Controlling Several Atoms in a Cavity

We treat control of several two-level atoms interacting with one mode of the electromagnetic field in a cavity. This provides a useful model to study pertinent aspects of quantum control in infinite dimensions via the emergence of infinite-dimensional system algebras. Hence we address problems arising with infinite-dimensional Lie algebras and those of unbounded operators. For the models considered, these problems can be solved by splitting the set of control Hamiltonians into two subsets: The first obeys an abelian symmetry and can be treated in terms of infinite-dimensional Lie algebras and strongly closed subgroups of the unitary group of the system Hilbert space. The second breaks this symmetry, and its discussion introduces new arguments. Yet, full controllability can be achieved in a strong sense: e.g., in a time dependent Jaynes-Cummings model we show that, by tuning coupling constants appropriately, every unitary of the coupled system (atoms and cavity) can be approximated with arbitrarily small error.


Introduction
Exploiting controlled dynamics of quantum systems is becoming of increasing importance not only for solving computational tasks or quantum-secured communication, but also for simulating other physical systems [1,2,3,4,5,6]. An interesting direction in quantum simulation applies many-body correlations to create 'quantum matter'. E.g., ultra-cold atoms in optical lattices are versatile models for studying large-scale correlations [7,6]. Tunability and control over the system parameters of optical lattices allows for switching between several low-energy states of different quantum phases [8,9] or in particular for following real-time dynamics such as the quantum quench from the super-fluid to the Mott insulator regime [10].
Thus manipulating several atoms in a cavity is a key step to this end [11] at the same time posing challenging infinite-dimensional control problems. While in finite dimensions controllability can readily be assessed by the Lie-algebra rank condition [12,13,14,15,16], infinite-dimensional systems are more intricate [17]. As exact controllability in infinite dimensions seemed daunting in earlier work [18,19,20,21], it took a while before approximate control paved the way to more realistic assessment [22,23,24], for a recent (partial) review see, e.g., also [25] and references therein.
Here we explore systems and control aspects for systems consisting of several twolevel atoms coupled to a cavity mode, i.e. the Jaynes-Cummings model [26,27,28,29]. We build upon our previous symmetry arguments [30,31] and moreover, we apply appropriate operator topologies for addressing two controllability problems in particular: (i) to which extent can pure states be interconverted and (ii) can unitary gates be approximated with arbitrary precision. In particular by treating the latter, we go beyond previous work, which started out by a finite-dimensional truncation of a two-level atom coupled to an oscillator [32] followed by generalisations to infinite dimensions [33,34,35] both being confined to establishing criteria of pure-state controllability. Note that [35] also treats one atom coupled to several oscillators.
The general aim of this paper is twofold: On the one hand we study control problems for atoms interacting with electromagnetic fields in cavities. On the other hand, we address quantum control in infinite dimensions. Therefore, the purpose of Section 2 is to provide enough material for a non-technical overview on the second subject in order to understand the results on the first (where the difficulties come from). Mathematical details are postponed to Sections 4 and 5, while results on cavity systems are presented in overview in Section 3.

Controllability
The control of quantum systems poses considerable mathematical challenges when applied to infinite dimensions. Basically, they arise from the fact that anti-selfadjoint operators (recall that according to Stone's Theorem [36,VIII.4], they are generators of strongly continuous, unitary one-parameter groups) do neither form a Lie algebra nor even a vector space. Or seen on the group level, the group of unitaries equipped with the strong operator topology is a topological group yet not a Lie group. So whenever strong topology has to be invoked, controllability cannot be assessed via a system Lie algebra. Thus we address these challenges on the group level by employing the controlled time evolution of the quantum system in order to approximate unitary operators, the action of which is measured with respect to arbitrary, but finite sets of vectors. This is formalized in the notion of strong controllability (see Section 2.3) introduced here as a generalisation of pure-state controllability already discussed in the literature. Central to our discussion are abelian symmetries. Assuming that all but one of our Hamiltonians observe such an ablian symmetry, we systematically analyze the infinite-dimensional control system in its block-diagonalized basis. We obtain strong controllability (beyond pure-state controllability) if one of the Hamiltonian breaks this abelian symmetry and some further technical conditions are fullfilled.

