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Abstract
We investigate a single-atom interferometer (SAI) in an optical dipole trap (ODT) with photon momentum kicks. An echo sequence is used for the SAI. We find experimentally that interference visibilities of a counter-propagating Raman type SAI decay much faster than the co-propagating case. To understand the underlying mechanism, a wave-packet propagating simulation is developed for the ODT-guided SAI. We show that in state dependent dipole potentials, the coupling between external dynamics and internal states makes the atom evolve in different paths during the interfering process. The acquired momentum from counter-propagating Raman pulses forces the external motional wave packets of two paths be completely separated and the interferometer visibility decays quickly compared to that of the co-propagating Raman pulses process. Meanwhile, the echo interference visibility experiences revival or instantaneous collapse which depends on the π pulse adding time at approximate integer multiples or half integer multiples of the trap period.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Matter-wave interferometers have attracted considerable attention over the latest decades because of its extensive applications in precision metrology [1,2], spatial field investigation [3] and testing of foundational knowledge in physics [4–8], especially, they can provide high accuracy in inertial sensing [9–14]. Versatile matter-wave interferometers have been realized with electrons [15], neutrons [16], atoms [17–20] and molecules [21–24]. To enhance sensitivity and improve statistical signal, a common practice is to use large ensembles of atoms or molecules. However, single-particle interferometers are unique in studying fundamental issues like quantum manipulation [25], quantum metrology, building up of quantum interference [26], as well as sensitive applications in precision measurements [27]. So far, for neutral atoms, only a few works on single-atom interferometers (SAIs) have been realized in optical lattice [28], free-space [29] and blue detuned bottle trap [30]. The main obstacle for SAIs is the fast decay of the interference visibility, i.e, quantum superpositions destroyed by unwanted coupling to the environment, especially under the bounded experimental condition.
On the other hand, integrated matter-wave interferometers in bounded space such as atom waveguides [31,32], BEC [33,34], optical lattice [35,36], quantum chip [33,37,38], etc, have important meaning for their potential engineering applications. However, interacting with trapped potentials makes the interferometers decoherence quickly. The decoherence mechanism has been a long time pursuit goal [39–42]. In previous works we have studied the decoherence of a SAI with co-propagating Raman pulses [43,44]. In practical applications, SAIs with counter-propagating Raman pulses have higher sensitivity in atom nonclassical state preparation [45], quantum control and manipulation [46,47] and sensing of atom motion [48] because they can deliver an effective momentum of 2 photons. But how the counter-propagating Raman type SAIs keep the coherence is an open problem.
Here we investigate the effect of an echo sequence to a neutral $^{87}$Rb atom interferometer trapped in a state-dependent optical dipole trap (ODT). We experimentally find that the spin echo [49,50] Ramsey spectra with counter-propagating Raman pulses show faster decay than the cases with co-propagating ones [43,44] under the same trap depth. By a model analysis, we show a coupling of the internal and external states of the SAI in ODT. The additional decoherence mechanism in counter-propagating Raman SAI is attributed to the separation of the external wavefunction in phase space induced by the inevitable back-action of Raman pulses on the atom. With the momentum kicks, the effect of the echo sequence to a trapped SAI is closely related to the trap period.
The article is organized as follows. In Sec. II, we describe our experimental processes and results of the single neutral $^{87}$Rb atom interferometer in the ODT with co-propagating and counter-propagating Raman pulses. In Sec. III, we give theoretical analysis about the single atom interferometer in dipole trap and show the self-revival behavior as a result of the coupling between atom’s spin states and motional states in trap. Finally, Sec. IV gives a conclusion.
