Synchronous Spin-Exchange Optical Pumping
We describe a new approach to precision NMR with hyperpolarized gases designed to mitigate NMR frequency shifts due to the alkali spin exchange field. The electronic spin polarization of optically pumped alkali atoms is square-wave modulated at the noble-gas NMR frequency and oriented transverse to the DC Fourier component of the NMR bias field. Noble gas NMR is driven by spin-exchange collisions with the oscillating electron spins. On resonance, the time-average torque from the oscillating spin-exchange field produced by the alkali spins is zero. Implementing the NMR bias field as a sequence of alkali 2-pulses enables synchronization of the alkali and noble gas spins despite a -fold discrepancy in gyromagnetic ratio. We demonstrate this method with Rb and Xe, and observe novel NMR broadening effects due to the transverse oscillating spin exchange field. When uncompensated, the spin-exchange field at high density broadens the NMR linewidth by an order of magnitude, with an even more dramatic suppression (up to 70x) of the phase shift between the precessing alkali and Xe polarizations. When we introduce a transverse compensation field, we are able to eliminate the spin-exchange broadening and restore the usual NMR phase sensitivity. The projected quantum-limited sensitivity is better than 1 nHz/.
The ability to produce highly magnetized noble gases via spin-exchange collisions with spin-polarized alkali atoms Walker and Happer (1997) has greatly impacted scientific studies of magnetic resonance imaging Walkup and Woods (2014), high-energy nuclear physics with spin-polarized targets Singh et al. (2015), and chemical physics Jiménez-Martínez et al. (2014). Applications in precision measurements began with NMR gyros Meyer and Larsen (2014) and have continued with fundamental symmetry tests using multiple cell free induction decay Vold et al. (1984), dual-species masers Glenday et al. (2008); Rosenberry and Chupp (2001), self-compensating co-magnetometers Smiciklas et al. (2011), NMR oscillators Yoshimi et al. (2012), and free spin-precession co-magnetometers Bulatowicz et al. (2013); Tullney et al. (2013); Allmendinger et al. (2014); Sheng et al. (2014).
Some of these approaches Meyer and Larsen (2014); Smiciklas et al. (2011); Yoshimi et al. (2012); Bulatowicz et al. (2013); Sheng et al. (2014) take advantage of enhanced NMR detection by the embedded alkali magnetometer. The alkali and noble-gas spin ensembles experience enhanced polarization sensitivity due to the Fermi-contact interaction during collisions between the two species. The effective Fermi-contact fields experienced by the two species are
where are the electron and nuclear spin operators, the atomic densities, the nuclear magnetic moment of the noble gas, the Bohr magneton, and the atomic g-factor . The frequency-shift enhancement factor Grover (1978); Schaefer et al. (1989) was recently measured to be Ma et al. (2011) for RbXe. Thus the detected NMR field is 500 larger for the embedded magnetometer than for any external sensor. This seemingly decisive advantage comes with the cost of similarly enhancing the field due to the spin-polarized alkali atoms, 190 G at cm. In typical longitudinally polarized NMR this field produces large frequency shifts of order 0.1 Hz. One approach for mitigating this effect is to compare two Xe isotopes Bulatowicz et al. (2013); Meyer and Larsen (2014), for which the enhancement factors are equal to about 0.1% Bulatowicz et al. (2013). Another recent strategy nulls the alkali field by saturating the alkali electron spin resonance during free precession intervals Sheng et al. (2014).
This Letter describes a new approach, synchronous spin-exchange optical pumping, in which both the alkali and noble gas spins are polarized transverse to the bias magnetic field. The essential components are shown in Fig. 1. This bias magnetic field is applied as a sequence of short alkali pulses, allowing the alkali atoms to be polarized perpendicular to the bias field Korver et al. (2013). By periodically reversing the alkali polarization direction at the noble gas resonance frequency, transverse nuclear polarization is resonantly generated by spin-exchange collisions. On resonance, the alkali polarization is in phase with the noble-gas precession, so there is no time-averaged torque from the alkali field and therefore no frequency shift of the NMR resonance. However, when the alkali polarization is modulated somewhat off-resonance, we find that the torque from the alkali field suppresses the phase shift of the NMR precession and generates a longitudinal component of the noble-gas polarization. This results in a novel broadening of the NMR resonance that is readily suppressed by adding an AC compensation field 180 out of phase with and along the same direction as the Rb polarization modulation. When fully compensated, the broadening effect becomes negligible and this approach thus maintains the SNR advantages of NMR detection by the embedded alkali magnetometer, while suppressing the Fermi contact field shifts. This is a promising approach for ultra sensitive NMR using hyper polarized Xe and He in the future.
