Method, system, ion trap, computer program and computer-readable storage medium for determining a shape of two permanent magnets incorporated in an ion trap

Abstract

A method for determining a shape of two permanent magnets (2) incorporated in an ion trap (1) is provided, comprising - providing a first function characteristic for an arbitrary shape of the two permanent magnets (2) in lateral directions and having a predetermined thickness(6) in vertical direction, and - providing a second function dependent on the first function characteristic for a magnetic field established by the two permanent magnets (2), - providing a first constraint characteristic for a predetermined point (7) in lateral direction and vertical direction where the magnetic field has to be vanished, - determining an extremum of a functional dependent on the second function dependent on the first constraint, and - determining the shape of the two permanent magnets (2) dependent on the extremum such that a gradient of magnitudes of the magnetic field is maximized above the two permanent magnets (2) with the magnetic field being vanished at the predetermined point (7). Further, a system, an ion trap (1), a computer program and a computer-readable storage medium is provided.

Description

[0001] P2023,1262 WO N September 12, 2024 - 1 - Description METHOD, SYSTEM, ION TRAP, COMPUTER PROGRAM AND COMPUTER- READABLE STORAGE MEDIUM FOR DETERMINING A SHAPE OF TWO PERMANENT MAGNETS INCORPORATED IN AN ION TRAP The present disclosure relates to a method, a system, a computer program and a computer-readable storage medium for determining a shape of two permanent magnets incorporated in an ion trap as well as an ion trap. An object to be solved is to provide a method, with which a shape of two magnets of an ion trap is optimized with respect to their magnetic field. Furthermore, an ion trap with such magnets as well as a system, a computer program and a computer-readable storage medium for executing such a method are to be provided. The object is solved by the subject matter of the independent claims. Advantageous embodiments, implementations and further developments are the subject matter of the respective dependent claims. The method for determining a shape of two permanent magnets incorporated in an ion trap is described. In particular, the ion trap is configured to confine at least one ion and / or to modify an electronic state of the at least one ion, in particular to perform a quantum calculation with at least two ions. The ion trap can be a Paul trap, a linear ion trap, a surface ion trap and / or a multi-layer ion trap. Exemplarily, the ion trap comprises a set of electrodes. For example, a radio frequency, RF, voltage is applied to the electrodes such that a time-varying electric field is provided P2023,1262 WO N September 12, 2024 - 2 - configured to confine and / or to modify the ion. In particular, the ion or the ions are located along a trapping axis. For example, the ion, in particular each of the ions, intersects with the trapping axis and / or oscillates around the trapping axis. Exemplarily, the ions are arranged in an ion chain along the trapping axis. The two permanent magnets are spaced apart from the trapping axis. The two permanent magnets are configured to produce a magnetic field, in particular, at a region around the trapping axis, spaced apart from the two permanent magnets. According to at least one embodiment of the method, a first function is provided characteristic for an arbitrary shape of the two permanent magnets in lateral directions and, wherein the two permanent magnets have a predetermined thickness in vertical direction. The first function comprises variables, and / or parameters being characteristic for the arbitrary shape, wherein the arbitrary shape has in particular a continuous and closed outer edge in lateral directions. The first function is defined in a three-dimensional Euclidean space with an x-axis, a y-axis and a z-axis, being orthogonal to one another. The lateral directions are formed by the x- axis and the y-axis and the vertical direction is formed by the z-axis. In particular, the two permanent magnets are formed of a first permanent magnet and a second permanent magnet, being spaced apart from one another in lateral directions. A first sub function is, for example, characteristic for a shape of the first permanent magnet in lateral directions. A second sub function is, for example, characteristic for a shape of the second permanent magnet, being in particular mirror P2023,1262 WO N September 12, 2024 - 3 - symmetric to the shape of the first permanent magnet. The first sub function and the second sub function form the first function. The permanent magnets each have a top surface and a bottom surface opposite the top surface. Exemplarily, the top surface and the bottom surface solely extend in lateral directions. The top surface and the bottom surface are connected by a side surface arranged perpendicular to the top surface and the bottom surface. The side surfaces are formed in lateral directions according to the arbitrary shape. In particular, the permanent magnets are planar permanent magnets. In particular, the predetermined thickness of the first permanent magnet equals the predetermined thickness of the second permanent magnet. The predetermined thickness is at least 1 micrometer, and at most 500 micrometers, approximately 10 micrometers. According to at least one embodiment of the method, a second function is provided dependent on the first function, wherein the second function is characteristic for a magnetic field established by the two permanent magnets. Exemplarily, the second function comprises further