Optically pumped gradient magnetometer

Abstract

A method is provided for sensing a magnetic field in a magnetic gradiometer of the kind in which pump light and light constituting an optical carrier traverse first and second atomic vapor cells that contain host atoms and that are separated from each other by a known distance. According to such method, the host atoms are prepared in a coherent superposition of two quantum states that differ in energy by an amount that is sensitive to an ambient magnetic field. Modulation of the optical carrier in the respective cells gives rise to sidebands that interfere to generate a beat frequency indicative of the magnetic field gradient. The host atoms are prepared at least in a mode that allows measurement of ambient magnetic field components perpendicular to the axis of the pump light. In such mode, the host atoms are spin-polarized by pump light while subjected to a controlled magnetic field directed parallel to the pump beam, and then the controlled magnetic field is adiabatically extinguished.

Claims

1. A method of sensing a magnetic field, comprising: in a first atomic vapor cell and in a second atomic vapor cell situated at a distance from the first atomic vapor cell, preparing host atoms in a coherent superposition of two quantum states, the two quantum states differing in energy by an amount that depends upon a strength of an ambient magnetic field; modulating an optical carrier in the first cell and in the second cell, thereby to impose on the carrier at least one first-order sideband from the first cell and at least one first-order sideband from the second cell, the sidebands from the respective cells having frequencies that depend on values of the ambient magnetic field at the respective cells where they arose; impinging light comprising the sidebands onto a photodetector; and measuring a beat frequency generated by interference between the sidebands from the respective cells, wherein: the preparing of the host atoms is performed in one or more modes, wherein each mode comprises spin-polarizing host atoms and placing the spin-polarized host atoms in a coherent superposition of states; and the preparing of the host atoms is performed at least in a mode here denominated the B-Perpendicular mode, which comprises: spin-polarizing the host atoms with a pump beam of light while subjecting the host atoms to a controlled magnetic field directed parallel to the pump beam, and then adiabatically extinguishing the controlled magnetic field, whereby ambient magnetic field components perpendicular to the pump beam can be sensed.

2. The method of claim 1, wherein the first and second atomic vapor cells are both filled with nitrogen buffer gas.

3. The method of claim 1, further comprising filtering the modulated optical carrier with a polarization-selective element so that light that effectively contains only the sidebands is impinged on the photodetector.

4. The method of claim 1, further comprising filtering the modulated carrier with a narrow-band frequency filter or an atomic filter cell so that light that effectively contains only the sidebands is impinged on the photodetector.

5. The method of claim 1, wherein the host atoms are atoms of rubidium-87.

6. The method of claim 1, wherein the host atoms have a D1 atomic transition and a D2 atomic transition, the pump beam is tuned to one of the said transitions, and the probe carrier is tuned to the other of the said transitions.

7. The method of claim 1, wherein the atomic ensembles are prepared in a coherent superposition of |F=2, m.sub.F=2> and |F=1, m.sub.F=1> levels, wherein F=2 and F=1 represent different hyperfine ground states, and m.sub.F=2 and m.sub.F=1 are Zeeman sublevels of the hyperfine ground states.

8. The method of claim 1, wherein the coherent superposition is between Zeeman sublevels of two distinct hyperfine manifolds.

9. The method of claim 1, wherein the preparing the host atoms comprises, after spin-polarizing the host atoms, subjecting the spin-polarized host atoms to a microwave π/2 pulse having a frequency chosen to resonate with a pair of magnetically sensitive hyperfine energy levels.

10. The method of claim 1, wherein: the atomic vapor cells are filled with respective fills of buffer gas that differ in pressure, in composition, or in both pressure and composition; and the respective fills of buffer gas are selected such that absent an ambient magnetic field, the sidebands from the respective atomic vapor cells have different optical frequencies, whereby a non-zero beat frequency is generated both when an ambient magnetic field is present and when an ambient magnetic field is absent.

11. The method of claim 1, wherein: the modulating of the optical carrier comprises, within each of the atomic vapor cells, using one of two atomic transitions to modulate the optical carrier and thereby produce a decremental sideband; the modulating of the optical carrier further comprises, within each of the atomic vapor cells, using the other of the two atomic transitions to modulate the optical carrier and produce an incremental sideband; the decremental and incremental sidebands have different sideband frequencies; the measuring of the beat frequencies comprises measuring a decremental beat frequency generated by interference between the decremental sidebands generated within the first and second cells; the measuring of the beat frequencies further comprises measuring an incremental beat frequency generated by interference between the incremental sidebands generated within the first and second cells; and the method further comprises obtaining a subtractive difference between the decremental and incremental beat frequencies, thereby to obtain a magnetic field measurement with reduced temperature sensitivity.

