A complicated Duffing oscillator in the surface-electrode ion trap

Abstract

The oscillation coupling and different nonlinear effects are observed in a single trapped ⁴⁰Ca⁺ ion confined in our home-built surface-electrode trap (SET). The coupling and the nonlinearity are originated from the high-order multipole potentials, such as hexapole and octopole potentials, due to different layouts and the fabrication asymmetry of the SET. We solve a complicated Duffing equation with coupled oscillation terms by the multiple-scale method, which fits the experimental values very well. Our investigation in the SET helps for exploring multi-dimensional nonlinearity using currently available techniques and for suppressing instability of qubits in quantum information processing with trapped ions.

Appendices

Appendix 1: The nonlinear coefficients in Eq. (4)

Our calculation is based on Eq. (4), in which the nonlinear coefficients originate from the hexapole and octopole potentials. By setting e and m to be the electric quantity and the mass of a single calcium ion, we have the nonlinear coefficients as

$$ \alpha_{3}=\frac{36e^{2}r_{0}^{2}\left( U_{13}^{\ast }\right) ^{2}M_{13}^{2}}{m^{2}r_{0}^{8}\varOmega ^{2}}+\frac{140eM_{21}\left[ mr_{0}^{4}(V_{21}^{\ast })\varOmega ^{2}+8er_{0}^{2}\left( U_{21}^{\ast }\right) \left( U_{7}^{\ast }\right) M_{7}\right]}{m^{2}r_{0}^{8}\varOmega ^{2}}, $$

(10)

$$\alpha_{2}=\frac{6er_{0}M_{13}\left[mr_{0}^{4}(V_{13}^{\ast})\varOmega ^{2}+6er_{0}^{2}\left( U_{7}^{\ast }\right) \left(U_{13}^{\ast }\right) M_{7}\right] }{m^{2}r_{0}^{8}\varOmega ^{2}}, $$

(11)

$$ \begin{aligned} \alpha_{21} &=\frac{3e\left[ 70er_{0}^{2}M_{6}M_{21}(U_{6}^{\ast })(U_{21}^{\ast })+42er_{0}^{2}M_{7}M_{20}(U_{7}^{\ast })(U_{20}^{\ast }) \right] }{m^{2}r_{0}^{8}\varOmega ^{2}} \\ &\quad+\frac{3e\left[ 24er_{0}^{2}M_{12}M_{13}(U_{12}^{\ast })(U_{13}^{\ast })+7mr_{0}^{4}\varOmega ^{2}M_{20}(V_{20}^{\ast })\right] }{m^{2}r_{0}^{8} \varOmega ^{2}}, \end{aligned} $$

(12)

$$ \begin{aligned} \alpha_{22} &=\frac{3e\left[ 70er_{0}^{2}M_{8}M_{21}(U_{8}^{\ast })(U_{21}^{\ast})+42er_{0}^{2}M_{7}M_{22}(U_{7}^{\ast })(U_{22}^{\ast}) \right] }{m^{2}r_{0}^{8}\varOmega ^{2}} \\ &\quad+\frac{3e\left[ 24er_{0}^{2}M_{13}M_{14}(U_{13}^{\ast})(U_{14}^{\ast })+7mr_{0}^{4}\varOmega ^{2}M_{22}(V_{22}^{\ast })\right] }{m^{2}r_{0}^{8} \varOmega ^{2}}, \end{aligned} $$

(13)

$$ \begin{aligned} \alpha_{4} &=\frac{e\left[ 32er_{0}^{2}\left( U_{12}^{\ast }\right) ^{2}M_{12}^{2}-18er_{0}^{2}\left( U_{13}^{\ast }\right) ^{2}M_{13}^{2}-6er_{0}^{2}\left( U_{13}^{\ast }\right) \left( U_{15}^{\ast }\right) M_{13}M_{15}+21er_{0}^{2}\left( U_{6}^{\ast }\right) \left( U_{20}^{\ast}\right) M_{6}M_{20}\right] }{m^{2}r_{0}^{8}\varOmega ^{2}} \\ &\quad+\frac{e\left[ -60mr_{0}^{4}(V_{21}^{\ast})\varOmega ^{2}M_{21}-14mr_{0}^{4}(V_{23}^{\ast})\varOmega ^{2}M_{23}-240er_{0}^{2}(U_{7}^{\ast})\left( U_{21}^{\ast}\right) M_{7}M_{21}-56er_{0}^{2}(U_{7}^{\ast})\left( U_{23}^{\ast}\right) M_{7}M_{23}\right] }{m^{2}r_{0}^{8}\varOmega ^{2}}, \end{aligned} $$

