Chaotic Bohmian trajectories for stationary states

Bohmian mechanics describes actual point-particles moving under the influence of the wave function [2–5]. Denoting the particle positions by ${{\bf{x}}}_{k}$, k = 1,..., n, and the configuration by $x=({{\bf{x}}}_{1},...,{{\bf{x}}}_{n})$, the dynamics is given by the guidance equation

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}x}{{\rm{d}}t}={v}^{\psi }(x,t),\end{eqnarray} \tag{ 1 }$$

where the velocity field ${v}^{\psi }=({{\bf{v}}}_{1}^{\psi },\ldots ,{{\bf{v}}}_{n}^{\psi })$ is given by

$$\begin{eqnarray}&&{{\bf{v}}}_{k}^{\psi }(x,t)=\displaystyle \frac{{\hslash }}{{m}_{k}}{\mathfrak{I}}\left(\displaystyle \frac{{{\boldsymbol{\nabla }}}_{k}\psi (x,t)}{\psi (x,t)}\right)=\displaystyle \frac{{\hslash }}{{m}_{k}}{{\boldsymbol{\nabla }}}_{k}S(x,t),\end{eqnarray} \tag{ 2 }$$

with $\psi =| \psi | {{\rm{e}}}^{{\rm{i}}S}$ and m_k the mass of the kth particle. The wave function ψ satisfies the usual Schrödinger equation. This theory reproduces the predictions of standard quantum theory provided that for an ensemble of systems all with the same wave function, the particle configuration is distributed according to $| \psi (x,0){| }^{2}$. This will be the case for typical initial configurations of the Universe [4, 5].

If the spectrum is non-degenerate then the energy eigenstates can be chosen real and the Bohmian particles do not move (since then S = 0). So in order to allow for chaos we need to consider a degenerate spectrum. We will consider stationary superpositions of the form

$$\begin{eqnarray}&&\psi ({\bf{x}},t)={{\rm{e}}}^{-{\rm{i}}{Et}}\phi ({\bf{x}})={{\rm{e}}}^{-{\rm{i}}{Et}}\displaystyle \sum _{i}{\phi }_{i}({\bf{x}}),\end{eqnarray} \tag{ 3 }$$

where the energy eigenstates ${\phi }_{i}$ are eigenstates of a complete set of commuting observables so that they constitute an orthonormal basis of the Hilbert space.

We will consider two types of states ${\phi }_{i}$. The first type is of the form

$$\begin{eqnarray}&&\phi ({\bf{x}})=f(x)g(y)h(z),\end{eqnarray} \tag{ 4 }$$

with ${\bf{x}}=(x,y,z)$ and f, g and h real functions. Such states will be considered for the 3d harmonic oscillator (${\phi }_{{n}_{x},{n}_{y},{n}_{z}}^{3{\rm{d}}}$). We will also consider systems in 2d such as the particle in a 2d-box (${\phi }_{{n}_{x},{n}_{y}}^{\mathrm{box}}$) and the 2d harmonic oscillator (${\phi }_{{n}_{x},{n}_{y}}^{2{\rm{d}}}$). In that case the z-coordinate should be dropped. The second type is of the form

$$\begin{eqnarray}&&\phi ({\bf{x}})=f(r)g(\theta ){{\rm{e}}}^{{\rm{i}}m\varphi },\end{eqnarray} \tag{ 5 }$$

where $(r,\theta ,\varphi )$ are spherical coordinates, with θ the polar angle and φ the azimuthal angle, and f and g real. These states are eigenstates of the angular momentum operator ${\widehat{L}}_{z}=-{\rm{i}}{\partial }_{\phi }$ with eigenvalue m. Such states will be considered for the 3d harmonic oscillator (${\phi }_{k,l,m}^{\mathrm{sph}}$). The explicit expressions of the states are given in appendix B. In terms of spherical coordinates the guidance equation reads

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}r}{{\rm{d}}t}=\displaystyle \frac{\partial S}{\partial r},\qquad \displaystyle \frac{{\rm{d}}\theta }{{\rm{d}}t}=\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{\partial S}{\partial \theta },\qquad \displaystyle \frac{{\rm{d}}\varphi }{{\rm{d}}t}=\displaystyle \frac{1}{{r}^{2}{\sin }^{2}\theta }\displaystyle \frac{\partial S}{\partial \varphi }.\end{eqnarray} \tag{ 6 }$$

For systems in two dimensions and using polar coordinates $(r,\varphi )$, one should put $\theta =\pi /2$, like for the 2d harmonic oscillator (${\phi }_{{n}_{r},{n}_{l}}^{\mathrm{pol}}$).

For chaotic motion to be possible, the number of dimensions needs to be at least 3. This follows from the Poincaré–Bendixson theorem [32]. For a non-stationary dynamics this number needs to be at least 2 (since such a dynamics is equivalent to a stationary one by introducing an independent variable τ, treating time as a dependent variable $t(\tau )$ and introducing the equation of motion ${\rm{d}}t/{\rm{d}}\tau =1$). For a stationary wave function the velocity field is stationary too. Hence for such a state we need at least 3 spatial dimensions for chaos to be possible.

Constants of motion reduce the effective number of degrees of freedom. For example, for states of the form (5), the phase does not depend on r and θ and hence the guidance equation (6) implies that r and θ are constant. The only effective degree of freedom is φ. The possible trajectories are circles [3] and there is no chaos.

There are a couple of types of constant of motion that we will encounter. The first type arises for a superposition of the form

$$\begin{eqnarray}&&\psi ({x}_{1},\ldots ,{x}_{n},t)={f}_{1}({x}_{1}){g}_{1}({x}_{2}){\chi }_{1}({x}_{3},\ldots ,{x}_{n},t)+{f}_{2}({x}_{1}){g}_{2}({x}_{2}){\chi }_{2}({x}_{3},\ldots ,{x}_{n},t),\end{eqnarray} \tag{ 7 }$$

with ${x}_{1},\ldots ,{x}_{n}$ Cartesian coordinates, and where f_i and g_i, i = 1, 2, are real. For such a wave function, the guidance equations for x₁ and x₂ are of the form

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{x}_{1}}{{\rm{d}}t}=({f}_{1}{\partial }_{{x}_{1}}{f}_{2}-{f}_{2}{\partial }_{{x}_{1}}{f}_{1}){g}_{1}{g}_{2}\,\displaystyle \frac{{\mathfrak{I}}({\chi }_{1}^{* }{\chi }_{2})}{| \psi {| }^{2}},\quad \displaystyle \frac{{\rm{d}}{x}_{2}}{{\rm{d}}t}=({g}_{1}{\partial }_{{x}_{2}}{g}_{2}-{g}_{2}{\partial }_{{x}_{2}}{g}_{1}){f}_{1}{f}_{2}\,\displaystyle \frac{{\mathfrak{I}}({\chi }_{1}^{* }{\chi }_{2})}{| \psi {| }^{2}}.\end{eqnarray} \tag{ 8 }$$

Dividing these velocity components and separating the variables x₁ and x₂, we obtain

$$\begin{eqnarray}&&f({x}_{1})\displaystyle \frac{{\rm{d}}{x}_{1}}{{\rm{d}}t}=g({x}_{2})\displaystyle \frac{{\rm{d}}{x}_{2}}{{\rm{d}}t},\end{eqnarray} \tag{ 9 }$$

with

$$\begin{eqnarray}&&f=\displaystyle \frac{{f}_{1}{f}_{2}}{{f}_{1}{\partial }_{{x}_{1}}{f}_{2}-{f}_{2}{\partial }_{{x}_{1}}{f}_{1}},\qquad g=\displaystyle \frac{{g}_{1}{g}_{2}}{{g}_{1}{\partial }_{{x}_{2}}{g}_{2}-{g}_{2}{\partial }_{{x}_{2}}{g}_{1}}.\end{eqnarray} \tag{ 10 }$$

Integration over time yields the constant of motion

$$\begin{eqnarray}&&C({x}_{1},{x}_{2})=F({x}_{1})-G({x}_{2}),\end{eqnarray} \tag{ 11 }$$

with ${\rm{d}}F({x}_{1})/{\rm{d}}{x}_{1}=f({x}_{1})$ and ${\rm{d}}G({x}_{2})/{\rm{d}}{x}_{2}=g({x}_{2})$.

