# Patent

## Title
Ion trap

## Abstract

An ion trap 1 comprises a magnetic field generator 2 arranged to generate a magnetic field and an array 3 of electrodes arranged to generate an electrostatic field including a turning point in electrical potential at a location where the magnetic field is substantially homogeneous. The array of electrodes 3 is planar and parallel to the direction of the magnetic field at the location, with the result that the ion trap can be described as a coplanar waveguide Penning trap.

## Background
FIELD OF THE DISCLOSURE
The disclosure relates to an ion trap, and to a mass spectrometer and to a quantum electrodynamic circuit incorporating the ion trap.
BACKGROUND TO THE DISCLOSURE
An ion trap is an apparatus used to confine or isolate a charged particle, such as an electron. One such apparatus is known as a Penning trap. A Penning trap uses a magnetic field and an electrostatic field together to trap a charged particle. The magnetic field causes the charged particle to perform a rotational movement with the direction of the magnetic field being the axis of the rotation. This effectively confines the charged particle to a plane normal to the direction of the magnetic field. The electrostatic field is arranged to confine the charged particle at a location along the direction of the magnetic field, by providing a potential well for the particle at the desired location.
In order for a Penning trap to confine a charged particle effectively and to be useful for performing measurements on the trapped particle, it is important both for the magnetic field to be homogeneous and for the electrostatic field to be harmonic at the location the charged particle is to be trapped. This places constraints in the design of a conventional Penning trap and its variants, generally making them complex and expensive.
SUMMARY OF THE DISCLOSURE
According to a first aspect of the disclosure, there is provided an ion trap comprising:
a magnetic field generator arranged to generate a magnetic field; and
an array of electrodes arranged to generate an electrostatic field including a turning point in electrical potential at a location where the magnetic field is substantially homogeneous,
wherein the array of electrodes is planar and parallel to the direction of the magnetic field at the location.
By arranging the array of electrodes to be planar, the electrodes can be fabricated much more easily than in a conventional Penning trap, in which the electrodes are typically arranged as cylinders or hyperboloids of revolution. Furthermore, by arranging the array of electrodes parallel to the direction of the magnetic field, desired variation in the electrical potential along the direction of the magnetic field can be provided straightforwardly and reliably. In particular, the turning point at the location can be well defined. The combination of a well defined turning point and a homogeneous magnetic field at the location can make the ion trap useful for trapping ions. Indeed, it will be appreciated that, in use, ions can be trapped at the location.
It is possible for the electrodes of the array to be at different heights within the array or to have contoured surfaces. However, it is preferred that the electrodes of the array each have surfaces facing the location where the magnetic field is substantially homogeneous and that these surfaces are substantially coplanar.
The array of electrodes may comprise a row of three or more electrodes, which row is arranged to be parallel to the direction of the magnetic field at the location where the magnetic field is substantially homogeneous. Typically, the row comprises five electrodes.
Usually, the lengths of the electrodes along the direction of the row are such that an electrode in the middle of the row is shortest and electodes at the ends of the row are longest. This facilitates the lines of equipotential in the electric field to be hyperbolic along the length of the row above an electrode in the middle of the row.
The electrode array may comprise a guard electrode on each side of the row. Typically, the guard electrodes are coupled to ground, but it is also possible for an electric potential to be applied to the guard electrodes. The guard electrodes can help to ensure that the electric field has a well defined turning point above the electrode in the middle of the row in a direction across the width of the row. Moreover, the guard electrodes can help to shape the electric field such the an ion finds an equilibrium position in the electric filed at a non-zero height above the electrode in the middle of the row.
The guard electrodes can overlap the electrodes of the row. This can reduce the effect of gaps between the electrodes on the shape of the electric field, as insulation between the electrodes can be provided primarily normal to the plane of the electrode array.
Helpfully, the array of electrodes may be provided on a substrate and the magnetic field generator may be provided on the same substrate. Indeed, according to a second aspect of the disclosure there is provided an ion trap comprising:
a substrate;
a magnetic field generator provided on the substrate and arranged to generate a magnetic field; and
an array of electrodes provided on the same substrate and arranged to generate an electrostatic field including a turning point in electrical potential at a location where the magnetic field is substantially homogeneous,
wherein the array of electrodes is planar and parallel to the direction of the magnetic field at the location.
This significantly simplifies fabrication of the ion trap, by allowing the magnetic field generator to be provided close to the electrode array. This can help to eliminate the need for expensive superconducting coils to generate a sufficiently strong magnetic field from a remote location, as in conventional Penning traps, as the magnetic field close to the magnetic field generator can be utilised. It can also mean that the ion trap can be manufactured as a small scale integrated electronic device, for example using thin or thick film fabrication techniques.
The magnetic field generator may comprise a row of magnetic elements, which row extends in the same direction as the row of electrodes. This can help to ensure that the magnetic field is sufficiently homogeneous at the desired location, as the different magnetic elements in the row can compensate for one another.
In one example, the magnetic elements each comprise a wire arranged to conduct an electric current. Wires made of conducting material can be sufficient for many applications, but the wires may alternatively be superconducting wires. In another example, the magnetic elements each comprise a permanent magnet. Again, the permanent magnets may be high temperature superconductors.
A primary magnetic element may arranged to generate a first component of the magnetic field and the other magnetic elements may be arranged to generate compensating components of the magnetic field that reduce the gradient and curvature of the first component of the magnetic field at the location where the magnetic field is substantially homogeneous. The compensating components of the magnetic field can be selected by choosing appropriate dimensions for the magnetic elements and, where the magnetic elements are wires, choosing appropriate electric currents in the wires or, where the magnetic elements are permanent magnets, choosing the strength of the magnets appropriately.
The electrode array may be provided on a top surface of the substrate and the magnetic field generator may be provided below the electrode array.
The ion trap has a number of useful applications. In particular, there is provided a mass spectrometer comprising the ion trap. There is also provided a quantum electrodynamic circuit comprising the ion trap. In the context of an electrodynamic circuit, the ion trap may be arranged to store information based on spin of an electron captured in the trap or based on cyclotron motion of an electron captured in the trap. Alternatively, the ion trap may find application as a quantum processor or matter-wave beam-splitter.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a schematic illustration of an ion trap according to a first preferred embodiment.
FIG. 2 is an exploded schematic illustration of the ion trap omitting a substrate.
FIG. 3 is a graphical illustration of electrostatic potential along the length of a row of electrodes of the ion trap.
FIG. 4 is a graphical illustration of electrostatic potential with height above the centre of an electrode in the middle of the row of electrodes of the ion trap.
FIG. 5 is a graphical illustration of a magnetron ellipse of an ion captured in the ion trap.
FIG. 6 is a graphical illustration of an optimal tuning ratio for the ion trap.
FIG. 7 is a graphical illustration of an optimal tuning ratio of the ion trap as a function of a trapping height y0.
FIG. 8 is a graphical illustration of variation of the trapping height y0 as a function of voltage ratios Tc, Te.
FIG. 9 is a graphical illustration of anharmonicity C012 along a compensated path of the ion trap.
FIG. 10 is a graphical illustration of quadratic axial frequency shifts produced by anharmonicities C006, C014.
FIG. 11 is a graphical illustration of a shift produced by a vertical anharmonicity C008.
FIG. 12 is a graphical illustration of optimal trapping positions y0012 and y0016 as a function of a length lc of compensation electrodes of the ion trap.
FIG. 13 is a graphical illustration of an optimal tuning ratio of the ion trap as a function of the length lc of compensation electrodes.
FIG. 14 is a graphical illustration of axial dip of a single electron trapped in the ion trap for three different temperatures of the electron.
FIG. 15 is another graphical illustration of axial dip of a single electron trapped in the ion trap for three different temperatures of the electron.
FIG. 16 is a schematic illustration of a mass spectrometer incorporating the ion trap.
FIG. 17 is a schematic illustration of a waveguide incorporating the ion trap.
FIG. 18 is a schematic illustration of the waveguide coupled to an external superconducting microwave cavity.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
Referring to FIGS. 1 and 2, an ion trap 1 comprises a magnetic field generator 2 and an electrode array 3 provided on a substrate 4. The ion trap 1 is effectively a variant of a conventional Penning trap, and can be referred to as a Penning trap in the context of the general principles of its operation, although not in terms of its specific structure, which differs significantly. The term “coplanar waveguide Penning trap” has been coined to describe the ion trap 1.
It should be noted that the ion trap 1 is described in the context of a trap for negatively charged ions, in particular an electron, for simplicity. However, the skilled person will recognise that the trap can equally be used for trapping positively charged ions by reversing the polarity of the electrode array 3.
In this embodiment, the substrate 4 is a dielectric material suitable for forming a microwave transmission line, for example for microwaves in the range 3 to 30 GHz. Suitable materials are, for example, sapphire or quartz.
The electrode array 3 comprises a ring electrode 6, two compensation electrodes 7, 8 and two end cap electrodes 9, 10 arranged in a row 5. The ring electrode 6 is in the middle of the row 5 and the end cap electrodes 9, 10 are at the ends of the row 5. The compensation electrodes 7, 8 are each between a respective end cap electrode 9, 10 and the ring electrode 6. In other embodiments, the compensation electrodes 9, 10 are omitted and the row 5 includes only the ring electrode 6 and end cap electrodes 9, 10. On each side of the row 5 is a guard electrode 11, 12. Wires (not shown) are provided so that electric potential can be applied to the electrodes 6, 7, 8, 9, 10, 11, 12. In this embodiment, the two compensation electrodes 7, 8 are electrically coupled to one another by the wires, the two end cap electrodes 9, 10 are electrically coupled to one another by the wires and the two guard electrodes 11, 12 are electrically coupled to one another by the wires so that respective electric potentials Vc, Ve, Vg can be applied to the pairs of compensation electrodes 7, 8, end cap electrodes 9, 10 and guard electrodes 11, 12.
The electrodes 6, 7, 8, 9, 10, 11, 12 can be made of any conducting material, such as gold or copper. Alternatively, they may be superconducting. Typically a conducting material is sufficient when the ion trap 1 is used as a mass spectrometer, whereas a superconducting material is appropriate when the ion trap is used in quantum electrodynamic circuit (circuit-QED) applications.
The electrode array 3 has a length lz in the direction of the row 5 of electrodes 6, 7, 8, 9, 10 and a width a0. Within the length lz of the electrode array 3, the ring electrode 6 has a length ln the compensation electrodes 7, 8 each have length lc and the end cap electrodes 9, 10 each have length le. In other words, the lengths lc of the compensation electrodes 7, 8 are the same, and the lengths le of the end cap electrodes 9, 10 are the same. The ring electrode 6, compensation electrodes 7, 8 and end cap electrodes 9, 10 each also have the same width S0. Typically, the length lr of the ring electrode 6 is greater than or equal to 0.1 mm, the length lc of the compensation electrodes 7, 8 is less than or equal to 2 mm and the length of the end cap electrodes 9, 10 is between 0.5 mm and 10 mm inclusive. The electrodes 6, 7, 8, 9, 10 in the row 5 are each spaced apart from one another by gaps. In this embodiment, the gaps all have the same length q. The electrodes 6, 7, 8, 9, 10 are arranged in the row 5 such that the row 5 is symmetrical about an imaginary line that bisects its length. In other words, the row 5 is symmetrical lengthwise about a centre line of the ring electrode 6.
The electrode array 3 is planar, in the sense that the electrodes 6, 7, 8, 9, 10, 11, 12 of the electrode array 3 are all arranged side by side facing in the same direction. In this embodiment, the electrodes 6, 7, 8, 9, 10, 11, 12 of the electrode array 3 are provided on a top surface of the substrate 4 and themselves each have top surfaces that are in the same plane. In other words, the top surfaces of the electrodes 6, 7, 8, 9, 10, 11, 12 are coplanar. However, in other embodiments, the top surfaces of the electrodes 6, 7, 8, 9, 10, 11, 12 are at different heights or are contoured, whilst the electrode array 3 still remains planar overall.
The magnetic field generator 2 comprises a row 13 of a primary magnetic element 14 and four shim magnetic elements 15, 16, 17, 18. The primary magnetic element 14 is in the middle of the row 13 and the shim magnetic elements 15, 16, 17, 18 are positioned symmetrically on each side of the primary magnetic element 14. The primary magnetic element 14 has a length lp in the direction of the row 13, the shim magnetic elements 15, 16 adjacent the primary magnetic element 14 each have length ls1 in the direction of the row 13 and the shim magnetic elements 17, 18 at the ends of the row 13 each have length ls2 in the direction of the row 13. The primary magnetic element 14 is spaced apart from each of the shim magnetic elements 15, 16 adjacent the primary magnetic element 14 by gaps having length lg1 and the shim magnetic elements 15, 16 adjacent the primary magnetic element 14 are spaced apart from the shim magnetic elements 17, 18 at the ends of the row 13 by gaps having length lg2.
In this embodiment, the magnetic elements 14, 15, 16, 17, 18 are each superconducting wires coupled so as to convey an electric current perpendicular to the length of the row 13 and parallel to the plane of the electrode array 3. In other embodiments, the magnetic elements 14, 15, 16, 17, 18 are each conducting wires, permanent magnets or high temperature superconducting permanent magnets. In any case, the magnetic elements 14, 15, 16, 17, 18 are arranged to generate a magnetic field that has a direction substantially parallel to the row 5 of electrodes 6, 7, 8, 9, 10 and that is substantially homogeneous at a location above the ring electrode 6.
Ignoring the outer edges of the electrode array 3 and instead assuming that the outer edges extend to infinity, for simplicity, the electric field generated by the electrode array 3 can be expressed as

