Supercomputer Modeling of Stochastic Dynamics of the Mercury Ion Array in an Optical Lattice

Abstract

Simulations of the resonant ions stochastic dynamics in the polychromatic optical field are presented. We prove the possibility of long-term four- and nine-particle ionic Coulomb planar clusters (crystals) by all-optical method. An estimate of lifetime of a single particle in an optical lattice is also carried out. Our analysis is based on the numerical solution of the stochastic diferential equations with multiplicative noise using MVS-100K and MVS-10P supercomputers.

Keywords

• Stochastic diferential equations
• Dissipative optical lattice
• Coulomb clusters
• Parallel Monte Carlo method
• Distributed computing

Introduction

Electromagnetic ion traps have many important applications in quantum informatics, high resolution spectroscopy of ions, metrology, physics of cold collisions and many-body physics [ 1–4 ]. A new and interesting trend in this field of research is the so-called all-optical confinement of ions, i.e. optical ion trapping without applying additional radiofrequency or electrostatic and magnetic fields [ 5–9 ]. In particular, it is assumed that development of all-optical methods of ion trapping can be useful for creation of ionic clock with better characteristics [ 10 ].

In our previous papers [ 7–9 ] the solution of all-optical ion trapping problem was proposed, based on using the rectified gradient forces that act on ions in the polychromatic field [ 11–15 ]. We demonstrated, by the numerical simulations of stochastic ion motion in the 3D polychromatic optical super-lattice (OSL), the long-term all-optical trapping of two- and three-ion ytterbium clusters in OSL. In the present work, we carried out numerical simulations of dynamics of four- and nine mercury ions in OSL and demonstrated the long-term all-optical trapping of ordered ion array (planar Coulomb cluster) in OSL. Note that a large array of trapped cold ions has attracted special interest from researchers because of its very useful applications [ 1, 3 ]. Now there is a broader interest in the form of arrays of ion traps in the context of quantum computings [ 4 ]. Stochastic dynamics of a single particle in an optical lattice is also considered.

The mathematical model is a system of stochastic differential equations (SDEs) for positions and velocities of each ionic particle. We take into account four acting forces in the model: the trapping, friction, Coulomb, and stochastic forces. The last force arises due to quantum fluctuations of the optical forces [ 11 ]. The Monte Carlo method with parallelization among computing cores is used to evaluate different average characteristics of this physical problem. 2

Equations of Stochastic Motion

Our study is based on the system of stochastic differential equations with multiplicative noise which can be written in the following dimensionless form r (, ) = v (, ), v (, ) = F (r1, . . . , r , v ) ∈ T = [0, ] , +

︀√ 2 (v ) ∘ W (, ) , F (r1, . . . , r , v ) ≡ F (r , v ) + F (r , v ) + F (r1, . . . , r ) , r (0, ) = r0 ( ), v (0, ) = v0 ( ), 3 are the position r and velocity v , )T, v = ( , of the center-of-mass , )T; is time; is are the trapping, friction and Coulomb forces interpreted in the Stratonovich sense [ 16 ]. acting on the -th ion, respectively. The symbol ∘ means that this SDEs are

W (, ) is a standard three-dimensional vector Wiener process [ 17 ]. Refunction

W (, ) is continuous on T; 3) for ∀ > call some of its properties: 1) W (0, ) ≡ 0; 2) for fixed ∈ 1 and ∀{ } ∈ T (

= the vectorrandom vectors W 1 (, ) and W 2 (, ) are independent. 1, . . . , ) such that 0 < 1 < 2 < · · · < , the random vectors W ( 1, ), W ( 2, ) − W ( 1, ), . . . , W ( , ) − W ( −1, ) are independent; 4) for ∀ 1, 2 ∈ T, ( 1 < 2), the random vector W ( 2, ) − W ( 1, ) has a Gaussian distribution with mean 0 and dispersion matrix ( 2 − 1) 3, where 3 is an unit matrix of the 3-th order. Notice also that for ∀ 1, 2, (1 6 1 < 2 6 ), the The argument

∈ emphasizes that r , v , and W are the random vector functions in corresponding probability space (, F, P). Below the letter will be omitted.

