Analytical Approximate Solution of a Coupled Two Frequency Hill's Equation
aa r X i v : . [ phy s i c s . p l a s m - ph ] A ug Analytical Approximate Solution of a Coupled TwoFrequency Hill’s Equation
Varun Saxena a, ∗ a School of Engineering, Jawaharlal Nehru University, New Mehrauli Road, Delhi 110067,(India)
Abstract
A coupled two frequency Hill’s equation is solved. Analytically approximatesolution correct up-to first order is derived using modified Lindstedt-Poincareperturbation method. For a wide range of controlling parameters we comparethe numerical and analytical solutions. The solution is the first step towardsdeveloping a comprehensive understanding of the electrodynamics of chargedparticles in a combinational ion trap utilizing both electrostatic DC and RFfields along with a constant static magnetic field with prospects of confiningantimatter such as anti hydrogen for a reasonably long durations of time.
Keywords:
Hill’s equation, Perturbation methods, Modified LindstedtPoincare method, Paul trap, Penning trap
1. Introduction
Hill’s equation [1] is a second order differential equation with periodic coef-ficients. The equation can be described as ¨ x + f ( t ) x = 0 (1)where f ( t ) is a period function, often a combination of several cosine and sinefunctions. Hill’s equations finds application in several diverse areas of appliedsciences. The differential equation appears in several settings, such as, in theanalysis of lunar stability [2], modeling of quadrupole mass spectrometer [3], thedynamics of an electron in a crystal using one dimensional Schrodinger equation[4], in a two level system in quantum optics [5] and electromagnetic ion traps[6] in which electrostatic DC and RF fields are used to confine charged particlesin a limited space in a perturbation free environment.A well known equation arising out of Eq. (1) is the Mathieu equation ¨ x + ( a − q cos(2 t )) x = 0 (2) ∗ Corresponding author
Email address: [email protected] (Varun Saxena )
Controlling parameters, a , q determine the stability of the solution of Eq. (2).For example, if a = 0 , the solutions up-to q = 0 . are stable. Stability of Eq.(2) is well documented in the literature. The equation governs the dynamics ofcharged particles inside an electromagnetic ion trap, namely, Paul trap, whereincharged particles are under the influence of electrostatic DC and RF fields only.The coefficients a and q are proportional to the applied voltage strengths.In this paper, we attempt to derive analytical approximate solution for acoupled two frequency Hill’s equation [7] which can be written as ¨ x − p ˙ y + ( a − q cos(2 η − t ) − q cos(2 t )) x = 0¨ y + p ˙ x + ( a − q cos(2 η − t ) − q cos(2 t )) y = 0 (3)Such a coupled system has recently gained importance to study the electro-dynamics of charge particles relevant to particle confinement using two radiofrequencies [7, 15] in a combinational trap utilizing features of both Paul andPenning trap. In context to such a trapping, coefficients a , q , q are propor-tional to the applied electrostatic DC and RF (radio frequency) and p is theproportional to the applied magnetic field.To get a better understanding of how Eq. (3) relates to the trapping ofparticles inside a combinational trap, consider the quadrupole potential in adual frequency Paul trap given by Φ( x, y, z, t ) = ( U + V cos( ω t ) + V cos( ω t )) (cid:0) ( x + y − z ) /r (cid:1) (4)The electric field generated by this potential is −→ E ( x, y, z, t ) = −−→∇ Φ( x, y, z, t ) .Since there exists a magnetic field −→ B = B ˆ k due the features attributed toa Penning trap, the net force experienced by a charged particle of charge Q and mass M moving with a velocity −→ v is given by the Lorentz force equation −→ F = − Q −→∇ Φ + Q ( −→ v × −→ B ) . If −→ v = v x ˆ i + v y ˆ j + v z ˆ k , the components of force inthe three orthogonal directions, i. e., F x , F y , F z are given by F x = − ( U + V cos( ω t ) + V cos( ω t )) (cid:0) x/r (cid:1) + Qv y B (5) F y = − ( U + V cos( ω t ) + V cos( ω t )) (cid:0) y/r (cid:1) − Qv x B (6) F z = ( U + V cos( ω t ) + V cos( ω t )) (cid:0) z/r (cid:1) (7)Here, U , V , are the applied DC and RF voltages respectively, ω , ω arethe primary and secondary RF frequencies, respectively, and r is the trapdimension. Upon substituting ω t = 2 τ , F x = M ¨ x , F y = M ¨ y , a = 8 QU /M r , q , = − QV , /M r , p = 2 QB /ω M , v x = ˙ x , v y = ˙ y and ω /ω = η in Eq.