φ(x,y,z)=Vr×fr(x,y,z)+Vc×fc(x,y,z)+Ve×fe(x,y,z)+fgaps(x,y,z|Vr,Vc,Ve)  (1)

where Vr, Vc and Ve represent DC voltages applied to the ring electrode 6, the compensation electrodes 7, 8 and the end cap electrodes 9, 10 respectively. The functions fr, fc and fe represent the contribution to the electrostatic field of the ring electrode 6, the compensation electrodes 7, 8 and the end cap electrodes 9, 10 respectively. These functions fr, fc and fe depend only on the dimensions of the electrodes 6, 7, 8, 9, 10. The function fgaps represents the contribution to the electrostatic field of the gaps between the electrodes 6, 7, 8, 9, 10 and depends on both the dimensions of the gaps and the DC voltages Vr, Vc and Ve appliedto the electrodes 6, 7, 8, 9, 10.
The basic functioning of the ion trap 1 can be illustrated by computing an example using equation (1). For this purpose, we choose lr=0.9 mm, lc=2.0 mm, le=5.0 mm, η=0.1 mm and S0=7.0 mm and the voltages applied to the electrodes 6, 7, 8, 9, 10, 11, 12 are Vr=−1V, Vc=−1.15V, Ve=−4V and Vg=0V. These voltages allow for capturing electrons or any negatively charged particles around a position directly above the centre of the ring electrode 6, which is defined in a Cartesian reference frame at x=0, y=y0, z=0, where y0 is known as the trapping height. It is noteworthy that the end cap electrodes 9, 10 are not grounded. The relationship between the voltages applied to the electrodes 6, 7, 8, 9, 10 in the row 5 can generally be defined as

|Ve|>|Vc|≧|Vr|  (2)

in order that the trapping height y0 has a non-zero value and there is an equilibrium position at a distance y0>0 above the surface of the ring electrode 6. Referring to FIG. 3, the electric field 0 at the trapping height y0 (≈1.37 mm in this example) above the ring electrode 6 varies in direction z along the length of the row 5 of electrodes 6, 7, 8, 9, 10, having a maxima 19 above the centre of the ring electrode 6 and mimima 20 on each side of the ring electrode 6. In FIG. 3, the dashed line 21 shows how the voltages Vr, Vc and Ve applied to the ring electrode 6, compensation electrodes 7, 8 and end cap electrodes 9, 10 vary with the direction z. The variation of the electric field φ in the direction y normal to the electrode array 3 is shown in FIG. 4. It can be seen that the electric field φ has a minima 22 above the centre of the ring electrode 6 at the trapping height y0.
The trapping height y0 can be determined by the equality




  
    
      
        
          
            
              
                ∂
                
                  φ
                  ⁡
                  
                    (
                    
                      0
                      ,
                      y
                      ,
                      0
                    
                    )
                  
                
              
              
                ∂
                y
              
            
            
          
          
            y
            =
            
              y
              0
            
          
        
        =
        0
      
    
    
      
        (
        3
        )
      
    
  




If the length η of the gaps between the ring electrode 6, compensation electrodes 7, 8 and end cap electrodes 9, 10 is vanishingly small, i.e. η→0, then fgaps→0 and, with this approximation, the equation for calculating the trapping height y0 is




  
    
      
        
          
            
              
                
                  
                    
                      
                        
                          
                            ∂
                            
                              f
                              r
                            
                          
                          
                            ∂
                            y
                          
                        
                        
                      
                      
                        y
                        =
                        
                          y
                          0
                        
                      
                    
                    +
                    
                      
                        T
                        c
                      
                      ·
                      
                        
                          ∂
                          
                            f
                            c
                          
                        
                        
                          ∂
                          y
                        
                      
                    
                  
                  
                
                
                  y
                  =
                  
                    y
                    0
                  
                
              
              +
              
                
                  T
                  e
                
                ·
                
                  
                    ∂
                    
                      f
                      r
                    
                  
                  
                    ∂
                    y
                  
                
              
            
            
          
          
            y
            =
            
              y
              0
            
          
        
        =
        0
      
    
    
      
        (
        4
        )
      
    
  




This equation introduces a compensation tuning ratio




  
    T
    c
  
  =
  
    
      V
      c
    
    
      V
      r
    
  




and an end cap to ring tuning ratio





  
    T
    e
  
  =
  
    
      
        V
        e
      
      
        V
        r
      
    
    .
  