Time evolution
We treat control problems of the forṁ where the H k with k ∈ {1, . . . , d} are selfadjoint control Hamiltonians on an infinitedimensional, separable Hilbert space H and the controls u k : R → R are piecewiseconstant control functions. Since H is infinite-dimensional, the operators H k are usually only defined on a dense subspace D(H k ) ⊂ H called the domain of H k , the only exceptions being those H k which are bounded. However, in this context, control problems where all H k are bounded are not very interesting from a physical point of view. In other words, there is no way around considering those domains and many difficulties of control theory in infinite dimensions arises from this fact ‡ .
We will also assume that Eq. (1) will have unique solutions for all initial states ψ 0 ∈ H and all times t. So for each pair of times t 1 < t 2 there is a unitary propagator U(t 1 , t 2 )ψ 0 = T t 2 t 1 exp(−itH(t))ψ 0 , where T denotes time ordering. Observe that this condition is usually not satisfied, not even if the H k share a joint domain of essential selfadjointness. Fortunately, the systems we are going to study do not show such pathological behavior. Yet, a minimalistic way to avoid this problem would be to restrict to control functions where only one u k is different from 0 at each time t. In this case the propagator U(t 1 , t 2 ) is just a concatenation of unitaries exp(itH k ) which are guaranteed to exist due to selfadjointness of the H k .

Pure-state controllability
A key-issue in quantum control theory is reachability: Given two pure states ψ 0 , ψ ∈ H, we are looking for a time T > 0 and control functions u k such that ψ = U(0, T )ψ 0 . In infinite dimensions, however, this condition is too strong, since there might be states which can be reached only in infinite time, or not at all. Yet, one may find a reachable state "close by" with arbitrary small control error. Therefore we will call ψ reachable from ψ 0 if for all ǫ > 0 there is a finite time T > 0 and control functions u k such that ψ − U(0, T )ψ 0 < ǫ holds. Accordingly, we will call the system (1) pure-state controllable, if each pure state ψ can be reached from one ψ 0 (and, by unitarity, also vice versa).
Since pure states are described by one-dimensional projections, two state vectors describe the same state if they differ only by a global phase. Hence the definition just given is actually a bit too strong. There are several ways around this problem, like using the trace norm distance of |ψ ψ| and |ψ 0 ψ 0 | rather then the norm distance of ψ and ψ 0 . For our purposes, however, the most appropriate method is to assume that the unit operator ½ on H is always among the control Hamiltonians. This may appear somewhat arbitrary, but it helps to avoid problems with determinants and traces on infinite-dimensional Hilbert spaces, which otherwise would arise. ‡ Note that domains of unbounded operators are not just a mathematical pedantism. The domain is a crucial part of the definition of an operator and contains physically relevant information. A typical example is the Laplacian in a box which requires boundary conditions for a complete description. Up to a certain degree, domains can be regarded as an abstract form of boundary conditions (possibly at infinity).

