Planar, strong magnetic field source for a chip ion trap

No non-linear effects have been observed around the maximum operation currents 10 A-12 A.

“However, no such non-linear effects have been observed around the maximum operation currents 10 A-12 A.”

The magnetic field source has remained undamaged after multiple quenches as a result of the active and passive quench protection measures.

“We have experienced multiple quenches, but despite the aforementioned limitations, the magnetic field source has remained undamaged as a result of the active and passive quench protection measures.”

The authors fabricated a magnetic field source with four pairs of concentric current loops implemented with NbTi superconducting wires.

“We have fabricated a magnetic field source, as introduced in Fig. 1, with a set of four pairs of concentric current loops implemented with NbTi superconducting wires.”

In the source, for every n-th loop in one section, its current runs in the opposite sense to that of the same current in the mirror-symmetric loop of the other section.

“For every single n-th loop in one section, its current In runs in opposite sense-clockwise/anticlockwise or vice versa-to that the same current in the mirror-symmetric loop of the other section.”

The devised magnetic field source consists of two symmetric sections, each with four coplanar and concentric rectangular closed loops of current.

“It consists of two symmetric sections, each with four coplanar and concentric rectangular closed loops of current.”

The effective cross-sectional areas A_n effective obtained from the fits are in very good agreement with those expected from the dimensions of the source and the number of turns of NbTi wire used.

“The values obtained for A n effective are in very good agreement with the ones expected from the dimensions of the source and the number of turns of NbTi wire used.33”

• cites [33] J. H. Lacy, “Development of a planar magnetic field source for the geonium chip Ppnning trap,” Ph.D. thesis, University of Sussex, Falmer, UK, 2019.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• AssumptionThe authors state this as an assumption: The theoretical formulas of Eq. (12) assume a homogeneous current density distribution across the normal section of the source, J = I/A.
• AssumptionThe authors state this as an assumption: The Meissner effect can be neglected for fits at distances ≥1.5 mm above the ∼100 μm thin superconducting wires.

The widths a0, a1, a2, and a3 were optimized, within fixed Lx and Lz, for minimizing the amplitude of the currents required for a homogeneous field at y0 = 1.6 mm.

“The values of a 0 , a 1 , a 2 , and a 3 have been optimized (within the constraint of given fixed Lx and Lz) for minimizing the amplitude of the currents required for an homogeneous field at y 0 = 1.6 mm (measured from the magnetic field source's surface).”

The fits indicated an average tilt angle between the vertical axes of the source and calibration board of (6.6 ± 1.8)°.

“The fits indicated an average tilt angle between both vertical axes of the source and calibration board of (6.6 ± 1.8) ○ .”

• AssumptionThe authors state this as an assumption: The theoretical formulas of Eq. (12) assume a homogeneous current density distribution across the normal section of the source, J = I/A.
• AssumptionThe authors state this as an assumption: The Meissner effect can be neglected for fits at distances ≥1.5 mm above the ∼100 μm thin superconducting wires.
• AssumptionThe authors state this as an assumption: In the rest of the article, a fixed, effective tilt angle is assumed for currents I0, I1, and I3 and zero for I2.

We have demonstrated fields of up to 0.5 T with our basic planar magnetic field source.

“We have demonstrated fields of up to 0.5 T with our basic planar magnetic field source.”

The calibration matrix Γ at y0 = 1.45 mm above the geonium chip surface has the values given in Eq. (13).

“As a concrete example, we have obtained Γ at y 0 = 1.45 mm above the geonium chip surface [see Fig. 8(c)].The result is given in the following equation:”

• AssumptionThe authors state this as an assumption: The theoretical formulas of Eq. (12) assume a homogeneous current density distribution across the normal section of the source, J = I/A.
• AssumptionThe authors state this as an assumption: The Meissner effect can be neglected for fits at distances ≥1.5 mm above the ∼100 μm thin superconducting wires.
• AssumptionThe authors state this as an assumption: In the rest of the article, a fixed, effective tilt angle is assumed for currents I0, I1, and I3 and zero for I2.

Fitting the datasets of Fig. 10 with a polynomial gives significantly less accurate results than using Eq. (12).

“However, when fitting the datasets of Fig. 10 with a polynomial, the results are significantly less accurate than when using Eq. ( 12).”

The fitted curve of Fig. 15 shows that all experimental data smoothly follow the fitted polynomial.

“The fitted curve of Fig. 15 shows that all experimental data smoothly follow the fitted polynomial.”

For magnetic fields as strong as in Fig. 13, sensors 2 and 5 appear to suffer a small offset with respect to the others.

“As will be explained in Sec.IV F, for magnetic fields as strong as in Fig. 13, those two sensors appear to suffer a small offset with respect to the others.”

The acquired difference coefficients ΔBz,0, ΔBz,010, ΔBz,020, ΔBz,030 fully agree with the results obtained from subtracting the values of Eq. (17) minus the values of Eq. (18).

“The second information gained from the polynomial fit of Fig. 15 is that the acquired difference coefficients, ΔBz ,0 , ΔBz ,010 , ΔBz ,020 , ΔBz ,030 , fully agree with the results obtained from subtracting the values of Eq. ( 17) minus the values of Eq. ( 18).”

