Entangling two atoms of different isotopes via Rydberg blockade

TL;DR

This paper demonstrates the first experimental entanglement of two different isotope neutral atoms using Rydberg blockade, enabling advanced quantum simulations and computing with multi-species systems.

Contribution

It reports the first realization of heteronuclear entanglement between different isotope neutral atoms, including a heteronuclear C--NOT gate and entangled state with high fidelity.

Findings

Successfully entangled ${}^{87} ext{Rb}$ and ${}^{85} ext{Rb}$ atoms.

Implemented a heteronuclear C--NOT gate with high fidelity.

Achieved raw fidelities of 0.73 and 0.59 for entanglement and gate operations.

Abstract

Quantum entanglement is crucial for simulating and understanding exotic physics of strongly correlated many-body systems, such as high--temperature superconductors, or fractional quantum Hall states. The entanglement of non-identical particles exhibits richer physics of strong many-body correlations and offers more opportunities for quantum computation, especially with neutral atoms where in contrast to ions the interparticle interaction is widely tunable by Feshbach resonances. Moreover, the inter-species entanglement forms a basis for the properties of various compound systems, ranging from Bose-Bose mixtures to photosynthetic light-harvesting complexes. So far, the inter-species entanglement has only been obtained for trapped ions. Here we report on the experimental realization of entanglement of two neutral atoms of different isotopes. A ${}^{87}\mathrm{Rb}$ atom and a ${}^{85}\mathrm{Rb}$ atom are confined in two single--atom optical traps separated by 3.8 $\mu$m. Creating a strong Rydberg blockade, we demonstrate a heteronuclear controlled--NOT (C--NOT) quantum gate and generate a heteronuclear entangled state, with raw fidelities $0.73 \pm 0.01$ and $0.59 \pm 0.03$, respectively. Our work, together with the technologies of single--qubit gate and C--NOT gate developed for identical atoms, can be used for simulating any many--body system with multi-species interactions. It also has applications in quantum computing and quantum metrology, since heteronuclear systems exhibit advantages in low crosstalk and in memory protection.

Full text

• 1

State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences - Wuhan National Laboratory for Optoelectronics, Wuhan 430071, China

• 2

Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China

• 3

University of Chinese Academy of Sciences, Beijing 100049, China

• 4

LPTM, UMR8089 of CNRS and Univ. Cergy–Pontoise, F–95302 Cergy–Pontoise, France

• 5

LPTMS, UMR8626 of CNRS and Univ. Paris–Sud, F–91405 Orsay, France

• 6

SPEC, CEA & CNRS, Univ. Paris–Saclay, CEA Saclay, F–91191 Gif–sur–Yvette, France

• 7

Russian Quantum Center, Novaya Street, Skolkovo, Moscow Region R–143025, Russia

• 8

Van der Waals–Zeeman Institute, Institute of Physics, Univ. Amsterdam, The Netherlands

Trapped neutral atoms offer unique possibilities for quantum simulation, thanks to an excellent control of the interaction strength over 12 orders of magnitude[12] and to the creation of tunable arrays of single atoms for the simulation of spin systems[13]. Important experiments have been performed towards quantum simulation using identical neutral atoms[18, 19, 14, 15, 16, 17], and theoretical proposals aim at universal simulators[20]. Mixed–species architectures further enlarge the set of systems that can be simulated to encompass new phenomena ranging from heteronuclear Efimov effects[21] to exotic superfluid pairing mechanisms in quantum fluid mixtures[22, 23].

Heteronuclear qubits are also helpful for solving fundamental issues in quantum information processing, such as low–crosstalk individual manipulation [4]. The two different Rubidium isotopes used in our experiment exhibit different level structures, allowing us to implement a novel technique to manipulate individual atom states by using a difference between the transition frequencies of the two atoms [10]. This feature is a fundamental difference compared to previous experiments on identical atoms [24, 25], where individual addressing relied on the spatial separation between the atoms. In our setup, the atoms do not need to be spatially separated, and all laser beams cover both atoms. We implement this technique for the first time with neutral atoms and show that the fidelities of the created CNOT quantum gate and entangled state are on par with recent homonuclear results [24, 25]. Our analysis shows that the fidelities are mainly limited by technical reasons and by the thermal motion of the atoms.

