Matter-wave gravimeter with microwave separation at the magic field

Abstract

The general field of the invention is that of matter-wave gravimeters. The gravimeter according to the invention comprises at least: means for generating, for capturing and for cooling a cloud of ultra-cold atoms; means of transferring the atoms into a superposition, with equal weights, of a first internal electronic state called state |1>) and of a second internal electronic state called state |2> comprising the application of at least a first microwave field and of a radiofrequency field; means for separating the atoms into two wave packets for a given period of time under the effect of at least a second microwave field, the said separation leading to a phase-shift associated with the local gravitational field; calibration means allowing a “magic” magnetostatic field to be determined for which the difference in energy between the first internal electronic state and the second internal electronic state is independent, to a first order, of the fluctuations of the magnetostatic field.

Claims

1. A matter-wave gravimeter using first, second, and third microwave fields for the measurement of a local gravitational field, comprising: an electronic chip comprising a measurement axis, the electronic chip further comprising at least a first central conducting wire and two lateral waveguides disposed symmetrically on either side of the first conducting wire, the first conducting wire having a first current flowing through it, the first lateral waveguide having a first waveguide current flowing therethrough modulated at a first waveguide microwave frequency to generate the second microwave field and the second lateral waveguide having a second waveguide current flowing therethrough modulated at a second waveguide microwave frequency to generate the third microwave field; a vacuum vessel and an assembly of six laser beams combined with a magnetic field gradient generated by coils external to the vacuum vessel configured to capture and for cool a cloud of ultra-cold atoms; the first conducting wire configured to generate a magnetostatic field to thereby trap the cloud of atoms; a first short duration pulse combining the first microwave field and a radiofrequency field generated by the first conducting wire on the electronic chip to thereby transfer the atoms into a superposition with equal weights of a first internal electronic state called state |1> and of a second internal electronic state called state |2>, driving each atom into the resultant state equal to (|1>+|2>)/√2; the second microwave field generated by the first lateral waveguide and/or the third microwave field generated by the second lateral waveguide being configured to separate the atoms into two wave packets during a given period of time under the effect of the second microwave field and/or the third microwave field, the separation into two wave packets resulting in a phase-shift associated with the local gravitational field; the cloud of atoms being positioned, prior to separation, above the first conducting wire, the magnetostatic field generated by the first conducting wire being the “magic” magnetostatic field in which the difference in energy between the first internal electronic state and the second internal electronic state is independent, to a first order, of the fluctuations of the magnetostatic field; the atoms being recombined by the elimination of the applied microwave fields, the phase shift due to the local gravitational field being converted into a difference of population between the internal states by application of a second pulse of short duration combining a first microwave field and a radiofrequency field generated by the first conducting wire on the electronic chip; and an imaging sensor associated with a quasi-resonant laser beam configured to detect the atomic cloud using the technique of imaging by absorption.

2. The matter-wave gravimeter of claim 1 wherein the magnetostatic field oscillating around predetermined values in the presence of the second microwave field and/or the third microwave field, the gravimeter comprising a spectroscopy instrument for measuring the variations of difference in energy between the first internal electronic state and the second internal electronic state, the magic field corresponding to a minimum variation of difference in energy.

3. The matter-wave gravimeter of claim 1 wherein the atoms are rubidium 87 and the first internal electronic state and the second internal electronic state correspond to the two hyperfine levels of rubidium 87 which are denoted {F=2, m.sub.F=+1} and {F=1, m.sub.F=−1}.

4. The matter-wave gravimeter of claim 3 wherein: the level {F=1, m.sub.F=−1} being coupled to the level {F=2, m.sub.F=−1} by the second microwave frequency (ω.sub.1), the level thus obtained being called first cross-coupled level |a>, with this first coupling is associated a first Rabi frequency (Ω.sub.1), proportional to the modulus of the microwave field at the second microwave frequency (ω.sub.1); and the level {F=2, m.sub.F=+1} being coupled to the level {F=1, m.sub.F=+1} by the third microwave frequency (ω.sub.2), the level thus obtained being called second cross-coupled level |b>, with this second coupling is associated a second Rabi frequency (Ω.sub.2), proportional to the modulus of the microwave field at the third microwave frequency (ω.sub.2).

