A method and system for reducing the amplitude of an oscillating electric field at the equilibrium position of a trapped ion

Abstract

A method of reducing the magnitude of a quasi-static electric dipole field at the null position of an oscillating electric quadrupole field of an ion trap, comprising: trapping at least one ion in a t

Description

Field 5 The present disclosure relates to a method, system and software instructions for reducing the magnitude of a quasi-static dipole electric field at the position of a null of an oscillating electric quadrupole field. The application also relates to an optical clock; a quantum computing system; a quantum simulator system; a trapped ion electric field sensor; a 10 trapped ion quantum network node and a trapped ion force sensor comprising the system. In particular, this disclosure relates to taking advantage of the electric field dependence of the equilibrium position of a trapped ion in order to identify imperfections, in the form of a dipole electric field at the null of an oscillating quadrupole field, in the trapping electric field and to use identified imperfections in order to reduce the magnitude of the dipole field at 15 the said null. Summary According to a first aspect of the present disclosure, there is provided a method of reducing 20 the magnitude of a quasi-static electric dipole field at the null position of an oscillating electric quadrupole field of an ion trap, the method comprising: trapping one or more ions in a trapping electric field, wherein the trapping electric field comprises the oscillating electric quadrupole field and wherein the trapping electric field comprises an electric field amplitude which is a function of an electric field amplitude 25 of the oscillating electric field; inducing a change in an equilibrium position of one of the one or more trapped ions and measuring said change using an interferometry sequence comprising: applying at least two coherent laser pulses to the one of the one or more trapped ions using two or more laser fields with different wavevectors; 30 changing at least one of the driving laser field and the trapping electric field amplitude between successive coherent laser pulses, wherein the driving laser field is changed between successive coherent laser pulses at least once during the interferometry sequence; and measuring a state of the one of the one or more trapped ions after the 35 application of the at least two coherent laser pulses; repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the one or more trapped ions; and 11 04 23 determining a probability that the one or more trapped ions change state during the interferometry sequence based on the plurality of measurements of the state of the ion. In one or more embodiments, the method may further comprise adjusting the trapping 5 electric field based on the determined probability in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating electric quadrupole field of the ion trap. In one or more embodiments, the method may further comprise adjusting one or more parameters of the system in which the one or more trapped ions are implemented to account for unwanted electric field effects. In either approach, the 10 probability that the trapped ion, or the ions, changes state during the interferometry sequence is indicative of the magnitude of the quasi-static electric dipole field at the null of the oscillating electric quadrupole field of the ion trap and the effects on the system in question can be reduced by either adjusting the trapping electric field or adjusting the parameters of the system. In yet other embodiments, a plurality of probabilities may be 15 determined and the effects on the system in question may be reduced based on this determined plurality of probabilities. In one or more embodiments, determining the probability of the trapped ion being in a given state may comprise calculating the statistical likelihood of the ion moving from a first 20 state to a second state during the interferometry sequence. In one or more embodiments, the state of the ion may refer to the electronic state in which an unpaired valence electron is situated. In one or more embodiments, the state of the ion may refer to the electronic state of one or more valence electrons in an atomic ion, or a molecular orbital state of a molecular ion. In one or more embodiments, the state of the ion may refer to an atomic 25 hyperfine state of an atomic ion, or a molecular hyperfine state of a molecular ion. In one or more embodiments, adjusting the electric field may comprise one or more of: altering the voltage applied to one or more compensation electrodes; moving one or more electrodes configured to generate the trapping electric field; and changing the voltage on 30 one or more electrodes configured to generate the trapping electric field. In one or more embodiments, the trapping electric field may further comprise a static electric field and wherein the trapping electric field amplitude is additionally comprised of an electric field amplitude of the static electric field. In one or more embodiments, repeating the interferometry sequence may be performed the plurality of times by one or a combination of: 11 04 23 performing the interferometry sequence on the same trapped ion a plurality of times; and trapping a plurality of ions in the oscillating electric field and performing the interferometry sequence on each of the ions. 5 In one or more embodiments, the first laser pulse may comprise a resonant pi / 2 pulse and the final laser pulse may comprise a resonant pi / 2 pulse and, in sequences comprising more than two laser pulses, the other laser pulses may comprise resonant pi pulses. 10 In one or more embodiments, controlled phase differences between the coherent laser pulses may be chosen such that the probability of the ion changing state depends on the strength of the offset electric field. This may be achieved by using a total control phase, which is a function of the controlled phase differences between the pulses, of pi / 2. In one or more embodiments, a single probability may reveal information about the offset field. In 15 one or more embodiments, a plurality of probabilities may be used to learn about the offset field. In one or more embodiments, the trapping electric field amplitude may be changed between two different trapping electric field amplitude values during the pulse sequence. 20 In one or more embodiments, each laser pulse may be provided at least a predetermined delay after the previous laser pulse. In one or more embodiments, the steps of: 25 repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion; 30 may be performed a first plurality of times wherein, for each repeat of these steps in the first plurality of times, different controlled phase differences between the coherent laser pulses are used; and wherein adjusting the trapping electric field may be based on the first plurality of measurements of the probability. In one or more embodiments, the steps of: 11 04 23 repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion; 5 may be performed a second plurality of times wherein, for each repeat of these steps in the second plurality of times, a different set of trapping electric field amplitudes is used during the pulse sequence; and wherein adjusting the trapping electric field may be based on the second plurality of measurements of the probability. 10 In one or more embodiments, the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the 15 interferometry sequence based on the plurality of measurements of the state of the ion; may be performed a third plurality of times wherein, for each repeat of these steps in the third plurality of times, a different number of coherent laser pulses is applied; and wherein adjusting the trapping electric field may be based on the third plurality of measurements of the probability. 20 In one or more embodiments, for each of the first plurality of times the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the 25 interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the 30 interferometry sequence based on the plurality of measurements of the state of the ion may be repeated the second plurality of times such that a plurality of probabilities are obtained at different combinations of controlled phase differences and different sets of trapping electric field amplitudes; and wherein adjusting the trapping electric field may be based on all of the determined 35 probabilities. In one or more embodiments, for each of the first plurality of times the steps of: 11 04 23 repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, 5 the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion 10 may be repeated the third plurality of times such that a plurality of probabilities are obtained at different combinations of controlled phase differences and different numbers of coherent laser pulses; and wherein adjusting the trapping electric field may be based on all of the determined probabilities. 15 In one or more embodiments, for each of the second plurality of times the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the 20 interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the 25 interferometry sequence based on the plurality of measurements of the state of the ion may be repeated the third plurality of times such that a plurality of probabilities are obtained at different sets of trapping electric field amplitudes and different numbers of coherent laser pulses; and wherein adjusting the trapping electric field may be based on all of the determined 30 probabilities. In one or more embodiments, for each of the first plurality of times the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and 35 determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: 11 04 23 repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion 5 may be repeated the second plurality of times; and that for each of the second plurality of times the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the 10 interferometry sequence based on the plurality of measurements of the state of the ion, the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the 15 interferometry sequence based on the plurality of measurements of the state of the ion may be repeated the third plurality of times such that a plurality of probabilities are obtained at different combinations of controlled phase differences and different sets of trapping electric field amplitudes and different numbers of coherent laser pulses; and wherein adjusting the trapping electric field may be based on all of the determined 20 probabilities. In one or more embodiments, a first time the steps of: repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and 25 determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are performed, the method may comprise probing the offset electric field E along a first direction; and a subsequent time the steps of: 30 repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are performed, the method may comprise probing the electric field E along a second direction, 35 different to the first direction wherein adjusting the trapping electric field may be based on the first and subsequently determined probabilities. It will be appreciated that the plurality of repetitions of the 11 04 23 interferometry sequence of the first time and the plurality of repetitions of the interferometry sequence of the subsequent time may be performed either sequentially, i.e., wherein all of the repetitions of the first time are performed followed by all of the repetitions of the subsequent time, or may be performed in an interleaved matter, i.e., one or more of the 5 first repetitions of the first time may be taken followed by one or more of the repetitions of the subsequent time followed by one or more of the repetitions of the first time and so on. In yet other embodiments, a plurality of probabilities may be determined in each case and the trapping electric field may be adjusted based on these determined pluralities of probabilities. 