⇦Chinese Physics Letters, 2017, Vol. 34, No. 7, Article code 076701 Coupled Two-Dimensional Atomic Oscillation in an Anharmonic Trap * Dong Hu(胡栋), Lin-Xiao Niu(牛临潇), Jia-Hua Zhang(张家华), Xin-Hao Zou(邹新昊), Shu-Yang Cao(曹书阳), Xiao-Ji Zhou(周小计)** Affiliations School of Electronics Engineering and Computer Science, Peking University, Beijing 100871 Received 9 February 2017 *Supported by the State Key Development Program for Basic Research of China under Grant No 2016YFA0301501, and the National Natural Science Foundation of China under Grant Nos 61475007, 11334001 and 91336103.
**Corresponding author. Email: xjzhou@pku.edu.cn Citation Text: Hu D, Niu L X, Zhang J H, Zou X H and Cao S Y et al 2017 Chin. Phys. Lett. 34 076701 Abstract In atomic dynamics, oscillation along different axes can be studied separately in the harmonic trap. When the trap is not harmonic, motion in different directions may couple together. In this work, we observe a two-dimensional oscillation by exciting atoms in one direction, where the atoms are transferred to an anharmonic region. Theoretical calculations are coincident to the experimental results. These oscillations in two dimensions not only can be used to measure trap parameters but also have potential applications in atomic interferometry and precise measurements. DOI:10.1088/0256-307X/34/7/076701 PACS:67.85.Hj, 67.85.Jk, 37.10.Jk © 2017 Chinese Physics Society Article Text Atomic oscillation is a fundamental technique, which has various applications in many aspects, including trap frequency measuring and atom momentum manipulation.[1-4] When the atom is excited along an axis of the trap near the center, we usually carry out a Taylor expansion in the trap center, and accurate to quadratic terms, assuming that the atom would oscillate harmonically in one direction with the corresponding trapping frequency.[5] In another case, when the atom reaches regions far from the trap center, there is also an anharmonic region, where high order terms of the trap begin to show influences,[6,7] and motions in different directions may couple together. Thus a two-dimensional oscillation will appear by exciting atoms only in one direction. The anharmonicity of trapping potential is prominent in many cases, including the large-momentum-transfer (LMT) beam splitter based either on the Bloch oscillation[8,9] or the high order Bragg scattering,[10] and the Bloch oscillation[11] in coordinate space with a large displacement. In such cases, the atom would obtain a large momentum and reach regions far from the trap center during the experiment. In addition, in precise measurement, the physical parameters must be measured with high accuracy, and the high order terms also need to be considered. When atoms are excited to a position far from the trap center, but still can be captured by the trap, the anharmonicity of the trapping potential begins to appear. To the best of our knowledge, for dynamics in this region, the anharmonicity has not been measured in detail before. In this study, by applying the nonholonomic coherent control method,[12,13] we excite atoms to high momentum states, and observe a coupled oscillation in two axes. This effect shows the influence of high order terms in the trapping field. In addition, we find that this oscillation mode is highly related to the trapping parameters, which shows a beat signal of trap frequency in two directions, and from the amplitude of oscillation we can obtain the offset between optical and magnetic field centers with an accuracy of 1 μm. In this Letter, we first analyze the structure of the hybrid trap and give a criterion of the anharmonic region. Then, the experimental results of atomic oscillation in this anharmonic region are shown by the excited atoms in the axial direction. Moreover, we analyze the observation mainly in the radial direction with both classical and quantum theories, and two theories are compared to emphasize the influence of the distribution of atoms. Finally, we discuss the breathing mode we observed and show the application of our observation. Before introducing the experimental measurement, we first analyze the configuration of our hybrid trap. The hybrid trap is composed of a quadrupole magnetic trap and an optical dipole trap, as shown in Fig. 1(a), where $\hat{x}$, $-\hat{y}$ and $\hat{z}$ denote the axis direction of the optical trap laser, gravity and the probe beam, respectively. The red horizontal line stands for an optical dipole trap (ODT) in the $\hat{x}$ direction, along with a pair of quadrupole coils placed beside the vacuum cell in the $\hat{z}$ axial direction. The yellow line represents a resonant probing light along $\hat{z}$, used for detecting atoms by the time of flight (TOF) method. The magnetic-optical hybrid trap combines the benefits of both magnetic and optical traps.