Fast state and trap rotation of a particle in an anisotropic potential

TL;DR

This paper develops a method to rapidly rotate the state of a particle in a 2D anisotropic harmonic trap by transforming the system into normal modes, enabling efficient state manipulation.

Contribution

It introduces a symplectic transformation approach to decouple the dynamics and design fast trap-rotation protocols for arbitrary initial states.

Findings

Normal modes are decoupled via symplectic transformations.

Fast trap-rotation processes are feasible when normal frequencies are commensurate.

The method applies to both quantum and classical particles.

Abstract

We study the dynamics of a quantum or classical particle in a two-dimensional rotating anisotropic harmonic potential. By a sequence of symplectic transformations for constant rotation velocity we find uncoupled normal generalized coordinates and conjugate momenta in which the Hamiltonian takes the form of two independent harmonic oscillators. The decomposition into normal-mode dynamics enables us to design fast trap-rotation processes to produce a rotated version of an arbitrary initial state, when the two normal frequencies are commensurate.

Tables

Table 1. Table 1: Some numerical values of the trapping frequencies ω 1 subscript 𝜔 1 \omega_{1} and ω 2 subscript 𝜔 2 \omega_{2} , rotation angular velocity θ ˙ ˙ 𝜃 \dot{\theta} and time duration T 𝑇 T of the rotation operation calculated according to Eqs. ( 26 - 29 ). θ f = π / 2 subscript 𝜃 𝑓 𝜋 2 \theta_{f}=\pi/2 , n 1 = 1 subscript 𝑛 1 1 n_{1}=1 , and n 2 = 2 subscript 𝑛 2 2 n_{2}=2 .

${\omega }_1$	${\omega }_2$	$\dot{\theta }$	$T_{1,2}={\theta }_f/\dot{\theta }$
$2\pi \times 1$ kHz	$2\pi \times 1.79$ kHz	$2\pi \times 0.23$ kHz	$1.08$ ms
$2\pi \times 2$ kHz	$2\pi \times 3.59$ kHz	$2\pi \times 0.46$ kHz	$0.54$ ms
$2\pi \times 5$ kHz	$2\pi \times 8.96$ kHz	$2\pi \times 1.16$ kHz	$0.22$ ms
$2\pi \times 10$ kHz	$2\pi \times 17.93$ kHz	$2\pi \times 2.32$ kHz	$0.11$ ms

Equations

$$H$$

$$q_j=\sqrt{\frac{m{\omega }_j}{\hslash }}{\tilde{q}}_j,p_j=\frac{{\tilde{p}}_j}{\sqrt{m\hslash {\omega }_j}},$$

$$H$$

$$H=v^{T}Av,$$

$$A$$

$$J$$

$$H=v^{T}Av=V^T\left(S^{T}AS\right)V.$$

$$\dot{\theta }<\text{min}({\omega }_1,{\omega }_2).$$

$$\dot{\theta }<{\omega }_1<{\omega }_2.$$

$$S_0=\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 1\end{array}\right),$$

$$A_1$$

$$S_1$$

$$A_2$$

$$S_2$$

$$A_3$$

$$S_3$$

$$\tan 2\alpha =\frac{4\dot{\theta }\sqrt{{\omega }_1^2-{\dot{\theta }}^2}}{{\omega }_1^2-{\omega }_2^2-4{\dot{\theta }}^2}.$$

$$A_4$$

$${\Omega }_1^2$$

$${\Omega }_2^2$$

$$S$$

$$\left(\begin{array}{c}{q}_1\\ q_2\\ p_1\\ p_2\end{array}\right)=S\left(\begin{array}{c}{Q}_1\\ Q_2\\ P_1\\ P_2\end{array}\right).$$

