Deflection of a Molecular Beam Using the Bichromatic Stimulated Force
We demonstrate that a bichromatic standing-wave laser field can exert a significantly larger force on a molecule than ordinary radiation pressure. Our experiment measures the deflection of a pulsed supersonic beam of CaF molecules by a two-frequency laser field detuned symmetrically about resonance with the nearly closed transition. The inferred force as a function of relative phase between the two counterpropagating beams is in reasonable agreement with numerical simulations of the bichromatic force in this multilevel system. The large magnitude of the force, coupled with the reduced rate of spontaneous emission, indicates its potential utility in the production and manipulation of ultracold molecules.
pacs:37.10.Mn, 37.10.Vz, 37.20.+j
Present address: ]Department of Physics, Yale University, New Haven, CT 06520, USA. ††thanks: Deceased
Radiative forces on atoms have been the major tool enabling laser cooling and trapping Metcalf1999 and the myriad of applications which have resulted, such as precision spectroscopy, quantum degenerate gases, ultracold collisions, and quantum simulations. There are two general types of radiative force, the spontaneous force, also known as radiation pressure, and the stimulated force, also known as the dipole force. Radiation pressure is the result of repeated absorption/spontaneous emission cycles, while the stimulated force arises from absorption followed by stimulated emission. The latter requires an intensity gradient and can be thought of as a coherent redistribution of photons between various propagation directions. Laser cooling, trapping, and manipulation of atoms is a well-developed field, but more recently, these techniques have been increasingly applied to molecules Bohn2017 . This extension is nontrivial due to the complicated internal structure of molecules caused by their vibrational and rotational degrees of freedom DiRosa2004 . Radiation pressure has been used to slow, cool, and trap molecules that fortuitously have near-cycling transitions Barry2012 ; Hemmerling2016 ; Truppe2017 ; Hummon2013 ; Barry2014 ; Norrgard2016 ; Anderegg2017 ; Truppe2017a ; Kozyryev2017 ; Lim2018 . Stimulated forces have also been used to manipulate molecules Hill1975 ; Voitsekhovich1994 ; Stapelfeldt1997 ; Bishop2010 , but on a much more limited scale. Compared to radiation pressure, stimulated forces have two significant advantages for molecules: (1) radiation pressure is limited by the spontaneous emission rate, while stimulated forces can greatly exceed this saturated value; and (2) radiation pressure relies on spontaneous emission which can quickly optically pump the molecules into “dark” states which no longer interact with the laser field.
A specific type of stimulated force, the bichromatic force (BCF) Soding1997 ; Metcalf2017 , is particularly promising for manipulating molecules Chieda2011 . As the name implies, the BCF involves two frequencies which are tuned symmetrically above and below a resonant frequency by . The two frequencies are both present in two oppositely-directed beams, which gives rise to counterpropagating trains of beat notes with a fixed relative phase, . In a simplified picture of the BCF, if each beat is considered an effective -pulse which inverts the population, a molecule can be excited by a beat from one direction and then rapidly returned to the ground state by a beat from the other direction. Each absorption/stimulated emission cycle imparts an impulse of and repeats at a rate determined by the beat frequency, which is set by the detuning, . For a given detuning, the Rabi frequency must be maintained at in order to maintain the effective -pulse condition and properly cycle the population. The rate of momentum transfer, i.e. force, can thus increase arbitrarily with detuning, provided there is sufficient laser intensity to maintain the correct Rabi frequency. The direction of force is controlled by , since this determines the sequence of beat-pulse arrivals. There are several beneficial aspects of the BCF: it can be much larger than the radiation pressure force; it has a large range of velocities over which the force is relatively constant; and it does not rely on spontaneous emission and therefore does not inherently suffer from the problem of optical pumping into dark states. All of these are important for slowing a beam of molecules. The large force means that the beam can be slowed over a short distance, resulting in less spreading and higher brightness. The large capture range means that higher velocities and broader velocity distributions can be slowed. The relative unimportance of spontaneous emission reduces the need for multiple lasers to retrieve population which has been lost to optical pumping. This makes the BCF more applicable than the spontaneous force in manipulating diatomic and polyatomic molecules which lack near-cycling transitions due to poor vibrational overlap.
