Predefined one-body potentials

Table 20: Predefined potentials and parameters.

whichpot	Description	Potential	Parameters
HO1D	$1$D harmonic oscillator	$V(x)=\frac12 p_1^2 x^2$	$p_1\equiv$trap frequency.
HO2D	$2$D harmonic oscillator	$V(x,y)=$ $\frac12 (p_1^2 x^2 + p_2^2 y^2)$	$p_{1/2}\equiv$trap frequency in $x/y$
HO3D	$3$D harmonic oscillator	$V(x,y,z)=$ $\frac12 (p_1^2 x^2 + p_2^2 y^2 + p_3^2 x^2)$	$p_{1/2/3}\equiv$trap frequency in $x/y/z$
h+d	$1$D harmonic oscillator plus Gaussian central barrier	$V(x)= \frac12 (x-p_1)^2$ $+ p_2 exp(- \frac{(x-p_1)^2}{2 p_3^2})$	$p_1\equiv$ diplacement, $p_2\equiv$ height of Gaussian, $p_3\equiv$ width of Gaussian
h2D+d	$2$D harmonic oscillator plus Gaussian central barrier	$V(x)= \frac12 (p_1^2 x^2+p_2^2 y^2)$ $+ p_3 exp(- (\frac{(x)^2}{2 p_4^2}+\frac{(y)^2}{2 p_5^2}))$	$p_{1/2}\equiv$ trap frequency in $x$-/$y$-direction, $p_3\equiv$ height of Gaussian, $p_{4/5}\equiv$ width of Gaussian in direction $x$/$y$
HO<X>D+td_gauss	<X>D harmonic oscillator plus time-dependent [height $A(t)$] Gaussian central barrier in $x$-direction	$V(x)= \frac12 p_1^2 ( (x-p_2)^2 + y^2)$ $+ A(t) exp(- \frac{(x-p_5)^2}{2 p_6^2})$	$p_{1}\equiv$ trap frequency in $x$- and $y$-direction, $p_2\equiv$ displacement of the minimum of the trap w.r.t. the barrier, $p_5\equiv$ displacement of Gaussian w.r.t. $x=0$, $p_{6}\equiv$ width of Gaussian barrier in direction $x$. Height $A(t)= \begin{cases} \frac{t p_3}{p_4} & t\leq p_4 \\ p_3 & t > p_4 \end{cases}$
Tilt	Tilted triple well potential	$V(x)=-p_1 x + p_2 \sin(2 x)^4 + (x/2.2)^{20}$	$p_1\equiv$ tilt parameter, $p_2\equiv$ depth of the wells.
Tiltinit	single well potential to initialize system for a propagation in the tilted triple well potential Tilt.	$V(x) = V(x)=-p_1 x + p_2 \sin(2 x)^4 + [(x-\frac{p_3 \pi}{2})/0.7]^{20}$	$p_1\equiv$ tilt parameter, $p_2\equiv$ depth of the wells, $p_3$ is used to select displacement from the origin.
OL_HW_1D	lattice in one dimension with hard wall boundaries	$V(x)= \begin{cases} p_1\sin(p_2 x)^2 & p_3 < x < p_4 \\ 1000 & \text{else} \end{cases}$	$p_1\equiv$ depth of lattice, $p_2\equiv$ frequency of lattice, $p_3,p_4$ are the boundaries for the hard walls.
TDHIM	Harmonic potential with time-dependent frequency for benchmarks with the time-dependent harmonic interaction Hamiltonian.	$V(x)=\frac{1}{2}(1+\sin(t) \cos(2t)$ $\sin(\frac{1}{2}t) \sin(0.4t)) x^2$	No parameters, use with which_interaction ='TDHIM'
tun	Potential for tunneling to open space dynamics	$V(x) = \begin{cases} \frac{1}{2} x^2 & x\leq2 \\ 2.2662969 exp(-2 (x-2.25)^2) & x > 2 \end{cases}$	no parameters, initial wavefunction should be localized at $x=0$, use whichpot='ini' for relaxation of initial state.
thr	Tunneling to open space with a threshold	$V(x)=\begin{cases} \frac{1}{2}x^2 & x \leq 2 \\ Ax^3+Bx^2 +Cx+D & 2 < x \leq 4 \\ p_1 & x > 4 \end{cases}$	Parameter $p_1$ defines the threshold and the polynomial coefficients $A=1-\frac{1}{4} p_1$, $B=2.25 p_1 - 9.5$, $C=28-6p_1$, $D=5T-24$.