12. A method of sensing a magnetic field, comprising: in a first atomic vapor cell and in a second atomic vapor cell situated at a distance from the first atomic vapor cell, preparing host atoms in a coherent superposition of two quantum states, the two quantum states differing in energy by an amount that depends upon a strength of an ambient magnetic field; modulating an optical carrier in the first cell and in the second cell, thereby to impose on the carrier at least one first-order sideband from the first cell and at least one first-order sideband from the second cell, the sidebands from the respective cells having frequencies that depend on values of the ambient magnetic field at the respective cells where they arose; impinging light comprising the sidebands onto a photodetector; and measuring a beat frequency generated by interference between the sidebands from the respective cells, wherein: the preparing of the host atoms is performed in one or more modes, wherein each mode comprises spin-polarizing host atoms and placing the spin-polarized host atoms in a coherent superposition of states; and the preparing of the host atoms is performed in at least one mode here denominated the B-Parallel mode, which comprises: spin-polarizing the host atoms with a pump beam of light to place the host atoms in a first hyperfine state; using an initial microwave pulse to transfer the host atoms to a second hyperfine state; and xusing a subsequent microwave pulse, which is a π/2 pulse, to place the host atoms in a coherent superposition of the second hyperfine state with a third hyperfine state characterized in that the second and third hyperfine states have the same value of the hyperfine angular momentum quantum number m.sub.F, whereby the modulating of the optical carrier is effective at least when the optical carrier is propagating in a direction parallel to the ambient magnetic field.

13. The method of claim 12, wherein the host atoms are transferred from the first hyperfine state to the second hyperfine state by a process of adiabatic rapid passage (ARP).

14. The method of claim 13, wherein the pump beam and the optical carrier traverse parallel paths through the atomic vapor cells.

15. The method of claim 14, wherein the B-parallel mode and the B-perpendicular mode are performed in alternation.

16. The method of claim 1, wherein source light from a single light source provides, sequentially, both the pump beam and the optical carrier.

17. The method of claim 16, wherein, in a sequence, the pump light is provided initially as a strong pulse for optical pumping, and is then attenuated and detuned to provide the optical carrier.

18. The method of claim 12, wherein: the atomic vapor cells are filled with respective fills of buffer gas that differ in pressure, in composition, or in both pressure and composition; and the respective fills of buffer gas are selected such that absent an ambient magnetic field, the sidebands from the respective atomic vapor cells have different optical frequencies, whereby a non-zero beat frequency is generated both when an ambient magnetic field is present and when an ambient magnetic field is absent.

19. The method of claim 12, wherein: the modulating of the optical carrier comprises, within each of the atomic vapor cells, using one of two atomic transitions to modulate the optical carrier and thereby produce a decremental sideband; the modulating of the optical carrier further comprises, within each of the atomic vapor cells, using the other of the two atomic transitions to modulate the optical carrier and produce an incremental sideband; the decremental and incremental sidebands have different sideband frequencies; the measuring of the beat frequencies comprises measuring a decremental beat frequency generated by interference between the decremental sidebands generated within the first and second cells; the measuring of the beat frequencies further comprises measuring an incremental beat frequency generated by interference between the incremental sidebands generated within the first and second cells; and the method further comprises obtaining a subtractive difference between the decremental and incremental beat frequencies, thereby to obtain a magnetic field measurement with reduced temperature sensitivity.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a simplified schematic diagram of a system for an atomic magnetic gradiometer of the kind described here. In the example shown in the figure, a pair of atomic vapor cells are shown as containing different pressures of buffer gas, in accordance with one of the new approaches described herein.

(2) FIGS. 2 and 3 are partial transition diagrams for .sup.87Rb illustrating state preparation (FIG. 2) and sideband-generation (FIG. 3) processes according to example embodiments.

(3) FIG. 4 shows an example of a beat note produced from a gradiometer of the kind described herein.

(4) FIG. 5 is a timing diagram illustrating cyclic, two-state operation of an example gradiometer. In the figure, notional graphs of signal evolution illustrate a state-preparation stage followed by a beat-note stage in each cycle of operation.

(5) FIG. 5A is a flowchart illustrating a mode of operating a gradiometer with a frequency offset between the vapor cells.