In our experiment, we fully control two heteronuclear atom qubits represented by a single ${}^{87}\mathrm{Rb}$ atom and a single ${}^{85}\mathrm{Rb}$ atom, and exploit the heteronuclear Rydberg interaction to deterministically entangle the two different atoms. The control qubit is encoded in the ground hyperfine states $|F=1,m_{F}=0\rangle=|\downarrow\rangle$ and $|2,0\rangle=|\uparrow\rangle$ of 87Rb, whereas the target qubit is encoded in the states $|2,0\rangle=|\Downarrow\rangle$ and $|3,0\rangle=|\Uparrow\rangle$ of 85Rb (Fig. 1a). For both atoms, the Rydberg state is $|r\rangle=|79D_{5/2},m_{j}=5/2\rangle$. We exploit the difference in the resonance frequencies of the two atoms to ensure a negligible crosstalk during state measurements and qubit operations (see Methods).

The experimental apparatus and the single–atom trapping procedure for 87Rb and 85Rb atoms have been described in our recent work[11]. We trap a single 87Rb atom in the dipole trap–1 and a single 85Rb atom in the dipole trap–2 located $3.8\,\mathrm{\mu m}$ away (see Fig. 1b), and then optically pump the atoms to the $|\uparrow\rangle$ and $|\Uparrow\rangle$ states, respectively. After that the trapping potentials are adiabatically lowered from 0.6 mK to 0.1mK. Both microtraps have trapping frequencies $\omega_{y}/2\pi=1.39\pm 0.01\,\mathrm{kHz}$ in the longitudinal direction and $\omega_{r}/2\pi=16.9\pm 0.1\,\mathrm{kHz}$ in the radial direction (see Fig. 1b). We measure the atom temperatures $T_{87}=8\pm 1\,\mu\mathrm{K}$ and $T_{85}=9\pm 1\mu\mathrm{K}$ using release and recapture methods. Next, we combine Rydberg excitation pulses and single qubit operations with Raman lasers in order to demonstrate the heteronuclear Rydberg blockade, implement the C–NOT gate, and entangle the two heteronuclear atoms. At the end of each sequence, we detect the qubit state by using a resonant laser to “blow away” $|\uparrow\rangle$ and $|\Uparrow\rangle$ atoms, so that the survival probabilities refer to the atoms in the $|\downarrow\rangle$ and $|\Downarrow\rangle$ states (see Fig. 1c).

We first calculate the expected Rydberg blockade shift. If both atoms are in the $|r\rangle$ state, their interaction is dominated by the Förster resonance between the two–atom states in the $(79d_{5/2},79d_{5/2})$, $(80p_{3/2},78f)$, and $(81p_{3/2},77f)$ manifolds. We restrict the Förster interaction Hamiltonian to a subspace spanned by 436 states corresponding to distinguishable atoms. Taking the initial state $|r\!\Uparrow\!>$ we account for its coupling to the Förster states and calculate the time evolution of the probability for double excitation, $P_{85}(y,t)=1-|<\!r\!\Uparrow|e^{-iHt/\hbar}|r\!\Uparrow\!>|^{2}$ and its average over time, $P_{85}(y)$. The latter depends on the offset $y=|y_{2}-y_{1}|$ of the two atoms along the $y$ direction. The blockade shift[26] $\Delta E(y)$ is deduced from the relation $P_{85}(y)=(\hbar\Omega_{85})^{2}/((\hbar\Omega_{85})^{2}+{\Delta}E^{2})$, where $\Omega_{85}$ is the effective Rabi frequency for ${}^{85}\mathrm{Rb}$. At zero temperature, for the distance $z=3.8~{}\mathrm{\mu m}$ between the microtraps, assuming a spatial offset of $y=1\,\mu\mathrm{m}$, the effective Rydberg interaction between the atoms is close to the strongly–interacting Förster regime[12]. Accordingly, the numerical results yield $P_{85}\approx 10^{-6}$ and a very large blockade shift $\Delta E/h=600\,\mathrm{MHz}$ (see Methods and Supplemental Material). The finite temperature of the atoms causes them to explore larger values of the offset, $y\gtrsim 10\,\mathrm{\mu m}$, leading to the mean double–excitation probability $<P_{85}>\approx 0.013$ for our temperatures $T_{87}=8\,\mu$K and $T_{85}=9\,\mu$K.