2. Experimental measurements
Our experimental setup is depicted in Fig. 1(a) and has been described in detail in Ref.[43,44]. A single $^{87}$Rb atom is confined in a microscopic optical dipole trap with a TEM$_{00}$ mode. The trap laser beam has a waist radius of $2.1\textrm { }\mu \textrm {m}$ and a Rayleigh length of $16.7\textrm { }\mu \textrm {m}$. Laser induced fluorescence of one atom is detected with an avalanche photodiode. A bias magnetic field of $B_{0}=2\textrm { G}$ is applied along the z axis. We define the quantum states as $|1\rangle \equiv |5S_{1/2}, F=1,m_{F}=0\rangle$ and $|2\rangle \equiv |5S_{1/2}, F=2,m_{F}=0\rangle$. Using an optical pumping laser beam, the initial state $|2\rangle$ is prepared [43]. The manipulations between states $|1\rangle$ and $|2\rangle$ are performed by two-photon stimulated Raman transitions. Two Raman laser beams are red-detuned by about $60\textrm { GHz}$ from the $D1$ transition and separated by $6.8\textrm { GHz}$ utilizing an acoustic-optical modulator. The Raman pulses propagate along the axial direction of the optical dipole trap and are focused in the location of the atom. In our apparatus, we can choose the co-propagating Raman pulses ($I_{R1}$ and $I_{R2}$) or counter-propagating Raman pulses ($I_{R1}$ and $I_{R3}$).
Fig. 1. Schematic diagrams of experimental setup and single atom interferometer. (a) Experimental setup; (b) $\pi /2 - \pi - \pi /2$ Raman pulse sequence; (c) Experimental time sequence for free space interference.
For the counter-propagating pulses, a typical $\pi /2-\pi -\pi /2$ Raman type SAI process is shown in Fig. 1(b). Figure 1(c) shows the corresponding experimental time sequence of the atom interference in free space. After the initial state is prepared, we turn off the dipole laser field and apply Raman pulses sequence. Typically, our $\pi$-pulse duration is $1.1\textrm { }\mu \textrm {s}$. An electro-optical modulator (EOM) is applied to change the relative phase of the Raman laser beams. In the final state detection, a probe laser beam is applied to push out atom in state $|2\rangle$. However, atom in state $|1\rangle$ is not influenced by this laser and remains in the trap. Figure 2 shows the building up of interference pattern using counter-propagating Raman pulses. The Raman phase offset is varied from $-4\pi$ to $4\pi$ and the measured values are averaged with N repeated single atom events. The interference’s building-up becomes more obvious with increasing of repeated measurement time N. As shown in Fig. 2(g) and Fig. 2(h), when N=100, a maximum population of more than 90% is detected in F=1.
Fig. 2. Building up of the interference pattern of the single atom interferometer with counter-propagating Raman pulses. Data points (solid dots or squares) are experimentally measured values averaged with N repeated single atom events under the trap depth $U_{0}/k_{B}=36.0\textrm { }\mu \textrm {K}$. N = 1(a), 2(b), 5(c), 10(d), 20(e), 50(f), 100(g). All data are put together in (h) for comparison. Solid red line in (g) is a sine fit to the data points. Solid straight broken lines are used to link neighbor points for visual guidance purpose. The corresponding statistical error is so small as to be neglectful.
Firstly, the spin echo measurements with co-propagating Raman pulses are experimentally carried out in the ODT. The experimental time sequence is depicted in Fig. 3(a), and the dipole laser field keeps existent throughout the process. We varied the adding time of the second $\pi /2$-pulse around $t=2\tau$ and obtained the fringe visibilities. As shown in Fig. 3(b), the curves are theoretical simulated results using the method depicted in Sec. III under the specific trap depth and initial practical conditions, i.e., atom temperature, wave packet’s width and an appropriate dissipative item. The data points are our experimental results. They decay as exponential functions and are fitted similarly as mentioned in [39,51]. The homogeneous dephasing time of our experimental and theoretical results are listed in Table 1, and they are well consistent.
Fig. 3. Experimental time sequence and visibilities of the spin echo fringes. (a, b) for the co-propagating Raman pulses; (c, d) for the counter-propagating Raman pulses. (i), (ii), (iii), (iv) represent the trap depths of $8.6\textrm { }\mu \textrm {K}$, $15.0\textrm { }\mu \textrm {K}$, $24.0\textrm { }\mu \textrm {K}$ and $36.0\textrm { }\mu \textrm {K}$, respectively. The solid lines in (b) and (d) are our numerical results calculated in Sec.III. The inset figures in (b) and (d) show examples of the corresponding fringes for trap depth of $24.0\textrm { }\mu \textrm {K}$.