We use the dimensionless variables measuring the positions in units of = , time in units of −1 (where

= ~ 2/ is the wave number, and velocities in units of 0 = √︀ / is the ionic mass), is a period of the OSL cell along -axis ( = , , where ), = ~ ′/2 is the characteristic temperature determining the so-called Doppler cooling is the photon recoil frequency, temperature limit [ 11 ]. The dimensional time ˘ = / describing the simulation results; = 0/ number (a small parameter) for the problem under consideration. ≪ 1 is an analog of the Knudsen is also calculated for where = ︂( − ( , ) + 2 ( ) )︂ ,

= , , , (, ) = ( )[ + cos(2 )]/(1 + ) 3 (4 + 3), 1 = [2 2/(4 + 3)2] 0. are the friction coefficients, ( ) = 0ℒ (/ 0) + 1ℒ (/ 1), 0 = 2 1 ︀⧸ The velocity diffusion coefficient can be written in the form [ 15 ] ( ) = + ( ) , where = 2 ( (2 2/9 )(16 3 + 40 2 + 33 + 9)/(4 + 3)3, 1 = (16 2/9)( + 1)/ (4 + 3)3.

An amplitude of the noise √︀ 2 (v ) in Eq. (2) is calculated on the basis of + 1)/3(4 + 3), ( ) = 0ℒ (/ 0) + 1ℒ (/ 1), 0 = diffusion coefficients (9).

The long-range Coulombic interaction F sionless Coulomb energy ( ) = 2/(4 distance in the following way 0 can be expressed via the dimen

) of the ions separated by the Components of the trapping force vector F

= (

, , )T are [ 7 ] (5) (6) (7) (8) (9) (10) ,

= , , , ︁̂ 0ℒ ︁( )︁ 0 + ̂︁ 1ℒ ︁( )︁ )︂ 1 , wavelength. where ̂︁ 0 = 4 1

︀⧸ 9 ( + 1)(4 + 3); ︁̂ 1 = 2 ̂︁ 0 ℒ ( ) = 1/(1 + 2) is the Lorentzian function; (/ 6 )(4 + 3)2 are the squares of the so-called capture velocities. The physical constants 1, , , , , 1, , determine various ion and force characteristics ︀⧸ (4 + 3); = / ; 2 0 = (/ 6 ) 2 and 2 1 = of the three-dimensional OSL; it is supposed that ≫ , where is the light Components of the friction force vector F are defined as F (r1, . . . , r ) = − ∑︁

′=1 ′̸= (|r − r ′ |) .

r

Pay attention, the phases of optical fields forming OSL are set so that friction coefficients (8) reach maximum at the center of OSL cell (unlike the case of our previous articles [ 7, 8 ]). 3

Numerical Algorithm

For solution of Eqs. (1)–(4), we developed the numerical algorithm which is a combination of two other computational approaches: 1) the velocity Verlet method [ 18 ] for integrating Newton’s ordinary differential equations of particle motion, and 2) the numerical scheme (so called an “integrator”) published by R. Mannella et al. [ 19–21 ] for solution of the Langevin stochastic equation. The integrator of Mannella has the following advantages: a) ability to reproduce the equilibrium behaviour and properties of dynamical system with a high accuracy; b) ability to well reproduce long-time dynamics of phenomena, such as large rare fluctuations [ 20 ], and hence correctly to describe decay of metastable states.

Recall the velocity form of the Verlet algorithm: r +1 = r + v ℎ + F ℎ 22

F +1 = F ({r +1}),

v +1 = v + (F +1 + F ) ℎ + O(ℎ 3), = 1, . . . , ,

+ O(ℎ 2) . 2 ℎ 2

However our numerical algorithm for solving stochastic system of differential equations (1)–(4) turns out considerably more complicated: r +1 = r + v ℎ + F (r , v ) ℎ 2

2 + g Z2, +

S1Z3, + O(ℎ 5/2),

Y1, , Y2, , Y3, ∈ R3 are three uncorrelated random vectors with normal distribution (0, 1) (mean zero and standard deviation one) [ 19 ]; ˆf is a given function of physical parameters; ℎ is a time step. By definition, we assume here that for any vectors A, B the record AB gives the vector ( , , )T, and √︀ 2 (v ) ≡ ︁( √︀ 2 ( ), ︁√ 2 ( ), √︀ 2 ( ) ︁) T . The Monte Carlo method is used to evaluate both the average of the solution and the average for different functions from the solution (the size of the cluster, the cluster lifetime, kinetic energy, temperature, etc.). Due to the slow convergence of the Monte Carlo method, the volume of independent samples can be very large. We set different values (from 214 to 216 ) in different variants. The number of time steps reached ∼ 6 · 107 . Use was made of 128–256 processing cores and the run time reached 12 hours. We used the uniform random number generator with period length ≈ 1038 from [ 22, 23 ].

To implement parallel computing the DVM-system developed in Keldysh Institute of Applied Mathematics of RAS was used. The calculations were carried out using the MVS-100K and MVS-10P supercomputers at the Joint Supercomputer Center of RAS.

