Derivation of a Hamiltonian for Formation of Particles in a Rotating System Subjected to a Homogeneous Magnetic ﬁeld

Derivation of a Hamiltonian for formation of particles in a rotating system subjected to a homogeneous magnetic ﬁeld.

Sergio Manzetti 1,2 and Alexander Trounev 3 1. Uppsala University, BMC, Dept Mol. Cell Biol, Box 596, SE-75124 Uppsala, Sweden. 2. Fjordforsk A/S, Bygdavegen 155, 6894 Vangsnes, Norway. 3. A & E Trounev IT Consulting, Toronto, Canada.

December 17, 2018

1 Abstract

Bose-einstein condensates have received wide attention in the last decades given their unique properties. In this study, we apply supersymmetry rules on an existing Hamiltonian and derive a new model which represents an inverse scattering problem with similarities to the circuit and the harmonic oscillator equations. The new Hamiltonian is analysed and its numerical solutions are presented. The

1Please cite as: Sergio Manzetti and Alexander Trounev. (2019) ”Derivation of a Hamilto- nian for formation of particles in a rotating system subjected to a homogeneous magnetic ﬁeld” In: Modeling of quantum vorticity and turbulence with applications to quantum computations and the quantum Hall eﬀect. Report no. 142020. Copyright Fjordforsk A/S Publications. Vangsnes, Norway. www.fjordforsk.no

1 results suggest that the SUSY Hamiltonian describes the formation of vortices in a homogeneous magnetic ﬁeld in a rotating system.

2 Introduction

Hamiltonian operators are probably the most fundamental part in quantum physics and form the basis for calculating the physical properties for elementary particles, their energy and observables. Operators possess properties, such as self-adjointness and unboundness, which allows them to give sensible physical answers to quantized systems, where operators such as the Hamiltonian in the anyon-model of Laughlin [1] are of particular appeal. Supersymmetry is also an appreciable aspect of matter and energy, and by its models developed in the last decades, a concise foundation for its application in various fields of physics shows potential towards problems in the areas of conductance, magnetism and gravity fluctuations [2, 3, 4, 5]. A possible application of supersymmetry is also in quantum computing algorithms and can be based on solving computa- tional chains of commands using elementary particles, their spin identity and/or charge and a supersymmetric rule in interpreting their signal. Supersymmetry has previously been investigated in relation to particles with emphasis on the spin-population in terms of non-relativistic supersymmetric terms by Valenzuela et al [6]. Brink et al. [7] have also provided new insight to the field of research on supersymmetry and particles and reported a system based on supersymmetry on a particular operator to include fermions and anyons where they obtained a solution of the supersymmetric Calogero model and a supersymmetric extension of the deformed Heisenberg algebra. Hlousek and Spector [8] also contributed to this field and developed a supersymmetric model for anyons where the interplay between supersymmetry and physical properties of anyons (such as rotations and confinement) was analyzed. Hlousek and Spector constructed an own supersymmetric generalization of the Hopf term, which generates the anyon model as supersolitons. This led to the proposition that the transition from unbroken gauge symmetry to the Higgs phase is simultaneously a transition from anyon

2 to canonical spin and statistics. In the present paper one rather focuses on an approach where we simplify an existing Hamiltonian [1] into a supersymmetric counter-part which we study its operator properties, and from which we develop a master equation using the Gross-Pitaevskii model [9, 10].

Supersymmetry A supersymmetric state evolves from factorizing the Schr¨odingerequation into a super-pair of the Hamiltonian. This can be easily represented by splitting the kinetic term into two ﬁrst-order derivative operators, A and A†. This well- known procedure [11] gives the superpotential

√ 2 0 V (x) = W (x) − (~/ 2m)W (x), (1)

with its solution:

√ 0 ψ0(x) W (x) = −~/ 2m , (2) ψ0(x)

where ψ0(x) is the original wavefunction for the non-supersymmetric Schr¨odinger equation. The charges of the particles are also represented by supersymmetric terms, using the operators:

    † 0 0 † 0 A Q =   Q =   , A 0 0 0 which commute by elementary superalgebra [9]. The supercharges have an advantage over regular charge-representation for elementary particles in classi- cal physics in that they can be considered as operators which change bosonic degrees of freedom into fermionic degrees of freedom and vice-versa [11]. In order to develop a supersymmetric analogue of the Laughlin Hamiltonian, we ﬁrst revisit the Laughlin state of anyons [1], to then devise the supersymmetric sign duality (see above) on the factorized components of the Laughlin Hamiltonian.