Table 1. Temperatures of single atoms and dephasing times for trap depths of $8.6\textrm { }\mu \textrm {K}$, $15.0\textrm { }\mu \textrm {K}$, $24.0\textrm { }\mu \textrm {K}$ and $36.0\textrm { }\mu \textrm {K}$.
In cases of counter-propagating Raman pulses, the interference fringes in the optical trap are obtained by changing the relative phase of the Raman laser beams. The experimental time sequence is shown in Fig. 3(c). Experimental measurement results and visibility curves of theoretical simulation are shown in Fig. 3(d). Analogizing the fitted parameters of the Gaussian equation in [39], the simulated dephasing time is also extracted, as listed in Table 1. The theoretical simulations are well consistent with the pure experimental results.
Finally, the temperatures of single atoms in the traps are measured by using release and recapture method. The corresponding values are listed in Table 1. Comparing Fig. 3(d) with Fig. 3(b), it is obvious that the coherence time reduces significantly under same trap depth conditions. To model the underlying mechanism, we recall that the major difference in the two cases is the photon recoil. For co-propagating Raman pulses, the recoil momentum acquired is negligible, and the external states are same for different internal states of the atom. However, the counter-propagating Raman pulses inevitably bring an extra momentum change, $\hbar (\vec {k_{1}}+\vec {k_{2}})=\hbar (|k_{1}|+|k_{2}|)\approx 2\hbar k$ [52,53], to the motional states of different spin states. This makes the external and internal states be coupled. Once the external states miss overlap, the interference processes is disturbed.
3. Theoretical analysis and discussion
In order to understand the interference process of the trapped SAI more clearly, we simulate the whole interference process by solving the schrödinger equation using coherent superposition of internal states together with external motional states. According to the experiment, the absorption and emission of photons due to the Raman pulses are along the axial direction, so we can safely simplify the system to one dimension space. In the center of mass reference frame, the Hamiltonian governing the process has the form [54]:
The atom is initially prepared in state $|\Psi \rangle =|1\rangle \otimes |\psi \rangle =|1,\psi \rangle$, here $|\psi \rangle$ represents the motional part of atomic wave function and can be written as:
For the interferometer process with co-propagating Raman pulses, the photon recoil is negligible. Firstly, a $\pi /2$ pulse (indicated by the white double arrows in Fig. 4(a) at $t=0$) excites the atom into a coherent superposition state $|\Psi \rangle =\frac {1}{\sqrt {2}}|1,\psi \rangle +\frac {i}{\sqrt {2}}|2,\psi \rangle$. The state-dependent light shift makes atoms feel different trapping potentials $V_{1}$ and $V_{2}$ and evolve in different paths. Then a $\pi$-pulse at time $\tau$ reverses two spin states. Here $\tau =T_{tr}/2$, and $T_{tr}$ is atom’s oscillation period in dipole trap. Lastly, the detecting $\pi /2$-pulse at $t=2\tau$ is used to obtain the population probabilities of atom in different spin states. The population probability of atom in $|2\rangle$ can be given as [56]:
Fig. 4. Evolution of probability distribution of external states for a single-atom interferometer. (a) Co-propagating Raman pulses and (b) Counter-propagating Raman pulses in the ODT with trap depth $U_{0}/k_{B}=36.0\textrm { }\mu \textrm {K}$. The $\pi$-pulse is added at $t=T_{tr}/2$. Color bar represents the value of probability distribution.
As seen from Eq. (3), the phase evolution of atom interferometer during the period of $\pi$ pulse and the second $\pi /2$ pulse compensates the phase difference accumulating before. As a result, the echo revival occurs at $t=2\tau$, and the coherence time is extended [39,43,55]. The simulated probability distributions of the external states for different spin states are shown in Fig. 4(a). There is no evident separation between the probability distributions of the external states in this evolution process.