(6) FIG. 6 is a partial transition diagram showing two transitions that can each give rise to a coherent superposition of states. The respective coherent superpositions that are shown have a substantially equal and opposite dependence on the magnetic field. For convenience, the corresponding transitions are referred to here as an “incremental” transition and as a “decremental” transition, respectively.

(7) FIG. 7 is a theoretical graph illustrating the temperature dependence of a beat-note signal produced by a magnetic gradiometer of the kind described herein.

(8) FIG. 7A is a flowchart illustrating a mode of operating a gradiometer with reduced sensitivity to temperature.

(9) FIG. 8 is a schematic diagram of a magnetic gradiometer in a configuration adapted for coaxial propagation of the pump and probe beams.

(10) FIG. 9 is a timing diagram that provides an example of a sequence of optical and microwave pulses useful for state preparation in a mode of operation of a magnetic gradiometer in which the ambient magnetic field is perpendicular to the pump beam.

(11) FIG. 9A is a flowchart illustrating a B-perpendicular mode of operating a gradiometer, which is advantageous when the ambient magnetic field is perpendicular to the pump axis.

(12) FIGS. 10-12 illustrate a mode of operation of a magnetic gradiometer, referred to here as the “B-parallel” mode, in which the ambient magnetic field is parallel to the pump beam. FIG. 10 shows the orientations of the laser axis, the ambient field B, and the coil field B.sub.C. FIG. 11 is a partial transition diagram showing the spin states that are involved in an example. FIG. 12 is a timing diagram that provides an example of the sequence of the pulses used for state preparation in the B-parallel mode of operation. FIG. 12A is a flowchart illustrating, in an example, the B-parallel mode of operating a gradiometer.

(13) FIG. 13 is a flowchart illustrating a procedure for operating a magnetometer that uses a single source laser.

DETAILED DESCRIPTION

(14) The host atomic system for the illustrative embodiment described here is the 5S.sub.1/2 ground state of .sup.87Rb. Possible alternative host systems include .sup.85Rb, .sup.133Cs, .sup.39K, .sup.41K, helium (He), and artificial atoms such as nitrogen vacancy centers.

(15) It should be noted that whereas some Zeeman magnetometers of the prior art operate with transitions involving a pair of Zeeman sublevels of the same hyperfine manifold (i.e., having the same total angular momentum quantum number F), the hyperfine gradiometer described here utilizes a pair of Zeeman sublevels of two distinct hyperfine manifolds (e.g., F=1 and F=2). This is noteworthy, not least because the transition frequency between two distinct hyperfine manifolds is usually in the range of hundreds of megahertz to several gigahertz, whereas the transition frequencies between the Zeeman sublevels in the same hyperfine manifold are typically on the order of hundreds of kilohertz in the terrestrial magnetic field.

(16) In our magnetometer, the magnetic energy of a microwave pulse induces Rabi oscillations between the two Zeeman sublevels with angular frequency Ω=(μ.sub.B/h)B, where μ.sub.B is the Bohr magneton and h is Planck's constant. By turning off the pulse at a time when the atomic populations have an equal probability of being in the two states, Ωt=π/2, a coherent superposition is induced between them. The coherence oscillates at the resonance frequency of the microwave radiation, modulating the atomic susceptibility and hence the refractive index of the medium near the resonance.

(17) When probed by a weak beam, the oscillating refractive index generates sidebands in a process known as parametric frequency conversion. This process generates sidebands separated from the carrier by the frequency of the microwave magnetic coupling. In our illustrative embodiment, we use the 5s.sup.2S.sub.1/2 ground state and F=|J+I|=1, 2 hyperfine ground state levels in a warm ensemble of .sup.87Rb.

(18) In example implementations of our techniques as discussed below, the sidebands used are sidebands of first order.

(19) As shown in FIG. 1, we use two vapor cells 101, 102, each configured as a one-centimeter cube. The two cells are physically separated by the wavelength of the incoming microwave radiation, which in our example is 4.4 cm. We chose this separation so that both cells would experience the same microwave phase.

(20) Both cells are filled, for example, with nitrogen (N.sub.2) buffer gas. Possible alternative buffer gases include argon and neon, or even gas mixtures.

(21) With the ambient field 105 along the direction of the pump, we set the quantization axis to be along the direction corresponding to the z axis in the figure.