10 In one or more embodiments, each of the first and second direction may have one of: a directional vector entirely in the plane of the oscillating electric field; and a directional vector having a component out of the plane of the oscillating electric field. In these embodiments, the method may be performed with a system arranged as a 15 linear Paul trap. In one or more embodiments, the first and second directions may be relatively orthogonal directions. 20 In one or more embodiments, the method may further comprise: measuring a detuning of a laser from a transition resonance frequency using interferometry by: applying a first laser pulse to the trapped ion when the electric field amplitude comprises a fixed electric field amplitude and the first laser pulse has a 25 first phase; applying a second laser pulse to the trapped ion when the electric field amplitude comprises the fixed electric field amplitude and the second laser pulse is driven by the same laser field with the same wavevector as the first laser pulse and the second laser pulse has a second phase different to the first phase; and 30 measuring a state of the ion after the application of the first and second laser pulses; repeating the process of measuring the detuning of the laser a plurality of times in order to obtain a plurality of measurements of the state of the ion; determining a fixed electric field amplitude probability of the trapped ion being in 35 the given state based on the plurality of measurements of the state of the ion; wherein detuning of the laser may be accounted for based on the fixed electric field amplitude probability. 11 04 23 In one or more embodiments, the method may comprise alternating between conducting interferometry sequences for probing the offset field E and conducting interferometry sequences for probing the detuning of the laser field from a transition resonance 5 frequency. In one or more embodiments, the same predetermined delays may be used in both types of interferometry sequence. In one or more embodiments, the average of the square of the amplitude of the oscillating electric field of the ion trap over the course of the pulse sequence, comprising any 10 initialisation step; the coherent pulses and the measurement, may be equal to the square of the amplitude of the oscillating electric field of the ion trap during an operational mode. In one or more embodiments, the state of the ion may be measured by fluorescence detection. In one or more embodiments, the fluorescence detection may be preceded by a quantum logic transfer step of quantum logic spectroscopy experiments. In one or more 15 embodiments, the ion trap comprises a linear Paul trap or a ring Paul trap. According to a second aspect of the present disclosure, there is provided a system configured to reduce the magnitude of a quasi-static electric dipole field at the null position of an oscillating electric quadrupole field of an ion trap comprising: 20 a plurality of electrodes configured to generate a trapping electric field for trapping one or more ions wherein the trapping electric field comprises the oscillating electric quadrupole field and wherein the trapping electric field comprises an electric field amplitude which is a function of an electric field amplitude of the oscillating electric field; a first laser beam and a second laser beam configured to apply laser pulses 25 to the trapped ion and a detector; the system configured to induce a change in equilibrium position of one of the one or more trapped ions and use interferometry to measure said change using an interferometry sequence, by controlling: the laser beams to apply coherent laser pulses to the one of the one or 30 more trapped ions and to change at least one of: the laser beam used to drive the coherent pulses and the trapping electric field amplitude between application of successive coherent laser pulses, wherein the laser beam used to drive the coherent pulses is changed between application of successive coherent laser pulses at least once during the interferometry sequence; and 35 the detector to measure the state of the one of the one or more trapped ions after the application of the coherent laser pulses; wherein the system is further configured to: 11 04 23 repeat the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the one or more trapped ions; determine a probability that the one or more trapped ions change state during the interferometry sequence based on the plurality of measurements of the state of the ion; 5 and adjust the trapping electric fields based on the probability in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating electric quadrupole field. In yet other embodiments, a plurality of probabilities may be determined and the trapping electric field may be adjusted based on this determined 10 plurality of probabilities. According to a third aspect, there is disclosed a computer readable medium having stored thereon software instructions that, when executed by a processor, cause the processor to generate control signals to cause a system of the second aspect to perform the method of 15 the first aspect. According to a fourth aspect of the present disclosure, there is provided an optical clock comprising the system of the second aspect. 20 According to a fifth aspect of the present disclosure, there is provided a quantum computing system comprising the system of the second aspect. According to a sixth aspect of the present disclosure, there is provided a quantum simulator system comprising the system of the second aspect. 25 According to a seventh aspect of the present disclosure, there is provided a trapped ion electric field sensor comprising the system of the second aspect. According to an eighth aspect of the present disclosure, there is provided a trapped ion 30 force sensor comprising the system of the second aspect. Brief Description of the Drawings One or more embodiments will now be described by way of example only with reference 35 to the accompanying drawings in which: 11 04 23 Figures 1A - 1B show an example of the electric quadrupole arrangement configured to generate the oscillating electric field of an ion trap; Figure 2 shows an example of the time averaged effect of the oscillating electric field of Figures 1A and 1B of the ion trap; 5 Figure 3 shows an example embodiment of a method for reducing the magnitude of a quasi-static electric dipole field at the null position of the oscillating electric quadrupole field of an ion trap; Figure 4 shows an example pulse sequence of laser pulses and the corresponding electric field amplitude during said pulse sequence; 10 Figure 5 shows a second example pulse sequence of laser pulses and the corresponding electric field amplitude during said pulse sequence; Figure 6 shows an example system comprising a linear ion trap; Figure 7 shows the example system comprising an ion trap in a view looking along the axial direction; 15 Figure 8 shows experimental results of changing the amplitude of the trapping electric field on the Rabi frequency with which a transition is driven; and Figure 9 shows experimental results of changing an electric field offset on the phase [(d>la - ¢^) - ($2a - ¢^)]. 20 Detailed Description Ions can be trapped in a variety of configurations of ion traps by using arrangements of electric fields. These arrangements can comprise at least one oscillating electric field 25 (such as a radio frequency, RF, quadrupole field) and, in some examples, may also include one static electric field. Figures 1A - 1B show an example of how an ion is trapped in an oscillating electric field. In this example, at a first time instant, diagonally opposing electrodes are charged with the same polarities. As shown in Figure 1A, a first pair of electrodes 101 A, 101 Bare positively charged and a second pair of electrodes 101C, 101D 30 are negatively charged. Assuming a positively charged ion 102, an ion 102 near the centre of the trap is attracted towards the negatively charged electrodes 101C, 101D and it is repulsed from the positively charged electrodes 101 A, 101B. As shown in Figure 1B, at a second time instant, the charges of the electrodes 101A, 101B, 101C, 101D may be inverted such that the first pair of electrodes 101 A, 101B are negatively charged and the 35 second pair of electrodes 101C, 101D are positively charged. This causes the ion to then be attracted to electrodes 101 A, 101B and repulsed from electrodes 101C, 101D. By alternating between the configuration of Figure 1A and the configuration of Figure 1B, the 11 04 23 time-averaged force acting on the ion may be towards the centre of the trap such that the ion is dynamically trapped, as is simplistically shown in Figure 2. It will be appreciated that the examples shown in Figures 1A - 1B show a simplified two-5 dimensional example of an ion trap. In order to provide for trapping in the third dimension, out of the plane in the examples of Figures 1A - 1B, several different options can be used. In the arrangement of a linear ion trap, a static (DC) electric field may be generated by electrodes having a like-charge arranged on either side of the ion in the third dimension. 10 The time-averaged position of a trapped ion shown in Figure 2 is referred to as the trapped ion’s equilibrium position. At a trapped ion’s equilibrium position the time-averaged electric field is zero. Usually ion traps are configured with the aim of having the equilibrium position of a trapped ion coincide with a null of the oscillating trapping electric field. Trap imperfections and external field sources may give rise to a slowly varying (quasi-static) 15 unwanted dipolar electric field near the centre of an ion trap. A quasi-static offset field at the null position of the oscillating field causes the equilibrium position of a trapped ion to be shifted from the null of the oscillating field by qEt Eq 1 Ut »----r mail Where orthogonal directions defined by the ion’s secular motion are indexed by i, ut is the displacement of the ion equilibrium position from the null position of the oscillating field in 20 the i direction, q is the ion charge, Et is the component of the quasi-static offset field in the i direction, m is the ion mass, cot is the frequency of the ion’s secular motion in the i direction. As a result, a trapped ion will experience an oscillating electric field at its equilibrium position. This unwanted oscillating field will cause the ion to exhibit additional motion at the frequency of the oscillating field, called excess micromotion. This unwanted 25 field will also exacerbate the Stark effect on the energy levels of the ion. If effects of the unwanted offset field on the ion can be accurately measured, then information about the unwanted offset electric fields may be determined and, therefrom, it may be possible to make changes to the system in order to account for the unwanted offset 30 electric field. It may also be possible that if effects of the unwanted offset field on the ion can be accurately measured, changes may be made to the system to reduce these effects, and this may include reduction of the magnitude of the unwanted offset field. It is possible to measure excess micromotion using techniques such as the measurement 35 of modulation of ion fluorescence as a result of micromotion, and measuring the strength 11 04 23 of resonance sidebands in transition spectra as a result of micromotion. The latter of these techniques was employed in Ref. D2. The techniques mentioned above, however, suffer from the disadvantage of lower resolution compared to that of the systems and methods that are described below. One of the disadvantages of the ion fluorescence measurement 5 is that the measurement results are sensitive to the radiation pressure of the laser field used. One of the disadvantages of the sideband method is that a measurement result in one direction gives information about the magnitude of a component of the offset field, but not about the sign of the component of the offset field. These disadvantages do not apply to the technique disclosed below. 