Patent Publication Number: US-11639976-B2

Title: Time-multiplexed dual atomic magnetometry

Description:
RELATED APPLICATIONS 
     This application is a continuation of U.S. patent application Ser. No. 17/386,081, filed Jul. 27, 2021, which claims priority to U.S. Provisional Patent Application No. 63/057,815, filed Jul. 28, 2020 and titled “Time-Multiplexed Dual Magnetometry”. Each of these applications is incorporated herein by reference in its entirety. 
    
    
     BACKGROUND 
     In atom magnetometry, spin-polarized atoms in an external magnetic field precess at the Larmor frequency. This precession is typically measured via Faraday rotation, i.e., the spin-polarized atoms rotate the linear polarization of a weak laser beam that passes through the atoms. This rotation of the laser polarization can be detected with a polarimeter, and the resulting electronic signal can be processed into a value of the external magnetic field. Since the gyromagnetic ratio of the atoms (i.e., the Larmor frequency per unit magnetic-field strength) is determined primarily by the energy-level structure of the atoms, atomic magnetometry benefits from high accuracy, as compared to other forms of magnetometry (e.g., SQUID, fluxgate, Hall effect, magnetoresistance, etc.). 
     In addition, recent advances in the development of millimeter-size vapor cells has enabled the placement of atom-vapor-based magnetometer heads close to the sample to be measured (e.g., within one centimeter of the sample). This improves sensitivity since the magnetic field strength generated by a sample drops at least as 1/r, where r is the distance between the sample and the magnetometer head. In some cases (e.g., a magnetic dipole), the magnetic field drops as rapidly as 1/r 3 , further emphasizing the need to miniaturize sensor heads so that they can be placed closer to the sample. 
     SUMMARY 
     An atomic magnetometer is typically operated in an alternating sequence of pumping and probing stages. In each pumping stage, a pump laser beam is directed through a vapor cell to spin-polarize gaseous atoms therein. An external magnetic field may be applied to establish a quantization axis. The polarization (e.g., circular or linear), propagation direction, and modulation (e.g., AM or FM) of the pump laser beam is selected such that the optical pumping results in a ground-state coherence (i.e., a coherence between two or more magnetic ground-state sublevels of the atoms). During the probing stage, a linear polarized probe laser beam passes through the vapor cell. The spin-polarized atoms rotate the polarization of the probe laser beam synchronously with the Larmor precession. The rotated polarization is then measured with a polarimeter. 
     The duration of each probing stage is limited by a dephasing time of the atoms (i.e., the transverse spin relaxation time T 2 ). Specifically, collisions between atoms disrupt the ground-state coherences, washing out the Larmor precession, and hence the measured signal. Although techniques exist for increasing the dephasing time (e.g., spin-relaxation coatings and buffer gasses), dephasing times are still typically on the order of milliseconds. Furthermore, as the size of the vapor cell decreases, the rate of collisions between the atoms and the walls increases, leading to shorter dephasing times (even if the walls have a spin-relaxation coating). 
     The duration of each pumping stage is typically between a few hundred microseconds and a few milliseconds, depending on the available power of the pump laser beam and the transition strengths of the atomic species used. Thus, for each cycle of one pumping stage followed by one probing stage, the atoms may be measured for as little as 50% of the time. That is, half of the time is wasted preparing the atoms, which limits the signal-to-noise ratio (SNR). Gaps in the measured time record of the atoms can also introduce aliasing and other deleterious signal-processing artifacts that mask the true magnetic signal to be measured. 
     To solve these problems, the present embodiments feature systems and methods for time-multiplexed atomic magnetometry performed with two vapor cells located adjacent to (e.g., on opposite sides of) the sample to be measured. The first vapor cell is operated according to a first sequence of alternating pumping and probing stages. Similarly, the second vapor cell is operated according to a second sequence of alternating pumping and probing stages. However, the second sequence is delayed relative to the first sequence such that the second vapor cell is pumped while the first vapor cell is probed, and the first vapor cell is pumped while the second vapor cell is probed. With this time-multiplexed operation, the magnetic field generated by the sample can be measured without any time gaps. More specifically, and as described in more detail below, the signals from the two vapor cells can be interleaved to form a single gapless time sequence that represents the time-varying magnetic field generated by the sample over the entire time sequence. 