Equation 4 shows that the trapping height y0 depends only the compensation tuning ratio Tc and the end cap to ring tuning ratio Te. This formal dependence holds also for the less restrictive case that the length η of the gaps is just very small or sufficiently small to be discounted, η<<lr,lc,le,S0. Equation 4 cannot be solved analytically fory0, only numerical values can be obtained.
The series expansion of φ(x, y, z) around the equilibrium position (0, y0, 0), including terms up to the 4th order, has the following form




  
    
      
        
          φ
          ⁡
          
            (
            
              x
              ,
              y
              ,
              z
            
            )
          
        
        =
        
          
            φ
            ⁡
            
              (
              
                0
                ,
                
                  y
                  0
                
                ,
                0
              
              )
            
          
          +
          ...
          +
          
            
              
                
                  C
                  002
                
                ⁢
                
                  z
                  2
                
              
              =
              
                
                  
                    C
                    200
                  
                  ⁢
                  
                    x
                    2
                  
                
                +
                
                  
                    
                      C
                      020
                    
                    ⁡
                    
                      (
                      
                        y
                        -
                        
                          y
                          0
                        
                      
                      )
                    
                  
                  2
                
              
            
            
              }
              
                φ
                quad
              
            
          
          +
          
            
              
                
                  C
                  012
                
                ⁢
                
                  
                    z
                    2
                  
                  ⁡
                  
                    (
                    
                      y
                      -
                      
                        y
                        0
                      
                    
                    )
                  
                
              
              =
              
                
                  
                    C
                    210
                  
                  ⁢
                  
                    
                      x
                      2
                    
                    ⁡
                    
                      (
                      
                        y
                        -
                        
                          y
                          0
                        
                      
                      )
                    
                  
                
                +
                
                  
                    
                      C
                      030
                    
                    ⁡
                    
                      (
                      
                        y
                        -
                        
                          y
                          0
                        
                      
                      )
                    
                  
                  3
                
              
            
            
              }
              
                odd
                ⁢
                
                  
                
                ⁢
                anharmonicities
              
            
          
          +
          
            
              
                
                  
                    
                      
                        C
                        202
                      
                      ⁢
                      
                        z
                        2
                      
                      ⁢
                      
                        x
                        2
                      
                    
                    +
                    
                      
                        C
                        022
                      
                      ⁢
                      
                        
                          
                            z
                            2
                          
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                0
                              
                            
                            )
                          
                        
                        2
                      
                    
                    +
                    
                      
                        C
                        220
                      
                      ⁢
                      
                        
                          
                            x
                            2
                          
                          ⁡
                          
                            (
                            
                              y
                              -
                              
                                y
                                0
                              
                            
                            )
                          
                        
                        2
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        C
                        004
                      
                      ⁢
                      
                        z
                        4
                      
                    
                    +
                    
                      
                        C
                        400
                      
                      ⁢
                      
                        x
                        4
                      
                    
                    +
                    
                      
                        
                          C
                          040
                        
                        ⁡
                        
                          (
                          
                            y
                            -
                            
                              y
                              0
                            
                          
                          )
                        
                      
                      4
                    
                  
                
              
            
            
              }
              
                even
                ⁢
                
                  
                
                ⁢
                anharmonicities
              
            
          
        
      
    
    
      
        (
        5
        )
      
    
  




The expansion coefficients are defined by




  
    
      
        
          
            
              C
              ijk
            
            =
            
              
                
                  ∂
                  
                    i
                    +
                    j
                    +
                    k
                  
                
                ⁢
                
                  φ
                  ⁡
                  
                    (
                    
                      x
                      ,
                      y
                      ,
                      z
                    
                    )
                  
                
              
              
                
                  ∂
                  
                    x
                    i
                  
                
                ⁢
                
                  ∂
                  
                    y
                    j
                  
                
                ⁢
                
                  ∂
                  
                    z
                    k
                  
                
              
            
          
          
        
        
          (
          
            0
            ,
            
              y
              0
            
            ,
            0
          
          )
        
      
    
    
      
        (
        6
        )
      
    
  




The symmetry of the electric filed φ(x, y, z) implies that all expansion coefficients Cijk with odd i and/or odd k vanish. The expansion coefficients Cijk define to a great extent the performance of the ion trap 1. As for equation (4), if the length η of the gaps is sufficiently small, then the expansion coefficients Cijk scale linearly with the voltage Vr applied to the ring electrode 6, i.e. Cijk Vr·cijk where cijk=cijk(Tc,Te) depends only on Tc and Te.
Plugging the series expansion given in equation (5) into Laplace equation, ∇2φ(x, y, z)=0, the following equalities can be obtained

C200+C020+C002=0

3C030+C210C012=0

6C400+C220+C202=0

6C040+C220+C022=0

6C004+C202+C022=0

In the case of a three dimensional hyperbolic or cylindrical trap, the coordinates x and y are undistinguishable, i.e. C200=C020. Thus, the first equality in (7) reduces to C200=2C020. From this, the electric potential of an ideal classical Penning trap arises, φ=C200(z2−(x2+y2)/2). For the ion trap described herein, x and y are distinguishable and the curvatures C200 and C020 are not identical, i.e. C200≠C020. Hence the general form of the pure quadrupole potential, i.e. including terms only up to the second order, is:




  
    
      
        
          
            φ
            quad
          
          ⁡
          
            (
            
              x
              ,
              y
              ,
              z
            
            )
          
        
        =
        
          
            
              C
              002
            
            ·
            
              (
              
                
                  z
                  2
                
                ⁢
                
                  
                    
                      x
                      2
                    
                    +
                    
                      
                        (
                        
                          y
                          -
                          
                            y
                            0
                          
                        
                        )
                      
                      2
                    
                  
                  2
                
              
              )
            
          
          +
          
            
              1
              2
            
            ⁢
            
              C
              002
            
            ⁢
            
              ɛ
              ·
              
                (
                
                  
                    x
                    2
                  
                  -
                  
                    
                      (
                      
                        y
                        -
                        
                          y
                          0
                        
                      
                      )
                    
                    2
                  
                
                )
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




The ellipticity parameter ∈ is given by




  ɛ
  =
  
    
      
        
          C
          200
        
        ⁢
        0
        ⁢
        
          C
          020
        
      
      
        C
        002
      
    
    .
  




In general the ellipticity parameter ∈ is non-zero, i.e. ∈≠0, and the ion trap 1 is therefore an elliptical Penning trap.

