J. Nig. Soc. Phys. Sci. 3 (2021) 38–41 Journal of the Nigerian Society of Physical Sciences Velocity distribution of 43Ca+ ion cloud in the low temperature limit in a quadrupole Penning Trap Dyavappa B. M.∗ Department of Physics, Government First Grade College for Women, Kolar, India Abstract Penning trap has electric field created by DC voltage applied between ring and end cap electrodes and magnetic field is applied along symmetry axis, as the electric field confines ions in the axial direction through an electric potential minimum and the magnetic field confines the ions in the radial direction. The trapping potential created by the DC voltage applied between the end cap and ring electrodes in the low temperature limit is cancelled by Coulomb interaction of ions and the total energy is mainly kinetic energy of ions. The velocity distribution of 43Ca+ ions along axial direction, in radial plane and total velocity distribution due to resulting motion of both axial and radial motion of ions in low temperature limit in a Quadrupole Penning trap are presented here. These results reveal the properties of 43Ca+ ion cloud and are useful to study confining techniques for different types of ions in low temperature limit and a qubit can be encoded in the hyperfine ground states of 43Ca+ isotope for ion trap quantum computation. DOI:10.46481/jnsps.2021.132 Keywords: Quadrupole Penning trap, Velocity distribution function, 43Ca+ cloud. Article History : Received: 08 August 2020 Received in revised form: 24 January 2021 Accepted for publication: 29 January 2021 Published: 27 February 2021 c©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: B. J. Falaye 1. Introduction 43Ca+ ion has only one valence electron and has simple en- ergy level structure, has long-lived D-states, therefore it can be used for quantum computation [1, 2, 3], for building an ion clock [4] and also for laser cooling. The large fine-structure splitting of ∆νFS = 6.7T Hz between the P1/2 and P3/2 states in 43Ca+ion as shown in Figure 1 allows a large detuning of the Raman light fields from the P− levels and thus high fidelity gate operations, as spontaneous emission processes are largely suppressed, which is requirement for ion trap quantum compu- tation. And encoding the qubit in the hyperfine ground states ∗Corresponding author tel. no: +91 9483113600 Email address: dyavappabm@gmail.com (Dyavappa B. M. ) ensures that decay from spontaneous emission is completely avoided and thus, very long coherence times may be achieved potentially and the qubits will ideally depend only in second or- der on the external magnetic field [5]. A thorough knowledge of different properties of ions including velocity distribution along axial direction, in radial plane and the total velocity distribution is necessary. The quadrupole Penning trap is made with two end-cap electrodes and a ring electrode. The equation of ring electrode is r 2 r20 − z2 z20 = +1 and equations of two similar end-cap electrodes is r 2 r20 − z2 z20 = −1, where r0 is the inner radius of the ring electrode in the radial plane and z0 is half of the vertical dis- tance between the two end-cap electrodes such that r0 = √ 2 z0. The trap potential created by the DC voltage applied between 38 Dyavappa / J. Nig. Soc. Phys. Sci. 3 (2021) 38–41 39 the end cap and ring electrodes is given by [6, 7]. V (r, z) = V0 r20 + 2z 2 0 ( 2z2 − r2 ) (1) The vector potential of the magnetic field B is [4, 5] A = 1 2 (B × r) = 1 2 B(−yx̂ + xŷ) (2) The Lorenz