Supersymmetric States of Quantum Droplets

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Supersymmetric states of quantum droplets

Technical Report · May 2020 DOI: 10.13140/RG.2.2.19602.68806

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Sergio Manzetti 1,2∗ and Alexander Trounev 2,3 1. Uppsala University, BMC, Dept Mol. Cell Biol, Box 596, SE-75124 Uppsala, Sweden. 2. Fjordforsk A/S, Midtun, 6894 Vangsnes, Norway. Email: [email protected] 3. A & E Trounev IT Consulting, Toronto, Canada.

May 25, 2020

Abstract

1 A dropleton, or quantum droplet made of bosonic particles interacting via the long range dipole-dipole interaction is a collective state found in rare-earth metal gases. The properties of dropleton are particularly interesting, as droplets composed of a higher number of atoms with large enough magnetic moment may assume a supersymmetric ground state. This effect results when laser electromagnetic fields affect the quantum gas, and reduce the number of available symmetries for the system of magnetic dipoles. Here, we devise a recently developed supersymmetric equation to identify the lowest energy state of dropleton composed of N atoms. We also inspect on the existence of preferred symmetries for various values of N and by modulating the strength and property of the external vector potential. We finally extend our analysis identify the fate of droplets and formed torii during the evolution of time. The results are impor-

1Please cite as: Manzetti Sergio and Trounev Alexander (2020). Supersymmetric states of quantum droplets. Technical Reports, 2024:1-21. Fjordforsk AS, www.fjordforsk.no

1 tant for advanced applications in computing and data-processing units based on dropletons, which may form the basis for CPU technologies of the future.

1 Introduction

Quantum information processing and computing are dependent on a generation of processors and computational units which radically differs from the older generation. The emerging technologies in quantum computing devise the spin of electrons as ideals for generating computational signals for the execution of chains of command in the CPU [1, 2, 3, 4]. Alternative methods to generate computational signals can also be generated by using other for fermionic systems, for instance anyons [5] or quantum liquids, where the re-arrangement of quasi-/particle symmetry by an external magnetic field generates the computational signal. One particular quantum fluid of interest is represented by liquid-like quantized quasiparticle ensembles knows as dropletons, formed in metal lattices which, considered incompressible, assume often a lower energy state when composed of several quasiparticles compared to states for one single quasiparticle [6]. Dropletons are formed in rare-earth metal lattices under the influence of an imposed dipolar electromagnetic fields, as recently observed in 164Dy and erbium atoms [7, 6, 8, 9, 10, 11, 12, 13]. The appealing property of dropletons is their existence at ultra-low atom densities (108 times lower than in a helium droplet) which follows strict symmetry-reorganization by symmetry breaking [11]. Determining the symmetry of dropletons is experimentally chal- lenging, and models for the analysis have been developed [6, 14, 7] using the classical Bose-Einstein equation for Bose-Enstein condensates. Previously, we developed a Hamiltonian for the description of vorticity in quantized as well as continuum systems [15, 16, 17, 18], which of we also developed analytical solutions to [19]. Using this operator, we studied the impact of homogeneous and in- homogeneous magnetic fields on vortex and quasiparticle dynamics, and derived the preferred symmetries under the influence of magnetic fields [15, 16, 17]. As dropletons are formed by a balance of the mutual attractive and repulsive forces

2 that derive from a nuclear potential [6, 8, 9, 10, 11], theoretical studies with alternating (rotating, oscillating or translocating) magnetic ﬁelds are of fundamental importance for the study of dropletons. Furthermore, the formation of dropletons is extremely sensitive to physical parameters, such as temperature [13], and theoretical models that best describe ultrasensitive phenomena require the use of non-self adjoint operators [20, 21, 22, 23]. Non-self adjoint operators have the following advantages, a) they pertain a wide spectrum of eigenvalues and b) generate often real solutions in the lower or upper limit, by using various methods i.e. transformation or perturbation methods [20]. The non-self-adjoint supersymmetric operator we developed: 2 e2 H = − ~ ∇2 − A~2, (1) SUSY 2m c2 has no upper limit and yields analytical solutions in the lower limit [17, 19, 24] in the form of the supersymmetric wave-equation

HSUSY Ψ = EΨ. (2)

In this study, we devise this equation to study the formation of dropletons in an electron gas, where we particularly investigate the preferred symmetries assumed by the system composed of N particles, their respective energy, their wavefunction and the type of symmetry arising during phase-transitions. To describe the evolution of the wave function of a system of bosons, we use the nonlinear equation

∂ψ 1 2 ∂ψ ∂ψ 2 3 ~ 2 (3) ∂t = 2 ∇ ψ − iΩ(x ∂y − y ∂x ) − g|ψ| ψ − gqf |ψ| ψ + (A) ψ

