⇦Chinese Physics Letters, 2022, Vol. 39, No. 1, Article code 013401 Microwave Induced Ultralong-Range Charge Migration in a Rydberg Atom Huihui Wang (王慧慧)1,2, Yuechun Jiao (焦月春)1,2, Jianming Zhao (赵建明)1,2*, Liantuan Xiao (肖连团)1,2, and Suotang Jia (贾锁堂)1,2 Affiliations 1State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006, China 2Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China Received 13 November 2021; accepted 6 December 2021; published online 29 December 2021 *Corresponding author. Email: zhaojm@sxu.edu.cn Citation Text: Wang H H, Jiao Y C, Zhao J M et al. 2022 Chin. Phys. Lett. 39 013401    Abstract A microwave induced superposition of the $40S_{1/2}$ and $40P_{1/2}$ states of a Cs atom has been investigated in detail. Ultralong-range charge migration which spans a region more than 200 nm has been discovered. As far as we know, this is the first time to discover charge migration in such a long range. This leads to a large dipole moment which oscillates periodically. The present discovery may stimulate new applications such as quantum simulation of many body physics dominated by periodic interactions. In addition, we find an interesting phenomenon that Cs atoms in the superposition of $40S_{1/2}$ and $40P_{1/2}$ have a much larger blockade radius than those of Cs ($40S_{1/2}$) or Cs ($40P_{1/2}$) atoms.
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 DOI:10.1088/0256-307X/39/1/013401 © 2022 Chinese Physics Society Article Text Light induced electron density redistribution has fundamental importance in various processes of physics, chemistry and biology. The redistribution of electron density may lead to the phenomena of charge transfer[1,2] or charge migration.[3–5] Charge migration in molecules typically means ultrafast electron dynamics in a superposition of more than one electronic states. Extensive investigations of laser control of charge migration[5–8] have been preformed due to its important potential applications. For recent surveys in literature on charge migration, one can see Ref. [9]. In particular, attosecond charge migration in HCCI$^{+}$ has been observed experimentally,[5] which serves as a milestone in the research field of ultrafast charge migration. Charge migration may occur only when more than one electronic states are populated. Consequently, nuclear motions will induce decoherence,[10–12] typically in about 10 fs. A method to fight with decoherence was proposed very recently.[13] Alternatively we can decrease the influence of decoherence of charge migration using systems which naturally have long coherence time. Accordingly charge migration in atoms will have much longer coherence time than in molecules. In particular, the coherence time of a Rydberg atom[14–16] reaches microseconds (µs). Ultralong-range charge migration in a Rydberg atom and the associated extremely large dipole moment is expected to offer a good model system for quantum simulation and manipulation. Rydberg atoms are highly excited atoms with larger sizes, greater transition dipole moments, and stronger interactions.[17] Accordingly, ultracold Rydberg atoms have emerged as a competitive platform for various applications.[18–21] Rydberg levels can be tuned to Förster resonance by weak electric fields, resulting in strong dipole-dipole interactions and blockade effects.[22–24] The Rydberg blockade effect has been employed to study quantum logic gates,[18] single-photon sources,[19] transistors,[20] and quantum simulation. Large polarizability makes Rydberg atoms quite sensitive to external electric fields. Therefore, Rydberg atoms have also been used for precision measurements of external fields.[25,26] In particular, Rydberg atoms have been widely used for quantum simulation of condensed phase dynamics dominated by dipole interactions,[27,28] Coulomb interactions,[29] strong coupling regime,[30] and so on.[31,32] In this work, we discover ultralong-range charge migration in a Cs Rydberg atom and the resultant fascinating properties. Our results turn out to enhance the competitiveness of Rydberg atoms for state-of-the-art applications. Specifically, we investigate charge migration in the superposition of the $40S_{1/2}$ and $40P_{1/2}$ states of a Cs atom. Preparation of an initial superposition state $\vert\psi(t'=0)\rangle=\frac{1}{\sqrt{2}} \vert40S_{1/2}\rangle+\frac{1}{\sqrt{2}}e^{i\delta}\vert40P_{1/2}\rangle$ can be carried out by more than one labs in the world with the experimental techniques detailed in the Supplementary Information. The initial state $\vert\psi(t'=0)\rangle$ will propagate freely, leading to the time-dependent state $\mathinner{\vert\psi(t')\rangle}$ and the electron density $\rho(\boldsymbol{r},t')=\mathinner{\langle\psi(t')\vert\delta(\boldsymbol{r}-\boldsymbol{r}')\vert\psi(t')\rangle}$. The electron density $\rho(\boldsymbol{r},t')$ turns out to oscillate around its average value $\mathinner{\langle\rho(\boldsymbol{r})\rangle}_{T}$ periodically with a period $T=\frac{2\pi}{\omega}=15.7$ ps. We only need to focus on the migrating part of the density $\Delta\rho(\boldsymbol{r},t')=\rho(\boldsymbol{r},t') -\mathinner{\langle\rho(\boldsymbol{r})\rangle}_{T}$. For convenience, we define the starting time $t=0$ for observing electron density as $t=t'-\frac{\delta}{\omega}$. Then, $\Delta\rho(\boldsymbol{r},t)$ will have largest amplitude at $t=0$. Large amplitude charge migration mainly occurs along the polarization of the field, which is defined as the $z$-axis. Consequently it is convenient to focus on the one-dimensional reduced density $\Delta\rho(z,t)$ along $z$-axis, which can be obtained by integrating $\Delta\rho(x,y,z,t)$ over $x$ and $y$. Accordingly the propagation of $\Delta\rho(z,t)$ is shown in Fig. 1 for one period $T$.