     By using two vapor cells, the present embodiments advantageously have twice the signal-to-noise ratio of conventional atomic magnetometers that use only one vapor cell (assuming equal vapor pressures, vapor cell sizes, atomic species, etc.). However, the present embodiments also offer advantages over simply doubling the size or pressure of one vapor cell. For example, doubling the vapor cell size results in the extra atoms being located farther from the sample, where they are less sensitive to the magnetic field. As a result, doubling the vapor cell size does not necessarily double the SNR. On the other hand, in the present embodiments the two vapor cells may be located on opposite sides of the sample, in which case both vapor cells are located proximate to the sample, ensuring equal sensitivity to the magnetic field. Increasing the vapor pressure inside the cell can help, although the resulting pressure broadening can reduce T 2 . More than two vapor cells can be used to achieve even greater increases in signal-to-noise ratio. 
     In embodiments, a time-multiplexed dual atomic magnetometer operates as a pair of free-induction-decay atomic magnetometers. In these embodiments, the signal from each of the two vapor cells is continuously recorded over several oscillations. For a single probing stage, the resulting signal is approximately equal to an exponentially decaying sinusoid, which can be fitted to extract a center frequency which equals the average Larmor frequency over the probing phase. The Larmor frequency may then be converted into a corresponding value of the magnetic field. Repeating this process over several consecutive cycles produces a time sequence of magnetic field values. The bandwidth of this approach is limited by the duration of one cycle. However, the sequence can be used to identify low-frequency components spanning over several cycles. 
     To increase the bandwidth, instantaneous-phase retrieval may be implemented on the recorded signals. This technique was recently demonstrated for atomic magnetometers in “Wide-bandwidth atomic magnetometry via instantaneous-phase retrieval” by N. Wilson et al. (arXiv:2003.04526v1), although it has been used in geosciences for several decades. For example, see “The calculation of instantaneous frequency and instantaneous bandwidth” by A. E. Barnes (Geophysics 57, 1520-1539, 1992). In addition to these references, details about instantaneous phase and frequency can be found in “Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals” by B. Boashash (Proc. IEEE 80, 520-538, 1992). With instantaneous-phase retrieval, the present embodiments are expected to operate at bandwidths exceeding 10 kHz. 
     The present embodiments may be used to enhance magnetometry in a host of applications, including geosciences, magnetic communication, threat detection, the measurement of bio-magnetic signals (e.g., magnetoencephalography), nuclear magnetic resonance (NMR), and magnetic resonance imaging (MRI). In another application, the present embodiments are used to measure the time-varying magnetic field generated by molecules in an aqueous solution. It is hypothesized that the motion of these molecules (e.g., stretching, rotating, translating, etc.) causes electric charges therein to accelerate, which produces magnetic fields at the femtotesla level. When the time-varying magnetic field is later “played” to cells (e.g., by applying electric currents to coils to replicate the time-varying magnetic field), the cells may behave as if the original molecules were present. In this regard, the time-varying magnetic field may be used to replicate the pharmacological effects of a compound on the cells, but without physically exposing the cells to the actual compound. As such, the present embodiments may be used to identify new therapies for treating cancer. Examples of molecules whose time-varying magnetic fields can be measured and subsequently used for such therapeutic purposes include small interfering RNA (siRNA) and messenger RNA (mRNA) from genes. 
     In embodiments, a time-multiplexed dual atomic magnetometer includes first and second vapor cells positioned such that an external magnetic field induces Larmor precession of atoms within the first and second vapor cells. The dual atomic magnetometer also includes a first polarimeter that measures a first polarization of a first probe beam after the first probe beam propagates through the first vapor cell. The first polarimeter outputs a first polarization signal indicative of the first polarization. The dual atomic magnetometer also includes a second polarimeter that measures a second polarization of a second probe beam after the second probe beam propagates through the second vapor cell. The second polarimeter outputs a second polarization signal indicative of the second polarization. The time-multiplexed dual atomic magnetometer also includes a signal processor that processes alternating data blocks of the first and second polarization signals to generate a single gapless temporal sequence that represents the magnetic field generated by the sample. 
     In other embodiments, a method for time-multiplexed dual atomic magnetometry includes inducing, with an external magnetic field, Larmor precession of atoms that are located within first and second vapor cells. The method also includes measuring, with a first polarimeter, a first polarization of a first probe beam after the first probe beam propagates through the first vapor cell. The first polarimeter outputs a first polarization signal indicative of the first polarization. The method also includes measuring, with a second polarimeter, a second polarization of a second probe beam after the second probe beam propagates through the second vapor cell. The second polarimeter outputs a second polarization signal indicative of the second polarization. The method also includes processing alternating data blocks of the first and second polarization signals to generate a single gapless temporal sequence that represents the magnetic field generated by the sample. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is a top sectional view of a dual atomic magnetometer that measures magnetic fields generate by a sample, in an embodiment. 