Here A~ is a dimensionless vector potential and q, qqf , Ω - parameters of the model describing the particle interaction and the angular velocity, respectively. Eq. (3) is diﬀerent from the standard eGPE [7, 6], because we explicitly take into account the rotation of the system and describe the electromagnetic ~ interaction using the vector potential. For a system of magnetic dipoles di in an external ﬁeld B~ , we have N ~ ~ X [di × ~ri] [B × ~r] A~ = + (4) r3 2 i=1 i

3 As is known, a system of magnetic dipoles in an external magnetic ﬁeld acquires a magnetic moment

Mi = χikBk. (5)

On the other hand, if the body acquires a magnetic moment, then it begins to rotate at an angular velocity

e Ω = g χ−1M . (6) l 2mc lk ik i here glk are gyromagnetic coefficients. This effect, first discovered by Einstein and de Haas [25], should also be observed in quantum systems. Consider the local model of the electromagnetic interaction of dipoles with two parameters

g [M~ × ~r] [B~ × ~r] A~ = dd + . (7) 2 2 3/2 (r + r0) 2 Using (7), one can solve equation (2), ﬁnd the eigenvalues and eigenfunctions. Then the solution of equation (2) is used as the initial data for equation (3). By these solutions of equation (3) we therefore aim to determine the number of quantum droplets, their stability and the physical parameters that determine their formation, existence and lifetime.

2 Numerical analysis and results

2.1 Droplet formation

Consider a sphere of radius R = 4, put B~ = M~ = (0, 0, 1), r0 = 0.3, gdd = 1 and find the first 10 eigenvalues of equation (2) - see Figure 1. Figure 1 shows the eigenfunctions of (2), which yields ten symmetric fluid-like states with parallel and concentric amplitudes. The positive and negative eigenvalues give all wave-amplitudes which locate both at the radial boundaries of the system as well as in the center. Particular features are found for eigenvalues E=- 0.135926 and E=-0.29154 which show similarities to liquid droplets similar to the droplet simulated by devising Taylor expansions in water simulation models

4 [26]. When we study the other eigenstates of (2), we ﬁnd a preferred C2 symmetry (belonging to the cyclic permutation group in group-theory) for eigenstates E=0.0989539 and E=0.0992089, E=0.245759 and E=0.24609, E-0.115858, E=- 0.117923 (Fig 1). The symmetry of the states of E=-0.135926, E=0.29154 and

E=0.203948 are also C2 symmetries with a clear rotation by π about the z-axis. We find however no symmetry for E=0.205016. Figure 2 shows the system potential −A~2 at z = 0 and five eigenfunctions corresponding to the energy levels E = 0.0989539, 0.24609, −0.135926, −0.117923, 0.203948 (indicated above the figures). The potential is not infinite and vanishes at the origin (denoted by a red peak emerging from the center of the potential), forming a combina- tion of repulsion and attraction, as is observed in droplet-experiments [9]. The first eigenfunction (Fig 1) describes 12 droplets located on two circles parallel to plane (x, y), with an apparent icosahedral Ih-symmetry (Fig 2 top middle) similar to an atomic g-orbital. The ninth eigenfunction (Fig 1) describes 10 droplets located on a circle in plane (x, y) with a Ih symmetry as well (Fig 2, top left), also similar to an atomic orbital of type fundamental (f-). The fifth eigenfunction (Fig 1) describes 3 concentric torii (Fig 2, lower left), which has 2 similarities to the 3dy atomic orbital in the arrangement of the torii. The fourth and sixth eigenfunctions from figure 1 describes droplets in shapes similar to a cucumber (Fig 2, lower middle and right). Neither of these shapes have been observed before in neither atomic orbitals nor liquid /crystal shapes according to our knowledge, however both pertain C2 symmetries, as their parental eigenfunctions.

5 . Figure 1 The ﬁrst ten eigenfunctions of (2) in a case of spherical region (with eigenvalues indicated above the ﬁgures) in plane (x, z).

. Figure 2 The potential of the system (top left) for z = 0 and ﬁve eigenfunctions in a case of spherical region (eigenvalues are indicated above the ﬁgures).