Strong controllability
Next, the analysis shall be lifted to the level of operators, i.e. to unitaries U from the group U(H) of unitary operators on the Hilbert space H such that a time T > 0 and control functions u k exist with U = U(0, T ). As in the last paragraph, this has to be generalized to an approximative condition again. The best choice-mathematically as well as from a practical point of view-is approximation in the strong sense: We look for unitaries U such that for each set of (not necessarily orthonormal or linearly independent) vectors ψ 1 , . . . , ψ f ∈ H and each ǫ > 0, there exists a time T > 0 and control functions u k such that In other words, we are comparing U and U(0, T ) only on a finite set of states, and the worst-case error one can get here is bounded by ǫ. We will call the control system (1) strongly controllable if each unitary U can be approximated that way. (NB, in strong controllability, one again has the choice of one single joint global phase factor.) Clearly, strong controllability implies pure-state controllability. To see this, choose an arbitrary but fixed ψ 0 ∈ H. For each ψ ∈ H, there is a unitary U with Uψ 0 = ψ.
They form a neighborhood base for the strong topology, and we will call them (strong) ǫ-neighborhoods. The condition in Eq. (2) can now be restated as: Any ǫ-neighborhood of U contains a time-evolution operator U(0, T ) for appropriate time T and control functions u k . In turn, this can be reformulated as: U is an accumulation point of the setG of all unitaries U(0, T ). The set of all accumulation points ofG (which containsG itself) is a strongly closed subgroup § of U(H), which we will call the dynamical group G generated by control Hamiltonians H k with k ∈ {1, . . . , d}. If we choose the controls as described in Subsection 2.1 (i.e. piecewise constant and only one u k different from zero at each time), G is just the smallest strongly closed subgroup of U(H) that contains all exp(itH k ) for all k ∈ {1, . . . , d} and all t ∈ R. Note that it contains in particular all unitaries that can be written as a strong limit s-lim T →∞ U(0, T ). In finite dimensions, § There is a subtle point here: The group U(H) is not strongly closed as a subset of the bounded operators B(H). Actually its strong closure is the set of all isometries; cf. [37,Prob. 225]. Hence whenever we talk about strongly closed groups of unitaries, this has to be understood as the closure in the restriction of the strong topology to U(H) (which coincides with the restriction of the weak topology).
G can be calculated via its system algebra, i.e. the Lie algebra l generated by the iH k , since each U ∈ G can be written as U = exp(H) for an H ∈ l.
In infinite dimensions, however, several difficulties can occur. First, unbounded operators H k are only defined on a dense domain D(H k ) ⊂ H. The sum H k + H j is therefore only defined on the intersection D(H k )∩D(H j ) and the commutator even only on a subspace thereof. There is no guarantee that D(H k ) ∩ D(H j ) contains more than just the zero vector. In this case, the Lie algebra cannot even be defined.
The minimal requirement to get around this difficulty is the existence of a joint dense domain D, i.e. D ⊂ D(H j ) and H j D ⊂ D for all j. However, even then we do not know whether G can be generated from l in terms of exponentials. In general, it is impossible to define some exp(H) for all H ∈ l.
There are several ways to deal with these problems. One is to consider cases where the H k generate (i) a finite-dimensional Lie algebra and admit (ii) a common, invariant, dense domain consisting of analytic vectors [18,20]. In this case the exponential function is defined on all of l, and we can proceed in analogy to the finite-dimensional case. The problem is that the group G will become a finite-dimensional Lie group and its orbits through a vector ψ ∈ H are finite-dimensional as well. Hence, we never can achieve full controllability. This approach is well studied; cf. [18,20] and references therein.
Another possibility which includes the possibility to study an infinite-dimensional Lie algebra l is to restrict to bounded generators H k . In this case, one can define l as a norm-closed subalgebra of the Lie algebra B(H) of bounded operators, and one ends up with a Banach-space theory which works almost in the same way as the finitedimensional analog; cf. [38] for details. Although this is a perfectly reasonable approach from the mathematical point of view, it is not very useful for physical applications, since in most cases at least some of the H k are unbounded.
In this paper, we will thus consider a different approach which splits the generators into two classes. The first d−1 generators H 1 , . . . , H d−1 admit an abelian symmetry and can be treated-with Lie-algebra methods-along the lines outlined in the next subsection. Secondly, the last generator H d breaks this symmetry and achieves full controllability with a comparably simple argument. The details will be explained in Section 4 and 5.

