Implementation of STIRAP in degenerate systems by dimensionality reduction
We consider the problem of the implementation of Stimulated Raman Adiabatic Passage (STIRAP) processes in degenerate systems, with a view to be able to steer the system wave function from an arbitrary initial superposition to an arbitrary target superposition. We examine the case a -level atomic system consisting of ground states coupled to a common excited state by laser pulses. We analyze the general case of initial and final superpositions belonging to the same manifold of states, and we cover also the case in which they are non-orthogonal. We demonstrate that, for a given initial and target superposition, it is always possible to choose the laser pulses so that in a transformed basis the system is reduced to an effective three-level system, and standard STIRAP processes can be implemented. Our treatment leads to a simple strategy, with minimal computational complexity, which allows us to determine the laser pulses shape required for the wanted adiabatic steering.
Destructive quantum interference allows the control of the properties of quantum systems as well as their evolution in time. Many important features can be understood by considering a three-level atom consisting of two ground states coupled to a common excited state by two laser fields. Whenever the detuning between the two laser fields matches the ground state splitting, the system is prepared into a superposition of the ground state which is decoupled from the laser radiation - the so called dark state moi (); ari76 (); ari (). This also allows the control of the absorptive and dispersive properties of a medium consisting of three-level atoms harris_rev ().
Additional interesting features appear for time dependent laser fields. In this case the dark state becomes time-dependent, and this allows one to control the quantum state of the atom via adiabatic following of the dark state - the so called Stimulated Raman Adiabatic Passage (STIRAP) rmp98 (). STIRAP is not directly applicable to degenerate systems as in this case the system may have several dark states, so that the non-adiabatic coupling between them is not negligible and the adiabatic theorem does not directly apply. Several strategies have been developed for the adiabatic steering of degenerate quantum systems in different configurations. This led to a number of schemes for the creation and manipulation of superpositions shore95 (); martin95 (); kis01 (); kis02 (); kis04 (); boozer (); shapiro (), as well as schemes for the implementation of quantum gates based on STIRAP renzoni02 (); goto04 (); jauslin05 (). Of particular relevance for the work presented here, is previous work dealing with an atomic system consisting of a multiplet of degenerate ground states coupled to a common excited state by laser pulses. Solutions for the steering of arbitrary superposition were identified by using numerical optimal control techniques kis02 (). Analytic solutions were also found for specific configurations kis01 (). Analytic solutions of the nondegenerate quantum control problem in the case of arbitrary initial and final superpositions belonging to different manifold of states were given in Ref. shapiro ().
In this work we consider the problem of steering the atomic wave function by STIRAP in an -level atomic system consisting of ground states coupled to a common excited state by laser pulses. We analyze the general case of initial and final superpositions belonging to the same manifold of states, and we cover also the case in which they are non-orthogonal. We demonstrate that, for a given initial and target superposition, it is always possible to choose the laser pulses so that in a transformed basis the system is reduced to an effective three-level system, and standard STIRAP processes can be implemented. Our treatment leads to a simple strategy, with minimal computational complexity, which allows us to determine the laser pulse shapes required for the wanted adiabatic steering.
This work is organized as follows. In Sec. II we define the system of interest, and state the problem under consideration. In Sec. III we derive the conditions for the reduction of the system to an effective three-level system. We then specify the conditions on the laser pulses for the transfer from a given initial superposition to a wanted final superposition. In Sec. IV we demonstrate the validity of our approach with numerical simulations. Conclusions are drawn in Sec. V.
Ii Statement of the Problem
We consider an level atomic system with degenerate ground states coupled to a common excited state by laser fields of equal frequency , taken to be equal to the atomic transition frequency. This is the same model considered in Refs. kis02 (); kis01 () to understand the mechanism of STIRAP processes in systems with a degenerate dark state subspace. The scheme finds direct application in the creation and manipulation of atomic systems. For it directly describes an atomic system with three degenerate ground states coupled to a common excited state by fields of different polarizations. The procedure identified in this work also applies, for larger , to level schemes including non degenerate ground state sublevels, e.g. sublevels of different hyperfine states. In this case each level is individually resonantly coupled to a common excited state by a laser field of appropriate frequency and polarization. This gives rise to a degenerate dark space for which the procedure of implementation of STIRAP identified in this work applies.