The BCF has been observed and studied in various atomic systems Soding1997 ; Williams1999 ; Cashen2001 ; Partlow2004 ; Liebisch2012 ; Chieda2012 ; Feng2017 . Here we demonstrate the effectiveness of the BCF in a molecular system via the transverse deflection of a beam of CaF, a polar diatomic molecule, as shown in Fig. 1(a). We find that the force exhibits the expected dependence on the relative phase, , and is larger, at our achieved laser intensity, than the maximum radiation pressure force by a factor of 4. Significantly, the deflection does not require any measures to counteract rovibrational optical pumping. Overall, the measurements are in reasonable agreement with numerical simulations of the multilevel molecular system. We note that similar results on the BCF have been recently reported with triatomic SrOH molecules Kozyryev2018 , a system which required an additional laser for rovibrational repumping.
The basic theory of the BCF in a two-level system has been dealt with from various perspectives, including the -pulse picture Voitsekhovich1989 ; Voitsekhovich1991 ; Soding1997 ; Cashen2003 , numerical simulations of the optical Bloch equations Williams1999 , and the doubly-dressed-atom picture Grimm1994 ; Yatsenko2004 . Extensions to multilevel systems, which are necessary for applying the BCF to molecules, have also been carried out Aldridge2016 ; Yang2017 . We have recently reported Aldridge2016 on such numerical calculations of the time-dependent density matrix for CaF, which we briefly summarize here. The BCF is applied using the branch at 531 nm on the rotational transition, as shown in Fig. 1(b) and Fig. 1(c). This transition is nearly closed with respect to vibration–Franck-Condon factor (FCF) = 0.9987–and selection rules prevent loss to other rotational levels. The decay rate of the excited state is MHz). The resonant frequency, , is taken to be between the centers of mass of the excited-state and ground-state hyperfine manifolds. The former splitting is negligible Tarbutt2015 , while the latter extends over 146 MHz. As verified in the calculations Aldridge2016 , this hyperfine structure does not require sidebands for repumping due to the relatively large Rabi frequencies and detunings used. The linearly-polarized electric field is taken to have four components of equal amplitude, : two counterpropagating beams (along ), each containing two frequencies, and . The phase difference between the electric fields of the counterpropagating beat notes is . We incorporate a total of 16 states, which includes the rotation, fine structure, and hyperfine structure with the associated magnetic sublevels. Rabi frequencies are defined for each possible transition between ground and excited states, and spontaneous decay is included. To prevent population from being trapped in dark sublevels, a skewed magnetic field of magnitude 29.2 G and oriented at with respect to the laser polarization, as employed in the experiment, is included as well. The transverse velocity enters via the substitution in the expression for the electric field, and the velocity-dependent force is obtained from the density matrix elements by applying Ehrenfest’s theorem. Examples of the BCF vs. velocity for various phases, , are shown in Fig. 2(a). Of particular note is the enhancement of the BCF relative to the saturated radiation pressure force, as well as the broad range of velocities over which the BCF is relatively constant. The dependence of the velocity-averaged force on is striking: it is maximized for , effectively vanishes for , and is inverted for . Care must be taken in calculating the velocity-averaged force. The force vs. velocity profile exhibits many narrow spikes which should not be included because such large forces act only for a short time before the velocity change pushes the molecule off the spike. The averaging which accounts for this effect is done by calculating the mean and standard deviation, , for the distribution and identifying those points whose magnitude exceeds the mean by . These points are reset to the mean, and the process is repeated until convergence is achieved. Points whose magnitude is less than the mean are not adjusted since these less-than-average forces result in small velocity changes which can act for extended periods. We note that excluding the spikes in this way typically reduces the average force by , as shown in Fig. 2(a).
