Download Accurate determination of the rotational constants of ultracold

Accurate determination of the rotational
constants of ultracold molecules using double
photoassociation spectroscopy
Jie Ma,1,* Yuqing Li,1 Wenliang Liu,1 Peng Chen,1 Guosheng Feng,1 Chenyang Hu,1
Jizhou Wu,1,2 Liantuan Xiao,1 and Suotang Jia1
1
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, College of
Physics and Electronics Engineering, Shanxi University, Taiyuan, 030006, China
2
[email protected]
*
[email protected],
Abstract: We report on an accurate determination of the rotational
constants of the ultracold long-range Cesium molecules in near dissociation
domain. The scheme relies on a precise reference of the frequency
difference in a double photoassociation spectroscopy induced by two laser
beams based on an acoustic-optical modulator. The rotational constants are
obtained by fitting a non-rigid rotor model into the frequency intervals of
the neighboring rotational levels deduced from the reference.
©2014 Optical Society of America
OCIS codes: (140.3320) Laser cooling; (300.6320) Spectroscopy, high-resolution; (300.6390)
Spectroscopy, molecular; (230.1040) Acousto-optical devices; (120.7280) Vibration analysis.
References and links
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
O. Dulieu and C. Gabbanini, “The formation and interactions of cold and ultracold molecules: new challenges
for interdisciplinary physics,” Rep. Prog. Phys. 72(8), 086401 (2009).
S. Ospelkaus, K. K. Ni, D. Wang, M. H. de Miranda, B. Neyenhuis, G. Quéméner, P. S. Julienne, J. L. Bohn, D.
S. Jin, and J. Ye, “Quantum-state controlled chemical reactions of ultracold potassium-rubidium molecules,”
Science 327(5967), 853–857 (2010).
M. Schnell and G. Meijer, “Cold Molecules: Preparation, Applications, and Challenges,” Angew. Chem. Int. Ed.
Engl. 48(33), 6010–6031 (2009).
K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne, “Ultracold photoassociation spectroscopy: Long-range
molecules and atomic scattering,” Rev. Mod. Phys. 78(2), 483–535 (2006).
J. H. Gurian, P. Cheinet, P. Huillery, A. Fioretti, J. Zhao, P. L. Gould, D. Comparat, and P. Pillet, “Observation
of a Resonant Four-Body Interaction in Cold Cesium Rydberg Atoms,” Phys. Rev. Lett. 108(2), 023005 (2012).
J. Ma, L. R. Wang, Y. T. Zhao, L. T. Xiao, and S. T. Jia, “Absolute frequency stabilization of a diode laser to
cesium atom-molecular hyperfine transitions via modulating molecules,” Appl. Phys. Lett. 91(16), 161101
(2007).
C. Drag, T. B. Laburthe, B. T’Jampens, D. Comparat, M. Allegrini, A. Crubellier, and P. Pillet,
“Photoassociative Spectroscopy as a Self-Sufficient Tool for the Determination of the Cs Triplet Scattering
Length,” Phys. Rev. Lett. 85(7), 1408–1411 (2000).
J. Ma, J. Wu, G. Chen, Q. Fan, H. Feng, X. Dai, W. Sun, L. Xiao, and S. Jia, “Experimental Determination of the
Rotational Constants of HighLying Vibrational Levels of Ultracold Cs2 in the 0g− Purely Long-Range State,” J.
Phys. Chem. Lett. 4(21), 3612–3617 (2013).
M. Pichler, H. Chen, and W. C. Stwalley, “Photoassociation spectroscopy of ultracold Cs below the 6P1/2 limit,”
J. Chem. Phys. 121(4), 1796–1801 (2004).
J. Ma, L. R. Wang, Y. T. Zhao, L. T. Xiao, and S. T. Jia, “High sensitive photoassociation spectroscopy of the
Cs molecular 0u+ and 1g long-range states below the 6S1/2 + 6P3/2limit,” J. Mol. Spectrosc. 255(2), 106–110
(2009).