Synchronous spin-exchange is accomplished by a collisional variant of the Bell-Bloom method of synchronous optical pumping Bell and Bloom (1961). The nuclei are polarized by spin-exchange collisions with alkali-metal atoms whose spin, oriented transverse to a bias magnetic field , oscillates at the nuclear Larmor frequency. The combined effects of Larmor precession, transverse spin-relaxation, and spin-exchange collisions are described by the Bloch equation for the transverse spin components :
The noble-gas gyromagnetic ratio is , the transverse relaxation rate of the nuclei is , and the spin-exchange collision rate is . A notable omission from Eq. 3 is the torque from the magnetic field produced by the alkali spins, which we temporarily assume has been eliminated as will be explained in detail below.
The alkali spin-polarization is modulated at a frequency near the -th submultiple of the time-averaged noble-gas resonance frequency . In the rotating-wave approximation we neglect all but the near-resonant co-rotating Fourier component , which produces a transverse noble-gas polarization
with the usual amplitude and phase response of a driven oscillator. On resonance, the transverse polarization in the rotating frame is . Off-resonance, the rotating frame magnetic field causes the polarization to tilt in the x-y plane by an angle . In comparison to the usual longitudinal spin-exchange Walker and Happer (1997), the maximum polarization attainable depends on a competition between and rather than the longitudinal relaxation rate . The polarization is also reduced by the Fourier amplitude for our roughly square wave polarization modulation. When combined with a feedback system to actively null the phase shift , we observe that the resonance frequency is an accurate measure of the Larmor frequency: . Thus synchronous spin-exchange is attractive for accurate absolute magnetometry and rotation sensing.
The synchronously oscillating transverse alkali polarization, an impossibility in a DC magnetic field due to the fold larger magnetic moment as compared to Xe, is enabled by replacing the usual DC bias field by , a sequence of short (4 s) pulses. Each pulse produces precession of the alkali atoms Korver et al. (2013), i.e. . Here the Rb gyromagnetic ratio is for the Rb isotope we use. The application of a sequence of 2 pulses causes no time-averaged precession of the Rb spins, thereby allowing the Rb to be optically pumped as if in zero field. Meanwhile, the polarized Xe nuclei precess only mrad per pulse (for Xe) and effectively see only the DC average .
The apparatus includes a nominally spherical 8 mm diameter glass cell that contains Rb vapor (density cm), 32 Torr of Xe gas, 4 Torr of Xe gas, and 300 Torr of N. The cell is held inside a ceramic oven that is heated by running 20 kHz AC current through commercial Kapton flex-circuit heater strips that are configured to minimize stray fields. Outside the oven are Helmholtz field coils for fine-shimming of the 3 magnetic field components, plus a set of 4 coils for generating the pulses. The oven and coils are inside a 3 layer magnetic shield with optical access ports for the lasers. The 2 coils are designed to maximize uniformity, maintain low inductance, and minimize stray fields that couple to the magnetic shields. They are driven by a custom MOSFET circuit designed to minimize pulse-to-pulse charge fluctuations.
Optical pumping is performed using two free-running 40mW 795 nm distributed feedback diode lasers that are combined on a non-polarizing beamsplitter. One of the lasers is tuned above the Rb D1 resonance, the other below. Since the AC Stark field changes sign at resonance, we can reduce the effective field from about 2 mG (for a single pumping frequency) to G by adjusting the relative detunings and intensities of the two lasers. The two output beams from the polarizing beamsplitter, each containing both pump frequencies, enter the cell and propagate along along the directions; doing this substantially improves the uniformity of the optical pumping of the optically thick alkali vapor. The last optical elements before the cell are wave plates and liquid crystal variable retarders that allow the circular light polarizations to be reversed in about 200 s. At low Rb density, the laser detunings are selected to produce about 70% Rb polarization while nulling the AC Stark shifts. At high densities the polarization drops to about 40% without significantly degrading the AC Stark cancellation.
An off-resonant 30 mW probe laser, propagating along , serves to observe the z-component of the alkali polarization using low-noise Faraday rotation detection. Because the alkali atoms are optically pumped along , the Faraday rotation detects the -component of the Xe field: . Note that the magnetometer sensitivity reverses sign as the alkali spin direction is reversed due to the synchronous pumping. In order to detect , we add a 1 mG field at Hz. Demodulation of at then gives an output equal to . Characterization of the magnetic response implies a relaxation rate /s with a magnetic response bandwidth of kHz and a dynamic range of mG. The noise floor of the magnetometer is typically 1 nG/, limited by photon-shot-noise.
