Transient Non-Confining Potentials for Speeding Up a Single Ion Heat Pump

We propose speeding up a single ion heat pump based on a tapered ion trap. If a trapped ion is excited in an oscillatory motion axially the radial degrees of freedom are cyclically expanded and compressed such that heat can be pumped between two reservoirs coupled to the ion at the turning points of oscillation. Through the use of invariant-based inverse engineering we can speed up the process without sacrificing the efficiency of each heat pump cycle. This additional control can be supplied with additional control electrodes or it can be encoded into the geometry of the radial trapping electrodes. We present novel insight how speed up can be achieved through the use of inverted harmonic potentials and verified the stability of such trapping conditions.


Introduction
Trapped ions are an established platform for realizing high-fidelity quantum information processing [1,2], quantum simulation [3,4], and precision metrology experiments [5,6]. Recently a single ion, trapped in a tapered trap, was employed to realize a single ion heat engine [7,8]. Due to the controllability of the environment this system implements a formidable model experiment for studying thermodynamics at the single particle limit towards the quantum regime. In this paper we study the reverse process, a single ion heat pump, and how this process can be sped up through the shortcut to adiabaticity technique involving the use of invariant-based inverse engineering [9,10]. In the following, as in the single ion heat engine, the ion is confined in a harmonic potential and the motional radial degrees of freedom serve as the working agent, where we consider temperature only in the radial directions. A thermal state that is adiabatically transported along the taper (see Fig. 1) into a region with lower trap frequency attains a lower temperature due to the reduced energy level spacing in the harmonic potential. This mechanism could be used to couple to a reservoir, such as neighbouring ions, to affect cooling by absorbing heat. Thus, a subsequent adiabatic transport back to the starting position at higher confinement results in an increased  Figure 1. Tapered ion trap. The tapered electrodes are supplied symmetrically with radiofrequency voltage for the radial confinement. Endcaps are used to supply axial confinement with dc voltages. During shortcut to adiabaticity protocol the radiofrequency is switched off and the voltage on the endcaps is used to realize the axial confinement or anti-confinement respectively. Due to the short duration stable trapping conditions can be maintained.
temperature. Dumping heat to another reservoir allows one to recool the working agent for starting a new cycle of the heat pump. Speeding this procedure up to increase the heat pumping rate through the use of e.g. bang-bang transport is typically limited by the condition of performing the change of the radial trapping frequency adiabatically. In the following, we will describe how shortcuts to adiabaticity can be employed to go beyond this limitation [11,12,13,14,15], in particular the invariant-based inverse engineering approach will allow the design of protocols by controlling the radial trapping frequency with external electrode voltages. One possibility is to control the radial trapping frequency by varying the radiofrequency amplitude which is symmetrically supplied to the tapered electrodes of the ion trap. The speed up in this case is limited by the fact that the radial confinement should be sustained. A further speed up would be possible if the trapping potential can be inverted. This is achieved by switching off the radiofrequency confinement for a short period and using the radial DC potentials generated by the end-cap electrodes to supply a specially designed time varying radial quadratic potential. Due to Laplace's equation, a confining potential in one direction leads necessarily to repelling potentials in the other two directions, or vice versa. We present a shortcut of short duration, which helps both to achieve high cooling rates and avoid losing the ion from the trap. For typical trapping frequency changes from 3 MHz to 1 MHz, a shortcut duration of 20 ns can be achieved, through the use of nonconfining potentials. In order to avoid instability due to micromotion, it is necessary that the radiofrequency period is shorter than the shortcut duration. ‡ Numerical simulations confirm stable trapping conditions despite the inverted trapping potentials over short time periods. It is important to note that the speed up is only limited by the ‡ Note that the switching should be synchronized to the radiofrequency phase maximal voltages and the currents which can be applied to the electrodes. The single ion heat pump could be an important method for lowering temperatures in a trapped ion based quantum information processor and the speed up described could help to compete against deleterious heating rates.

Invariant-based inverse engineering for mixed states
Closed quantum systems follow a unitary dynamics described by the Liouville equation of motion whereρ(t) is the density matrix describing the system andĤ(t) the Hamiltonian controlling its dynamics. Related to any Hamiltonian there are dynamical invariants of motion [16] i with constant expectation values, -i.e. quantities preserved by the dynamics generated by (1). The invariant expands an orthonormal basis |φ n (t) with constant eigenvalues λ n ,Î (t) = n |φ n (t) λ n φ n (t)|.
In this basis the density matrix elements ρ lk ≡ φ l (t)|ρ(t)|φ k (t) are calculated from [17] ρ where the populations remain constant and the off-diagonal elements depend on the difference of time derivatives of two Lewis-Riesenfeld phases [16]. A simpler derivation of the Lewis-Riesenfeld relation for pure states is done in Appendix A. From Eq. (4) we observe that a system initialized in an eigenstate of the invariant will remain in the same instantaneous eigenstate without transitions, imposing the so-called frictionless conditions [Ĥ(0),Î(0)] = [Ĥ(t f ),Î(t f )] = 0, we ensure that the system starts and ends as an eigenstate of the Hamiltonian without unwanted excitations. A perfect state transfer fromĤ(0) toĤ(t f ) is designed by first choosing properlyÎ(t) and then reverse engineering the dynamics to deduceĤ(t). In particular, for an effectively 1D time dependent harmonic potential an associated dynamical invariant (2) reads [18] where b(t) is a free function of time satisfying the Ermakov equation [19] b being ω 0 the initial frequency of the oscillator at time t = 0. The frictionless conditions fulfilling the previous six boundary conditions will produce a perfect control, see Eq. (7) driving each Fock state |n(0) , to the corresponding Fock state |n(t f ) independently of the process time t f . More details in Appendix B can be found. Note that typically for ultrafast processes, very short t f values, ω 2 0 /b 4 <b/b and the trapping parabola becomes a repeller potential. The stability and experimental implementation of such scenario will be deeply analyzed in the following sections.

Coherent states
The previous protocol (9) is not only valid to connect single |n to |n Fock states but also coherent states [20] These are pure states forming a linear superposition. As at initial time the frictionless conditions guarantee thatĤ andÎ share a common basis |φ n (0) = |n(0) and according to Eq. (4), or simply (A.4) as the system is pure, this initial state |ψ(0) = |α(0) will evolve to [21] |ψ ,Î(t f )] = 0 guarantees |φ n (t f ) = |n(t f ) , thus the system ends as also a coherent state with frequency ω f .

Thermal states
From the set of Eqs. (4) we observe that any system that initially is diagonal in the basis expanded by the eigenstates of the invariant will keep its populations constant during the whole process. Moreover, imposing [Î(t b ),Ĥ(t b )] = 0 at t b = 0, t f the initial and final states will be also diagonal in the energy basis expanded byĤ(0) andĤ(t f ), which is the case for thermal states. Considering the time-dependent harmonic oscillator (5) and if initially the system is assumed to be the thermal stateρ(0) = exp(−β 0Ĥ (0))/Z, with Z a normalization constant, initial inverse temperature β 0 , and ω(t = 0) = ω 0 , by changing ω(t) according to Eqs. (9) and (8) the system will evolve reaching the final thermal stateρ(t f ) = exp(−β fĤ (t f ))/Z , corresponding to aĤ(t f ) with frequency ω(t f ) = ω f and a cooling/heating β f = γ 2 β 0 .

Quantum dynamical evolution of Gaussian states
Note that both coherent and thermal states are Gaussian states, -i.e. the symmetric Wigner function is Gaussian, x ≡ (q, p) corresponds to the eigenvalues of the quadrature operatorŝ x ≡ (q,p). Consequently, the density operatorρ has a one-to-one correspondence with the first and second-order statistical moments of the state,ρ ≡ρ(x, V) [22]. The first moments are called the displacement vector, or simply the mean valuē and the second moment, called covariant matrix, with generic element where ∆x i =x i − x i and {Â,B} =ÂB +BÂ. In particular, for coherent and thermal states of a harmonic oscillator these momentsx and V are constructed from the set of operatorsX ≡ (q,p,q 2 ,p 2 ,qp +pq), and the Wigner function is reconstructed, with x T , the transpose of x and V −1 the inverse matrix of V. In order to describe the dynamical evolution ofρ, or equivalently W (x), it is enough to describe the evolution of the set of observablesX to reconstruct the state using Eqs. (15) and (16), avoiding the use of wave packet propagation. This is done within the Heissenberg representation . Note that the set of five operatorsX form a closed Lie algebra, as the Hamiltonian (5) of a harmonic oscillator is a linear combination of someX i elements, the dynamical equation of motion (17) is also closed to the algebra. Consequently, the evolved stateρ(t) remains Gaussian during the whole evolution.

Robustness improvements
The main source of imperfection in the experimental implementation of the shortcut is produced by the time variation of the control ω 2 (t). Controlling this by the pseudopotential through dynamic change of the amplitude of the radio-frequency voltage has the disadvantage that non-confining potentials cannot be supplied. Amplitude control of this voltage is technologically more involved and intrinsically limited by the period of the radiofrequency. Thus the biggest speed up potential and controllability is obtained by controlling the DC potentials by low-noise high-speed arbitrary waveform generators. If radiofrequency confinement is kept on very accurate timing and high voltages are needed. In order to allow for a reliable control of the confinement, we therefore switch off the radiofrequency drive during the control period. This can be efficiently achieved by a solid state radiofrequency toggle switch [24] directly after a high voltage rf generator [25]. In many cases the high voltage rf generator is replaced by a low voltage radiofrequency generator with a subsequent radiofrequency amplifier with 50Ω impedance. Impedance matching is then achieved with a helical responators which additionally transforms the radiofrequency voltages. In these cases an ultra low resistance toggle switch has to be used directly after the helical resonator with one terminal connected with the trap electrodes and the other connected with a circuit of equivalent impendance. Anharmonicities of the trapping potentials can be neglected as the ion is kept at the extremal point of the harmonic confinement at all the time.
Thanks to the freedom in the construction of the shortcut protocol at intermediate time more constraints such as minimizing dω 2 /dt ≡ ∂ t (ω 2 ) due to experimental limits can be realized. This is originated from the slew rate and bandwidth limit of digital analog converters and power amplifiers. The minimization of ∂ t (ω 2 ) can then be performed by optimal control techniques but the boundary conditions for b could violated. Discontinuities inḃ,b would be unfeasible due to the requirement of instantaneous jumps in the control voltages.
As an example, minimizing max |∂ t (ω 2 )|, the maximum value of ∂ t (ω 2 ) in the interval t ∈ [0, t f ], will reduce the power employed by the control protocol improving the heat extraction process. Defining C(t) = ω 2 (t), the extreme condition that minimizes dC/dt = 0 is satisfied by the useless control C(t) = ω 2 opt (t) = const. The mean value theorem provides a useful bound for the instantaneous maximum value of the control. and C(t f ) = ω 2 f the maximum of its derivative must be where the equality holds for the ω 2 (t) = ω 2 0 + (ω 2 f − ω 2 0 )t/t f control. However, the resulting b(t) deduced from Eq. (7) does not satisfy the six frictionless boundary conditions (8). As result discontinuities inḃ andb at t = 0 and t f should be applied requiring instantaneous switches in the controls. In order to avoid discontinuities hardly resolved experimentally we use the non-uniqueness of b(t) to add extra-parameters a i in the interpolation of b(t) = i a i t i to ensure (8) and using Eq. (9) create controls ω 2 (t; a i ) such that the value of ∂ t (ω 2 ) is controlled through thee extra-parameters a i [17,26]. By using gradient descent methods ω 2 (t; a i ) is optimized. As an example, for an expansion process of 20 ns see Fig. 2, the addition of the extra-coefficient a 6 t 6 in the interpolation of b(t) allows a reduction of max |∂t(ω 2 opt )| max |∂t(ω 2 )| ∼ 0.78 in contrast with a standard 6 order interpolation, see Appendix B. Additionally, this design also reduces the value of max |ω 2 |, thus the protocol improves both the slew rate and power of the required controls. Other sophisticated designs are also possible due to the freedom to interpolate b(t) at intermediate times.

Proposed experimental implementation
In the following, we will consider the 3D-Hamiltonian corresponding to an ion trap symmetrically driven with radiofrequency and end-cap geometry. In order to fulfill Laplace's equation the Hamiltonian describing the trapped ion becomes: withp = (p x ,p y ,p z ), ω z (t) the frequency along the axial z-direction, and ω ⊥ (t) = ω x (t) = ω y (t) = Ω(t) + ∆(t) the radial frequencies produced by the RF and DC voltages in conjunction §. This Hamiltonian has a symmetric radial confinement in the x and ydirections that will be employed as working fluid to produce the heat pump processes. In the following we disregard the effect of control voltages on the longitudinal confinement because the ion is always kept at the extremal point of the longitudinal confinement and we use the longitudinal degrees of freedom as a classical piston being driven. Under this prescription the radial Hamiltonian reads, withp ⊥ = (p x ,p y ). Definingr ⊥ = (x,ŷ) we observe that this radial Hamiltonian has the same structure as Eq. (5), consequently the radial frequency can be modified from the frictionless boundary conditions (8) with a radial expansion/compression ratio γ ⊥ = (ω ⊥,0 /ω ⊥,f ) 1/2 , the same for both the x and y axes.
The shortcut to adiabaticity will be implemented by common voltages on the endcap electrodes of an ion trap, while the dominant radiofrequency saddle potential has been momentarily turned off. The differential voltage on the end-caps can be used to control the axial movement of the ion, but can be disregarded here. The radial confinement caused by the radial frequency is only relevant at the turning points of the axial transport, when the ion is coupled to the reservoirs. Alternatively, a linear trap design could be used without a taper, with the radial frequency being switched to different amplitudes in between. The radial trapping potential during the shortcut is applied by a common voltage on the end-cap electrodes, and needs § Note that the trapping frequency caused by the pseudopotential and the DC potentials cannot be simply added especially when large voltages are involved (see equation 11  to be matched to the initial and final confinement provided by the pseudopotential. Laplace's equation and the geometric symmetry specifies that ω 2 is inverted with half the magnitude. We have compared three expansion protocols; shortcut, linear and smooth ramp ω(t) = (ω 0 e Γt 0 + ω f e Γt )/(e Γt 0 + e Γt ) for the cooling of thermal and coherent states, see Fig. 3 .
The initial thermal state is characterized by the statistical momentsX 1 (0) = X 2 (0) =X 5 (0) = 0 and with l 0 = /(2mω ⊥,0 ) and k 0 = m ω ⊥,0 /2 corresponding to aĤ(0) with a frequency ω ⊥ (0) = ω ⊥,0 and inverse temperature β 0 . The target state has similar statistical moments corresponding to a final frequency ω ⊥,f and inverse temperature β f = γ 2 ⊥ β 0 . In Fig. 4a we plot the fidelity F(ρ(t f ),ρ target ) of the evolved stateρ(t f ) compared to the target thermal stateρ target corresponding toĤ(t f ) having a frequency ω ⊥,f . We observe how the shortcut by construction ensures fidelity one independently of the time employed to produce the expansion of the harmonic trap whereas the linear and smooth ramp protocols fail as the process is no longer adiabatic, see insets of Fig. 3.
Similarly, we analyze the three previous protocols for the expansion of a coherent state in the trapping potential (20). The initial state has the statistical moments X 3 (0) =X 2 1 (0) + l 2 0 ,X 4 (0) =X 2 2 (0) + k 2 0 ,X 5 (0) = 4 Re(α 0 )Im(α 0 ) associated withĤ(0) and ω ⊥,0 . AtĤ(t f ) the target state has similar statistical moments with ω ⊥ (t f ) = ω ⊥,f and photon number α f = α 0 e −igω ⊥,0 with g = t f 0 dt /ρ 2 . As for the case of thermal states we observe in Fig. 4b how the shortcut drives the initial system until the desired target state independently of the expansion time t f .  Figure 5. Experimental control sequence. The radial radiofrequency drive is switched off during the application of the shortcut to adiabaticity protocol on the dc electrodes, the shortcut changes the radial confinement. Figure 5 shows the whole control sequence responsible for the shortcut to adiabaticity protocol which includes antitrapping potentials for short compression cycles. The radiofrequency is switched off during that time such that the DC control potentials can be kept at lower voltages. By construction the protocol keeps the fidelity at 1, but stable trapping conditions have to be maintained due to the anti-trapping potentials involved. In Fig. 6 we have verified that indeed phase stable trapping can be maintained due to the shortness of the anti-trapping potentials. We have included in the dynamics the whole experimental control sequence Fig. 5, where the trapping potential is given by Eq. (20) and the micromotion exerted on the ion due to the rfdriving has been taken into account. To include this micromotion, a simulation based on the velocity Verlet method was performed. Both the radiofrequency drive ω RF /(2π) and the axial trapping ω z /(2π) frequencies were set to 100 kHz. In order to avoid instability due to micromotion, the corresponding radiofrequency period is shorter than the shortcut duration produced in 20 ns. For this expansion time (see Fig. 3b), the adiabaticity parameter goes beyond the adiabatic regime for the linear and smooth ramps, thus making the shortcut necessary to ensure a perfect driving. Note, due to the zero-crossings of ω 2 , the adiabaticity parameter diverges, but this does not compromise the effectiveness of the shortcut. This is also apparent in Fig. 6a, where one can observe that a phase relation is maintained before and after the shortcut. In contrast, in Fig.  6b, although the ion remains trapped after the linear ramp the final evolved state is excited. The excitations modify the ion oscillations rotating the axis of the ellipse with respect to the original direction that corresponds to the final unexcited state.

Discussion
Making use of shortcuts to adiabaticity we have improved the efficiency of a heat pump for a single ion. The expansion protocol allows ultra-fast and high-fidelity processes through the use of transient non-confining potentials. The stability of the potential has been analyzed and the experimental feasibility discussed. The shortcut control has been improved according to experimental constrains, in particular minimizing the required power and thus reducing the effect of noise produced by the controls. These improved controls could be useful since efficient heat pump extraction protocols provide new cooling mechanisms and constitute the basis of stroke heat engines/refrigerators [28] allowing us to test the laws of thermodynamics and get closer to the absolute zero temperature [29] in the single particle domain. The possibility to design different refrigerators based on the Otto cycle according to the performance of each stroke offers a new venue to design new heat pump protocols. As example, not only optimizing the compression/expansion strokes but also designing efficient trapping potentials at the isochores for the thermalization processes by controlling the trap frequencies ω(t).
Additionally, using the temperature of the bath as a control could lead to new shortcut to adiabaticity such that the optimal performance of the heat pump would be achieved. These extensions are of additional interest also to different refrigerators types like the continuous refrigerator where the ion is in continuous contact with the bath [30], which might be easier to implement experimentally.

Appendix A. Invariant-based inverse engineering for pure states
Related to any HamiltonianĤ(t) there are invariants of motion [16] i with constant expectation values for any wave function satisfying the time-dependent Schrödinger equation The invariant expands an orthonormal basis |φ n (t) with constant eigenvalues λ n , These states can be used to express the dynamical wave function as a linear superposition of the "dynamical modes" c n being the constant time-independent coefficients of the expansion with the Lewis-Riesenfeld phases defined as [16] α n (t) = 1 t 0 dt φ n (t ) i ∂ ∂t −Ĥ(t ) φ n (t ) . (A.5) Suppose that we want to drive the system by changing a control parameter (t) from an initial HamiltonianĤ( (t = 0)) with (t = 0) = 0 to a final configuration governed bŷ H( (t = t f )), where (t = t f ) = f in such a way that the populations in the initial and final instantaneous basis are the same but transitions at intermediate times are allowed †. Our aim is to deduce the time dependency of the control (t) that enables us to perform this task. We assume that the structure of the Hamiltonian controlling the dynamics of the system is known, i.e., the dependency ofĤ =Ĥ( ) as a function of is known but not the time dependency of = (t), which is our target. OnceĤ( ) is known, a related invariant can be found using Eq. (A.1) and subsequently its eigenvectors |φ n ( ) ‡ and eigenvalues deduced. Then the state of the system at any time will be described by Eqs. (A.4) and (A.5) evolving during the whole process as a linear combination of the dynamical modes. Generally, notice thatÎ(t = 0) does not commute withĤ(t = 0), then the eigenstates of the invariant do not coincide with those of the Hamiltonian. A similar situation occurs at t = t f . Imposing the frictionless conditions [Î(0),Ĥ(0)] = 0 and [Î(t f ),Ĥ(t f )] = 0 will allow us to deduce a control strategy = (t) that guarantees a perfect state evolution without final excitations such that the initial and final states are compatible with the initial/final Hamiltonians [9,10].
H(t). Then any b(t) fulfilling the previous six conditions at the extremes will produce the desired driving between the states ofĤ(0) andĤ(t f ) independently of the expansion/compression time t f . In order to satisfy (8) we interpolate b(t) = 5 i=0 a i t i with at least the same number of coefficients a i as conditions over b. Solving for the coefficients we find b(t) = 6(γ − 1)s 5 − 15(γ − 1)s 4 + 10(γ − 1)s 3 + 1 where s := t/t f . We can take advantage of the non-uniqueness of b(t) at intermediate times to design more sophisticated b(t) functions and additionally minimize or impose possible experimental constraints [26,17,34,35,36].