In [19], two energy eigenstates of the form ${\phi }_{{n}_{x},{n}_{y},{n}_{z}}^{\mathrm{box}}+{\rm{i}}{\phi }_{{n}_{y},{n}_{z},{n}_{x}}^{\mathrm{box}}$ where considered for a particle in a square box. It was claimed that for $({n}_{x},{n}_{y},{n}_{z})=(3,2,1)$, the dynamics was regular, while for $({n}_{x},{n}_{y},{n}_{z})=(103,102,101)$, the dynamics was chaotic in certain regions. However, there are two independent constants of motion of the form (11) which depend on different pairs of coordinates, so that there is only one effective degree of freedom and there cannot be chaos, regardless of the choice of quantum numbers $({n}_{x},{n}_{y},{n}_{z})$.

For spherical coordinates, we can have similar constants of motion (examples are found in [33]). For example, consider a superposition of the form

$$\begin{eqnarray}&&\psi ({r}_{1},{\theta }_{1},{\varphi }_{1},\ldots ,{r}_{n},{\theta }_{n},{\varphi }_{n},t)={f}_{1}({r}_{1}){g}_{1}({\theta }_{1}){\chi }_{1}({\varphi }_{1},\ldots ,t)+{f}_{2}({r}_{1}){g}_{2}({\theta }_{1}){\chi }_{2}({\varphi }_{1},\ldots ,t),\end{eqnarray} \tag{ 12 }$$

where f_i and g_i are real, i = 1, 2. The guidance equations for r₁ and ${\theta }_{1}$ are of the form

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{r}_{1}}{{\rm{d}}t}=({f}_{1}{\partial }_{{r}_{1}}{f}_{2}-{f}_{2}{\partial }_{{r}_{1}}{f}_{1}){g}_{1}{g}_{2}\,\displaystyle \frac{{\mathfrak{I}}({\chi }_{1}^{* }{\chi }_{2})}{| \psi {| }^{2}},\quad \displaystyle \frac{{\rm{d}}{\theta }_{1}}{{\rm{d}}t}=\displaystyle \frac{1}{{r}_{1}^{2}}({g}_{1}{\partial }_{{\theta }_{1}}{g}_{2}-{g}_{2}{\partial }_{{\theta }_{1}}{g}_{1}){f}_{1}{f}_{2}\,\displaystyle \frac{{\mathfrak{I}}({\chi }_{1}^{* }{\chi }_{2})}{| \psi {| }^{2}}.\end{eqnarray} \tag{ 13 }$$

In this case, there is the constant of motion

$$\begin{eqnarray}&&C({r}_{1},{\theta }_{1})=F({r}_{1})-G({\theta }_{1}),\end{eqnarray} \tag{ 14 }$$

with ${\rm{d}}F({r}_{1})/{\rm{d}}{r}_{1}=f({r}_{1})$ and ${\rm{d}}G({\theta }_{1})/{\rm{d}}{\theta }_{1}=g({\theta }_{1})$, and

$$\begin{eqnarray}&&f=\displaystyle \frac{1}{{r}_{1}^{2}}\displaystyle \frac{{f}_{1}{f}_{2}}{{f}_{1}{\partial }_{{r}_{1}}{f}_{2}-{f}_{2}{\partial }_{{r}_{1}}{f}_{1}},\qquad g=\displaystyle \frac{{g}_{1}{g}_{2}}{{g}_{1}{\partial }_{{\theta }_{1}}{g}_{2}-{g}_{2}{\partial }_{{\theta }_{1}}{g}_{1}}.\end{eqnarray} \tag{ 15 }$$

Finally, consider a superposition of the form

$$\begin{eqnarray}\psi ({x}_{1},\ldots ,{x}_{n},t) & = & {f}_{1}({x}_{1}){f}_{2}({x}_{2}){f}_{2}({x}_{3}){\chi }_{1}({x}_{4},\ldots ,{x}_{n},t)\\ & & +{f}_{2}({x}_{1}){f}_{1}({x}_{2}){f}_{2}({x}_{3}){\chi }_{2}({x}_{4},\ldots ,{x}_{n},t)\\ & & +{f}_{2}({x}_{1}){f}_{2}({x}_{2}){f}_{1}({x}_{3}){\chi }_{3}({x}_{4},\ldots ,{x}_{n},t),\end{eqnarray} \tag{ 16 }$$

with ${x}_{1},\ldots ,{x}_{n}$ Cartesian coordinates, and where f₁ and f₂, are real. One can show similarly as above that there is the constant of motion

$$\begin{eqnarray}&&C({x}_{1},{x}_{2},{x}_{3})=F({x}_{1})+F({x}_{2})+F({x}_{3}),\end{eqnarray} \tag{ 17 }$$

with ${\rm{d}}F(x)/{\rm{d}}x=f(x)$ and

$$\begin{eqnarray}&&f=\displaystyle \frac{{f}_{1}{f}_{2}}{{f}_{1}{\partial }_{x}{f}_{2}-{f}_{2}{\partial }_{x}{f}_{1}}.\end{eqnarray} \tag{ 18 }$$

It is often reported that for a stationary state, the energy of the Bohmian particles

$$\begin{eqnarray}&&{ \mathcal E }(x,t)=\displaystyle \sum _{k}\displaystyle \frac{| {{\bf{v}}}_{k}^{\psi }(x,t){| }^{2}}{2}+V(x)+Q(x,t)\end{eqnarray} \tag{ 19 }$$

is a constant of motion, see e.g. [29, 30]. The quantity V is the usual potential energy and Q is the quantum potential, defined as $Q=-{\sum }_{k}{{\rm{\nabla }}}_{k}^{2}| \psi | /2| \psi |$. It appears as an extra potential in Newton's equation

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}}^{2}{{\bf{x}}}_{k}}{{\rm{d}}{t}^{2}}=-{{\boldsymbol{\nabla }}}_{k}(V+Q),\end{eqnarray} \tag{ 20 }$$

which follows from taking the time derivative of the guidance equation. Along a trajectory, we have that ${\rm{d}}{ \mathcal E }/{\rm{d}}t=\partial Q/\partial t$, so that ${ \mathcal E }$ is generically not conserved. For a stationary state $\psi (x,t)={{\rm{e}}}^{-{\rm{i}}{Et}}\phi (x)$ it is conserved, since $| \psi |$ and hence Q are time-independent. However, we also have that ${ \mathcal E }$ takes the same value for all possible trajectories, namely ${ \mathcal E }=E$. Hence this is a trivial constant of motion and it does not reduce the effective number of degrees of freedom. Therefore the Poincaré–Bendixson theorem does not exclude the possibility of chaos. In section 4, we will provide some examples of stationary states in 3d that lead to chaos.

In this section, we present simple quantities which we will compare to the average Lyapunov exponent later on.