Y. C. Zhang, J. Ma, Y. Q. Li, J. Z. Wu, L. J. Zhang, G. Chen, L. R. Wang, Y. T. Zhao, L. T. Xiao, and S. T. Jia,
“Direct measurement of laser-induced frequency shift rate of ultracold cesium molecules by analyzing losses of
trapped atoms,” Appl. Phys. Lett. 101(13), 131114 (2012).
B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules (Longman Group Press, 1983).
D. Comparat, C. Drag, A. Fioretti, O. Dulieu, and P. Pillet, “Photoassociative Spectroscopy and Formation of
Cold Molecules in Cold Cesium Vapor: Trap-Loss Spectrum versus Ion Spectrum,” J. Mol. Spectrosc. 195(2),
229–235 (1999).
J. Z. Wu, J. Ma, Y. C. Zhang, Y. Q. Li, L. R. Wang, Y. T. Zhao, G. Chen, L. T. Xiao, and S. T. Jia, “High
sensitive trap loss spectroscopic detection of the lowest vibrational levels of ultracold molecules,” Phys. Chem.
Chem. Phys. 13(42), 18921–18925 (2011).
#202889 - $15.00 USD
Received 11 Dec 2013; revised 20 Jan 2014; accepted 31 Jan 2014; published 10 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003754 | OPTICS EXPRESS 3754
15. A. Fioretti, D. Comparat, C. Drag, C. Amiot, O. Dulieu, F. Masnou-Seeuws, and P. Pillet, “Photoassociative
spectroscopy of the Cs2 0g− long-range state,” Eur. Phys. J. D 5(3), 389–403 (1999).
16. J. Ma, J. Z. Wu, Y. T. Zhao, L. T. Xiao, and S. T. Jia, “Determination of the rotational constant of the Cs2 0g- (6s
+ 6p3/2) state by trap loss spectroscopy,” Opt. Express 18(16), 17089–17095 (2010).
1. Introduction
Recently, there have been some extensive studies in the production and manipulation of the
ultracold molecules, which have shown significantly promising applications in areas ranging
from precision mmeasurement to quantum information storage and processing [1–3].
Photoassociation (PA) of the ultracold atoms has been established as a versatile tool in the
formation of the ultracold molecules [4]. The PA process is dependent on the collisional
interaction properties. On the other hand, PA spectroscopy (PAS) delivers the detail
knowledge of the structure of the formed molecules. Specially, PAS provides a unique
opportunity to obtain information on the molecular long-range state. In particular, PAS has
been used vastly to perform precise measurements of the high-lying vibrational levels that are
essential to determine the molecular parameters near the dissociation limits and often difficult
to access by the traditional bound-bound molecular spectroscopy.
Numerous experimental studies have been reported about the ultracold Cesium molecule
(Cs2) formed by PA, for instance, resonant four-body interaction in cold Cesium Rydberg
atoms [5], absolute frequency stabilization to Cesium atom molecular hyperfine transitions
[6], and precise measurement of Cesium atomic triplet state scattering length [7]. As PA
spectra deeply covers abundant rotational data, reasonable investigations are required to
precisely determine the rotational constants of the long-range molecules, which are crucial in
predicting the unexplored levels and obtaining the precise potential energy curve [8].
Recently, some efforts have been made to obtain the rotational constants of the low-lying
vibrational levels in the Cs2 0g− (6S1/2 + 6P1/2) long-range state [9]. However, the rotational
constants of the high-lying vibrational levels, which are important to precisely define the
near-dissociation part of the potential energy curves, are difficult to determine by
conventional direct measuring method, i.e. using a high resolution wavelength meter. The
small frequency intervals between neighbouring rotational levels preclude the accurate
measurements. An available method is to employ a Fabry-Perot reference cavity to monitor
the changes of the relative frequency. However, the sensitivity is not sufficiently high. In
particular, the measuring accuracy of a marker interferometer is greatly suppressed by the low
frequency 1/f noise which results from the long scanning time in PA experiments. An active
locking could be carried out to remedy the noise for some situations (with sub-MHz
accuracy), nevertheless, powerful calibration schemes with quick and expedient implement
still need to be widely researched.