3 Supersymmetric representation of the Laughlin state Laughlin [1] deﬁned the Hamiltonian operator:

2 H = |~/i∇ − (e/c)A~| , (3)

~ 1 where A = 2 H0[xyˆ − yxˆ], is the symmetric gauge vector potential, where

H0 is the magnetic ﬁeld intensity. However, presenting this treatise in a ﬁrst quantized formalism, we use the regular form:

2 H = [~/i∇ − (e/c)A~] , (4)

where the Hamiltonian is simply the normal square of the covariant derivative. Supersymmetry rules introduced above are here adapted to form a supersymmetric model for the states of the quantum hall particles. We do this by looking at the Hamiltonian in eqn. (4), which is composed of the two ﬁrst-order diﬀerential operators:

H = [~/i∇ − (e/c)A~] × [~/i∇ − (e/c)A~]. (5)

Applying SUSY [11] rules on the two factorized components in eqn. (5), we in- vert the sign of the second ﬁrst-order diﬀerential operator, and get the superpair of the Laughlin Hamiltonian:

~ ~ HSUSY = [~/i∇ − (e/c)A] × [−~/i∇ − (e/c)A]. (6)

The SUSY counter-part of the Hamiltonian from (5) becomes:

∗ HSUSY = TT , (7)

with

T = [~/i∇ − (e/c)A~], (8)

4 and

∗ T = [−~/i∇ − (e/c)A~], (9)

where T and T ∗ are the superpair in the SUSY Hamiltonian and one an- other’s complex conjugate.

We consider then T and T ∗, with γ = (e/c)A~, in one dimension:

h d T = − γ · I˜ i dx

h d T ∗ = − − γ · I˜ . i dx which commute by the relation:

[TT ∗] − [T ∗T ] = 0 (10)

making T and T ∗ two commutating complex operators in Hilbert space H . Both T and T ∗ are unbounded operators given the condition:

||T x|| c||x||, (11)

T and T ∗ are also non-linear given the γ constant, being a constant trans- lation from the origin. T and T ∗ are non-self-adjoint in H as the following condition is not satisﬁed:

hT φ, ψi = hφ, T ∗ψi, (12)

From this, it follows that:

HSUSY : D(HSUSY ) −→ H ,

and

D(HSUSY ) ⊂ H ,

5 where HSUSY is an unbounded nonlinear and non-self-adjoint operator com- 2 plex on Hilbert space H = L [−∞, +∞]. However, HSUSY is self-adjoint in its 2 domain, D(HSUSY ) on L [a, b].

From supersymmetry theory in quantum mechanics [11] it follows that H†Ψ = HΨ† = EΨ = EΨ†, therefore we can assume that:

HSUSY Ψ = EΨ, (13)

where E is the energy of the system, which generates the boundaries of 2 D(HSUSY ) on L [0,L], where the interval of quantization (L) is contained in 2 2 2 n ~ π the zero-point energy term E= 2mL2 . We are primarily interested in studying the system in eqn (13) at singularity, which yield the form:

HSUSY Ψ = 0, (14) and follow by developing an analytical solution to eqn. (13). In the one- dimensional case, we represent (13) in the form

0 0 −γψ(x) + i~ψ (x) = ψ1(x), Eψ(x) = −γψ1(x) − i~ψ1(x) (15)

This system of equations has periodic solutions at E < γ2 and exponential for E > γ2. Consequently, if we consider special solutions for E = 0, then they will be periodic everywhere, where the vector potential describing the magnetic field is nonzero. We shall consider the behavior of a large number of particles in a homogeneous magnetic field in a rotating system. The vector potential in this case has the form A~ = [B~r~ ]/2. Magnetic field has one component Bz only, 2 2 2 2 therefore A = Bz (x + y )/4. For normal charge particles these potential acts like a trap. But in a case of SUSY Hamiltonian the potential change sign, as we can see from eqn. (15), so we have opposite action and special kind of dynamics lids to the vorticity formation.