(5), Eq. (6) and rearranging the terms, one obtains the two coupled equationsgiven in Eq. (3). Since τ is actually a dummy variable, without loss of generality,it can be replaced by t in the subsequent equations. The confinement in the x − y plane is through a set of coupled differential equations given by Eq. (3),whereas, along the z axis, the trapping is on account of a combination of DCand RF voltages, exactly like it is in a dual frequency Paul trap.The Lorentz force due to the magnetic field acts inwards. This increasesthe stability of the charged particles simultaneously being trapped by the ap-plication of a static and a dynamic electric field in combination with a constantmagnetic field. In recent years, the trap employing dual frequency has gainedimportance since it is being viewed as a promising option to trap anti-hydrogen.In general, charged particles with varied charge to mass ratio can be trappedeffectively inside a dual frequency Paul trap [7].To produce anti-hydrogen, positron and antiproton are to be trapped and amagnetic field is required to trap the resulting neutral particle, anti-hydrogen.The limitation of a conventional single frequency Paul trap in trapping twospecies with different charge to mass ratio is that the weakly confined speciesis pushed away from the trap center [8]. The ALPHA experiment [9, 10] andATRAP experiment [11, 12] rely on a variation of Penning trap using staticmagnetic field for their initial confinement. However it is not possible to trapoppositely charged particles in a Penning trap on account of the presence ofonly DC electric field along with a static magnetic field. Hence a combinationaltrap inheriting features of both a dual frequency Paul trap and a Penning trapholds a lot of potential in confinement of oppositely charged species with a largecharge to mass variation and will most certainly be a significant improvementwhen compared to earlier methods utilizing both electric and magnetic fields ina conventional single frequency Paul trap [13,14].Dynamics governed by the differential equations given in Eq. (3) is thereforeof great interest. It offers a starting point to the understanding of the electro-dynamics that will emerge inside a combinational trap. In Sec. 2, we derive thetime evolution of position of the confined particle in x and y direction. In Sec. 3,comparison of the analytical approximate solution with the numerical solutionfor a wide range of control parameters shows the robustness of the solution todepict the particle dynamics. Sec. 4 contains a conclusion and a discussion onthe importance of the analytical solution.
2. Analytical Approximate Solution
We begin expressing the equations in a concise form by writing A ( t ) = a − q cos(2 η − t ) − q cos(2 t ) . The coupled differential equations in Eq. (3)can now be written as ¨ x − p ˙ y + A ( t ) x = 0 (8) ¨ y + p ˙ x + A ( t ) y = 0 (9)Multiplying Eq. (8) by imaginary j and adding to Eq. (9) gives ¨ z − jp ˙ z + A ( t ) z = 0 (10)Where z = y + jx . Let z = w ( t ) exp( jpt/ . The function w ( t ) is actually acomplex function which can further be substituted as w = X + jY . Hence uponsubstituting z = ( X + jY ) exp( jpt/ and after some basic manipulations, Eq.