(22) The pump laser 110, tuned in our example to the D1 line at 795 nm, is circularly polarized by a quarter-wave plate (QWP) 115 before being directed through the vapor cells in the z direction, which is parallel to the quantization axis. The probe laser 120 is tuned in our example to the D2 line at a carrier wavelength of 780 nm. In the view of FIG. 1, the probe beam is transmitted sequentially through the vapor cells in they direction, which is orthogonal to the pump axis. The microwave π/2 pulse is applied from a suitable microwave emitter 125, which for illustrative purposes only is represented in the figure as a microwave horn. Alternatively, an antenna may be used, and may be preferable in order to make the assembly compact.

(23) Optionally, a mirror 130 is used to reflect the once-modulated carrier back through the vapor cells to be subjected to modulation for a second time for signal enhancement before impingement on a photodetector 135 for readout of the beat note. Because the sidebands are polarized orthogonally to the unmodulated carrier, a polarization selector 140 can be used to filter the modulated carrier so that light that effectively contains only the sidebands will be impinged on the photodetector.

(24) Alternatively to a polarization filter, or in addition thereto, a narrow-band frequency filter or an atomic filter cell can be used for optical filtration.

(25) By measuring the frequency of the beat note signal using detection electronics, the value of the magnetic field gradient between the two sets of atoms can be obtained. Examples of detection electronics include, without limitation, a frequency counter and a data-acquisition system with a signal processor to compute frequency. To ease measurement of the frequency of the beat note signal, a bias magnetic field gradient can be added such that the frequency of the beat note is at an easily measurable value.

(26) FIGS. 2 and 3 are partial transition diagrams for .sup.87Rb illustrating the state preparation (FIG. 2) and sideband-generation (FIG. 3) processes. As seen in the diagram, the quantizing static magnetic field splits the 5s.sup.2S.sub.1/2 level into a ground-state manifold of eight sublevels.

(27) As indicated in FIG. 2, we apply a circularly polarized (σ.sup.+) 795-nm pump laser on the 5s.sup.2S.sub.1/2.fwdarw.5p.sup.2P.sub.1/2 D1 line. The atoms absorb angular momentum from the σ.sup.+ light and make Δm.sub.F=+1 transitions to the excited state, resulting in a net migration of atoms to the |F=2, m.sub.F=2> end state. The |F=2, m.sub.F=2> state is a dark state, that is, the atomic population is trapped and cannot be excited by the σ.sup.+ pump light.

(28) The pump is square-wave modulated at 200 kHz between the F=1 and F=2 ground states, clearing the atomic population that isn't in the dark end state. An acousto-optic modulator (AOM) is used as an optical switch for the pump beam.

(29) With reference to FIG. 3, sideband generation is initiated by sending in a weak (12 μW) linearly polarized 780-nm carrier beam on the 5s.sup.2S.sub.1/2.fwdarw.5p.sup.2P.sub.3/2 D2 line directionally orthogonal to the pump.

(30) As shown in FIG. 3, the π/2 pulse of magnetic energy between the |F=1, m.sub.F=1> and |F=2, m.sub.F=2> sublevels brings the atoms into a coherent superposition.

(31) The coherence of each atomic ensemble modulates the refractive index of the atoms at a modulation frequency. The modulation frequency is the sum of the hyperfine frequency of the clock transition, ν.sub.HF, plus the Zeeman splitting νz=γB, where γ is the gyromagnetic ratio. As noted above, the gyromagnetic ratio for .sup.87Rb is 7 Hz/nT.

(32) Since the coherence is established on the stretched states (i.e., the states of maximum m.sub.F) of the F=1 and F=2 levels, the total Zeeman frequency shift is 3×7 Hz/nT for a total splitting of ν.sub.HF+21 Hz/nT×ΔB. This is the frequency offset of the sidebands from the carrier beam.

(33) We perform this process in both vapor cells, thereby generating two sets of sidebands.

(34) The sideband light exits the cells orthogonally polarized to the carrier. Accordingly, we use a polarizing beam splitter (PBS) to block the carrier light while allowing the sidebands to pass.

(35) The sidebands are detected at the photodiode as a beat note. The frequency of the beat indicates the value of the magnetic gradient between the two atomic vapor cells.

(36) In order to measure a beat frequency at zero field, we can use the fact that buffer gases, including nitrogen, shift the hyperfine frequency of atomic ensembles by an amount that depends on gas density, among other factors. Thus, for example, each cell can be filled with a different pressure of nitrogen buffer gas: E.g., 30 torr in the cell nearest the microwave radiation source and 15 torr in the other cell. This would offset the sidebands by 7 kHz at zero field. This approach is discussed in greater detail below.