6 3 Numerical solution to the SUSY-Hamiltonian

We use Gross-Pitaevskii model [9, 10] and Sobolev gradient method [12]. Put ψ = ψ(x, y, t), then in nondimensional variables master equation has a form:

∂ψ 1 2 ∂ψ ∂ψ 2 2 2 2 (16) ∂t = 2 ∇ ψ + iΩ(x ∂y − y ∂x ) − β|ψ| ψ + γ (x + y )ψ Note that nonlinear Schr¨odingerequation (Gross-Pitaevskii model) follows from (16) when replacing t → it, γ → iγ. Eqn. (16) describes evolution of the wave function from some initial state ψ(x, y, 0) = ψ0(x, y) and up to stationary state with E = 0. The energy approximates zero asymptotically with time, with some relatively small positive or negative deviation (Fig 1). We can stop calculation at E = 0, or at t = topt, where solution comes to stationary state with E > 0 or E < 0, but close to E = 0.

. Figure 1. The change in the energy of the system as a function of time (top) and the wave function at diﬀerent instants of time with the corresponding energy levels (bottom).

Since we investigate periodic solutions of equation (16), the initial data can also be given in the form of a periodic function, where we select,

ψ(x, y, 0) = (cos(xπ/2L) cos(yπ/2L))n, n = 1, 2, 3, 4, (17)

which is presented in ﬁg. 2 where the simulation data is shown from equation

7 (16) with the initial conditions in eqn. (17) and zero boundary conditions. The parameters of equation (16) were given the following: Ω = 20, β = 1000, γ2 = 1/2, where Ω is the angular momentum and β and γ2 are arbitrary constants. As it follows from the data, shown in Fig. 2, vorticity is formed in all cases with an equal number of vortices (twelve holes/vortices). Externally, the final distribution of the amplitude of the wave function looks the same for n = 1,2,3,4, although the initial data differ quite strongly. The vortices are formed around a quantum void, depicted in blue in the figures.

. Figure 2. The amplitude of the wave function at diﬀerent instants of time for the initial data (17) with n = 1, 2, 3, 4 and Ω = 20, β = 1000, γ2 = 1/2 . Black: holes/vortices; conﬁned blue region: quantum void.

We also investigate the behaviour of the wavefunction at energy level n=4, with angular momentum Ω=15 and 20 (Fig. 3).

8 . Figure 3. The amplitude of the wave function at diﬀerent instants of time for the initial data (17) with n = 4 and Ω = 15, 18, 20, 30; β = 1000, γ2 = 1/2 .

We note the essential diﬀerence between this problem and the analogous problem for the Bose-Einstein condensate [13]. The vortices arise in a com- pletely symmetric fashion during the decay of the symmetry of the periodic boundary conditions. Also, we note that the value of Ω is an order of magni- tude higher than in the analogous problem [13]. We found out what caused such a big diﬀerence. For this, we took as the initial data the ground states described in [13], namely

2 2 √ ψ(x, y, 0) = ψv(x, y) = a exp[−(x + y )/2], a = 1, 1/ π (18) ψ(x, y, 0) = ψv1(x, y) = (x + iy)ψv(x, y). The simulation data of the amplitude of this wave function with initial data in the form of ψv with a = 1 is hereby shown in figure 4. In this case, 4 vortices arise even at Ω = 0.95, β = 100, while for Ω = 1.4 their number is 24. This agrees with the results of [13], however there is also a difference in the geometry of the vortex distribution. This can be seen in figure 5, which shows the 3D

9 distributions of the amplitude of the wave function corresponding to the data in Fig. 4. Hence, figure 5 shows that individual vortices are represented as holes. We furthermore show in fig. 6 shows the simulation data of the amplitude of the wave function with initial data in the form of ψv1 with a = 1 and Ω = 1.4, β = 100. Comparing the data shown in Fig. 4 and 6, we find a great difference in the final states with equal values of the parameters, based on the smoothness of the initial state. The difference of the two initial states in (16) and (18) is therefore of paramount importance to study the SUSY Hamiltonian further.

. Figure 4. The amplitude of the wave function at diﬀerent instants of time for the initial data (18) in form ψv with a = 1 and Ω = 0.95, 1.1, 1.2, 1.4; β = 100, γ2 = 1/2 .

10 . Figure 5. 3D distributions of the amplitude of the wave function at diﬀer- ent instants of time for the initial data (18) in form ψv with a = 1 and Ω = 0.95, 1.1, 1.3, 1.4; β = 100, γ2 = 1/2 .

