Quantum control of elementary particle at nucleon had already been investigated for a couple of decades at the frontier area both in quantum physics and system control. At the nucleus, a kind of control for particles motion had been considered in the form of Klein-Gordon-Schrödinger equation, which is a system describing the Yukawa interaction between nucleon (e.g. proton, neutron) and meson. The aim of this paper is trying to find an external force to act at one of the particles to change the particle from its ground state to reach its exciting state. A lot of very interesting results had been obtained and make a clear picture with numerical approximate and computational approach. For interpreting, the meson can escape from the cloud of proton 'p' to join the neutron 'n', it turned the proton 'p' to a new neutron, and neutron 'n' to a new proton. The Yukawa interaction can be incident naturally, whether it can be controlled? The control either be accelerated or slow-down the speed of meson exchange effect. Corresponding to theoretical results, two dimension control of heavy particles had been utilized in experimental aspect, it needs to seek control input to make it happened. For instance, using shaped laser pulse, particles beam, etc. Taking this opportunity, it is delighted to report the detailed conclusion of nuclei control issues and explore future research direction.
Quantum control, Klein-Gordon-Schrödinger equation, Nucleus, Numerical approach
In the physics, chemistry and mathematics fields, quantum control had been considered in literatures and contributed papers world widely. By the investigation of existing research works, it is easily find for us that the recent development and progress in the quickly growing areas. In fact, there are various breakthroughs had been made for control of particles at the quantum physics subject. In the frontier realms, one can list the most highlighted milestone works such as trapped cooling atom with laser technology see [1-4] controlling molecules rearrangement and dissociation see [5] controlling molecules within chemistry reaction (e.g. break the weak bond); Iteration learning controlling (e.g. finding optimal set instead of one particular shaped laser wave functions [6]), so forth. It is also extended to control of motions of atoms and molecules; Control of their structures; Control quantum qubit NOT gate; Control of logic quantum qubit computations. With these amounts of outstanding research achievements (cf. [7-10]), a great deal results had already been obtained in the past several decades (cf. [11-13]). Certainly, it would be a convinced standpoint to boost the field to the future.
Eventually, at the view of physics, it extremely needs to solve the problems which still exist in the majority number. Control of nucleus had been considered for a long time in a variety methodologies at the real laboratory, It is apparently could not be directly realized at present experimental equipment's. However, theoretical control nuclei could be predictively worked at the academic level. It keens to have continuously and mutated study to seek a pathway or shortcut to control elementary particles anticipatively. A physical model of control theory of quantum system and computational simulation of numerical experiments would be taken account into this work.
Firstly, let us introduce Klein-Gordon-Schrödinger (KGS) equation in the nucleus with real atomic units. Suppose Ω is an open bounded set of spatial space R3 and set Q = (0, T ) × Ω for time T > 0. The i denote the unit of imaginary part of complex space. For (x,t) ∈ Q, x = {x1,x2,x3}, the description of nucleon particle has the form of Schrödinger equation (cf. [14])
iℏ∂ψ∂t + ℏ22M∂2ψ∂x2 + ϕψ + euψ = 0, (1)
Where ψ denote complex-valued function representing probability density of nucleon field with initial ground state ψ(0) = ψ0. ℏ is reduced Planck constant, M is the mass of nucleon, and e is the charge of proton. The simultaneous equation of (1) is the Klein- Gordon dynamics, which described the motion of meson by
1c2∂2ϕ∂t2 - ℏ22m∂2ϕ∂x2 + (mcℏ)2ϕ = |ψ|2 + e′vϕ, (2)
Where ϕ denote real-valued function representing probability density of meson field with initial ground states given by ϕ(0) = ϕ0 and ϕt(0) = ϕ1. c is speed of light. m is mass of meson, e′ is charge of meson. Definitely, the physics model consists of (1) and (2) represents the motions of two heavy particles in the presence of Yukawa interaction see [7-10]. Two functions u and v represent control inputs corresponding to external forces such as ultrashort (e.g. femtosecond/attosecond) infrared laser pulse (e.g. terahertz) in real laboratory experiments as in [15,16]. The shaped laser pulse might be suitable to be manipulated for the purpose (cf. [17,18]).