In the experiment, we subject a supersonic beam of CaF to the BCF and measure the force via the transverse deflection of the beam. The pulsed molecular beam is generated by ablating a Ca rod situated in front of a 1 mm diameter nozzle from which a mixture of Ar (98%) and SF (2%) at a backing pressure of 30 psig emerges. The ablation employs a 10 ns, 10 mJ pulse from a frequency-doubled Nd:YAG laser focused to 1 mm diameter. The rod is slowly rotated and translated to reduce shot-to-shot signal variations. Various species result from the ablation and subsequent reactions, one of which is the CaF of interest. A pulsed valve (Lasertechnics LPV piezo-electric valve) reduces the gas load into the chamber and yields a pulse of molecules with an average longitudinal velocity m/s and a FWHM of 75 m/s, as measured by time-of-flight. A 1 mm diameter aperture located 131 mm downstream from the nozzle collimates the beam to a FWHM of 13.7 mrad.
The BCF is applied in the interaction region, situated 251 mm from the source. The light for the BCF derives from a distributed-Bragg-reflector (DBR) diode laser at 1062 nm which is amplified in a fiber amplifier and then frequency doubled in a single pass through a periodically-poled lithium niobate crystal, providing a maximum of 1.3 W at 531 nm. The two frequencies for the BCF, at , where MHz), are generated by two acousto-optical modulators (AOMs) and combined on a beamsplitter to provide matched intensities. At this detuning, the total peak intensity is 60 W/cm in each of the beam frequency components, corresponding to a Rabi frequency of MHz) Aldridge2016T . A sample of the 531 nm light at the central (resonant) frequency, , is used in an I saturated absorption spectrometer to lock the frequency of the DBR laser. The two-frequency beam is focused to an elliptical radius (waist) of 0.7 mm horizontally by 0.5 mm vertically at its perpendicular intersection with the molecular beam. The counterpropagating two-frequency beam is provided by retroreflection from a plane mirror located outside the vacuum chamber. The distance of this mirror from the molecular beam determines the phase difference, . For a detuning MHz), is obtained for a distance of 98.6 mm–a path length difference of 197.2 mm. For larger phase differences, the intensity imbalance due to the expansion of the retroreflected beam must be accounted for. The interaction time is sufficiently short that optical pumping out of the nearly-closed transition is negligible.
The deflection caused by the BCF is measured by molecular fluorescence. At a distance 223 mm downstream from the interaction region, light from a cw dye laser perpendicularly (along ) intersects the molecular beam immediately in front of a 0.76 mm slit which is scanned along in front of a fixed PMT (Hamamatsu H10721-20) detector. The resulting fluorescence is collected and collimated by a 20 mm focal-length, 25 mm diameter aspherical lens and imaged 1:1 with a matching lens onto the 8 mm diameter active area of the detector. The vertical () aperture of the detector is restricted to 1.5 mm in order to reduce both scattered light from the walls of the vacuum chamber and the signal from molecules which did not pass through the central high-intensity region of the BCF beams. The detection laser is locked to the , transition at 583 nm using I saturated absorption. Although this nondiagonal transition has a poor FCF of , it can nevertheless be saturated with the utilized power of 300 mW focused to an elliptical waist of 0.4 mm horizontally by 1.2 mm vertically. This transition has the significant advantage that the strongest fluorescence, , is separated in wavelength by nm from the excitation transition, allowing background scattered light to be greatly reduced with appropriate filtering. The tradeoff is that each molecule only emits one photon before ending up in the dark state.
Data are obtained by integrating the fluorescent signal during a 675 s gate centered on the arrival time of the peak of the pulsed molecular beam, which occurs ms after the pulsed valve opens. A second gate, centered 2.25 ms after the molecular peak, is used to measure the background scattered light which is then subtracted from the signal occurring in the first gate. At each position of the slit, 5000 shots, each yielding detected photons, are averaged for each deflection condition and repeated in a randomized order to reduce systematic errors. Error bars are statistical in nature, and are dominated by shot-to-shot fluctuations in the molecular source.
Scans of the undeflected beam and the deflected beam for various phases, , are shown in Fig. 3. Also shown are the differences between the deflected and undeflected curves, which clearly indicate the directions of the force: for , 0 for , and for . In order to compare with theory, we calculate the deflection as the difference between the average values of for the two curves. Because of uncertainties in the fluorescence collection efficiency in the wings of the molecular beam profiles, the deflection calculations only include the central regions, indicated by the dashed vertical lines in Fig. 3. The deflection vs. is plotted in Fig. 4. These points represent the average deflection over a number of runs, typically 9 per phase, and the error bars are the corresponding standard deviations.