. Figure 6. 2D and 3D distributions of the amplitude of the wave function at diﬀerent instants of time for the initial data (18) in form ψv1 with a = 1 and Ω = 1.4; β = 100, γ2 = 1/2 .

In Fig. 7 shows 3D distribution of the amplitude of the wave function in the ﬁnal state in Fig. 6 in two angles, illustrating the depth of penetration of holes. 2D and 3D distributions of the amplitude square of the wave function at

11 √ different instants of time for the initial data (18) in form ψv with a = 1/ π and Ω = 1.4; β = 100, γ2 = 1/2 are shown in Fig. 8. Thus, we have shown that the supersymmetric Hamiltonian has a wide field of application including the simulation of vorticity in quantum systems in a magnetic field and possibly for quantum rotation. The results obtained by us convincingly testify to the existence of periodic solutions of equation (16) containing vortices in a wide range of parameters and under different initial conditions. The numerical model for equation (16) is much simpler than in the case of a similar problem formulated for the Bose-Einstein condensate. This is explained by the fact that equation (16) has steady-state solutions that correspond to states with zero energy for the initial model. To find these solutions, we used standard package for PDE solving in the system Mathematica version 10.x-11.x.

. Figure 7. 3D distributions of the amplitude of the wave function at t = 2 for 2 the initial data (18) in form ψv1 with a = 1 and Ω = 1.4; β = 100, γ = 1/2 .

12 . Figure 8. 2D and 3D distributions of the amplitude square of the wave function √ at diﬀerent instants of time for the initial data (18) in form ψv with a = 1/ π and Ω = 1.1, 1.4; β = 100, γ2 = 1/2 .

Finally, we note that the SUSY Hamiltonian does not differ in its application from the well-known Hamiltonian (4) defined by Laughlin [1]. Equation (16) can be regarded as a natural extension of model [1] for the numerical study of the dynamics of quasiparticles in magnetic fields.

4 Discussion

The SUSY Hamiltonian (6) describes a system composed of a damping part and an angular motion part combined together, which is similar to the damped harmonic oscillator [14]. The Hamiltonian can, in its given form (eqn. (6) therefore describe a scattering phenomenon [15] where the oscillating particle

13 has a small imaginary part in its mass and experiences decay (with negative value of γ) =(m) 6=0, releasing energy, or the particle experiences gain and ab- sorbs energy (with a positive value of γ). This system is also a retrospective inverse scattering problem [15], which can describe physical parameters (i.e. electromagnetic ﬁeld intensity or conductance) from observations of the evolution of the wavefunction. In its form given in eqn. (6), it has a non-unitary time-evolution, which requires the generation of a master equation in order to explain a quantum-event with a fully developed time-description [16, 17]. We have generated a master eqn. in eqn (16) which yields a supersymmetric model and thus drives the general state in eqn. (6) into a pure quantum state [18]. This gives a description of a physical properties of a new equilibrium system possibly occurring in quantum Halls and in superconductors and/or alkali met- als. In its general form however (eqn. (6)), the SUSY Hamiltonian can be made more general given its retrospective time-properties [16, 19], and because the physical law described by it is currently unknown, it can be applied also to other phenomena, for instance non-equilibrium phenomena in quantum electro- dynamics and quantum ﬁeld theory.

5 Conclusions

A supersymmetric version of the ”Laughlin Hamiltonian” for bose-einstein condensates has here been developed and its properties discussed. Given the emerg- ing interest in anyons in the last decades and the putative application in quantum computing and quantum algorithms [20, 21, 22, 23, 24, 25], the approach of considering the supersymmetric variant of this Hamiltonian is expected to be a resource to the quantum physics community. The SUSY Hamiltonian describes a problem which is similar to the harmonic oscillator equations and the circuit equations and can, therefore, be of importance to interpret particles in one-dimensional systems in the future. Further work will evolve on studying the evolution of the SUSY Hamiltonian model and the diﬀerence between smooth

14 and less smooth initial states, as well as the evolution of angular momenta under and formation of vortices.

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18 6 Acknowledgements and correspondence

A sincere gratitude is owed to A. Prof. Michael Rogers at the Oxford College of Emory University for critical discussion on the manuscript. The author would also like to thank Professor Ronald Friedman at Indiana University Purdue University Fort Wayne and Prof. Thors Hans Hansson at Stockholm University for useful comments, and Prof Anna Fino from the Mathematics Department, University of Torino, for inspiring to write this paper.

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