         FIG.  2    is a side section view of the dual atomic magnetometer of  FIG.  1   . 
         FIG.  3    shows first and sequence timing sequences that illustrate time-multiplexed operation of the dual atomic magnetometer of  FIGS.  1  and  2   , in an embodiment. 
         FIG.  4    shows how instantaneous-phase retrieval is implemented with each data block of  FIG.  3    to obtain a corresponding instantaneous-phase block, in an embodiment. 
         FIG.  5    shows how several consecutive frequency blocks may be concatenated into the single gapless temporal sequence, in an embodiment. 
         FIG.  6    is a side section view showing the dual atomic magnetometer of  FIGS.  1  and  2    being used with a bias field oriented perpendicularly to the propagation direction of probe beams, in an embodiment. 
         FIG.  7    is a side sectional view of components of the dual atomic magnetometer of  FIGS.  1  and  2    in which a charge moves along the +z direction within a bias field oriented in the +y direction. 
         FIG.  8    illustrates how an alternating sequence of the data blocks can be processed to generate a single gapless temporal sequence for the scenario depicted in  FIG.  7   , in an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
       FIGS.  1  and  2    are top and side sectional views of a dual atomic magnetometer  100  that measures magnetic fields generated by a sample  110 . The dual atomic magnetometer  100  includes first and second vapor cells  104 ( 1 ),  104 ( 2 ) filled with first and second atomic vapors  106 ( 1 ),  106 ( 2 ), respectively. The sample  110  is placed inside a sample cell  102  that is located between the first and second vapor cells  104 ( 1 ),  104 ( 2 ) in the x direction (see the right-handed coordinate system  120 ). A first pump beam (e.g., see first pump beam  640 ( 1 ) in  FIG.  6   ) spin-polarizes the first atomic vapor  106 ( 1 ) by optically pumping the atoms of the first atomic vapor  106 ( 1 ) into one or more ground-state magnetic sublevels such that the atoms precess at a Larmor frequency. The spin-precessing atoms, in turn, rotate the polarization of a linearly polarized first probe beam  130 ( 1 ) that passes through the first vapor cell  104 ( 1 ). A first polarimeter  140 ( 1 ) measures the polarization of the first probe beam  130 ( 1 ) after exiting the first vapor cell  104 ( 1 ), outputting a first polarization signal  142 ( 1 ). 
     Similarly, a second pump beam (e.g., see second pump beam  640 ( 2 ) in  FIG.  6   ) spin-polarizes the second atomic vapor  106 ( 2 ) by optically pumping the atoms of the second atomic vapor  106 ( 2 ) into one or more ground-state magnetic sublevels such that the atoms also precess at a Larmor frequency. These spin-polarized atoms rotate the polarization of a linearly polarized second probe beam  130 ( 2 ) that passes through the second vapor cell  104 ( 2 ). A second polarimeter  140 ( 2 ) measures the polarization of the second probe beam  130 ( 2 ) after exiting the second vapor cell  104 ( 2 ), outputting a second polarization signal  142 ( 2 ). 
     The polarization of the first probe beam  130 ( 1 ) oscillates at an instantaneous Larmor frequency f L (t), assuming that magnetic field gradients are negligible (i.e., the atoms in the first atomic vapor  106 ( 1 ) interacting with the first probe beam  130 ( 1 ) are subjected to the same magnetic field). The instantaneous Larmor frequency f L (t) depends on the scalar magnitude of the magnetic field, which has two components: a time-varying signal field {right arrow over (B)} (s) (t)=(B x   (s) (t), B y   (s) (t), B z   (s) (t)) arising from the sample  110 , and a constant (i.e., time-independent) bias field {right arrow over (B)} (0) =(B x   (0) , B y   (0) , B z   (0) ). Thus, the instantaneous Larmor frequency f L (t) can be represented mathematically as f L (t)=γ|{right arrow over (B)} (s) (t)+{right arrow over (B)} (0) |(2π), where γ is the gyromagnetic ratio of the species of the first atomic vapor  106 ( 1 ). Assuming |{right arrow over (B)} (s) (t)&lt;&lt;|{right arrow over (B)} (0) |, the instantaneous Larmor frequency f L (t) is approximated by f L (t)≈γ|{right arrow over (B)} (0) |/(2π), which is time-independent. Therefore, the bias field) {right arrow over (B)} (0)  sets a nominal Larmor frequency f L   (0) ≈γ|{right arrow over (B)} (0) |/(2π) that is subsequently modified by the signal field {right arrow over (B)} (s) (t). In  FIGS.  1  and  2   , the bias field {right arrow over (B)} (0)  is oriented along the +z direction (i.e., {right arrow over (B)} (0) =(0, 0, B z   (0) ), for which f L   (0) =γB z   (0) /(2π). However, the bias field {right arrow over (B)} (0)  may point in other directions, as discussed in more detail below. The same argument holds for the second probe beam  130 ( 2 ) and the second atomic vapor  106 ( 2 ). It is assumed herein that the bias field {right arrow over (B)} (0)  is the same at both of the vapor cells  104 ( 1 ) and  104 ( 2 ) and the sample cell  102 . 