10 It is possible to measure the transition frequency between two energy levels, this is termed spectroscopy. Because the unwanted offset electric field causes the trapped ion to experience an unwanted oscillating electric field at its equilibrium position, which in turn causes the energy levels of the ion to be shifted via the Stark effect, it is possible to use 15 spectroscopy measurements to infer information about the unwanted offset electric field. This approach may be particularly effective when highly polarizable states are employed, such as the Rydberg states in Ref. D1. It is noted that in the laser pulse sequence of Ref. D1 the laser pulses were not phase coherent, and so, that laser pulse sequence does not constitute an interferometry sequence. The spectroscopy technique suffers from the 20 disadvantage of lower resolution compared to that of the systems and methods that are described below. The technique disclosed below may provide for determination of the unwanted offset electric field at a higher accuracy and in a shorter interrogation time than achieved using 25 prior techniques. Another effect sensitive to the unwanted offset field is the change of the trapped ion equilibrium position when the amplitude of the trapping fields is changed. A change of the amplitude of the trapping fields causes the ion’s secular frequencies to change a)u -> o)2i 30 and the ion’s equilibrium position to change: qE^ 1 1 \ Eq. 2 = u2i - uu «-- ------ m \a2i2 This change in equilibrium position Au can be detected using an imaging system. The resolution of such a technique in the object plane is limited by the resolution with which the position of the trapped ion can be determined, where that resolution is the diffraction limit. Further, this technique is less sensitive to movement of the ion out of the object 35 plane, which gives rise to defocussing. This technique is less sensitive to a change of ion 11 04 23 equilibrium position Au, and to the offset electric field E that causes it, than the technique presented herein. Figure 3 shows a method 300 of reducing the magnitude of an offset field at the null 5 position of an oscillating electric field used for trapping an ion. The oscillating electric field may oscillate at RF frequencies. However, it will be appreciated that the oscillating electric field may oscillate at any frequency suitable to maintain the trapping of the trapped ion. For example, the oscillating electric field may 10 oscillate at frequencies between 10kHz and 10GHz. In some examples, the oscillating electric field may oscillate at frequencies between 1 and 100 MHz. In some examples, the oscillating electric field may oscillate at frequencies between 5 and 20 MHz. The oscillating quadrupole electric field may be generated by four electrodes, as described 15 with reference to Figure 1, where a first pair of diametrically opposed electrodes of the four electrodes are each configured to have a first voltage and a second pair of diametrically opposed electrodes of the four electrodes are each configured to have a second voltage different from the first voltage. The electrodes are configured such that the polarity of the first electrode pair is opposite of that of the second electrode pair, i.e., when the first pair 20 of electrodes have a positive charge thereon, the second pair of electrodes have a negative charge thereon and vice versa. The magnitude of the potential at the electrodes may be varied during operation, as is described further below. In one or more examples the voltage on the first pair of electrodes may be set to a fixed value while an RF voltage is applied to the second pair of electrodes. 25 It will be appreciated that, while quadrupole is often used herein to refer to the structure of the type of electric field, other arrangements of electric field may be implemented. For example, an octupole electric field, or even higher order, may be implemented instead. In yet other examples, combinations of quadrupole, octupole or higher order electric fields 30 may be implemented. Further, any reference to quadrupole, octupole or other order of electric field arrangement does not preclude the use of one or more compensation electrodes configured to contribute to the electric field and, thereby, reduce the dipole electric field magnitude at the position of the null of the oscillating electric field. 35 By reducing the magnitude of the dipole offset electric field at the position of the null of the oscillating electric field, the dependence of the ion’s equilibrium position on the electric field amplitude is reduced. Reducing the dependence of the equilibrium position on the 11 04 23 electric field amplitudes advantageously reduces the excess micromotion of the ion and also reduces the undesired Stark effect on the states of the ion. Herein, ion may refer to any of an atomic ion or a molecular ion where, in either case, the 5 ion may comprise a single valence electron or a plurality of valence electrons. The ion may have a hyperfine structure. The method may comprise trapping 301 at least one ion in a trapping electric field wherein the trapping electric field comprises an oscillating electric quadrupole field and may further 10 comprise a static quadrupole electric field. The trapping electric field comprises an electric field amplitude which is a function of an electric field amplitude of the oscillating electric field and the electric field amplitude of the static electric field. As such, it will be appreciated that references to changing the trapping electric field amplitude may refer to changing one or both of the oscillating electric field amplitude and the static electric field amplitude. The 15 oscillating and static electric field amplitudes may be changed by changing the amplitude of the voltages applied to the electrodes configured to generate those fields. A change of the trapping electric field amplitude would cause the ion’s secular frequencies to change (¾ ^21- As such, with reference to Equation 2, this would cause the ion equilibrium position to change by Au. 20 The method further comprises inducing a change in an equilibrium position of the at least one trapped ion and measuring said change using an interferometry sequence. As can be seen from Equation 2, changing the ion’s secular frequencies, a>n -> W2 / , results in a shift Au in the equilibrium position of the trapped ion if there is an offset electric field E. The 25 secular frequencies are changed by changing the amplitude of the voltages applied to the electrodes to generate the trapping fields. Performing the interferometry sequence comprises applying 303 a first pi / 2 laser pulse to the trapped ion and subsequently applying 305 M-1 pi laser pulses to the trapped ion and 30 subsequently applying 307 a second pi / 2 laser pulse to the trapped ion. The index j describes the order in which the pulses are applied. The first pi / 2 pulse has index j = 1 and the final pi / 2 pulse has index j = M + 1. The / W+1 laser pulses are phase coherent. The different laser pulses are driven by at least two different laser fields with different wavevectors. The application of a laser pulse causes a change in the state of the ion. 35 More particularly, two different states of the ion may be considered where the laser field resonantly couples the two states. In other words, the laser field is resonant to the transition between the two states. In some examples, the ion might be prepared in a 11 04 23 plurality of initial states and the laser might drive transitions between the plurality of initial states to a corresponding plurality of final states. The method further comprises measuring 309 a state of the ion after the application of the 5 / W+1 coherent laser pulses. It will be appreciated herein that measuring the state is performed after the application of all of the coherent laser pulses (i.e., after the whole pulse sequence has completed), and not individually after each laser pulse such that multiple measurements are obtained. The final state of the ion after interferometry can be measured using any suitable technique. In one or more embodiments, the final state of 10 the ion may be measured using a fluorescence measurement. The final state of the trapped ion may also be measured using a technique used in quantum logic spectroscopy, whereby the state of the ion is coupled to the state of a second ion, a subsequent fluorescence measurement of the state of the second ion reveals the state of the ion. The phase differences of the laser fields during the coherent laser pulses experienced by the 15 ion determines the ion’s final state with: 1 3 Pe = 2 (1 + C°S where pe is the probability of finding the ion in an excited state, e, and the total phase QT satisfies: &T = ©i + Z7=2(-1)7+1 ■ 207 + 0m+i + ?r. if M is even 07. = 0^^2(-1)-^-20,-0^+1, if Mis odd where 07 is the phase of the laser field experienced by the ion during the jth laser pulse. It will be appreciated that this describes the idealised relationship but that experimental 20 imperfections may include errors in pulse areas and decoherence, as such, the probability variation may differ from the presented equation in true experimental conditions. The phase Qj has the components: Qj = Qj + <pj Eq. 5 Where the component Qj can be controlled using, for example, an acousto-optic modulator, while the component depends on both the position of the ion when the jth pulse is 25 applied and on the laser beam which supplies the jth pulse. And thus 1 Eq 6 Pe = - [i + cos(er + cpTy\ Where if M is even Eq. 7 if M is odd 11 04 23 M x 0T = 01 + ^(~1)J + 1 ■ + 0M+1 7=2 M Sy = 6i + lp+1 ■ 26y + 6m+1 + Tt J=2 7 M >. 