(10) can be written as ¨ w + (cid:0) A ( t ) + p / (cid:1) w = 0 (11)Where, ¨ w = ¨ X + j ¨ Y . Writing a a + p / , q = q r q , Ω = 2 η − , Ω = 2 ,Eq. (11) can be expressed as ¨ w + ( a − q cos(Ω t ) − q r q cos( η Ω t )) w = 0 (12)Applying Modified Lindstedt-Poincare method [16] in Eq. (12), we begin bywriting a ν + q α + q α x = x + q x + q x (13)Substituting the values of a and x from Eq. (13) in Eq. (12) and solvingequations, one at a time for O ( q ) , O ( q ) , O ( q ) , we get X = D φ ( t ) + E ψ ( t ) (14) Y = D φ ( t ) + E ψ ( t ) (15)Where D , and E , , are real constants that depend on the initial positionsand velocities of the charged particle, i.e., x , y , v x , v y . Moreover, one canexpress φ ( t ) and ψ ( t ) , correct up-to first order as φ ( t ) = cos( νt )+ a cos( ν − Ω ) t + a cos( ν +Ω ) t + a cos( ν − η Ω ) t + a cos( ν + η Ω ) t (16) ψ ( t ) = sin( νt )+ a sin( ν − Ω ) t + a sin( ν +Ω ) t + a sin( ν − η Ω ) t + a sin( ν + η Ω ) t (17)Where a = q / ( ν − ( ν − Ω ) ) , a = q / ( ν − ( ν + Ω ) ) , a = q r q / ( ν − ( ν − η Ω ) ) , a = q r q / ( ν − ( ν + η Ω ) ) and ν is the slow frequency given by ν = q [( a + p /
4) + (2 q / Ω ) (1 + q r /η )] (18)In Eq. (13), the constants a and x are written up-to second order, even thoughindependent solutions of Eq. (16), Eq. (17) are written up-to first order. Thishas been done to evaluate the slow frequency by deriving expressions for α and α . The value of α , to eliminate secular terms for O ( q ) comes out to be α = 0 . Similarly, the value of α , to eliminate secular terms for O ( q ) comesout to be α = ( − /Ω )(1 + q r /η ) . Backtracking from X and Y , the timeevolution of position x ( t ) and y ( t ) for the charged particle is x = Y cos( pt/
2) + X sin( pt/ (19) y = X cos( pt/ − Y sin( pt/ (20)Its worth observing that ˙ φ = ˙ φ ( t = 0) = 0 and ψ = ψ ( t = 0) = 0 . If onewrites φ = φ ( t = 0) and ˙ ψ = ˙ ψ ( t = 0) , the values of constants D , and E , come out to be, D = y /φ , D = x /φ , E = ( v y + px / / ˙ ψ and E = ( v x − py / / ˙ ψ .
3. Comparison of Analytical Solution with Numerical Solution
The solutions are obtained by varying the controlling parameters, namely, p , q , q and η . In Fig. 1, a comparison of the numerical and analytical solutionis shown with parameter values p = 0 . , q . , η = 45 in sub figures(a) q . , (b) q . , (c) q . and with parameter values p = 0 . , q . , η = 45 in sub figures (d) q . , (e) q . , (f) q . .In Fig. 2, a comparison of the numerical and analytical solution is shown withparameter values p = 0 . , q . , η = 45 in sub figures (a) q . ,(b) q . , (c) q . and with parameter values p = 0 . , q . , q . in sub figures (d) η = 5 , (e) η = 35 , (f) η = 55 . The values of q and q are proportional to the applied RF voltages V and V respectively, p isproportional to the applied magnetic field strength B and η is the ratio of thesecondary voltage frequency ω and primary voltage frequency ω .
4. Conclusion and Discussion
For particle trajectory in x − y plane, the analytical approximate solutioncorrect up-to first order, is derived for the coupled two frequency Hill’s equationusing modified Lindstedt-Poincare method. The analytical solution matcheswell with the numerical solution obtained by numerical simulating the systemof coupled differential equations given in Eq. (3). The analytical solution has alimited number of harmonic terms, i.e., ( ν ± Ω ) and ( ν ± η Ω ) terms, whereasthe numerical solution encompasses the effect of all the harmonic terms whichmake up the complete solution. Therefore, the matching is observed for somerange of controlling parameters only. If the order of the analytical solutionis increased, the range of operating parameters for which the two solutionsmatch will widen. However, the derivation of such higher order terms will bemathematically challenging. It is important to see that the solution describedby Eq. 16 and Eq. 17 will blow up when η ∼ . To obtain single frequencysolutions one can simply substitute q r = 0 and keep η away from . In mostof the practical settings [7,15], the value of η is substantially higher than ,a regime wherein the analytical solutions are a good match to the numericalsolutions.Experience guides us that analytical solution correct up-to