Neutral impurities in a Bose-Einstein condensate for simulation of the Fröhlich-polaron
- Michael Hohmann†^{1}Email author,
- Farina Kindermann†^{1},
- Benjamin Gänger^{1},
- Tobias Lausch^{1},
- Daniel Mayer^{1, 2},
- Felix Schmidt^{1, 2} and
- Artur Widera^{1, 2}
Received: 30 July 2015
Accepted: 30 October 2015
Published: 14 November 2015
Abstract
We present an experimental system to study the Bose polaron by immersion of single, well-controllable neutral Cs impurities into a Rb Bose-Einstein condensate (BEC). We show that, by proper optical traps, independent control over impurity and BEC allows for precision relative positioning of the two sub-systems as well as for dynamical studies and independent read-out. We furthermore estimate that measuring the polaron binding energy of Fröhlich-type Bose polarons in the low and intermediate coupling regime is feasible with our experimental constraints and limitations discussed, and we outline how a parameter regime can be reached to characterize differences between Fröhlich and Bose-polaron in the strong coupling regime.
Keywords
Bose-Einstein condensate single atom impurity Bose polaron Fröhlich polaron1 Introduction
The immersion of single, controllable atoms into a Bose-Einstein condensate (BEC) realizes a paradigm of quantum physics - individual quantum objects interacting coherently with a single or few mode bath. This system allows to experimentally address various questions of quantum engineering, including local, non-demolition measurement of a quantum many-body system [1]; cooling of qubits while preserving internal state coherence [2, 3]; or engineering bath-mediated, long-range interaction between two or more impurities [4]. An impurity strongly interacting with the quantum gas will loose its single particle properties and it is rather described in terms of quasi particles, which are known as Bose polarons [5–7]. Particularly for strong interaction, such systems have been predicted to show remarkable properties such as self-trapping [5] or polaron clustering [8]. Recently, the dynamic as well as interaction effects on polaronic phenomena in Bose gases have attracted much interest [9–13]. Experimentally, impurities in BECs have been introduced, for example, as many atoms of different internal state [14, 15], different atomic species [16, 17], as individual ions [18, 19], or electrons [20].
1.1 The Fröhlich polaron
2 Experimental realization
Our experimental approach to realizing the Bose polaron aims at immersing single or few neutral Caesium (^{133}Cs) atoms into a Rubidium (^{87}Rb) BEC. Using single impurities allows us to study the dynamics and properties of individual polarons. This is in contrast to solid-state systems, where only averaged polaronic effects can be measured by probing macroscopic properties of the material [25, 32]. While for single impurities polaron-polaron interaction effects [33] are absent, we can study these effects by choosing a specific number of multiple impurities. Moreover, tight external control over individual impurities allows to prepare and study polaron dynamics in steady state or in non-equilibrium states, which is challenging in typical condensed matter systems. Our combination of species features several advantages, facilitating the realization, control and characterization of the Bose polaron. First, due to the relatively high nuclear charge of Rb and Cs the fine structure splitting of both species is also relatively large and allows to tune dipole traps in between the two fine structure lines of the first excited P-level with moderate unwanted photon scattering [34, 35]. As a consequence the atoms of this element do not experience a dipole potential. In order to improve the control over both species independently, we employ this fact constructing a species-selective conveyor-belt lattice allowing for trapping and controlled transport of impurity atoms, only. The lattice offers a unique way to study nonequilibrium phenomena, such as polaron transport [36], coherence properties [17], or Bloch oscillations [37], being inaccessible in solid state systems.
Second, for a dipole trap wavelength of \(\lambda= 1{,}064\mbox{ nm}\), the trapping frequency ω and thus the gravitational sag \(g/\omega^{2}\) with g the gravitational acceleration, is equal to the percent level for the two species. Therefore, a high spatial overlap is ensured when both species are trapped in the same dipole potential, even for ultracold temperatures. Third, with Cs representing the minority component, three-body losses limiting the lifetime of the polaron are due to Rb-Rb-Cs collisions rather than Cs-Cs-Rb collision, where the loss coefficient \(L_{3}\) of the former is an order of magnitude smaller than for the latter [38]. While three-body losses will still be a limitation of the atomic lifetime, the lifetime of the polaron is expected to be significantly larger than the lifetime in balanced Rb-Cs mixtures. Furthermore, fluorescence imaging of such a small number of atoms in an optical lattice allows for single-site resolved detection of the impurities with standard optical systems [39], enabling the tracking of single impurity dynamics with high precision.