The cyclotron frequency ωp (=2πvp), magnetron frequency ωm(=2πvm) and axial frequency ωz (=2πvz) of a particle with charge q and mass m in an ideal elliptical Penning trap are given by




  
    
      
        
          
            ω
            p
          
          ⁢
          
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      ω
                      c
                      2
                    
                    -
                    
                      ω
                      z
                      2
                    
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  
                    
                      
                        ω
                        c
                        2
                      
                      ⁢
                      
                        ω
                        1
                        2
                      
                    
                    +
                    
                      
                        ɛ
                        2
                      
                      ⁢
                      
                        ω
                        z
                        4
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            ω
            m
          
          ⁢
          
            
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      ω
                      c
                      2
                    
                    -
                    
                      ω
                      z
                      2
                    
                  
                  )
                
              
              +
              
                
                  1
                  2
                
                ⁢
                
                  
                    
                      
                        ω
                        c
                        2
                      
                      ⁢
                      
                        ω
                        1
                        2
                      
                    
                    +
                    
                      
                        ɛ
                        2
                      
                      ⁢
                      
                        ω
                        z
                        4
                      
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            ω
            z
          
          ⁢
          
            
              2
              ⁢
              
                C
                002
              
              ⁢
              
                q
                m
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




with




  
    
      ω
      c
    
    =
    
      
        
          
            q
            m
          
          ·
          B
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        
          ω
          1
        
      
      =
      
        
          
            ω
            c
            2
          
          -
          
            ω
            z
            2
          
        
      
    
  
  ,




where B is the strength of the magnetic field provided by the magnetic field generator 2 at the trapping height y0. When the ellipticity parameter ∈ is zero, i.e. ∈=0, expressions for the cyclotron frequency ωp, magnetron frequency ωm and axial frequency ωz of a circular Penning trap can be recovered. For the ion trap 1 described herein, the ellipticity parameter ∈ is approximately equal to 0.41. According to equation (9), the cyclotron frequency ωp, magnetron frequency ωm and axial frequency ωz of a trapped electron are therefore ωp=2π·14 GHz, ωz=2π·28 MHz and ωm=2π·26 kHz respectively, based on a magnetic field strength B approximately equal to 0.5 Tesla.

The radial motion in an ideal elliptical trap is given by

(x(t),y(t)−y0)=(Apξp cos(ωpt)+Amξm cos(ωmt),Apηp sin(ωpt)+Amηm sin(ωmt)

where the amplitudes are given by




  
    
      
        
          
            
              A
              p
            
            =
            
              
                1
                
                  ω
                  p
                
              
              ⁢
              
                
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      E
                      p
                    
                  
                  
                    
                      γ
                      p
                    
                    ⁢
                    m
                  
                
              
            
          
          ,
          
            
              γ
              p
            
            =
            
              
                1
                -
                
                  
                    ω
                    z
                    2
                  
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ω
                      p
                      2
                    
                  
                
              
              ≃
              
                1
                ⁢
                
                  
                
                ⁢
                and
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              A
              m
            
            =
            
              
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  
                    E
                    m
                  
                
                
                  
                    (
                    
                      
                        ω
                        m
                        2
                      
                      -
                      
                        
                          ω
                          z
                          2
                        
                        /
                        2
                      
                    
                    )
                  
                  ⁢
                  
                    m
                    ′
                  
                
              
            
          
          ,
          
            

          
          ⁢
          
            and
            ⁢
            
              
            
            ⁢
            by
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          ξ
          
            p
            ,
            m
          
        
        =
        
          
            
              
                ω
                c
                2
              
              +
              
                
                  ɛω
                  z
                  2
                
                ±
                
                  
                    
                      
                        ω
                        c
                        2
                      
                      ⁢
                      
                        ω
                        1
                        2
                      
                    
                    +
                    
                      
                        ɛ
                        2
                      
                      ⁢
                      
                        ω
                        z
                        4
                      
                    
                  
                
              
            
            
              2
              ⁢
              
                
              
              ⁢
              
                
                  ω
                  p
                
                /
                
                  ω
                  1
                
              
              ⁢
              
                
                  
                    
                      ω
                      c
                      2
                    
                    ⁢
                    
                      ω
                      1
                      2
                    
                  
                  +
                  
                    
                      ɛ
                      2
                    
                    ⁢
                    
                      ω
                      z
                      4
                    
                  
                
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  
  
    
      
        
          η
          
            p
            ,
            m
          
        
        =
        
          
            
              
                ω
                c
                2
              
              +
              
                
                  ɛω
                  z
                  2
                
                ±
                
                  
                    
                      
                        ω
                        c
                        2
                      
                      ⁢
                      
                        ω
                        1
                        2
                      
                    
                    +
                    
                      
                        ɛ
                        2
                      
                      ⁢
                      
                        ω
                        z
                        4
                      
                    
                  
                
              
            
            
              2
              ⁢
              
                
              
              ⁢
              
                
                  ω
                  p
                
                /
                
                  ω
                  1
                
              
              ⁢
              
                
                  
                    
                      ω
                      c
                      2
                    
                    ⁢
                    
                      ω
                      1
                      2
                    
                  
                  +
                  
                    
                      ɛ
                      2
                    
                    ⁢
                    
                      ω
                      z
                      4
                    
                  
                
              
            
          
        
      
    
    
      
        
      
    
  




The symbols Ep, Em represent the cyclotron and magnetron energies respectively. The orbit of the reduced cyclotron motion is only slightly affected by the ellipticity, ξp≅ηp≅1. It very nearly follows the circular shape of conventional Penning traps. On the contrary, the magnetron motion becomes an ellipse, as shown in FIG. 5, where the orientation of the major and minor axes (along x or y) depends on the sign of the ellipticity parameter ∈. Moreover, that motion is stable for −1<∈<1 and, at the limit |∈|→1 it becomes very slow, ωm→0. In that case, the magnetron ellipse tends towards a line, with an increasingly wide major axes and a vanishing minor one (if ∈→+1ξm→∞, ηm→0 and vice versa). For values |∈|≧1 the magnetron becomes an unbounded hyperbolic motion and trapping is not possible.
The ideal trap model is only valid for vanishing amplitudes of the particle's motion. In real experiments the electric anharmonicities must be taken into account. These generate energy-dependent fluctuations/deviations of the trapping frequencies.
All anharmonicities, even and odd, up to the 4th order in the expansion of φ, 3≦i+j+k≦4 (see equation (5)), produce frequency shifts which scale linearly with the energies of the particle. Hence, they can be expressed in matrix form:




  
    
      
        
          (
          
            
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    v
                    p
                  
                
              
            
            
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    v
                    z
                  
                
              
            
            
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    v
                    m
                  
                
              
            
          
          )
        
        =
        
          
            
              
                (
                
                  
                    
                      
                        M
                        
                          1
                          ,
                          1
                        
                      
                    
                    
                      
                        M
                        
                          1
                          ,
                          2
                        
                      
                    
                    
                      
                        M
                        
                          1
                          ,
                          3
                        
                      
                    
                  
                  
                    
                      
                        M
                        
                          2
                          ,
                          1
                        
                      
                    
                    
                      
                        M
                        
                          2
                          ,
                          2
                        
                      
                    
                    
                      
                        M
                        
                          2
                          ,
                          3
                        
                      
                    
                  
                  
                    
                      
                        M
                        
                          3
                          ,
                          1
                        
                      
                    
                    
                      
                        M
                        
                          3
                          ,
                          2
                        
                      
                    
                    
                      
                        M
                        
                          3
                          ,
                          3
                        
                      
                    
                  
                
                )
              
              }
            
            
              M
              =
              
                frequency
                -
                
                  shifts
                  ⁢
                  
                    
                  
                  ⁢
                  matrix
                
              
            
          
          ·
          
            (
            
              
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      E
                      p
                    
                  
                
              
              
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      E
                      z
                    
                  
                
              
              
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      E
                      m
                    
                  
                
              
            
            )
          
        
      
    
    
      
        (
        12
        )
      
    
  




Each perturbation to φquad appearing in equation (5) delivers such a frequency-shifts matrix. In total, the ion trap 1 requires nine Mijk matrices, corresponding to each Cijk perturbative hamiltonian. The overall frequency shifts matrix M is the sum of all of them:

M=M012+M210=M030+M220+M022=M004+M400+M040  (13)

For a single electron captured in the ion trap 1, with the voltages given above the magnetic field strength B being equal to 0.5 Tesla, the overall frequency shifts matrix M is




  
    
      
        M
        =
        
          
            (
            
              
                
                  
                    5
                    ·
                    
                      10
                      
                        -
                        6
                      
                    
                  
                
                
                  0.5
                
                
                  
                    -
                    0.9
                  
                
              
              
                
                  
                    1
                    ·
                    
                      10
                      
                        -
                        3
                      
                    
                  
                
                
                  203
                
                
                  
                    -
                    411
                  
                
              
              
                
                  
                    
                      -
                      2
                    
                    ·
                    
                      10
                      
                        -
                        6
                      
                    
                  
                
                
                  
                    -
                    0.4
                  
                
                
                  2
                
              
            
            )
          
          ⁢
          
            Hz
            /
            
              K
              .
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




Accurate measurement of the axial frequency ωz is important as, in most cases, determination of the electron's eigenfrequencies or spin state depend upon it. Thus,




  
    M
    
      2
      ,
      2
    
  
  =
  
    
      Δ
      ⁢
      
        
      
      ⁢
      
        v
        z
      
    
    
      Δ
      ⁢
      
        
      
      ⁢
      
        E
        z
      
    
  




is the most relevant of the frequency shifts in the frequency shifts matrix M. In the present example, it amounts to 203 Hz/K. Dependence of vz on the axial energy Ez, which is not constant but fluctuating, would render the detection of the electron almost impossible, even at cryogenic temperatures.