We demonstrate the Rydberg blockade by applying a Rydberg $\pi$ pulse on 87Rb, waiting for 0.3 $\mu$s, and applying a Rydberg pulse of variable duration on ${}^{85}\mathrm{Rb}$ (Fig. 2a). We measure the Rabi oscillations between the 85Rb $|\Uparrow\rangle$ and $|r\rangle$ states as a function of the second pulse duration (Fig. 2b). The Rydberg states are detected through the atom loss with an efficiency of $\sim$90%, and the Rydberg excitation efficiency for 87Rb and 85Rb is $\sim$96% (see Methods). The lifetime of the $|r\rangle$ state is over 180 $\mu$s, providing a long enough blockade for 85Rb. We do not record the experimental data when 87Rb is still in the trap after the sequence, so as to eliminate unblockaded events when 87Rb is not excited to the $|r\rangle$ state. The peak to peak amplitude of 85Rb Rabi oscillations between the $|\Uparrow\rangle$ and $|r\rangle$ states is 0.91 $\pm$0.02 in the absence of 87Rb in trap–1 (Fig. 2b). In its presence, the experimental data show a strong Rydberg blockade which suppresses the oscillation amplitude to 0.03 $\pm$0.01, in accordance with our theoretical prediction. The remaining weak oscillations of 85Rb are mainly due to not perfect experimental conditions, including the loss of 87Rb and transitions to other Rydberg states.

We use the Rydberg blockade to generate a heteronuclear C–NOT gate following the protocol of Ref.[27]. This involves three Rydberg pulses (Fig. 3a): (i) a $\pi$ pulse on 87Rb between the $|\uparrow\rangle$ and $|r\rangle$ states, (ii) a $2\pi$ pulse on 85Rb between $|\Uparrow\rangle$ and $|r\rangle$, and (iii) a $\pi$ pulse on 87Rb between $|r\rangle$ and $|\uparrow\rangle$. Then, combining two Hadamard gates realized using Raman $\pi/2$ pulses between the $|\Uparrow\rangle$ and $|\Downarrow\rangle$ states, we demonstrate the heteronuclear C–NOT gate shown in Fig. 3. Its intrinsic coherence is illustrated by measuring the oscillation of the output probabilities as a function of the relative phase between the two Hadamard gates (Fig. 3b). Setting the relative phase to 0 ($\pi$), the C–NOT gate will flip the target qubit if the control qubit is $|\uparrow\rangle$ ($|\downarrow\rangle$).

The fidelity of the CNOT gate is determined by measuring its truth table probabilities (Fig. 3c). We add an extra Raman $\pi$ pulse before acting with the “blow away” laser to transfer the $|\uparrow\rangle$ state ${}^{87}\mathrm{Rb}$ atoms to $|\downarrow\rangle$ and the $|\Uparrow\rangle$ state ${}^{85}\,\mathrm{Rb}$ atoms to $|\Downarrow\rangle$, in order to exclude other atom losses as in Ref.[25]. The raw fidelity of the C–NOT gate is $F=\operatorname{Tr}[|U_{\mathrm{ideal}}^{T}|U_{\mathrm{CNOT}}]/4=0.73(1)$. It is mainly limited by technical reasons and can be made higher by stabilizing the Raman pulse powers and by improving the Rydberg excitation efficiency (see Methods).

Eventually, we deterministically generate a heteronuclear entangled state of ${}^{87}\mathrm{Rb}$ and ${}^{85}\mathrm{Rb}$. Starting with the two–atom state $(|\uparrow\rangle+i|\downarrow\rangle)|\Downarrow\rangle/\sqrt{2}$, we apply the C–NOT gate to create the entangled state $(|\uparrow\Uparrow\rangle+|\downarrow\Downarrow\rangle)/\sqrt{2}$. In order to quantify the entanglement of our created Bell state, we measure the coherence $C$ between the $|\uparrow\Uparrow\rangle$ and $|\downarrow\Downarrow\rangle$ states by studying the response of the system to the simultaneous rotation of the two qubits[28]. For that purpose, we apply to both atoms $\pi/2$ pulses carrying the same phase $\phi_{1}$ relative to the initial pulses (Fig. 4a) and measure the oscillations of the parity signal $P=P_{\uparrow\Uparrow}+P_{\downarrow\Downarrow}-P_{\uparrow\Downarrow}-P_{\downarrow\Uparrow}$ as a function of $\phi_{1}$ (Fig. 4c). This gives us access[28, 24] to the coherence $|C|=0.16\pm 0.01$ which, combined with the populations $P_{\uparrow\Uparrow}=0.41$ and $P_{\downarrow\Downarrow}=0.44$ (Fig. 4b), leads to the entangled state fidelity $F=(P_{\uparrow\Uparrow}+P_{\downarrow\Downarrow})/2+|C|=0.59\pm 0.03$. The obtained fidelity is clearly above the threshold of $0.5$ ensuring the presence of entanglement. We obtain it without any corrections for atom or trace losses. It is lower than the fidelity of our C–NOT gate mainly because of the motion of 87Rb. Following Ref.[24] we evaluate that at our temperatures and C-NOT gate fidelity the upper bound of the entanglement fidelity is $F_{\mathrm{ent-max}}=0.65$, which is slightly above our experimental result.