variables and / or further parameters being characteristic for the magnetic field produced by the permanent magnets with the arbitrary shape according to the first function. Exemplarily, the permanent magnets have a homogeneous magnetisation. The magnetic field is produced dependent on a shape of the permanent magnets. For example, a gradient of magnitudes of the magnetic field for different positions on P2023,1262 WO N September 12, 2024 - 4 - the trapping axis are characteristic for a magnetic field gradient along the trapping axis, in particular dependent on the shape. Exemplarily, a part of the magnetic field of a small (dx, dy) segment of the permanent magnets can be calculated as a field of a magnetic dipole. A total magnetic field of the permanent magnets in any point in space x, y and z is calculated, for example, as an integral over all small segments of the permanent magnets. The magnetic field gradient can be calculated as an integral similarly. According to at least one embodiment of the method, a first constraint is provided characteristic for a predetermined point in lateral direction and vertical direction where the magnetic field has to be vanished. In particular, the predetermined point is arranged in vertical direction above the top surfaces of the permanent magnets. Exemplarily, the predetermined point is arranged between the two permanent magnets in lateral direction. This predetermined point does not overlap with the permanent magnets in lateral directions in top view. The magnetic field has in particular a magnitude being zero at the predetermined point. A predetermined distance in vertical direction between the predetermined point and the top surfaces is, for example, at least three times larger than the predetermined thickness of the permanent magnets. Exemplarily, the predetermined distance of the predetermined point is at least 20 micrometers, and at most 1 mm, approximately 100 micrometers. P2023,1262 WO N September 12, 2024 - 5 - According to at least one embodiment of the method, an extremum of a functional is determined dependent on the first constraint, wherein the functional is dependent on the second function. Exemplarily, a Lagrangian function is defined which is dependent on the functional and the first constraint. In particular, Euler-Lagrange equations are calculated based on the Lagrangian function, being a set of differential equations. The Euler-Lagrange equations are used to find a result function characteristic for the shape of the permanent magnets that extremizes the functional under the consideration of the first constraint. This is that the extremum is, exemplarily, characteristic for the result function. The first constraint is, for example, included using a method of Lagrange multipliers. Exemplarily, the Euler-Lagrange equation can be solved analytically and the result function is obtained in an implicit form represented by f(x,y) = 0. According to at least one embodiment of the method, the shape of the two permanent magnets is determined dependent on the extremum such that a gradient of magnitudes of the magnetic field is maximized above the two permanent magnets with the magnetic field being vanished at the predetermined point. In particular, the shape is determined based on the result function that extremizes the functional. With such a shape of the two permanent magnets, the gradient of magnitudes of the magnetic field is maximized along the trapping axis. P2023,1262 WO N September 12, 2024 - 6 - The method described herein above is, exemplarily, performed in the order indicated. The method described herein above is, exemplarily, a computer implemented method. In particular, magnetic field gradients are applied for certain types of quantum computers and in other fields, e.g. quantum metrology. An idea of the method described herein is, inter alia, to determine the shape of the permanent magnets such that a maximal gradient results at a distance above the permanent magnets. Advantageously, the method works for an arbitrary magnetization direction of the permanent magnets. The method further advantageously works to optimize the gradient of a particular magnetic field component, or the gradient of a projection of the magnetic field on an arbitrary direction. According to at least one embodiment of the method, the two permanent magnets with the determined shape are incorporated in the ion trap. This is that the gradient of magnitudes of the magnetic field is maximized at the trapping axis. Advantageously, a resonance frequency of each of the ions at the trapping axis, on which the magnetic field gradient acts, is unique for each trapped ion due to the gradient. According to at least one embodiment of the method, a magnetization direction of the permanent magnets are equal to one another. The magnetization direction is characteristic for a predominant orientation of magnetic moments of a material of the permanent magnets. Exemplarily, the material of the permanent magnets is equal to one another and the material is equally magnetised. In particular, the magnetization directions point in the same direction. P2023,1262 WO N September 12, 2024 - 7 - Exemplarily, the magnetization direction of the first permanent magnet and the magnetization direction of the second permanent magnet are perpendicular to lateral directions, i.e. are oriented along the vertical direction. For example, the magnetization direction points away from the permanent magnets in direction to the trapping axis. According to at least one embodiment of the method, a second constraint is provided characteristic for a predetermined maximal extension of the at least two permanent magnets in lateral directions, and the extremum