Dynamics of small trapped one-dimensional Fermi gas under oscillating magnetic fields

Deterministic preparation of an ultracold harmonically trapped one-dimensional Fermi gas consisting of a few fermions has been realized by the Heidelberg group. Using Floquet formalism, we study the time dynamics of two- and three-fermion systems in a harmonic trap under an oscillating magnetic field. The oscillating magnetic field produces a time-dependent interaction strength through a Feshbach resonance. We explore the dependence of these dynamics on the frequency of the oscillating magnetic field for noninteracting, weakly interacting, and strongly interacting systems. We identify the regimes where the system can be described by an effective two-state model and an effective three-state model. We find an unbounded coupling to all excited states at the infinitely strong interaction limit and several simple relations that characterize the dynamics. Based on our findings, we propose a technique for driving transition from the ground state to the excited states using an oscillating magnetic field.

Figure 1

(a) The 1D coupling constant $g$ as a function of the magnetic field $B$, calculated using Eqs. (4) and (5). (b) $g\left(t\right)$ as a function of time for an oscillating magnetic field $B\left(t\right)=\overline{B}+bcos\left(\omega t\right)$ with $\overline{B}=550\mathrm{G}$ and $b=6\mathrm{G}$. The solid line is calculated from Eqs. (4, 5, 6, 7). The dots are calculated from the first-order expansion given in Eq. (8).

Figure 2

Dynamics of the (1,1) system under an oscillating magnetic field in the $\overline{g}\to 0$ limit. The driving strength is $d=0.01a_z{E}_{\text{ho}}$ for all cases. The figures remain almost unchanged for different driving strength $d$ for $d<0.2a_z{E}_{\text{ho}}$. (a) The on-resonance case where $\omega =2{\omega }_z$. The solid, dashed, dotted, dash-dotted, and dash-dot-dotted lines show the probability $P_q\left(\tau \right)$ of occupying the state with $q=0$, 1, 2, 3 and any other state as functions of the scaled dimensionless time $\tau =t/(2\pi \hslash a_z/d)$. (b) and (c) The slightly off-resonance cases, where the detunings $\delta$ are $0.15d/\left(\hslash a_z\right)$ and $0.3d/\left(\hslash a_z\right)$, respectively. The solid, dashed, dotted, and dash-dot-dotted lines show $P_q\left(\tau \right)$ for $q=0$, 1, 2, and any other state as a function of $\tau$. (d) The energy spectrum of the (1,1) system near the $\overline{g}\to 0$ limit for even-parity states. The solid, dashed, dotted, and dash-dotted lines show the energy of the four lowest states as a function of $g/\left(a_z{E}_{\text{ho}}\right)$, which are labeled by $q=0$, 1, 2, and 3.

Figure 3

Dynamics of the (2,1) system under an oscillating magnetic field in the $\overline{g}\to 0$ limit. The driving strength is $d=0.01a_z{E}_{\text{ho}}$ for all cases. The figures remains almost unchanged for different driving strength $d$ for $d<0.2a_z{E}_{\text{ho}}$. (a) The on-resonance case where $\omega =2{\omega }_z$. The solid, dashed, dotted, dash-dotted, and dash-dot-dotted lines show the probability $P_n\left(t\right)$ of occupying the odd-parity states with relative eigenenergies $E_n/\hslash {\omega }_z=2$, 4, 6, 6, and any other state as functions of the scaled dimensionless time $\tau =t/(2\pi \hslash a_z/d)$. Note that there are two degenerate states with $E_n/\hslash {\omega }_z=6$ that are affected by the zero-range interaction in the $\overline{g}\to 0$ limit. (b) and (c) The slightly off-resonance cases, where the detunings $\delta$ are $0.15d/\left(\hslash a_z\right)$ and $0.3d/\left(\hslash a_z\right)$, respectively. The solid, dashed, dotted, dash-dotted, and dash-dot-dotted lines show the probability $P_n\left(t\right)$ of occupying the state with $E_n/\hslash {\omega }_z=2$, 4, 6, 6, and any other state as functions of $\tau$. (d) The energy spectrum of the (2,1) system near the $\overline{g}\to 0$ limit for states with odd relative parity. The solid, dashed, dotted, and dash-dotted lines show the states with relative eigenenergies $E_n/\hslash {\omega }_z=2$, 4, 6, and 6. Please note that the dotted and dash-dotted lines are very close to each other. There are one and two additional states with $E_n/\hslash {\omega }_z=4$ and 6, respectively, that are not affected by the interaction and are not shown in the figure.

