Analysis of the Trapping Potential of the Linear Radio
Frequency Ion Traps for Quantum Computing
The dynamics of an ion in a linear radio frequency ion trap is established by classical motion equation. For an radio frequency electric potential,
The x and y equations become Mathieu equations;
d2x / dℓ2 + (a – 2qcos (2ℓ)) x = 0 (2)
and, d2y/dℓ2 – (a – 2qcos (2ℓ)) x = 0 (3)
where, ℓ = Ωt/2, is time which is dimensionless, a = (4QV)/mr20Ω2 and q = (2QV)/mr20Ω2 are DC and RF dimensionless. Q and m are ion and mass.
In the assumptions regarding psuedopotential, q <<1, the motion of ion along the i axis is decomposed into secular motion of large and slow amplitude at the frequency, w and small and fast amplitude micromotion at the radio frequency Ω 1.
φ (x,y, z,t) = φRF (x,y, z)cos(Ωt) + φDC(x,y, z) (4)
This defines the arbitrary electric potential.
Secular motion is described as; ψsec(x,y, z) = Q2/4mQ2 |∇φRF (x,y, z)|2 + QφDC(x,y, z) (5)
Atomic ions trapped in ion traps are most preferred method for quantum computer realization, due to the increased lifespan of coherence of internal qubit and later-mediated interaction of Coulomb 2; 3; 4. Several quantum gates procedures have been put forth by researchers3 and many traps have been demonstrated with ions, though the number of remained small. The challenge still remains in front of the researchers to increase the number of ions in the trap where the system’s quantum behavior can’t be competently modeled through conventional means. The linear radio frequency ion trap with the ions cooled by laser and detained in 1D crystal, is the working platform for quantum computing of ion trap. Nonetheless, leveling the linear radio frequency ion trap to measurable numbers of ion still poses crucial difficulties for the researchers.
The combination of dc and rf fields in the trap puts the ion in radial motion which is complex and is described by differential equations. These equations when solved leads to the stability diagram which enables one to evaluate the efficiency of trap, provides values of various critical parameters like voltage components and rf amplitude, trap size, ion mass and the rf 5. Till the time these values are adhered to, the ion retains its position at the device’s axis. Moreover the magnitude of the pseudo potential of the restoring forces, responsible for holding ions in the direction has direct proportional relationship with distance from center. The secular motion of ion becomes distorted in the event of misaligned of the pseudo potential and the radio frequency field minima in the trap. Misalignments arising from the asymmetry in the construction of trap or patch of minimal dc potentials on the surface of electrodes lead to the ion getting displaced and out from radio frequency field. This is unrelated to the cause of misalignment. Along with micro-motion of the ions causing them to heat up and secular motion of ion; there is the vibration of ion in axial direction.
In the case when the number of confined ions is few, they align themselves in the linear position along the axis however, increasing the dc voltage or increase in the number of ions causes instability in the trap. This happens due to the close association of ions with each other, leading to ions squeezing against each other. The radial restoring force becomes weak in comparison to the Coulomb repulsion between adjacent ions causing ions to move in zig-zag pattern. The situation worsens with the addition of more ions in the trap and the zig-zag pattern takes the complex structure of three-dimensional helix 6. Ions move farther from the axis and the micro-motion heating initiates which have to be avoided in the experiment.
The major problem of radio frequency ion traps is that the charged particles like ions are comparatively sensitive to stray electric fields in the vicinity. Such fields adversely impact the motions of these ions and becoming time-dependent, this result in heating of ions. The heating rates for two ions in the trap are of 1 ms order. The rate of heating increases with the increase in the number of ions and is dependent on the number of particles coupled with stray fields along with the increase in the quantum of degree of freedom. Ion traps are required to perform the sufficient number of gate operations and to deploy the ions in excited state for qubits, it is extremely crucial for the avoidance of impulsive emission that they should be metastable. This entails that these states are driven by weal optical transitions and low Rabi frequencies correspondingly. The number of difficulties such as restrained frequency of trap and heating augments with the number of ions in trap elucidating the inability of trap to allow a sufficiently greater number of ions.
Despite of its limitations, linear rf ion traps have demonstrated several fundamental concepts of quantum information. The major demonstration has been the test of a Bell inequality as demonstrated by Rowe et al 7 and decoherence free subspace demonstration by Kielpinski et al8. Due to the increase in the ion number in trap, it becomes increasingly difficult to trigger the ion string with a photon. Four options have been proposed to overcome these difficulties, these are; splitting the ion string in minute portions and facilitate the movement of ions 9, coupling of ions through photons and cavities 10, wiring of ion traps and usage of image chargers 11, and utilizing the radial modes of string of ion 12. Presently, the first proposal appears as the most promising as demonstrated by 8. deployment of segmented ion traps is permitted the movement by changing of voltages on the electrodes of trap. The ion strings can also be split and merged which permits the flexibility to tailor the ion string size as per the register size of quantum. Rowe et al 7 and Barrett et al 13demonstrated this procedure successfully.
The difficulties of the quantum computing by linear radio frequency ion traps, there are further challenges to be addressed. To achieve the quantum computing at universal level, the algorithms are required to be tolerant of faults and have to be implementing in the similar manner. This invariably requires the error correction of quantum computing associated with iron traps. The attempt of implementation of quantum error rectification with repeated efforts to ensure prolonged coherence time and rectification of errors introduced by gate operations have become a crucial research area. The major obstacle experienced while performing the successful quantum error rectification protocol is the restricted operational fidelity. The major areas requiring improvement are the coherence times of qubit which are of single or dual order of magnitude greater than the primary gate operations. Several cases have demonstrated coherence times as much as five order of magnitude larger than gate time 14. Trap electrodes cooled down to cryogenic temperature allows strong suppression of motional decoherence 15. The accuracy of initializations achieved at 0.999 is required to be improved upon requirement. As demonstrated by Knill et al, operation of single qubit is capable of being carried out with fidelities beyond 0.995 16 and also facilitates further improvement in fidelity with the introduction of increasingly stable laser fields at the position of ions 17. Two qubit gate operations fidelity achievement of 0.9 by Benhelm et al marks a distinct success1819. However, the fidelity of the operations is dependent on the type of gate. The major sources of error identified are off-resonant scattering, insufficient cooling, error in addressing standalone qubits, intensity of noise and frequency of laser. Myerson et al demonstrated that single qubit read out can be executed with fidelity of 0.999. Moreover, as demonstrated by Blakestad et al, ion strings can be merged, split and shuttled with low decoherence and high fidelity20.
The major challenge emerged from the analysis is the two qubit gate operations which is the restricting factor. The last decade has been significant in terms of the achievement in fidelities. Figure 1 depicts the progress in the last decade in decreasing the error rate of operations of two qubit gates. This also demonstrates the progress made with increase in the number of tangled ions. Open circles in the figure represent the experiments conducted on two qubit gate operations and the diamonds represents the results. Stars mark the largest number of ions obtained and numbers underneath the reference shows the trap cycle’s number used for the operation. Trend is indicated by the dotted and dashed lines.
Benhelm et al established two-qubit operations of gate large enough to permit the principle quantum computation which is fault tolerant as per the method by Knill1816. Linear radio frequency ion traps are crucial for the quantum computations and the development in their technique promises further observations in relation to the quantum computations, the single qubit operational gates have been significantly evolved in the last decade, however the two qubit gates is still the focus of researchers to achieve the desired results.
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