       FIG.  1    also shows a signal processor  144  that implements time-multiplexed operation of the dual atomic magnetometer  100 . The signal processor  144  is a circuit that acquires and processes the polarization signals  142 ( 1 ) and  142 ( 2 ) into a magnetic-field sequence {B j }. Although not shown in  FIG.  1   , the signal processor  144  may include a computing device with a processor and a memory storing machine-readable instructions that, when executed by the processor, control the signal processor  144  to implement the functionality described herein. Alternatively, the signal processor  144  may be a chip or circuit (e.g., a field-programmable gate array) that has been previously programmed to implement the functionality described herein. When the polarization signals  142 ( 1 ) and  142 ( 2 ) are analog electronic signals, the signal processor  144  may include analog-to-digital converters that convert the polarization signals  142 ( 1 ) and  142 ( 2 ) into digital electronic signals that are subsequently processed. Alternatively, each of the polarimeters  140 ( 1 ) and  140 ( 2 ) may include an analog-to-digital converter, wherein the polarization signals  142 ( 1 ) and  142 ( 2 ) are received by the signal processor  144  as digital electronic signals. A reference oscillator  148  establishes common timing for data acquisition, time tagging, and laser-timing control. The signal processor  144  may output the magnetic-field sequence {B j } to data storage (e.g., a memory card or hard drive), a computer monitor or screen for display to a user, or another computer system (e.g., via Ethernet or Wi-Fi) for additional signal processing and storage. 
     In some embodiments, the signal processor  144  also serves as a controller that outputs one or more timing signals  146  that control when the first and second pump beams and the first and second probe beams  130 ( 1 ),  130 ( 2 ) pass through the vapor cells  104 ( 1 ) and  104 ( 2 ). For example, the timing signals  146  may be used to gate (i.e., turn on and off) each of the pump beams and probe beams  130 ( 1 ),  130 ( 2 ) by driving a corresponding acousto-optic modulator, electro-optic modulator, or mechanical shutter. The timing signals  146  may also be used to change the frequency of one or more of the pump beams and the probe beams  130 ( 1 ),  130 ( 2 ). In other embodiments, a controller separate from the signal processor  144  implements timing control of the first and second pump beams and the first and second probe beams  130 ( 1 ),  130 ( 2 ). 
       FIG.  3    shows first and second timing sequences  300 ( 1 ),  300 ( 2 ) that illustrate time-multiplexed operation of the dual atomic magnetometer  100  of  FIGS.  1  and  2   . The first timing sequence  300 ( 1 ) corresponds to operation of the first vapor cell  104 ( 1 ), the first probe beam  130 ( 1 ), the first pump beam, and the first polarimeter  140 ( 1 ). Similarly, the second timing sequence  300 ( 2 ) corresponds to operation of the second vapor cell  104 ( 2 ), the second probe beam  130 ( 2 ), the second pump beam, and the second polarimeter  140 ( 2 ). 
     The first timing sequence  300 ( 1 ) is formed from a first repeating frame  302 ( 1 ) that has: (i) a first pumping stage  304 ( 1 ) with a first pumping duration T p   (1) , (ii) a first measurement stage  306 ( 1 ) with a first measurement duration T m   (1) , and (iii) a first dead stage  308 ( 1 ) with a first dead-time duration T d   (1) . During the first pumping stage  304 ( 1 ), the first probe beam  130 ( 1 ) is blocked while the first pump beam spin-polarizes the first atomic vapor  106 ( 1 ). During the first measurement stage  306 ( 1 ), the first pump beam is blocked while the first probe beam  130 ( 1 ) propagates through the first atomic vapor  106 ( 1 ). The first polarimeter  140 ( 1 ) measures the polarization of the first probe beam  130 ( 1 ) to obtain a first data block  340 ( 1 ) of the first polarization signal  142 ( 1 ). During the first dead stage  308 ( 1 ), no first polarization signal  142 ( 1 ) is obtained (e.g., both the first probe beam  130 ( 1 ) and the first pump beam are blocked, or the output of the first polarimeter  140 ( 1 ) is ignored). The first timing sequence  300 ( 1 ) is therefore periodic with a first period T 1 =T p   (1) +T m   (1) +T d   (1) , and has a measurement duty cycle η 1 =T m   (1) /T 1 . 