$$\{\begin{array}{ccc}{Q}_1& =& q_2\frac{\sqrt{\delta }\sin \alpha -\dot{\theta }\cos \alpha }{\sqrt{\delta {\omega }_2}}+p_1\sqrt{\frac{{\omega }_1}{\delta }}\cos \alpha \\ Q_2& =& q_2\frac{\sqrt{\delta }\cos \alpha +\dot{\theta }\sin \alpha }{\sqrt{\delta {\omega }_2}}-p_1\sqrt{\frac{{\omega }_1}{\delta }}\sin \alpha \\ P_1& =& -q_1\frac{\sqrt{\delta }\cos \alpha +\dot{\theta }\sin \alpha }{\sqrt{{\omega }_1}}+p_2\sqrt{{\omega }_2}\sin \alpha \\ P_2& =& q_1\frac{\sqrt{\delta }\sin \alpha -\dot{\theta }\cos \alpha }{\sqrt{{\omega }_1}}+p_2\sqrt{{\omega }_2}\cos \alpha \end{array},$$

$$H$$

$$\frac{{\Omega }_2}{{\Omega }_1}=\frac{n_2}{n_1}$$

$$T=\frac{2\pi n_1}{{\Omega }_1}=\frac{2\pi n_2}{{\Omega }_2}.$$

$$T=\frac{{\theta }_f}{\dot{\theta }}=\frac{2\pi n_1}{{\Omega }_1(\dot{\theta },{\omega }_1,{\omega }_2)}=\frac{2\pi n_2}{{\Omega }_2(\dot{\theta },{\omega }_1,{\omega }_2)}.$$

$${\omega }_1$$

$${\omega }_2$$

$${\kappa }_{\pm }={\left(-1+\frac{2{\pi }^2{\delta }_{+}}{{\theta }_f^2}\pm \frac{2\sqrt{{\pi }^4{\delta }_{-}^2-2{\pi }^2{\delta }_{+}{\theta }_f^2+{\theta }_f^4}}{{\theta }_f^2}\right)}^{1/2}$$

$$T_{n_1,n_2}$$

Full text

I Introduction

Motivated by existing or developing quantum technologies, much work is currently being devoted to control the motional dynamics of quantum systems. Basic operations such as shuttling, expansions/compressions, merging and separation of atom or ion chains, or rotations of the quantum particles are needed to implement interferometers, quantum information applications, or quantum thermodynamical devices. Performing fast operations that do not leave residual excitations is generically of interest not only to save time but to avoid decoherence as well.

Shortcuts to adiabaticity (STA) are proposed as a set of efficient techniques to design such operations Torrontegui2013 . For two or more effective dimensions, shortcut design, by inverse engineering the control parameters using invariants of motion, is much facilitated by finding dynamical normal modes Palmero2014 . These modes are independent harmonic motions in the regime of small oscillations corresponding in general to time-dependent harmonic oscillators. Studies on different operations on trapped ions Palmero2013 ; Palmero2014 ; Palmero2015a ; Palmero2015b ; Lu2015 ; Palmero2016 ; Palmero2017 made clear that it is not always possible to find a point transformation111In a point transformation the new coordinates only depend on the old ones and not on old momenta. that leads to independent normal modes. The condition that allows to find a point transformation was finally given in Lizuain2017 for two-dimensional (2D) Hamiltonians: the effective potential can be scaled or translated but it should not rotate. Thus the rotation of a 2D anisotropic trap is the paradigmatic model in which such point transformation cannot be made and it was left as an open question if more general transformations could be used to speed up the rotation Lizuain2017 . The inertial effect due to the trap rotation can be formally compensated by an effective angular momentum term Masuda ; Lizuain2017 to leave the particle at rest in the rotating frame. This term though may be difficult to implement, for example if the particle is not charged, so we consider in this paper that the only manipulation available is the rotation of the trap itself, without any additional force. STA for simple 1D-trap rotations, without compensation terms, were described in Palmero2016 but STA for the more realistic 2D anisotropic trap had not been described.