For the design of the experimental apparatus, additional considerations have to be made. In a combined system of quantum gas and single atoms, the respective ways to experimentally extract information from averages differ: For a quantum gas, a single realization yields an ensemble average of typically \(10^{3} \ldots10^{5}\) atoms. For single atoms, in contrast, averages have to be formed as time averages of typically \(10^{2} \ldots10^{3}\) repetitions for identical parameter values. For a combination, the statistics is clearly limited by probing the single impurity, while usually the time scale of a single experimental run is limited by the production of a BEC. Therefore we produce a BEC all-optically in a crossed dipole trap with high trap frequencies, resulting in cycling times smaller than 7 seconds.
Our system therefore combines several advantages over previous experiments [38], such as spatial control and high resolution imaging of the impurity atoms combined with a short cycle time which enables us to study polaronic effects in yet unexplored regimes. In the following we first discuss the experimentally relevant parameter ranges and constraints for our system of single Cs impurities in a Rb BEC, before we turn to the presentation of our experimental apparatus.
2.1 The Rb-Cs Bose-polaron
For the experimental characterization of the Bose-polaron, we focus on the binding energy \(E_{\mathrm{p}} = h \nu_{\mathrm{p}}\) in the following. This energy can be measured by radio or microwave spectroscopy, driving a Zeeman or hyperfine transition between two impurity states, where one state is interacting with the bosonic bath forming the polaron, whereas the other state is non-interacting or only weakly interacting. The polaron binding energy manifests itself as a shift of the transition’s RF spectrum compared to an impurity without the bosonic bath, similar to Fermi-polarons [28, 29].
The method of RF spectroscopy is a standard tool for high-precision, but it is limited by the lifetime of the polaron. In our system the polaron decays predominantly by three-body losses, i.e. molecule formation, occurring with rate \(\gamma_{\mathrm{p}} = n_{\mathrm {BEC}}^{2} L_{3}\) with the loss coefficient \(L_{3}\) and the BEC density \(n_{\mathrm {BEC}}\). The decay limits the time during which the rf transition occurs to \(\tau =\frac{1}{\gamma_{\mathrm{p}}}\), implying a lower bound for the linewidth of the measured polaron spectroscopy peak. In order to clearly resolve the polaron peak at a frequency shift of \(\nu_{\mathrm{p}}\), the ratio \(\beta= \nu_{\mathrm{p}}/\gamma_{\mathrm{p}}\) should be significantly larger than 1 yielding a figure of merit for the determination of optimum experimental parameters.
From this we calculate the polaron coupling constant α and polaronic binding energy \(E_{\mathrm{p}}\) using a simple analytic expression for an impurity with infinite mass from [21], see Figure 1(b). Furthermore, the polaronic decay rate \(\gamma_{\mathrm{p}}\) is calculated from the BEC peak density and theoretical values for \(L_{3}\) in vicinity of the Feshbach resonance [40]. For \(\beta> 1\), we can reliably resolve polarons spectroscopically within the discussed parameter ranges, indicated by the vertical shaded regions in Figure 1.
Furthermore, we use the parameter \(\epsilon= 2\pi^{3/2} (1+m_{\mathrm{B}}/m_{\mathrm{I}}) a_{\mathrm {IB}} \sqrt{a_{\mathrm {BB}} n_{\mathrm {BEC}}}\) from [7] in Figure 1(c) to indicate where the Bose polaron realized can be described by means of the Fröhlich model discussed above, where \(\epsilon\ll1\) corresponds to a good description. One can see that ϵ lies well below 1 for the experimentally directly accessible range (blue), placing our setup mainly in the weak and intermediate coupling regime. For reference, the value of \(\epsilon <0.3\) given in [7] is marked as horizontally shaded area in Figure 1.
In conclusion, realizing a Bose-polaron is well possible with our experimental setup. Also the regime of Fröhlich-type polarons in the weak and intermediate coupling regime is well accessible and new polaron physics can be explored. For the observation of a strongly coupled Fröhlich polaron, the regime of \(\beta>1\) can be expanded towards higher \(a_{\mathrm {IB}}\), by lowering the 3-body lossrate. This can be achieved by reducing the BEC density, and thus decreasing ϵ, too, for example by increasing the trapping volume and levitating magnetically [44]. Employing Feshbach resonances with more favorable three-body loss properties, or the application of optical tuning methods [45] are being considered.