[14-17] To avoid the Majorana transition,[18] the center of the optical dipole trap in the $y$ direction is lower than that of the magnetic trap with the distance $y_{\rm m}$, and the combined potential can be written as $$ U(x,y,z)=U_{\rm B}(x,y,z)+U_D(x,y,z)+mgy,~~ \tag {1} $$ where the magnetic potential $U_{\rm B}(x,y,z)=\mu B' \sqrt{x^2+(y-y_{\rm m})^2+4z^2}$ with magnetic field gradient $B'$ and the Bohr magneton $\mu$, the optical trap $U_D(x,y,z)=-U_0e^{-2(y^2+z^2)/w_0^2}$ is a Gaussian beam along $\hat{x}$, with depth $U_0=\frac{2P}{\pi w_0^2}$ determined by beam waist $w_0$ and light intensity $P$, $m$ is the atomic mass, and $g$ is the acceleration of gravity. In regions near the coordinate origin satisfying $|y|\approx 0$, $|z|\approx 0$ and $|x|,|z| \ll |y-y_{\rm m}|$, the potential can be approximately considered as a harmonic trap $U=\frac{1}{2}(\omega_{x} x^{2}+\omega_{y} y^{2}+\omega_{z} z^{2})$, with $w_x=\sqrt{\frac{uB'}{my_{\rm m}}}$, $w_y=\sqrt{\frac{4U_0}{mw^2_0}}$, $w_z=\sqrt{\frac{4U_0}{mw^2_0}-\frac{4uB'}{my_{\rm m}}}$. The harmonic approximation is fulfilled when atoms are located in the small area near the center of optical potential, as shown in Fig. 1(b). The potential is parabolic in three different directions, where both $\frac{\partial U^{2}}{\partial x\partial y}$ and $\frac{\partial U^{2}}{\partial y\partial x}$ are zero. Under this condition the movements of atoms in these two directions can be separated from each other. In regions far from the trap center along the $x$ direction, with a displacement comparable with or even larger than $|y-y_{\rm m}|$, the $U_{\rm B}({\boldsymbol r})$ term cannot be approximated as parabolic anymore and high order terms must be considered. The $\frac{\partial U^{2}}{\partial x\partial y}$ and $\frac{\partial U^{2}}{\partial y\partial x}$ terms will appear, which means that atomic movement in the axial and radial direction would couple together. As shown in Fig. 1(c), the trap is still near parabolic in the $\hat{x}$ direction, while in the $\hat{y}$ direction the trap becomes asymmetric.
cpl-34-7-076701-fig1.png
Fig. 1. (Color online) (a) The structure of our hybrid trap. The trap consists of a quadrupole magnetic trap and an optical dipole trap. We choose the coordinate origin as the center of the optical trap. Both (b) and (c) are the trap potential, which show the harmonic region (b) near the trap center, and the anharmonic region (c) viewed from a larger scale. (d) The time sequence to prepare the high momentum state. (e) A typical experimental picture after the time of flight.
In the experiment, we obtain a Bose–Einstein condensation (BEC) of $^{87}$Rb atoms with about $2.0\times 10^5$ atoms in the hybrid trap.[19] After the preparation of condensates, to force atoms far away from the trap center along the $\hat{x}$ direction, we prepare condensates into high momentum states by a nonholonomic coherent loading process which is nonadiabatic. Normally this loading process consists of several pulses, which are generated by two counter-propagation red-detuned laser beams with wavelength 852 nm and with lattice intensity of $60E_{\rm r}$. Here $E_{\rm r}={(\hbar k_{\rm L})^2}/{2m}$ is the atomic recoil energy, with $k_{\rm L}$ being the lattice wave vector. To excite an atom to $\pm4\hbar k$, we apply two pulses of lattice light, and the sequence has three parameters, as shown in Fig. 1(d), the pulse durations $\tau_1$, $\tau_3$ for two lattice pulses and the interval $\tau_2$ in between. The optimal parameters are found numerically. We calculate the $\pm4 \hbar k$ momentum component for different time sequences by searching each value of three parameters from 0 μs to 100 μs with step 0.1 μs, and obtain the optimal parameters $\tau_1=4.0$ μs, $\tau_2=51.4$ μs and $\tau_3=1.8$ μs. When most of the atoms are excited to high momentum states with the designed pulse sequence, we keep the atoms evolving in the hybrid trap for a time $t$ and take pictures by the TOF method after 28 ms free expansion. Figure 1(e) gives a typical experimental picture where more than 95% of the atoms are located in the $\pm4 \hbar k$ states, which is in accordance with our theoretical calculation.
cpl-34-7-076701-fig2.png
Fig. 2. (Color online) Atomic oscillation in the axial direction. (a) The TOF pictures for atoms evolving in the hybrid trap in the axial direction for different time. (b) The center position of the atoms and the fitting axial direction (black dots for $4 \hbar k$, red dots for $-4 \hbar k$, and the blue line is the fitting result of $4 \hbar k$).