(14)

$$ \begin{aligned} \alpha_{5} &=\frac{e\left[ 32er_{0}^{2}\left( U_{14}^{\ast }\right) ^{2}M_{14}^{2}-18er_{0}^{2}\left( U_{13}^{\ast }\right) ^{2}M_{13}^{2}+6er_{0}^{2}\left( U_{13}^{\ast }\right) \left( U_{15}^{\ast }\right) M_{13}M_{15}+21er_{0}^{2}M_{8}M_{22}(U_{8}^{\ast })(U_{22}^{\ast }) \right] }{m^{2}r_{0}^{8}\varOmega ^{2}} \notag \\ &\quad+\frac{e\left[-60mr_{0}^{4}(V_{21}^{\ast })\varOmega ^{2}M_{21}+14mr_{0}^{4}(V_{23}^{\ast })\varOmega^{2}M_{23}-240er_{0}^{2}(U_{7}^{\ast })\left( U_{21}^{\ast }\right) M_{7}M_{21}+56er_{0}^{2}(U_{7}^{\ast })\left( U_{23}^{\ast }\right) M_{7}M_{23}\right] }{m^{2}r_{0}^{8}\varOmega ^{2}}, \end{aligned} $$

(15)

$$ \begin{aligned} \alpha_{6} &=\frac{e\left[ 6er_{0}^{2}\left( U_{13}^{\ast }\right) \left(U_{11}^{\ast }\right) M_{13}M_{11}+14mr_{0}^{4}(V_{19}^{\ast})\varOmega ^{2}M_{19}+21er_{0}^{2}\left( U_{6}^{\ast}\right) \left( U_{22}^{\ast }\right) M_{6}M_{22}\right] }{m^{2}r_{0}^{8}\varOmega ^{2}} \\ &\quad+\frac{e\left[ 64er_{0}^{2}(U_{12}^{\ast })\left(U_{14}^{\ast}\right) M_{12}M_{14}+56er_{0}^{2}\left( U_{7}^{\ast }\right) \left( U_{19}^{\ast }\right) M_{7}M_{19}+21er_{0}^{2}M_{8}M_{20}(U_{8}^{\ast })(U_{20}^{\ast }) \right] }{m^{2}r_{0}^{8}\varOmega ^{2}}, \end{aligned} $$

(16)

$$\alpha_{7}=\frac{e\left[6er_{0}^{3}\left( U_{13}^{\ast }\right) \left(U_{6}^{\ast }\right) M_{13}M_{6}+8mr_{0}^{5}(V_{12}^{\ast})\varOmega ^{2}M_{12}+32er_{0}^{3}\left( U_{12}^{\ast }\right) \left( U_{7}^{\ast }\right) M_{12}M_{7}\right] }{m^{2}r_{0}^{8}\varOmega ^{2}}, $$

(17)

$$ \alpha_{8}=\frac{e}{m^{2}r_{0}^{8}\varOmega ^{2}}\left[ 8mr_{0}^{5}(V_{14}^{ \ast })\varOmega ^{2}M_{14}+32er_{0}^{3}\left( U_{14}^{\ast }\right) \left( U_{7}^{\ast }\right) M_{14}M_{7}+6er_{0}^{3}M_{8}M_{13}(U_{8}^{\ast })(U_{13}^{\ast })\right] , $$

(18)

with $\omega_{0z}^{2}=e\left[ 8er_{0}^{4}\left( U_{7}^{\ast }\right)^{2}M_{7}^{2}+4mr_{0}^{6}(V_{7}^{\ast })\varOmega^{2}M_{7}\right] /m^{2}r_{0}^{8}\varOmega^{2},$ and the scaling factor r ₀ (see definition in [24]).