First, given a decomposition $\psi ={\sum }_{i=1}^{N}{c}_{i}{\phi }_{i}$ in an orthonormal basis ${\phi }_{i}$, i = 1,..., N, the participation ratio (PR) [34] is defined by

$$\begin{eqnarray}&&\mathrm{PR}(\psi )=\displaystyle \frac{1}{{\displaystyle \sum }_{i=1}^{N}| \langle {\phi }_{i}| \psi \rangle {| }^{4}}=\displaystyle \frac{1}{{\displaystyle \sum }_{i=1}^{N}| {c}_{i}{| }^{4}}.\end{eqnarray} \tag{ 21 }$$

It quantifies the number of basis states on which the state ψ is delocalized. The PR is equal to 1 when ψ is a basis element ${\phi }_{i}$ and takes the maximum value N when ψ is an equally weighted superposition of all the basis states.

For many-particle systems, entanglement will couple the dynamics of the different particles. So the entanglement will generically play a role in the generation of chaos. Here, we will consider three measures of entanglement: the Meyer–Wallach measure (Q), the geometric measure (E_G) and the three-tangle (${\tau }_{3}$). For 3-qubit states, the Meyer–Wallach measure of entanglement $Q(\psi )\in [0,1]$ is defined as [35]

$$\begin{eqnarray}&&Q(\psi )=\displaystyle \frac{1}{3}\ \displaystyle \sum _{k=1}^{3}\,2(1-\mathrm{tr}{\widehat{\rho }}_{k}^{2}),\end{eqnarray} \tag{ 22 }$$

with ${\widehat{\rho }}_{k}$ the reduced density matrix for the kth qubit. It is the average of the linear entropies $2(1-\mathrm{tr}{\widehat{\rho }}_{k}^{2})$ of each qubit [35].

The geometric measure of entanglement ${E}_{{\rm{G}}}(\psi )\in [0,1[$ is defined as [36]:

$$\begin{eqnarray}&&{E}_{{\rm{G}}}(\psi )=1-\mathop{\max }\limits_{| \phi \rangle \,\mathrm{sep}.}| \langle \phi | \psi \rangle {| }^{2},\end{eqnarray} \tag{ 23 }$$

where the maximum is taken over all possible separable (i.e. non-entangled) states. It can be interpreted as the distance of a state $| \psi \rangle$ to the set of separable states.

For a three-qubit state

$$\begin{eqnarray}&&| \psi \rangle =\displaystyle \sum _{i,j,k=0,1}{c}_{{ijk}}| {ijk}\rangle ,\end{eqnarray} \tag{ 24 }$$

with $| {ijk}\rangle$ the basis states, the three-tangle is defined as [37]:

$$\begin{eqnarray}&&{\tau }_{3}(\psi )=4| {d}_{1}-2{d}_{2}+4{d}_{3}| ,\end{eqnarray} \tag{ 25 }$$

with

$$\begin{eqnarray*}{d}_{1} & = & {c}_{000}^{2}{c}_{111}^{2}+{c}_{001}^{2}{c}_{110}^{2}+{c}_{010}^{2}{c}_{101}^{2}+{c}_{100}^{2}{c}_{011}^{2},\\ {d}_{2} & = & {c}_{000}{c}_{111}{c}_{011}{c}_{100}+{c}_{000}{c}_{111}{c}_{101}{c}_{010}+{c}_{000}{c}_{111}{c}_{110}{c}_{001}\\ & & +{c}_{011}{c}_{100}{c}_{101}{c}_{010}+{c}_{011}{c}_{100}{c}_{110}{c}_{001}+{c}_{101}{c}_{010}{c}_{110}{c}_{001},\\ {d}_{3} & = & {c}_{000}{c}_{110}{c}_{101}{c}_{011}+{c}_{111}{c}_{001}{c}_{010}{c}_{100}.\end{eqnarray*}$$

It can be interpreted as the amount of entanglement between one qubit and the remaining two qubits not accounted for by the amount of entanglement between pairs of qubits. All these entanglement measures vanish if the state ψ is separable. Conversely, when the measure vanishes this implies that the state ψ is separable in the case of the first two measures, but not in the case of the three-tangle.

For the harmonic oscillator we can consider the complete set of energy eigenstates ${\phi }_{{n}_{x},{n}_{y},{n}_{z}}^{3{\rm{d}}}(x,y,z)$, see (65), or ${\phi }_{k,l,m}^{\mathrm{sph}}(r,\theta ,\varphi )$, see (67). These states are respectively of the form (4) and (5). For each choice, we will consider a superposition of three states that gives rise to chaotic Bohmian motion.

As a first example, consider the superposition

$$\begin{eqnarray}&&{\phi }^{\mathrm{sph}}(r,\theta ,\varphi )={\phi }_{0,3,1}^{\mathrm{sph}}(r,\theta ,\varphi )+{{\rm{e}}}^{{\rm{i}}\pi /3}{\phi }_{0,3,0}^{\mathrm{sph}}(r,\theta ,\varphi )+{{\rm{e}}}^{{\rm{i}}\pi /7}{\phi }_{1,1,0}^{\mathrm{sph}}(r,\theta ,\varphi )\end{eqnarray} \tag{ 30 }$$

with energy 9/2. This state has nodal lines, which are however difficult to find analytically. Figure 1 illustrates a typical Bohmian trajectory for this wave function. The trajectory starting outside the bulk of the wave packet moves towards it and eventually moves inside it. The Poincaré section in the plane z = 0 shown on the middle panel of figure 1 indicates that the dynamics is not regular. This is confirmed by the Lyapunov exponent $h\approx 0.06$ (see the right panel of figure 1).

Zoom In Zoom Out Reset image size

As a second example, consider now the stationary state

$$\begin{eqnarray}&&{\phi }^{3{\rm{d}}}(x,y,z)={\phi }_{1,1,1}^{3{\rm{d}}}(x,y,z)+{{\rm{e}}}^{{\rm{i}}\pi /3}{\phi }_{3,0,0}^{3{\rm{d}}}(x,y,z)+{{\rm{e}}}^{{\rm{i}}\pi /7}{\phi }_{1,2,0}^{3{\rm{d}}}(x,y,z),\end{eqnarray} \tag{ 31 }$$

which also has energy 9/2. In this case there is a nodal plane at x = 0. In the limit of x going to zero, the velocity does not blow up but rather becomes tangential to the plane x = 0. The Bohmian trajectories do not cross this plane, they tend to be repelled by it. The other nodal lines are given by

$$\begin{eqnarray}&&\left\{(x,y,z)\,:x=\pm \sqrt{\displaystyle \frac{3}{2}+\sin \left(\displaystyle \frac{\pi }{7}\right)-2{y}^{2}\sin \left(\displaystyle \frac{\pi }{7}\right)},y\ne 0,\right.\\ &&\quad \left.| y| \leqslant \displaystyle \frac{1}{2}\sqrt{\displaystyle \frac{3+2\sin (\pi /7)}{\sin (\pi /7)}},z=\displaystyle \frac{(1-2{y}^{2})[3\cos (\pi /7)-\sqrt{3}\sin (\pi /7)]}{6\sqrt{2}y}\right\}.\end{eqnarray} \tag{ 32 }$$

These nodal lines end in the plane x = 0. A typical Bohmian trajectory for this system is illustrated in figure 2. The trajectory first spirals around a nodal line. When the particle arrives at the end of the nodal line, the trajectory continues freely until it reaches the vicinity of another nodal line. Such behavior has been reported before [19, 22]. Most of the Poincaré section in the plane z = 0 is composed of regions where the points look randomly distributed which constitutes a strong clue that the motion is chaotic. This is confirmed by the calculated Lyapunov exponent which is again approximately equal to 0.06. The right panel of figure 2 illustrates the exponential divergence of two initially close trajectories.

