TIME-MULTIPLEXED DUAL ATOMIC MAGNETOMETRY

Abstract

Time-multiplexed atomic magnetometry uses first and second atomic vapor cells located adjacent to a sample to be measured. Each vapor cell operates according to a sequence of alternating pumping and probing stages. However, the sequences are temporally offset from each other 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. The Hilbert transform of the signals may be taken to obtain their instantaneous phases, which may then be interleaved to form a single gapless time sequence that represents the magnetic field of the sample over a time window that lasts for several continuous pumping/probing stages.

Claims

1. A time-multiplexed dual atomic magnetometer, comprising: first and second vapor cells located adjacent to a sample cell such that a magnetic field generated by a sample within the sample cell induces Larmor precession of atoms within the first and second vapor cells; 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 outputting a first polarization signal indicative of the first polarization; 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 outputting a second polarization signal indicative of the second polarization; and 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.

2. The time-multiplexed dual atomic magnetometer of claim 1, wherein the signal processor: sequentially performs instantaneous-phase retrieval on each of the alternating data blocks to obtain a corresponding magnetic-field sequence; and appends each magnetic-field sequence to the single gapless temporal sequence.

3. The time-multiplexed dual atomic magnetometer of claim 2, wherein the signal processor outputs the single gapless temporal sequence.

4. The time-multiplexed dual atomic magnetometer of claim 1, further comprising a controller that gates the first and second probe beams such that: the first probe beam only propagates through the first vapor cell during a first measurement stage to generate a first data block of the alternating data blocks; and the second probe beam only propagates through the second vapor cell during a second measurement stage to generate a second data block of the alternating data blocks, the second measurement stage beginning when the first measurement stage ends.

5. The time-multiplexed dual atomic magnetometer of claim 4, wherein a first duration of the first measurement stage is similar to a second duration of the second measurement stage.

6. The time-multiplexed dual atomic magnetometer of claim 4, wherein the controller gates a first pump beam and a second pump beam such that: the first pump beam only propagates through the first vapor cell during a first pumping stage prior to the first measurement stage, the first pump beam spin-polarizing the atoms in the first vapor cell; and the second pump beam only propagates through the second vapor cell during a second pumping stage prior to the second measurement stage, the second pump beam spin-polarizing the atoms in the second vapor cell.

7. The time-multiplexed dual atomic magnetometer of claim 6, wherein a first duration of the first pumping stage is similar to a second duration of the second pumping stage.

8. The time-multiplexed dual atomic magnetometer of claim 1, further comprising at least one magnetic field coil that applies a magnetic bias field to the first vapor cell, the second vapor cell, and the sample cell.

9. The time-multiplexed dual atomic magnetometer of claim 8, wherein: the first and second probe beams propagate along a common propagation direction; and the at least one magnetic field coil is positioned such that the magnetic bias field is parallel to the common propagation direction.

10. The time-multiplexed dual atomic magnetometer of claim 8, wherein: the first and second probe beams propagate along a common propagation direction; and the at least one magnetic field coil is positioned such that the magnetic bias field is perpendicular to the common propagation direction.

11. A method for time-multiplexed dual atomic magnetometry, comprising: generating, by a sample within a sample cell, a magnetic field that induces Larmor precession of atoms within first and second vapor cells located adjacent to the sample cell; 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 outputting a first polarization signal indicative of the first polarization; 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 outputting a second polarization signal indicative of the second polarization; and 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.

12. The method of claim 11, wherein said processing includes: sequentially performing instantaneous-phase retrieval on each of the alternating data blocks to obtain a corresponding magnetic-field sequence; and appending each magnetic-field sequence to the single gapless temporal sequence.

13. The method of claim 12, further comprising outputting the single gapless temporal sequence.

14. The method of claim 11, further comprising gating the first and second probe beams such that: the first probe beam only propagates through the first vapor cell during a first measurement stage to generate a first data block of the alternating data blocks; and the second probe beam only propagates through the second vapor cell during a second measurement stage to generate a second data block of the alternating data blocks, the second measurement stage beginning when the first measurement stage ends.

15. The method of claim 14, wherein said gating the first and second probe beams includes gating the first and second probe beams such that a first duration of the first measurement stage equals a second duration of the second measurement stage.