M2,2 is given by the sum of M2,2004 and M2,2012. Taking into account that vm<<vz<<vp, we have




  
    
      
        
          M
          
            2
            ,
            2
          
          004
        
        =
        
          
            -
            
              q
              
                16
                ⁢
                
                  
                
                ⁢
                
                  π
                  4
                
                ⁢
                
                  m
                  2
                
              
            
          
          ⁢
          
            3
            
              v
              z
            
          
          ⁢
          
            C
            004
          
        
      
    
    
      
        (
        15
        )
      
    
  
  
    
      
        
          M
          
            2
            ,
            2
          
          012
        
        ≃
        
          
            
              q
              2
            
            
              32
              ⁢
              
                
              
              ⁢
              
                π
                6
              
              ⁢
              
                m
                3
              
            
          
          ⁢
          
            
              η
              m
              2
            
            
              v
              z
              5
            
          
          ⁢
          
            C
            012
            2
          
        
      
    
    
      
        
      
    
  




M2,2012 is always positive, since it is proportional to the square of C012, while M2,2004 can be positive or negative, depending on the sign of C004. Hence, if an appropriate optimal tuning ratio Tcopt can be found, such that the latter matrix element cancels the former TcoptM2,2012+M2,2004=0, then the linear dependence of vz upon the axial energy Ez can be eliminated.
The existence of the optimal tuning ratio Tcopt cannot be universally guaranteed, however it turns out that this is often the case. For the ion trap 1 described herein, it can be seen in FIG. 6, where




  
    Δ
    ⁢
    
      
    
    ⁢
    
      v
      z
    
  
  
    Δ
    ⁢
    
      
    
    ⁢
    
      E
      z
    
  




is plotted as a function of the applied tuning ratio Tc. One value, Tcopt=1.13440 eliminates M2,2.

Since vz∞C0021/2, both frequency shifts in equation (15) equally scale with Vr−1/2. The equation M2,2(Tcopt)=0 is independent of the actual value of the voltage Vr applied to the ring electrode 6 and is solely defined by the voltage ratios Tc and Te. A similar argument applies to the mass m and charge q of the particle. Thus, the optimal tuning ratio Tcopt is a well defined quantity, independent of the voltage Vr applied to the ring electrode 6 and of the mass m and charge q of the trapped particle. It does change with the tuning ratio Te, but this is simply equivalent to an inevitable dependence upon the trapping height y0. The appearance of ηm in M2,2012 also implies that the optimal tuning ratio Tcopt theoretically varies with the strength of magnetic field B. However, that dependence is negligible, being approximately equal to −2.10−6 Tesla−1 in the present example.
M2,2012 results from the slight dependence of the trapping height y0 on the axial energy Ez, i.e. y0=y0(Ez). Indeed, for vanishing axial energy (Ez=0), y0 is the solution to the implicit equation C010(y0)=0. For Ez>0, that equation must be modified into C010(y0′)+z2·C012 (y0′)=0. Here, z2 represents the time average of Az2 cos2(ωzt). Thus, the real trapping height, y0′=y0+Δy, depends on the axial amplitude, hence, on EzΔy, and can be estimated as follows, assuming the approximation C012(y0′)≅C012(y0):




  
    
      
        
          
            {
            
              
                
                  C
                  010
                
                ⁡
                
                  (
                  
                    y
                    0
                    ′
                  
                  )
                
              
              +
              
                
                  〈
                  
                    z
                    2
                  
                  〉
                
                ⁢
                
                  
                    C
                    012
                  
                  ⁡
                  
                    (
                    
                      y
                      0
                      ′
                    
                    )
                  
                
              
            
            }
          
          -
          
            
              C
              010
            
            ⁡
            
              (
              
                y
                0
              
              )
            
          
        
        =
        
          
            0
            →
            
              
                
                  
                    
                      
                        
                          C
                          010
                        
                        ⁡
                        
                          (
                          
                            
                              y
                              0
                            
                            +
                            
                              Δ
                              ⁢
                              
                                
                              
                              ⁢
                              y
                            
                          
                          )
                        
                      
                      -
                      
                        
                          C
                          010
                        
                        ⁡
                        
                          (
                          
                            y
                            0
                          
                          )
                        
                      
                    
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      y
                    
                  
                  
                    }
                    
                      =
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        
                          C
                          020
                        
                      
                    
                  
                
                ⁢
                Δ
                ⁢
                
                  
                
                ⁢
                y
              
              +
              
                
                  〈
                  
                    z
                    2
                  
                  〉
                
                ⁢
                
                  
                    C
                    012
                  
                  ⁡
                  
                    (
                    
                      y
                      0
                    
                    )
                  
                
              
            
          
          =
          
            
              0
              ⇒
              
                Δ
                ⁢
                
                  
                
                ⁢
                y
              
            
            =
            
              
                
                  -
                  
                    1
                    2
                  
                
                ⁢
                
                  
                    C
                    012
                  
                  
                    C
                    020
                  
                
                ⁢
                
                  〈
                  
                    z
                    2
                  
                  〉
                
              
              =
              
                
                  (
                  
                    
                      
                        A
                        z
                        2
                      
                      ⁢
                      
                        〈
                        
                          
                            cos
                            2
                          
                          ⁡
                          
                            (
                            
                              
                                ω
                                z
                              
                              ⁢
                              t
                            
                            )
                          
                        
                        〉
                      
                    
                    =
                    
                      
                        E
                        z
                      
                      
                        m
                        ⁢
                        
                          
                        
                        ⁢
                        
                          ω
                          z
                          2
                        
                      
                    
                  
                  )
                
                =
                
                  
                    
                      -
                      
                        1
                        
                          8
                          ⁢
                          
                            π
                            2
                          
                          ⁢
                          m
                          ⁢
                          
                            
                          
                          ⁢
                          
                            v
                            z
                            2
                          
                        
                      
                    
                    ·
                    
                      
                        C
                        012
                      
                      
                        C
                        020
                      
                    
                  
                  ⁢
                  
                    E
                    z
                  
                
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




At y0+Δy, the axial potential is modified with respect to y0. In particular, the Ez=0 axial curvature, C002 (y0), changes to C002′(y0′). This subsequently forces the variation of the axial frequency ωz as a function of the axial energy Ez




  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            ω
            z
          
        
        =
        
          
            
              
                
                  ∂
                  
                    ω
                    z
                  
                
                
                  ∂
                  y
                
              
              ·
              Δ
            
            ⁢
            
              
            
            ⁢
            y
          
          =
          
            
              
                
                  q
                  m
                
                ⁢
                
                  1
                  
                    
                      
                        
                          2
                          q
                        
                        ⁢
                        
                          C
                          002
                        
                      
                      m
                    
                  
                
                ⁢
                
                  
                    
                      ∂
                      
                        C
                        002
                      
                    
                    
                      ∂
                      y
                    
                  
                  
                    }
                    
                      C
                      012
                    
                  
                
                ⁢
                Δ
                ⁢
                
                  
                
                ⁢
                y
              
              ⇒
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    v
                    z
                  
                
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    E
                    z
                  
                
              
            
            =
            
              
                -
                
                  
                    
                      q
                      2
                    
                    ⁢
                    
                      C
                      012
                      2
                    
                  
                  
                    32
                    ⁢
                    
                      π
                      6
                    
                    ⁢
                    
                      m
                      3
                    
                    ⁢
                    
                      v
                      z
                      5
                    
                  
                
              
              ⁢
              
                (
                
                  
                    C
                    002
                  
                  
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    
                      C
                      020
                    
                  
                
                )
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




The model can be tested by computing numerically the radial motion of an electron in the ion trap 1, using φ (x, y, z) of equation 1 without approximations. The numerical calculation shows a vertical shift of the radial ellipse relative to the ideal one, as shown in FIG. 5. This is computed assuming Ez=4.2 K, Em=−Ez(vm/vz) and Ep ̃0. The shift predicted by equation (16) amounts to Δy=−0.355 μm and is in good agreement with the numerical result of Δy=−0.325 μm. For low ellipticity,




  ɛ
  →
  0
  ⇒
  
    
      η
      m
      2
    
    →
    
      
        1
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        
          
            C
            002
          
          
            2
            ⁢
            
              
            
            ⁢
            
              C
              020
            
          
        
      
      →
      
        -
        1
      
    
  




(see equation (6)). In this case, equations (15) and (17) become identical. The former is actually accurate only for moderate values of the ellipticity parameter ∈, below approximately 0.5. Otherwise, it delivers shifts increasingly bigger than those predicted by the rigorous equation 15. In the limit ∈→+1ηm→0 and, as predicted by that equation, the effect of C012 upon vz is theoretically cancelled.