     The second timing sequence  300 ( 2 ) is similar to the first timing sequence  300 ( 1 ) except that it is delayed with respect to the first timing sequence  300 ( 1 ) by a second dead-time duration T d   (2)  of a second dead stage  308 ( 2 ). Specifically, the second timing sequence  300 ( 2 ) is formed from a second repeating frame  302 ( 2 ) that has: (i) a second pumping stage  304 ( 2 ) with a second pumping duration T p   (2) , (ii) a second measurement stage  306 ( 2 ) with a second measurement duration T m   (2) , and (iii) the second dead stage  308 ( 2 ). During the second measurement stage  306 ( 2 ), the first polarimeter  140 ( 2 ) measures the polarization of the second probe beam  130 ( 2 ) to obtain a second data block  340 ( 2 ) of the second polarization signal  142 ( 2 ). The second timing sequence  300 ( 2 ) therefore is periodic with a second period T 2 =T p   (2) +T m   (2) +T d   (2) , and has a measurement duty cycle η 2 =T m   (2) /T 2 . 
     The duration T d   (2)  is selected such that the second pumping stage  304 ( 2 ) ends when the first measurement stage  306 ( 1 ) ends. This allows the second measurement stage  306 ( 2 ) to begin immediately when the first measurement stage  306 ( 1 ) ends, eliminating any gap between the data blocks  340 ( 1 ) and  340 ( 2 ). Similarly, the duration T d   (1)  is selected such that the first pumping stage  304 ( 1 ) ends when the second measurement stage  306 ( 2 ) ends. This allows the first measurement stage  306 ( 1 ) to resume immediately when the second measurement stage  306 ( 2 ) ends, eliminating any gap between the second data block  340 ( 2 ) and a subsequent third data block  340 ( 3 ). 
     The polarization signal  142  within each data block  340  approximates an exponentially-decaying sine wave at the instantaneous Larmor frequency. The time constant of the exponential decay is determined by transverse spin relaxation of the atoms in the vapors  106 . The vapor cells  104  may be filled with a buffer gas (e.g., N 2  or  4 He) and/or lined with an anti-relaxation coating (e.g., paraffin) to reduce spin relaxation and increase the time constant. Dephasing times T 2  are typically between a fraction of a millisecond and several tens of milliseconds, depending on the geometry and size of the vapor cells  104 , the pressures of the vapors  106  and buffer gas (when included), the choice of atomic species for the vapors  106  (e.g., Rb, Cs, K, Na, etc.), the choice of species for the buffer gas (when included), the type of anti-relaxation coating (when included), etc. The dephasing time T 2  is the primary determinant of the measurement durations T m   (1)  and T m   (2) , as the signal-to-noise ratio decays with T 2 . 
     If general, the first and second measurement durations T m   (1) , T m   (2)  do not need to be equal. Similarly, the first and second pumping durations T p   (1) , T p   (2)  do not need to be equal. In some embodiments, the first and second measurement durations T m   (1) , T m   (2)  are equal, as shown in  FIG.  3   . In these embodiments, T d   (1) , T d   (2) , T m   (1) , and T m   (2)  are all similar. In some embodiments, the first and second pumping durations T p   (1) , T p   (2)  are similar. 