In this paper, we develop a robust technique, namely “double PA spectroscopy”, for
frequency measurement with high sensitivity and demonstrate the experimental determination
of the rotational constants for the high-lying vibrational levels in the molecular long-range
state. The rotational constants of the vibrational levels [vD] – v = 15 ~42 in the Cs2 0g− longrange state below the (6S1/2 + 6P1/2) limit have been measured.
2. Experimental section
The experimental setup is schematically depicted in Fig. 1(a), basically consists of three parts:
laser system, double pass and PA system. PA is induced by a cw widely tunable Ti: Sapphire
laser (Coherent MBR-110, ~1 W, line-width ~0.1 MHz) driven by a 10W Verdi pump laser.
The long-time frequency drift of Ti: Sapphire laser system is less than 500 kHz by locking to
its self-reference cavity. The absolute laser frequency measurement is performed and
monitored by a wavelength meter (HighFinesse-Angstrom WS/7R, absolute accuracy ~60
MHz). The wavelength meter is calibrated against the Cesium atomic hyperfine resonance
transition, 6S1/2 (F = 4) → 6P3/2 (F’ = 5), corresponding to wavenumber of 11732.176 cm−1.
Ultracold Cesium atoms are produced in a typical vapour-loaded MOT [10]. The
temperature is measured to be about 120 μK by time-of-flight method. Atomic number is
#202889 - $15.00 USD
Received 11 Dec 2013; revised 20 Jan 2014; accepted 31 Jan 2014; published 10 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003754 | OPTICS EXPRESS 3755
measured to be about 5 × 107 using absorption method, and the dimension (FWHM) of the
atomic cloud is ~800 µm, yielding a peak density on the order of 1011 cm−3. Both the trapping
laser and the repumping laser are frequency stabilized diode laser systems (Toptica DL100,
~160 mW with line-width 0.8 MHz).
Fig. 1. (a). Experimental setup. M: high reflective mirror; OI: optical isolator; H: half wave
plate; PBS: polarization beam splitter; L: lens; S: Shutter; Q: quarter wave plate; AOM:
acousto-optic modulator; BT: beam trap; BPF: band pass filter; APD: Avalanche photodiode;
PD: photodiode; CCD: Charge-coupled Device. Figure 1 (b). Schematic diagram of the double
PA spectroscopy. The dashed inset demonstrates the sequence diagram. ts, ~10 ms, represents
the response time in which the laser beam can be turned on/off completely.
Formation of the ultracold molecules by PA and the double PA Spectroscopy are
schematically demonstrated in Fig. 1(b). Initially an excitation from a free scattering state of
two atoms to a bound excited molecular level is driven by resonant laser with frequency ν.
This excited molecule spontaneously decays into ground state. PAS directly illustrates the
number of produced excited molecules as a function of ν. PA spectra below the dissociation
limit (6S1/2 + 6P1/2) are recorded by monitoring the atomic fluorescence as PA laser scans.
Modulation technology is implemented to improve the signal-to-noise ratio of PA spectra
[10].
The double PA spectroscopy directly offers a precise reference of the frequency difference
and enables measuring small frequency interval of the adjacent rotational levels for high-lying
vibrational states. As shown in the dashed inset of Fig. 1(b), the double PA spectrum is
obtained through alternately illuminating the cold atomic sample by two PA laser beams (I
and II). Partial output from the PA laser is frequency shifted by a double-passed acousto-optic
modulator (AOM, MT110-B50A1-IR, AA Optoelectronic) to provide a calibration beam with
an fixed offset of Δ0 = ν1 – ν2 = 220 MHz. The modulated laser beam (Beam II) is diffracted
from the double pass setup and well overlapped in space with the main output of the PA Laser
(Beam I). Beam I at first induces trap loss of the cold atoms and a ro-vibrational spectrum
including rotational levels from J = 0 to J = n is acquired. Later, Beam I is quickly switched
off while simultaneously Beam II interacts with the atoms in turn. A selected spectral feature
#202889 - $15.00 USD
Received 11 Dec 2013; revised 20 Jan 2014; accepted 31 Jan 2014; published 10 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003754 | OPTICS EXPRESS 3756
containing rotational level J = n is thus scanned twice (denoted as J’ = n). A series of the
double PA spectra are therefore recorded. As PA laser scanning with stringent linearity and
unvaried speed, frequency interval between the resonance peaks J = n and J’ = n exactly
equals to Δ0 = 220 MHz, i.e. the fixed frequency difference between Beam I and Beam II.
This known precise separation of Δ0 is used to calibrate the double PA spectra, frequency
intervals ΔυJ (J = 0, 1, …, n-1) of the neighboring rotational levels are accurately deduced. It
should be noted that the intensities of Beam I and II are set identical in order to avoid effects
of laser induced frequency shifts [11]. The continuity and linearity of the frequency scans are
assured by monitoring the PA laser frequency real time with a wavelength meter (WS/7R).
Ultimately, the rotational constants are obtained by fitting the data of ΔυJ to the rotor model
[12].
3. Results and discussion
Fig. 2. (a) PA spectrum (red curve) vs. double PA spectrum (green curve) for the high-lying
vibrational level ([vD] – v = 17) of the Cs2 0g− (6S1/2 + 6P1/2) long-range state. The PA spectrum
is directly inverted in order to make curves comparable. PA laser frequency is tuned near
11174.91cm−1. (b) Dependence of the frequency intervals ΔυJ on J for [vD] – v = 17 of the Cs2
0g− (6S1/2 + 6P1/2) long-range state. The circle symbols represent experimental data, while the
lines are the fits to the nonlinear non-rigid and linear rigid rotor models.
Trap-loss spectroscopy [13] conveniently maps out the binding energies of the vibrational
levels. The vibrational spectroscopy of the Cs2 0g− (6S1/2 + 6P1/2) long-range state has been
reported in the previous works based on the direct trap-loss detection [9]. However, the rovibrational spectrum of the high-lying vibrational levels (specially [vD] – v < 23) with
enhanced resolution has still remain to be achieved by direct trap-loss detection. Here, we
employ a lock-in method based on modulating the fluorescence of the ultracold atoms in
order to availably improve sensitivity of the trap-loss detection [14]. A typical PA rovibrational spectrum (red line), which specifies the high-lying vibrational level ([vD] – v = 17)
of the Cs2 0g− (6S1/2 + 6P1/2) long-range state, is acquired as the atomic sample is illuminated
only by Beam I with an intensity of 43.5 W/cm2, as shown in Fig. 2(a). As the response time
of the atoms against the PA Beam is typically long (~2s), the PA laser scanning speed should
be small to enhance the resolution. The PA Laser is slowly scanned at a rate of 5.0 MHz/s. In
addition, the response time (~10 ms) for switching the two shutters (S1 and S2) is shorter than
the loading time (~1 s) of MOT, thus the resonance peaks are clearly demonstrated without
affecting each other. The resolved rotational structure with rotational progressions J as high as
6 is therefore clearly observed. It should be noted that the frequency intervals ΔυJ for the
neighbouring rotational progressions of the high-lying vibrational levels (such as [vD] – v =
17) are rather small comparing to those for the low-lying levels. As the value of the fixed
#202889 - $15.00 USD
Received 11 Dec 2013; revised 20 Jan 2014; accepted 31 Jan 2014; published 10 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003754 | OPTICS EXPRESS 3757
frequency offset can be set through the AOM, whose accuracy is on the order of kHz, the
slight ΔυJ can be directly measured using this precise reference,.