Let Uad be a closed and convex admissible set of U. Without lost of generality, control variables usually taken as time depended only functions, denote as u(t) = (u(t), v(t)) for simplification, its belonging space denoted as U = L2(0, T )2. Introduce two Hilbert spaces H = L2(Ω), V = H10(Ω) with usual norm and inner products (cf. [15,19]). Hence, two embeddings in Gelfand triple space V ↪ H ↪ V' are continuous, dense and compact, where V′ i.e. H−1(Ω) is conjugate space of V (cf. [15,20]). For meeting the realistic simulation, taking both real spaces for real part and imaginary part of wave function ψ, and taking real space for wave function ϕ as usual in here. For complex valued function ψ, if necessary, consider real part only for simplifying. Denote ' = ∂∂x for wave function.
Definition 1: If functions (ψ,ϕ) belong to the space defined by
W(0,T;V,V′) = {(ψ,ϕ)|ψ ∈ L2(0,T;V),ψ′ ∈ L2(0,T;V′),ϕ ∈ L2(0,T;V),ϕ′ ∈ L2(0,T;H),ϕ′′ ∈ L2(0,T;V′)}.
It is solution space. If (ψ,ϕ),(ψ,ϕ) ∈ W(0,T;V,V′), then inner product is defined by
((ψ,ϕ),(ψ,ϕ))
= (ψ,ψ)V + (ψ′,ψ′)V + (ϕ,ϕ)V + (ϕ′,ϕ′)H + (ϕ′′,ϕ′′)V′.
Then, inner product induced norm of solution space W(0,T;V,V′) defined by
∥(ψ,ϕ)∥W(0,T;V,V′)
= (∥ψ∥2L2(0,T;V) + ∥ψ′∥2L2(0,T;V′) + ∥ϕ∥2L2(0,T;V) + ∥ϕ′∥2L2(0,T;H) + ∥ϕ′′∥2L2(0,T;V′))12.
Hence, W(0,T;V,V′) become a Hilbert space equipped with above inner product and norm.
Definition 2: Let T > 0, the pairing ( ψ, ϕ) are weak solutions of (1) and (2), if ψ, ϕ,
W(0,T;V,V′) and satisfy
⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪∫0T∫Ω[ - iℏψηt - ℏ22MψXηX + ϕψη]dxdt= ∫0T∫Ωuηdxdt + ∫Ωiψ0η(0)dx,∫0T∫Ω[1c2ϕρtt + ℏ22mϕXρX + m2c2ℏ2ϕρ - |ψ|2ρ]dxdt∫0T∫Ωvρdxdt + ∫Ωϕ1ρ(0)dx - ϕ0ρt(0)dx (3)
for all η¯ ∈ C1(0,T;V),ρ¯ ∈ C2(0,T;V) and such that η(T) = 0,ρ(T)= ρt(T) = 0, a.e. t∈ [0,T].
The cost function associated with KGS states system (1)-(2) is taken as
J(u) = ε1∥ψf(u) - ψtarget∥2V + ε2∥ϕf(u) - ϕtarget∥2V + (u,u)U, (4)
for all u∈Uad, where ψtarget, ϕtarget ∈ V are target exciting states, ψf(u) and ϕf(u) are final observed quantum states, respectively. Here, ε1 and ε2 are penalty weighted coefficients for balancing the evaluates of system and running criterias.
As is well known, quantum optimal control is to solve two fundamental problems:
i). Find an element u* = (u*, v*) such that
infu ∈ Uad J(u) = J(u∗).
ii). Characterization of u*.
Such u* is called quantum optimal control for Klein-Gordon-Schrödinger system (1)-(2) subject to cost function (4).