cpl-39-1-013401-fig1.png
Fig. 1. The migrating part of the one-dimensional reduced electron density $\Delta\rho(z,t)$ in units of $a_0^{-1}$.
In the following, we use charge to represent negative charge contributed by electron distribution. Charge migration along $z$-axis can be clearly identified in Fig. 1. In the beginning stage, more density is accumulated in the region of $z < 0$. Consequently there is charge migration from $z < 0$ to $z>0$, namely along the positive direction of $z$-axis. At $t=\frac{T}{4}$ the density $\Delta\rho(z,t)$ is exactly zero for all $z$. Therefore, the density distribution at $t=\frac{T}{4}$ is just the average one $\mathinner{\langle\rho(z)\rangle}_{T}$. The charge keeps migrating from $z < 0$ to $z>0$ until $t=\frac{T}{2}$. Then more density is accumulated in the region of $z>0$. Consequently, the direction of charge migration changes for the next half period. The charge is migrating from $z>0$ to $z < 0$ for $\frac{T}{2} < t < T$. On closer inspection of $\Delta\rho(z,t)$, the maximum amplitudes of the density oscillation are located around $\pm1700 a_{0}$. Many small oscillations of $\Delta\rho(z,t)$ can be found along $z$, due to the oscillation nature of Rydberg wavefunctions. More details of the oscillation characteristics can be unraveled by a two-dimensional representation of $\Delta\rho(\boldsymbol{r},t)$. Accordingly the complete information of $\Delta\rho(\boldsymbol{r},t)$ is provided in the Supplementary Information in terms of the cylindrical coordinates. Charge migration is noticeable in the region $-2500a_{0} < z < 2500a_{0}$, which spans more than 200 nm. The order of magnitude for the migration speed can be estimated to be $2\times200\,{\rm nm}/15.72\,{\rm ps}\approx 25\,{\rm km/s}$, while the corresponding order of magnitude for the case of HCCI$^{+}$ is $2\times0.25~{\rm nm}/1.85~{\rm fs}\approx270~{\rm km/s}$.[5,33] The latter is one order of magnitude larger, because the kinetic energy of valence electrons are much larger than that of Rydberg electrons.
cpl-39-1-013401-fig2.png
Fig. 2. The electron flux $F_{z}(z,t)$ along $z$-axis in units of ps$^{-1}$.
A more convenient tool to study charge migration is the electron flux density or flux.[34–36] For one-dimensional charge migration, the electron flux $F_{z}(z,t)$ along $z$-axis is sufficient.[12] The electron flux $F_{z}(z,t)$ can be obtained either by integrating the flux density in each plane perpendicular to $z$-axis or from the continuity equation. The corresponding electron flux $F_{z}(z,t)$ is shown in Fig. 2 for one period $T$. Positive (or negative) values of $F_{z}(z,t)$ means charge migration along the positive (or negative) direction of $z$-axis. Apparently there is unidirectional charge migration along the positive direction of $z$-axis for the first half period $0 < t < \frac{T}{2}$. The electron flux reaches its maximum at $t=\frac{T}{4}$ and then decreases to zero at $t=\frac{T}{2}$. The direction of charge migration changes to the negative direction of $z$-axis for the next half period $\frac{T}{2} < t < T$. For any given time $t\ne 0,\frac{T}{2},T$, the electron flux $F_{z}(z,t)$ has its maximum amplitudes at $z=0$. This is just because all the electron density migrating from $z < 0$ to $z>0$ or the reverse must pass the position $z=0$. This fact leads to a natural way of evaluating the total migrating charge $\Delta Q_{m}$, which migrates between the two regions $z < 0$ and $z>0$. The total migrating charge $\Delta Q_{m}$ can be obtained by integrating $F_{z}(z,t)$ at $z=0$ for the first half period. More intuitively, an equivalently way to obtain $\Delta Q_{m}$ is to compare $\Delta\rho(z,t)$ at different times in the $z < 0$ or $z>0$ regions. In the Supplementary Information, four equivalent calculations all lead to the migrating charge $\Delta Q_{m}=-0.35e$. The total migrating charge $\Delta Q_{m}$ is similar to the value of the corresponding migrating charge in molecules.[12,37] The amplitude and range of charge migration are directly related to the value of the system dipole moment. The present superposition of the $40S_{1/2}$ and $40P_{1/2}$ states leads to an extremely large dipole moment even when the external field is exactly zero. At the time $t=0$, the dipole moment is 1305.7 Debye. The dipole moment changes to $-1305.7$ Debye at $t=\frac{T}{2}$. When all the external fields are switched off, the system dipole moment oscillates between 1305.7 and $-1305.7$ Debye. This leads to large dipole-dipole interactions between two such atoms. It is worth noting that the dipole-dipole interaction has a distance dependence of $\frac{1}{R^{3}}$. However, the Van der Waals interaction between two normal Rydberg atoms has a distance dependence of $\frac{1}{R^{6}}$. Consequently the interaction between two