In a frame rotating at frequency , the Hamiltonian in the rotating-wave approximation (RWA) can be written as
where is the time-dependent Rabi frequency for the transition . The Rabi frequencies are taken as real without loss of generality, as any complex phase can be re-absorbed into a re-definition of the basis states. The interaction scheme is represented in Fig. 1. The system has a subspace of superposition of ground states decoupled from the laser fields (”dark state subspace”) of dimension kis02 ().
We aim to determine a set of laser pulses which drives the atomic system from an arbitrary initial ground state superposition
to an arbitrary final target ground state superposition
We restrict our analysis to the case of temporal evolution determined by the adiabatic following of a dark state, without any mixing with the excited state.
Iii Theoretical analysis
For clarity, we consider separately the two cases of orthogonal and non-orthogonal initial and target states. We first discuss the orthogonal case in Sec. III.A while the general case is discussed in Sec. III.B
iii.1 Case I: orthogonal initial and target states
We consider the case of orthogonal initial and target states
We introduce a new atomic basis for the ground state subspace in which the first two states are the initial and the target states and . The basis is then completed by linear combinations of the original ground states, as it can be obtained by standard Gram - Schmidt orthogonalization procedure:
where the coefficients are determined by the orthogonalization procedure. For notational convenience we rewrite this as
The matrix is orthogonal and the first two rows correspond to the coefficients and of the initial and the target superpositions, respectively.
In the new basis the Hamiltonian reads as
where the transformed pulses are defined as
Notice how the transformed Hamiltonian has the same structure of the initial one. We can thus easily find the dark states associated with (11). We first parametrize the laser pulses in terms of a total amplitude
and angles as
By forming the state
the dark states can be easily obtained (apart a normalization factor) as kis01 ():
Of particular relevance for the following is the first dark state, which reads
In the basis with couplings it is immediate to see that the system can be reduced by an appropriate choice of the laser pulses to a three-level system consisting of the states , so that adiabatic transfer from to can be implemented.
In this subspace, the energy eigenstate corresponding to a eigenvalue is the first dark state . A sufficient condition to remain in this energy eigenstate throughout the evolution is (see e.g. Eq. (6) in schaller2006b ())
and may now serve to find optimized control functions with the side constraints that initially and finally . It should be noted that fully adiabatic evolution may be a rather strict criterion, as it is for practical application only required that the final state of the system corresponds to the final dark state. In situations where the adiabatic preparation time is an issue, it may therefore be favorable to find more optimal control functions minimizing only the final occupations of the other two eigenstates.
Taking further into account typical experimental implementations, it is convenient to choose the transformed Rabi frequencies as
where has to be compatible with the conditions above. As it will be shown, this choice leads to physical laser pulses which are linear combinations of delayed pulses, and are easy to implement. Condition (25c) determines the reduction of the system to an effective three-level system, with the two states (i.e. coupled via a common excited state. The remaining states are spectator ground states not involved in the process. The reduction to an effective system for an appropriate choice of laser pulses is shown in Fig. 1(b).
We can then implement a standard STIRAP process by taking a pair of pulses which satisfy the standard requirements of STIRAP in terms of smoothness, duration, strength and overlap. The system has a dark state, Eq. (23). The temporal dependence of the angle is determined by the temporal-dependence of , . In the specific case
which implies that and . The dark state has thus the properties
Therefore as in standard STIRAP in a three-level system, adiabatic evolution along the dark state will lead to the transfer of the system from to .
The conditions (25) for the transformed pulses are translated for the physical laser pulses as
which is a linear system easily solvable because the coefficients matrix is orthogonal. We stress that each pulse is a linear combination of pulses and . Specific examples will be given in Section IV, which is devoted to numerical solutions of the adiabatic evolution. We also notice that our derivation remains valid for more general parametrizations of transformed pulses, e.g. , , in which case the evolution would remain adiabatic whenever .
iii.2 Case II: non-orthogonal initial and target states
We consider the case in which the initial and target states are not orthogonal:
We introduce a new basis () for the ground state subspace, with the first two basis vectors defined as
and the remaining basis states determined by standard Gram- Schmidt orthogonalization procedure, so to complete the basis. In this new basis, the final target state is expressed as
As for the case analyzed previously, we rewrite this as
where the matrix is orthogonal. We proceed as before and introduce the new set of laser couplings Eq. (12), and we parametrize them in terms of a total amplitude Eq. (13) and angles Eq. (14). The expressions for the dark states in terms of the parameters , Eqs. 22 and in particular Eq. 23, remain valid.