Theorem 3: For given ψ0 ∈ V, ϕ0 ∈ V, ϕ1 ∈ H, if Uad is convex and closed (bounded) subset of U, then there exist a unique weak solution (ψ(u),ϕ(u)) of KGS system (1)-(2) at space W(0,T;V,V′), and has inclusive W(0,T;V,V′) ⊂ C(0,T;H).
Indeed, using Faedo-Galerkin method and citing [21,22], a prior estimates for ψ and ϕ can be obtained by routing manner at solution space.
For ∀u ∈ U, via the virtual of Theorem 3, there exists a unique weak solution (ψ(u),ϕ(u)) of KGS system (1)-(2) in the solution space W (0,T;V,V). Hence there is a solution mapping from control space to solution space u→(ψ(u),ϕ(u)): U → W (0,T;V,V), which is continuous mapping. In here, (ψ(u),ϕ(u)) is called the states of KGS control system (1)-(2).
Theorem 4: Given ψ0 ∈ V, ϕ0 ∈ V, ϕ1 ∈ H, if space Uad ⊂ U, is bounded closed convex space, then there is at least one quantum optimal control u∗ ∈ Uad, of Klein-Gordon Schrödinger system (1)-(2) subject to cost function (4).
Follow article [22], straightforwardly to get the proof of Theorem 2 at real spaces.
Theorem 5: Given ψ0,ϕ0 ∈ V and ϕ1 ∈ H. For Uad ⊂ U, be a bounded closed convex space, then quantum optimal control u* = (u*, v*) for cost function (4) subject to states system (1)-(2) is characterized by simultaneously optimality (Euler-Lagrange) system:
⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪iℏψt + ℏ22MψXX + ϕψ + eu∗ψ = 0 in Q,ψ(0) = ψ0 on Ω,1c2ϕtt - ℏ22mϕXX + m2c2ℏ2ϕ = |ψ|2 + e′v∗ϕ in Q,ϕ (0) = ϕ0, ϕt(0) = ϕ1 on Ω. (5)
⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪iℏpt +ℏ22MpXX +ψq + ϕp = 0 in Q,ipf = ψf(u∗) - ψtarget on Ω,1c2qtt - ℏ22mqXX + m2c2ℏ2q = 2|ψ|ϕp in Q,qf = ϕf(u∗) - ϕtarget, q′f = 0 on Ω. (6)
(u∗,u - u∗)U + ∫Qp(u∗)(u - u∗) + q(u∗)(v - v∗)dxdt ≥ 0,∀u = (u,v) ∈ Uad. (7)
Where (p,q) ∈ W(0,T;V,V′) are weak solutions of adjoint systems (6) corresponding to solutions (ψ,ϕ) in states system (5) respectively. Clearly, inequality (7) is well known as the necessary optimality condition.
Using the definitions of weak solution and weak form (3) in the defined solution space W(0,T;V,V′), one can quickly get full proof as analogical manipulation citing [11,13,21,22].