The goal of this work is to perform a rotation as represented schematically in Fig. 1 in the lab frame of coordinates $x,y$: The trap is at rest for $t<0$; then it is rotated up to time $T$; and finally it remains again at rest for $t>T$. The trap rotation must be designed such that the state at time $T$ is exactly the rotated version of the initial state at time $t=0$, for all possible initial states. Equivalently, from the point of view of the rotating frame, the objective is to get at time $T$ the same state that was prepared at time [math], regardless of what that state may be.

Rotations of condensates or of a few particles are of interest for different reasons, such as reordering chains, redirecting, squeezing Palmero2016 , or creating artificial magnetism Fetter2009 ; Dalibard2011 . Here we treat the simplest case of a single particle in a rotating 2D trap. The operation would be instrumental in driving atoms through corners and junctions in a scalable quantum processor 20 ; 21 , and may be regarded as a first step towards the more difficult problem of rotating ion chains 17 ; 20 ; 22 , which would facilitate scalability in linear traps, and be useful to rearrange the chain, e.g. to locate a cooling ion at the right position in the chain 17 . Rotor states have other applications in sensing, metrology, and fundamental physics studies Urban2019 .

The treatment and transformations are done first in a classical setting. However, since we deal with a harmonic anisotropic trap the results can be translated into quantum mechanics rather directly. After setting the model in Sec. II, the independent normal modes will be first defined and characterized by normal frequencies in Sec. III. Section IV analyzes the fast rotations that may be achieved at certain process times for configurations in which the normal frequencies are commensurate. The minimal time is identified, examples are given, and a stability analysis is carried out. Finally, Sec. V discusses some open questions.

II Physical model

Our starting point is the Hamiltonian of a particle of mass $m$ in a 2D anisotropic harmonic potential with axial (angular) frequencies $\omega_{1}$ and $\omega_{2}$, which rotates around the $z$ axis perpendicular to the trap plane by an angle $\theta$ with an angular velocity $\dot{\theta}$, see Fig. 1. In the rotating frame of coordinates $\{\tilde{q}_{1},\tilde{q}_{2}\}$ and momenta $\{\tilde{p}_{1},\tilde{p}_{2}\}$ the Hamiltonian is given by, see Appendix A,

[TABLE]

where $L_{z}=\tilde{q}_{1}\tilde{p}_{2}-\tilde{q}_{2}\tilde{p}_{1}$. $H$ has the form of two harmonic oscillators coupled by an angular momentum $L_{z}$ that accounts for the inertial effects Goldstein .

By introducing the dimensionless coordinates and momenta

[TABLE]

the Hamiltonian (1) can be written ($\hbar=1$ hereafter) as

[TABLE]

where $\eta=\sqrt{\omega_{1}/\omega_{2}}$.

This rotating frame Hamiltonian depends only on the angular velocity $\dot{\theta}$ as a control parameter. We shall consider throughout the work a constant rotation velocity, i. e., a linear-in-time angle $\theta(t)=\dot{\theta}t$ from $t=0$ to $t=T$. Thus the Hamiltonian in the rotating frame is time independent during the rotation.

The Hamiltonian (3) can be written in compact matrix representation as the quadratic form

[TABLE]

where $v^{T}=(q_{1},q_{2},p_{1},p_{2})$ and $A$ is the symmetric $4\times 4$ matrix

[TABLE]

Our first goal is to find a transformation to a frame in which the corresponding effective Hamiltonian is uncoupled in both coordinates and momenta, or, using the four-dimensional matrix formalism, it is characterized by a diagonal matrix. To do so we will use the symplectic approach to canonical transformations.