16. The method of claim 14, further comprising gating a first pump beam and a second pump beam such that: the first pump beam only propagates through the first vapor cell during a first pumping stage prior to the first measurement stage, the first pump beam spin-polarizing the atoms in the first vapor cell; and the second pump beam only propagates through the second vapor cell during a second pumping stage prior to the second measurement stage, the second pump beam spin-polarizing the atoms in the second vapor cell.

17. The method of claim 16, wherein said gating the first and second pump beams includes gating the first and second pump beams such that a first duration of the first pumping stage is similar to a second duration of the second pumping stage.

18. The method of claim 11, further comprising applying a magnetic bias field to the first vapor cell, the second vapor cell, and the sample cell.

19. The method of claim 18, wherein said applying includes applying the magnetic bias field such that a direction of the magnetic bias field is parallel to a common propagation direction of the first and second probe beams.

20. The method of claim 18, wherein said applying includes applying the magnetic bias field such that a direction of the magnetic bias field is perpendicular to a common propagation direction of the first and second probe beams.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0014] FIG. 1 is a top sectional view of a dual atomic magnetometer that measures magnetic fields generate by a sample, in an embodiment.

[0015] FIG. 2 is a side section view of the dual atomic magnetometer of FIG. 1.

[0016] 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.

[0017] 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.

[0018] FIG. 5 shows how several consecutive frequency blocks may be concatenated into the single gapless temporal sequence, in an embodiment.

[0019] 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.

[0020] 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.

[0021] 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 OF THE EMBODIMENTS

[0022] 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).

[0023] 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).

[0024] The polarization of the first probe beam 130(1) oscillates at an instantaneous Larmor frequency f.sub.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.sub.L(t) depends on the scalar magnitude of the magnetic field, which has two components: a time-varying signal field {right arrow over (B)}.sup.(s)(t)=(B.sub.x.sup.(s)(t), B.sub.y.sup.(s)(t), B.sub.z.sup.(s)(t)) arising from the sample 110, and a constant (i.e., time-independent) bias field {right arrow over (B)}.sup.(0)=(B.sub.x.sup.(0), B.sub.y.sup.(0), B.sub.z.sup.(0)). Thus, the instantaneous Larmor frequency f.sub.L(t) can be represented mathematically as f.sub.L(t)=γ|{right arrow over (B)}.sup.(s)(t)+{right arrow over (B)}.sup.(0)|/(2π), where γ is the gyromagnetic ratio of the species of the first atomic vapor 106(1). Assuming |{right arrow over (B)}.sup.(s)(t)|<<|{right arrow over (B)}.sup.(0)|, the instantaneous Larmor frequency f.sub.L(t) is approximated by f.sub.L(t)≈γ|{right arrow over (B)}.sup.(0)|/(2π), which is time-independent. Therefore, the bias field {right arrow over (B)}.sup.(0) sets a nominal Larmor frequency f.sub.L.sup.(0)≈γ|{right arrow over (B)}.sup.(0)|/(2π) that is subsequently modified by the signal field {right arrow over (B)}.sup.(s)(t). In FIGS. 1 and 2, the bias field {right arrow over (B)}.sup.(0) is oriented along the +z direction (i.e., {right arrow over (B)}.sup.(0)=(0, 0, B.sub.z.sup.(0)), for which f.sub.L.sup.(0)=γB.sub.z.sup.(0)/(2π). However, the bias field {right arrow over (B)}.sup.(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)}.sup.(0) is the same at both of the vapor cells 104(1) and 104(2) and the sample cell 102.

[0025] 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.sub.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.sub.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.

[0026] 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).

[0027] 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).

[0028] 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.sub.p.sup.(1), (ii) a first measurement stage 306(1) with a first measurement duration T.sub.m.sup.(1), and (iii) a first dead stage 308(1) with a first dead-time duration T.sub.d.sup.(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.sub.1=T.sub.p.sup.(1)+T.sub.m.sup.(1)+T.sub.d.sup.(1), and has a measurement duty cycle η.sub.1=T.sub.m.sup.(1)/T.sub.1.

[0029] 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.sub.d.sup.(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.sub.p.sup.2, (ii) a second measurement stage 306(2) with a second measurement duration T.sub.m.sup.(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.sub.2=T.sub.p.sup.(2)+T.sub.m.sup.(2)+T.sub.d.sup.(2), and has a measurement duty cycle η.sub.2=T.sub.m.sup.(2)/T.sub.2.

[0030] The duration T.sub.d.sup.(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.sub.d.sup.(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).