To conclude, we have realized a C–NOT gate between two non–identical single atoms and demonstrated a negligible crosstalk between the two atomic qubits. The gate is based on a strong heteronuclear Rydberg blockade, and the raw fidelity is 0.73 $\pm$ 0.01. The entanglement of two different atoms is then deterministically generated with the raw fidelity $0.59\pm 0.03$. Our work makes a significant step towards the manipulation of heteronuclear atom systems. Unlike identical atoms, we use a difference in the transition frequencies to individually address a single atom. In this case, the two atoms can be put at a short separation while maintaining individual addressing to explore the physics in a very strong Rydberg interaction regime. Many atoms representing different isotopes can be trapped in an array with an arbitrary geometry[16, 17] to realize a Rydberg quantum simulator of exotic spin models, such as the Kitaev toric code, color code, or coherent energy transfer. Our results pave a way towards quantum computing with heteronuclear systems and towards the realization of a high fidelity state detection, which has recently been predicted not to have any fundamental limit even at room temperature[29].

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• Acknowledgements   

This work was supported by the National Key Research and Development Program of China under Grants No. 2016YFA0302800, the National Natural Science Foundation of China under Grants No. 11674361, the Strategic Priority Research Program of the Chinese Academy of Sciences under Grant No. XDB21010100 and Youth Innovation Promotion Association CAS No. 2017378. GVS acknowledges support from IFRAF. DJP and GVS emphasize that the research leading to their results in this paper has received funding from the European Research Council under European Community’s Seventh Framework Programme (FR7/2007-2013 Grant Agreement no. 341197).

• Author Contributions   

Y.Z. and P.X. contributed equally to this paper; Y.Z., P.X., X.D.H., Y.Y.L., M.L., J.W. and M.S.Z. designed and performed the experiment and analyzed the experimental data; D.J.P. and G.V.S. performed the theoretical modelling and numerical calculations; P.X., D.J.P., G.V.S., and M.S.Z wrote the manuscript; All authors discussed the manuscript. M.S.Z. supervised the project.

• Competing Interests   

The authors declare that they have no competing financial interests.

• Correspondence   

Correspondence and requests for materials should be addressed to P.X. (email: etherxp@wipm.ac.cn) and M.S.Z. (email: mszhan@wipm.ac.cn)

Methods

In the first paragraph, we describe the laser system used in our experiment to realize a coherent Rydberg excitation. In the second paragraph, we describe our procedure demonstrating that the crosstalk between the control and target qubits is negligible. The third paragraph is dedicated to the discussion of the C-NOT gate and entanglement fidelities. Finally, in the fourth paragraph, we summarize our calculations of the Rydberg blockade shift, which is described in greater detail in the Supplemental Material.