To null stray DC magnetic fields, we temporarily turn off the 2 pulses while retaining the pump modulation, effectively running the magnetometer as a zero-field magnetometer with polarization modulation. After field nulling with the zero-field magnetometer, we turn on the 2 pulses. The area of the 2 pulses is set to better than % by nulling the magnetometer signal generated by . The 25 kHz repetition rate of the 2 pulse sequence and the modulation waveform for the optical pumping are generated by direct digital synthesis using an FPGA phase-locked to a commercial atomic Rb clock. With stray fields nulled and the 2 pulse area set, the synchronously pumped Xe signals are readily observed when the pumping modulation frequency is brought near either the Hz Xe resonance frequency or the Hz 3rd subharmonic of the Xe resonance frequency. A magnetometer waveform is shown in Fig. 2, and a sample spin-exchange resonance curve is shown in Fig. 3b. The resonance curve is taken by lock-in detection of the 2 magnetometer signal. The in-phase lock-in output is proportional to while the quadrature is proportional to . The signal amplitude, typically 180 G for both isotopes at cm when driven at , is consistent with expectations from independent measurements of , , and .
So far we have ignored possible effects from the alkali field; indeed the resonance in Fig. 3b was taken with a compensated alkali field as will be explained below. However, when uncompensated the alkali field can have a dramatic effect. Under the conditions of Fig. 3a, the in-phase response is broadened by a factor of 4 as compared to Fig. 3b. Even more dramatic is the observed suppression of the quadrature signal by 10x, while a dispersive z-polarization is acquired. At higher densities we have observed broadening of up to 10 and quadrature suppression of 75 for Xe.
The broadening from the spin-exchange field can be understood as follows. First, when the pumping is off-resonance, the phase shift between the nuclear precession and the alkali field produces a DC torque and so the Xe spin develops a z-polarization. The z-polarization, which can be a substantial fraction of the transverse polarization (as illustrated in Fig. 3a), then couples with the alkali field to produce a transverse torque that is 90 degrees out of phase with the pumping. This can be considered as a negative feedback mechanism that tries to null the relative phase between the Xe precession and the pumping. The net effect is to generate a large z-polarization and to suppress the phase shift between the pumping and the Xe precession. Under our conditions the alkali field, when uncompensated, is sufficiently strong to make at large detunings.
The Bloch equation for the z-polarization is
which leads to a steady-state
The effective magnetic field seen by the Xe nuclei is the sum of the alkali field and an anti-parallel square-wave modulated external compensation field: . The -th Fourier component of this field is . The size of the spin-exchange torque is characterized by the parameter .
The z-polarization of the Xe now allows the transverse fields to apply a torque, so the transverse components of the Bloch equations become
where we have again made the rotating wave approximation. Notice that the torque from the alkali field is 90 out of phase with the pumping torque, acting to suppress the phase shift between the pumping and the precession. The steady-state polarization in the rotating frame is
where the spin-exchange broadened linewidth is
and the detuning is . For a large spin-exchange field, i.e. , these equations account for the broadening, small response, and large response. They also predict that with the application of a compensation field 180 out of phase with the pumping, the linewidth should narrow to , the magnitude of should be restored, and should be suppressed, thus bringing the response into agreement with Eq. 4. All these features are illustrated in the data of Fig. 3, with fits to Eqs 9-11.
We have studied the spin-exchange broadening over a range of alkali field strengths for both Xe isotopes. Figure 4 shows the predicted narrowing of the resonance width with application of the compensation field. For Xe-129 there is an order of magnitude narrowing of the width over the field range, despite having reduced by a factor of 3 by pumping at the third subharmonic of the Larmor frequency. We have confirmed that at a given density the compensation fields for Xe and Xe are the same to within 1%. Figure 4 also shows an example of how the quadrature slope dramatically increases as the compensation field approaches the optimum. We have observed as much as a factor of 75 ratio between the compensated and uncompensated slopes.
At higher densities than reported in this paper, the broadening slows as a function of compensation field. We believe that this arises because of feedback from the alkali field: at sufficiently high densities, the tipping of the alkali field off of the -axis becomes important and needs to be taken into account. Such non-linearities are likely closely related to the studies of strongly coupled alkali and noble gas spin from Ref. Kornack and Romalis (2002).
In the future, two-species operation will allow full realization of the spectroscopy potential of synchronous spin-exchange for fundamental symmetry tests. With two-species, one species can be used to remove magnetic field noise, leaving the other sensitive to non-magnetic interactions. This strategy is used in various forms in many experiments Meyer and Larsen (2014); Glenday et al. (2008); Rosenberry and Chupp (2001); Bulatowicz et al. (2013); Tullney et al. (2013); Allmendinger et al. (2014); Sheng et al. (2014). The resulting effective frequency noise resulting from some non-magnetic interaction is of order
Our current apparatus is limited by probe photon shot noise at about the G/ level, but even this relatively modest magnetometer performance projects to nHz/. Supposing we can reach the G/ level of a good magnetic shield, the noise level would be 3 nHz/, competitive with the Ne-Rb-K co-magnetometer of Ref. Smiciklas et al. (2011). At the quantum projection noise limit of the magnetometer, the noise level becomes sub-nHz/. These numbers are very promising for experiments such as sensitive tests of Lorentz violation Smiciklas et al. (2011) and limits on short-range nuclear interactions Bulatowicz et al. (2013); Tullney et al. (2013).