A lot of successful algorithms for finding optimal control of quantum system, for example, there are iteration learning algorithm; Genetic algorithm; evolutional algorithm, etc. To calculate quantum optimal pairing u* and v*, in this paper, is for seeking quantum optimal u*, a reasonable semi-discrete algorithm (spatial variable x discrete, time t continuous) is employed for control in nucleus. In there, the finite element method is used in computational approach in the form of variational method at Hilbert spaces. Quadratic base function is adopted in finite element approximate. Further, a modified nonlinear conjugated gradient method (CGM) is utilized in the system optimality minimization. Particularly, the wave-particles duality and uncertainty principle of Heisenberg [23] allow the ideal and reliable scheme. The convergence order is guaranteed (cf. [6]) for first order O(h). Roughly introduce the structure of numerical algorithm. Let 0 = t0 < t0 < • • • < tN < tN+1 = T be a partition of the interval [0, T] into subintervals Ie = [te-1, te) of length dte = te - te-1, e = 1, 2, • • • , N + 1. The corresponding spatial interval would be Je = [xe-1, xe] of length he = xe1 - xe−11 = xe2 - xe−12 = xe3 - xe−13, e = 1, 2, • • • , N + 1. Let Vh be a approximate space expanded by quadratic basis functions bei (i = 1, 2, 3, e = 1, 2,..., N + 1), which continuous on each interval Je given by
⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪be1(x) = (1 - x - xehe)(1 - 2(x - xe)he),be2(x) = 4(x - xe)he(1 - x - xehe),be3(x) = - (x - xe)he(1 - 2(x - xe)he). Using bei to construct the total approximate solution for j-th nucleon and j′−th meson as
⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ψjh(t,x) = ∑e = 1Nψe,jh(t,x) = ∑e = 1N∑i = 13ξe,ji(t)bei(x) ∈ Vh,ϕj′h(t,x) = ∑e = 1Nϕe,j′h(t,x) = ∑e = 1N∑i = 13ζe,j′i(t)bei(x) ∈ Vh,
Where ξe,ji, ζe,j′i are indeterminate coefficients, which will be solved by fourth order Runge-Kutta method as in [11]. Assume that uh = (uh,vh) is valid at one iteration step, its corresponding approximate solutions are ψjh and ϕj′h. Denote nc,n′c are the total number of involved nucleon and meson, respectively. Consider minimization problem for discrete objective functions:
Jh - J(uh) - ε1∑j = 1nc∫Ω(ψjh(T) - ψjtarget)2dx + ∫0T|uh|2dt (8) + ε2∑j′ = 1n′c∫Ω(ϕj′h(T) - ϕj′target)2dx + ∫0T|vh|2dt.
A modified nonlinear conjugate gradient method (CGM) is used for minimization problem in (8). The detail computing procedure can be find in [11] for one dimension.
Theorem 6: For finite element problems (8), there exists at least a minimizer {u∗} of sequence uh.
Theorem 7: For minimizing the control sequence {uh}, extract a subsequence, again denoted by {uh} which convergent to u∗ in L2(Ω) × L2(Ω), to minimize the discrete problem (8).
By extended to two dimension case, the convergence of iteration procedure in minimizing Jh is guaranteed in [6], and the approximate control uh converges to u∗ in the order of O(h) as h → 0.
Set physics constant ℏ = M = m = c = 1, e = e′ = 1. Nucleon and meson particles motions are considered in two dimensional spatial space [0, X] × [0, Y], where X = 50(μm) and Y = 50(μm). Take t0 = 0.0(fs), T = 20.0(fs) and time interval ∆t = 1.0(fs). Assume the stop criteria ε = 10-35 in minimization scheme. Denote center coordinate x0 of domain. Define two appendix functions for initial states configuration:
Ψ(x,t,v1) = 32–√41 - v21−−−−√sec2121 - v21−−−−√(x - v1t - x0) × exp(i(v1x + (1 - v21 + v41)2(1 - v21)t)),
Φ(x,t,v2) = 34(1 - v22)sec2121 - v22−−−−√(x - v2t - x0),
Where, v1 and v2 are velocities of nucleon and meson, respectively.
Use Mathematica for three sets of simulation experiments.
Set x coordinate location center x0 = 25.0(μm), two particles locations at y coordinate are y10 = 15.0(μm) and y20 = 35.0(μm). Precisely, nucleon located at (25,15) and meson located at (25,35) in the (x,y) plane. In physics, it will be n + π+ → p, one proton will be gotten. Their wave propagation velocities v1 = 5/12 and v2 = -5/12. Therefore, initial ground states can be given by
ψ0 = Ψ(x - x0)2 + (y - y10)2,−−−−−−−−−−−−−−−−√0,v2),
ϕ0 = Φ(x - x0)2 + (y - y20)2,−−−−−−−−−−−−−−−−√0,v1).