(37) Since the sidebands are generated from population in the end state of the manifold in each cell, the frequency offset of the sidebands due to the magnetic field is 21 Hz/nT×ΔB, hence the beat-note frequency in the present example, with offset, would be given by f.sub.b=7 kHz+21 Hz/nT×ΔB.

(38) FIG. 4 shows an example of a beat note produced from the gradiometer. In order to measure its frequency, we fit it with a high-order polynomial and extract the frequency value.

(39) It will be evident from FIG. 4 that the amplitude of the sidebands decays over time. That is because the optical sidebands are generated only for the duration of time when the atoms are in a coherent state. For .sup.87Rb atoms confined in a 1-cm-cube vapor cell, this duration is typically on the order of 1 ms. Coherence is lost through various mechanisms including wall collisions, power broadening, and buffer-gas collisions. Because the generated beat-note signal also decays over time, the atoms must be periodically re-pumped to the coherent state.

(40) An illustrative but non-limiting, repetition rate for a measurement is 500 Hz. At that rate, the pump may be turned on for, e.g., 1 ms in each repetition and then turned off, with the microwave π/2 pulse arriving a few microseconds later.

(41) Thus, as shown in FIG. 5, the gradiometer will generally be operated in two stages. The state-preparation stage 1 extends along the time axis from 0 to t.sub.s, and the beat-note stage 2 extends from t.sub.s to t.sub.m. After the beat-note signal has decayed substantially, the measurement process is restarted by preparing the atomic ensembles again in the coherent state at t=0.

(42) A Method to Add a Frequency Offset Between the Respective Hyperfine Frequencies of the Two Vapor Cells.

(43) If the beat frequency were allowed to go to zero in the absence of an ambient field, it would be difficult to confirm that the magnetometer was operating properly, and there would not be a signal for small changes in the gradient around a gradient of zero. It would also be difficult to observe effects due to the alignment of the magnetometer relative to the ambient field. Therefore, as briefly discussed above, it is desirable to provide an offset between the sidebands produced by the respective vapor cells that is independent of the ambient magnetic field.

(44) Any of various values can be chosen for such an offset. By way of example, we have found a frequency offset of 7 kHz to be useful.

(45) The buffer gas within the vapor cell produces a shift in the hyperfine frequency that is dependent on the buffer gas density, among other factors. In examples, the frequency offset is produced by filling the two cells with the same gas, but at different pressures.

(46) Alternatively, or in addition, each cell may be filled with a different buffer gas or a different buffer gas mixture. Buffer gas mixtures, in particular, may be of interest for reducing temperature sensitivity or for giving the same optical linewidth while maintaining a frequency offset.

(47) More specifically, the shift Δν=(1/2π)Δω in the hyperfine frequency is approximated to second order in temperature by the expression Δν=P.sub.s(β.sub.s+δ.sub.sΔT+γ.sub.sΔT.sup.2), where P.sub.s is the buffer-gas pressure, ΔT is the gas temperature in Celsius degrees, and β.sub.s, δ.sub.s, and γ.sub.s are coefficients whose values for selected gases are listed in Table 1, which was taken from Vanier et al., J. Appl. Phys. Vol. 53, No. 8, August 1982, at page 5388.

(48) TABLE-US-00001 β.sub.s (Hz/torr) δ.sub.s (Hz/° C./torr) γ.sub.s (Hz/° C..sup.2/torr) Argon −59.7 −0.32 −0.00035 Krypton −593.5 −0.57 Nitrogen 547.9 0.52 −0.0013

(49) In experimental trials at an operating temperature of about 100° C. for the vapor cells, we produced a zero-field offset frequency of 7 kHz using nitrogen-filled cells at respective pressures of 15 torr and 30 torr.

(50) More generally, a typical range of operating temperatures for the vapor cells is 80° C. to 120° C.

(51) FIG. 5A is a flowchart illustrating a mode of operating a gradiometer with a frequency offset between the vapor cells: At 501, the respective vapor cells are prepared with different fills of buffer gas. At 502, the host atoms are pumped and the coherent superposition of states is prepared. At 503, the probe interrogates the host atoms and the sidebands are generated. At 504, the sidebands are isolated from the carrier. At 505, the beat note between the sidebands is obtained, with an offset due to the different fills of buffer gas. At 506, the beat frequency is measured, and a measurement of the magnetic field gradient is obtained from the beat frequency.