III Symplectic diagonalization

In the $4\times 4$ matrix representation presented above, a canonical transformation will be defined by the transformation $v=SV$ to a new set of canonical coordinates $V^{T}=(Q_{1},Q_{2},P_{1},P_{2})$ provided $S$ is a $4\times 4$ symplectic matrix. A symplectic matrix $S$ satisfies $S^{T}JS=J$, where $J$ is the skew-symmetric matrix Goldstein

[TABLE]

Note that its inverse is simply $J^{-1}=J^{T}=-J$. As well, $S^{-1}=J^{-1}S^{T}J$. $4\times 4$ real symplectic matrices form the ten-dimensional symplectic group $Sp(4,\mathbb{R})$ simon1999 . Applying a symplectic (i. e., canonical) transformation to $H$ amounts to rewrite it as

[TABLE]

Given the matrix $A$ (5) we want to find a symplectic matrix $S\in Sp(4,\mathbb{R})$ so that $S^{T}AS$ is a diagonal matrix. Such a diagonalizing symplectic matrix $S$ will always exist as long as $A$ is a positive definite matrix. This result is known as Williamson’s Theorem Williamson1936 ; Pirandola2009 ; Gosson . The positivity of $A$ imposes an upper bound for the allowed rotation velocity in order to end up with an uncoupled effective Hamiltonian. In particular, the rotation velocity must satisfy

[TABLE]

For simplicty, and without loss of generality, we will consider $\omega_{1}<\omega_{2}$ (or $0<\eta<1$) throughout this work. Therefore, the three (angular) frequencies in our model satisfy the conditions

[TABLE]

III.1 Constructing the $S$ matrix

We will construct the $S$ matrix after a four-step sequence of symplectic transformations.

i) The first transformation brings the matrix $A$ (5) to a block diagonal form. This is achieved by the symplectic matrix

[TABLE]

which leads to

[TABLE]

This transformation is not a point transformation since $S_{0}$ mixes coordinates and momenta as already noted in Manko1996 .

ii) The second transformation diagonalizes one of the two blocks in $A_{1}$. We choose the lower one in this case (the “momenta block”)222If $\omega_{1}>\omega_{2}$ had been assumed, at this point the upper block should be diagonalized instead of the lower one. This transformation is performed by the symplectic matrix

[TABLE]

and leads to

[TABLE]

iii) The third step transforms the block that it is already diagonal (the lower block in our case) into the identity. This is achieved by the symplectic matrix

[TABLE]

which transforms $A_{2}$ into

[TABLE]

The transformation requires $\omega_{1}>\dot{\theta}$, which is consistent with Eq. (9).

iv) Finally, a (formal) rotation of an angle $\alpha$ brings the upper block to a diagonal form, leaving the lower block unaltered,

[TABLE]

with the angle of rotation $\alpha$ given by

[TABLE]

This last transformation, leads to our objective, a diagonal matrix

[TABLE]

where the $\Omega_{1,2}$ are the normal mode frequencies with squares

[TABLE]

see a plot of these frequencies as a function of $\dot{\theta}$ in Fig. 2. These eigenfrequencies have been found before by Bialynicki-Birula using a different approach BB97 . Our four-step method is sketched in Manko1996 , although the eigenfrequencies and explicit transformations were not given there.

III.2 Uncoupled Hamiltonian and normal modes

After the sequence of four different transformations, the symplectic matrix we were looking for can be written as (the product of symplectic matrices is symplectic)

[TABLE]

$S$ diagonalizes the initial $A$ matrix by the relation $A_{4}=S^{T}AS$ and relates old coordinates and momenta in the rotating-frame and new coordinates and momenta in the transformed frame by the transformation $v=SV$ or

[TABLE]

By inverting this relation, we can give explicit expressions for the new frame coordinates and momenta in terms of the original ones,

[TABLE]

with $\delta=\omega_{1}^{2}-\dot{\theta}^{2}$, which makes clear that this is not a point transformation. The Hamiltonian written in normal-mode coordinates and momenta takes the simple form of two independent harmonic oscillators with normal frequencies $\Omega_{1,2}$,

[TABLE]

As discussed in Appendix B, these transformations are identical for a quantum Hamiltonian and can be related to quantum unitary transformations. Therefore, the Hamiltonian (23) can be quantized by substituting the generalized coordinate and momenta by the corresponding operators. Since the Hamiltonian is quadratic, we may equivalently rely on a phase-space description of the quantum state dynamics in Wigner representation. The dynamics of the Wigner function is governed by a classical Liouville equation; equivalently, a phase-space point is driven by classical Hamiltonian dynamics.