The graphics of ψ(0), ϕ(0) and ψ(0) + ϕ(0) are plotted in Figure 1, respectively.
Take target quantum excited states
ψtarget = Ψ(x - x0)2 + (y - y20)2,−−−−−−−−−−−−−−−−√T,v2),
ϕtarget = Φ(x - x0)2 + (y - y10)2,−−−−−−−−−−−−−−−−√T,v1).
Assume the iteration step nn, and define Gaussian enveloped function s(nn) = exp( - π2(t - (nn - 1)Δt)2). Set initial control input u(0) = s(nn)sin(300t + π6) for nucleon and v(0) = s(nn)sin(240t + π3) for meson. Notice that the last iteration wave function is not used in next step control: ψ(nn + 1,0) ≠ ψ(nn,Δt),ϕ(nn + 1,0)≠ϕ(nn,Δt). It means that quantum dot at spatial location is not depended on time. Wave function has the same fashion at each iteration. Figure 2 given the states transform of nucleon.
Figure 3 given the states transform of meson.
Figure 4 use Mathematica VQM package1, [24] 3D plot of wave function ψ + ϕ.
Figure 5 is the probability density plot of ψ + ϕ.
At Figure 4 and Figure 5, control function u and v act at the nucleon and meson, and force particles reach optimal states at final step. They replaced their spatial position and changed the representing color. It means that neutron absorb meson become proton (e.g. n + π+ → p), similarly simulation can be obtained for proton emit meson become neutron (e.g. p + π- → n). Observe imaginary part coefficients of used control function at each iteration, the gauge is agree with realistic atom unit. Figure 6 is the optimal state of ψ∗ + ϕ∗.
The graphics of u*(t) and v*(t) at all iterations are plotted in Figure 7 and Figure 8.
Notable, there is a gap in the plot of v* at Figure 8. Before the gap, meson is out of the nucleon cloud, after the Yukawa interaction, meson motion moved into (absorbed by) the cloud of one nucleon, but they had neither been completely reached the nucleon core itself, therefore, the probability distribution of meson will be likely surrounding the nucleon, and its (external) control input cannot be acted at this area. In the spatial dimension, the empty gap definitely appeared. Statistically, that is to say, it cannot find the meson at gap space. As is known that the radius of neutron is rn = 1.11492 × 10-13 m, the radius of proton is rp = 1.113386 × 10-15 m, the radius of pion meson is rπ = 1.53472 × 10-18 m, and the electron radius is re = 2.817939 × 10-15 m. The guess amplitude of the gap is small than the radius of nucleon. Very interesting, in Figure 7 they are no gap at nucleon control function at all, owing to its control input can be acted at the nucleon itself either not absorbed (no emission) or absorbed (emission) meson. Both in Figure 7 and Figure 8, the amplitudes are clearly changed after n = 3, it reflects that the positions of nucleon and meson are slightly changed.
Using quark calculation to know that neutron n = udd (spin 1/2, charge 0), proton p = uud (spin 1/2, charge +1), in here up quark u23 , down quark d−13, and meson pion π+ = ud¯ (no spin, charge +1), where anti-quark d¯ = d13. Then to have u23d−13d−13 + u23d13 → u23u23d−13. In resultant, there is one quark u and one quark d not changed, they combine a new quark u of π+ to compose proton, meanwhile, destroy (annihilation) a quark d and anti-quark d¯. The mass of up quark u23 is 5MeV/c2, and the mass of down quark d−13 is 10MeV/c2, where 1MeV/c2 =1.782677 × 10−30 kg. It means that d is bigger than u, then neutron n = udd is bigger than proton p = uud. In scientific calculating, the mass of neutron n is 939.6MeV/c2, and proton p is 938.3MeV/c2. Symbolically, 2 big quantum dots and 1 small quantum dot ⇝ 2 small quantum dots and 1 big quantum dot. Confidently, it is accordance with the real physical facts.