However, the counter-propagating Raman pulses during the Ramsey interferometer process inevitably bring an extra momentum change about $2\hbar k$, to atomic spin dependent external motion, thus making the coupling of the internal and the motional states. After the first $\pi /2$ counter-propagating Raman pulses (indicated by the white arrows in Fig. 4(b) at $t=0$), the atom has a “Schrödinger cat”-like superposed state $|\Psi \rangle =\frac {1}{\sqrt {2}}|1,\psi \rangle +\frac {i}{\sqrt {2}}|2,\psi +2\hbar k\rangle$. Since the interaction time between atom and Raman pulses is very short compared to the free evolution time of dark period, this momentum change process can be seemed as an instantaneous process. After a dark period of time $\tau$, the atom state evolves as:
Then a $\pi$-pulse is added, and a $\pi /2$-pulse later at $t=2\tau$ acts as a detector to obtain the population probability of atom in state $|2\rangle$:
Following the interference process and using the same parameters as the experiment shown in Fig. 3, we calculate the visibility of the spin echo fringes for the co- and counter-propagating atom interferometer, which are shown with the solid lines in Fig. 3(b) and Fig. 3(d), respectively. The calculated curves fit the the experimentally measured results very well.
For clearly showing the effect of the spin echo, we also calculate the probability of atom lastly staying in state $|2\rangle$ with and without the spin echo pulse during the interference process. The trap depth is $U_{0}/k_{B}=36.0 \textrm { }\mu \textrm {K}$ and the simulated dephasing time $T^{\ast }_{2}$ is extracted similarly as mentioned in [39,51]. Figure 5 shows the Ramsey spectrum and echo spectrum with and without the photon momentum kicks under the same initial conditions. Figure 5(a) and Fig. 5(b) present the co-propagating Raman processes, the coherence time of the trapped SAI with the echo sequence ($T^{\ast }_{2}=30\textrm { ms}$) is longer than the case without the echo sequence ($T^{\ast }_{2}=26.4\textrm { ms}$). For the counter-propagating Raman processes in Fig. 5(c) and Fig. 5(d), $T^{\ast }_{2}$ of the trapped SAI without the echo sequence is 5.2 ms, while $T^{\ast }_{2}=7.7\textrm { ms}$ with the echo sequence. For these two cases, the echo sequence both extend the coherence time.
Fig. 5. The simulated Ramsey and echo spectra for a SAI in ODT. (a) The co-propagating Raman process without spin echo. The simulated dephasing time $T^{\ast }_{2}$ is 26.4 ms. (b) The co-propagating Raman process. The $\pi$ pulse is added at $t=20\textrm { ms}$ and $T^{\ast }_{2}=30$ ms. (c) The counter-propagating Raman process without the spin echo. $T^{\ast }_{2}=5.2$ ms. (d) The counter-propagating Raman process. The $\pi$ pulse is added at $t=2T_{tr}$=3.6 ms. An echo revival appears at $t=4T_{tr}$ and $T^{\ast }_{2}$ is extended to 7.7 ms. The trap depth $U_{0}/k_{B}=36.0 \textrm { }\mu \textrm {K}$ and the other parameters are the same as in Fig. 3.
Especially, in Fig. 5(c) and Fig. 5(d), the interference spectra are divided. The initial Ramsey fringe quickly collapses within a few tens of microseconds and revivals at the time of the first trap period $t=T_{tr}=1.8\textrm { ms}$($T_{tr}$ is the atom’s oscillatory period for $U_{0}/k_{B}=36.0 \textrm { }\mu \textrm {K}$). There are a series of self-revivals when the atom oscillates to the time of each integer multiples of cycle in the dipole trap. Each self-revival falls rapidly and appears at the next integer multiple of the trap period. This is the result of wave functions separation and periodic evolution in the ODT.