Appendix 2: Details of the steady-state solution Eq. (5)

Equation (5) is obtained by the standard steps of the multiple-scale method. Starting from Eq. (4), we assume that the driving frequency is a perturbative expansion of harmonic oscillator frequency [28], i.e., $\omega_{z}=\omega_{0z}+\epsilon^{2}\sigma,$ with a small dimensionless parameter $\epsilon.$ Following the method in [29], we rewrite Eq. (4) by setting $z=\epsilon u, x=\epsilon p, y=\epsilon q,$ the damping term as $2\epsilon^{3}\mu \dot{u}$ and the driving term as $\epsilon^{3}k_{z}\cos (\omega_{z}t).$ Introducing a new parameter $T_{i}=\epsilon^{i}t (i=0,1,2),$ we rewrite u, p, q as,

$$ \begin{aligned} u&=\epsilon ^{0}u_{0}\left( T_{0},T_{1},T_{2}\right) +\epsilon ^{1}u_{1}\left( T_{0},T_{1},T_{2}\right) +\epsilon ^{2}u_{2}\left( T_{0},T_{1},T_{2}\right) , \\ p&=\epsilon ^{0}p_{0}\left(T_{0},T_{1},T_{2}\right) +\epsilon ^{1}p_{1}\left( T_{0},T_{1},T_{2}\right) +\epsilon ^{2}p_{2}\left( T_{0},T_{1},T_{2}\right) , \\ q&=\epsilon ^{0}q_{0}\left( T_{0},T_{1},T_{2}\right) +\epsilon ^{1}q_{1}\left( T_{0},T_{1},T_{2}\right) +\epsilon ^{2}q_{2}\left( T_{0},T_{1},T_{2}\right) . \end{aligned} $$

(19)

Then we compare the coefficients of $\epsilon ^{0}, \epsilon ^{1}$ and $\epsilon ^{2},$ which yields,

$$ D_{0}^{2}u_{0}+\omega_{0z}^{2}u_{0}=0, $$

(20)

$$ D_{0}^{2}u_{1}+\omega_{0z}^{2}u_{1}=-2D_{0}D_{1}u_{0}- \alpha_{2}u_{0}^{2}-\alpha_{7}q_{0}u_{0}-\alpha_{8}p_{0}u_{0}, $$

(21)

$$ \begin{aligned} D_{0}^{2}u_{2}+\omega_{0z}^{2}u_{2} &=-[-k_{z}\cos \left( \omega_{0z}T_{0}+\sigma T_{2}\right)+q_{0}^{2}\alpha_{4} u_{0}+p_{0}^{2}\alpha_{5}u_{0}+p_{0}q_{0}\alpha_{6}u_{0}+2\mu D_{0}u_{0} \\ &\quad+D_{1}^{2}u_{0}+2D_{0}D_{2}u_{0}+q_{0} \alpha_{21}u_{0}^{2}+p_{0}\alpha_{22}u_{0}^{2}+\alpha_{3}u_{0}^{3}+ 2D_{0}D_{1}u_{1} \\ &\quad+\alpha_{7}q_{0}u_{1}+\alpha_{8}p_{0}u_{1}+2\alpha_{2} u_{0}u_{1}+\alpha_{7}q_{1}u_{0}+ \alpha_{8}p_{1}u_{0}], \end{aligned} $$