0T = 01 + ■ 20y - 0M + 1 7=2 M 0^01 + ^(-1)7^.26^-6^, 7=2 7 Because depends on the ion position, ¢T contains information about the ion position or about changes of the ion position. We refer to QT as the total control phase. It will be appreciated that because the phase contributions Qj are controlled, the total control phase Qt can be accounted for. During the jth pulse, when the ion is at the position indexed by p, position up, and when the laser field wavevector is indexed by a, wavevector ka, the phase takes the value <$>pa. The difference between the position-dependent phases experienced at two positions up and uq, when the laser field wavevector is ka, satisfies 10 where kai is the component of the laser field wavevector ka in the / direction, upi is the component of the position up in the i direction, and uqi is the component of the position uq in the i direction. Changing the trap stiffness from a)pi -> a)qi when there is an offset electric field E affecting 15 the trapped ion system causes the ion position to change up uq. By combining Equation 2 and Equation 8 we see the phase difference (Ppa - <t>qa is related to the unwanted electric field E by y qkaiEi / 1 1 \ Eq. 9 pa L m The phase difference <t>p a - <t>q a is sensitive to E along the direction d, where Eq. 10 If the trap stiffness is changed (Dpi —* <Dqi between application of the coherent laser pulses of an interferometry sequence, the phase difference $>pa - <&qa will affect the probability pe of measuring the ion in the state e, as is described by Equation 6. Because $>pa - <&qa 5 depends on E, the probability pe of measuring the ion in the state e will depend on the unwanted electric field E. 11 04 23 At position up the phase difference between two laser fields, indexed by a and / 3, with wavevectors ka and kp, is ($pa - ^pp). If the two laser fields are provided by two laser 10 beams which are derived from the same source, this phase difference depends on the path length different between the beams from the position where they are separated to the position Up. At position uq the phase difference between the two laser fields is (¢^ - ¢^). As can be appreciated from Equation 8, these phase differences are related by 15 (^pa — ^p^) — — '^ai — ^i)(upi — uqi) Eq. 11 i If the change of ion position up -> uq is due to a change of trap stiffness from wpi -> a)qi when there is an unwanted electric field E affecting a trapped ion system, then the phase difference ($pa - OPjS) - (¢^ - ¢^) is related to the unwanted electric field E by (%« - % / j) - (%« - Ev-12 Zq(kai - k^Ej ( 1___ Equation 12 is found by substituting Equation 2 into Equation 11. 20 The phase difference ($pa - Op / 5) - (¢^ - ¢^) is sensitive to E along the direction d, where 11 04 23 By applying interferometry sequences in which different coherent laser pulses are driven by laser fields with at least two different wavevectors, and in which the trap stiffness may be changed between application of the coherent laser pulses, the probability pe of 5 measuring the ion in the state e, or a plurality of probabilities, can be used to determine phases which reveal information about the unwanted electric field E, as can be appreciated from Equation 12 and Equation 9. Each time the interferometry sequence is performed, the state of the ion in that instance 10 will be determined. Repeating the process of performing the interferometry sequence with the at least one trapped ion and measuring the final state of the at least one trapped ion a plurality of times will allow a probability of the trapped ion being in a given state to be calculated. Repeating the process of performing the interferometry sequence with the at least one trapped ion and measuring the final state of the at least one trapped ion a plurality 15 of times may be represented as a loop repeated N times of the steps: initializing 302 the state of the ion; applying 303 the first pi / 2 laser pulse; optionally changing 304 the trapping electric field amplitude between a first trapping electric field amplitude and a second trapping electric field amplitude; applying a sub-loop M-1 times involving applying 305 a pi laser pulse and then optionally changing 306 the trapping electric field amplitude between 20 a first trapping electric field amplitude and a second trapping electric field amplitude; applying 307 the second pi / 2 laser pulse; setting 308 the trapping electric field amplitude to the first trapping electric field amplitude; and measuring 309 the state of the ion. The laser fields driving each laser pulse may have different wavevectors. Between successive coherent laser pulses at least one of: the laser field’s wavevector and the trapping electric 25 field amplitude will be changed. The laser field’s wavevector will be changed between successive coherent laser pulses at least once during the interferometry sequence. The method 300 also includes determining 310 the probability of the trapped ion being in a given state. This may, for example, comprise calculating the fraction of the plurality of 30 measurements in which the state of the ion is e. 11 04 23 It will be appreciated that, generally, one may discuss exciting the ion into a higher state from a lower state. However, the ion may equally be driven into a lower state from a higher state. Additionally, regardless of whether the ion is driven into a higher or lower state, the probability of the ion being in either of those states may be determined at step 310. 5 Having calculated the probability of the ion being in a given state, the method can calculate one or more pieces of information such as or the component of E in a direction d. The method may comprise adjusting 311 the trapping electric field in order to reduce the magnitude of E in the direction d, as discussed in detail below, based on the probability of 10 the ion being in the given state and / or one or more pieces of the calculated information. This may thereby reduce the magnitude of the unwanted offset electric field E at the position of the null of the oscillating trapping electric field. It will be appreciated that one may implement interferometry sequences to gain information 15 about E, which comprise more than two trap stiffness settings, and thus involve more than two equilibrium positions of a trapped ion. Adjusting 311 the trapping electric field may comprise one or more of: applying or changing a static voltage at one or more electrodes configured to 20 generate the trapping electric field; moving one or more of the electrodes configured to generate the trapping electric field relative to the trapped ion; adjusting the trapping electric field by applying or changing a voltage at one or more compensation electrodes; and 25 - moving one or more of the compensation electrodes relative to the trapped ion. Taking any of the above actions will change the local electric field around the ion and, if done based on the probability of the ion being in a given state as described, will advantageously reduce the magnitude of the unwanted offset electric field E at the position 30 of the null of the oscillating trapping electric field. The methods shown in Figure 4 and Figure 5 illustrate an advantage that can be obtained, over the previously disclosed interferometry methods for probing E, by employing interferometry sequences in which the coherent laser pulses are driven using at least two 35 different laser fields with different wavevectors. These methods allow similar sensitivities to E to be attained as attained using the previously disclosed methods, while requiring fewer changes of the trap stiffness during the pulse sequence. Because trap stiffness 11 04 23 changes can cause heating of the ion’s motion, which can give rise to pulse area errors, which in turn can cause a systematic error in estimates of the phases ¢T which are sensitive to the ion position, and thus cause a systematic error in the estimate of E, reducing the number of trap stiffness changes during a pulse sequence can be 5 advantageous. Furthermore, sequences with fewer trap stiffness changes may be implemented in less time than sequences with more trap stiffness changes. In the method shown in Figure 4, the amplitude of the trapping electric field is kept fixed during each implementation of a pulse sequence 401. The sequence 401 is applied first 10 with the fixed trap stiffness a)u 408, and state of the ion is measured 407. Then the sequence 401 is applied with the fixed trap stiffness o)2i 409, and the state of the ion is measured 407. The process is repeated, to determine probabilities pei and pe2 corresponding to the trap stiffnesses and o)2i. First the state of the ion is initialised 402. The pulse sequences are comprised of / W+1 coherent pulses, with a total pulse area of 15 / W*pi. The first coherent pulse 403 has pulse area pi / 2, the final pulse has pulse area pi / 2 406, and the M-1 pulses between the first pulse and the final pulse each has pulse area pi 404 and 405. A laser field with wavevector ka provides the pulses with odd-valued indices, while a laser field with wavevector kp provides the pulses with even-valued indices. As can be appreciated from Equation 7 — ¢1 / 3), when the trap stiffness is a)u Eq. 14 — ¢2 / 3), when the trap stiffness is 6j21 20 Because pe depends on <f)T according to Equation 6, the phase differences ($la - OljS) and (o2« - ¢2^) can be determined from the probabilities pei and pe2- Therefrom the phase difference ($la - OljS) - ($2a - O2jS) can be determined. This phase difference contains information about the unwanted electric field E, according to Equation 12. 25 It will be appreciated that the example pulse sequence in Figure 4 works for sequences with odd-valued M. By making small adjustments, the sequence works for even-valued M. In the method shown in Figure 5 the phase difference ($la - OljS) - ($2a - O2jS) is determined, just as in the method shown in Figure 4. In the method shown in Figure 5 this 30 phase difference is determined from a single probability pe, which is itself determined from a single set of measurements. This is unlike the method shown in Figure 4, in which (ola - $ljS) is determined from a first probability pei and ($2a - O2jS) is determined from a second probability pe2- While the method shown in Figure 4 involves no changes of the trap stiffness during the pulse sequence, the method shown in Figure 5 involves two 11 04 23 changes of the trap stiffness during the pulse sequence. The method shown in Figure 5 involves M+1 coherent laser pulses, with a total pulse area of / W*pi, where M is a multiple of four. The method proceeds as follows: The ion is initialised 502 in state g, with the trap stiffness set 507 to a first value a)^. A pi / 2 pulse is applied 503 using a laser field with 5 wavevector ka. The following M / 2-2 pulses are pi pulses, the laser field driving the pulses alternates between the laser field with wavevector kp 504 and the laser field with wavevector ka 505. Another pi pulse is applied 504 driven by the laser field with wavevector kp. The trap stiffness is changed to the second value 508. The following M / 2 pulses are pi pulses, the laser field driving the pulses alternates between the laser field with 1 o wavevector kp 504 and the laser field with wavevector ka 505. The trap stiffness is changed to the first value a)u 507. A final pi / 2 pulse is applied using the laser field with wavevector ka 503. The state of the ion is then measured 506. As can be appreciated from Equation 7, the sum of the position-dependent phases is ¢T = y [(¢^ - ¢^) - (¢2^ _ $2 / ?)]