Figure 2(a) gives the oscillation in the axial direction for each evolving time $t$, where each picture is obtained from three times averaged and the data is measured by 1 ms step. Obviously, when $t$ increases from 0 to 60 ms, the momentum of atoms changes periodically, from $\pm4 \hbar k$ to 0, then to $\mp4 \hbar k$, and then changes back. The movement of atoms along the axial direction looks like a simple harmonic oscillation. From Fig. 2(a), we draw the oscillation of the center of atoms in Fig. 2(b), where the position is chosen as the peak of atoms distribution in the axial direction. The black dots and red dots are corresponding to $4 \hbar k$ and $-4 \hbar k$, respectively, and the line is a theoretical fitting. The center of mass oscillation of $\pm4 \hbar k$ atoms along $\hat{x}$ is fitted with $$ x(t)=A'\sin(2\pi w_x t+\psi)+c,~~ \tag {2} $$ and gets $A'=615(7)$ μm, $w_x=26.4(1)$ Hz which is in accordance with the trapping frequency, while $A'$ is the displacement after 28 ms TOF, which is much larger than the displacement before expansion $A$. For an atom with momentum of $4\hbar k$ we have $A=129$ μm, which is comparable with $y_{\rm m}$. This means that atoms have moved into the anharmonic region of the hybrid trap, as shown in Fig. 1(c), where the potential is not parabolic anymore and the movement along the radial direction may be raised.
cpl-34-7-076701-fig3.png
Fig. 3. (Color online) Atomic oscillation in the radial direction. (a) The experimental TOF pictures for atoms evolving in the radial direction for different times. (b) Center of mass position of the atom in the radial direction. The experimental data (blue dots) and classical dynamics (red solid line) are presented, while results of GPE simulation with and without the interaction term are shown in the brown dashed line and green dash-dotted line, respectively. The black triangles are experimental points with small axial excitations.
At the same time, Fig. 3(a) gives the position of $4 \hbar k$ atoms along the radial direction after the free expansion. It also shows an oscillation with a period of about 18 ms. Meanwhile, the amplitude is increasing from 0 μm to about 150 μm. In Fig. 3(b), the center of mass position of atoms shown in Fig. 3(a) is given in blue dots, which shows a clear oscillation. In comparison, the atoms are excited by a magnetic field with about $0.5\hbar k$ in the axial direction in another experiment, as shown in the black triangles, and there is nearly no change in the radial direction during the evolving process. To analyze this movement in the radial direction, we theoretically simulate the process with both classical dynamics and quantum methods. In classic dynamics, if we excite atoms in the axial direction along the optical beam axis, a sinusoidal oscillation $x=A_0\sin(2\pi w_xt +\psi_0)+x_0$ will appear, where $A_0$, $\psi_0$ and $x_0$ are amplitude, phase and initial position of the oscillation, respectively. Therefore, the force on atoms along the radial direction is $$\begin{align} F_y =\,&\frac{-\mu B(y-y_{\rm m})}{\sqrt{x^2+(y-y_{\rm m})^2+4z^2}}\\ &-\frac{2U_0y}{w_0^2e^{2(y^2+z^2)/w_0^2}}-mg .~~ \tag {3} \end{align} $$ Substituting the equation of motion along the axial direction into Eq. (3), we can obtain the center of mass evolution in the radial direction. Comparing experimental and theoretical results, we can adjust parameters in the equation, such as the light intensity or the magnetic field strength. Furthermore, by manipulating factors such as distance $y_{\rm m}$ between the optical and magnetic centers, we can obtain the movements in the radial direction precisely as the red solid line shown in Fig. 3(b), which fits well with the experiment. In the quantum method, we can use a two-dimensional Gross–Pitaevskii (GP) equation to simulate an atom's evolution, $$ i\hbar \frac{\partial \psi(x,y,t)}{\partial t}=\Big[-\frac{\hbar^2}{2m}\nabla^2+U+\delta|\psi|^2\Big] \psi,~~ \tag {4} $$ where $\delta$ is the interaction term, and $U$ is the external potential. As the movement of atoms along the axial direction is clear, the equation is simplified to the one-dimensional one by assuming the motion in $\hat{x}$ as an ordinary harmonic oscillation, and the potential in the radial direction is changing with time, when the atoms oscillate to different positions. Solving the simplified GP equation, we obtain the wave function $\psi(y,t)$ of the system, and by a fourier transformation we can obtain the wave function in momentum space $|\psi(k,t)|=\int\psi(y,t)e^{-iky}dy$, where $|\psi(k,t)|^2$ denotes position evolution of atoms along $\hat{y}$. The brown dashed line shown in Fig. 3(b) is the calculated result. To study the influence of the interaction on the coupled oscillation in the $\hat{y}$ direction, we set the interaction term in Eq. (4) as zero, and the simulation results are shown as the green dash-dotted line in Fig. 3(b). We can see that the atom position calculated with and without interaction are nearly the same, which means that the interaction term has a slight influence on this dynamics, and the classical oscillation can explain well the motion in a quasi-harmonic trap with large kinetic energy. In the classical version, the dynamics in the radial direction can be seen as a forced oscillation, and the kinetic energy in the radial direction comes from the potential energy in the same direction. The oscillation amplitude is amplified first, then after 60 ms, it would be reduced when the driving force is out-of-phase with the motion. In the experiment we do not see this because the atom in our case would drop out of the trap afterwards. We attribute the difference between classical and quantum simulation to the influence of the atom cloud's shape during the oscillation, for the GPE simulation the atom is considered as a wave function while in the classical simulation an atom is taken as a point particle. As shown in Fig. 3, during this large distance oscillation, we have also measured the width of atom cloud in the TOF images, by calculating the full width at half maximum of the atoms in the radial direction, and observed the monopole oscillation (breathing mode),[20] with an oscillation frequency 101(5) Hz as shown in Fig. 4(a), which is consistent with the theoretical value $\sqrt5\omega_0=97.9$ Hz, with $\omega_0$ being the geometry mean of the trapping frequency.