0.1 Laser system for coherent Rydberg excitations

The narrow linewidth and stabilized laser source required to realize a coherent Rydberg excitation are challenging to set up. In our experiment, the Ryd480 Ryderg laser is generated from a TA–SHG pro with a seed laser whose wavelength is 960 nm. The frequencies of the lasers Ryd480 and Ryd780 are locked to a Fabry–Perot cavity with high finesse (58000 for $960\,\mathrm{nm}$ and 91000 for $780\,\mathrm{nm})$. We then reduce the linewidth to $\sim$10 kHz for Ryd780 and to $\sim$20 kHz for Ryd480. The long–term drift of both lasers is less than 50 kHz. The frequency of Ryd480 is set to $625253.6\,\mathrm{GHz}$, and we expand the beam waist of Ryd480 to $\sim 12.8\,\mathrm{\mu m}$, so that it covers both atoms. The Ryd780 laser light is divided into two beams with the frequency difference $1127\,\mathrm{MHz}$, corresponding to the difference in the excitation frequencies of 85Rb (Ryd780-85) and 87Rb (Ryd780-87). The frequencies of the Ryd780-87 and Ryd780-85 lasers are $384223.2\mathrm{GHz}$ and $384224.3\,\mathrm{GHz}$, respectively. The beam waist of Ryd780-87 laser is $\sim 7.1\,\mathrm{\mu m}$ , and Ryd780-85 laser has the beam waist of $\sim 7.8\,\mathrm{\mu m}$. We use PID controllers with holding function to lock the laser power of Ryd480 to 51 mW, and the power of Ryd780-87 and Ryd780-85 to 5.6 $\mathrm{\mu W}$. The pulse area fluctuations of the Ryd480 and Ryd780 laser pulses are suppressed to less than 1%. Using the method from Ref.[26], we estimate $\Omega_{780-87}=2\pi\cdot$ 226 MHz, $\Omega_{780-87}=2\pi\cdot$ 206 MHz, and $\Omega_{480-85}$= $\Omega_{480-87}$= 2$\pi\cdot$ 28 MHz;

Coherent Rabi oscillations between the ${}^{85}\mathrm{Rb}$ $|\Uparrow\rangle$ and $|r\rangle$ states and between the ${}^{87}\mathrm{Rb}$ $|\uparrow\rangle$ and $|r\rangle$ states are shown in Fig. 2b and in Extended Data Fig. 1b. For 87Rb, the peak to peak Rabi amplitude is 0.82 $\pm$ 0.02. The survival probability of 87Rb after a $\pi$ pulse is 13%. This includes the 4% probability of populating the $|\uparrow\rangle$ state, the rest being the result of spontaneous emission from the Rydberg state during the detection. Thus, the Rydberg excitation efficiency for 87Rb is $\sim$96% and the detection efficiency for the Rydberg state is $\sim$90%. The corresponding efficiency for 85Rb is almost the same.

0.2 Crosstalk

The crosstalk of the two atomic qubits is crucial for our setup because all lasers cover both atoms, and the individual addressing of a single atom relies on the difference between the resonance frequencies of 87Rb and 85Rb rather than on the spatial distribution. During qubit state measurements, the 85Rb resonant laser may cause unwanted scattering of 87Rb as it is detuned 1.1GHz from its resonance frequency, and vice versa. We check this influence by adding a 85Rb “blow away” pulse between the 87Rb ground state Rabi oscillation and the 87Rb “blow away” pulse. We then compare the Rabi oscillations of 87Rb with and without the 85Rb pulse as shown in Extended Data Fig. 1a. The amplitudes of the Rabi oscillations are equal to each other within the measurement uncertainty, which shows a negligible crosstalk in the state measurement. For the excitation to Rydberg states, we use two–photon transitions with the total Rabi frequency of about 1MHz. Thus, the GHz spectral difference can provide enough protection for the qubit operations with each single atom. We also observe almost no excitation of 85Rb when adding the 87Rb Rydberg excitation laser as shown in Extended Data Fig. 1b. All experimental data show a negligible crosstalk between the two atomic qubits, which represents an important advantage of heteronuclear atom systems.

0.3 The CNOT gate and entanglement fidelity

The fidelity of the CNOT gate is mainly limited by technical reasons. One of them is the long–term drift ($\sim$10%) of the Raman pulse powers, which reduces the accuracy of Raman $\pi/2$ and $\pi$ pulses and causes the fidelity loss of $\sim$9%. Another reason is the $\sim$96% Rydberg excitation efficiency, which causes about 12% of two–atom loss. By using the power stabilization to suppress the long term drift and employing the compensating stray electrical field to improve the Rydberg excitation efficiency [19], one should get a significantly higher fidelity of the C–NOT gate.