2.2 Rb Bose-Einstein condensate
In order to optimize the statistics of single atom probing, the experimental setup aims at a short BEC production time. This is realized by, first, a short initial laser cooling stage, where a 3D magneto-optical trap (MOT) is loaded from a 2D MOT in ≈2.5 s at a rate of 10^{9} atoms/s and, second, evaporation in a steep optical dipole trap which is formed by a horizontal and a vertical beam within ≈4 s. In order to avoid perturbation from cooling and trapping of single Cs atoms in a MOT, the Rb cloud is prepared in the magnetic field insensitive \(F=1\), \(m_{F} =0\) state while evaporating. We typically prepare a BEC with \(2.5\times10^{4}\) atoms at a peak density of \(1.2 \times10^{14}\mbox{ cm}^{-3}\) and a critical temperature of approximately 120 nK. The BEC’s decay rate of 0.4 Hz is dominated by two- and three-body collisions at this density but decreases to 0.08 Hz for a lower number of atoms. For technical details of the BEC production and state preparation, see Section 3.
2.3 Single atoms
In order to provide position control over the single Cs atoms independently from the Rb BEC trap, we apply a species selective optical conveyor belt lattice, formed by two counter propagating, linearly polarized laser beams with wavelength \(\lambda_{\mathrm{lat}} = 790\mbox{ nm}\) and a waist of 29 μm. For this wavelength between the Rb D-Lines, the resulting potential cancels out for Rb atoms [34, 35], but at the same time the frequency is blue detuned for Cs, providing confinement along the lattice axis in the nodes of the standing wave with depths up to \(7{,}300 E_{r}^{\mathrm{Cs}} = 850\mbox{ }\mu\mbox{K}\times\mathrm{k_{B}}\), with \(E_{r}^{\mathrm{Cs}} = \hbar^{2} k^{2} /2 m_{\mathrm{Cs}}\) the single photon recoil energy, \(k=2 \pi/\lambda_{\mathrm{lat}}\), kB the Boltzmann constant, and \(m_{\mathrm{Cs}}\) the mass of a Cs atom. While the lattice provides tight axial confinement for Cs atoms, it does not confine the atoms radially. We therefore superpose the lattice axis with one beam of the dipole trap (see Section 3) and obtain a maximum trap depth of 1.45 mK radially, resulting in trap frequencies of \(2\pi\times4\mbox{ kHz}\) radially and \(2\pi\times460\mbox{ kHz}\) axially. The lifetime of atoms in the lattice at full depth is limited to \(\tau \approx1.24\mbox{ s}\) by phase fluctuations. If in addition an optical molasses is used to cool the atoms, the lifetime can be extended up to \(\tau= 71\mbox{ s}\), limited by background pressure.
An important quantity characterizing the lattice is the selectivity \(s = E_{r}^{\mathrm{Rb}}/E_{r}^{\mathrm{Cs}}\) for a given intensity and wavelength. By performing Raman-Nath [47–49] scattering on the BEC for various lattice wavelengths, we have identified the optimal wavelength around 780 nm and find a selectivity of \(5{,}000:1\) for Cs. In order to transport the Cs atoms by a defined distance, a precisely controlled relative detuning δ between the lattice beams is used, which causes the standing wave interference pattern to move at a velocity \(v=\lambda_{\textrm{lat}} \delta/2\) for a specific amount of time [50]. For details also see Section 3.
2.4 Combining single atoms with the quantum gas
We see that the overlap in occupied trap volume of the two species decreases upon further cooling of the Rb cloud, leading to an increasing chance that the two species do not interact. For experiments involving colder Rb clouds, especially a BEC, the experiment will therefore be enhanced by an intermediate cooling step for Cs.
3 Methods
Our experiments take place in a two-chamber vacuum system consisting of a low pressure (\({\approx}10^{-10}\mbox{ mbar}\)) and a high pressure region (\({\approx}10^{-7}\mbox{ mbar}\)), separated by a differential pumping section of length 83 mm and diameter increasing from 1.8 mm to 4 mm. The low pressure region is formed by a glass cell, whereas the high pressure region is located in a titanium chamber and contains a 2D MOT for ^{87}Rb which is loading atoms from the background gas. The distance between the 3D MOT region and the 2D MOT is approximately 30 cm.