- By repeatedly applying the sequence, the probability pe of finishing in the state e can be 15 determined. From pe the phase <f)T can be determined, using Equation 6. This phase difference contains information about the unwanted electric field E, according to Equation 12. A further method is now described, which illustrates a further advantage that can be 20 obtained, over the previously disclosed interferometry methods for probing E, by employing interferometry sequences in which the coherent laser pulses are driven using at least two different laser fields with different wavevectors. In the interferometry methods, the position-dependent phase ¢T reveals information about 25 the offset field E along a direction d. A disadvantage of the previously-disclosed methods is that in a linear Paul trap configuration, when the radial trapping frequencies a>x and are non-degenerate, with (1)X <z, that the phase ¢T can become much more sensitive to the component of E along the x-direction than to the component of E along the y-direction. As a result, the accuracy with which the component of E along the y-direction is measured 30 may be reduced. This problem can be overcome using the following method, which allows the direction d in which ¢T is sensitive to E to be flexibly varied, This method involves a pulse sequence composed of a pi / 2 pulse, followed by M-1 pi pulses, followed by a pi / 2 pulse. The total number of coherent pulses is M+1, and the total pulse area is M*pi. The pulses are indexed by j, which describes the order in which the 35 pulses are applied, j values range from 1 to M+1. The complete set of M + 1 pulses include 11 04 23 four subsets of pulses. The four subsets of pulses are not applied consecutively, rather pulses from the four subsets are interleaved. The pulses within the first subset of pulses are driven by the laser field with wavevector ka, when the trap stiffness is at the first trap stiffness value a)u. The first subset has total 5 pulse area Ma*pi / 2, and the pulses have odd-valued indices j. As can be appreciated from Equation 7 these pulses contribute the phase Ma*<bla to $T. The pulses within the second subset of pulses are driven by the laser field with wavevector ka, when the trap stiffness is at the second trap stiffness value o)2i. The second subset has total pulse area Ma*pi / 2, and the pulses have even-valued indices j. As can be 10 appreciated from Equation 7 these pulses contribute the phase -Ma*$>2a to $t- The pulses within the third subset of pulses are driven by the laser field with wavevector kp, when the trap stiffness is at the first trap stiffness value The third subset has total pulse area MjS*pi / 2, and the pulses have odd-valued indices j. As can be appreciated from Equation 7 these pulses contribute the phase Mp*<t>1p to $T. 15 The fourth subset of pulses are driven by the laser field with wavevector kp, when the trap stiffness is at the second trap stiffness value ay2i. The fourth subset has total pulse area Mp*^\l2, and the pulses have even-valued indices j. As can be appreciated from Equation 7 these pulses contribute the phase -Mp*<b2p to $T. With this sequence the sum of the position-dependent phases is given by: $7=^(^-^ + ^^-^) Eq15 20 By repeatedly applying the sequence and measuring the state of the ion the probability pe can be determined. From pethe phase $T can be determined using Equation 6. The phase $T is sensitive to E along the direction , / 1 1 \ Eq. 16 di = (Maka + Mpkp)(—-—} '^li m2i ' 25 By changing the relative sizes of Ma and Mp, the direction d in which $T is sensitive to E can be varied. If the sequence is changed, such that the pulses within the third subset of pulses have 30 even-valued indices j, while the pulses within in the fourth subset of pulses have oddvalued indices j, then $7 = Eq. 17 And the phase $T is sensitive to E along the direction 11 04 23 , / 1 1 \ Eq. 18 di = (Maka-Mpk^(—-—} And thus the direction d can be varied even further. It will be appreciated that the sequence in Figure 5 displays the special case when Ma = Mp. It will be appreciated that this sequence can be extended to allow d to be varied in three 5 dimensions, by including a fifth subset of pulses and a sixth subset of pulses driven by a laser field with a third wavevector kY. While it is described herein that measurements are sensitive to the offset electric field along a first direction, this direction may be changed by changing the pulses used within a 10 particular pulse sequence or it may be changed by changing the directions of propagation of the laser beams used. This will be appreciated from Equations 13, 16 and 18. Figures 6 and 7 shows an example system 600 comprising a linear ion trap and laser geometry configured to carry out the above-described method. The example ion trap 15 represented in Figures 6 and 7 comprise a linear ion trap 601 using a quadrupole electric field arrangement. It will be appreciated, however, that any suitable ion trap arrangement may be used such as, but not limited to, a hyperbolic ion trap, linear ion trap, a ring ion trap, a cylindrical ion trap, a planar ion trap, a wheel ion trap, or a sandwich ion trap. In this example, a 88Sr+ ion 602 is confined in the linear ion trap 601 and the subsequently 20 described experimental details are suitable for trapping a 88Sr+ ion. It will be appreciated that other ions may be used in this arrangement, and the same or different experimental parameters may be used to trap the ion such as different electrode configurations, different laser arrangements and different laser frequencies. 25 In this example, voltages are applied to four gold-coated blade electrodes 603 to confine the ion in the x and y radial directions. The voltages on these four blade electrodes 603 are configured to generate the oscillating electric field in the x, y plane. Static voltages are applied to two gold-coated endcap electrodes 604 to confine the ion in the, z, axial direction. Any other suitable electrode design may be used. The co-ordinate axes defined 30 in the figures are defined by the ion’s secular motion and the electrode 603, 604 geometry. As such, the electrodes 603 are configured to generate the oscillating electric field in the x, y plane. The electrodes 603 may also generate a static electric field which produces the non-degeneracy of the radial modes, and wy. The electrodes 604 configured to generate the static electric field which provides confinement along the z direction are 35 arranged in diametric opposition along the z direction. It will be appreciated that, while the 11 04 23 electrodes 604 configured to generate the static electric field may start in a diametrically opposed arrangement, the exact relative position of these electrodes 604 may be adjusted in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating electric quadrupole field. The relative position of the electrodes 603 may 5 also be adjusted in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating electric quadrupole field. In the following section, directional vectors will be referred to in the format (x, y, z). In this example, three 674 nm laser beams 605 are provided to illuminate the ion with 10 unnormalized propagation directions of (0, 0, 1), (-1, 1, -^2) and (-1, -1, 0). These independently controllable laser beams may be provided by a single laser or may be provided by a plurality of lasers. In one or more embodiments, the axial (0, 0, 1) laser beam 605 may be configured to propagate through one or more holes in the endcap electrodes. 15 A first laser beam, such as the laser beam arranged along (-1, -1, 0) may be configured to provide coherent laser pulses with a wavevector ka in the plane of the oscillating electric field. In one or more embodiment, a further laser may be provided along another direction kp in the plane, or having components in the plane, of the oscillating electric field which is 20 also configured to provide the coherent laser pulses of the interferometry sequence. As previously described, the probability of the ion 602 being in a given state can be calculated when a laser fields with wavevectors ka and kp are used and, therefrom, information about the electric field E in the d direction may be obtained. The system may be configured such that different pulse sequences are implemented so information about the offset electric 25 field E in different d directions may be determined, that a 2D or 3D measurement of offset electric field E can be determined. It will be appreciated that, where a measurement is sensitive to the offset field E in a direction d which has components du in the plane of the oscillating electric field and dL out of the plane of the oscillating electric field and a value of Ei is being sought which is in the plane of the oscillating electric field, it will be necessary 30 to resolve the components of the electric field in the direction du. The axial laser beam may also be used to measure unwanted electric fields in the z direction by using the method 300 of Figure 3. It will be appreciated that in an ideal linear Paul trap there is no oscillating field along the axial direction, but that imperfections, such as machining imperfections, may give rise to such an oscillating field along the axial direction. This 35 unwanted field may be probed using the technique disclosed herein, using a laser beam which has a wave vector with an axial component together with another laser beam with a 11 04 23 different wavevector. It will be appreciated that, where measurements of the unwanted electric field are sought in the x, y and z directions, it may be particularly beneficial for the laser beams that are configured to provide the coherent laser pulses and for the changes of the trapping field amplitudes and for the pulse sequences to be arranged such that the 5 different directions d are orthogonal to each other. It may also be beneficial for the laser beams that are configured to provide the coherent laser pulses and the changes of the trapping field amplitudes and the pulse sequences to be arranged such that the different directions d are aligned with the x, y and z axes which are defined by the ion’s secular motion. 10 A Doppler cooling laser may be provided, such as the laser in the (1,-1, ^2) direction 605 in Figure 6. The Doppler cooling laser may be configured to provide a laser configured to slow the movement of the ion and, thereby, increase the degree to which the interferometry results are described by Equation 3. An additional Doppler cooling laser beam may be 15 employed when radial secular frequencies of the system are degenerate, such as the laser in the (1,1,0) direction 605 in Figure 7. One of the lasers 605 of the system 600 may comprise a repump laser. The repump laser may be configured to counter optical pumping to an unwanted state driven by the Doppler 20 cooling laser beam. One of the lasers of the system 600 may comprise a quench laser. The quench laser 605 may be configured to transfer an ion from an excited state to the ground state. The Doppler cooling, repump and quench laser beams 605 may be configured to copropagate. Due to experimental set-up restrictions, optical access to the ion may be restricted. Providing for copropagating laser beams 605 may allow the beams 25 to be focussed together and thereby make efficient use of available set-up space. By way of specific example for illustrative purposes, a strontium 88 ion may be initialised in a particular sublevel of its ground state 5Si / 2 by optical pumping. A pulse of 674nm laser light may transfer the ion from state 5Si / 2 mj=1 / 2 to 4D5 / 2 mj=-3 / 2. A pulse of the 1033nm 30 laser light may transfer the ion from 4D5 / 2 to 5P3 / 2, from which it may decay to either sublevel of 5S1 / 2. The process does not affect the ion if it was initially in state 5S1 / 2 mj=-1 / 2. By repeating this process (typically 10 times) if the ion was initially in state 5S1 / 2 mj=1 / 2, it will likely finish in 5S1 / 2 mj=-1 / 2. The 1092nm laser field is turned on during this process to prevent optical pumping to state 4D3 / 2. 