[0031] 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.sub.2 or .sup.4He) and/or lined with an anti-relaxation coating (e.g., paraffin) to reduce spin relaxation and increase the time constant. Dephasing times T.sub.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.sub.2 is the primary determinant of the measurement durations T.sub.m.sup.(1) and T.sub.m.sup.(2), as the signal-to-noise ratio decays with T.sub.2.

[0032] If general, the first and second measurement durations T.sub.m.sup.(1), T.sub.m.sup.(2) do not need to be equal. Similarly, the first and second pumping durations T.sub.p.sup.(1), T.sub.p.sup.(2) do not need to be equal. In some embodiments, the first and second measurement durations T.sub.m.sup.(1), T.sub.m.sup.(2) are equal, as shown in FIG. 3. In these embodiments, T.sub.d.sup.(1), T.sub.d.sup.(2), T.sub.m.sup.(1), and T.sub.m.sup.(2) are all similar. In some embodiments, the first and second pumping durations T.sub.p.sup.(1), T.sub.p.sup.(2) are similar.

[0033] 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)}.sup.(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.sub.l.sup.(i)(t) is obtained mathematically as the argument of an analytic phase ϕ.sub.a.sup.(i)(t):

ϕ.sub.l.sup.(i)(t)=arg(ϕ.sub.a.sup.(i)(t))=arg(φ.sup.(i)(t)+i{φ.sup.(i)(t)}),  (1)

where φ.sup.(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.sub.l.sup.(i)(t) after unwrapping. The derivative of the instantaneous phase ϕ.sub.l.sup.(i)(t), after unwrapping, gives the instantaneous Larmor frequency for the data block 340(i):

[00001]$\begin{array}{cc}{f}_L^{\left(i\right)}\left(t\right)=\frac{1}{2\pi }\frac{d{\varphi }_I^{\left(i\right)}\left(t\right)}{\mathrm{dt}}.& \left(1\right)\end{array}$

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.sub.L.sup.(i)(t), as described previously.

[0034] In some embodiments, and as shown in FIG. 4, the unwound instantaneous phase ϕ.sub.l.sup.(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.sub.i that represents the average value of dϕ.sub.l.sup.(i)(t)/dt (i.e., the average Larmor frequency) over the measurement duration T.sub.m.sup.(i) of the corresponding data block 340(i). Dividing mi by the gyromagnetic ratio γ gives a single corresponding magnetic-field value B.sub.i. A sequence of several consecutive magnetic-field values {B.sub.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.sub.i} can be calculated to identify components of {right arrow over (B)}.sup.(s)(t) with frequencies less than the measurement duration T.sub.m.

[0035] 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.sub.L.sup.(i)(t) 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.sup.(i)={ϕ.sub.1, ϕ.sub.2, . . . , ϕ.sub.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.sup.(i)={f.sub.j=(ϕ.sub.j+1−ϕ.sub.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.sub.1 and ϕ.sub.N such that the frequency sequence F.sup.(i) and the phase sequence Φ.sup.(i) have the same number of points, and the points are aligned in time.

[0036] 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.sub.j} that approximates the time-varying total magnetic-field strength |{right arrow over (B)}.sup.(s)(t)+{right arrow over (B)}.sup.(0)|. The magnetic-field sequence {B.sub.j} may be subsequently analyzed (e.g., Fourier transform) to identify features associated with the sample 110.

[0037] 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.sub.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).

[0038] 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)}.sup.(0)=(0, 0, B.sub.z.sup.(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)}.sup.(0) is greatest. The bias field {right arrow over (B)}.sup.(0)=(0, 0, B.sub.z.sup.(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.

[0039] The instantaneous Larmor precession frequency f.sub.l(t) is given mathematically by

[00002]$\begin{array}{cc}\begin{array}{c}{f}_L\left(t\right)=\frac{1}{2\pi }.Math.{\stackrel{.fwdarw.}{B}}^{\left(s\right)}\left(t\right)+{\stackrel{.fwdarw.}{B}}^{\left(0\right)}.Math.\\ =\frac{\gamma }{2\pi }\sqrt{{\left(B_x^{\left(0\right)}-B_x^{\left(s\right)}\left(t\right)\right)}^2+{\left(B_y^{\left(0\right)}-B_y^{\left(s\right)}\left(t\right)\right)}^2+{\left(B_z^{\left(0\right)}-B_z^{\left(s\right)}\left(t\right)\right)}^2}.\end{array}& \left(3\right)\end{array}$