Single atom MOT
The position of the Cs MOT is overlapped with the Rb MOT as both use the same coil system. A set of 6 diaphragms with variable aperture mounted in the coil holders is used to align the MOT beams to precisely overlap in the middle of the glass cell. The cooling light is 10 MHz red detuned to the \(F=4 \rightarrow F'=5\) transition of the Cs D2 line and has a total power of typically 500 μW. The repumping light is on resonance to the \(F=3 \rightarrow F'=3\) transition with a total power of typically 15 μW. In every beam pair a piezo driven mirror at 110 Hz frequency destroys phase coherence between orthogonal beam pairs and therefore avoids interference effects, which lead to an instable MOT position. To keep the MOT loading time short, a low magnetic field gradient of 40 G/cm axially and 20 G/cm radially is applied for 20 ms. During this sufficiently short time the trap volume is large enough to load on average one atom [54]. In a next step the magnetic field gradient is increased in 8 ms to 275 G/cm in axial and 140 G/cm in radial direction. This effectively pins the atom number and avoids additional atom loading during the imaging process. The duration of the low gradient stage depends on the vapor pressure of the Cs atoms and the required atom number. For an efficient transfer of the trapped atoms into the optical dipole trap, a low MOT temperature is needed. Therefore we increase the red detuning of the cooler light to 72 MHz for 50 ms, while setting its power so low that we just do not lose the atoms. We release the atoms into the dipole trap by switching off the MOT beams. The cooler is switched off 2 ms after the repumper ensuring that the atoms remain in their lowest fine-structure state \(|{F=3}\rangle \).
Optical dipole trap and evaporative cooling
Our BEC is produced in a crossed beam optical dipole trap at 1,064 nm. The trap is formed by a horizontal beam with a focal waist of \(w_{0} = 22\mbox{ }\mu\mbox{m}\) at 4 W and a vertical beam with a focal waist of \(w_{0} = 165\mbox{ }\mu\mbox{m}\) at 12 W of power, yielding trap frequencies of 3.7 kHz radial to the horizontal beam and 120 Hz along the horizontal beam. The beam setup allows both for a forced evaporation scheme [55] as well as standard, passive evaporation [56]. The laser light is generated by a Nufern NUA-1064-PD-0050-D0 fiber amplifier and transported to the experiment by two optical fibers (LMA-PM-15 by NKT Photonics and Liekki Passive-10/125-PM). A PID controller and AOMs are employed to stabilize the dipole trap beams’ power with a bandwidth of 110 kHz.
After the Rb MOT has been loaded, both dipole trap beams are switched on at full power during a CMOT phase [57, 58]. The repumping light is switched off 200 μs before the cooling light, so that Rb is pumped to the \(F=1\) state and approx. 10^{6} atoms are transferred to the dipole trap. After 300 ms of self-evaporation we decrease the horizontal beam’s power exponentially. As soon as the increase in density within the crossing region causes atom loss, we ramp down the power of the vertical beam as well until a BEC with \(2.5\times10^{4}\) atoms forms after a total evaporation time of 4.3 s. The entire BEC sequence, including 3D MOT loading, CMOT phase and evaporation ramps has been optimized with an evolutionary algorithm, that is implemented as a part of our timing software [59].
Internal state preparation
The Rb atoms are prepared in the magnetically insensitive state \(|{F=1,m_{F}=0}\rangle \) during evaporation so that the magnetic field gradient of the single atom MOT does neither heat nor destroy the cold Rb cloud: During self-evaporation, a magnetic field of 1.4 G is switched on to lift the degeneracy of the Zeeman substates. Then the atoms are pumped to the \(|{F=1,m_{F}=1}\rangle \) state by a 900 μs light pulse resonant to the \(F=1 \rightarrow F'=1\) \(\sigma ^{+}\)-transition and a 500 μs light pulse resonant to the \(F=2 \rightarrow F'=2\) π-transition of the \(D_{2}\) line. Both beams are turned on simultaneously. Their respective powers are 300 nW and 480 μW with red detunings of 2.6 MHz and 1 MHz at equal beam waists of 1.2 mm.
By applying a magnetic field gradient of typically 5.7 G/cm during a 13 ms time of flight experiment, we perform a Stern-Gerlach experiment to measure the population of magnetic substates. Without optical pumping we observe an almost equally distributed spin mixture, while with optical pumping approximately 90% of the atoms are in the \(m_{F}=1\) Zeeman substate and approximately 10% remain in \(m_{F}=0\). Optical pumping does not deplete the number of atoms in the condensate.
After optical pumping we transfer the population to the \(m_{F}=0\) state with a Landau-Zener sweep: We apply a radio frequency at 10.176 MHz for 100 ms while we increase the magnetic field linearly. The magnetic field ramp is chosen such that the detuning of the radio frequency with respect to the \(|{m_{F}=1}\rangle \rightarrow |{m_{F}=0}\rangle \) transition falls from approximately +550 kHz to −550 kHz. We observe a Rabi-frequency of \(\Omega= 2\pi\times 2.1\mbox{ kHz}\) with a transfer efficiency of 95%, with no negative impact on the number of atoms in the BEC.