       FIGS.  4  and  5    illustrate how an alternating sequence of the data blocks  340  can be processed to generate a single gapless temporal sequence  502  that represents the signal field {right arrow over (B)} (s) (t). In  FIG.  4   , instantaneous phase retrieval is implemented with each data block  340 ( i ) to obtain a corresponding instantaneous-phase block  440 ( i ). Details about instantaneous-phase retrieval can be found in “Wide-bandwidth atomic magnetometry via instantaneous-phase retrieval” by N. Wilson et al. (arXiv:2003.04526v1), which is incorporated herein by reference in its entirety. Specifically, the instantaneous phase ϕ I   (i)  (t) is obtained mathematically as the argument of an analytic phase ϕ a   (i) (t): 
                         ϕ   I     (   i   )       ⁡     (   t   )       =       arg   ⁡     (       ϕ   a     (   i   )       ⁡     (   t   )       )       =     arg   ⁡     (         φ     (   i   )       ⁡     (   t   )       +     i   ⁢           ⁢   ℋ   ⁢     {       φ     (   i   )       ⁡     (   t   )       }         )           ,           (   1   )               
where φ (i) (t) is the measured polarization angle of the data block  340 ( i ), and  { } indicates a Hilbert transform. In  FIGS.  4  and  5   , each phase block  440 ( i ) shows ϕ I   (i) (t) after unwrapping. The derivative of the instantaneous phase ϕ I   (i) (t), after unwrapping, gives the instantaneous Larmor frequency for the data block  340 ( i ):
 
     
       
         
           
             
               
                 
                   
                     
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     The magnetic field sensed by the atoms in the vapor  106  during the data block  340 ( i ) is directly proportional to the instantaneous Larmor frequency f L   (i) (t), as described previously. 
     In some embodiments, and as shown in  FIG.  4   , the unwound instantaneous phase ϕ I   (i) (t) of each instantaneous-phase block  440 ( i ) is fit to a straight line (e.g., via linear regression) to obtain a corresponding slope m i  that represents the average value of dϕ I   (i) (t)/dt (i.e., the average Larmor frequency) over the measurement duration T m   (i)  of the corresponding data block  340 ( i ). Dividing m i  by the gyromagnetic ratio γ gives a single corresponding magnetic-field value B i . A sequence of several consecutive magnetic-field values {B i } can then be used to identify changes in the magnetic field between data blocks  340 ( i ). For example, the Fourier transform of the sequence {B i } can be calculated to identify components of {right arrow over (B)} (s) (t) with frequencies less than the measurement duration T m . 
     In some embodiments, and as shown in  FIG.  5   , the time-derivative of each phase block  440 ( i ) is calculated to obtain a corresponding frequency block  540 ( i ) that numerically represents the instantaneous Larmor frequency f L   (i)  of the phase block  440 ( i ). For example, when each phase block  440 ( i ) is represented as a temporal phase sequence of N instantaneous-phase values Φ (i) ={ϕ 1 , ϕ 2 , . . . , ϕ N } equally spaced in time by a point spacing Δt, then the corresponding frequency block  540 ( i ) can be represented as a temporal frequency sequence of N−2 values F (i) ={f j =(ϕ j+1 −ϕ j−1 /(2Δt)} for j=2 to N−1. Other methods of numerical differentiation may be used to calculate the temporal frequency sequence from the temporal phase sequence (e.g., the method of finite difference coefficients) without departing from the scope hereof. Such methods may also be used to obtain frequencies points corresponding to ϕ 1  and ϕ N  such that the frequency sequence F (i)  and the phase sequence Φ (i)  have the same number of points, and the points are aligned in time. 
     As shown in  FIG.  5   , several consecutive frequency blocks  540 ( i ) may be concatenated into the single gapless temporal sequence  502 . Equivalently, each frequency block  540  may be sequentially appended to the temporal sequence  502  to extend the temporal sequence  502  in time. Each point of the temporal sequence  502  may then be divided by the gyromagnetic ratio γ to obtain a magnetic-field sequence {B j } that approximates the time-varying total magnetic-field strength |{right arrow over (B)} (s) (t)+{right arrow over (B)} (0) |. The magnetic-field sequence {B j } may be subsequently analyzed (e.g., Fourier transform) to identify features associated with the sample  110 . 
     In other embodiments, the instantaneous-phase blocks  440 ( i ) are concatenated together to form the single gapless temporal sequence  502 . The time derivative of the temporal sequence  502  may then be calculated, after which each point is divided by the gyromagnetic ratio γ to obtain the magnetic-field sequence {B j }. In these embodiments, concatenating before the time derivative may improve estimates of the instantaneous frequency at the boundaries of the phase blocks  440 ( i ). 
     In the example of  FIGS.  1  and  2   , the dual atomic magnetometer  100  may also include a solenoid  114  to generate the bias field {right arrow over (B)} (0) =(0, 0, B z   (0) ) along the propagation direction of the probe beams  130 ( 1 ),  130 ( 2 ). The vapor cells  104 ( 1 ),  104 ( 2 ) and sample  110  are located along an axis of the solenoid  114  where the homogeneity of the bias field {right arrow over (B)} (0)  is greatest. The bias field {right arrow over (B)} (0) =(0, 0, B z   (0) ) may be alternatively generated with one or other magnetic coils, such as a pair of Helmholtz coils. Furthermore, one or more layers of magnetic shielding  112  may surround the solenoid  114  (or other magnetic coils), the sample cell  102 , and the vapor cells  104  to block external magnetic fields. 