IV Fast rotations

IV.1 Commensurate anisotropic oscillator

The time evolution generated by the Hamiltonian (23) is governed by two independent harmonic oscillators. In this frame, the corresponding classical trajectories will be given by Lissajous-like orbits, that will only be closed when the ratio between the $\Omega_{1,2}$ frequencies is a rational number, i. e., when they are commensurate. Let us suppose that $n_{1,2}$ are two integers ($n_{1}<n_{2}$). Then, if the condition

[TABLE]

is satisfied, the full period of the dynamics is given by

[TABLE]

If a rotation is performed in a time $T$, the system will end up in the same initial state in the rotating frame: the first oscillator performs $n_{1}$ oscillations, and the second one $n_{2}$ full oscillations.

To perform a rotation of an angle $\theta_{f}=\dot{\theta}T$ (assuming an initial angle $\theta_{i}=0$) at a constant angular velocity $\dot{\theta}$ in time $T$, the above relation may be written as

[TABLE]

For some fixed values of $\theta_{f}$, $n_{1}$ and $n_{2}$, these equalities do not have a unique solution since there are two equations but three different parameters (rotation velocity $\dot{\theta}$, and frequencies $\omega_{1}$ and $\omega_{2}$). Using Eq. (26) we may write two of the frequencies in terms of a third one, for instance

[TABLE]

where

[TABLE]

with $\delta_{\pm}=n_{1}^{2}\pm n_{2}^{2}$. Once one of the frequencies is fixed, the remaining two will be determined by Eq. (27). In the following, the value of the smallest axial frequency $\omega_{1}$ will be fixed, but a similar analysis could be done if any of the two remaining ones is fixed.333Also of interest is the setting where $\omega_{1}$ and $\omega_{2}$ are given, i.e., we do not assume that their values can be controlled. Then $\dot{\theta}$ for different $n_{1},n_{2}$ should be adjusted to satisfy the last equality in Eq. (26). Since $T$ is fixed by the last two ratios in Eq. (26), only a set of discrete values of $\theta_{f}$ are allowed in this scenario.

For a given value of $\omega_{1}$, relation (27) determines $\dot{\theta}$ and $\omega_{2}$, and using Eq. (26) the time duration of the rotation operation is

[TABLE]

which, for some fixed values of $\omega_{1}$ and $\theta_{f}$, is just a function of the integers $n_{1}$ and $n_{2}$. See some numerical values of $T_{1,2}$ for a $\pi/2$ rotation in Table 1.

IV.2 Fast rotations

In principle, the values of $n_{1}$ and $n_{2}$ can be chosen arbitrarily as long as $n_{1}<n_{2}$: The time duration of a given rotation (for given $\theta_{f}$ and $\omega_{1}$) will be completely determined by the factor $\kappa_{-}$. As it is shown in Fig. 3a-b, the fastest possible rotation (minimum value of $\kappa_{-}$) is found with the values $n_{1}=1$ and $n_{2}\rightarrow\infty$. This means that the minimum rotation time $T_{\rm{min}}$ corresponds to a single oscillation of the first (slow) normal mode oscillator and to infinitely many oscillations of the second one,

[TABLE]

This minimal time corresponds to the $\omega_{1}\ll\omega_{2}$ limit (i. e., an infinitely narrow trap) as shown in Fig. 3c.

Of course this limit is an idealization and in practice $\omega_{2}$ will have some maximal value. To illustrate features of a generic case ($n_{2}\neq\infty$) we choose $n_{1}=1$ and $n_{2}=2$ in numerical calculations.