An optimal tuning ratio Tcopt can be found within a continuous interval of trapping heights y0, however it varies smoothly as a function of trapping height y0. This is shown in FIG. 7, where the plot of optimal tuning ratio Tcopt versus trapping height y0 is presented. A useful interval exists 0.8 mm≦y0≦2 mm, where M2,2 can be eliminated. Beyond the upper and lower bounds of that limit, the optimal tuning ratio Tcopt does not exist.
As shown in FIG. 8, the tuning ratios Tc and Te can be tuned independently and multiple combinations can be found to obtain one particular trapping height y0. However, the compensated interval is determined by a univocal relationship, y0(Te, Tcopt), as featured in that FIG. 8. It must be noted also that the end cap to ring tuning ratio Te is the main parameter for changing y0, while 1.2≧Tc≧1, and the compensation tuning ratio Tc is basically used for compensation.
FIG. 9 shows the variation of C012 with the trapping height y0. It can be seen that at y0 approximately equal to 0.95 mm, C012 vanishes. In this case, the compensation




  
    Δ
    ⁢
    
      
    
    ⁢
    
      v
      z
    
  
  
    Δ
    ⁢
    
      
    
    ⁢
    
      E
      z
    
  




of also implies that C004 will disappear. Hence, within the compensated interval, an optimal trapping position can be found, y0012, for which C012=C004=0.

For a circular Penning trap, the symmetry of x and y implies that C202=C022. Thus if C004=0, the constraints of equation (7) force all other octupole terms, C202, C022, C400, C040 and C220, to vanish simultaneously. For an elliptical Penning trap though, x and y are not indistinguishable. If C004=0, that only implies that C202=−C022, but in general they will not disappear. The same argument applies to the case C012=0 and the coefficients C210 and C030. For these reasons, even when C004=C012=0, the overall frequency shifts matrix M will always contain some non-zero elements. An ideal elliptical trap cannot be exactly approximated with the ion trap 1 described herein. The non-vanishing elements are basically the shifts produced by Ep (first column of frequency shifts matrix M) and those due to Em (third column).
It can be shown for all trapping heights y0 within the compensated interval, the frequency shifts produced by the cyclotron and magnetron energies are of the same order of magnitude as in the example of equation (14). Thus, if Ep is in the cryogenic domain (Ep approximately equal to 4.2 K), the corresponding deviations of vp and vm are in the range of a few μHz, and of a few mHz for vz. The resulting relative uncertainties in any of the frequencies are negligible. Increasing the strength of the magnetic field B would make those shifts even smaller. Therefore, the first column of frequency shifts matrix M can always be ignored.
The numbers in the third column of equation (14) are apparently more significant. However, applying sideband cooling, the magnetron energy takes the value Em=−E(vm/vz), which is in the range of a few mK, (if Ez is approximately equal to 4.2 K). Since the magnetron energy does not fluctuate in time, the resulting frequency shifts represent mere offsets, which can be corrected for without affecting the “visibility” of the trapped particle. In the example, the biggest relative offset would be approximately 10−7 for vz, propagating to approximately 10−12 for vc. Increasing the strength of magnetic field B, or the voltage Vr applied to the ring electrode 6, reduces further the third column of frequency shifts matrix M.
After eliminating the linear dependence of the axial frequency ωz upon the axial energy Ez, non-linear shifts might still be important, specially when trapping height y0 is small. The next most significant even anharmonicities, whose effect can be calculated by first order perturbation theory, are C006 and C008. These produce the following quadratic and cubic shifts, respectively:




  
    
      
        
          
            Δ
            ⁢
            
              
            
            ⁢
            
              v
              z
            
          
          =
          
            
              
                15
                ⁢
                q
              
              
                128
                ⁢
                
                  π
                  6
                
                ⁢
                
                  m
                  3
                
              
            
            ⁢
            
              
                
                  C
                  006
                
                
                  v
                  z
                  5
                
              
              ·
              
                
                  (
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      E
                      z
                    
                  
                  )
                
                2
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            Δ
            ⁢
            
              
            
            ⁢
            
              v
              z
            
          
          =
          
            
              
                140
                ⁢
                q
              
              
                2048
                ⁢
                
                  π
                  8
                
                ⁢
                
                  m
                  4
                
              
            
            ⁢
            
              
                
                  C
                  008
                
                
                  v
                  z
                  7
                
              
              ·
              
                
                  (
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      E
                      z
                    
                  
                  )
                
                3
              
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




The next most significant odd anharmonicities, after those included in equation (5), are C014, C212, C032, C410, C230 and C050 (i+j+k=5). The calculation of the corresponding frequency shifts, with rigorous second order perturbation theory, would be extremely cumbersome. Instead, following the derivation of equations (16) and (17), we obtain




  
    
      
        
          
        
        ⁢
        
          
            Δ
            ⁢
            
              
            
            ⁢
            
              v
              z
            
          
          =
          
            
              -
              
                
                  3
                  ⁢
                  q
                
                
                  128
                  ⁢
                  
                    π
                    6
                  
                  ⁢
                  
                    m
                    3
                  
                  ⁢
                  
                    v
                    z
                    5
                  
                
              
            
            ⁢
            
              
                
                  
                    C
                    012
                  
                  ·
                  
                    C
                    014
                  
                
                
                  C
                  020
                
              
              ·
              
                
                  (
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      E
                      z
                    
                  
                  )
                
                2
              
            
          
        
      
    
    
      
        (
        19
        )
      
    
  
  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          
            v
            z
          
        
        =
        
          
            -
            
              q
              
                128
                ⁢
                
                  π
                  6
                
                ⁢
                
                  m
                  3
                
                ⁢
                
                  v
                  z
                  3
                
              
            
          
          ⁢
          
            
              
                
                  C
                  012
                
                ·
                
                  C
                  212
                
              
              
                C
                020
              
            
            ·
            Δ
          
          ⁢
          
            
          
          ⁢
          
            
              E
              z
            
            ⁡
            
              (
              
                
                  
                    
                      ξ
                      p
                      2
                    
                    
                      
                        γ
                        p
                      
                      ⁢
                      
                        v
                        p
                        2
                      
                    
                  
                  ⁢
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    E
                    p
                  
                
                +
                
                  
                    
                      ξ
                      m
                      2
                    
                    
                      
                        v
                        m
                        2
                      
                      -
                      
                        
                          v
                          z
                          2
                        
                        /
                        2
                      
                    
                  
                  ⁢
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    E
                    m
                  
                
              