Interestingly, the model we describe here reports the same property of dropleton energy as observed experimentally by Wenzel et al [6], where dropletons composed of several quasiparticles may assume a lower energy than a single particle. Here in Figure 2 we see an example of this, where the energy of

6 the dropleton composed of twelve quasi-particles (Fig 2, top, middle) has a lower energy (E=0.0989539) than the dropleton composed of ten quasiparticles (E=0.24609), or for that matter, lower than the energy of the dropleton composed of six quasiparticles (E=0.203948, Fig 2, lower right), indicating that the symmetry is pivotal to lower the energy. By using the eigenfunctions shown in Fig. 1, 2 as initial data for equation (3) we now set the parameters to Ω = 0, g = 10, gqf = 20 and model the stability of the droplets with time. This is shown in figure 3 and figure 4 (respectively for 12 and 10 droplets), by the evolution of the wave function in the z = 0 plane and the distribution of 12 droplets at the moment t = 1 (top right). Figure 4 shows similar data for 10 droplets. We see that distribution of droplets is stable over a time-period with a stable magnetic momentum, in the nonlinear interaction of atoms compared with linear theory. Note that the droplets are generated by using the eigenfunctions (fig 1) of (2) as initial data for the master- equation in (3), so preserved symmetries between figure 1,2 and figure 3,4 are to be expected.

7 . Figure 3 The modulus of the wave function for z = 0 and for diﬀerent time (shown above), and the distribution of 12 droplets at t = 1 (top right).

8 . Figure 4 The modulus of the wave function for z = 1.5 and for diﬀerent time (shown above), and the distribution of 10 droplets at t = 1 (top right).

2.2 Droplet annihilation

Now we consider the formation of quantum droplets in a cylinder shape, which can describe droplet phenomena in non-stationary states. Here we model the system by a radius R = 1 and a height of H = 4, put B~ = M~ = (0, 0, 1), r0 =

0.3, gdd = 1 and find the first 10 eigenvalues of equation (2) - Figure 5. Figure 6 shows the system potential −A~2 at z = 0 and five eigenfunctions corresponding to the energy levels E=-2.4505, 3.30123, 3.41416, 4.37332, 6.59275, which have both C2 and C4 symmetries. A distinction can be made by the eigenvalues in that the increasing eigenvalues give eigenstates with a larger number of amplitudes, something that was not observed for the spherical system (Fig 1). This set of eigenstates yields, when applied as initial conditions for the master

9 equation (3) - fig. 7, 8, an ensemble of droplet formed in narrow channels, also defined by a potential that vanishes at the origin. The distinction between the droplet states is made on the increasing number of droplets formed at higher eigenvalues of eqn. (2). The analysis in figure 6 suggests that the evolution of the ensemble of droplets by higher energy in a channel-like system leads to the formation of a higher number of dropletons packing along the column, both aligned in parallel as well as stacked one after the other (Fig 6).

. Figure 5 10 eigenfunctions in a case of cylindrical region (eigenvalues are indicated above the ﬁgures) in plane (x, z).

. Figure 6 The potential of the system (top left) for z = 0 and ﬁve eigenfunctions in a case of cylindrical region (eigenvalues are indicated above the ﬁgures).

10 We set the parameters to Ω = 0, g = 10, gqf = 20 and model the stability of the droplets shown in figures 5, 6 with time using equation (3). This is shown in figure 7 and figure 8 (respectively for 4 and 2 droplets). Here we observe in fig. 7 the annihilation of three of four droplets. Figure 8 shows the initial stage of the annihilation of one of the 2 droplets. The last stage is shown in Fig. 9. This disappearance is explained by the fact that equation (3) describes the ground state of the cylindrical system with the lowest energy, which corresponds to one droplet - see figure 5, which differs from the dropleton formation in spherical systems as also known by the results of [6].

. Figure 7 The modulus of the wave function for y = 0 and for diﬀerent time (shown above), and the distribution of 4 droplets at t = 1 (top right).

11 . Figure 8 The modulus of the wave function for y = 0 and for diﬀerent time (shown above), and the distribution of 2 droplets at t = 1 (top right).

. Figure 9 The modulus of the wave function for y = 0 and for diﬀerent time (shown above), and the distribution of droplet at t = 2 (top right).

12 2.3 Torus annihilation

It was found that not only droplets undergo annihilation, but also toroidal states

- Fig. 10-12. We set in equation (3) the parameters to Ω = 1, g = 0, gqf = 0 and

R = 1,H = 4, B~ = (0, 0, 1), M~ = (0, 0, 3), r0 = 0.3. Figure 10 shows the system potential −A~2 at z = 0 and ﬁve eigenfunctions corresponding to the energy levels E = −0.315586, −0.338904, 0.850144, −0.963126, 1.03871 (indicated above the ﬁgures). We used the state with E = −0.963126 as the initial data for equation

(3) with Ω = 1, g = 10, gqf = 20. Figure 11 shows the initial and ﬁnal state of the quantum system with the indicated parameters. We see that 4 tori in the initial state (see Fig. 11 on the left) turned into one torus (Fig. 11 on the right).