In order to compare the measured deflections with the results of the numerical simulations, several factors must be accounted for. In particular, the BCF has a significant dependence on intensity, as seen in Fig. 2(b). The slit in the detection region has a vertical () extent of 1.5 mm, which extrapolated back to the interaction region, corresponds to 0.79 mm. Since the incident elliptical BCF beam has a FWHM of 0.56 mm along and 0.82 mm along , not all molecules detected at a given value of will have experienced the same intensity. Furthermore, as a molecule propagates along through the BCF beams, it experiences a time-varying BCF intensity. For a given value of , we first calculate the -momentum transfer by integrating the intensity-dependent BCF force, , over time : . Since the transverse spread of velocities in the undeflected beam is significantly less than the range of velocities affected by the BCF (see Fig. 2(a)), we use an average value of the force over the relevant velocity range, as discussed earlier and shown in Fig. 2(b). We then repeat this for various values of and perform a weighted average over , accounting for the variation of the undeflected beam profile with . For this, we assume a profile, , which is radially symmetric, i.e. with . The function is taken to be a symmetrized version of the measured distribution of the undeflected beam. The signal, , corresponding to the deflected beam profile is given by:
where m is the molecular mass and is the time-of-flight from the interaction region to the detection region. From this calculated signal as a function of , we can extract the average deflection . As with the experimental data, we only use the central region of the profiles for these calculations. These average values of are shown as a function of , together with the corresponding measured values, in Fig. 4. We show three versions of the calculations. In the first (solid curve), we assume that the two counterpropagating beams have matched intensities. In the second, we allow for a constant fractional intensity imbalance, equal to the imbalance of the peak intensities resulting from the measured expansion of the retroreflected beam. This reduction of the retroreflected peak intensity ranges from 6.33% at to 30.71% at . In the third version, shown only for , we explicitly account for the and dependencies of the imbalance, obtained from the measured beam profiles, when calculating the force. This is nontrivial because of the ellipticity of the BCF beams and imbalance reversals in certain regions. This more realistic accounting for imbalance shows the best agreement with the data point, but still overestimates the deflection by 2.2 standard deviations. This could be due to imperfect overlap of the BCF beams and/or deflection due to radiation pressure. We also calculated the deflection in the extreme case of 100% intensity imbalance, i.e. no reflected intensity, shown as the dashed horizontal line. This corresponds to simple radiation pressure and demonstrates that the BCF can provide a significantly larger force.
We have conducted control experiments to confirm that the observed deflections are indeed due to the BCF. These results are also shown in Fig. 4. First, we blocked the retroreflected beam, which results in radiation-pressure deflection. The measured deflection is consistent with that calculated from radiation pressure (dashed horizontal line), which is significantly less than that from the maximum BCF. We then restored the retroreflected beam (at ), but blocked the frequency, yielding a simple standing wave of frequency . The deflection in this case is consistent with zero, as expected.
In conclusion, we have observed the action of the bichromatic force (BCF) on a molecule by measuring the deflection of a beam of CaF. The inferred force is significantly larger than the ordinary radiation pressure force, demonstrating that this stimulated force, with its broad velocity capture range and non-reliance on spontaneous emission, will be a welcome addition to the arsenal of molecule manipulation techniques. In particular, it will be useful in rapidly decelerating molecular beams for efficient trap loading, especially if chirping is incorporated to compensate for the changing Doppler shift Chieda2012 . For such applications, the use of a cryogenic buffer-gas-cooled molecular beam Lu2011 ; Kozyryev2018 would be very beneficial. Further improvements in the force are expected by utilizing higher detunings/intensities and employing polychromatic Galica2013 instead of bichromatic fields.
This work was supported by the National Science Foundation and the US-Israel Binational Science Foundation. The paper is dedicated to coauthor Edward Eyler for his vision of the bichromatic force as a useful means of manipulating molecules.
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