35 The system 600 may also comprise one or more compensation electrodes. The compensation electrodes 606 may comprise additional electrodes to which a different 11 04 23 voltage can be applied in order to cause a change in the overall trapping electric field of the system 600. Alternatively, the compensation electrodes 606 may be moved in order to cause a change in the overall trapping electric field of the system 600. Each of the one or more compensation electrodes 606 may comprise a pair of rods to which a voltage is 5 applied. Each of the rods of a compensation electrode 606 may be arranged adjacent to one of the electrodes configured to generate the oscillating electric field. In some embodiments, the first rod in a pair of rods of a compensation electrode may be arranged adjacent to a first oscillating electric field generating electrode and the second rod in the pair of rods of the compensation electrode may be arranged adjacent to a second 10 oscillating electric field generating electrode. It will be appreciated that a first pair of oscillating electric field generating electrodes may be configured to have a fixed voltage applied thereto and a second pair of oscillating electric field generating electrodes may be configured to have a time-varying voltage applied thereto. The first oscillating electric field generating electrode may be one of those configured to have a fixed voltage applied 15 thereto and the second oscillating electric field generating electrode may be one of those configured to have a time-varying voltage applied thereto. The system 600 may further comprise a photon-collection device (not shown) configured to provide a measure indicative of the number of photons emitted by the ion 602 during 20 the measurement step 309 of the interferometry sequence. In one or more embodiments, the photon-collection device may comprise a photomultiplier tube (PMT) or it may comprise another photon-collection device. As has been described already, the interferometry sequence is repeated a plurality of times 25 in order to obtain a plurality of measurements of the state of the ion. Then a probability of the trapped ion being in a given state can be determined. Determining the state of the same ion a plurality of times may comprise repeatedly performing the interferometry sequence on a single trapped ion in an ion trap. Repeatedly performing the interferometry sequence on a single trapped ion may comprise performing the interferometry sequence 30 on the same trapped ion a plurality of times or may comprise performing the interferometry sequence on a first trapped ion in the trap, removing that trapped ion from the trap and trapping a new ion in the same trap and then repeating the interferometry sequence. A combination of these approaches may be taken. Alternatively, where a plurality of ions are trapped in the ion trap, performing the interferometry sequence a plurality of times may be 35 achieved by performing the method on each, or a subset, of the plurality of trapped ions to simultaneously obtain a plurality of measurements. It will further be appreciated that the method may be repeated a plurality of times on a plurality of trapped ions, thereby utilising 11 04 23 a combination of said techniques. In a two-level system {|^>, |e>}, initialized in state \g), application of a laser field resonant to the \g) e) transition couples the states, and causes the state of the system to oscillate between \g), superposition states of \g) and |e>, and state |e). A pi / 2 pulse causes state \g) or state |e) to evolve to a superposition state 5 with equal \g) and |e) components. Application of a pi pulse to a superposition state which has equal \g) and |e) components may cause the system to evolve to a different superposition state which has equal \g) and |e) components or it may remain in the same superposition state, whichever of these outcomes that occurs depends on the phase relation between the superposition state and the laser field. Application of a pi / 2 pulse to 10 a superposition state which has equal \g) and |e) components may cause the system to evolve to state \g), or to state |e>, or to a different superposition state of \g) and |e>, or it may even remain in the same superposition state, whichever of these outcomes that occurs depends on the phase relation between the superposition state and the laser field. 15 A predetermined delay may be provided between the coherent laser pulses. The predetermined delay may be a sufficient time for the trapping electric field to be changed from the first electric field amplitude to the second electric field amplitude. In one or more embodiments, the predetermined time delay may be between 1ps and 20ms. The predetermined time delay may between 5 ps and 20 ps. The predetermined time delay 20 may be 10 ps. In one or more embodiments, the contribution from the controlled phase shifts (which is referred to herein as the total control phase) QT may satisfy QT = n / 2. It will be appreciated that Qt = n / 2 may be achieved using different combinations of controlled phase shifts Qj. 25 It will be appreciated that other values of QT may be used, for instance, QT = 3n / 2 may be used and 0T = 0 may be used. More specifically, in one or more embodiments, the first laser pulse may comprise a resonant pi / 2 pulse and the final laser pulse may also comprise a resonant pi / 2 pulse. 30 Experiments involving two pi / 2 pulses may be referred to as Ramsey interferometry experiments. Ramsey interferometry was employed in Ref. D2 to investigate the coherence time of a trapped ion qubit. Ramsey interferometry was also employed in Ref. D3 to characterise the stability of the amplitude of the RF drive of an ion trap via its effect on the secular motion frequency of a trapped ion. When \<pT\ « 2n and when control 35 phases QT = n / 2 or QT = 3n / 2 are used, the probability of the ion being in the given state may respond most strongly to changes in <f)T because, at this point, the magnitude of the rate of change of the probability pe with respect to ¢T is greatest. Absent of an offset electric field E, a probability of 0.5 may be expected when a phase difference 0T of pi / 2 or 3*pi / 2 is used. In some embodiments, <f)T and therefrom information about the unwanted electric field E, may be estimated based on measurements of the probability of the ion 5 being in the given state when the phase combination 0T is pi / 2 and 3*pi / 2. Where the probability of the ion being in the given state when 0T is pi / 2 is denoted as pe and the probability of the ion being in the given state when QT is 3*pi / 2 is denoted as pe’, the phase < / >T can be estimated from: 11 04 23 Estimation of < / >T using two sets of measurements in this fashion may have the advantage 10 of robustness to errors in the pulse area and to errors caused by decoherence. If N / 2 repetitions are conducted to determine pe and N / 2 repetitions are conducted to determine pe’, when |0T | « 2n and pe “ pe’ “ 0.5, then the uncertainties 2Jpe “ Ape “ 1 A / (2 / V) (due to quantum projection noise, using the normal approximation) and statistical uncertainty A¢T « 1h!N. 15 It will be appreciated that, <f)T and therefrom information about the unwanted electric field E, may be estimated based on measurements of the probability of the ion being in the given state when other phase combinations are used. Where the probability of the ion being in the given state when 0T is 0 is denoted as pe and the probability of the ion being 20 in the given state when 0T is 3*pi / 2 is denoted as pe’, the phase <f)T can be estimated from: / 1 1\ Eq. 20 <pT = arctan2 \pe--,pe~ -J The estimate using the approach described by Equation 20 may be more precise than the estimate using Eq. 19 if the condition \<pT\ « 2n is not satisfied. It will be appreciated that herein, where a phase difference is referred to, the unit of that 25 phase difference will be radians. The omission of “radians” herein is provided for ease of readability and in line with the practice of those skilled in the art and is not intended to imply that the phase differences referred to herein are measured in any other units. Figure 8 shows an example dataset which represents the results of how the Rabi 30 frequency for the \g) |e) transition decreases as the number of changes of the amplitude of the trapping electric field between a first trapping electric field amplitude and a second 11 04 23 trapping electric field amplitude is increased. This is caused by heating of the ion’s motion. If an interferometry sequence used to probe the offset electric field E involves many changes of the amplitude of the trapping electric field, the change in Rabi frequency with the number of changes of the amplitude of the trapping electric field can cause the results 5 to deviate from the description given by Equation 3, and this can cause a systematic shift in the estimate of E. And thus, it may be advantageous to use interferometry methods which require few changes of the amplitude of the trapping electric field, such as in the methods shown in Figure 4 and Figure 5. 10 Figure 9 shows an example dataset which represents how the phase [(¢^ - ¢^) -($2« - $2^)] changes with changes to the voltage applied to a compensation electrode, and thus to changes to the offset electric field E, when the method described in Figure 4 is used. At each voltage value the phase ($la - O1 / 5) was determined by repeatedly conducting the sequence to obtain a plurality of measurements of the state of the ion, using 15 two different values of 0T, and using Equation 20. At each voltage value the phase ($2« - ¢2 / 5) was likewise determined by repeatedly conducting the sequence to obtain a plurality of measurements of the state of the ion, using two different values of 0T, and using Equation 20. At each voltage value the phase [(¢^ - ¢^) - (¢2^ _ ^2 / 5)] was then determined. The phase [(¢^ - ¢^) - (¢2^ _ ^2 / 5)] depends linearly on the voltage 20 applied to the compensation electrode, and on a component of the offset field E, which is consistent with Equation 12 and with the linear dependence of the component of the offset field E on the compensation electrode voltage. The zero-offset voltage was independently determined using another technique, namely the resolved sideband technique. The method breaks down at offset voltages around ±3.2V, due to the periodicity of Equation 6. 25 To obtain estimates of [(¢^ - ¢^) - (¢2^ - ^2 / 5)] over a larger range of compensation electrode voltage values, a smaller change of the trap drive amplitude can be used. Datasets with different values of M allow OT, and therefrom E, to be estimated with different accuracies and from within different ranges. It may be advantageous to combine 30 to use results of measurements using different values of M, to achieve a high accuracy from within a broad range. Such an approach may be similar to methods used in quantum phase estimation techniques. Thus, in some embodiments the method may comprise repeating the interferometry 35 sequence and determining a probability that the trapped ion changes state during the interferometry sequence a first plurality of times. For each repeat of this group of steps in 11 04 23 the first plurality of times, a different set of control phases Qj may be used. Adjusting the trapping electric field may be based on the first plurality of measurements of the probability of the trapped ion being in the given state. 5 Further, in some embodiments the method may comprise repeating the interferometry sequence, and determining a probability that the trapped ion changes state during the interferometry sequence a second plurality of times. For each repeat of this group of steps in the second plurality of times, a different set of trapping electric field amplitudes during the pulse sequence may be used. Adjusting the trapping electric field may be based on 10 the second plurality of measurements of the probability of the trapped ion being in the given state. Further, in some embodiments the method may comprise repeating the interferometry sequence, and determining a probability that the trapped ion changes state during the 15 interferometry sequence a third plurality of times. For each repeat of this group of steps in the third plurality of times, a different value of M may be used. Adjusting the trapping electric field may be based on the third plurality of measurements of the probability of the trapped ion being in the given state. 