IV.3 Time evolution of states and observables

In the reference system of the normal modes, $\{Q_{1},Q_{2}\}$, a general wave function takes the form

[TABLE]

where the $c_{ij}$ are constant coefficients set by the initial conditions and $\phi^{(1,2)}_{j}(Q_{1,2})$ are the usual stationary eigenfunctions of the harmonic oscillators. If the rotation continues indefinitely, at a time $t+T$ with $T$ given in Eq. (25), one gets

[TABLE]

i. e., the wave function one period $T$ earlier, with an overall phase that depends on $n_{1}+n_{2}$. The quantum system is said to experience “exact revivals” at intervals of $T$ Razi2000 . Here we are interested in setting $t=0$ and the corresponding revival at $T$.

We may use Eq. (31) to find the wavefunction in the rotating frame. To perform the transformation back and forth between the rotating (coupled) and decoupled frames, mixing positions and momenta, a good strategy is to work in a mixed rotating-frame representation and use $\langle Q^{\prime}_{1},Q^{\prime}_{2}|p_{1},q_{2}\rangle=\delta[Q^{\prime}_{1}-Q_{1}(p_{1},q_{2})]\delta[Q^{\prime}_{2}-Q_{2}(p_{1},q_{2})]$. Of course the dynamics may also be solved entirely in the rotating frame by numerical integration in a finite basis,444Specifically in a truncated Fock space for the interaction-free part (two harmonic oscillators) enlarged until converge is achieved. or using the Wigner representation by solving individual trajectories or a system of equations for the moments.

IV.3.1 Periodic orbits in the rotating frame

In the normal-mode frame, the classical trajectories or corresponding center of a wavepacket describe closed Lissajous orbits for commensurate normal frequencies. In the rotating frame we find also corresponding closed orbits.

To visualize them let us suppose that the system is initially in the two-mode coherent state $|\psi(0)\rangle=|\alpha_{1},\alpha_{2}\rangle$. The state $|\alpha_{1},\alpha_{2}\rangle$ may be expanded in terms of the harmonic oscillators with frequencies $\omega_{1,2}$,

[TABLE]

with $\alpha_{j}=|\alpha_{j}|e^{i\varphi}$ being complex quantities characterizing the (square root) of the average excitation number and phase of the coherent state. The state-ket $|n_{1},n_{2}\rangle$ refers to the eigenstate of the 2D harmonic oscillator with a normalized spatial representation

[TABLE]

where $H_{n}(q)$ is the the $n$th order Hermite polynomial. The time-evolved two-mode coherent state in coordinate representation will be given by the wave function

[TABLE]

By integrating the probability density over a full period $T$,

[TABLE]

a track of the wave-packet is found, see Fig. 4, which is more intense where the motion is slow. The center of the wave-packet follows the classical closed Lissajous-like orbits, ending in its initial configuration after a full rotation is performed.

IV.3.2 Mean number of excitations, survival probability.

We will now consider the mean vibrational number as a function of time in the rotating frame,

[TABLE]

where the creation and annhilation operators in each direction are defined in terms of position and momentum operators as usual,

[TABLE]

for $j=1,2$. In the first column of Fig. 5, the time evolution of the mean number of excitations during $\pi/2$ rotations designed without final excitation using the first row of Table 1 ($n_{1}=1,n_{2}=2$) is shown for different initial states: the ground state of the non-rotating trap, an entangled state, and a coherent state. Interestingly, Fig. 5 (first column) demonstrates that the mean excitation can actually decrease, at least transitorily, with respect to the initial value. Of course for all states the final value coincides with the initial value.

The rotation process has been chosen so that the survival probability $P(t)=|\langle\psi(t)|\psi(0)\rangle|^{2}$ satisfies the condition $P(0)=P(T)$, due to commensurability. In the second column of Fig. 5, the probability of finding the system in its initial state is calculated for different initial quantum states. The revivals are seen clearly in all three cases. The survival of the coherent and entangled states decays at intermediate times much more severely than the one for the ground state. Indeed, a classical particle set initially at rest at the bottom of the trap would not be affected by the trap rotation.