first and secondorder are usually sufficient to provide deeper insights to both individual parti-cle as well as collective dynamics inside the trap [17-19]. The relevance of ananalytical solution cannot be understated when one has to study the collectivedynamics inside such combinational traps. Since the fields are spatially linearin this set up, one has to see if a distribution function can be constructed forthe particles by the method of inversion [17]. It is well known that RF heatingon account of applied RF fields will increase the temperature of the chargedparticles. The analytical tracking of temperature variation for each species in-side such a trap is therefore important [18-19]. Temperature can be evaluatedas the second order moment of the distribution function. To the best of myknowledge, such analytical work on collective dynamics for combinational traps(a) t (arb. units) -1.5-1-0.500.511.5 x ( a r b . un it s ) Numerical, q2=0.15Analytical, q2=0.15 (d) t (arb. units) -2-1.5-1-0.500.511.52 y ( a r b . un it s ) Numerical, q2=0.19Analytical, q2=0.19 (b) t (arb. units) -2-1.5-1-0.500.511.52 x ( a r b . un it s ) Numerical, q2=0.2Analytical, q2=0.2 (e) t (arb. units) -2-1.5-1-0.500.511.52 y ( a r b un it s ) Numerical, q2=0.23Analytical, q2=0.23 (c) t (arb. units) -2-1.5-1-0.500.511.52 x ( a r b . un it s ) Numerical, q2=0.24Analytical, q2=0.24 (f) t (arb. units) -2-1.5-1-0.500.511.52 y ( a r b . un it s ) Numerical, q2=0.27Analytical, q2=0.27
Figure 1: The plots (a), (b), (c), show a comparison of the numerical and theanalytical solutions for parameter values p = 0 . , q . , η = 45 . Plots(d), (e), (f) show a comparison of the numerical and the analytical solutions for p = 0 . , q . , η = 45 .(a) t (arb. units) -2-1.5-1-0.500.511.52 y ( a r b . un it s ) Numerical, q2=0.15Analytical, q2=0.15 (d) t (arb. units) -2-1.5-1-0.500.511.52 y ( a r b . un it s ) Numerical, η = 5Analytical, η = 5 (b) t (arb. units) -2-1.5-1-0.500.511.52 y ( a r b . un it s ) Numerical, q2=0.17Analytical, q2=0.17 (e) t (arb. units) -2-1.5-1-0.500.511.52 y ( a r b . un it s ) Numerical, η = 35Analytical, η = 35 (c) t (arb. units) -2-1.5-1-0.500.511.52 y ( a r b . un it s ) Numerical, q2=0.2Analytical, q2=0.2 (f) t (arb. units) -2-1.5-1-0.500.511.52 y ( a r b . un it s ) Numerical, η = 55Analytical, η = 55 Figure 2: The plots (a), (b), (c), show a comparison of the numerical and theanalytical solutions for parameter values p = 0 . , q . , η = 45 . Plots(d), (e), (f) show a comparison of the numerical and the analytical solutions for p = 0 . , q . , q . . EFERENCES
References [1] W. Magnus, S. Winkler, Hill’s Equations, Dover, New York (1979), p. 4[2] G. W. Hill, On the part of the motion of lunar perigee which is a functionof the mean motions of the sun and moon, Acta Math., 8, (1886), p. 1[3] Edmond de Hoffmann, Vincent Stroobant, Mass spectroscopy:Principlesand Applications, 2nd Edition, John Wiley & Sons Ltd, p. 65[4] Karl W. Boer, Udo W. Pohl, Qantum Mechanics of electrons in crystal,Semiconductor physics, Springer, (2018)[5] Christopher Gerry, Peter Knight, Introduction to Quantum Optics, Cam-bridge university press, (2004)[6] W. Paul, Electromagnetic traps for charged and neutral particles, Rev.Mod. Phys., 69, (1990), p. 531[7] N. Leefer, K. Krimmel, W. Bertsche, et al., Investigation of two-frequencyPaul traps for anti hydrogen production, Hyperfine Interact, 238, (2017),p. 12[8] D. Offenberg et. al., Translation cooling and storage of protonated proteinsin an ion trap at sub-kelvin temperatures, Phys. Rev. A, 78, (2008), p.0061401[9] G. B. Anderson et.al., Trapped anti hydrogen, Nature, 468, (2010), p. 673[10] G. B. Anderson et. al., Confinement of antihydrogen for 1,000 seconds,Nature Phys., 7, (2011), p. 263401[11] C. H. Story et.al., First laser controlled antihydrogen production, Phys.Rev. Lett., 93, (2004), p. 263401[12] G. Gabrielse, Trapped antihydrogen in its ground state, Phys. Rev. Lett.,108, (2012), p. 113002