force in an electromagnetic field with an electric field E and magnetic field B on 43Ca+ion of mass m, charge q, moving with a velocity v is given by [8] F = q [E + (v × B)] (3) F = q [( 2V0 d2 xx̂ + 2V0 d2 yŷ − 4V0 d2 z ẑ ) + B (ẏx̂ − ẋŷ) ] (4) The axial, pure cyclotron, reduced cyclotron and magnetron fre- quencies of 43Ca+ion are given respectively by [9, 10, 11] fz = 1 2π √ 4qV0 md2 , fc = qB 2πm f ′c = fc + √ f 2c − 2 f 2z 2 , fm = fc − √ f 2c − 2 f 2z 2 (5) The velocities of 43Ca+ion in the axial direction, radial plane and in space are vz = √ kBT m , vr = √ 2kBT m , v = √ 3kBT m (6) Figure 1: Energy level scheme of 43Ca+ isotope, a qubit can be encoded in the hyperfine ground states for ion trap quantum computation [5] 2. Theory We assume that the 43Ca+ion cloud is in thermal equilib- rium through electrostatic Coulomb interaction between ions. The rotation of 43Ca+ion cloud in magnetic field is equivalent to neutralization by opposite charge of ions and the distribution of magnetically confined ions in thermal equilibrium without rotation can be treated as ions confined and neutralized by a Figure 2: (i): Penning trap with 43Ca+ions confined in trap space, B: Magnetic Field, V0: storage voltage, (ii) Magnified view of motion of a 43Ca+ions con- fined in trap space in a Penning trap showing reduced cyclotron, magnetron and axial motions cylinder of opposite charge. The probability of velocity distri- bution in thermal-equilibrium is [12] dP ( vr,φ,z ) = 2π (πkT )3/2 ( m 2 )1/2 v × exp [ −m ( v2 −ωφvφ/2 2kBT )] drdφdzdvr dvφdvz (7) where ωφ is the rotational frequency of 43Ca+ion cloud as a whole determined by the temperature of the ion cloud. The Energy of single 43Ca+ion is [12] E = m 2 [ v2r + ( vφ r − qBr 2mc )2 + v2z ] +q [ VT (r, z) + Vq (r, z) ] (8) The momenta in radial plane, azimuthal and axial directions are [12] Pr = mvr = m dr dt , Pφ = mr 2 dφ dt + qB 2c r2 = mrvφ − ωc 2 mr2, Pz = m dz dt = mvz (9) 2.1. Velocity distribution of 43Ca+ ion cloud at the low temper- ature limit In a Penning trap 43Ca+ ions are confined in the low tem- perature limit when the electrostatic potential energy [12] qV q (r, z) � kBT (10) qV T (r, z) + qV q (r, z) + m 2 ωφ ( ωc −ωφ ) r2 = 0 (11) The energy along the axial direction in the low temperature limit is [12] Ez = 1 2 mv2z = 1 2 kBT (12) The energy in the radial plane in the low temperature limit is [12] Er = 1 2 mv2r + mv2φr2  + ( ω2c 8 ) r2 + ωc 2 vφ (13) 39 Dyavappa / J. Nig. Soc. Phys. Sci. 3 (2021) 38–41 40 If we neglect Coulomb interaction potential Vq (r, z) then the probability of velocity distribution in the low temperature limit is [12] dP ( Evr,vφ,vz ) = A′′′ex p [ −m ( v2z + v 2 r 2kBT )] dvr d ( vφ r ) dvz (14) The probability of velocity distribution in Z-direction in the low temperature limit is [12] dP (vz) = ( 2 πmkBT )1/2 ex p [ − ( mv2z 2kBT )] vzdvz (15) The probability density of velocity distribution in Z-direction in the low temperature limit is [12] ρz (vz) = ( 2 πmkBT )1/2 [ ex p ( − 1 2 mv2z kBT )] vz (16) The probability density of velocity distribution in axial direc- tion increases sharply up to √ mv2z /2kBT = 0.854 at ρz = 3.019× 1025, decreases abruptly and remains almost a constant in low temperature limit as shown in Figure 3. Figure 3: The axial probability density of velocity of 43Ca+ion cloud in the low temperature limit The probability of velocity distribution in the radial plane in the low temperature limit is [12] dP ( vr,φ ) = ( 1 kBT ) ex p ( − 1 2 mv2r kBT ) mvr