IV.4 Stability

As already pointed out in Sec. IV.2, the fastest allowed rotations are found for $n_{1}=1$ and $n_{2}\gg 1$, which imply very narrow quasi-1D traps with $\omega_{2}\gg\omega_{1}$. However, fast rotations come with a price, since as $n_{2}$ increases the ideal result becomes more unstable. This can be intuitively understood: for larger $n_{2}$ the second normal oscillator oscillates faster so it is easier to miss the exact final state due to some small timing error. This is confirmed in Fig. 6, which depicts the survival probability as a function $\epsilon$, a small deviation from the nominal operation time $T$. For larger $n_{2}$ the survival becomes less robust.

This effect can be quantified by approximating the survival probability to second order in $\epsilon$ as

[TABLE]

with

[TABLE]

$\Delta H^{2}$ depends on the considered initial state as Fig. 5 (right column) illustrates. The survival probability of the ground state, in particular, decays with $\epsilon^{2}$ at a rate

[TABLE]

which, for a given $\omega_{1}$, increases for faster rotations (larger $\dot{\theta}$ and $\omega_{2}$).

V Discussion

Controlling the motion of quantum particles is needed to manipulate them for fundamental science studies and to develop different quantum technologies. In particular, operations which are fast, robust, and do not leave residual excitations are typically preferred. Here we focused on rotating arbitrary states of a single particle in an anisotropic harmonic trap using the rotation speed and rotation time as the only control parameters. By “rotating” a state here we mean to end at a time $t=T$ with a particle/trap configuration which is identical to the one at time $t=0$ but rotated by some angle $\theta_{f}$ in the laboratory frame. As an inverse problem, even such a simple system and operation involves considerable complexities. Since normal modes cannot be found by a point transformation, we have first performed a non point (but canonical) transformation to find the normal modes for constant rotation speed. Based on the normal mode analysis we apply a shortcut-to-adiabaticity protocol in which any initial state becomes its rotated version in the final trap. Minimal times are found and a stability analysis with respect to time errors is performed.

We may envision several worthwhile and natural extensions of this work such as considering anharmonicities, two or more interacting particles in the trap, or, to achieve further flexibility in the rotation times, time-dependent rotation speeds $\dot{\theta}(t)$. This time-dependence makes the $A(t)$ matrix in the Hamiltonian of the rotating frame time dependent, and following the steps in the main text and Appendix B we may perform a time-dependent symplectic transformation and find that the interaction picture effective Hamiltonian will be given by $H_{I}=v^{T}[S^{T}(A-\dot{G})S]v$, see Eq. (48). Finding the time-dependent symplectic transformation $S$ that makes the $4\times 4$ matrix $A^{\prime}=S^{T}(A-\dot{G})S$ diagonal is a challenging open question.

We cannot fail to point out an analogy between the structure of $A^{\prime}$ and the effective Hamiltonian used in superadiabatic iterations to achieve shortcuts Demirplak2008 ; Ibanez2013 . If $S$ is set to diagonalize $A$, rather than the whole matrix $A^{\prime}$, two uncoupling strategies are: to ignore the inertial term ${\cal I}=-S^{T}\dot{G}S$ because it is small (this is analogous to an adiabatic approximation), or to compensate it exactly with $-{\cal I}$ (this is analogous to counter-diabatic driving). However implementing such a compensating term is often challenging in practice, in this case it implies crossed operator terms. A third route is to apply the next “superadiabatic” iteration, i.e., to find an $S^{\prime}$ that makes ${S^{\prime}}^{T}A^{\prime}S^{\prime}$ diagonal, which produces a term ${\cal I}^{\prime}=-S^{\prime T}\dot{G}S^{\prime}$ in the new Hamiltonian. Further iterations would repeat the same scheme but they do not need to converge so there may be an optimal iteration. Alternatively the coupling term may be approximated to achieve convergence Theis2018 . All this is very intriguing and will be explored elsewhere.

Acknowledgements.

We are grateful to K. Takahashi for discussions in early stages of the work. This work was supported by the Basque Country Government (Grant No. IT986-16).