(22)

where $D_{i}=\partial/\partial T_{i}, \left( i=0,1,2\right).$

Hence, we may solve $u_{0}=A\left( T_{2}\right) \exp (i\omega_{0z}T_{0})+\bar{A}\left( T_{2}\right) \exp (-i\omega_{0z}T_{0})$ and $u_{1}=\alpha_{2}[A^{2}\exp (2i\omega_{0z}T_{0})-6A\bar{A}+\bar{A}^{2}\exp (-2i\omega_{0z}T_{0})]/3\omega_{0z}^{2}$ from Eqs. (20) and (21) by eliminating the secular term. Similarly, from equations of the oscillations in x axis and y axis, which are similar to Eq. (4) but without the driven term, we may solve the variables p ₀, q ₀, p ₁ and q ₁ as

$$ p_{0}=C\left( T_{2}\right) \exp (i\omega_{0x}T_{0})+\bar{C}\left( T_{2}\right) \exp (-i\omega_{0x}T_{0}), $$

(23)

$$ q_{0}=B\left( T_{2}\right) \exp (i\omega_{0y}T_{0})+\bar{B}\left( T_{2}\right) \exp (-i\omega_{0y}T_{0}), $$

(24)

$$ p_{1}=\alpha_{2x}[C^{2}\exp(2i\omega_{0x}T_{0})- 6C\bar{C}+\bar{C}^{2}\exp (-2i\omega_{0x}T_{0})]/(3\omega_{0x}^{2}), $$

(25)

$$ q_{1}=\alpha_{2y}[B^{2}\exp (2i\omega_{0y}T_{0})-6B\bar{B}+\bar{B}^{2}\exp (-2i\omega_{0y}T_{0})]/(3\omega_{0y}^{2}), $$

(26)

where α _2x and α _2y correspond to the quadric nonlinearity of the ion motion equation in x and y directions, respectively. ω _0x /2π and ω _0y /2π represent the harmonic frequencies in x direction and y direction. $A=\frac{1}{2}a\exp (i\beta ), B=\frac{1}{2}b\exp (i\varsigma)$ and $C=\frac{1}{2}c\exp (i\eta),$ where $a,b,c,\beta ,\varsigma$ and η are real functions of $T_{2}, \beta ,\varsigma$ and η represent the phases of different dimensions. $\bar{A}, \bar{B}$ and $\bar{C}$ are conjugate terms of A, B and C. Substituting u ₀ and u ₁ into Eq. (22), we obtain an equation regarding the secular term, from which, in combination with Eqs. (23–26) with the expressions of A, B and C, we obtain

$$ -\frac{1}{2}k_{z}\sin \gamma +a\mu \omega_{0z}+\omega_{0z}\frac{\hbox{d}a}{\hbox{d}T_{2}} =0, $$

(27)

$$ \frac{1}{2}k_{z}\cos \gamma-\frac{3\alpha_{3}a^{3}}{8}- \frac{1}{4}\alpha_{4}ab^{2}-\frac{1}{4}\alpha_{5}ac^{2}+ \frac{5\alpha_{2}^{2}a^{3}}{12\omega_{0z}^{2}}+\frac{\alpha_{8} \alpha_{2x}ac^{2}}{4\omega_{0x}^{2}}+\frac{\alpha_{7}\alpha_{2y} ab^{2}}{4\omega_{0y}^{2}}+a\omega_{0z}\left(\sigma -\frac{\hbox{d}\gamma}{\hbox{d}T_{2}}\right) =0, $$

(28)

with γ = σ T ₂ − β. We assume the steady-state motion corresponding to $\frac{\hbox{d}\gamma}{\hbox{d}T_{2}}=\frac{\hbox{d}a}{\hbox{d}T_{2}}=0.$ So we have

$$ \left( \frac{3\alpha_{3}}{8}a^{3}-\frac{5\alpha_{2}^{2}}{12\omega_{0z}^{2}} a^{3}+\frac{\alpha_{4}}{4}ab^{2}+\frac{\alpha_{5}}{4}ac^{2}-\frac{ \alpha_{7}\alpha_{2y}}{4\omega_{0y}^{2}}ab^{2}-\frac{\alpha_{8} \alpha_{2z}}{ 4\omega_{0x}^{2}}ac^{2}-a\omega_{0z}\sigma \right) ^{2}+\left( a\omega_{0z}\mu \right) ^{2}=\frac{1}{4}k_{z}^{2}. $$

(29)

which is actually Eq. (5).