5. The matter-wave gravimeter of claim 3 wherein the magic magnetostatic field is in the range between 3.23 Gauss and 3.55 Gauss, the magic magnetostatic field increasing in a constant manner with the microwave power of the second microwave field and/or of the third microwave field.

6. The matter-wave gravimeter of claim 5 wherein the separation distance of the atoms into two wave packets is substantially equal to 20 microns.

7. The matter-wave gravimeter of claim 4 wherein the first Rabi frequency is equal to the second Rabi frequency.

8. The matter-wave gravimeter of claim 1 wherein the imaging sensor is a CCD camera.

9. The matter-wave gravimeter of claim 1 wherein the imaging sensor is a photodiode.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The invention will be better understood and other advantages will become apparent upon reading the description that follows presented by way of non-limiting example and by virtue of the appended figures, amongst which:

(2) FIG. 1 shows a gravimeter according to the invention during the phase for cooling the ultra-cold atoms;

(3) FIGS. 2, 3 and 4 show the development of the atomic cloud during the phases for creation of the atomic cloud, of superposition of the atomic states and of spatial separation of the said atomic states;

(4) FIG. 5 shows a gravimeter according to the invention during the measurement phase;

(5) FIG. 6 is a simplified representation of the hyperfine levels of rubidium 87;

(6) FIG. 7 shows, in the absence of a microwave field, the variation of the clock states |1> and |2> as a function of the magnetostatic field;

(7) FIG. 8 shows, in the absence of a microwave field, the variation of the difference in energy between the clock states |1> and |2> as a function of the magnetostatic field;

(8) FIG. 9 shows the variation of the magnetic magic field as a function of the normalized microwave power for a set of parameters detailed hereinafter.

DETAILED DESCRIPTION OF THE INVENTION

(9) The architecture of a gravimeter 1 according to the invention is shown on the face view in FIG. 1. The central part is composed of a vacuum vessel 10 all the walls of which are transparent, with the possible exception of the top side 11 which is composed of a chip 12 onto which conducting wires have been deposited.

(10) The atoms 20, initially in the gaseous phase at the ambient temperature in the cell, are trapped and cooled by means of six laser beams 30 disposed symmetrically two by two on three perpendicular axes in pairs combined with a magnetic field gradient generated by external magnetic coils 40. The six laser beams are disposed symmetrically on three perpendicular axes. In FIG. 1, which is a cross-sectional view, only four of the six laser beams are shown represented by arrows, the two missing beams are perpendicular to the plane of the sheet. The assembly of the laser beams and of the magnetic coils is called a three-dimensional magneto-optical trap or “3D MOT”.

(11) A the end of the cooling and trapping phase, the atoms are transferred into a purely magnetic conservative trap created in the neighbourhood of the wires of the chip 12 and prepared in an internal state, for example |1>. A the end of this phase, the atoms are situated at an initial spatial position h above the electronic chip 12.

(12) As can be seen in FIG. 2, the electronic chip 12 comprises at least a first central conducting wire 13 and two lateral waveguides 14 disposed symmetrically on either side of the first conducting wire, the cloud of atoms 20 being situated above the first conducting wire 13, the said first conducting wire having a first current flowing through it and generating a magnetostatic field, the first waveguide having a second current flowing through it modulated at a second microwave frequency generating a second microwave field and the second waveguide having a third current flowing through it modulated at a third microwave frequency, generating a third microwave field. This disposition allows the atomic cloud to be separated and recombined magnetically. The method of separation-recombination is detailed hereinbelow.

(13) In a first step illustrated in FIG. 3, the atoms are transferred into a superposition, with equal weights, of the internal states |1> and |2>, by a pulse of short duration, referred to as π/2 pulse, combining a microwave field and a radiofrequency field generated, for example, by the conducting lines of the chip 12. Each atom is then in a resultant intermediate state denoted (|1>+|2>)/√2.

(14) In a second step illustrated in FIG. 4, the atoms are separated into two wave packets associated with the internal states |1> and |2>, by virtue of a microwave potential MW depending on the internal state. The microwave field used for the separation is generated by the two coplanar waveguides or CPW 14. The separation is for example in the vertical direction so as to be the as sensitive as possible to the local gravitational field. The separation distance s of the atoms is of the order of one or of several tens of micrometers. The separation for a time T.sub.s leads to a phase-shift between the two wave packets associated with the local gravitational field.