(52) A Method to Reduce the Temperature Sensitivity of the Hyperfine Frequency of the Vapor Cells.

(53) In some implementations, the respective vapor cells can have different temperature coefficients (see Table 1). This difference can arise, for example, because the cells contain different buffer gases or different buffer-gas pressures. Unless compensated in some way, this is undesirable because it complicates the interpretation of the beat frequency.

(54) We have found a way to compensate for this temperature dependence. One way to reduce the temperature dependence is to use a temperature insensitive buffer gas mixture as is often done in vapor cell atomic clocks. Our new approach relies on the fact that if a coherent superposition of states can be based on a Δm.sub.F=1 transition, then, generally speaking, another coherent superposition can be based on a Δm.sub.F=−1 transition.

(55) For example, FIG. 6 is a partial transition diagram showing two transitions that can each give rise to a coherent superposition of states with a substantially equal and opposite dependence on the magnetic field. The incremental transition is the |1,1>.fwdarw.|2, 2> transition discussed above. The decremental transition is the |1, −1>.fwdarw.|2, −2> transition.

(56) Modulation by the incremental transition in the two vapor cells will produce a beat-note frequency equal to the relative temperature shift in the zero-field hyperfine frequency between the two cells, plus the quantity 3γ×(B.sub.1-B.sub.2), which we have discussed above.

(57) Modulation by the decremental transition in the two vapor cells will produce a beat-note frequency equal to the same relative temperature shift in the zero-field hyperfine frequency between the two cells, minus the quantity 3γ×(B.sub.1-B.sub.2), which we have discussed above.

(58) Hence, subtracting one beat frequency from the other produces the temperature-independent value, Δν=6γ×(B.sub.1-B.sub.2).

(59) Amore precise expression for Δν can be obtained from the well-known Breit-Rabi formula for the energy of the hyperfine transition. Expanding the formula to third order in the ambient magnetic field, and subtracting the resulting frequencies for the respective beat notes yields the expression, Δν=6γ(B.sub.1-B.sub.2)+(12γ.sup.3/ν(T).sup.2)(B.sub.1.sup.3−B.sub.2.sup.3), where the temperature dependence is contained in the second term.

(60) An example of the temperature shift in the beat-note difference Δν is provided by FIG. 7, in which the theoretically calculated shift is plotted as a function of temperature. In the system modeled in FIG. 7, both cells are filled with nitrogen at a pressure of 30 torr. One cell is held at constant temperature, and the temperature of the other cell is varied. It will be evident that the shift depends only very weakly on temperature.

(61) Each of the two beat frequencies can be measured by, for example, alternately interrogating first one of the two end states, and then the other. One way to do this is by switching the direction of the magnetic field that provides the quantization axis between interrogation cycles. Alternatively, the sense of the circular polarization of the pump beam could be switched between cycles.

(62) In addition to cancelling temperature dependence, the method described here can also cancel small shifts in the energy splitting of the hyperfine states due to the light from the probe laser.

(63) The method described does not require the respective cells to contain buffer gases at identical pressures or compositions.

(64) FIG. 7A is a flowchart illustrating a mode of operating a gradiometer with reduced sensitivity to temperature: At 701, the host atoms are pumped and the coherent superposition of states is prepared. At 702, the probe interrogates the host atoms and the sidebands are generated. As explained above, the process of state preparation can be varied so that one, and then the other, of the possible end states is interrogated. Accordingly, the incremental sidebands and the decremental sidebands can be produced in alternation.

(65) At 703, the sidebands, which as noted alternated between incremental and decremental sidebands, are isolated from the carrier. At 704, the beat note between the (incremental or decremental) sidebands is obtained. At 705, the difference is taken between the beat frequency from the incremental sidebands and the beat frequency from the decremental sidebands. At 706, a measurement of the magnetic field gradient is obtained from the difference between the respective beat frequencies.

(66) A Method to Allow Co-Propagation of the Pump and Probe Laser Beams, while Allowing the Measured Magnetic Field to have any Orientation Relative to the Laser Beams.

(67) We found that by adding certain steps involving adiabatic transitions to the state preparation, it is possible to overcome the limitations that selection rules place on the pump and probe directions, relative to the ambient magnetic field that is to be measured. In addition, to enable a compact sensor, it is useful to have the pump and probe copropagating.