              )
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




Referring to FIGS. 10 and 11, the shift predicted by equation (19) has a similar magnitude as the one produced by C006. This can be taken into account when designing the ion trap 1. It is noteworthy that equations (19) and (20) vanish at optimal trapping height y0012.
Equation (20) predicts a frequency shift scaling with the products ΔEz·ΔEp and ΔEz·ΔEm. In the former case, the shift is proportional to 1/vp2. Hence, it is normally negligible. In the latter case, ΔEm is very small and the corresponding value of Δvz is also inappreciable. Thus, C212 can be ignored. The same arguments apply to C032, which produces a shift very similar to equation 20 (ξp,m must be simply substituted by ηp,m). The remaining fifth order coefficients, C410, C230 and C050, generate only deviations of vp and vm with products of ΔEp and ΔEm. Therefore, they can be ignored too. Finally, similar arguments apply to all sixth order coefficients which have not been considered so far, namely C222, C204, C024, C420, C402, C042, C240, C600 and C060. They are all irrelevant.
FIG. 10 reveals the existence of one particular optimal trapping height, y0006, approximately equal to 0.83 mm, at which C006=0. The question which arises is whether the ion trap 1 can be designed to make y0006 and y0012 coincident. The answer is affirmative and is illustrated in FIGS. 12 and 13. FIG. 12 It shows the variation of y0012 and y0006 when changing the length lc of the compensation electrodes 7,8 while keeping all other dimensions of the ion trap 1 constant. For the example, when the length lc of the compensation electrodes 7,8 is approximately equal to 1.75 mm, the optimal trapping heights y0012=y0006≅0.82 mm. In this example, C004=C012=C006=0, at y0012, as shown in FIG. 13.
With the dependence vz=vz(Ez,Ez2,Ez3) given in equations (15), (18) and (19), a realistic simulation of the axial signal of a trapped electron can be performed. The simulations are shown in FIGS. 14 and 15. The curves are obtained by averaging the axial dip (with vz=vz(Ez,Ez2,Ez3) over a Boltzmann distribution of the axial energy E. Three different values of the axial temperature Tz are analyzed.
FIGS. 14 and 15 show the reduction of the “visibility” of the dips with increasing axial temperature. A random position, y0=1.6 mm (but with optimized tuning ratio), has been chosen for FIG. 14. In this case, C006 and C014 produce the increasing deterioration of the dip, while C008 is negligible. In FIG. 15, y0=y0012=y0006=0.82 mm. Now, C004=C012=C006=0. However, C008 still diminishes the quality of the signal with Tz. It can be concluded, that the detection of a single electron at 4.2 K (or lower) should be always possible within the compensated interval. However, for increasing temperatures, the non-linear anharmonicities make its observation significantly more difficult, even for relatively modest values of Tz.
So, it can be appreciated that there exists a trapping interval where the linear dependence of the axial frequency upon the axial energy can be eliminated by appropriately tuning the voltage of the compensation electrodes; and a particular position within the compensated interval, y0012, where the main anharmonicities, C012 and C004, simultaneously vanish. Moreover, optimization of the compensation electrodes 7, 8 permits the further elimination of C006 at the optimal trapping position. These properties exist for a wide range of possible dimensions and aspect ratios of the electrode array 3.
Considering now the magnetic field generator 2, the primary magnetic element 14 is a conducting wire of rectangular cross section placed underneath the electrode array 3. The primary magnetic element 14 and shim magnetic elements 15, 16, 17, 18 are arranged to conduct electrical current in a direction perpendicular to the length of the row 5 of electrodes 6, 7, 8, 9, 10 and parallel with the plane of the electrode array 3. Typically, the magnetic elements 14, 15, 16, 17, 18 extend perpendicular to the length of the row 5 of electrodes 6, 7, 8, 9, 10 for a distance greater than the width S0 of the electrodes 6, 7, 8, 9, 10 in the row 5, and to both side of the row 5. The primary magnetic element 14 also has a width lp in a direction along the length of the row 5 of electrodes 6, 7, 8, 9, 10 that is greater than the length lr of the ring electrode 6, and typically lp is at least twice lr.
The magnetic elements 14, 15, 16, 17, 18 are arranged such that the current flows only in the direction x perpendicular to the length of the row 5 of electrodes 6, 7, 8, 9, 10. Substantially no current flows in the magnetic elements 14, 15, 16, 17, 18 in the directions y and z perpendicular to the length of the row 5 of electrodes 6, 7, 8, 9, 10. This can only be achieved perfectly if the magnetic elements 14, 15, 16, 17, 18 are infinitely long. However, as real magnetic elements 14, 15, 16, 17, 18 are not infinitely long and have to be coupled so as to form a circuit, a magnetic shield (not shown) can be provided between the magnetic elements 14, 15, 16, 17, 18 and wires (not shown) that complete the electric circuit that provides electric current to the magnetic elements 14, 15, 16, 17, 18.
Considering the magnetic elements 14, 15, 16, 17, 18 as infinite straight wires in the direction x for simplicity, we can make a series expansion based on Maxwell's equations of the magnetic vector potential around the trapping position (0, y0, 0):




  
    
      
        
          A
          ⁡
          
            (
            
              y
              ,
              z
            
            )
          
        
        =
        
          
            A
            ⁡
            
              (
              
                y
                0
              
              )
            
          
          +
          
            
              
                ∂
                A
              
              
                ∂
                y
              
            
            ⁢
            
              (
              
                y
                -
                
                  y
                  0
                
              
              )
            
          
          +
          
            
              
                ∂
                A
              
              
                ∂
                z
              
            
            ⁢
            z
          
          +
          
            
              
                1
                
                  2
                  !
                
              
              ·
              
                
                  
                    ∂
                    2
                  
                  ⁢
                  A
                
                
                  ∂
                  
                    y
                    2
                  
                
              
            
            ⁢
            
              
                (
                
                  y
                  -
                  
                    y
                    0
                  
                
                )
              
              2
            
          
          +
          
            
              
                1
                
                  2
                  !
                
              
              ·
              
                
                  
                    ∂
                    2
                  
                  ⁢
                  A
                
                
                  ∂
                  
                    z
                    2
                  
                
              
            
            ⁢
            
              z
              2
            
          
          +
          
            
              
                1
                
                  
                    1
                    !
                  
                  ·
                  
                    1
                    !
                  
                
              
              ·
              
                
                  
                    ∂
                    2
                  
                  ⁢
                  A
                
                
                  
                    ∂
                    y
                  
                  ⁢
                  
                    ∂
                    z
                  
                
              
            
            ⁢
            
              z
              ⁡
              
                (
                
                  y
                  -
                  
                    y
                    0
                  
                
                )
              
            
          
          +
          
            
              
                1
                
                  
                    2
                    !
                  
                  ·
                  
                    1
                    !
                  
                
              
              ·
              
                
                  
                    ∂
                    3
                  
                  ⁢
                  A
                
                
                  
                    ∂
                    
                      y
                      2
                    
                  
                  ⁢
                  
                    ∂
                    z
                  
                
              
            
            ⁢
            
              
                z
                ⁡
                
                  (
                  
                    y
                    -
                    
                      y
                      0
                    
                  
                  )
                
              
              2
            
          
          +
          
            
              
                1
                
                  
                    1
                    !
                  
                  ·
                  
                    2
                    !
                  
                
              
              ·
              
                
                  
                    ∂
                    3
                  
                  ⁢
                  A
                
                
                  
                    ∂
                    y
                  
                  ⁢
                  
                    ∂
                    
                      z
                      2
                    
                  
                
              
            
            ⁢
            
              
                z
                2
              
              ⁡
              
                (
                
                  y
                  -
                  
                    y
                    0
                  
                
                )
              
            
          
          +
          
            
              
                1
                
                  3
                  !
                
              
              ·
              
                
                  
                    ∂
                    3
                  
                  ⁢
                  A
                
                
                  ∂
                  
                    y
                    3
                  
                
              
            
            ⁢
            
              
                (
                
                  y
                  -
                  
                    y
                    0
                  
                
                )
              
              3
            
          
          +
          
            
              
                1
                
                  4
                  !
                
              
              ·
              
                
                  
                    ∂
                    4
                  
                  ⁢
                  A
                
                
                  ∂
                  
                    z
                    4
                  
                
              
            
            ⁢
            
              z
              4
            
          
          +
          
            
              
                1
                
                  4
                  !
                
              
              ·
              
                
                  
                    ∂
                    4
                  
                  ⁢
                  A
                
                
                  ∂
                  
                    y
                    4
                  
                
              
            
            ⁢
            
              
                (
                
                  y
                  -
                  
                    y
                    0
                  
                
                )
              
              4
            
          
          +
          
            
              
                1
                
                  
                    2
                    !
                  
                  ·
                  
                    2
                    !
                  
                
              
              ·
              
                
                  
                    ∂
                    4
                  
                  ⁢
                  A
                
                
                  
                    ∂
                    
                      y
                      2
                    
                  
                  ⁢
                  
                    ∂
                    
                      z
                      2
                    
                  
                
              
            
            ⁢
            
              
                z
                2
              
              ⁡
              
                (
                
                  y
                  -
                  
                    y
                    0
                  
                
                )
              
            
          
          +
          ...
        
      
    
    
      
        (
        21
        )
      
    
  




This can yield the following magnetic vector potential terms



  
    
      
        A
        →
      
      0
    
    =
    
      
        -
        
          
            B
            z
            0
          
          ⁡
          
            (
            
              y
              -
              
                y
                0
              
            
            )
          
        
      
      ⁢
      
        
          u
          ^
        
        x
      
    
  




  
    
      
        A
        →
      
      1
    
    =
    
      
        1
        
          2
          !
        