(15) In a third step, the atoms are recombined by the elimination of the applied microwave fields. The phase-shift is subsequently converted into a difference of population between the internal states by means of a second “π/2” pulse.

(16) Finally, as shown in FIG. 5, the atomic cloud is detected by using the technique of imaging by absorption which consists in measuring by means of a CCD camera 51 the absorption of a quasi-resonant laser beam 50 by the atomic cloud. Access is thus gained, by optical spectroscopy, to the populations of the two internal states hence to the phase-shift sought. Lastly, the local gravitational field g is calculated.

(17) The magnetic trapping of the neutral atoms is based on the interaction of the magnetic moment p, of a particle with an external magnetic field B(r). The potential energy of the particle is: E.sub.B=−μ.Math.B, and the magnetic moment μ is in rapid precession around B at the Larmor frequency. In a conventional approach, μ may have any given orientation with respect to B(r). In quantum mechanics, the projection of μ onto B(r) can take a set of discrete values given by the quantum number m.sub.F. The potential energy of the atoms in a field B(r), in the limit of low magnetic fields, is then written:
E.sub.F,mf=μ.sub.Bg.sub.Fm.sub.F|B(r)|, with μ.sub.B the Bohr magneton and, g.sub.E the Landé factor corresponding to the angular momentum F.

(18) Since the Maxwell's equations prohibit the existence of a local maximum of magnetic field in vacuum or in a region with no source, only the atoms verifying that the product m.sub.Fg.sub.F is positive may be trapped by a magnetostatic field. FIG. 6 is a simplified representation of the hyperfine levels |F, m> of the fundamental level 5.sup.2S.sub.1/2 of rubidium 87. The clock states |1> and |2> are indicated by circles.

(19) The model described hereinabove indicating a linear variation of the potential energy E.sub.B as a function of the magnetic field B=|B| is only valid to the first order if μ.sub.BB/E.sub.hfs<<1. A more complete description of the energy levels of the hyperfine states (5.sup.2S.sub.1/2) of .sup.87Rb is given by the of Breit-Rabi formula (Equation 2) hereinbelow. The parameters g.sub.J and g.sub.I are the Landé factors respectively corresponding to the nuclear and electron angular momenta.

(20) $\begin{array}{cc}\{\begin{array}{c}{E}_{F=1,m_F}=E_{\mathrm{hfs}}\left(-\frac{1}{8}+\frac{g_I{m}_F}{g_J-g_I}\zeta -\frac{1}{2}\sqrt{1+m_F\zeta +{\zeta }^2}\right)\\ E_{F=2,m_F}=E_{\mathrm{hfs}}\left(-\frac{1}{8}+\frac{g_I{m}_F}{g_J-g_I}\zeta +\frac{1}{2}\sqrt{1+m_F\zeta +{\zeta }^2}\right)\\ \zeta =\frac{{\mu }_B\left(g_J-g_I\right)B}{E_{\mathrm{hfs}}}\end{array}& \left(\mathrm{Equations}2\right)\end{array}$

(21) These energies are shown in FIG. 7 as a function of the parameter ζ linearly dependant on the magnetic field B. In this FIG. 7, the energies are re-dimensioned with respect to the energy E.sub.hfs.

(22) For typical fields of a few Gauss, it is demonstrated that:

(23) $\zeta =\frac{{\mu }_B\left(g_J-g_l\right)}{E_{\mathrm{hfs}}}\approx {10}^{-3}.$
It is therefore legitimate to carry out a limited development of the roots in ξ in the equations 2, which gives, to the second order:

(24) $\sqrt{1+\xi +{\xi }^2}\approx 1+\frac{1}{2}\xi +\frac{3}{8}{\xi }^2$$\mathrm{and}$$\sqrt{1-\xi +{\xi }^2}\approx 1-\frac{1}{2}\xi +\frac{3}{8}{\xi }^2$

(25) The difference in energy between the levels {F=2, m.sub.F=+1} and {F=1, m.sub.F=−1} is therefore written:

(26) $\Delta E=2{\mu }_B{g}_{I}B+E_{\mathrm{hfs}}+\frac{3E_{\mathrm{hfs}}}{8}{\xi }^2=2{\mu }_B{g}_{I}B+E_{\mathrm{hfs}}+\frac{3{{\mu }_B^2\left(g_J-g_I\right)}^2}{8E_{\mathrm{hfs}}}{B}^2$

(27) This difference in energy ΔE between the clock states |1> and |2> is shown as a function of the magnetic field B in FIG. 8. The expression is seen to take the form of a parabolic arc.