The fidelity of the entangled state is lower than that of the heteronuclear C–NOT gate. This is mainly due to the motion of the 87Rb atom [24]. Single 87Rb atoms accumulate stochastic phases $\Phi=\textbf{k}\cdot\textbf{v}\delta t$ during the time $\delta t$ separating two Rydberg–${\pi}$ pulses. Here, $|\textbf{k}|=2\pi/\lambda_{480}-2\pi/\lambda_{780}$, and v is the atomic velocity. These phases vary from shot to shot. A simple estimation of the average yields $\langle e^{i\Phi}\rangle=e^{-\langle\Phi^{2}\rangle/2}=e^{-T|\textbf{k}|^{2}\delta t^{2}/m_{87}}$, where $m_{87}$ is the mass of 87Rb, and we took into account that $\langle{\bf v}^{2}\rangle=2T/m_{87}$. With $T_{87}=8\mu K$ and $\delta t=3.6\mu s$, we find $\langle e^{i\Phi}\rangle=0.78$, implying a maximum fidelity of $F_{\langle e^{i\Phi}\rangle}=0.89$. We combine this value with the C–NOT fidelity to obtain the maximum entanglement fidelity $F_{\mathrm{ent-max}}=0.65$, which is slightly above the upper limit of our experimental result. According to this calculation, an increase of the fidelity will rely on decreasing the time gap between two Rydberg–${\pi}$ pulses by increasing the intensity of the lasers, and on lowering the atom temperatures by using adiabatic cooling.

0.4 Calculation of the heteronuclear Rydberg blockade shift

Our theoretical model for the Rydberg blockade involves three steps, detailed in the Supplemental Material. First, we characterise the Förster resonance assuming that both atoms are immobile. The two–atom interaction Hamiltonian $H_{F}$ accounts for the dipole–dipole interaction between the atoms, the Rydberg energy defects, and the Zeeman interaction of each atom with the static magnetic field. If both atoms are excited to Rydberg states with energies close to the $79d_{5/2}$ state, their interaction is dominated by the Förster resonance[12] involving the states in the $(79d_{5/2},79d_{5/2})$, $(80p_{3/2},78f)$, and $(81p_{3/2},77f)$ manifolds. This amounts to restricting $H_{F}$ to a subspace spanned by 436 states.

Second, we calculate the blockade shift for fixed atoms(see Extended Data Fig.2a). Taking the initial two–atom state $|r\!\Uparrow\!>$ one sees that it is coupled to the doubly–excited Förster states via the operator $W$ which is proportional to $\Omega_{85}$. The probability for finding the atom pair in a doubly–excited state after a Rydberg pulse of duration $t$ on ${}^{85}\mathrm{Rb}$ is $P_{85}(y,t)=1-|<\!r\!\Uparrow|e^{-iHt/\hbar}|r\!\Uparrow\!>|^{2}$, where $H=H_{F}+W$ and the offset is $y=|y_{2}-y_{1}|$. We numerically calculate its average over time, $P_{85}(y)$. Following Ref.[26], we then define the blockade shift as $\Delta E(y)=\hbar\Omega_{85}(1/P_{85}(y)-1)^{1/2}$. For $y\lesssim 1\,\mathrm{\mu m}$, we find very large blockade shifts $\Delta E/h\gtrsim 600\,\mathrm{MHz}$ due to a strong Förster resonance with an effective interaction scaling as $1/R^{3}$, where $R$ is the internuclear distance. The blockade shift decreases for larger offsets and is of the order of a few $\mathrm{MHz}$ for $y\gtrsim 10\,\mathrm{\mu m}$.

Finally, we evaluate the impact of finite temperatures by averaging $P_{85}(y)$ over the probability density $p(y)$ for the atoms to have the offset $y=|y_{2}-y_{1}|$ (see Extended Data Fig.2b). In the conditions of our experiment the motion of atoms is classical, and $p(y)$ is Gaussian with the standard deviation $\sigma=[k_{B}T/(m_{\mathrm{red}}\omega_{y}^{2})]^{1/2}$, where the reduced mass is $m_{\mathrm{red}}=m_{87}m_{85}/(m_{87}+m_{85})$, and the average temperature $T$ satisfies the relation $T/m_{\mathrm{red}}=T_{87}/m_{87}+T_{85}/m_{85}$. The temperatures and trapping frequencies only enter our model through the combination $T/\omega_{y}^{2}$, which characterises the spatial extent of the classical motion of the atoms along $y$. For the experimental values $T_{87}=8\,\mathrm{\mu K}$, $T_{85}=9\,\mathrm{\mu K}$, and $\omega_{y}/2\pi=1.39\,\mathrm{kHz}$, we find the average value $<\!P_{85}\!>\approx 0.013$, which is of the same order as the observed quenched Rabi oscillation amplitude (Fig.2b).