3.1 Single atom imaging
To observe the single atoms we use fluorescence imaging. Near resonant light from the MOT beams is scattered in the whole solid angle. We collect 3.3% of the photons corresponding to 0.012 pW/atom for a saturation parameter \(S_{0}=I/I_{s}=1\) by a custom made objective. It has a numerical aperture (NA) of 0.36 and is placed beneath the glass cell at a distance of 30.3 mm to the atoms’ position (see Figure 5(c)). Stray light protection is crucial and the whole beam path is located inside black anodized lens tubes and mirror housings.
3.2 Optical lattice
3.3 Lattice transport
For the single atom transport in the conveyor belt optical lattice [50], a relative detuning δ between the two lattice beams is employed to create a standing wave pattern, moving at velocity \(v=\lambda_{\textrm{lat}} \delta/2\). For a typical transport, the detuning between the beams is ramped in a trapezoid shape: a linear ramp from zero to the maximum detuning yields an acceleration of the atoms in the lattice, followed by a plateau of constant detuning where the atoms move at constant velocity. After this plateau, the detuning is ramped back to zero and the atoms are decelerated again. The transport distance is given by the integral over the velocity. Thereby large transport distances in the range of millimeters can be realized, which are only limited by trap size. This is in contrast to phase shifting transport approaches, where the maximum transport distance is given by the maximum phase shift [60]. The absolute maximum acceleration \(a_{\max} = k U_{0} / m_{\mathrm{Cs}} \approx4\cdot10^{5}\mbox{ m/s$^{2}$}\) in the lattice is determined by the Cs mass \(m_{\mathrm{Cs}}\), the laser wave number k and the potential depth \(U_{0} \approx 1.5\mbox{ mK}\) of the standing wave potential.
In order to control the detuning, each lattice beam is frequency shifted by a common value of \(f_{1} \approx f_{2} = 160\mbox{ MHz}\) in an AOM double-pass setup. The relative detuning between the two beams then is given by \(\delta= f_{1} - f_{2}\) and can be controlled by the two RF frequencies supplied to the AOMs. Both RF signals have to be phase-stable compared to each other and the offset-frequency of both channels has to be equal up to a fraction of a Hertz to yield a stable standing wave pattern when both beams interfere for \(\delta= 0\mbox{ Hz}\). Therefore, we use a driver electronics based on direct digital synthesis (‘DDS’) with an amplifier chain. The DDS chips employed (AD9954) sample the output sine wave in a digital circuit and have an analog digital converter stage for signal output. By supplying both DDS chips with the same clock signal which is locked to a Rb frequency standard, the relative phase stability between the signals is guaranteed by the digital sampling of the output signal. The frequency of the output sine wave is set by a 32 bit control parameter, yielding a frequency resolution of 0.09 Hz which allows for very small frequency variations compared to the output frequency of 80 MHz. To supply output powers of up to ∼30 dBm a two stage amplifier chain combined with a voltage controlled attenuator for output power control is applied. The RF power level can be directly used to adjust and stabilize the light intensity of the lattice beams. The DDS chips are controlled by a microcontroller which provides exact timing of frequency changes and stores the data for the frequency ramps.
4 Conclusion
We have presented a cold gas experimental system to study the Bose polaron by immersion of single neutral Cs impurities into a Rb BEC. This single-impurity approach yields the possibility of investigating individual polarons in a highly controlled system. By tuning the impurity-bath interaction, a wide range of coupling strengths from the weak to the strong coupling regime is accessible. We have estimated typical parameters, characterizing Fröhlich-type Bose-Polarons in our system. Our estimations show the feasibility of spectroscopically measuring the binding energy of Fröhlich-polarons in the weak and intermediate coupling regime with our current setup and how the strong coupling regime can be reached. With an all-optical approach to BEC production, we reach short cycle times needed to obtain good statistics for single-impurity measurements. Experimentally, we have demonstrated trapping and transporting of impurity atoms in a species-selective conveyor-belt lattice, as well as imaging with high optical resolution. Furthermore, we have shown the successful immersion of single impurities into a cold gas and their detection after a defined interaction time. The combination of high-resolution imaging and position control in a quantum gas will allow a systematic study of static and dynamical properties of individual polarons as well as interaction effects of multi-polaron systems.
Notes
Declarations
Acknowledgements
The project was financially supported partially by the European Union via the ERC Starting Grant 278208 and partially by the DFG via SFB/TR49. D.M. is a recipient of a DFG-fellowship through the Excellence Initiative by the Graduate School Materials Science in Mainz (GSC 266), F.S. acknowledges funding by Studienstiftung des deutschen Volkes, and T.L. acknowledges funding from Carl-Zeiss Stiftung.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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