     The instantaneous Larmor precession frequency F I (t) is given mathematically by 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                   ) 
                 
               
             
           
         
       
     
     For B x   (0) =B y   (0) =0, Eqn. 3 simplifies to 
                         f   L     ⁡     (   t   )       =         γ     2   ⁢   π       ⁢     B   z     (   0   )       ⁢       1   +                  B   →       (   s   )       ⁡     (   t   )            2         (     B   z     (   0   )       )     2       -     2   ⁢         B   z     (   s   )       ⁡     (   t   )         B   z     (   0   )                 ≈       γ     2   ⁢   π       ⁢       B   z     (   0   )       ⁡     (     1   -         B   z     (   s   )       ⁡     (   t   )         B   z     (   0   )         +   ⋯     )             ,           (   4   )               
where the Taylor expansion in Eqn. 4 assumes |{right arrow over (B)} (s) (t)|&lt;&lt;B z   (0) . Eqn. 4 shows that f L (t) approximately equals the nominal Larmor frequency f L   (0) ≈γB z   (0) /(2π), but is modulated primarily (i.e., to first order) by the z-component B z   (s) (t) of the signal field {right arrow over (B)} (s) (t). Equivalently, B x   (s) (t) and B y   (s) (t) only modulate the instantaneous Larmor frequency f L (t) to second order in the Taylor expansion, and are therefore suppressed relative to B z   (s) (t). Accordingly, the setup shown in  FIGS.  1  and  2    preferentially measures the z component B z   (s) (t) of the signal field {right arrow over (B)} (s) (t).
 
       FIG.  6    is a side section view of the dual atomic magnetometer  100  of  FIGS.  1  and  2    being used with a bias field {right arrow over (B)} (0)  oriented perpendicularly to the propagation direction of the probe beams  130 ( 1 ) and  130 ( 2 ). Specifically, the bias field B z   (0)  points in the +y direction, i.e., {right arrow over (B)} (0) =(0, B y   (0) , 0). The instantaneous Larmor precession frequency f I (t) is given mathematically by 
                       f   L     ⁡     (   t   )       =         γ     2   ⁢   π       ⁢     B   y     (   0   )       ⁢       1   +                  B   →       (   s   )       ⁡     (   t   )            2         (     B   y     (   0   )       )     2       -     2   ⁢         B   y     (   s   )       ⁡     (   t   )         B   y     (   0   )                 ≈       γ     2   ⁢   π       ⁢         B   y     (   0   )       ⁡     (     1   -         B   y     (   s   )       ⁡     (   t   )         B   y     (   0   )         +   ⋯     )       .                 (   5   )               
Now, B x   (s)  (t) and B z   (s) (t) only modulate the instantaneous Larmor frequency f L (t) to second order in the Taylor expansion, and are therefore suppressed relative to B y   (s) (t). Accordingly, the setup shown in  FIG.  6    preferentially measures the y component B y   (s) (t) of the signal field {right arrow over (B)} (s) (t). Similar calculations show that when the bias field {right arrow over (B)} (0)  points in the x direction, B y   (s) (t) and B z   (s) (t) only modulate the instantaneous Larmor frequency f L (t) to second order, and are therefore suppressed. Thus, the dual atomic magnetometer  100  can be used to preferentially measure a component of the signal field {right arrow over (B)} (s) (t) by aligning the bias field {right arrow over (B)} (0)  along the direction of the component.
 
       FIG.  7    is a side sectional view of components of the dual atomic magnetometer  100  of  FIGS.  1  and  2    in which a charge  702  moves along the +z direction within a bias field {right arrow over (B)} (0)  oriented in the +y direction. For clarity in  FIG.  7   , the sample cell  102  and vapor cells  104 ( 1 ) and  104 ( 2 ) are not shown. Due to its motion, the charge  702  generates a magnetic field  730  that circles in the x-y plane. At the first probe beam  130 ( 1 ), the magnetic field  730  adds to the bias field {right arrow over (B)} (0) , causing the atoms probed by the first probe beam  130 ( 1 ) to precess at a Larmor frequency greater than γ|{right arrow over (B)} (0) |. At the at the second probe beam  130 ( 2 ), the magnetic field  730  subtracts from the bias field {right arrow over (B)} (0) , causing the atoms probed by the second probe beam  130 ( 2 ) to precess at a Larmor frequency less than γ|{right arrow over (B)} (0) |. Thus, due to the magnetic field  730 , atoms probed by the probe beams  130 ( 1 ) and  130 ( 2 ) precess at different rates. In this case, the data blocks  340  cannot be concatenated into the single gapless temporal sequence  502  since the Larmor frequency shifts with each data block  340 . 