In order to obtain the rotational constants for the high-lying vibrational levels, we
investigated the dependence of ΔυJ on J for high-lying vibrational level ([vD] – v = 17) of the
Cs2 0g− (6S1/2 + 6P1/2) long-range state, as shown in Fig. 2(b). The non-linear relationship
between Δυ and J is clearly exhibited. The main reason for the non-linearity is that the longrange interaction between the two constituent atoms of the long-range molecules is rather
weak comparing with the ordinary chemical bond. The red curve in Fig. 2(b) is the fit plotted
in accordance with a non-rigid rotor model [12]. The rotational frequency interval is
described as ΔυJ = 2B(J + 1)-4D(J + 1)3, where B is the rotational constant and D is the
centrifugal distortion constant. In order to test the appropriateness of this nonlinear model, the
fitting results (blue curve) using rigid rotor model are also given. The adjusted R2 for the nonrigid and rigid fittings are 0.99162 and 0.97249, respectively. Clearly, the non-rigid rotor
model is more precise than the rigid rotor model to obtain accurate rotational constants Bv = 17
= 25.73 ± 3.80 MHz and centrifugal distortion constants Dv = 17 = 0.07 ± 0.02 MHz.
Fig. 3. Six typical double PA spectra with the reference of frequency difference are provided
for typical vibrational levels of the Cs2 0g− (6S1/2 + 6P1/2) long-range state. The intensity of the
PA laser beams (I and II) is 43.5 W/cm−2 and keeps unchanged. The Ti:sapphire laser is slowly
scanned at a rate of 5.0 MHz/s. The colorful curves are the multipeak Lorentzian fitting results.
Figure 3 shows six typical double PA spectra for vibrational levels ([vD] – v = 15, 20, 22,
29, 39 and 41) of the Cs2 0g− (6S1/2 + 6P1/2) long-range state, whose detunings are ~2.24 cm−1,
~6.04 cm−1, ~7.88 cm−1, ~19.64 cm−1, ~41.76 cm−1, and ~47.45 cm−1 below the dissociation
limit 6S1/2 + 6P1/2 (~11178.151 cm−1), respectively. The resolved rotational structure (J = 0
~5) is clearly observed for all vibrational levels ([vD] – v = 15 ~43). In fact, some spectra are
rich in rotational structures (J = 0 ~7). Those large rotational levels can be rationalized due to
additional angular momentum contributions, such as from orbital angular momentum due to
p-wave or d-wave. All of the observed spectra demonstrate similar intensity undulation,
which is the result of the variation of the Franck-Condon factors for the photoassociation
#202889 - $15.00 USD
Received 11 Dec 2013; revised 20 Jan 2014; accepted 31 Jan 2014; published 10 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003754 | OPTICS EXPRESS 3758
transitions from initial scattering state of two free cold atoms to the final ro-vibrtional levels
of the long-range states.
For the higher-lying vibrational levels ([vD] – v < 15) of the Cs2 0g− long-range state
which are very closely located near the dissociation limit, the hyperfine structures and the
perturbation effects [15] play a devastating role that adversely hindered us from clearly
observing and distinguishing the ro-vibrational structure in the PA spectra. Thus, here we
focus on the vibrational levels ([vD] – v = 15 ~42), specially the high-lying vibrational levels
([vD] – v = 15 ~22) which have not fully investigated in the previous experimental reports
using traditional trap loss detection [9]. In current research, the double PA spectra for [vD] – v
= 15 ~22 of the Cs2 0g− (6S1/2 + 6P1/2) long-range state with explicitly resolved rotational
structure have been directly obtained based on the high-sensitive modulation spectral
technique, which facilitates reaching improved signal-to-noise ratio (SNR) as high as 21.5
(for [vD] – v = 29). By fitting the data of ΔυJ to the non-rigid rotor model, all the values of
rotational constants B and centrifugal distortion constants D for [vD] – v = 15 ~22 of Cs2 0g−
long-range state are listed in Table 1.