of the functional is determined dependent on the first constraint and the second constraint. In particular, with the second constraint being a predetermined maximal surface area of the permanent magnets and the predetermined thickness of the permanent magnets, the second constraint provides an additional condition of a limited volume of the permanent magnets. Exemplarily, the predetermined maximal extension is at least 10 micrometers, and at most 2 mm, approximately 300 micrometers. Advantageously, a maximal gradient of magnitudes of the magnetic field is achieved for a predetermined amount of magnetic material at a predetermined distance. According to at least one embodiment of the method, the extremum is characteristic for a result function dependent on at least one coefficient, and for determining the coefficient, the result function is substituted in the first constraint and / or the second constraint. P2023,1262 WO N September 12, 2024 - 8 - According to at least one embodiment of the method, the shape of the at least two permanent magnets is dependent on the result function and the at least one coefficient. Furthermore, a system for determining a shape of at least two permanent magnets incorporated in an ion trap is described. The system is configured to perform the method described herein. Therefore, all features and embodiments disclosed in connection with the method are also disclosed in connection with the system and vice versa. In particular, the system can be a personal computer with a graphical user interface, GUI. The GUI is configured to receive inputs such as at least one of the predetermined thickness of the permanent magnets, the predetermined distance of the predetermined point, material parameters of the permanent magnets. Additionally, an ion trap is described. The ion trap comprises the two permanent magnets, which shapes are determined by the method described herein. Therefore, all features and embodiments disclosed in connection with the method are also disclosed in connection with the ion trap and vice versa. According to an embodiment, the at least two permanent magnets are provided on a substrate of the ion trap. In addition a computer program is specified comprising instruction which, when the computer program is executed by a computer, causes the computer program to execute the method described herein. Further, a computer readable storage medium is specified on which the computer program described herein is stored. P2023,1262 WO N September 12, 2024 - 9 - In the following, the method and the ion trap are explained in more detail with reference to exemplary embodiments and the associated Figures. Figure 1 shows a flowchart of the method for determining a shape of two permanent magnets incorporated in an ion trap according to an exemplary embodiment. Figure 2 shows an ion trap according to an exemplary embodiment. Elements that are identical, similar or have the same effect are given the same reference signs in the Figures. The Figures and the proportions of the elements shown in the figures are not to be regarded as true to scale. Rather, individual elements may be shown exaggeratedly large for better representability and / or for better comprehensibility. Method stage S1 according to the exemplary embodiment of Figure 1 comprises that a first function is provided being characteristic for an arbitrary shape of two permanent magnets 2 in lateral directions and having a predetermined thickness 6 in vertical direction. In method stage S2, a second function is provided dependent on the first function, wherein the second function is characteristic for a magnetic field established by the two permanent magnets 2. A first constraint is provided in method stage S3, wherein the first constraint is characteristic for a predetermined point 7 in lateral direction and vertical direction where the magnetic field has to be vanished. P2023,1262 WO N September 12, 2024 - 10 - An extremum of a functional is determined, wherein the functional is dependent on the second function in method stage S4, wherein the determination is dependent on the first constraint. Finally, in method stage S5, the shape of the two permanent magnets 2 is determined, wherein the determination depends on the extremum such that a gradient of magnitudes of the magnetic field is maximized above the two permanent magnets with the magnetic field being vanished at the predetermined point 7. The ion trap 1 in connection with the exemplary embodiment of Figure 2 comprises the two permanent magnets 2 with the shapes determined by the method described in Figure 1. The two permanent magnets 2 have a first permanent magnet 3 and a second permanent magnet 4, both being planar permanent magnets having a main extension plane in lateral directions x, y. The first permanent magnet 3 and the second permanent magnet 4 are arranged on a substrate 5 of the ion trap 1 being spaced apart from one another in lateral directions x, y. In particular, the two permanent magnets 2 have an equal magnetisation direction indicated by the two arrows being perpendicular to the main extension plane of the permanent magnets 2. The predetermined point 7 is arranged above and between the first permanent magnet 3 and the second permanent magnet 4 at a predetermined distance in vertical direction z. The invention is not limited to the exemplary embodiments by their description. Rather, the invention encompasses any new P2023,1262 WO N September 12, 2024 - 11 - feature as well as any combination of features, which in particular includes any combination of features in the claims, even if this feature or combination itself is not explicitly indicated in the claims or exemplary embodiments.