Figure 4

Dynamics of the (1,1) system under an oscillating magnetic field in the weakly interacting regime. (a) The case where $\overline{g}=-0.1a_z{E}_{\text{ho}}$ and $d=0.02a_z{E}_{\text{ho}}$. The dynamics remains almost the same for different values of $\overline{g}$ and $d$ giving the same $\overline{g}/d$. Solid, dashed, and dash-dotted lines show the probability $P_q\left(t\right)$ of occupying the ground state, the lowest excited state with even parity, and any other state as a function of the scaled dimensionless time $\tau =t/(2\pi \hslash a_z/d)$. (b) The applicability of the two-state model. Circles show the maximum probability of occupying any state other than the two states in the Rabi oscillation $P_{\text{other}}$ as a function of $\overline{g}/d$. (c) The energy spectrum of the (1,1) system near $\overline{g}=-0.1a_z{E}_{\text{ho}}$ for even-parity states. Solid, dashed, and dotted lines show the energy of the three lowest states with even parity.

Figure 5

Dynamics of the (2,1) system under an oscillating magnetic field in the weakly interacting limit. The average coupling constant is $\overline{g}=-0.08a_z{E}_{\text{ho}}$, and the driving strength is $d=0.02a_z{E}_{\text{ho}}$. (a)–(c) Cases with driving frequency $\omega =6.0316{\omega }_z, 6.0266{\omega }_z$ and $6.0251{\omega }_z$, respectively. Solid, dashed, dotted, and dash-dotted lines show the probability $P_q\left(\tau \right)$ of occupying the ground state, $|{\psi }_b\rangle , |{\psi }_a\rangle$, and any other state as functions of the scaled dimensionless time $\tau =t/(2\pi \hslash a_z/d)$. See text for details. (d) The energy spectrum of the (2,1) system near $\overline{g}=-0.08a_z{E}_{\text{ho}}$ for states with odd parity. The solid, dashed, and dotted lines show the energy of ground state, $|{\psi }_b\rangle$, and $|{\psi }_a\rangle$, respectively. The rest of the states are represented by dash-dotted lines. Please note that the dashed and dotted lines are very close to each other. The states not affected by the zero-range interactions are not shown in the figure.

Figure 6

Dynamics of the (1,1) system under an oscillating magnetic field in the $\overline{g}\to \infty$ limit. The driving strength is $h=0.002/\left(a_z{E}_{\text{ho}}\right)$. The driving frequency is $\omega =2{\omega }_z$. (a) The solid lines show the probabilities $P_q\left(\tau \right)$ of occupying the 12 lowest eigenstates as functions of the scaled dimensionless time $\tau =t/[2\pi /\left(h{a}_z\hslash {\omega }_z^2\right)]$. The eigenstates with quantum number $q=1/2, 3/2, ...$ are shown successively by solid lines with darker gray (peaking at a shorter time) to lighter gray (peaking at a longer time). (b) Circles show the scaled peaking time ${\tau }_q$ as a function of $\sqrt{q}$. (c) Squares show the peak probability $P_q\left({\tau }_q\right)$ as a function of $1/\sqrt{q}$. (b) and (c) The lines are linear fits. (d) The energy spectrum for the (1,1) system near the $\overline{g}\to \infty$ limit. The states with lower (higher) energies are shown by solid lines with darker (lighter) gray.

Figure 7

Dynamics of the (1,1) system under an oscillating magnetic field in the $\overline{g}\to \infty$ limit. The driving strength is $h=0.002/\left(a_z{E}_{\text{ho}}\right)$. (a)The case of near-resonant driving frequency $\omega =1.998{\omega }_z$. The detuning is $\delta =1.0h{a}_z\hslash {\omega }_z^2$. Solid lines show the probability of occupying the eight lowest eigenstates as a function of the scaled time $\tau =t/[2\pi /\left(h{a}_z\hslash {\omega }_z^2\right)]$. The eigenstates with quantum number $q=1/2, 3/2, ...$ are shown successively by solid lines with darker to lighter gray. (b) The inverse revival period $1/\overline{\tau }$ for the ground state as a function of scaled detuning $\delta /\left(h{a}_z\hslash {\omega }_z^2\right)$. The solid line shows $1/\overline{\tau }=\left|\delta \right|/\left(h{a}_z\hslash {\omega }_z^2\right)$. (c) The energy spectrum for the (1,1) system near the $\overline{g}\to \infty$ limit. The states with lower (higher) energies are shown by solid lines with darker (lighter) gray.