20 It will be appreciated that varying the control phases Qj and varying the trapping electric field amplitudes and varying M are described as first and second and third pluralities of times or measurements respectively, however, this nomenclature is provided for convenience of description. The nomenclature does not require that measurements of the plurality of control phases must be performed before the measurements of the plurality of 25 trap stiffness changes, or that the measurements of the plurality of trap stiffness changes must be performed before the measurements of the plurality of M values or even that one set of measurements must be provided at all in order to perform, and obtain information from, another set of measurements. 30 It will be appreciated that to obtain the data shown in Figure 9 the pulse sequence was repeated the first plurality of times (two values of QT were used) and the second plurality of times (the trapping electric field amplitude was fixed during application of the pulse sequence, and two different fixed amplitude values were used). 35 In some embodiments, for each of the first plurality of times the steps of performing the interferometry sequence and determining the probability of the trapped ion being in a given state are performed, the same steps are repeated the second plurality of times. In this 11 04 23 way, a plurality of probabilities of the ion being in the given state are obtained at different combinations of control phases and different sets of trapping electric field amplitudes. Adjusting the trapping electric field may then be based on all of the determined probabilities or a subset of those probabilities. 5 In some embodiments, for each of the first plurality of times the steps of performing the interferometry sequence and determining the probability of the trapped ion being in a given state are performed, the same steps are repeated the third plurality of times. In this way, a plurality of probabilities of the ion being in the given state are obtained at different 10 combinations of control phases and different M values. Adjusting the trapping electric field may then be based on all of the determined probabilities or a subset of those probabilities. In some embodiments, for each of the second plurality of times the steps of performing the interferometry sequence and determining the probability of the trapped ion being in a given 15 state are performed, the same steps are repeated the third plurality of times. In this way, a plurality of probabilities of the ion being in the given state are obtained at different sets of trapping electric field amplitudes and different M values. Adjusting the trapping electric field may then be based on all of the determined probabilities or a subset of those probabilities. 20 In some embodiments, for each of the first plurality of times the steps of performing the interferometry sequence and determining the probability of the trapped ion being in a given state are performed, the same steps are repeated the second plurality of times; and that for each of the second plurality of times the steps of performing the interferometry 25 sequence and determining the probability of the trapped ion being in a given state are performed, the same steps are repeated the third plurality of times. In this way, a plurality of probabilities of the ion being in the given state are obtained at different combinations of control phases and different sets of trapping electric field amplitudes and different M values. Adjusting the trapping electric field may then be based on all of the determined 30 probabilities or a subset of those probabilities. It has been described and shown in the above equations that inducing a change in an equilibrium position Au of the at least one trapped ion and measuring said change using interferometry to determine a probability of the trapped ion being in a given state or a 35 plurality of probabilities of the trapped ion being in a given state provides information about the offset electric field, E, along a direction d. Because of this, in order to provide enhanced reduction of the magnitude of the offset electric field E, in one or more embodiments, the 11 04 23 method may comprise performing the method to determine said probability or probabilities sensitive to E along at least each of a first direction di and a second direction d2 different from the first direction. Adjusting the trapping electric field may then be based on the measurement of the probability or probabilities along the first direction, and the 5 measurement of the probability or probabilities along the second direction. The method may equally be implemented along a third direction d3 in order to determine a probability or probabilities along a third direction to provide for information in three dimensions. It will be appreciated, that the control phases 6j may be chosen to achieve phase 10 measurements which are robust against different types of error, caused by, for example, detuning of the laser field from the \g) |e) resonance, or pulse area errors. By measuring the offset field E at different times the drift of E over time can be measured. By taking into account this drift, the magnitude of the offset field E may be further reduced. 15 In some embodiments, where a plurality of probability measurements are being made, such as where a plurality of different probabilities are being measured using different control phases 0j or with different trap stiffness changes, the interferometry sequence may also be conducted using a fixed electric field amplitude and using coherent laser pulses 20 driven by a single laser field. Probabilities derived from such measurements may be used to correct for the time-varying detuning of the laser field from the transition resonance frequency. This will allow systematic offsets in the estimate of to be reduced or corrected. 25 In one or more embodiments, the average of the square of the amplitude of the oscillating electric field of the ion trap over the pulse sequence, comprising any initialisation step; the coherent pulses and the measurement, may be equal to the square of the amplitude of the oscillating electric field of the ion trap during an operational mode. An operational mode may be a mode of operation that the system is designed for, such as an optical clock or a 30 quantum computing system. It will be appreciated that the method defined here is provided to reduce the magnitude of the offset electric field E and consequently the magnitude of the oscillating electric field at the ion’s equilibrium position in order to reduce unwanted effects of excess micromotion and to reduce Stark shifts of energy levels. The trapped ion may then be implemented in any of a wide variety of applications. Such applications will 35 operate under a standard trapping electric field amplitude. It will be understood that the square of the amplitude of the oscillating electric field is proportional to the power dissipated in the system. With greater power dissipation comes greater heating of the ion 11 04 23 trap and changes in the temperature of the ion trap impact the trapping of the ion. As such, it may be beneficial to mitigate changes to the temperature when seeking to reduce the magnitude of the offset electric field E by having the average power dissipated in the system over the pulse sequence be equal to the average power dissipated in the system 5 during its normal mode of operation. There is also provided a computer readable medium having stored thereon software instructions that, when executed by a processor, cause the processor to generate control signals to cause a system such as that shown in Figures 6 and 7 to perform the above-10 described method. The method and system presented herein may be applicable to a wide variety of systems, such as any system where the accurate trapping of an ion in an electric field is required. Such applications may include but are not limited to: an optical clock; a quantum computing 15 system; a quantum simulator system; a trapped ion electric field sensor; and a trapped ion force sensor. In a trapped ion system, an offset electric field that varies in time causes the trapped ion to experience an amount of excess micromotion that varies in time. This in turn causes the 20 Doppler shift on a transition (due to excess micromotion) to vary in time. A varying offset electric field also causes the Stark shift to vary in time. By applying the described method from time to time in a system which utilises one or more trapped ions, the varying offset electric field can be kept under control, and the transition frequencies can be kept stable. Further, if the amount of excess micromotion changes, then the strengths with which 25 transitions can be driven using a laser field change in time. Trapped ion optical clocks require stable transition frequencies, and also accurate knowledge of resonance shifts. Thus, the presently disclosed method and apparatus may provide for improved trapped ion optical clocks. 30 In trapped ion quantum computers, simulators, network nodes and force sensors, different transitions need to be driven with high fidelity. If the strength of transitions changes in time, then the fidelity of operations decreases. Also, if the Doppler shifts and Stark shifts change in time, this causes resonances to shift and this also decreases the fidelity of operations. 35 This is detrimental to their operation. Thus, by way of the advantages described herein, the presently disclosed method and apparatus may provide for improved trapped ion quantum computers, simulators, network nodes and force sensors. 11 04 23 In some examples, it may not be necessary to adjust the trapping electric field based on the determined probability or the determined plurality of probabilities in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating 5 electric quadrupole field of the ion trap. Instead, the method may comprise determining a probability or plurality of probabilities that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, wherein the probability or plurality of probabilities is indicative of the strength of the quasi-static electric dipole field E at a null position of the oscillating electric quadrupole field of 10 the ion trap and, based on the probability or probabilities, parameters of a system in which the trapped ion is implemented may be varied to account for unwanted electric field effects. A system in which the trapped ion is implemented may be any relevant system, such as those discussed above including a trapped ion optical clock, a trapped ion quantum computer, simulator, network nodes or force sensor, for example. For example, in the 15 case of a trapped ion optical clock, instead of adjusting the trapping electric field, the frequencies of the laser fields and the frequency of the optical clock may be adjusted. Similarly, in the case of a trapped ion quantum computer or simulator, instead of adjusting the trapping electric field, the frequencies of the lasers may be adjusted and the pulse areas may be adjusted. Thus, it will be appreciated that the probabilities determined in 20 the plurality of interferometry sequences may be used to correct for effects that result from the offset field E, such as Doppler shifts of the frequencies of transitions, rather than being used to reduce the magnitude of E. Citations 25 D1 XP081170442, GERARD HIGGINS ET AL: “Highly-polarizable ion in a Paul trap”, ARXIV.ORG, CORNELL UNIVERSITY LIBRARY, (2019-04-17) D2 XP012239450, LEOPOLD T ET AL: “A cryogenic radio-frequency ion trap for quantum logic spectroscopy of highly charged ions”, REVIEW OF SCIENTIFIC INSTRUMENTS, vol. 90, no. 7, (2019-07-29), ISSN: 0034-6748, DOI: 10.1063 / 1.5100594 30 D3 US 2017 / 186595 A1 (2017-06-29) 11 04 23