Zoom In Zoom Out Reset image size

We have shown that the Bohmian trajectories of 3d systems with stationary states may be chaotic. We now study the effect of the particular form of the wave function on the amount of chaos. We will consider the harmonic oscillator and stationary wave functions of the form

$$\begin{eqnarray}&&\phi ({\bf{x}};\alpha ,\beta )=\displaystyle \sum _{i=1}^{4}\,{c}_{i}(\alpha ,\beta ){\phi }_{{k}_{i},{l}_{i},{m}_{i}}^{\mathrm{sph}}({\bf{x}}),\end{eqnarray} \tag{ 33 }$$

where ${c}_{i}(\alpha ,\beta )=| {c}_{i}(\alpha )| {{\rm{e}}}^{{\rm{i}}{\chi }_{i}(\beta )}$, so that the amplitude and the phase of the coefficients c_i are functions of respectively α and β. The quantum numbers ${k}_{i},{l}_{i},{m}_{i}$ are chosen such that $\phi ({\bf{x}};\alpha ,\beta )$ is a stationary state. We also assume the states to be normalized (i.e., ${\sum }_{i}| {c}_{i}{| }^{2}=1$). While this does not affect the Bohmian velocity field, it will be important when considering the PR. In order to quantify the amount of chaos, we compute the average Lyapunov exponent $\bar{h}$ over 150 different initial positions, uniformly distributed in a cube of edge length 10 centered around the origin. The trajectories used in the determination of the Lyapunov exponent were computed from t = 0 to $t=5\times {10}^{4}$.

Consider first the states

$$\begin{eqnarray}\phi ({\bf{x}};\alpha ,\beta ) & = & { \mathcal N }(\alpha )[\cos (\alpha ){\phi }_{1,3,0}^{\mathrm{sph}}({\bf{x}})+{{\rm{e}}}^{{\rm{i}}(\beta +\pi /3)}\sin (\alpha ){\phi }_{1,3,1}^{\mathrm{sph}}({\bf{x}})\\ & & +{{\rm{e}}}^{{\rm{i}}(2\pi \cos (\beta )+\pi /5)}{\cos }^{2}(\alpha ){\phi }_{2,1,-1}^{\mathrm{sph}}({\bf{x}})+{{\rm{e}}}^{{\rm{i}}(-2\beta +\pi /7)}{\sin }^{2}(\alpha ){\phi }_{2,1,0}^{\mathrm{sph}}({\bf{x}})],\end{eqnarray} \tag{ 34 }$$

with $\alpha \in [0,\pi /2]$, $\beta \in [0,2\pi ]$ and ${ \mathcal N }(\alpha )$ a normalization constant. The energy of these states is 13/2.

Taking $\beta =0$, so that the phases of the coefficients are fixed, we have

$$\begin{eqnarray}\phi ({\bf{x}};\alpha ,0) & = & { \mathcal N }(\alpha )[\cos (\alpha ){\phi }_{1,3,0}^{\mathrm{sph}}({\bf{x}})+{{\rm{e}}}^{{\rm{i}}\pi /3}\sin (\alpha ){\phi }_{1,3,1}^{\mathrm{sph}}({\bf{x}})\\ & & +{{\rm{e}}}^{{\rm{i}}\pi /5}{\cos }^{2}(\alpha ){\phi }_{2,1,-1}^{\mathrm{sph}}({\bf{x}})+{{\rm{e}}}^{{\rm{i}}\pi /7}{\sin }^{2}(\alpha ){\phi }_{2,1,0}^{\mathrm{sph}}({\bf{x}})].\end{eqnarray} \tag{ 35 }$$

Figure 3(a) shows the average Lyapunov exponent as a function of α. For $\alpha =0$ and $\alpha =\pi /2$, $\phi ({\bf{x}};\alpha ,0)$ is a superposition of only two eigenstates and thus the Bohmian trajectories are regular, see section 2.3. This agrees with the calculated value of $\bar{h}$ which is zero. $\bar{h}$ is maximum around $\alpha =\pi /10$ and then decreases as α increases.

Zoom In Zoom Out Reset image size

In order to investigate the effect of the choice of eigenstates in $\phi ({\bf{x}};\alpha ,\beta )$ on the chaotic behavior of the Bohmian trajectories, we also compute the average Lyapunov exponent for the stationary state

$$\begin{eqnarray}\phi ^{\prime} ({\bf{x}};\alpha ,0) & = & { \mathcal N }(\alpha )[\cos (\alpha ){\phi }_{2,3,2}^{\mathrm{sph}}({\bf{x}})+{{\rm{e}}}^{{\rm{i}}\pi /3}\sin (\alpha ){\phi }_{2,3,-1}^{\mathrm{sph}}({\bf{x}})\\ & & +{{\rm{e}}}^{{\rm{i}}\pi /5}{\cos }^{2}(\alpha ){\phi }_{3,1,1}^{\mathrm{sph}}({\bf{x}})+{{\rm{e}}}^{{\rm{i}}\pi /7}{\sin }^{2}(\alpha ){\phi }_{3,1,0}^{\mathrm{sph}}({\bf{x}})],\end{eqnarray} \tag{ 36 }$$

which has energy 17/2. This state differs from $\phi ({\bf{x}};\alpha ,0)$ only in the quantum numbers of the four eigenstates in terms of which it is constructed. Figure 3(b) shows the average Lyapunov exponent for some values of α. Just as before, for $\alpha =0$ and $\alpha =\pi /2$, $\bar{h}$ should vanish. For $\phi ^{\prime} ({\bf{x}};\alpha ,0)$, $\bar{h}$ reaches the maximal value approximately equal to 0.2 near $\alpha =\pi /4$.

Although the states $\phi ({\bf{x}};\alpha ,0)$ and $\phi ^{\prime} ({\bf{x}};\alpha ,0)$ have the same coefficients in their decomposition in the basis of eigenstates of the harmonic oscillator, the evolution of the average Lyapunov exponent is significantly different. For both wave functions, $\bar{h}$ takes a maximum value of around 0.2 but for a different value of α. So, in particular, a small increase of the energy of the wave function does not necessarily lead to a greater amount of chaos in the Bohmian trajectories. For instance, one can see by comparing figures 3(a) and (b) that for $\alpha =\pi /10$, $\bar{h}$ is more than twice as high for $\phi ({\bf{x}};\alpha )$ as for $\phi ^{\prime} ({\bf{x}};\alpha ,0)$, even though the energy of the latter wave function is higher. When $\alpha =\pi /4$ this situation is reversed.

In order to bring to light the effect of the choice of phases of the coefficients, we consider the states $\phi ({\bf{x}};\pi /4,\beta )$, where the amplitudes of the coefficients are fixed and the phases may vary. Figure 3(c) illustrates the average Lyapunov exponent for some values of the parameter β. The average Lyapunov exponent takes values between 0.07 and 0.16. Thus, $\bar{h}$ is quite sensitive to the choice of the relative phase of the coefficients. The dependence on β also looks rather complicated.

Let us now compare the average Lyapunov exponent with the PR (21). It is immediately clear that the PR has two shortcomings, which make that it can not agree perfectly with the average Lyapunov exponent. First, the PR does not depend on the particular form of the basis states. Therefore, it takes the same value for the wave functions $\phi ({\bf{x}};\alpha ,0)$ and $\phi ^{\prime} ({\bf{x}};\alpha ,0)$ (see (34) and (36)). Figure 3(d) shows the PR for these states as a function of α. The PR shows a good qualitative agreement with the average Lyapunov exponent $\bar{h}$ for the wave function $\phi ^{\prime} ({\bf{x}};\alpha ,0)$, but a bit less so for $\phi ({\bf{x}};\alpha ,0)$. For both $\phi ({\bf{x}};\alpha ,0)$ and $\phi ^{\prime} ({\bf{x}};\alpha ,0)$, the PR reaches a minimum for $\alpha =0$ and $\alpha =\pi /2$, just as $\bar{h}$ (whose theoretical value is zero). The PR has a maximum at $\alpha =\pi /4$ as does the average Lyapunov exponent for the wave function $\phi ^{\prime} ({\bf{x}};\alpha ,0)$. However, for the wave function $\phi ({\bf{x}};\alpha ,0)$ the maxima of $\bar{h}$ and the ${\rm{PR}}$ do not coincide. And while the PR is symmetric around $\alpha =\pi /4$, like $\bar{h}$ for $\phi ^{\prime} ({\bf{x}};\alpha ,0)$, $\bar{h}$ for $\phi ({\bf{x}};\alpha ,0)$ is not.

