imath\frac{\pi}{2}\right).$

(39)

Thus, once we have expressions for $|\tilde{\psi}_{a}(v_{z}t\hat{e}_{z}-{\bf{r}},t)|$ and $\tilde{\gamma}_{a}(v_{z}t\hat{e}_{z}-{\bf{r}},t)$, $\tilde{\psi}_{b}({\bf{r}},t)$ is determined through Eq. (39). To eliminate $\tilde{\psi}_{b}({\bf{r}},t)$, we insert Eq. (39) into the coupled set of time-dependent mean-field equations. This yields

$\displaystyle\imath\hbar\frac{\partial}{\partial t}\tilde{\psi}_{a}({\bf{r}},t)=\hat{\tilde{H}}_{\text{hold}}\tilde{\psi}_{a}({\bf{r}},t),$

(40)

where

$\displaystyle\hat{\tilde{H}}_{\text{hold}}=\frac{\hat{\bf{p}}^{2}}{2m}+g|\tilde{\psi}_{a}({\bf{r}},t)|^{2}+g|\tilde{\psi}_{a}(v_{z}t\hat{e}_{z}-{\bf{r}},t)|^{2}.$

(41)

From Eq. (III.5), we find

	$\displaystyle|\tilde{\psi}_{a}(v_{z}t\hat{e}_{z}-{\bf{r}},t)|^{2}=$
	$\displaystyle|\tilde{\psi}_{a}({\bf{r}},t)|^{2}+\frac{2\alpha_{z}v_{z}tz}{\lambda_{z}^{3}(t)\lambda_{\rho}^{2}(t)}-\frac{\alpha_{z}v_{z}^{2}t^{2}}{\lambda_{z}^{3}(t)\lambda^{2}_{\rho}(t)}.$		(42)

Plugging Eq. (III.5) into Eq. (41), we obtain

$\displaystyle\hat{\tilde{H}}_{\text{hold}}=\frac{\hat{\bf{p}}^{2}}{2m}+2g|\tilde{\psi}_{a}({\bf{r}},t)|^{2}-F_{z}(t)z+C(t).$

(43)

Equation (43) implies that $C(t)$, which is independent of ${\bf{r}}$, contributes an overall phase to $\tilde{\psi}_{a}({\bf{r}},t)$ at each time $t$. The effective time-dependent force $F_{z}(t)$ along the negative $z$-direction,

$\displaystyle F_{z}(t)=-\frac{2g\alpha_{z}v_{z}t}{\lambda_{z}^{3}(t)\lambda_{\rho}^{2}(t)},$

(44)

is due to the relative motion of the two components with respect to each other and the mean-field interactions.

We now make the assumption that the effective force term in Eq. (43) does not notably affect the time evolution of the density $|\tilde{\psi}_{a}({\bf{r}},t)|^{2}$. Under this assumption, the time evolution of the scaling factors $\lambda_{z}(t)$ and $\lambda_{\rho}(t)$ is governed by the differential equations derived by Castin and Dum from the scaling ansatz for a single-component BEC castin1996 :

$\displaystyle\frac{d^{2}\lambda_{z}(t)}{dt^{2}}=\frac{\omega_{z}^{2}}{\lambda_{\rho}^{2}(t)\lambda_{z}^{2}(t)},$

(45)

and

$\displaystyle\frac{d^{2}\lambda_{\rho}(t)}{dt^{2}}=\frac{\omega_{\rho}^{2}}{\lambda_{\rho}^{3}(t)\lambda_{z}(t)}.$

(46)

The black solid lines in Fig. 5(a) show $\lambda_{\rho}(t)$ and $\lambda_{z}(t)$, obtained by solving Eqs. (45)-(46) numerically for the same parameters as those employed in Fig. 4. Using these solutions, the solid line in Fig. 5(b) shows the effective force $F_{z}(t)$ as a function of the dimensionless time $tE_{R}/h$. The magnitude of the effective force first increases and then decreases with increasing time. As shown below, the turn-around time is, to leading order, given by the inverse of the transverse trapping frequency $\omega_{\rho}$.