Besides, we tested the algorithm by comparing our simulation result (for twoion cluster [ 7 ]) with the analytical predictions of so-called renormalized model of the metastable (cluster) state of ions in the dissipative optical superlattice [ 9 ]. As a result, the very good agreement between the results of a numerical simulations and analytical results of renormalized model was obtained. 4

Results of Computation

In all computations we set = 1.46 · 108 s−1, = 199 amu, = 194 nm, = 0.3, = 2.2, = 1.2, 20 = 6.542, 21 = 358.1, 21 = 0.2, = 2.63 · 10−3, = 3.35 · 105 s−1, = = 1, = 0.5 (i.e. the OSL cell is a cuboid). The number of ions takes values 1, 4, and 9.

In Fig. 1 a general view of the computational domain and initial positions of particles at N=9 is shown. Fig. 2 also shows initial position of ions in more Z-axis 0 --01..55 -1 -0.5

X-axis 0 0.5 1 details.

The example of nine-ion Coulomb cluster formation is shown in Fig. 3. Here the positions of ions (averaged over 214 independent samples) in 0.8 seconds for two values of the parameter is presented. Average coordinates of the central ion coincide with its initial values.

Fig. 4 shows a behavior of a particle at one partial solution of the basic equations (without averagings).

-1 -0.5 0.5 1 1.5 -1 -0.5 0.5 1 1.5

In Fig. 5 dependence of the clusters lifetime and a single particle on parameter is shown. The present result, i.e. almost linear dependence of ln on , are in a very good agreement with theories of the metastabe states of stochastic dynamical system [26]. For the case of small noises they predict the exponential

, ︁√ 2 = 4 0 , where is a distance between ions and . And Coulomb coupling parameter is defined as [ 3 ] where is kinetic temperature, is an OSL period. dependence of the metastable state lifetime on the relative height of the energetic barrier / and ), therefore ln ∼

+ (Arrhenius law). Indeed, the ∼

(at fixed parameters , where and are almost independent on at fixed

and . The small deviation from the linear relation probably is caused by influence of a Coulomb interaction of ions on height of an energy barrier.

The relative root-mean-square interionic separation is defined as [24, 25] (13) (14) 403.43 is means that relative fluctuations of ion positions around their mean are small compared to the cluster size and interionic distance but ions are strongly coupled by the Coulomb interactions.

We see from Fig. 6 that for the long-time ion array the both necessary conditions (of Coulomb ion cluster formation) can be satisfied simultaneously. Note, that the case ≃ 1 corresponds to breakup of the cluster. 0.35 0.3 0.25

Note also that at ≥ 0.1 the lifetime of cluster is small (Figs. 5, 6). Cluster states quickly break up in view of a Coulomb interaction and quantum fluctuations of the optical forces. Pay attention that in the theory of clusters [24, 25] the value = 0.1 usually defines a point of the cluster melting according to Lindemann’s criterion. 5

Conclusion

So, our numerical experiments prove that dissipative optical superlattices are able to form a long-term (up to ∼ 1000 seconds) many-particle Coulomb cluster, which is the highly ordered array of mercury ions. Such cluster is characterized by the small values of the relative root-mean-square interionic distance, ≪ 1, and by the large magnitude of the Coulomb coupling parameter ≫ 1. Dependences of basic parameters of a Coulomb cluster on the period of a dissipative optical superlattice are investigated. They sufficiently well correspond to the known theories of metastable states of stochastic dynamical systems and clusters [25, 26]. Comparison of the obtained numerical results with results of theoretical paper [ 9 ] shows very good agreement. For numerical solution of stochastic equations, we give generalization of the well-known velocity Verlet scheme for accounting of a random force.

Our algorithm, Eqs. (11)–(12), allows to consider correctly key features of our stochastic model: a metastability of Coulomb clusters in OSL, non-conservatism of optical trapping forces, nonlinearity of friction coefficients (8), and a multiplicativity of stochastic noise. Parallel realization of this algorithm on supercomputers was performed. In future works, it is planned to increase the number of ions up to several dozens. 23. Mikhailov, G.A., Marchenko, M.A.: Parallel Realization of Statistical Simulation and Random Number Generators. Russ. J. Numer. Anal. Math. Modelling. 17, 113124 (2002) 24. Doye, J.P.K., Wales, D.J.: An Order Parameter Approach to Coexistence in Atomic

Clusters. J. Chem. Phys. 102, 9673–9688 (1995) 25. Berry, R.S., Smirnov, B.M.: Computer Simulation of the Phase Transitions in Clusters. Nanomechanics Sci. and Techn. 3, 167–192 (2012) 26. H¨anggi, P., Talkner, P., Borkovec, M.: Reaction-Rate Theory: Fifty Years after

Kramers. Rev. Mod. Phys. 62, 251–341 (1990)

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