Also in this case it is possible to reduce the system to an effective three-level system and implement the adiabatic evolution from to as a standard STIRAP process. The required form for the transformed pulses is:
As only , are non-zero, the system is reduced to a three-level system, as in the case analyzed previously. Furthermore, the specific choice of the pulse form, Eqs. (35,36) leads to the transfer of the atomic system from to , i.e. from to . This can be shown by noticing that
Thus and . Therefore, the dark state has the properties
and the adiabatic following of the dark state leads to the evolution of the system from to .
The physical laser pulses are obtained by solving the system of equations (29), and again each is a linear combination of and .
Iv Numerical analysis
In this Section we prove the validity of our approach with numerical simulations. We numerically study the time-evolution of the atomic system to verify that our choice for the pulse sequence does indeed lead to adiabatic transfer from the initial to the target state. We consider a five-level system, with four ground states and one excited state. As already stressed, the procedure to identify the required pulse shape for the wanted transfer has minimal computational complexity, as it simply requires the inversion of an orthogonal matrix, which corresponds to a transposition. Thus, the same procedure can be applied to larger atomic systems, with the same coupling structure, without any computational difficulty.
For fully adiabatic evolution, cf. Eq. (24), the evolution of the atomic system does not involve populating the atomic excited state. Thus, we can study the time-evolution of the atomic system by solving the Schrödinger equation. In all numerical simulations presented here we will take the transformed pulses , to have Gaussian shape
where is the pulse centre. The pulse delay between the delayed pulses is chosen so as to guarantee an overlap, and essential condition for the STIRAP process. The amplitude of the pulses , its width are chosen so to guarantee the adiabaticity of the process. We notice that we chose the same amplitude for the pulses for simplicity. However this is not a requirement for the STIRAP process in the effective three level system, and any combination of amplitudes for the two pulses , , such that the adiabaticity condition is satisfied, will lead to the correct implementation of the STIRAP process.
Figure 2 shows the solution of the time-dependent Schrödinger equation for the case of orthogonal initial and target states. Figure 2(a) reports the pulse shapes , in the transformed basis, while in the initial basis the required laser pulses to obtain the wanted transfer are then determined via Eq. (29), and are reported in Fig. 2(b). We notice that the required relative sign between Rabi frequencies can experimentally be easily implemented by introducing a relative phase between the corresponding electric fields. The resulting time-dependent populations () of the different atomic states are reported in Fig. 2(c), together with the fidelity of preparation of the wanted state , where describes the state of the system at time . Our numerical results show that the fidelity approaches unity after the pulse sequence, i.e. the system is effectively prepared in the wanted state .
An analogous numerical analysis was also carried out for the case of non-orthogonal initial and target states. The procedure differs from the case analyzed previously only in the definition of the transformed laser pulses. The non-orthogonality of the initial and target superpositions require a different transformation for , as given by Eq. (36). Our numerical results for this case, presented in Fig. 3, confirm the validity of our approach: the process leads to a complete transfer from to , without populating the excited state. Also in this case the amplitudes of the transformed fields were taken to be equal for simplicity. However, this is not a requirement for the STIRAP process, and arbitrary amplitudes can be used provided that they are large enough to guarantee the adiabaticity of the process.
In this work we considered the problem of the implementation of Stimulated Raman Adiabatic Passage processes in degenerate systems, with a view to be able to steer the system wave function from an arbitrary initial superposition to an arbitrary target superposition. We examined the case a -level atomic system consisting of ground states coupled to a common excited state by laser pulses. We analyzed the general case of initial and final superpositions belonging to the same manifold of states, and we cover also the case in which they are non-orthogonal. We demonstrated that, for a given initial and target superposition, it is always possible to choose the laser pulses so that in a transformed basis the system is reduced to an effective three-level system, and standard STIRAP applies. Our treatment leads to a simple strategy, with minimal computational complexity, which allows us to determine the laser pulses shape required for the wanted adiabatic steering.
Acknowledgements.This work was supported by the Royal Society and the DFG (BR 1528/7-1).
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