       FIG.  8    illustrates how the alternating sequence of the data blocks  340  can be processed to generate a single gapless temporal sequence  802  for the scenario depicted in  FIG.  7   . Here, each data block  340 ( i ) is processed into a corresponding instantaneous-phase block  440 ( i ), as described above (not shown in  FIG.  8   ). Each phase block  440 ( i ) is then fit to a straight line to obtain a corresponding best-fit slope m i , also as described before and shown in  FIG.  8   . The best-fit slope m i  is then used to obtain a corresponding residual block  840 ( i ) of the phase block  440 ( i ), which approximates the signal field {right arrow over (B)} (s) (t) without the static bias field {right arrow over (B)} (0) . For the data blocks  340  obtained from the second vapor cell  104 ( 2 ) (e.g., data blocks  340 ( l ), where l is even), the corresponding residual blocks  840  are inverted to account for the different directions of the magnetic field  730  relative to the bias field {right arrow over (B)} (0) . Here, “inverted” means that the value of each point in the residual block  840  is multiplied by −1, as represented in  FIG.  8    by a circle with “−1” inscribed therein. The inverted residual blocks  840  are then be interleaved with the uninverted residual blocks  840  obtained from the first vapor cell  104 ( 1 ) (e.g., data blocks  340 ( l ), where l is odd) to form the single gapless temporal sequence  802 . The time-derivative of the temporal sequence  802  is then calculated to obtain a corresponding frequency sequence, which is then divided by γ to obtain the magnetic-field sequence {B j }. 
     Optical pumping of the vapors  106 ( 1 ) and  106 ( 2 ) may be implemented using a technique known in the art. For example, when the bias field {right arrow over (B)} (0)  is parallel to the propagation direction of the probe beams  130 ( 1 ) and  130 ( 2 ) (e.g., the z direction in  FIGS.  1  and  2   ), linearly polarized first and second pump beams may be directed through the respective vapor cells  104 ( 1 ) and  104 ( 2 ) parallel to the probe beams  130 ( 1 ) and  130 ( 2 ) to pump the atoms into a dark superposition of ground-state magnetic sublevels (i.e., coherent population trapping). The resulting ground-state coherence between these magnetic sublevels varies in time, and is equivalent to a precession of the atoms in the reference frame of the probe beams  130 ( 1 ) and  130 ( 2 ). Alternatively, the pump beams may be oriented perpendicularly to the probe beams  130 ( 1 ) and  130 ( 2 ). For example, in  FIG.  6    circularly polarized pump beams  640 ( 1 ) and  640 ( 2 ) may be directed through the respective vapor cells  104 ( 1 ) and  104 ( 2 ) parallel to the bias field {right arrow over (B)} (0)  to optically pump the atoms into a stretched state that precesses along the z direction in the reference frame of the probe beams  130 ( 1 ) and  130 ( 2 ). Another optical pumping technique known in the art may be used without departing from the scope hereof. 
     In some embodiments, more than two vapor cells  104  are placed around the sample cell  102 . For example, in the dual atomic magnetometer  100  shown in  FIG.  2   , third and fourth vapor cells  104  may be placed above and below the sample cell  102  (in the y direction). In these embodiments, four polarization signals can be processed and interleaved, as described above, to form the single gapless temporal sequence  802  and the magnetic-field sequence {B j }. Furthermore, while  FIGS.  1  and  2    show the vapor cells  104 ( 1 ) and  104 ( 2 ) located on opposite sides of the sample cell  102  (along the x direction), the vapor cells  104 ( 1 ) and  104 ( 2 ) may be positioned otherwise without departing from the scope hereof. For example, the vapor cells  104 ( 1 ) and  104 ( 2 ) may be positioned proximate to adjacent perpendicular side faces of the sample cell  102 . 
     Some embodiments include only the signal processor  144 , wherein all other components (e.g., the vapor cells  104 ( 1 ) and  104 ( 2 ), the sample cell  102 , the polarimeters  140 ( 1 ) and  140 ( 2 ), etc.) are provided by a third party. Other embodiments exclude the signal processor,  144 , which is provided by a third party. 
     Changes may be made in the above methods and systems without departing from the scope hereof. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween.