Invariant-based optimal composite stimulated Raman exact passage

Shortcuts to adiabaticity have been put forward for accelerating slow adiabatic passages in various quantum systems with tremendous applications for performing quantum information processing tasks. In this paper, we propose a hybrid protocol to achieve stimulated Raman exact passage (STIREP) by combining invariant-based inverse engineering, optimal control, and composite pulse approaches. We first derive the general solution and the corresponding pulse shapes by invariant-based inverse engineering. Counterintuitive and optimal (intuitive) pulse sequences are formulated in this context and incorporated into composite sequences. Such composite STIREP not only features robustness against the fluctuation of laser intensity, but also reduces the operation time and energy cost.


Introduction
Coherent manipulation and preparation for quantum states with high fidelity are requisite for many applications ranging from quantum information processing to control of chemical interaction [1][2][3][4][5]. Among all the techniques, resonant pulses, rapid adiabatic passage (RAP), stimulated Raman adiabatic passage (STIRAP), and their variants provide full population transfer in quantum two-or three-level systems with different advantages and disadvantages [6][7][8][9][10][11]. Resonant pulses are usually fast but sensitive to parameter variations, while adiabatic passages are in principle robust but slow and inaccurate. To remedy this, different approaches, such as shortcuts to adiabaticity (STA), optimal control, and composite pulses [12][13][14][15][16][17], have been proposed to combine the best of the two worlds. Some techniques sharing the concept of STA [18,19] include counter-diabatic driving, fast-forward scaling, inverse engineering based on the Lewis-Riesenfeld dynamical invariant, and single-shot shaped pulses [20,21], also combined with optimal control [22].
Quantum two-or three-level systems are ubiquitous in atom, molecular, and optical physics and solid-state devices. Particular systems might require specific techniques for population transfer in the presence of different types of inevitable systematic error or noise. These processes have been widely studied, and there are substantial theoretical discussions and experimental demonstrations of their efficiency for producing complete population transfer. In STIRAP, the Stokes pulse drives the transition between the initially unpopulated levels | ñ 2 and | ñ 3 . It coherently prepares the system and partially precedes the overlapped pump pulse (driving the transition between the initially populated level | ñ 1 and the intermediate level | ñ 2 ). Such a sequence is termed counterintuitive. Population transfer from level | ñ 1 to level | ñ 3 is approximately achieved by adiabatic passage along a single instantaneous dressed (dark) state. Transient population arising in the intermediate state | ñ 2 during the adiabatic transfer is small or even negligible. This technique quickly attracted considerable attention as a promising control tool for quantum information processing (QIP) [23][24][25][26]. QIP demands very high fidelity of operations, with the permissible error at most 10 −4 , which is difficult to achieve with the standard STIRAP, because due to its adiabatic nature it approaches unit efficiency only asymptotically as the temporal pulse area increases. For the usual Gaussian shape pulse, the necessary area for the 10 −4 benchmark is so large that it may break various restrictions in a real experiment. Nevertheless, the adiabatic process is slow and is therefore likely to be affected by decoherence, dissipation, or systematic error and not lead to an exact transfer. In order to solve these problems, several scenarios have been proposed to speed up STIRAP with an ultrahigh fidelity [27][28][29][30][31][32].
Motivated by the current experimental demonstration of composite STIRAP in a rare-earth doped solid [33], we hybridize the composite method with pulses designed by invariant-based inverse engineering, including exact counterintuitive or optimal (intuitive) sequences, to derive an exact, fast and robust technique. The inverse engineering approach based on the Lewis-Riesenfeld theory allows one to define dynamical invariant and eigenmodes on which the dynamics can be decomposed [34]. We connect the exact single-mode dynamics to the (approximate) adiabatic passage along a dark state resulting from a counterintuitive pulse sequence (STIRAP). We derive from a multimode decomposition with a parametrization linear in time a dynamics featuring a counterintuitive pulse sequence, which is shown to be the opposite sequence with the same pulse shapes as the area-optimal sequence. This linear multimode driving and the area-optimal pulses are implemented in composite sequences. We find that the invariant-based composite STIREP takes less and less time compared to composite STIRAP for more and more sequences.
In section 2, we define the system, and review the technique to design pulses by the inverse engineering approach based on the Lewis-Riesenfeld dynamical invariant, for completeness. In section 3, we derive explicit solutions for pulses that are optimal with respect to energy (or equivalently to time for bounded pulses) or to pulse area. Section 4 is devoted to the comparison of three composite protocols. The discussion in section 4 concludes the paper.

The model and Hamiltonian
The traditional STIRAP process is controlled by two counterintuitive Gaussian pulses, as shown in figure 1(a). The pump pulse connects the ground initial state | ñ 1 and excited state | ñ 2 at frequency ω p . The Stokes pulse links the intermediate state | ñ 2 and final target state | ñ 3 at frequency ω s . The energies w  1 , w  2 , and w  3 correspond to the discrete states | ñ 1 , | ñ 2 , and | ñ 3 , respectively. The corresponding Hamiltonian, in the rotating-wave approximation at resonance, ω 1 +ω p = ω 2 =ω 3 +ω s , reads and Ω s (t) denote the pump and Stokes Rabi frequencies, respectively, considered real functions.

The invariant
The Lewis-Riesenfeld theory [34] allows one to expand exactly the solution | ( ) y ñ t on dynamical modes | ( ) f ñ t n , which are defined as eigenvectors of the dynamical invariant I(t), or equivalently as Its eigenvalues are real and time independent and the eigenvectors form a complete orthonormal basis, which allows an The Hamiltonian (1) features SU(2) dynamical symmetry: where K 1 , K 2 , and K 3 are the spin-1 angular-momentum operators satisfying the commutation relations The dynamical invariant is also taken as an element of this algebra: from which, using the definition of the invariant and denoting , (where the superscript T denotes the transpose), we derive with the Hermitian matrix To parameterize the three angles α i of expansion (9), we need two (time-dependent) angles j, θ and a constant α 0 in order to satisfy (12), a a a a + + =  Adiabatic passage assumes a slow evolution of the parameters such that the dynamics projects approximately along a single where θ is the mixing angle, q = W W tan p s , which connects the initial with the final states. We notice that | ( ) f ñ t D is a particular case of the single-mode state (16) with j=0. The initial state | ñ 1 can in principle connect to the dark state in a counterintuitive sequence, i.e. when the Stokes precedes the pump pulse (i.e. θ going from 0 to π/2), but the dynamics cannot permanently project exclusively on it since the dark state is connected to the other bright states through the nonadiabatic couplingq in the adiabatic frame. This non-adiabatic coupling can be made small for a slow variation of the parameters, which defines the adiabatic passage. The Lewis-Riesenfeld theory allows one to derive an alternative dynamical basis, on which one can project at all times the dynamics exactly from the initial condition.
One can define a single-invariant-mode driving, or in short a single-mode driving, by imposing at the initial time , which then drives the dynamics into the target state, by imposing the final boundary θ(t f )=π/2 and ( ) at time t f . These boundaries have to be taken into account in the definition of the pulses (14) to ensure the desired dynamics. Imposing the complete dynamics occurring along the dark state would imply j=0 at all times, which would correspond to infinite pulse amplitudes from equations (14). Single-mode dynamics thus needs j ¹ 0 during the dynamics. This single-mode dynamics can be interpreted as an exact passage as opposed to the adiabatic (i.e. approximate) passage along a single adiabatic state. In addition to producing an exact passage, such dynamics can also feature certain optimization criteria or specific properties as studied below by an appropriate time parametrization of the angles θ(t) and j(t).

Multimode driving
One can more generally derive multimode solutions [29]. We analyze for instance a three-mode solution, still using θ(t i )=0, and denoting the initial boundary ( ) j e = t i , according to (4): i and the Lewis-Riesenfeld phase ξ 0 =0. For simplicity we consider a constant j and the linear parametrization and θ(T)=π/2: t sin and at final time The complete population transfer is then finally achieved when ( ) The resulting pulses, derived from (14), feature a counterintuitive sequence. The smallest pulse energy required for the complete transfer in this case is for k=±1. We refer to this dynamics as a linear multimode driving. Alternatively, for a given peak Rabi frequency Ω 0 , the fields (for k=±1) can be written as One can emphasize that the derived sequence 'sin cos' (for the pump and Stokes, respectively) is counterintuitive, and is the opposite sequence with the same pulse shapes as the optimal pulses (with respect to energy) 'cos sin' derived below in section 3. This is of interest because counterintuitive sequences are known as featuring a relatively low transient population in the excited state, which is usually a lossy state, and a certain degree of robustness.
As shown below numerically, such multimode driving provides a fast and robust protocol. It can be explained by a combined resonant and adiabatic process; see appendix A.

Optimal pulses
In the resonantly driven three-level system (1), the optimal pulses leading to the transfer have been already addressed [35]. Here we rederive them adding the explicit formula. We denote , the total duration of the process, i.e. the transfer time. Optimality can be analyzed in terms of pulse energy (defined through u p and u s for simplicity) T p s 0 2 2 or in terms of generalized pulse area By using Pontryagin's maximum principle, see appendix B, we obtain for the energy-optimal pulses One can remark that minimizing the cost E is equivalent to minimizing the transfer time for bounded controls, Denoting by Ω 0 the peak of the pump and Stokes Rabi frequencies, which is 5 times smaller than the linear multimode transfer time. These pulses (33) represent time-optimal pulses. As emphasized in section 2.4, the derived intuitive sequence cos sin is the opposite sequence with the same pulse shapes of the linear multimode driving.
If one considers the minimization of the effective pulse area (30), defining the area-optimal pulses, one can remark that this quantity is time-reparametrization invariant as it leads to the same trajectory in the (θ, j) space for any time reparametrization, which is given by equation (B.15). The energy-optimal pulses (B.25) feature an example of an areaoptimal pulse with which one can determine the resulting optimal area: Another convenient parametrization, symmetric around t=T/2 (i.e. such that j(T − t)=j(t)), such that j(T/2)=π/3, is given by Using the (dimensionless) quantity α such that  a 1 allows us to start and end the process with pulses close to zero. One can choose for instance α=36. An alternative parametrization, leading to smooth pulses, which start and end exactly at zero, is given by ,0o t h e r w i s e ) also leads to complete transfer for the same pulse area A p = , with the transfer time p = W T 2 2 0 . As a consequence, one can remark that all these pulses of small pulse area (π or less for the optimal one) feature intuitive sequences (or fully overlapping pulses), with a relatively large transient population in the excited state.

Composite protocols
The implementation of the previously defined optimal pulses induces a dynamics sensitive to systematic errors. Composite pulses are designed to solve this issue. They use a sequence of pulses with appropriate phases, The composite STIRAP derived in [17] implements a high order fidelity, which is robust to variation of systematic errors, with universal phases, i.e. independent of the specific pulse shape, delay, and area. The dynamics with a sequence of N STIRAPs (with N an odd number) is represented by the propagator whereŨ is a backward STIRAP propagator from state | ñ 3 to state | ñ 1 , i.e. with the exchange of the order of the pump and Stokes pulses. In this section we compare the implementation of the composite STIRAP for three different pulse shapes: the Gaussian, area-optimal, and linear multimode pulses, as illustrated in figures 1(b)-(d).
Protocol 1. The Gaussian pulses are the most widely used form in STIRAP; see figure 1(b). They can be written as where τ is the delay, the σ is the pulse width, and Ω 0 is the peak amplitude of both Rabi frequencies. A positive delay defines a so-called counterintuitive sequence, since the Stokes precedes the pump pulse. We emphasize that such a counterintuitive sequence is of importance compared to an intuitive sequence when the upper state is lossy, since it leads to much less transient population in this state. The fidelities corresponding to the change of the pulse amplitude from Ω 0 to Ω 0 (1+η), via a systematic error η, are shown in figure 2 (solid red lines). We can see that the fidelities get better with respect to the systematic error η for a larger sequence number N. However, the Gaussian pulses feature a large area and a large energy compared to the other invariant-based and optimal pulses as analyzed below. Protocol 2. We use parametrization (36) with geodesic (B.20), (B.23), designed to minimize the pulse area (30). Setting the same peak Rabi frequency as for the Gaussian pulses of Protocol 1, we obtain the transfer time T area ≈ 2.8 μs for the peak Rabi frequency Ω 0 =4 MHz. With the area-optimal pulses shown in figure 1(c), the fidelities versus the variation of the systematic errors for different N are shown in figure 2 (dotted black line).
Protocol 3. In this protocol, we consider the counterintuitive linear multimode driving pulse sequence (26) with k=1, i.e. of minimum energy; see figure 1(d). The energy associated with these pulses is E , for a given interaction time T. We obtain m » T 3 s multi for Ω 0 =4 MHz. The fidelities versus variation of systematic errors are shown in figure 2 (dashed blue line). Even though, as expected, the robustness of area-optimal pulses is worse than the other two protocols for a single pulse, it becomes comparable already for a sequence number = N 5.