The second shortcoming of the PR is that it does not depend on the phases of the coefficients c_i. As we have seen, with the wave functions $\phi ({\bf{x}};\pi /4,\beta )$, the amount of chaos strongly depends on the value of the phases, see figure 3(c).

Therefore, in order to characterize the amount of chaos in Bohmian trajectories, a simple measure of superposition such as the PR is not enough. One needs to take into account the particular form of the eigenstates that appear in the superposition as well as their relative phases, since both influence the complexity of the wave function. Nevertheless, generically, the PR gives the general trend of the value of the Lyapunov exponent.

Using two single-particle basis states ${\phi }_{1}$ and ${\phi }_{2}$, the most general two-particle state reads

$$\begin{eqnarray}&&{\phi }_{2}^{2{\rm{p}}}({{\bf{x}}}_{1},{{\bf{x}}}_{2})={c}_{1}{\phi }_{1}({{\bf{x}}}_{1}){\phi }_{1}({{\bf{x}}}_{2})+{c}_{2}{\phi }_{1}({{\bf{x}}}_{1}){\phi }_{2}({{\bf{x}}}_{2})+{c}_{3}{\phi }_{2}({{\bf{x}}}_{1}){\phi }_{1}({{\bf{x}}}_{2})+{c}_{4}{\phi }_{2}({{\bf{x}}}_{1}){\phi }_{2}({{\bf{x}}}_{2})\end{eqnarray} \tag{ 37 }$$

where the superscript denotes the number of particles and the subscript the number of single-particle basis states involved in the superposition. For states ${\phi }_{i}(x,y)$ of the form (4) there are two constants of motion of the form (11) (even for a non-stationary superposition)⁶ , so that the effective number of degrees of freedom is 2 and no chaos is possible.

Consider states of the form (37), but now with the ${\phi }_{i}(r,\varphi )$ of the form (5)). Unlike the case where they are of the form (4), such states generically do not seem to yield constants of motion, so that chaotic motion may be possible. As an example, consider the harmonic oscillator with single-particle energy eigenstates ${\phi }_{{n}_{r},{n}_{l}}^{\mathrm{pol}}(r,\varphi )$, see (61), and the wave function

$$\begin{eqnarray}{\phi }_{2}^{2{\rm{p}}}({{\bf{x}}}_{1},{{\bf{x}}}_{2}) & = & {\phi }_{1,1}^{\mathrm{pol}}({{\bf{x}}}_{1}){\phi }_{1,1}^{\mathrm{pol}}({{\bf{x}}}_{2})+{{\rm{e}}}^{{\rm{i}}\pi /3}{\phi }_{1,1}^{\mathrm{pol}}({{\bf{x}}}_{1}){\phi }_{2,0}^{\mathrm{pol}}({{\bf{x}}}_{2})\\ & & +{{\rm{e}}}^{{\rm{i}}\pi /5}{\phi }_{2,0}^{\mathrm{pol}}({{\bf{x}}}_{1}){\phi }_{1,1}^{\mathrm{pol}}({{\bf{x}}}_{2})+{{\rm{e}}}^{{\rm{i}}\pi /7}{\phi }_{2,0}^{\mathrm{pol}}({{\bf{x}}}_{1}){\phi }_{2,0}^{\mathrm{pol}}({{\bf{x}}}_{2}).\end{eqnarray} \tag{ 40 }$$

Figure 4 illustrates a typical set of trajectories for this system. The Lyapunov exponent takes a value $h\approx 0.17$, which indicates chaos.

Zoom In Zoom Out Reset image size

As we have seen in the previous section, stationary superpositions of only two single-particle basis states ${\phi }_{i}(x,y)$ of the form (4) cannot give rise to chaos. For superpositions using three such states, we show that chaos is possible. We give two examples for states of the form

$$\begin{eqnarray}&&{\phi }_{3}^{2{\rm{p}}}({{\bf{x}}}_{1},{{\bf{x}}}_{2})={c}_{1}{\phi }_{1}({{\bf{x}}}_{1}){\phi }_{1}({{\bf{x}}}_{2})+{c}_{2}{\phi }_{2}({{\bf{x}}}_{1}){\phi }_{2}({{\bf{x}}}_{2})+{c}_{3}{\phi }_{3}({{\bf{x}}}_{1}){\phi }_{3}({{\bf{x}}}_{2}),\end{eqnarray} \tag{ 41 }$$

one for the harmonic oscillator and one for the square box. Generically such states do not yield constants of motion.

As a first example, we consider the state

$$\begin{eqnarray}&&{\phi }_{3}^{2{\rm{p}}}={\phi }_{1,1}^{2{\rm{d}}}{\phi }_{1,1}^{2{\rm{d}}}+{{\rm{e}}}^{{\rm{i}}\pi /3}{\phi }_{2,0}^{2{\rm{d}}}{\phi }_{2,0}^{2{\rm{d}}}+{{\rm{e}}}^{{\rm{i}}\pi /7}{\phi }_{0,2}^{2{\rm{d}}}{\phi }_{0,2}^{2{\rm{d}}}\end{eqnarray} \tag{ 42 }$$

for the 2d harmonic oscillator with the energy eigenstates ${\phi }_{{n}_{x},{n}_{y}}^{2{\rm{d}}}$ given in (59). Figure 5(a) illustrates a typical pair of Bohmian trajectories for this system. Again, the trajectories start outside the bulk of the wave packet move towards it and eventually move inside it. The Lyapunov exponent is $h\approx 0.08$, so that the motion is chaotic.

Zoom In Zoom Out Reset image size

As a second example, we consider the state

$$\begin{eqnarray}&&{\phi }_{3}^{2{\rm{p}}}={\phi }_{7,1}^{\mathrm{box}}{\phi }_{7,1}^{\mathrm{box}}+{{\rm{e}}}^{{\rm{i}}\pi /3}{\phi }_{1,7}^{\mathrm{box}}{\phi }_{1,7}^{\mathrm{box}}+{{\rm{e}}}^{{\rm{i}}\pi /7}{\phi }_{5,5}^{\mathrm{box}}{\phi }_{5,5}^{\mathrm{box}}\end{eqnarray} \tag{ 43 }$$

for the 2d square box with the eigenstates ${\phi }_{{n}_{x},{n}_{y}}^{\mathrm{box}}$ given in (56). Figure 5(b) shows a typical set of Bohmian trajectories for this system. We obtain the rather large value of around 25 for the Lyapunov exponent. This example also illustrates the non-local character of Bohmian mechanics. Although the particles are always in different regions of the box, their trajectories strongly influence each other which allows chaotic motion (remember that for a 2d system with stationary wave function, the Bohmian trajectories are always regular). Actually, the amount of chaos would not change if we moved one particle in a box arbitrarily far away.