The set of currents for a homogeneous 0.5 T field at y0 = 1.45 mm is given by Eq. (15).

“Using Eq. ( 10), we can now determine the set of currents for a homogeneous 0.5 T field at y 0 = 1.45 mm as follows:”

• AssumptionThe authors state this as an assumption: The theoretical formulas of Eq. (12) assume a homogeneous current density distribution across the normal section of the source, J = I/A.
• AssumptionThe authors state this as an assumption: The Meissner effect can be neglected for fits at distances ≥1.5 mm above the ∼100 μm thin superconducting wires.
• AssumptionThe authors state this as an assumption: In the rest of the article, a fixed, effective tilt angle is assumed for currents I0, I1, and I3 and zero for I2.

When the experimental datasets of Fig. 13(a) and Fig. 14(a) are subtracted, the scattering of the data around the fitted curve is greatly reduced.

“The first observation is that when subtracted, the scattering of the data around the fitted curve-clearly visible in both figures [Figs.13(a) and 14(a)]-is greatly reduced.”

The source is close to the critical current of the NbTi wire used.

“The source is close to the critical current of the NbTi wire used.”

Comparing the currents in Eq. (16) to those in Eq. (15), the largest current of the set has dropped by a factor of two or more.

“Comparing the currents in Eq. ( 16) to those in Eq. ( 15), it is clear that the largest current of the set has dropped by a factor of two or more.”

For target Breduced = (Bz,0 = 0.3 T, Bz,010 = 0, Bz,002 = 0), the measured residual inhomogeneities are given by Eq. (18).

“The measured residual inhomogeneities are given in the following equation:”

• AssumptionThe authors state this as an assumption: The theoretical formulas of Eq. (12) assume a homogeneous current density distribution across the normal section of the source, J = I/A.
• AssumptionThe authors state this as an assumption: The Meissner effect can be neglected for fits at distances ≥1.5 mm above the ∼100 μm thin superconducting wires.
• AssumptionThe authors state this as an assumption: In the rest of the article, a fixed, effective tilt angle is assumed for currents I0, I1, and I3 and zero for I2.

In the example of Fig. 16, the residual gradient and curvature are much higher than those of the examples of Figs. 13 and 14.

“Clearly, the residual gradient and curvature in that example are much higher than those of the examples of Figs. 13 and 14.”

To address the five quantities including Bz,002, the source would need to be expanded with one extra pair of current loops.

“In order to address this new set, containing now five quantities, we would need to expand the source of Fig. 1 with one extra pair of current loops.”

The position y0 = 1.6 mm corresponds to the optimal trapping position of the geonium chip.

“This y 0 corresponds to the optimal trapping position 16 of the geonium chip 17”

• cites [16] 2013, Int. J. Mass Spectrom. sourceNot checked — no full text of the source
• cites [17] J. Pinder, “The geonium chip: Engineering a scalable planar penning trap,” Ph.D. thesis, University of Sussex, Falmer, UK, 2017.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

For a homogeneous field beyond the long Lx approximation, Bz,020 and Bz,002 must be canceled separately.

“Therefore, for a homogeneous field, they must be canceled separately.”

A fiducial cross cut patterned on the PCB with a CNC machine allows for a positioning accuracy of ±100 μm.

“Furthermore, a fiducial cross cut has been patterned on the PCB with a computer numerical control (CNC) machine, 34 allowing for a positioning accuracy of ±100 μm.”

• cites [34] 2016Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The source is designed for a Penning trap experiment that is very sensitive to radio frequency noise.

“The source is designed for a Penning trap experiment, which is very sensitive to radio frequency noise.27,28”

• cites [27] 2012, Appl. Phys. B sourceNot checked — no full text of the source
• cites [28] 2013, Nucl. Instrum. Methods Phys. Res., Sect. A sourceNot checked — no full text of the source

The calibration matrix Γ(y0) is different when measured at 4 K or at 300 K.

“This together with the Meissner-effect result in the calibration matrix, Γ(y 0 ), being different when measured at 4 K or at 300 K.”

Within the long Lx approximation, creating a homogeneous magnetic field requires canceling only the axial inhomogeneities Bz,010 = Bz,020 = Bz,030 = ... = 0.

“Therefore, for the source of Fig. 1, in principle, creating a homogeneous magnetic field requires canceling only the inhomogeneities of the axial component Bz, merely along the vertical axis ûy → B z,010 = B z,020 = B z,030 = ⋯ = 0.”

• AssumptionThe authors state this as an assumption: Within a small volume around (0, y0, 0), the axial component of the vector potential can be neglected and Ax is invariant under x, i.e., the long Lx approximation holds.

The Hall sensors have been individually calibrated for cryogenic use at 4 K.

“They have been individually calibrated for cryogenic use at 4 K. 32”

• cites [32]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Outside the rectangular casings, incoming and outgoing wires are bundled in twisted pairs of counter-propagating currents such that the undesired magnetic field they create is canceled or strongly reduced.