cpl-34-7-076701-fig4.png
Fig. 4. (Color online) (a) The width oscillation of atoms along the radial direction, where blue dots are experimental results and the red solid line is numerical fitting. (b) The maximum oscillation amplitude in the radial direction with different excitation strengths.
Figure 4(b) shows the classical simulation for different excitation strengths. The distance of the atom in the trap is related to the momentum obtained initially. The simulation shows that the maximum oscillation amplitude in the other direction is proportional to the initial momentum the atoms obtained. From the figure, we find that, even for an initial momentum of $1\hbar k$ the atom would displace in the axial direction with 10 μm, which is comparable with the atom size in that direction. The above calculation shows that, even in regions near the trap center, the effect exists. This effect would induce a shift of atom position as well as a phase shift, which acts as a decoherence source for atomic interferometer with atom momentum states. From the experiment, we show that when an atom obtains a high momentum it would go into the nonlinear region of the trap, and its motion in two dimensions is well governed by the trap parameter. From the motion of two separate atom clouds we can adjust the trapping configuration carefully and obtain a more symmetric potential which would be helpful to elongate the coherent time of the atomic interferometer in this kind of traps. Moreover, we can observe a two-dimensional coupled oscillation through exciting atoms in one direction and from a coupled two-dimensional atom interferometer. This interferometer is different from two independent one-dimensional interferometers. It can be used to study the two-dimensional distribution characteristics of a potential field. In conclusion, we have analyzed the structure of a general magnetic-optical hybrid trap in detail, especially taking the high order terms at large displacement from the trap center into consideration. With the designed standing wave pulses, we stimulate atoms into the anharmonic region of the trap and observe a coupled oscillation in orthogonal directions, which help us to measure and adjust the trap configuration. Furthermore, these coupled oscillations by exciting atoms only in one direction may have potential applications in two-dimensional interferometry and precise measurements. References Enhanced oscillation lifetime of a Bose?Einstein condensate in the 3D/1D crossoverObservation of quantum dynamical oscillations of ultracold atoms in the F and D bands of an optical latticeDynamics of Bose-Einstein condensates in optical latticesMany-body physics with ultracold gasesA quantum Newton's cradleDynamics of a cold atom cloud in an anharmonic trapRapidly rotating Bose-Einstein condensates in anharmonic potentialsA Bose-Einstein condensate in an optical latticeLarge Momentum Beam Splitter Using Bloch OscillationsCoherent Splitting of Bose-Einstein Condensed Atoms with Optically Induced Bragg DiffractionInducing Transport in a Dissipation-Free Lattice with Super Bloch OscillationsRapid nonadiabatic loading in an optical latticeLong-time nonlinear dynamical evolution for P -band ultracold atoms in an optical latticeRapid production of R 87 b Bose-Einstein condensates in a combined magnetic and optical potentialSimple method for generating Bose-Einstein condensates in a weak hybrid trapProduction of 87 Rb Bose—Einstein condensates in a hybrid trapHigh-Performance Sodium Bose?Einstein Condensate Apparatus with a Hybrid Trap and Long-Distance Magnetic TransferAtomi orientati in campo magnetico variabileThe Mode Matching of Hybrid Trap by Frequency CalibrationTransverse Breathing Mode of an Elongated Bose-Einstein Condensate
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