Claims

1. A method of reducing the magnitude of a quasi-static electric dipole field at the null position of an oscillating electric quadrupole field of an ion trap, the method comprising:trapping one or more ions in a trapping electric field, wherein the trapping electric5 field comprises the oscillating electric quadrupole field and wherein the trapping electric field comprises an electric field amplitude which is a function of an electric field amplitude of the oscillating electric field;inducing a change in an equilibrium position of one of the one or more trapped ions and measuring said change using an interferometry sequence comprising:10 applying two or more coherent laser pulses to the one of the one or moretrapped ions, wherein each laser pulse is driven by at least one laser field from at least two different laser fields with different wavevectors, wherein at least one of the driving laser field and the trapping electric field amplitude is changed between successive coherent laser pulses, wherein the driving laser field is changed15 between successive coherent laser pulses at least once;measuring a state of the one of the one or more trapped ions after the application of the coherent laser pulses;repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the one or more trapped ions;20 determining a probability that the one or more trapped ions change state during theinterferometry sequence based on the plurality of measurements of the state of the one or more trapped ions; andadjusting the trapping electric field based on the determined probability in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the25 oscillating electric quadrupole field of the ion trap.

2. The method of claim 1 wherein the trapping electric field further comprises a static electric field and wherein the trapping electric field amplitude is additionally comprised of an electric field amplitude of the static electric field.

303. The method of claim 1 wherein repeating the interferometry sequence is performed the plurality of times by one or a combination of:performing interferometry sequence on a single trapped ion a plurality of times; and trapping a plurality of ions in the oscillating electric field and performing the35 interferometry sequence on each of the ions.11 04 234. The method of claim 1 or claim 3 wherein the first laser pulse comprises a resonant pi / 2 pulse and the final laser pulse comprises a resonant pi / 2 pulse and any other laser pulse comprises a resonant pi pulse.5 5. The method of claim 4 wherein controlled phase differences are introducedbetween the at least two coherent laser pulses; and a total control phase is pi / 2, where the total control phase is a function of the controlled phase differences between the coherent laser pulses.10 6. The method of any preceding claim in which the trapping electric field amplitude ischanged between two different trapping electric field amplitude values.

7. The method of any preceding claim wherein a predetermined delay is providedbetween the at least two coherent laser pulses.

158. The method of any of claims 1 - 3 or 5 or 7 wherein the steps of:repeating interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; anddetermining a probability that the trapped ion changes state during the20 interferometry sequence based on the plurality of measurements of the state of theion;are performed a first plurality of times wherein, for each repeat of these steps in the first plurality of times, a different combination of controlled phase differences is used; and25 wherein adjusting the trapping electric field is based on the first plurality ofmeasurements of the probability.

9. The method of any preceding claim wherein the steps of: repeating the interferometry sequence a plurality of times in order to obtain a 30 plurality of measurements of the state of the ion; anddetermining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion;are performed a second plurality of times wherein, for each repeat of these steps in the second plurality of times, a different set of electric field amplitudes is used during 35 pulse sequencewherein adjusting the trapping electric field is based on the second plurality of measurements of the probability.11 04 2310. The method of any preceding claim wherein the steps of:repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and5 determining a probability that the trapped ion changes state during theinterferometry sequence based on the plurality of measurements of the state of the ion;are performed a third plurality of times wherein, for each repeat of these steps in the third plurality of times, a different number of coherent laser pulses is used during the pulse sequence; and10 wherein adjusting the trapping electric field is based on the third plurality ofmeasurements of the probability.

11. The method of claims 8 and 9 wherein for each of the first plurality of times the steps of:15 repeating the interferometry sequence a plurality of times in order to obtain aplurality of measurements of the state of the ion; anddetermining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of:20 repeating the interferometry sequence a plurality of times in order to obtain aplurality of measurements of the state of the ion; anddetermining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are repeated the second plurality of times such that a plurality of probabilities are obtained at 25 different combinations of controlled phase differences and different trap stiffness changes; andwherein adjusting the trapping electric field is based on all of the determined probabilities.30 12. The method of claims 8 and 10 wherein for each of the first plurality of times thesteps of:repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; anddetermining a probability that the trapped ion changes state during the 35 interferometry sequence based on the plurality of measurements of the state of the ion, the steps of:11 04 23repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; anddetermining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are 5 repeated the third plurality of times such that a plurality of probabilities are obtained at different combinations of controlled phase differences and different numbers of coherent laser pulses; andwherein adjusting the trapping electric field is based on all of the determined probabilities.1013. The method of claims 9 and 10 wherein for each of the second plurality of times the steps of:repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and15 determining a probability that the trapped ion changes state during theinterferometry sequence based on the plurality of measurements of the state of the ion, the steps of:repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; and20 determining a probability that the trapped ion changes state during theinterferometry sequence based on the plurality of measurements of the state of the ion are repeated the third plurality of times such that a plurality of probabilities are obtained at combinations of different trap stiffness changes and different numbers of coherent laser pulses; and25 wherein adjusting the trapping electric field is based on all of the determinedprobabilities.

14. The method of claims 8to 10 wherein for each of the first plurality of times the steps of:30 repeating the interferometry sequence a plurality of times in order to obtain aplurality of measurements of the state of the ion; anddetermining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of:35 repeating the interferometry sequence a plurality of times in order to obtain aplurality of measurements of the state of the ion; and11 04 23determining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are repeated the second plurality of times; and:for each of the second plurality of times the steps of:5 repeating the interferometry sequence a plurality of times in order to obtain aplurality of measurements of the state of the ion; anddetermining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion, the steps of:10 repeating the interferometry sequence a plurality of times in order to obtain aplurality of measurements of the state of the ion; anddetermining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are repeated the third plurality of times such that a plurality of probabilities are obtained at 15 different combinations of controlled phase differences, different trap stiffness changes and different numbers of coherent laser pulses, andwherein adjusting the trapping electric field is based on all of the determined probabilities.20 15. The method of any preceding claim wherein:a first time the steps of:repeating the interferometry sequence a plurality of times in order to obtain a plurality of measurements of the state of the ion; anddetermining a probability that the trapped ion changes state during the25 interferometry sequence based on the plurality of measurements of the state of the ion are performed, the method comprises a pulse sequence sensitive to the electric field along a first direction; anddetermining a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion are 30 performed, the method comprises a pulse sequence sensitive to the electric field along a second direction, different to the first direction, wherein adjusting the trapping electric field is based on the first and subsequently determined probabilities.35 16. The method of claim 15 wherein the first and second directions are relativelyorthogonal directions.11 04 2317. The method of any preceding claim wherein the method further comprises: measuring a detuning of a laser from a transition resonance frequency using interferometry by:applying a first laser pulse to the trapped ion when the electric field 5 amplitude comprises a fixed electric field amplitude;applying a second laser pulse to the trapped ion when the electric field amplitude comprises the fixed electric field amplitude and the second laser pulse is driven by the same laser field with the same wavevector as the first laser pulse and the second laser pulse has a second phase different 10 to the first phase; andmeasuring a state of the ion after the application of the first and second coherent laser pulses;repeating the process of measuring the detuning of the laser a plurality of times in order to obtain a plurality of measurements of the state of the ion15 determining a fixed electric field amplitude probability of the trapped ion being inthe given state based on the plurality of measurements of the state of the ion;wherein detuning of the laser is accounted for based on the fixed electric field amplitude probability.20 18. The method of any preceding claim wherein the average of the square of theamplitude of the oscillating electric field during application of the extended pulse sequence, comprising any initialisation step; the coherent pulses and the measurement, is equal to the square of the amplitude of the oscillating field of the ion trap during an operational mode.2519. A system configured to reduce the magnitude of a quasi-static electric dipole field at the null position of an oscillating electric quadrupole field of an ion trap comprising:a plurality of electrodes configured to generate a trapping electric field for trapping at least one ion wherein the trapping electric field comprises the oscillating electric30 quadrupole field and wherein the trapping electric field comprises an electric field amplitude which is a function of an electric field amplitude of the oscillating electric field;a first laser beam configured to apply coherent laser pulses to the trapped ion and a second laser beam configured to apply coherent laser pulses to the trapped ion and a detector;35 the system configured to induce a change in equilibrium position of at leastone trapped ion and use interferometry to measure said change using an interferometry sequence, by controlling:11 04 23the laser beams to apply at least two coherent laser pulses to the trapped ion, and for successive coherent laser pulses to be applied using at least one of: different laser beams and different trapping electric field amplitudes, and for the laser beam to be changed at least once between successive coherent laser pulses during the interferometry5 sequence; andthe detector to measure the state of the ion after the application of the at least two coherent laser pulses;wherein the system is further configured to:repeat the interferometry sequence a plurality of times in order to obtain a plurality10 of measurements of the state of the ion;determine a probability that the trapped ion changes state during the interferometry sequence based on the plurality of measurements of the state of the ion; andadjust the trapping electric fields based on the probability in order to reduce the magnitude of the quasi-static electric dipole field at the null position of the oscillating 15 electric quadrupole field.

20. A computer readable medium having stored thereon software instructions that, when executed by a processor, cause the processor to generate control signals to cause a system of claim 19 to perform the method of any of claims 1 to 18.2021. An optical clock comprising the system of claim 19.

22. A quantum computing system comprising the system of claim 19.25 23. A quantum simulator system comprising the system of claim 19.

24. A trapped ion electric field sensor comprising the system of claim 19.

25. A trapped ion force sensor comprising the system of claim 19.30

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