Rydberg atoms which are both in the present superposition of the $40S_{1/2}$ and $40P_{1/2}$ states will be much larger than the interaction between two $40S_{1/2}$ (or $40P_{1/2}$) Rydberg atoms, provided that the distance is long enough. For example, if the distance between two Cs atoms is 5 µm (which corresponds to typical Rydberg blockade radius), the maximum interaction is as large as 4.1 MHz for the present superposition states. The interaction for two Cs $40S_{1/2}$ atoms at the same distance is only 0.04 MHz, and even smaller for the case of $40P_{1/2}$. This implies an ever larger blockade radius if we want to prepare ensembles of Rydberg atoms in the present superposition state rather than in the $40S_{1/2}$ (or $40P_{1/2}$) state. Experimental realization of the system is feasible. Here we design an experiment to prepare the superposition state employing well-developed techniques for Rydberg atom, see the Supplementary Information for more details. Ultracold cesium atoms can be first trapped in a magneto-optical trap (MOT) then loaded into a tightly focused optical tweeze to prepare a single atom. The MOT temperature can be decreased to a few µK. The Rydberg $\vert 40S_{1/2}(m_J = 1/2)\rangle$ state will be prepared from the $\vert6S_{1/2}(F=4, m_F = 4)\rangle$ state with a two-photon scheme using an 852 nm laser with the $\sigma^+$ polarization and a 510 nm laser with the $\sigma^-$ polarization via the $\vert6P_{3/2}(F'=5, m_F = 5)\rangle$ intermediate state. To prepare the state of interest, a 63.6 GHz microwave field will be employed to couple the $40S_{1/2}$ and $40P_{1/2}$ states. The microwave field is linearly polarized along $z$-axis so that the quantum number $m_J=1/2$ is unchanged during the transition. When the microwave field is switched off, the population of the $40S_{1/2}$ and $40P_{1/2}$ states will be ${\rm {\cos}}^{2}\gamma$ and ${\rm {\sin}}^{2}\gamma$, respectively. Thus the initial superposition state for charge migration is $\mathinner{\vert\psi(t'=0)\rangle}={\rm {\cos}}\gamma\mathinner{\vert40S_{1/2}\rangle} +{\rm {\sin}}\gamma e^{i\delta}\mathinner{\vert40P_{1/2}\rangle}$ with a phase difference $\delta$ undetermined. Here $m_J = 1/2$ is omitted. The relative error of $\gamma$ is only a few percent, mainly due to the time for switching on/off of the microwave field. However, the error of $\gamma$ will only lead to a different initial state which gives the same charge migration except for a different amplitude. A linearly polarized microwave field induced partial transition between two Rydberg states can lead to ultralong-range charge migration. As far as we know, this is the first time to discover charge migration through a region exceeding 200 nm. In addition, it is straight forward to increase the range of charge migration to micrometers, e.g., by preparing a superposition of the $80S_{1/2}$ and $80P_{1/2}$ states instead of the present superposition of the $40S_{1/2}$ and $40P_{1/2}$ states. The phenomenon of charge migration may be experimentally confirmed in terms of high-contrast time-domain Ramsey interferometry.[32,38] The extremely large dipole moments mediated by ultralong-range charge migration result in large dipole-dipole interactions between atoms in the same superposition state. The large dipole-dipole interactions may be experimentally detected.[28,31] Compared to Cs Rydberg atoms of the $40S_{1/2}$ or $40P_{1/2}$, the interactions between atoms in the superposition of $40S_{1/2}$ and $40P_{1/2}$ states are about two orders of magnitude larger for distances of typical Rydberg blockade radius. Consequently there exists an even larger blockade radius than typical Rydberg blockade radius. This even larger blockade radius may be measured based on the well-developed experimental techniques for the Rydberg blockade effect. For an ensemble of atoms prepared by the same microwave field, all the dipole moments will oscillate concertedly. If they are loaded in a one-dimensional lattice, the interactions will be periodic. This opens the door to a new type of quantum simulation, for a system dominated by periodic interactions. We would like to express our gratitude to Professor Jörn Manz (Berlin) for stimulating discussions and careful reading of the manuscript. This work was supported by the National Key Research and Development Program of China (Grant No. 2017YFA0304203), the Program for Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (Grant No. IRT_17R70), the National Natural Science Foundation of China (Grant No. 11904215), the 111 Project (Grant No. D18001), the Fund for Shanxi “1331 Project”, and the Hundred Talent Program of Shanxi Province. References On the Theory of Oxidation‐Reduction Reactions Involving Electron Transfer. 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