DQD	Double quantum dot potential	$V(x) = -p_1 exp(-p_2(x+\frac{1}{2}p_3)^2) - p_4 exp(-p_5(x-\frac{1}{2}p_3))$	$p_1\equiv$ depth of first quantum dot, $p_2\equiv$ width of first quantum dot, $p_3\equiv$ distance of the two dots, $p_4\equiv$ depth of second quantum dot, $p_5\equiv$ width of second quantum dot.
DQDLASER	Double quantum dot potential illuminated by a laser with time-dependent amplitude $V_{\text{las}}(x,t)$.	$V(x)=-p_1 exp(-p_2(x+\frac{1}{2}p_3)^2) - p_4 exp(-p_5(x-\frac{1}{2}p_3))+V_{\text{las}}(x,t) exp(-p_9 (x-p_{10})^2)$	$p_{1-5}\equiv$ same as in DQD, $V_{\text{las}}(x,t)= p_7 \cos(p_8 t) \sin(\pi \frac{t}{p_6}) x$ for $t\leq p_6$ and $V_{\text{las}}(x,t)=0$ else.
zero	No potential	$V=0$	No parameters. Boundary conditions determined by the discrete variable representation.
MQT_ini	Rectangular $1$D box	$V(x)=\infty \qquad \forall [x\notin (0,20-p_1)]$ and $V(x)=0 \qquad \forall [x\in )0,20-p_1(]$	$p_1\equiv$ barrier width in propagation MQT_prop
MQT_prop	Rectangular $1$D box, barrier and open space	$V(x)=\infty \qquad \forall [x < 0]$ and $V(x)=0 \qquad \forall [x\in )0,20-p_1(]$ and $V(x)=0 \qquad \forall [x>20]$ and $V(x)=0.05 \forall [x\in (20-p_1,20)]$	$p_1\equiv$ barrier width
qpl	$1$D lattice with $2$ frequencies/amplitudes	$V(x)=p_1 \cos(p_2 x) + p_3 \cos(p_4 x)$	$p_{1/3}\equiv$ amplitudes of the lattices, $p_{2/4}\equiv$ frequencies of the lattices.
rot2D	$2$D harmonic oscillator with rotating anisotropy	$V(x,t)=\frac12 ((1+p_a(t))(x \cos(p_1 t) + y\sin(p_1 t))^2 + (1-p_a(t))(y \cos(p_1 t) - x \sin(p_1 t)))$	$p_1\equiv$ the rotation frequency, $p_a\equiv$ the time-dependent anisotropy. $p_2\equiv$ anisotropy maximum, $p_3\equiv$ ramp-up and ramp-down time of the anisotropy, $p_4\equiv$ plateau time at which $p_a=p_2$.
stir2D	$2$D harmonic trap with rotating stirring rod	$V(x,t) = \linebreak\frac{1}{2} (x^2 + y^2)+ \linebreak p_3 exp[ \linebreak -\frac{1}{p_4}(x-p_2\cos(p_1 t))^2 + (y - p_2\sin(p_1 t))^2]$	$p_1\equiv$ stirring frequency, $p_2\equiv$ stirring radius, $p_3\equiv$ height of Gaussian rod, $p_4\equiv$ width of Gaussian rod.
ellipse2D	Elliptic two-dimensional hard-walled potential well	$V(x)=\begin{cases} 1000 & \sqrt{(\frac{x}{p_1})^2+(\frac{y}{p_2})^2} > p_3 \\ 0 & \text{else} \end{cases}$	$p_{1/2/3}\equiv$ parameters defining the ellipse.