For the $\pi /2-\pi -\pi /2$ process with counter-propagating Raman pulses, the coherence time is strongly influenced by the adding time of $\pi$ pulse. When adding $\pi$ pulse at integer multiples of the trap period, $nT_{tr}$(n is integer), an echo revival appears approximate the time $t=2nT_{tr}$, and the coherence time is extended. In Fig. 5(d), the $\pi$ pulse is added at $t=2T_{tr}$. There is an echo revival at $t=4T_{tr}$. Comparing it with Fig. 5(c), the simulated dephasing time $T^{\ast }_{2}$ can be extended to 7.7 ms. However, when adding the $\pi$ pulse at time of half integer multiples of the trap period, the Ramsey fringe collapses rapidly. As seen in Fig. 6(a), the fringe decays instantaneous when the $\pi$ pulse is added at $t=2.5T_{tr}$. This phenomenon is similar to the atomic ions interferometer created by a short train of intense laser pulses to impart a spin-dependent momentum transfer [48,58] and we can understand the process through the atom states evolution in the phase space [59]. As shown in Fig. 6(b) and Fig. 6(c), the first $\pi /2$ pulse makes the spin states $|1\rangle$ and $|2\rangle$ evolve to different positions in phase space as they have different momenta. The difference of motional states destroy the spin coherence of SAI, thus the interference contrast decays. If the $\pi$ pulse is added at the time of integer multiples of the trap period, $nT_{tr}$, after the second free evolution time, the two paths’ wave functions will overlap again at $t=2nT_{tr}$, and it makes the interference contrast revival and the coherence time extended, as plotted in Fig. 6(b). However, if the $\pi$ pulse is added at the time of half integer multiples of the trap period, the phase difference between the wave functions of two paths is exacerbated greatly after each Raman pulses, as seen in Fig. 6(c), thus the interference visibility of the SAI decays quickly.
Fig. 6. The simulated echo spectrum and phase-space plots of atom external states evolution with counter-propagating Raman pulses for different processes. (a) A SAI in the ODT with counter-propagating Raman pulses. The $\pi$ pulse is added at $t=2.5T_{tr}$. The trap depth $U_{0}/k_{B}=36.0 \textrm { }\mu \textrm {K}$ and the other parameters are same as in Fig. 3. (b)-(c): The circle represents spin states: hollow circle $|1\rangle$, solid circle $|2\rangle$. The black arrow represents Raman pulse manipulation process. (b) and (c) are echo interferometer processes that adding $\pi$ pulse at time of integer multiples and half integer multiples of the trap period respectively.
4. Conclusion
In summary, we have investigated the effect of the echo sequence to a SAI in ODT with photon momentum kicks. The photon recoil effect induced by the counter-propagating Raman pulses changes the external motion state and makes the Ramsey fringe collapse quickly. Under the state-dependent dipole potentials, the spin echo spectrum of the SAI with photon recoil effect experiences echo revival when $\pi$ pulse is added at the time of approximate integer multiples of trap period, while an instantaneous collapse occurs when $\pi$ pulse is added at the time of half integer multiples of trap period. The phase space theory of atom states evolution well explains these motion-spin entanglement phenomena. Similar echo sequence was used in the atom ensemble [57] to reduce the effect caused by the anharmonicity and keep high coherence, however, there the momentum recoil is too small and nearly can be neglected.
The SAI trapped in ODT provides coupling of atom external and internal states. It is possible to use Ramsey spectroscopy to sense classic dynamics of the system in time, which could be useful for future quantum computer manufacturing [48]. It is also useful to help us to determine the best time to conduct the detection process in order to maximize the output signal in many fields like atom waveguides [31,32], since the interference visibility is closely related to the trap period in this case. Furthermore, we may use it to explore the classical and quantum boundary [41,60,61] since classic dynamics and quantum states influence each other simultaneously in this process. Lastly, our results are helpful in studying the trajectory of atom in trap, quantum coherence manipulations [62–64], atom-chip clock and atom waveguide [65–67] which mainly use light fields to trap and manipulate atoms.
Funding
National Key Research and Development Program of China (2016YFA0302002, 2016YFA0302800, 2017YFA0304501); National Natural Science Foundation of China (11774389); The Strategic Priority Research Program of the Chinese Academy of Sciences (XDB21010100); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019325); Postdoctoral Applied Research Program (62350079311135).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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