(28) This expression reaches a minimum when its derivative with respect to B is zero. This minimum is equal to:

(29) $B_0=\frac{-8{\mu }_B{g}_l{E}_{\mathrm{hfs}}}{3{{\mu }_B^2\left(g_j-g_l\right)}^2}\approx 3.229\mathrm{Gauss}$

(30) When the magnetostatic field has this value of 3.229 G, the variation of ΔE as a function of B becomes zero to the first order. Experimentally, this “magic” point is very advantageous since it allows the fluctuations of the DC magnetic field to be avoided, which potentially increases the coherence time by several orders of magnitude, as demonstrated in the literature.

(31) In the presence of the DC field at the magic point, the MW field allows the state |1> to be coupled with other internal states of .sup.87Rb, which allows the potential of the atoms in the state |1> to be spatially modified and allows them to be spatially separated from atoms in the state |2>. Reference will be made to the article P. Böhi et al., Nature Physics 5, 592-597 (2009) on this point. The internal states |1> and |2> and the associated energies are modified under the effect of the couplings induced by the MW fields. This is sometimes referred to in the literature as “contamination” of the states |1> and |2> by other internal states. One of the effects of this contamination is to modify the magic field condition described previously.

(32) The object of the invention is to determine a new magic field in the presence of microwave fields so as to conserve the independence from the fluctuations of the DC magnetic field, characteristic of the magic field.

(33) The principle for implementing an atomic interferometer according to the invention is to modify the energies of the two levels used by coupling them to two other hyperfine levels by virtue of two microwave frequencies. More precisely, the level {F=2, m.sub.F=+1} will be coupled to the level {F=1, m.sub.F=+1} by the microwave frequency ω.sub.2, and the level {F=1, m.sub.F=−1} will be coupled to the level {F=2, m.sub.F=−1} by the microwave frequency ω.sub.1, with the energy levels known as Breit-Rabi levels depending on where the origin of the energies is taken half way between the levels {F=1, m.sub.F=0} and {F=2,m.sub.F=0}):

(34) Energy of the level {F=2, m.sub.F=+1}:

(35) $E_{F=2,mF=+1}={\mu }_B{g}_{I}B+\frac{E}{2}\sqrt{1+\xi +{\xi }^2}$

(36) Energy of the level {F=1, m.sub.F=+1}:

(37) $E_{F=1,mF=+1}={\mu }_B{g}_{l}B-\frac{E}{2}\sqrt{1+\xi +{\xi }^2}$

(38) Energy of the level {F=1, m.sub.F=−1}:

(39) $E_{F=1,mF=-1}=-{\mu }_B{g}_{I}B-\frac{E}{2}\sqrt{1-\xi +{\xi }^2}$

(40) Energy of the level {F=2, m.sub.F=−1}:

(41) 0$E_{F=2,mF=-1}=-{\mu }_B{g}_{l}B+\frac{E}{2}\sqrt{1-\xi +{\xi }^2}$

(42) The level coming from the coupling between {F=2, m.sub.F=+1} and {F=1, m.sub.F=+1} by the microwave frequency ω.sub.2 is denoted as “cross-coupled” level |b>. It is assumed that, at the start of the interferometric sequence, the microwave power is zero and that |a> is then in the state {F=2, m.sub.F=+1}. δ.sub.2=±1 denotes the sign of the de-tuning between the frequency ω.sub.2 and the energy of the atomic transition from the level {F=2, m.sub.F=+1} to the level {F=1, m.sub.F=+1} at the start of the interferometric sequence. Typically, the value of the constant magnetic field is then equal to the magic field without microwave previously calculated, being around 3.23 G. Under the hypothesis of adiabatic development of the cross-coupled state, that is assumed to be verified in the framework of the invention, the energy of the state |b> is then:

(43) $E_{|b>}={\mu }_B{g}_{I}B-{\delta }_2\frac{\hslash }{2}\sqrt{{\Omega }_2^2+{\left[{\omega }_2-\frac{E_{\mathrm{hfs}}}{\hslash }\left(1+\frac{\xi }{2}+\frac{3}{8}{\xi }^2\right)\right]}^2}$

(44) where Ω.sub.2 is the Rabi frequency, proportional to the modulus of the microwave field at the frequency ω.sub.2.