First we consider superpositions formed using two different single-particle states ${\phi }_{i}$. These states can be seen as three-qubit states. We focus our attention on the states

$$\begin{eqnarray}&&{\phi }_{\mathrm{GHZ}}^{3{\rm{p}}}({{\bf{x}}}_{1},{{\bf{x}}}_{2},{{\bf{x}}}_{3})={c}_{1}{\phi }_{1}({{\bf{x}}}_{1}){\phi }_{1}({{\bf{x}}}_{2}){\phi }_{1}({{\bf{x}}}_{3})+{c}_{2}{\phi }_{2}({{\bf{x}}}_{1}){\phi }_{2}({{\bf{x}}}_{2}){\phi }_{2}({{\bf{x}}}_{3}),\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&{\phi }_{W}^{3{\rm{p}}}({{\bf{x}}}_{1},{{\bf{x}}}_{2},{{\bf{x}}}_{3})={c}_{1}{\phi }_{1}({{\bf{x}}}_{1}){\phi }_{2}({{\bf{x}}}_{2}){\phi }_{2}({{\bf{x}}}_{3})+{c}_{2}{\phi }_{2}({{\bf{x}}}_{1}){\phi }_{1}({{\bf{x}}}_{2}){\phi }_{2}({{\bf{x}}}_{3})+{c}_{3}{\phi }_{2}({{\bf{x}}}_{1}){\phi }_{2}({{\bf{x}}}_{2}){\phi }_{1}({{\bf{x}}}_{3}),\end{eqnarray} \tag{ 45 }$$

which belong respectively to the Greenberger–Horne–Zeilinger (GHZ) and the W entanglement classes [39].

For states ${\phi }_{i}(x,y)$ of the form (4), there are 5 independent constants of motion of the form (11), for ${\phi }_{\mathrm{GHZ}}^{3{\rm{p}}}$. The states ${\phi }_{W}^{3{\rm{p}}}$ lead to 3 constants of motion of the form (11) for $({x}_{1},{y}_{1})$, $({x}_{2},{y}_{2})$ and $({x}_{3},{y}_{3})$, and 2 of the form (17) for $({x}_{1},{x}_{2},{x}_{3})$ and $({y}_{1},{y}_{2},{y}_{3})$. In both cases, the constants of motion make that there is no chaos, despite the entanglement. This remains true even if the states are non-stationary (and thus have moving nodes).

For states ${\phi }_{i}(r,\varphi )$ of the form (5), the state ${\phi }_{\mathrm{GHZ}}^{3{\rm{p}}}$ gives two independent constants of motion of the form (11) for $({r}_{1},{r}_{2})$ and $({r}_{2},{r}_{3})$. If we introduce the variable $\varphi ={\varphi }_{1}+{\varphi }_{2}+{\varphi }_{3}$, then the wave function depends on this variable and r₁, r₂ and r₃. So in terms of the variables $\varphi ,{\varphi }_{2},{\varphi }_{3},{r}_{1},{r}_{2},{r}_{3}$, the Bohmian dynamics for $\varphi ,{r}_{1},{r}_{2},{r}_{3}$ is independent of ${\varphi }_{2}$ and ${\varphi }_{3}$. The two constants of motion imply that the dynamics for $\varphi ,{r}_{1},{r}_{2},{r}_{3}$ is regular. Furthermore, the dynamics of ${\varphi }_{2}$ and ${\varphi }_{3}$ only depends on $\varphi ,{r}_{1},{r}_{2},{r}_{3}$, and therefore there is no chaos.

For states ${\phi }_{i}(r,\varphi )$ of the form (5), the state ${\phi }_{W}^{3{\rm{p}}}$ leads to one constant of motion of the type (17) for $({r}_{1},{r}_{2},{r}_{3})$ and trajectories can be chaotic. Indeed, consider the harmonic oscillator and the state

$$\begin{eqnarray}&&{\phi }_{W}^{3{\rm{p}}}={\phi }_{3,1}^{\mathrm{pol}}{\phi }_{4,0}^{\mathrm{pol}}{\phi }_{4,0}^{\mathrm{pol}}+{{\rm{e}}}^{{\rm{i}}\pi /3}{\phi }_{4,0}^{\mathrm{pol}}{\phi }_{3,1}^{\mathrm{pol}}{\phi }_{4,0}^{\mathrm{pol}}+{{\rm{e}}}^{{\rm{i}}\pi /7}{\phi }_{4,0}^{\mathrm{pol}}{\phi }_{4,0}^{\mathrm{pol}}{\phi }_{3,1}^{\mathrm{pol}}.\end{eqnarray} \tag{ 46 }$$

The Lyapunov exponent is approximately equal to 0.12, which indicates chaos. A particular set of trajectories is shown in figure 6.

Zoom In Zoom Out Reset image size

In the previous section we have seen that the states (44) and (45) do not lead to chaos for states using two states ${\phi }_{i}(x,y)$ of the form (4). Chaos is possible for states that are formed using three such states. More precisely, we consider the states

$$\begin{eqnarray}&&{\phi }_{3}^{3{\rm{p}}}({{\bf{x}}}_{1},{{\bf{x}}}_{2},{{\bf{x}}}_{3})={c}_{1}{\phi }_{1}({{\bf{x}}}_{1}){\phi }_{1}({{\bf{x}}}_{2}){\phi }_{1}({{\bf{x}}}_{3})+{c}_{2}{\phi }_{2}({{\bf{x}}}_{1}){\phi }_{2}({{\bf{x}}}_{2}){\phi }_{2}({{\bf{x}}}_{3})+{c}_{3}{\phi }_{3}({{\bf{x}}}_{1}){\phi }_{3}({{\bf{x}}}_{2}){\phi }_{3}({{\bf{x}}}_{3}).\end{eqnarray} \tag{ 47 }$$

In the case of the harmonic oscillator, we consider

$$\begin{eqnarray}&&{\phi }_{3}^{3{\rm{p}}}={\phi }_{3,1}^{2{\rm{d}}}{\phi }_{3,1}^{2{\rm{d}}}{\phi }_{3,1}^{2{\rm{d}}}+{{\rm{e}}}^{{\rm{i}}\pi /3}{\phi }_{4,0}^{2{\rm{d}}}{\phi }_{4,0}^{2{\rm{d}}}{\phi }_{4,0}^{2{\rm{d}}}+{{\rm{e}}}^{{\rm{i}}\pi /7}{\phi }_{2,2}^{2{\rm{d}}}{\phi }_{2,2}^{2{\rm{d}}}{\phi }_{2,2}^{2{\rm{d}}}.\end{eqnarray} \tag{ 48 }$$

The Lyapunov exponent is approximately 0.19, so that there is chaos. Figure 7(a) shows a particular set of trajectories for this system.

Zoom In Zoom Out Reset image size

As a second example, consider the square box and the state

$$\begin{eqnarray}&&{\phi }_{3}^{3{\rm{p}}}={\phi }_{5,5}^{\mathrm{box}}{\phi }_{5,5}^{\mathrm{box}}{\phi }_{5,5}^{\mathrm{box}}+{{\rm{e}}}^{{\rm{i}}\pi /3}{\phi }_{7,1}^{\mathrm{box}}{\phi }_{7,1}^{\mathrm{box}}{\phi }_{7,1}^{\mathrm{box}}+{{\rm{e}}}^{{\rm{i}}\pi /7}{\phi }_{1,7}^{\mathrm{box}}{\phi }_{1,7}^{\mathrm{box}}{\phi }_{1,7}^{\mathrm{box}}\end{eqnarray} \tag{ 49 }$$

which has the same coefficients as the state (48). A typical set of Bohmian trajectories is shown in figure 7(b). Just as in the two-particle case, the particles move in different quadrants of the box. The Lyapunov exponent is approximately 25, so that there is chaos.

In this section we explore the relation between the average Lyapunov exponent and entanglement measures. Here, we consider three measures of multipartite entanglement: the Meyer–Wallach measure Q (22), the geometric measure E_G (23) and the three-tangle ${\tau }_{3}$ (25). These entanglement measures vanish if the state ψ is separable. In that case, the motion of the Bohmian particles is uncorrelated. Conversely, when the measure vanishes this implies that the state ψ is separable in the case of the first two measures, but not in the case of the three-tangle.