“Outside the rectangular casings, any incoming and outgoing wires are bundled in twisted pairs of counter-propagating currents (input-output) such that the undesired magnetic field they create is canceled or strongly reduced.”

Beyond the long Lx approximation, compensating all inhomogeneities up to and including A4 in the series expansion of the vector potential requires ten independent current loops.

“Beyond that approximation, compensating all inhomogeneities up to (and including) ⃗ A 4 in the series expansion of the vector potential will require the use of ten independent current loops.22,23”

• cites [22] 2013, Ion trapNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [23] 2013, Ion trapNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The applied currents and the magnetic coefficients are univocally linked by a system of linear equations through the matrix Γ.

“The applied currents (I 0 , I 1 , I 2 , I 3 ) and the magnetic coefficients (Bz ,0 , Bz ,010 , Bz ,020 , Bz ,030 ) are univocally linked by a system of linear equations, 22,23”

• cites [22] 2013, Ion trapNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [23] 2013, Ion trapNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Atomic physics measurements with trapped ions at the ppb level of precision show that magnetic inhomogeneities up to Bz,004 might play a significant role at that level of accuracy.

“Atomic physics measurements with trapped ions at the ppb level of precision show that magnetic inhomogeneities up to Bz ,004 might play a significant role at that level of accuracy.25”

• cites [25] 2004, Phys. Rev. Lett. sourceNot checked — no full text of the source

Γ′ can be determined by measuring the magnetic field only along the y direction.

“However, Γ ′ can be determined by measuring the field only along ûy, as described in detail in Sec.IV B.”

It is necessary to filter the currents to prevent noise from coupling into the trapped particles' detection circuit.

“It is therefore necessary to filter the currents to prevent the noise from coupling into the trapped particles' detection circuit.”

With the source configuration, for any position along the vertical y axis, the resulting magnetic field is purely axial along z.

“Hence, with this configuration, for any position located along the vertical ûy axis, (0, y 0 , 0), the resulting magnetic field is purely”

The perceptible reduction of the data scattering strongly suggests that at high fields, sensors 2 and 5 are subject to some offset with respect to the others.

“The perceptible reduction of the data scattering strongly suggests that at high fields, sensors 2 and 5 are subject to some offset with respect to the others, as mentioned in Sec.IV E.”

The use of the theoretical functions of Eq. (12) does not bias the results obtained.

“The use of the theoretical functions of Eq. ( 12) does not bias the results obtained.”

The field strength is limited by the amount of current density achievable within the wires.

“The field strength is limited by the amount of current density achievable within the wires.”

It is more probable that the NbTi wire wrapped around the rectangular spools suffered a slight inclination with respect to the spools during the winding procedure than that there is a tilt angle between the source and the calibration board.

“However, rather than a tilt angle between the source and the calibration board, it is more probable that the NbTi wire wrapped around the rectangular spools of Fig. 3 suffered a slight inclination with respect to the spools during the winding procedure.17”

• cites [17] J. Pinder, “The geonium chip: Engineering a scalable planar penning trap,” Ph.D. thesis, University of Sussex, Falmer, UK, 2017.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

It is not possible to distinguish which sensors are the right ones.

“However, it is not possible to distinguish which sensors are the”

The current optimization has the downside of leaving Bz,030 (or Bz,020) floating.

“The current optimization has the downside of leaving Bz ,030 (or Bz ,020 ) floating.”

The small amount of sensors and their spacing of 0.5 mm cap the accuracy of the calibration.

“The small amount of sensors and their spacing of 0.5 mm cap the accuracy of the calibration discussed in Sec.IVC.”

The source provides a field of similar or even better spatial homogeneity than these prominent examples of Penning trap technology.

“Both these are prominent examples of the great accuracy of Penning trap technology, and our source provides a field of similar or even better spatial homogeneity.”

The Γ matrix obtained by polynomial fitting is inevitably less accurate.

“As a result, the so obtained Γ matrix is inevitably less accurate.”

For an infinitely long Lx source, the third and fourth rows of Γ' should be identical but with opposite signs.

“As discussed in Sec.II B 2, for an infinitely long Lx source, the third and fourth rows of Γ ′ should be identical, but with opposite signs.”

The coincidence of both results strongly supports the procedure for measuring Γ, Γ', and the residual inhomogeneities.

“Hence, the coincidence of both results strongly supports the procedure for measuring Γ, Γ ′ and the residual inhomogeneities discussed in Secs.IV B and IV E, respectively.”

The measured residual value of Bz,030 has a negligible effect on the trapped electron's motional eigenfrequencies.

“However, as discussed in Sec.IV E, the measured residual value of Bz ,030 has a negligible effect on the trapped electron's motional eigenfrequencies.16”

• cites [16] 2013, Int. J. Mass Spectrom. sourceNot checked — no full text of the source

The examples of compensated fields presented in Figs. 13 and 14 demonstrate the ability of the magnetic field source to provide a magnetic field tailored to the needs of a cryogenic ion trap with a single electron/ion.

“The examples of compensated fields presented in Figs. 13 and 14 demonstrate the ability of the magnetic field source to provide a magnetic field tailored to the needs of a cryogenic ion trap with a single electron/ion.”