[0002] P2023,1262 WO N September 12, 2024 - 12 - Reference signs 1 ion trap 2 permanent magnet 3 first permanent magnet 4 second permanent magnet 5 substrate 6 predetermined thickness 7 predetermined point S1..S6 method stages

Claims

P2023,1262 WO N September 12, 2024 - 13 - Claims 1. Method for determining a shape of two permanent magnets (2) incorporated in an ion trap (1), comprising - providing a first function characteristic for an arbitrary shape of the two permanent magnets (2) in lateral directions and having a predetermined thickness(6) in vertical direction, and - providing a second function dependent on the first function characteristic for a magnetic field established by the two permanent magnets (2), - providing a first constraint characteristic for a predetermined point (7) in lateral direction and vertical direction where the magnetic field has to be vanished, - determining an extremum of a functional dependent on the first constraint, wherein the functional is dependent on the second function, and - determining the shape of the two permanent magnets (2) dependent on the extremum such that a gradient of magnitudes of the magnetic field is maximized above the two permanent magnets (2) with the magnetic field being vanished at the predetermined point (7).

2. Method according to claim 1, wherein - the two permanent magnets (2) with the determined shape are incorporated in the ion trap (1).

3. Method according to one of the claims 1 or 2, wherein - a magnetization direction of the permanent magnets (2) are equal to one another.

4. Method according to one of the claims 1 to 3, whereinP2023,1262 WO N September 12, 2024 - 14 - - a second constraint is provided characteristic for a maximal extension of the at least two permanent magnets (2) in lateral directions, and - the extremum of the functional is determined dependent on the first constraint and the second constraint.

5. Method according to claim 4, wherein - the extremum is characteristic for a result function dependent on at least one coefficient, and - for determining the coefficient, the result function is substituted in the first constraint and / or the second constraint.

6. Method according to claim 5, wherein - the shape of the at least two permanent magnets (2) is dependent on the result function and the at least one coefficient.

7. System for determining a shape of at least two permanent magnets (2) incorporated in an ion trap (1), wherein the system is configured to perform the method according to one of the preceding claims.

8. Ion trap (1), comprising - the at least two permanent magnets (2) with the determined shape according to one of the claims 1 to 3, wherein - the at least two permanent magnets (2) are provided on a substrate (5) of the ion trap (1).

9. Computer program comprising instructions which, when the computer program is executed by a computer, cause the computer program to execute the method according to one of the claims 1 to 6.P2023,1262 WO N September 12, 2024 - 15 - 10. Computer-readable storage medium on which the computer program according to claim 9 is stored.

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