Table 1. Rotational Constants and Centrifugal Distortion Constants of Vibrational Levels
([vD] – v = 15 ~22)
[vD] –
v
B
(MHz)
D(MH
z)
15
21.84
± 3.97
0.06 ±
0.02
16
22.77
± 3.54
0.06 ±
0.03
17
25.73
± 3.80
0.07 ±
0.02
18
28.38
± 3.66
0.05 ±
0.04
19
32.55
± 3.05
0.07 ±
0.05
20
36.67
± 2.87
0.08 ±
0.05
21
39.06
± 2.45
0.11 ±
0.04
22
47.43
± 4.81
0.16 ±
0.03
Figure 4 depicts the rotational constants for the vibrational levels ([vD] – v = 15 ~43) of
the Cs2 0g− (6S1/2 + 6P1/2) long-range state as a function of the vibrational quantum number
[vD] – v. In particular, the rotational constants for the high-lying vibrational levels ([vD] – v =
15 ~22) are illustrated in the blue region as a “new extended region” comparing to the
previous experimental report [9]. Here, the uncertainty is mainly due to a possible statistical
error in the process of multi-peak fitting (~2 – 3 MHz) as well as the error in the
determination of the resonant peak position (~1 MHz). It is valuable to find that the rotational
constants decrease gradually as [vD] – v increasing in the case of high-lying vibrational levels
and reach almost zero at the range of the dissociation limit. This can be understood by
considering the molecular inter-nuclear separation as described in Ref. 16. In addition, the
rotational constants of the high-lying vibrational levels ([vD] – v = 15 ~22) are consistent with
those of the low-lying vibrational levels. The blue curve is the linear regression result. The
adjusted R2 of fitting is 0.982 and the regression coefficient is 2.67 MHz. For comparison we
also present in Fig. 4 the previous experimental values [9] of the rotational constants for the
different vibrational levels ([vD] – v = 23 ~42). The current experimental results demonstrate
a similar trend with the previous values.
#202889 - $15.00 USD
Received 11 Dec 2013; revised 20 Jan 2014; accepted 31 Jan 2014; published 10 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003754 | OPTICS EXPRESS 3759
Fig. 4. Dependence of rotational constants B (blue balls) on vibrational quantum number [vD] –
v of the Cs2 0g− (6S1/2 + 6P1/2) long-range state. The red circles represent the experimental
values of the rotational constants for different vibrational states listed in Ref. 9.
4. Conclusions
In conclusion, a double PA spectroscopy technique has been proposed that provides a precise
frequency reference to experimentally determine the rotational constants of the high-lying
vibrational levels of the molecular long-range state. A linear dependence of the rotational
constants on the vibrational quantum number is found. The proposed technique allows one to
reach all the data of the available experimental ro-vibrational levels below the dissociation
limit. Nevertheless, at the present stage of the experiment, the interpretation of the results
remains limited, due to not considering the perturbation effect that stems from the mixing
with the triplet sigma ungerade levels. Our scheme has several advantages. First, the
rotational constants of the high-lying vibrational levels of Cs2 long-range states have been
directly measured for the first time. Furthermore, a high quality cavity that necessitates
elaborate controls and additional expenditure is not required. Finally, the simple and robust
scheme can be adapted to investigate a number of other long-range molecular species.
Acknowledgments
The work was supported by the 973 Programs (No. 2012CB921603), the 863 Program (No.
2011AA010801), the PCSIRT (No. IRT13076), the International Science and Technology
Cooperation Program of China (No. 2011DFA12490), NSF of China (No. 61378014, No.
61308023 and No. 10934004), the NSFC Project for Excellent Research Team (No.
61121064), the SRFDPHE (No. 20131401120012) and the NSF for Young Scientists of
Shanxi Province (Grant No. 2013021005-1).
#202889 - $15.00 USD
Received 11 Dec 2013; revised 20 Jan 2014; accepted 31 Jan 2014; published 10 Feb 2014
(C) 2014 OSA
24 February 2014 | Vol. 22, No. 4 | DOI:10.1364/OE.22.003754 | OPTICS EXPRESS 3760