(68) It is noteworthy in this regard that because of the selection rules, there will be a dead zone when the ambient magnetic field B is perpendicular to the pump, and there will be a dead zone when B is parallel to the probe.

(69) In a system designed with the probe parallel to the pump, pointing the pump along B causes a dead zone because the probe is perforce also along B. Pointing the probe perpendicular to B causes a dead zone because the pump is also perforce perpendicular to B.

(70) In a system designed with the probe perpendicular to the pump, pointing the probe along B causes the pump to point perpendicular to B, and both effects lead to a dead zone for that pointing direction.

(71) The techniques described below are believed to be useful for mitigating the above problems.

(72) FIG. 8 is a schematic diagram of the magnetic gradiometer, in a configuration adapted for coaxial propagation of the pump and probe beams.

(73) A coil, not shown explicitly in the figure, generates a controllable applied magnetic field B.sub.C parallel to the laser axis, which is applied to the two vapor cells 801, 802. A series of additional optical elements represented in the figure as a 780-nm band-pass filter 805, a half-wave plate 810, a calcite polarization beam splitter 815, and a .sup.85Rb vapor cell 820 are included to assure that carrier light will be excluded from impingement on the photodetector 825. The use of the .sup.85Rb vapor cell is advantageous because, although the generated sidebands are orthogonally polarized to the carrier and can be separated from it on that basis, DC faraday rotation can still cause some of the carrier energy to reach the photodetectors. Polarization beam splitter 830 and wave plate 835, which is half-wave at the probe wavelength and quarter-wave at the pump wavelength, provide a linearly polarized probe beam and a circularly polarized pump beam.

(74) The configuration of FIG. 8 has two modes of operation, one of which is designed to give a maximum signal in the “B-perpendicular” regime, and the other of which is designed to give a maximum signal in the “B-parallel” regime. In the “B-perpendicular” regime, the laser axis is perpendicular to the ambient field. In the “B-parallel” regime, the laser axis is parallel to the ambient field.

(75) As explained above, state preparation is initiated by pumping the atomic ensembles. In some cases, it may be desirable to perform the pumping in the presence of an imposed, controllable magnetic field, whereas in some cases an imposed field may be unnecessary.

(76) The atoms absorb angular momentum from circularly polarized pump light and undergo transitions to an end state with Δm.sub.F=+1 or in some cases Δm.sub.F=−1. The quantization necessary for this process is provided by the ambient magnetic field component parallel to the pump axis. As a consequence, state preparation is ineffective when the ambient field is perpendicular to the combined axes of the pump and probe lasers.

(77) In the B-perpendicular mode of operation, we overcome this obstacle by using a coil to generate a magnetic field parallel to the laser axis. The field generated by the coil is used initially to provide the quantization axis. After the desired end state is reached, the coil is shut off along an adiabatic trajectory.

(78) FIG. 9 is a timing diagram that provides an example of the sequence of the pulses used for state preparation in the B-perpendicular mode of operation. As shown, the pump 900 and the coil 905 are initially maintained concurrently in an on state. At the time that the pump is turned off, the current in the coil is gradually decreased so that the applied magnetic field decays along an adiabatic trajectory. This has the effect of adiabatically rotating the quantization field from the coil field B.sub.C to the ambient field that is to be measured. In order for the rotation to be adiabatic, the rotation rate of the field must be less than the Larmor precession frequency.

(79) After the field has been fully rotated, the microwave π/2 pulse 910 is initiated and sidebands 915 in the probe beam are generated.

(80) FIG. 9A is a flowchart illustrating the B-perpendicular mode of operating a gradiometer. At 951, an external coil applies a magnetic field directed parallel to the axis of the pump beam. At 952, the host atoms are pumped to the end state. At 953, the external coil is slowly de-energized along a trajectory that adiabatically extinguishes the applied magnetic field. At 954, the microwave π/2 pulse is applied to produce the coherent superposition of states. At 955, the probe interrogates the host atoms and the sidebands are generated. At 956, the sidebands are isolated from the carrier. At 957, the beat note between the sidebands is obtained, and at 958 a measurement of the magnetic field gradient is obtained from the beat frequency.

(81) When the ambient field is parallel to the axis of the probe beam, sidebands can be generated in the probe beam only by microwave transitions in which Δm.sub.F=0. (More specifically, only σ.sup.+ or σ.sup.− optical transitions can be excited by the probe when B is parallel to the probe beam. With a Δm.sub.F=1 microwave excitation, a πoptical transition would be required, but that is forbidden by the selection rules.)