. Figure 10 The potential of the system (top right) for z = 0 and ﬁve eigenfunctions at y = 0 in a case of cylindrical region (eigenvalues are indicated above the ﬁgures).

13 . Figure 11 The wave function in the initial (left) and ﬁnal (right) state.

In details, this transition can be traced in Fig. 12, where the modulus of the wave function in the plane y = 0 at various time instants is shown. At the top right in Fig. 12, the change in the modulus of the wave function at point (x, y, z) = (−0.3, 0, 0) as a function of time is shown. We note that the transition process is a nonlinear eﬀect depending on the square and cube of the wave function module according to (3). Therefore, this eﬀect is not observed in systems with low density as in Figs. 3, 4. We also note that the rotation of the system does not prevent annihilation, as can be seen from the data in Fig. 12.

14 . Figure 12. Slice representation of the toroids from Fig. 11, shown over several instances in time. The states represent the modulus of the wave function for y = 0 at the point (x, y, z) = (−0.3, 0, 0) depending on time (top right plot).

3 Discussion

Our model shows in Fig. 3 and 4 that dropletons composed of several droplets (quasiparticles) can assume a lower energy than dropletons composed of fewer partcles, in a similar fashion to the results by Wenzel et al [6]. This particularly interesting property of dropletons, where a single quaisparticle can assume a higher energy than an ensemble of quasiparticles arranged in a speciﬁc symmetry is proposed to be described by the following theorem. Take a dropleton, composed of 4 droplets (quasiparticles) arranged in a special symmetry, de-

15 scribed by their individual wavefunction:

ΨDropleton = αψa + βψb + γψc + δψd A (8) α ' (−β) ' γ ' (−δ), which states that the wavefunction of the dropleton can be described by a superposition of the wavefunctions of the individual quasiparticles multiplied by their respective coeﬃcients, which, if similar and of opposite sign to one another, lower the total energy of the dropleton by aﬀecting the amplitude of its total wavefunction:

∞ X ΨDropletonA = ΛAψj. (9) j=α,β,...

Here we see that that ΛA 6 ΛB, even if dropleton A has a higher number of quasiparticles than dropleton B, iff the individual droplets in dropleton B do not respect the conditions given in (8). In simple terms, the description given above suggests that the individual wavefunctions of the quasiparticles in a droplet cancel out one another when assuming a particular symmetry in a similar fashion to quantum noise cancellation [27], and the ensemble assumes a lower total amplitude, lowering the total energy of the dropleton. A second phenomenon of interest we observe here is the annihilation of droplets in the cylindrical dropletons (Fig 7 and 8) and torii (Fig 11, 12). According to the data, the energy of the system remains constant, which thereby suggests than dropletons of several quasiparticles can form from a single particle (Fig 7 and 9) and still retain the same energy level. Droplet annihilation is not reported in any source according to our knowledge. When we investigated the annihilation of torii, we identified a similar trend as for cylindrical systems, where four torii turn into one. This mechanism is suggested to obey a highly symmetrical evolution during a time-period (Fig 12 - profile of torii), where three of four torii vanish (Fig 11, Fig 12, two larger torii one smaller from the central pair) while the single central torus grows during the evolution of the wavefunction and forms the single doughnut-shaped electromagnetic entity. By a look into

16 the literature, we ﬁnd one example where torus annihilates, in a study by [28] on the evolution of a toroid structure in geometry. Also another study, [29] reports the destruction of a toroid structure for a piecewise-smooth dynamical system. Our results indicate therefore that dropletons may annihilate, however this has not been observed empirically to date and annihilation may be some- what associated with condensation.

4 Conclusions

In this study, we model dropletons of several energy levels using a supersymmetric wave equation as initial condition for a nonlinear model with similarities to the extended Gross-Pitaevskii equation. The results show that dropletons can assume a lower energy even when composed of several droplets, as observed empirically. We propose a theory for this phenomenon by describing that the be- haviour of the wavefunction of a low-lying dropleton composed of many droplets may share similarities to quantum-noise cancellation. Our model shows also that droplets tend to annihilate if the system of droplets in a dropleton becomes too dense, and annihilation appears to follow particular symmetric pathways. This phenomenon can also be observed for toroid structures described by the same model. This study shows that dropletons dynamics obeys wave-mechanics in its most fundamental nature, however, the annihilation mechanism does not appear to alter the energy of the system, and retains it constant, even when several droplets vanish from the system. This last observation grants for fur- ther analysis in a subsequent paper.

17 5 Acknowledgements and correspondence

The authors declare that they have no competing ﬁnancial interests. Correspondence and requests should be addressed to Sergio Manzetti, [email protected]).

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