Finally, and in contrast to the co-magnetometer approach of Ref. Smiciklas et al. (2011), synchronous spin-exchange promises not only sensitivity but accuracy. Configured as a dual-species oscillator Meyer and Larsen (2014) to actively null , for example, the nuclear phase becomes an accurate integral of the inertial rotation rate. The above noise estimates suggest an achievable sensitivity of better than 100 deg/ for a synchronously pumped NMR gyro with a unity scale factor and a bandwidth that approaches the oscillation frequency.
We are grateful for many discussions with M. Larsen, R. Wyllie, B. Lancor, M. Ebert, and I. Sulai. This work was supported by NSF GOALI PHY1306880 and Northrop Grumman Corp.
- Walker and Happer (1997) T. G. Walker and W. Happer, Rev. Mod. Phys. 69, 629 (1997).
- Walkup and Woods (2014) L. L. Walkup and J. C. Woods, NMR Biomed. 27, 1429 (2014).
- Singh et al. (2015) J. T. Singh, P. A. M. Dolph, W. A. Tobias, T. D. Averett, A. Kelleher, K. E. Mooney, V. V. Nelyubin, Y. Wang, Y. Zheng, and G. D. Cates, Phys. Rev. C 91, 055205 (2015).
- Jiménez-Martínez et al. (2014) R. Jiménez-Martínez, D. J. Kennedy, M. Rosenbluh, E. A. Donley, S. Knappe, S. J. Seltzer, H. L. Ring, V. S. Bajaj, and J. Kitching, Nat. Commun. 5 (2014).
- Meyer and Larsen (2014) D. Meyer and M. Larsen, Gyroscopy and Navigation 5, 75 (2014).
- Vold et al. (1984) T. G. Vold, F. J. Raab, B. Heckel, and E. N. Fortson, Phys. Rev. Lett. 52, 2229 (1984).
- Glenday et al. (2008) A. G. Glenday, C. E. Cramer, D. F. Phillips, and R. L. Walsworth, Phys. Rev. Lett. 101, 261801 (2008).
- Rosenberry and Chupp (2001) M. A. Rosenberry and T. E. Chupp, Phys. Rev. Lett. 86, 22 (2001).
- Smiciklas et al. (2011) M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, Phys. Rev. Lett. 107, 171604 (2011).
- Yoshimi et al. (2012) A. Yoshimi, T. Inoue, T. Furukawa, T. Nanao, K. Suzuki, M. Chikamori, M. Tsuchiya, H. Hayashi, M. Uchida, N. Hatakeyama, S. Kagami, Y. Ichikawa, H. Miyatake, and K. Asahi, Phys. Lett. A 376, 1924 (2012).
- Bulatowicz et al. (2013) M. Bulatowicz, R. Griffith, M. Larsen, J. Mirijanian, C. B. Fu, E. Smith, W. M. Snow, H. Yan, and T. G. Walker, Phys. Rev, Lett. 111, 102001 (2013).
- Tullney et al. (2013) K. Tullney, F. Allmendinger, M. Burghoff, W. Heil, S. Karpuk, W. Kilian, S. Knappe-Grueneberg, W. Mueller, U. Schmidt, A. Schnabel, F. Seifert, Y. Sobolev, and L. Trahms, Phys Rev. Lett. 111, 100801 (2013).
- Allmendinger et al. (2014) F. Allmendinger, W. Heil, S. Karpuk, W. Kilian, A. Scharth, U. Schmidt, A. Schnabel, Y. Sobolev, and K. Tullney, Phys Rev. Lett. 112, 110801 (2014).
- Sheng et al. (2014) D. Sheng, A. Kabcenell, and M. V. Romalis, Phys Rev. Lett. 113, 163002 (2014).
- Grover (1978) B. C. Grover, Phys. Rev. Lett. 40, 391 (1978).
- Schaefer et al. (1989) S. R. Schaefer, G. D. Cates, T.-R. Chien, D. Gonatas, W. Happer, and T. G. Walker, Phys. Rev. A 39, 5613 (1989).
- Ma et al. (2011) Z. L. Ma, E. G. Sorte, and B. Saam, Phys. Rev. Lett. 106, 193005 (2011).
- Korver et al. (2013) A. Korver, R. Wyllie, B. Lancor, and T. G. Walker, Phys. Rev. Lett. 111, 043002 (2013).
- Bell and Bloom (1961) W. E. Bell and A. L. Bloom, Phys. Rev. Lett. 6, 280 (1961).
- Kornack and Romalis (2002) T. W. Kornack and M. V. Romalis, Phys. Rev. Lett. 89, 253002 (2002).