(82) By direct optical pumping, we are only able to populate |2, 2> or |2, −2> state and then generate a pair of states that differ by a microwave σ.sup.+ or σ.sup.− transition, i.e., a transition in which the magnetic quantum number m.sub.F changes by ±1.

(83) In the B-parallel mode of operation, we overcome this obstacle by adding an extra step to the process of state preparation. FIGS. 10-12 illustrate the modified process. FIG. 10 shows the orientations of the laser axis, the ambient field B, and the coil field B.sub.C. FIG. 11 is a partial transition diagram showing the spin states that are involved in an example. FIG. 12 is a timing diagram that provides an example of the sequence of the pulses used for state preparation in the B-parallel mode of operation.

(84) With joint reference to FIGS. 11 and 12, it will be seen that the pump laser 1200 is operated so as to populate, e.g., the |2, 2> state in the manner described previously. The pump is then turned off, and the magnetic coil 1205 is operated concurrently with the microwave source 1210 to induce adiabatic rapid passage (ARP) in the atomic ensembles. ARP transfers population from the |2, 2> state to the |1, 1> state. Note that this population transfer can be accomplished using a resonant microwave π-pulse.

(85) ARP is performed by applying a square, first pulse of microwave energy, and during the microwave pulse, applying a magnetic field ramp from the magnetic coil to sweep the energy splitting of the states through resonance with the microwave radiation. After the first microwave pulse ends and the current in the coil is turned off, a second microwave pulse induces a coherent superposition between the |1, 1> and |2, 1> states. A sideband signal 1215 is generated by this coherent superposition.

(86) Our two modes of operation can be used in alternation to determine which mode yields the strongest signal in any given situation.

(87) FIG. 12A is a flowchart illustrating, in an example, the B-parallel mode of operating a gradiometer. At 1251, the host atoms are pumped to the initial end state |2,2>. At 1252, ARP is performed to transfer the host atoms to the state |1,1>. At 1253, the microwave π/2 pulse is applied to produce a coherent superposition of the |1,1> and |2,1> states. At 1254, the probe interrogates the host atoms and the sidebands are generated. At 1255, the sidebands are isolated from the carrier. At 1256, the beat note between the sidebands is obtained. At 1257, the beat frequency is obtained, and at 1258, a measurement of the magnetic field gradient is obtained from the beat frequency.

(88) The example described above uses different light sources for the pump and the probe. However, it is also possible to use the same source laser to provide both the pump light and the probe light. The source laser provides circularly polarized light, which is also used as the “carrier” light for the probe. With a complex state-preparation scheme, it may be possible to achieve a coherent superposition of states that will generate sidebands of opposite circular polarization to the pump so that they can be isolated using a polarization-selective element. However, we expect that such an approach will be sensitive only to magnetic field components that lie along the direction of the source beam. Hence, a more generally applicable approach will isolate the sidebands from the carrier using a wavelength-selective optical element such as an etalon cavity. We believe, in this regard, that an etalon could be designed with a free spectral range that permits transmission of both the incremental and decremental sidebands, which would facilitate temperature-insensitive operation as discussed above.

(89) In an example procedure using a single light source, a strong pumping pulse is produced first, for optical pumping of the atoms. After the pumping interval, the source light is attenuated, but maintained, for a second interval, which we refer to as the probing period. Depending on the experimental conditions, sidebands tend to be generated most efficiently when the probe is detuned from the pump frequency to suppress resonant absorption. Hence, the light source is preferably detuned during the probing period. In examples, detuning is effectuated by a voltage applied to the laser controller, according to known techniques, during the probe phase. The voltage is calibrated to shift the optical frequency by the desired amount of detuning.

(90) FIG. 13 is a flowchart illustrating a procedure for operating a magnetometer that uses a single source laser. At 1301, a selection is made between the B-perpendicular mode and the B-parallel mode. In examples, these two modes are selected in alternation as the procedure is iterated. At 1302, the host atoms are pumped to an end state in accordance with the selected mode. At 1303, the coherent superposition is prepared in accordance with the selected mode. At 1304, the detuned source laser interrogates the host atoms and the sidebands are generated. At 1305, the sidebands are isolated from the carrier. At 1306, the beat note between the sidebands is obtained, and at 1307, a measurement of the magnetic field gradient is obtained from the beat frequency. The procedure may then be iterated until a desired number of iterations has been reached. At 1308, after the last iteration, a best measurement of the magnetic field gradient may be selected.