Discussion and conclusion
We can compare the transfer time and the energy of the studied protocols for area-optimal, Gaussian, and linear multimode pulses at the same peak Rabi frequency (see figure 3). They are all robust to systematic errors and all feature a high fidelity that can satisfy the quantum information processing constraint [30].
The transfer times of composite STIRAP and composite STIREP are shown in figure 3(a) for different sequence numbers. We can conclude that the area-optimal pulses and linear multimode pulses, even if they are not designed for optimizing the time transfer, are much shorter than the Gaussian shape pulses. The more sequence we use in the dynamics, the more time we can save. The linear multimode pulses are only a little longer than the area-optimal pulses. We remark that, obviously, we could save even more time by using the time-optimal pulses (33). We can observe that the area-optimal pulses need less energy than the linear multimode pulses, see figure 3(b), as expected for intuitive compared to counterintuitive sequences. This means we can combine the area-optimal pulses and composite pulses into a hybrid protocol which features pulses starting and ending at zero, a short transfer time, and a low energy cost (but at the price of high transient population in the upper state, see below).
STIRAP was initially developed for controlling atomic and molecular systems, in which the dark state (18) plays a major role to transfer population over a dissipative intermediate state, rendering the dynamics insensitive to the loss from it (in the adiabatic limit). In order to quantify it, we calculate the time area of the transient population in the excited state as ( ) for a single pulse pair. Such an area P e ≈0.264 for the (counterintuitive) linear multimode driving pulse is twice that, 0.138, for the (counterintuitive) Gaussian pulse, but half that, 0.494, for the (intuitive) area-optimal protocol. Therefore, the linear multimode pulses should find applications in quantum information processing when the upper state is relatively weakly lossy or the process times are sufficiently short.
We have used single-and two-photon resonances in the derived hybrid protocols since they are based on the search for optimal pulse sequences already for a single shot. Detuned excitation corresponding to lower coupling strength would not lead to optimal sequences. In a system featuring large inhomogeneous broadening as in rare-earth doped solids, the use of composite detuned sequences has been shown to be more efficient and more robust [33] since the resonant sequence is sensitive to single-photon inhomogeneous broadening. It would be of interest to reconsider the problem of optimality in Λ systems including this additional effect (through a Lindblad equation with the density matrix formalism) and a single-photon detuning.
Last but not least, recent experiments on STA for speeding up STIRAP demonstrate that our hybrid protocols should be implementable for fast and robust state preparation and also for producing quantum gates with various systems such as cold atoms, superconducting circuits, and solid-state spins [36][37][38].

Appendix A. Connection between multimode driving and resonant pulses
In this appendix, we interpret the linear multimode driving, which provides fast and robust state transfer, in term of a resonantly driven spin-1 particle. To this end, we transform the Hamiltonian (1) into the adiabatic frame , and the matrix elements for the spin-1 particle are In the adiabatic limiṫ  q 1, the precession axis isŴ = J z 0 , so the state vector does not precess, and the state evolution will be along the dark state without any transition. Regarding the multimode driving, we substitute the pulses (26)  In this case, the state vector starts to precess whenq > 0 and W ¹ J z 0 , resulting in the non-adiabatic evolution. The resonant condition, p W = T k 2 0 , which is fulfilled, allows the state vector to be driven back to the target dark state, that is, the ideal target state | ( ) f ñ T D . Therefore, the linear multimode driving can be interpreted as a resonant driving in the adiabatic frame, thus producing a fast and robust state transfer.

Appendix B. Optimal control theory
In order to formulate the problem, one uses the fact that this resonant three-state problem is of SU(2) symmetry. One can separate the dynamics into two independent sub-dynamics, each of them being associated with the real and imaginary parts of the initial condition, here taken as real | ( ) y á ñ = 1 0 1, leading to the system of equationṡ , where the superscript T denotes the transpose.
To determine optimal control fields ( ) t u of a dynamical system˙( ( ) ( )) = t t x f x u ; (of dimension N) with respect to the minimization of a given cost