(45) Similarly, the energy of the “cross-coupled” level |a>, coming from the coupling between {F=1, m.sub.F=−1} and {F=2, m.sub.F=−1} by the microwave frequency ω.sub.1 is given by:

(46) $E_{|a>}=-{\mu }_B{g}_{I}B+{\delta }_1\frac{\hslash }{2}\sqrt{{\Omega }_1^2+{\left[{\omega }_1-\frac{E_{\mathrm{hfs}}}{\hslash }\left(1-\frac{\xi }{2}+\frac{3}{8}{\xi }^2\right)\right]}^2}$

(47) where it has been assumed that, at the start of the interferometric sequence, |b> is in the state {F=1, m.sub.F=−1} and the frequency ω.sub.1 is out of tune with the energy of the atomic transition {F=1, m.sub.F=−1} to {F=2, m.sub.F=−1} whose sign is given by δ.sub.1=±1. The assumption of adiabatic development is also made, and Ω.sub.1 is defined as the Rabi frequency, proportional to the modulus of the microwave field at the frequency ω.sub.1.

(48) In the framework of the gravimeter according to the invention, the signs of δ.sub.1 and δ.sub.2 are necessarily opposite, in order to create microwave potentials of the same nature for the two cross-coupled levels, either attractive or repulsive.

(49) The method according to the invention consists therefore in choosing a value of magnetic field B.sub.m such that the curve E.sub.|b>(B)−E.sub.|a>(B) goes through a minimum at this point. The magnetic field fluctuations then have no effect to a first order on this difference in energy.

(50) By way of example, FIG. 9 shows the variation of the magic field B as a function of the parameter κ representative of the normalized microwave power. More precisely, these curves have been obtained for the following parameters:
δ.sub.1=−δ.sub.2=+1
ℏω.sub.1=E.sub.F=2,mF=−1(B.sub.0)−E.sub.F=1,mF=−1(B.sub.0)+ℏΔ.sub.1
ℏω.sub.2=E.sub.F=2,mF=+1(B.sub.0)−E.sub.F=2,mF=−1(B.sub.0)+ℏΔ.sub.2
Δ.sub.1=−Δ.sub.2=0.1μ.sub.BB.sub.0/2
Ω.sub.1=Ω.sub.2=Ω

(51) The normalized microwave power or level of contamination is defined by K=|Ω/Δ.sub.1| and varies between 0 and 1 in FIG. 9.

(52) As has been seen, it is possible, with a knowledge of all the parameters of the gravimeter, to determine by calculation the value of the magic field. However, it will be understood that certain parameters are difficult to define with a high precision. Accordingly, the gravimeter according to the invention comprises calibration means allowing a “magic” magnetostatic field to be determined for which the difference in energy between the first internal electronic state and the second internal electronic state is independent, to a first order, of the fluctuations of the magnetostatic field, the calibration means comprising firsts means allowing the magnetostatic field to be made to oscillate around predetermined values in the presence of the microwave fields and seconds means for measuring the variations of difference in energy between the first internal electronic state and the second internal electronic state, the magic field corresponding to the minimum variation of difference in energy.

(53) Based on the theoretical model described previously, the experimenter finds experimentally the optimum value of the magic field and thus calibrates the magnetostatic field. The values of B.sub.m as a function of K can also be determined. For this purpose, the experimenter performs the Ramsey spectroscopy of the two-photon transition between the states |1> and |2>, by measuring with precision the resonance frequency ν.sub.12, in a first instance without microwave field in order to determined B.sub.0, and then in the presence of a microwave field in order to estimate experimentally B.sub.m(K).