dvr (17) The probability density of velocity distribution in radial plane in the low temperature limit is [12] ρr (vr ) = ( 1 kBT ) ex p ( − 1 2 mv2r kBT ) (18) The probability density of velocity distribution in radial plane increases sharply up to √ mv2r /2kBT = 1.4072 at ρr = 4.89 × 1022, decreases abruptly and remains almost a constant in low temperature limit as shown in Figure 4 The total probability of velocity distribution is [12] dP (v) = ( 2m π ) 1 2 ( 1 kBT ) 32 ex p ( − 1 2 mv2 kBT ) vdv (19) Figure 4: The radial probability density of velocity of 43Ca+ion cloud in the low temperature limit Figure 5: The total probability density of velocity of 43Ca+ion cloud in the low temperature limit The total probability density of velocity distribution is [12] ρ (v) = ( 2m π ) 1 2 ( 1 kBT ) 3 2 ex p ( − 1 2 mv2 kBT ) v (20) The total probability density of velocity distribution in trapping region increases sharply up to √ mv2/2kBT = 1.4086 at ρ = 3.399 × 1032, decreases abruptly and remains almost a constant in low temperature limit as shown in Figure 5. The axial, radial and total probability densities of velocity distribution increase sharply up to √ mv2z /2kBT = 0.854, √ mv2r /2kBT = 1.4072 and √ mv2/2kBT = 1.4086 respectively, at ρz = 3.019 × 10 25, ρr = 4.89×10 22and ρ = 3.399×1032 respectively, beyond which decrease abruptly, remain almost constant for the low temper- ature limit as shown in the Figure 6. The radial probability density of velocity distribution is less than the axial probability density of velocity distribution, which in turn less than the total probability density of velocity distribution. 3. Conclusion The probability density of velocity distribution of 43Ca+ion cloud along axial direction and in radial plane together results 40 Dyavappa / J. Nig. Soc. Phys. Sci. 3 (2021) 38–41 41 Table 1: The values of axial, radial and total probability densities of velocity of 43Ca+ion cloud in the low temperature limit mv2 2kB T √ mv2 2kB T T (K) kBT ( 10−23 J ) ρz (vz) ( 1025 ) ρr (vr ) ( 1022 ) ρ (v) ( 1032 ) 0 0 ∞ ∞ 0 0 0 2 1.4142 4 5.52 2.69719 0.66645 3.3882 4 2 2 2.76 0.992243 4.9034 0.9171 6 2.4495 1.3 1.79 0.365019 2.7814 0.19158 8 2.8284 1 1.38 0.134288 1.3272 0.03359 10 3.1623 0.8 1.1 0.04940 0.61254 0.005714 12 3.4641 0.6667 0.92 0.018174 0.26943 0.0009229 14 3.7417 0.57 0.7866 0.006686 0.11593 0.000149 16 4 0.5 0.69 0.0024595 0.048617 0.00002253 18 4.2426 0.44 0.6 0.0009048 0.020568 0.0000035288 20 4.4721 0.4 0.55 0.00033286 0.0082545 0.000000657 Figure 6: The axial, radial and total probability density of velocity of 43Ca+ion cloud in the low temperature limit total probability density of velocity distribution under low tem- perature limit. The radial probability density of velocity dis- tribution is less than the axial probability density of velocity distribution, which in turn less than the total probability den- sity of velocity distribution. These results reveal the velocity properties of the 43Ca+ion cloud and are useful to design and carry out experiments on stored 43Ca+ ions, with velocity re- lated parameters under low temperature limit and also for ion trap quantum computation in Quadrupole Penning trap. References [1] A. Steane, “The ion trap quantum information processor”, Applied Physics B: Lasers and Optics 64 (1997) 623. [2] C. D. Bruzewicz et al., “Dual-species, multi-qubit logic primitives for Ca+/Sr+ trapped-ion Crystals”, npj. Quantum Information 5 (2019) 102. 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