These measures will not fully agree with the average Lyapunov exponent, as they do not depend on the choice of the basis states $| 0\rangle$ and $| 1\rangle$ (just like the PR), while this influences the possibility of chaos, as we have seen before. In addition, note that entanglement is not sufficient to imply chaos. As a simple example, consider real wave functions. The wave function can be arbitrarily much entangled, but the Bohmian configuration is static, so that there is no chaos.

In the following, we will consider stationary states belonging to the W entanglement class, i.e. states that are related by an invertible local operation to the W state $| W\rangle =(| 001\rangle +| 010\rangle +| 100\rangle )/\sqrt{3}$. These states can be parametrized as follows

$$\begin{eqnarray}&&| W(a,b,c)\rangle =\sqrt{a}| 001\rangle +\sqrt{b}| 010\rangle +\sqrt{c}| 100\rangle +\sqrt{1-(a+b+c)}| 000\rangle ,\end{eqnarray} \tag{ 50 }$$

where $a,b,c\geqslant 0$, $a+b+c\leqslant 1$, and $| 0\rangle$ and $| 1\rangle$ are single-particle basis states [39]. We will take $\langle {\bf{x}}| 0\rangle$ and $\langle {\bf{x}}| 1\rangle$ to be the eigenstates ${\phi }_{{n}_{r},{n}_{l}}^{\mathrm{pol}}(r,\varphi )$ of the 2d harmonic oscillator. For these states, the three-tangle vanishes for all values of $a,b$ and c. Since the amount of chaos generically varies with the choice of coefficients, this measure will not correlate with the amount of chaos.

We consider first the following one-parameter family of states

$$\begin{eqnarray}&&| W(a,1/2-a,1/2)\rangle =\sqrt{a}| 001\rangle +\sqrt{1/2-a}| 010\rangle +\sqrt{1/2}| 100\rangle ,\end{eqnarray} \tag{ 51 }$$

with $\langle {\bf{x}}| 0\rangle ={\phi }_{4,0}^{\mathrm{pol}}({\bf{x}})$ and $\langle {\bf{x}}| 1\rangle ={\phi }_{3,1}^{\mathrm{pol}}({\bf{x}})$ and $a\in [0,1/2]$, which have energy 15. Figure 8 shows from left to right the average Lyapunov exponent $\bar{h}$ (computed from 300 trajectories), the Meyer–Wallach and geometric measure of entanglement and the PR as a function of the parameter a. For the states (51), the three-tangle and the geometric measure of entanglement are independent of a; they have the value of 0 and 1/2 respectively. The average Lyapunov exponent $\bar{h}$ varies with a, is maximal at $a=1/4$ and minimal at a = 0 and $a=1/2$. Both the PR and the Meyer–Wallach measure of entanglement have the same general behavior as $\bar{h}$ with respect to a. This agreement in behavior can be understood by noticing that the states $| 001\rangle$, $| 010\rangle$ and $| 100\rangle$ from which $| W(a,1/2-a,1/2)\rangle$ is constructed have similar spatial variations. For a = 0 and $a=1/2$, $| W(a,1/2-a,1/2)\rangle$ has only two terms, with one particle decoupling from the others, which explains why the Lyapunov exponent is minimal for these values.

Zoom In Zoom Out Reset image size

Now consider the other one-parameter family of stationary states

$$\begin{eqnarray}&&| W(a,1/4,1/4)\rangle =\sqrt{a}| 001\rangle +\sqrt{1/4}| 010\rangle +\sqrt{1/4}| 100\rangle +\sqrt{(1/2-a)}| 000\rangle ,\end{eqnarray} \tag{ 52 }$$

with again $\langle {\bf{x}}| 0\rangle ={\phi }_{4,0}^{\mathrm{pol}}({\bf{x}})$ and $\langle {\bf{x}}| 1\rangle ={\phi }_{3,1}^{\mathrm{pol}}({\bf{x}})$ and $a\in [0,1/2]$, which have energy 15. Figure 9(a) shows $\bar{h}$, (c) the Meyer–Wallach and geometric measures of entanglement and (d) the PR as a function of a. The lower value of $\bar{h}$ for a = 0 may originate from the fact that the third particle decouples from the other ones. The lower value of $\bar{h}$ for $a=1/2$ may come from the fact that then the state is of the form (45) and as mentioned before admits then one constant of motion. These properties do not exclude chaos but reduce the possible amount of chaos. The agreement of $\bar{h}$ with the Meyer–Wallach and geometric measures of entanglement or the PR is not as good as in the previous example. The Meyer–Wallach measure and the geometric measure of entanglement are monotonously increasing functions of a, unlike $\bar{h}$ which has a maximum. The PR reaches a maximum, just like $\bar{h}$, but its maximum is reached for a different value of a. The discrepancy between the Lyapunov exponent and the measures of complexity of the wave function originates from the fact that the former depends on the particular form of the terms in the superposition, while the latter do not. In this case, unlike in the previous example, not all terms have the same form due to the presence of the term proportional to $| 000\rangle$.

Zoom In Zoom Out Reset image size

As a last example, we consider the state (52) with $\langle {\bf{x}}| 0\rangle ={\phi }_{4,2}^{\mathrm{pol}}({\bf{x}})$ and $\langle {\bf{x}}| 1\rangle ={\phi }_{3,3}^{\mathrm{pol}}({\bf{x}})$, $a\in [0,1/2]$. We denote this state as $| W^{\prime} (a,1/4,1/4)\rangle$. It has energy 21, which is higher than the energy of $| W(a,1/4,1/4)\rangle$. The Lyapunov exponent $\bar{h}$ for this state, plotted in figure 9(b), qualitatively displays the same behavior as in the case of $| W(a,1/4,1/4)\rangle$. However, it is interesting to see that for most values of a, $| W(a,1/4,1/4)\rangle$ gives higher values of $\bar{h}$, despite the fact that this state has lower energy than $| W^{\prime} (a,1/4,1/4)\rangle$. So, just as we have already seen before in section 4.2, increasing the energy of the system does not necessarily lead to a higher amount of chaos in the Bohmian trajectories. Similar fluctuations as a funtion of the mean energy where observed in [24].

In summary, not all measures of entanglement lead to a reasonable agreement with the amount of chaos. For the family of states (50), the three-tangle vanishes whereas the Bohmian trajectories display variable amount of chaos as a function of the parameters. Similarly, the geometric measure of entanglement of the states (51) is constant while the Lyapounov exponent varies as a function of the parameter a. The best agreement is obtained for the Meyer–Wallach measure of entanglement. However, even this measure will not always be suitable. For example, for the state ${\phi }_{\mathrm{GHZ}}^{3{\rm{p}}}$ (see equation (44)), $Q({\phi }_{\mathrm{GHZ}}^{3{\rm{p}}})=2(1-| {c}_{1}{| }^{4}-| {c}_{2}{| }^{4})$, while the Bohmian trajectories are regular for any ${c}_{1},{c}_{2}$ as shown in section 6.1. (Actually, also the three-tangle is non-zero in this case: ${\tau }_{3}({\phi }_{\mathrm{GHZ}}^{3{\rm{p}}})=4| {c}_{1}{| }^{2}| {c}_{2}{| }^{2}$.)