For B.sub.x.sup.(0)=B.sub.y.sup.(0)=0, Eqn. 3 simplifies to

[00003]$\begin{array}{cc}{f}_L\left(t\right)=\frac{\gamma }{2\pi }{B}_z^{\left(0\right)}\sqrt{1+\frac{{.Math.{\stackrel{.fwdarw.}{B}}^{\left(s\right)}\left(t\right).Math.}^2}{{\left(B_z^{\left(0\right)}\right)}^2}-2\frac{B_z^{\left(s\right)}\left(t\right)}{B_z^{\left(0\right)}}}\approx \frac{\gamma }{2\pi }{B}_z^{\left(0\right)}\left(1-\frac{B_z^{\left(s\right)}\left(t\right)}{B_z^{\left(0\right)}}+\mathrm{.Math.}\right),& \left(4\right)\end{array}$

where the Taylor expansion in Eqn. 4 assumes |{right arrow over (B)}.sup.(s)(t)|<<B.sub.z.sup.(0). Eqn. 4 shows that f.sub.L(t) approximately equals the nominal Larmor frequency f.sub.L.sup.(0)≈γB.sub.z.sup.(0)/(2π), but is modulated primarily (i.e., to first order) by the z-component B.sub.z.sup.(s)(t) of the signal field {right arrow over (B)}.sup.(s)(t). Equivalently, B.sub.x.sup.(s)(t) and B.sub.y.sup.(s)(t) only modulate the instantaneous Larmor frequency f.sub.L(t) to second order in the Taylor expansion, and are therefore suppressed relative to B.sub.z.sup.(s)(t). Accordingly, the setup shown in FIGS. 1 and 2 preferentially measures the z component B.sub.z.sup.(s)(t) of the signal field {right arrow over (B)}.sup.(s)(t).

[0040] 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)}.sup.(0) oriented perpendicularly to the propagation direction of the probe beams 130(1) and 130(2). Specifically, the bias field {right arrow over (B)}.sup.(0) points in the +y direction, i.e., {right arrow over (B)}.sup.(0)=(0, B.sub.y.sup.(0), 0). The instantaneous Larmor precession frequency f.sub.l(t) is given mathematically by

[00004]$\begin{array}{cc}{f}_L\left(t\right)=\frac{\gamma }{2\pi }{B}_y^{\left(0\right)}\sqrt{1+\frac{{.Math.{\stackrel{.fwdarw.}{B}}^{\left(s\right)}\left(t\right).Math.}^2}{{\left(B_y^{\left(0\right)}\right)}^2}-2\frac{B_y^{\left(s\right)}\left(t\right)}{B_y^{\left(0\right)}}}\approx \frac{\gamma }{2\pi }{B}_y^{\left(0\right)}\left(1-\frac{B_y^{\left(s\right)}\left(t\right)}{B_y^{\left(0\right)}}+\mathrm{.Math.}\right).& \left(5\right)\end{array}$

Now, B.sub.x.sup.(s)(t) and B.sub.z.sup.(s)(t) only modulate the instantaneous Larmor frequency f.sub.L(t) to second order in the Taylor expansion, and are therefore suppressed relative to B.sub.y.sup.(s)(t). Accordingly, the setup shown in FIG. 6 preferentially measures the y component B.sub.y.sup.(s)(t) of the signal field {right arrow over (B)}.sup.(s)(t). Similar calculations show that when the bias field {right arrow over (B)}.sup.(0) points in the x direction, B.sub.y.sup.(s)(t) and B.sub.z.sup.(s)(t) only modulate the instantaneous Larmor frequency f.sub.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)}.sup.(s)(t) by aligning the bias field {right arrow over (B)}.sup.(0) along the direction of the component.

[0041] 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)}.sup.(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)}.sup.(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)}.sup.(0)|. At the at the second probe beam 130(2), the magnetic field 730 subtracts from the bias field {right arrow over (B)}.sup.(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)}.sup.(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.

[0042] 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.sub.i, also as described before and shown in FIG. 8. The best-fit slope m.sub.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)}.sup.(s)(t) without the static bias field {right arrow over (B)}.sup.(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)}.sup.(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.sub.j}.

[0043] 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)}.sup.(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)}.sup.(0) to optically pump the atoms into a stretched state that processes 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.

[0044] 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.sub.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.

[0045] 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.

[0046] 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.