The optimal cost function value J(u*) = 0.747069, and minimization error eJ(u*) = 0.00563018. Quantum optimal control is calculated,
u∗ = e− π(t - 9.5)2[ - 0.320553 + 0.804574 sin(300t + π6)],
v∗ = e− π(t - 9.5)2[ - 0.00130171 + 0.748113cos(240t) + 0.431923sin(240t) - 0.0596983sin(240t + π3)].
Figure 9 is for their graphics.
The cost values J(u) and error values eJ(u) are shown in Figure 10 and Figure 11, respectively.
Non-control. For the comprising, non-control iteration data had also been obtained, and their values are shown in Table 1. Clearly, control is accelerating the meson exchange effect in the Yukawa interaction. The energy is not conserved at whole control process. The value of cost function, and its error at each iteration step, see Table 1.
The implementation is taken place at the device with CPU 1.3GHZ, MacOS GPU v3. Total computing time is 2950.12 second. The used maximum memory is 2262.107390 bytes.
Set proton (neutron and meson) located at (25,15) and neutron located at (25,35) in the (x, y) plane. Meson wave propagation velocities v1 = 5/12 or v2 = -5/12 between proton and neutron. Set initial control inputs u(0) = s(nn)sin(300t + π6) for proton, no control for neutron, and v(0) = s(nn)sin(240t + π3) for meson. Therefore, initial ground states can be given by
ψ0n = Ψ(x - x0)2 + (y - y10)2,−−−−−−−−−−−−−−−−√0,0),
ψ0p = Ψ(x - x0)2 + (y - y20)2,−−−−−−−−−−−−−−−−√0,0),
ϕ0 = Φ(x - x0)2 + (y - y10)2,−−−−−−−−−−−−−−−−√0,v1).
The graphics of ψn(0), ψp(0), and ϕ(0) are plotted in Figure 12, respectively.
Notice that neutron ψn(0) and meson ϕ(0) is located at same position. It means that proton is located at (25,15) initially. Physically, it will be n + π+ → p and p, two protons will be gotten. Figure 13 is their 3D plot of wave functions ψn + ψp + ϕ at each control iteration.
More precisely status, use Mathematica VQM package to execute 3D plot of wave functions ψn + ψp + ϕ at each control iteration Figure 14.
Figure 15 is the probability density plot of ψn + ψp + ϕ.
Wigner function for representing probability density |ψ|2 and |ϕ|2 are compared here Figure 16.
Take proton mass Mp = 1.67262 × 10-29 g, neutron mass Mn = 1.67493 × 10-29 g, and meson mass Mm = 2.305589 × 10-30. The speed of light c = 2.99792 × 108, the reduced Planck constant ℏ = 1.0545715964207855 × 10-34, electron charge e = 1.602176462 × 10-19. Particles located at the same position as Ex. b). In physics, meson speed vm = 0.998c between proton and neutron. It will be n → π-1 + p and p, meson will be appeared with neutron and proton, respectively. Notice that the domain of spatial variable is taken as Ex. b) for the visualization, and set meson wave propagation velocities v1 = 5/12, v2 = -5/12 for continuity of algorithm. For much more delicate physical control issues using this paradigm, it concerns future work beyond this section. Fortunately, current simulation is the foundation of realistic physical control of KGS system. Ex. c) will suggest that control can be actually executed at the physics field. Certainly, further demonstration in the viewpoint of physics area will appear sooner after. Set initial control inputs u(0) = 5.0 × 1021s(nn)sin(300t + π6) for neutron and proton, and v(0) = 3.0 × 1019s(nn)sin(240t + π3) for meson. Therefore, initial ground states can be given and expressed by
ψ0n = 5.0 × 10− 19Ψ(x - x0)2 + (y - y10)2,−−−−−−−−−−−−−−−−√0,0),
ψ0p = 3.0 × 10− 19Ψ(x - x0)2 + (y - y20)2,−−−−−−−−−−−−−−−−√0,0),
ϕ0 = 0.1 × 10− 21Φ(x - x0)2 + (y - y10)2,−−−−−−−−−−−−−−−−√0,v1).