B.1.1. Square box potential

For a particle moving in a square box with sides of length one [42], the energy eigenstates are

$$\begin{eqnarray}&&{\phi }_{{n}_{x},{n}_{y}}^{\mathrm{box}}(x,y)=2\sin ({n}_{x}\pi x)\sin ({n}_{y}\pi y)\end{eqnarray} \tag{ 56 }$$

with the ${n}_{i}=1,2,\,...$ $(i=x,y)$ and have energy

$$\begin{eqnarray}&&{E}_{{n}_{x},{n}_{y}}=\displaystyle \frac{{\pi }^{2}}{2}({n}_{x}^{2}+{n}_{y}^{2}).\end{eqnarray} \tag{ 57 }$$

These eigenstates are real. There is some degeneracy. States ${\phi }_{{n}_{1},{n}_{2}}$ and ${\phi }_{{n}_{2},{n}_{1}}$ have the same energy. Moreover, there may be some accidental degeneracy as for example for the states ${\phi }_{5,5}^{\mathrm{box}}$ and ${\phi }_{7,1}^{\mathrm{box}}$, or ${\phi }_{7,4}^{\mathrm{box}}$ and ${\phi }_{8,1}^{\mathrm{box}}$.

B.1.2. Harmonic oscillator potential

For the harmonic oscillator [23, 42], the Hamiltonian is given by

$$\begin{eqnarray}&&{\widehat{H}}_{2{\rm{d}}}=\displaystyle \frac{1}{2}({\widehat{p}}_{x}^{2}+{\widehat{p}}_{y}^{2})+\displaystyle \frac{1}{2}({\widehat{x}}^{2}+{\widehat{y}}^{2}),\end{eqnarray} \tag{ 58 }$$

which simply corresponds to the sum of the Hamiltonians of two 1d harmonic oscillators with the same frequency in the x and y direction.

The eigenstates of ${\widehat{H}}_{2{\rm{d}}}$ can be constructed as products of eigenstates of the 1d harmonic oscillator and are given by

$$\begin{eqnarray}&&{\phi }_{{n}_{x},{n}_{y}}^{2{\rm{d}}}(x,y)=\displaystyle \frac{1}{\sqrt{\pi {2}^{{n}_{x}+{n}_{y}}{n}_{x}!{n}_{y}!}}{{\rm{e}}}^{-({x}^{2}+{y}^{2})/2}{H}_{{n}_{x}}(x){H}_{{n}_{y}}(y),\end{eqnarray} \tag{ 59 }$$

where H_n(x) are the Hermite polynomials. These states have energy

$$\begin{eqnarray}&&{E}_{{n}_{x},{n}_{y}}^{2{\rm{d}}}={n}_{x}+{n}_{y}+1,\end{eqnarray} \tag{ 60 }$$

which is $({n}_{x}+{n}_{y}+1)$-fold degenerate. The states ${\phi }_{{n}_{x},{n}_{y}}^{2{\rm{d}}}(x,y)$ are real.

Alternatively, one can use energy eigenstates that are also eigenstates of the z-component of angular momentum ${\widehat{L}}_{z}$ [42, 43]. In polar coordinates, the common eigenstates of ${\widehat{H}}_{2{\rm{d}}}$ and ${\widehat{L}}_{z}$ are given by

$$\begin{eqnarray}&&{\phi }_{{n}_{r},{n}_{l}}^{\mathrm{pol}}(r,\varphi )={{ \mathcal N }}_{{n}_{r},{n}_{l}}{r}^{| {n}_{r}-{n}_{l}| }{{\rm{e}}}^{-{r}^{2}/2}{L}_{n}^{| {n}_{r}-{n}_{l}| }({r}^{2}){{\rm{e}}}^{{\rm{i}}({n}_{r}-{n}_{l})\varphi },\end{eqnarray} \tag{ 61 }$$

with

$$\begin{eqnarray}&&{{ \mathcal N }}_{{n}_{r},{n}_{l}}=\sqrt{\displaystyle \frac{n!}{\pi (n+| {n}_{r}-{n}_{l}| )!}}\end{eqnarray} \tag{ 62 }$$

and with $n=\min ({n}_{l},{n}_{r})$. The energy eigenvalue is

$$\begin{eqnarray}&&{E}_{{n}_{r},{n}_{l}}^{\mathrm{pol}}={n}_{r}+{n}_{l}+1\end{eqnarray} \tag{ 63 }$$

and the ${\widehat{L}}_{z}$ eigenvalue is $({n}_{r}-{n}_{l})$ with ${n}_{i}=0,1,2,\,...$ $(i=r,l)$. Again, the energy ${E}_{{n}_{r},{n}_{l}}^{\mathrm{pol}}$ is $({n}_{r}+{n}_{l}+1)$-fold degenerate.

B.2.1. Harmonic oscillator

The Hamiltonian corresponding to a particle in a 3d isotropic harmonic potential is

$$\begin{eqnarray}&&{\widehat{H}}_{3{\rm{d}}}=\displaystyle \frac{1}{2}({\widehat{p}}_{x}^{2}+{\widehat{p}}_{y}^{2}+{\widehat{p}}_{z}^{2})+\displaystyle \frac{1}{2}({\widehat{x}}^{2}+{\widehat{y}}^{2}+{\widehat{z}}^{2}),\end{eqnarray} \tag{ 64 }$$

which corresponds to the sum of the Hamiltonians of three 1d harmonic oscillators with the same frequency in the x, y and z direction.

The eigenstates of ${\widehat{H}}_{3{\rm{d}}}$ are simply product of eigenstate of the 1d harmonic oscillator [42]

$$\begin{eqnarray}&&{\phi }_{{n}_{x},{n}_{y},{n}_{z}}^{3{\rm{d}}}(x,y,z)=\displaystyle \frac{{{\rm{e}}}^{-({x}^{2}+{y}^{2}+{z}^{2})/2}{H}_{{n}_{x}}(x){H}_{{n}_{y}}(y){H}_{{n}_{z}}(z)}{\sqrt{\pi {2}^{{n}_{x}+{n}_{y}+{n}_{z}}{n}_{x}!{n}_{y}!{n}_{z}!}}\end{eqnarray} \tag{ 65 }$$

where H_n(x) are the Hermite polynomials. These states have energy

$$\begin{eqnarray}&&{E}_{{n}_{x},{n}_{y},{n}_{z}}^{3{\rm{d}}}={n}_{x}+{n}_{y}+{n}_{z}+\displaystyle \frac{3}{2}.\end{eqnarray} \tag{ 66 }$$

The energy ${E}_{{n}_{x},{n}_{y},{n}_{z}}^{3{\rm{d}}}$ is $(n+1)(n+2)/2$-fold degenerate, with $n={n}_{x}+{n}_{y}+{n}_{z}$. The states ${\phi }_{{n}_{x},{n}_{y},{n}_{z}}^{3{\rm{d}}}(x,y,z)$ are real.

Alternatively, one can use states that are also eigenstates of ${\widehat{{\bf{L}}}}^{2}$ and ${\widehat{L}}_{z}$:

$$\begin{eqnarray}&&{\phi }_{k,l,m}^{\mathrm{sph}}(r,\theta ,\varphi )={{ \mathcal N }}_{k,l}{\left(\displaystyle \frac{r}{\sqrt{2}}\right)}^{l}{{\rm{e}}}^{-{r}^{2}/2}{L}_{k}^{l\,+\,1/2}({r}^{2}){Y}_{l}^{m}(\theta ,\varphi ),\end{eqnarray} \tag{ 67 }$$

with

$$\begin{eqnarray}&&{{ \mathcal N }}_{k,l}=\sqrt{\sqrt{\displaystyle \frac{1}{4\pi }}\displaystyle \frac{{2}^{2k+2{kl}+3}k!}{(2k+2l+1)!!}}.\end{eqnarray} \tag{ 68 }$$

The energy is

$$\begin{eqnarray}&&{E}_{k,l}^{\mathrm{sph}}=n+\displaystyle \frac{3}{2},\end{eqnarray} \tag{ 69 }$$

where $n=2k+l$. The l quantum number can take the value $l=0,2,\ldots ,n-2,n$ if n is even and $l=1,3,\ldots ,n-2,n$ if n is odd, and $m=-l,-l+1,\ldots ,l-1,l$ [44].