Appendix A Laboratory frame

The Hamiltonian for a particle of mass $m$ in a two-dimensional anisotropic harmonic potential with axial frequencies $\omega_{1}$ and $\omega_{2}$ and with a time varying orientation angle $\theta(t)$ (i. e., which is rotating around the $z$ axis with angular velocity $\dot{\theta}(t)$) is given in laboratory $\{x,y\}$ frame by

[TABLE]

Defining the rotated coordinates and momenta by the relations

[TABLE]

with $R(t)$ being the usual rotation matrix

[TABLE]

the new Hamiltonian is given by

[TABLE]

with $L_{z}=\tilde{q}_{1}\tilde{p}_{2}-\tilde{q}_{2}\tilde{p}_{1}$. This last term, which couples coordinates and momenta, accounts for the inertial effects that arise due to the time-dependent canonical transformation applied.

Appendix B Quantum unitary transformations

It is also instructive to set a quantum description by means of a unitary transformation of the Hamiltonian. As it is well known from group theory, the generators of symplectic matrices are symmetric matrices in the sense that any symplectic matrix $S$ can be written in terms of its generator $G$ as $S=e^{2JG}$, $G$ being a symmetric matrix and $J$ the symplectic matrix (6). Let us define the unitary operator

[TABLE]

where $v^{T}$ is now regarded as a vector of operators $v^{T}=(\widehat{q}_{1},\widehat{q}_{2},\widehat{p}_{1},\widehat{p}_{2})$. The unitarily transformed, interaction picture Hamiltonian will be given by

[TABLE]

where the last term arises due to the possible time dependence of the unitary transformation. For a quadratic Hamiltonian with the form $H=v^{T}Av$, see Eq. (4), and the unitary operator $\mathcal{U}$ defined by (46), it can be shown that the above effective Hamiltonian is given by

[TABLE]

Details of this calculation are given in Appendix C.

In a time independent scenario, where $\dot{G}=\dot{S}=0$, we have an uncoupled (i. e., without cross terms) effective interaction picture Hamiltonian

[TABLE]

since $S^{T}AS$ is a diagonal matrix as shown in Sec. III. Indeed, the inverse unitary transformation $\mathcal{U}^{\dagger}(...)\mathcal{U}$ maps all the components $v_{j}$ to $V_{j}$,

[TABLE]

so that $H$ is recovered,

[TABLE]

In summary, the same symplectic transformation that diagonalizes the classical Hamiltonian matrix provides as well a quantum Hamiltonian written as a sum of quadratic operators without cross terms.

If the symplectic transformation $S$ depends on time, the extra term $-v^{T}\left(S^{T}\dot{G}S\right)v$ has to be included in the effective Hamiltonian to account for the inertial effects. For a time dependent transformation, one would have to symplectically diagonalize the full matrix $A-\dot{G}$.

Appendix C Detailed calculation of Eq. (48)

Let $\mathcal{A}$ and $\mathcal{B}$ be two real symmetric matrices. Taking into account that the position-momentum commutators $[q_{j},p_{k}]=i\delta_{jk}$ can be summarized as $[v_{j},v_{k}]=iJ_{jk}$, one can find the relation

[TABLE]

where $[.,.]_{n}$ denotes the $n$th nested commutator between the involved operators. Using this result, the two terms in the effective Hamiltonian (47) will be calculated separately:

(i) The first term $\mathcal{U}H\mathcal{U}^{\dagger}$ can be calculated using the Baker-Campbell-Hausdorff (BCH) formula and the previous result (52) to sum the series expansion. For the unitary operator defined in (46) and a Hamiltonian with the form (4) we have

[TABLE]

(ii) To calculate the second term $i\mathcal{\dot{U}}\mathcal{U}^{\dagger}$, we must be careful when computing the time derivative of $\mathcal{U}$, since it involves not-commuting operators Hall2015 ,

[TABLE]

Here, again, Eq. (52) has been used to sum the series expansion.

The sum of these two terms leads finally to the interaction picture effective Hamiltonian (48)

[TABLE]