The graphics of ψn(0), ψp(0) and ϕ(0) are plotted in Figure 17, respectively.
Figure 18 is normalized 3D plot of wave function ψn + ψp + ϕ at each control iteration.
Precisely, Mathematica VQM package to do 3D plot of wave functions ψn + ψp + ϕ at each control iteration as Figure 19.
Normalized 3D wave function plots Figure 20:
Figure 21 is the probability contour plot of ψn + ψp + ϕ.
Remark 8 For the robustness and tolerance occurred at control in the nucleus, it needs to discuss their theoretic and computational issues.
Cite [12] to know, perturbation problems always incident at the real laboratory control of particles, optical technique adopted in the physical experiment needs much more delicate control structure setting in system its self and each parameter selection. This induced noise and uncertainties which happened in controlling of particles. For example, in former poster [12], the result is for control of perturbed Klein-Gordon-Schrödinger system as the uncertainties occurred in the electric field, that conclusion is, if tolerance or perturbation is bounded (very tiny) comparing with control variable or quantum system gauge, then control is also valid, feasible and maintain efficiency. It means that quantum control for nuclei is worked under external noise (influence) is small enough. If no boundedness perturbation, then not valid.
At control of macrosystem, the robustness control put at the first place to be considered, such as the policy declaration, large scale complex system cybernetic, scientific management system configuration, the industrial and engineering system confinement, and financial system (rates) adjustment, etc. To say control, it means that there is an objective, a target to achieve, at a macro system, the aim might be quite clear and urgent. An obvious result is desired in a limit time duration. Therefore, robust control is necessary and often to be utilized sufficiently. On the other hand, these robust control concept used in the microsystem, for instance, in the Klein-Gordon-Schrödinger system, it is not so significant than macrosystem. The discussion of robustness as to a quantum system, first, it needs to define what is a robust control meaningful to a quantum particle? second, whether such a robust control is necessary to be surveyed? third, whether robust control worked for control of particles?
As is well known that, there are very famous experiments using the elementary particles, and lots of interesting results had been accomplished in the past century, those physical or chemical experimentation did not be called control of particles, they had been laying on the physics or chemistry experiments in each of their field. Because at those experiments' initial setup, control did not in the mind. Especially, these experiments are for getting the physics results, test fiy the particles detection, interaction and so forth. Modern control theory is proposed at later year in the field of aero engineering for launch of rocket at military purpose. Rapidly, they covered amount of fields, and extended to almost all the scientific areas.
Now, at this paper, let's put the further themes robust control aside, it mainly considers control at nucleus due to it needs us to do control at particles firstly. It is believed that robustness, fault tolerance, perturbation, and uncertainties would be one of the research directions in the future works. With these theoretical and computational results, together with sustainable and successful control of particles at real experiment, at that time, it would be adequate to discuss a great deal problems including robust control, processing control, and many control issues which already appeared at macrosystems, and had sophisticated theory for their application to microsystem.
This work present two dimension control of couple heavy particles in the Yukawa interaction at nucleus. The theoretic analysis and experiment demonstration evident that it is reasonable to execute the quantum controlling with the advanced optical technology. The study would provide a bright perspective for quantum nuclei controlling in realistic meaning. In the future work, the proposed computational approach would be developed to a numerical methodology not only in theoretical study but also in laboratory experiments for a wide class of control problem of quantum system in physical field. Particularly, it can be applied to control of elementary particles at nucleus with a kind of inventing instruments at laboratory.
The author really appreciates 258th ACS National Meetings & Exposition 2019.
1Visual Quantum Mechanics, Bernd Thaller, University of Graz, Austria (cf. [24]).