      
      ⁢
      
        
          ∂
          
            B
            z
          
        
        
          ∂
          y
        
      
      ⁢
      
        (
        
          
            z
            2
          
          -
          
            
              (
              
                y
                -
                
                  y
                  0
                
              
              )
            
            2
          
        
        )
      
      ⁢
      
        
          u
          ^
        
        x
      
    
  




  
    
      
        A
        →
      
      2
    
    =
    
      
        1
        
          3
          !
        
      
      ⁢
      
        
          
            ∂
            2
          
          ⁢
          
            B
            z
          
        
        
          ∂
          
            y
            2
          
        
      
      ⁢
      
        (
        
          y
          -
          
            y
            0
          
        
        )
      
      ⁢
      
        (
        
          
            3
            ⁢
            
              z
              2
            
          
          -
          
            
              (
              
                y
                -
                
                  y
                  0
                
              
              )
            
            2
          
        
        )
      
      ⁢
      
        
          u
          ^
        
        x
      
    
  




  
    
      
        A
        →
      
      3
    
    =
    
      
        1
        
          4
          !
        
      
      ⁢
      
        
          
            ∂
            3
          
          ⁢
          
            B
            z
          
        
        
          ∂
          
            y
            3
          
        
      
      ⁢
      
        (
        
          
            6
            ⁢
            
              
                
                  z
                  2
                
                ⁡
                
                  (
                  
                    y
                    -
                    
                      y
                      0
                    
                  
                  )
                
              
              2
            
          
          -
          
            z
            4
          
          -
          
            
              (
              
                y
                -
                
                  y
                  0
                
              
              )
            
            4
          
        
        )
      
      ⁢
      
        
          u
          ^
        
        x
      
    
  




  
    
      
        A
        →
      
      4
    
    =
    
      
        1
        
          5
          !
        
      
      ⁢
      
        
          
            ∂
            4
          
          ⁢
          
            B
            z
          
        
        
          ∂
          
            y
            4
          
        
      
      ⁢
      
        (
        
          y
          -
          
            y
            0
          
        
        )
      
      ⁢
      
        (
        
          
            10
            ⁢
            
              
                
                  z
                  2
                
                ⁡
                
                  (
                  
                    y
                    -
                    
                      y
                      0
                    
                  
                  )
                
              
              2
            
          
          -
          
            
              (
              
                y
                -
                
                  y
                  0
                
              
              )
            
            4
          
          -
          
            5
            ⁢
            
              z
              4
            
          
        
        )
      
      ⁢
      
        
          u
          ^
        
        x
      
    
  



It can be appreciated that all magnetic inhomogeneities are caused by the drop of the magnetic field, Bz, with the vertical distance y. If the derivatives of Bz with y are eliminated, then the only term of the magnetic vector remaining is {right arrow over (A)}0. In this case the magnetic field will be homogeneous. Accordingly, the shim magnetic elements 15, 16, 17, 18 are positioned to achieve this.
The primary magnetic element 14 generates a main magnetic field, which is everywhere inhomogeneous, due to the drop of its strength normal to the plane of the electrode array 3 (direction y). The pair of shim magnetic elements 15, 16 adjacent the primary magnetic element 14 generate a first compensating magnetic field. This has a gradient and a curvature. It can be shown that the gradient and curvature are controlled basically by the current I1 flowing along the shim magnetic elements 15, 16. The current I1 can be chosen to compensate for the first order magnetic vector {right arrow over (A)}1 of the magnetic field generated by the primary magnetic element 14 (at the trapping height y0). Hence, the overall gradient of the magnetic field is compensated towards zero. However, the compensating magnetic field generated by the pair of shim magnetic elements 15, 16 adjacent the primary magnetic element 14 does not compensate for the second order magnetic vector {right arrow over (A)}2 of the magnetic field generated by the primary magnetic element 14. This can be compensated for by the shim magnetic elements 17, 18 at the ends of the row 13 of magnetic elements. More specifically, a current I2 flowing in the shim magnetic elements 17, 18 at the ends of the row 13 of magnetic elements can be chosen to compensate for the second order magnetic vector {right arrow over (A)}2 of the magnetic field generated by the primary magnetic element 14 (at the trapping height y0). In other embodiments, further shim magnetic elements are provided to compensate for higher order components of the magnetic field, thereby improving the inhomogeneity of the magnetic field.
In the illustrated embodiment, the magnetic elements 14, 15, 16, 17, 18 are coplanar, but this is not strictly required. The shim magnetic elements 15, 16, 17, 18 can be positioned above the primary magnetic element 14. Usually the cross section of the shim magnetic elements 15, 16, 17, 18 is smaller than that of the primary magnetic element 14.
Referring to FIG. 16, a mass spectrometer 23 according to a preferred embodiment of the disclosure comprises the ion trap 1 located in a cryogenic vacuum chamber 24 capable of cooling the ion trap to a temperature of 4.2 K or lower. A DC voltage source 25 is provided to supply the voltages Vr,Vc,Ve to the ring electrode 6, compensation electrodes 7, 8 and end cap electrodes 9, 10. A microwave generator 26 and function generator 27 are provided for injecting microwaves into the ion trap for probing the trapped particles, and an oscilloscope 28 and Fourier transform analyser are provided for analysing microwaves exiting the ion trap 1. Multiple ion traps 1 can be provided in the cryogenic vacuum chamber 24, allowing the mass spectrometer 23 to analyse multiple trapped particles at the same time, under similar ambient conditions.
Referring to FIGS. 17 and 18, the ion trap 1 can provide a cavity 29 for microwaves. The cavity is equivalent to an LC circuit, as shown in FIG. 17, and can be coupled to a distance microwave cavity 30 via an external microwave transmission line to form a quantum electrodynamic circuit.
Other variations and modifications will be apparent to the skilled person. Such variations and modifications may involve equivalent and other features which are already known and which may be used instead of, or in addition to, features described herein. Features that are described in the context of separate embodiments may be provided in combination in a single embodiment. Conversely, features which are described in the context of a single embodiment may also be provided separately or in any suitable sub-combination.
It should be noted that the term “comprising” does not exclude other elements or steps, the term “a” or “an” does not exclude a plurality, a single feature may fulfil the functions of several features recited in the claims and reference signs in the claims shall not be construed as limiting the scope of the claims. It should also be noted that the Figures are not necessarily to scale; emphasis instead generally being placed upon illustrating the principles of the present invention.

## Claims

1. An ion trap comprising:
a magnetic field generator arranged to generate a magnetic field; and
an array of electrodes arranged to generate an electrostatic field including a turning point in electrical potential at a location where the magnetic field is substantially homogeneous,
wherein the array of electrodes is planar and parallel to the direction of the magnetic field at the location.

2. The ion trap of claim 1, wherein the electrodes of the array each have surfaces facing the location where the magnetic field is substantially homogeneous, which surfaces are substantially coplanar.

3. The ion trap of claim 1, wherein the array of electrodes comprises a row of three or more electrodes, which row is arranged to be parallel to the direction of the magnetic field at the location where the magnetic field is substantially homogeneous.

4. The ion trap of claim 3, wherein the row comprises five electrodes.

5. The ion trap of claim 3, wherein the lengths of the electrodes along the direction of the row are such that an electrode in the middle of the row is shortest and electodes at the ends of the row are longest.

6. The ion trap of claim 3, comprising a guard electrode on each side of the row.

7. The ion trap of claim 6, wherein the guard electrodes overlap the electrodes of the row.

8. The ion trap of claim 1, wherein the array of electrodes is provided on a substrate and the magnetic field generator is provided on the same substrate.

9. The ion trap of claim 8, wherein the magnetic field generator comprises a row of magnetic elements, which row extends in the same direction as the row of electrodes.

10. The ion trap of claim 9, wherein the magnetic elements each comprise a wire arranged to conduct an electric current.

11. The ion trap of claim 9, wherein a primary magnetic element is arranged to generate a first component of the magnetic field and the other magnetic elements are arranged to generate compensating components of the magnetic field that reduce the gradient and curvature of the first component of the magnetic field at the location where the magnetic field is substantially homogeneous.

12. The ion trap of claim 8, wherein the array of electrodes is provided on a top surface of the substrate and the magnetic field generator is provided below the array of electrodes.

13. A mass spectrometer comprising the ion trap of claim 1.

14. A quantum electrodynamic circuit comprising the ion trap of claim 1.

15. A method of trapping an ion, the method comprising:
using a magnetic field generator to generate a magnetic field; and
using an array of electrodes to generate an electrostatic field including a turning point in electrical potential at a location where the magnetic field is substantially homogeneous,
wherein the array of electrodes is planar and parallel to the direction of the magnetic field at the location.