Abelian symmetries
One way to avoid the problem described in the last subsection, arises if the control system admits symmetries. In this section, we will only sketch the structure, while the details are postponed to Sect. 4. Let us consider the case of a U(1)-symmetry , i.e. a (strongly continuous) unitary representation z → π(z) ∈ U(H) of the abelian group U(1) on H where U(H) denotes the group of unitaries on H. It can be written in terms of a selfadjoint operator X with pure point spectrum consisting of (a subset of) Z as U(1) ∋ z = e iα → π(z) = The generalization to multiple charges, i.e. a U(1) N , is straightforward. exp(iαX) ∈ U(H). If we denote the eigenprojection of X belonging to the eigenvalue µ ∈ Z as X (µ) (allowing the case X (µ) = 0 if µ is not an eigenvalue of X) we get a block-diagonal decomposition of H in the symmetry-adapted basis as and we can rewrite π(z) again as U(1) ∋ z = e iα → π(z) = ∞ µ=−∞ e iαµ X (µ) ∈ U(H). Here we will make two assumptions representing substantial restrictions of generality: (i) All eigenvalues of X are of finite multiplicity, i.e. the H (µ) are finite-dimensional. This is crucial for basically everything we will discuss in this paper.
(ii) All eigenvalues of X are non-negative. This assumption can be relaxed at certain points (e.g. all material in Sect. 4.1 can be easily generalized). However, it helps to simplify the discussion at a technical level and all examples we are going to consider in the next section are of this form.
The first important consequence of (i) concerns the space of finite particle vectors D X = {ψ ∈ H | X (µ) ψ = 0 for all but finitely many µ}, since it becomes (due to finite-dimensionality of H (µ) ) a "good" domain for basically all unbounded operators appearing in this paper. Moreover one gets the following theorem: Theorem 2. 1. Consider a strongly continuous representation π of U(1) on H and the corresponding charge-type operator X. Then the following statements hold: (i) A selfadjoint operator H commuting with X admits D X as an invariant domain, i.e. D X ⊂ D(H) and HD X = D X . Hence the space u(X) = {iH | H = H * commuting with X} is a Lie algebra with the commutator as its Lie bracket.
(ii) The exponential map is well defined on u(X) and maps it onto the strongly closed subgroup (iii) The subalgebra l ⊂ u(X) generated by a family of Hamiltonians iH 1 , . . . , iH d ∈ u(X) is mapped by the exponential map into the dynamical group G of the corresponding control problem. The strong closure of exp(l) coincides with G.
The basic idea behind this theorem, is that one can cut off the decomposition (4) at a sufficiently high µ without sacrificing strong approximations as described in Subsection 2. 3. One only has to take into account that the cut-off on µ has to become higher when the approximation error decreases. This strategy allows for tracing a lot of calculations back to finite-dimensional Lie algebras. We will postpone a detailed discussion of this topic-including the proof of Theorem 2.1-to Section 4.
The only additional material one needs at this point, since it is of relevance for the next section, is a subgroup of U(X) and its corresponding Lie algebra which relates unitaries with determinant one and their traceless generators. Since the iH ∈ u(X) are unbounded and not necessarily positive, it is difficult to give a reasonable definition of tracelessness, and the determinant of U ∈ U(X) runs into similar problems. However, the elements of U ∈ U(X) and iH ∈ u(X) are block diagonal with respect to the decomposition of H given in (4). In other words U = µ U (µ) and H = µ H (µ) are infinite sums of operators ¶, where U (µ) = X (µ) UX (µ) ∈ U(H (µ) ), H (µ) = X (µ) HX (µ) ∈ B(H (µ) ), and X (µ) denotes the projection onto the X-eigenspace H (µ) . Since all the U (µ) and H (µ) are operators on finite-dimensional vector spaces, one can define su(X) := {iH ∈ u(X) | tr(H (µ) ) = 0 for all µ ∈ Z}.
Obviously, SU (X) is a (strongly closed) subgroup of U(X) and su(X) is a Lie subalgebra of u(X). The image of su(X) under the exponential map therefore coincides with SU (X). Note that SU (X) is effectively an infinite direct product of groups SU(d (µ) ), if d (µ) = dim H (µ) and not the "special" subgroup of U(X).

Breaking the symmetry
To get a fully controllable system, one has to leave the group U(X), which can be thought of as being represented as block diagonal, see ± = X (µ) E ± which we require to be non-zero. For the exceptional case µ = 0 the relation X as its source and X (µ) − as its target projection. ¶ Two small remarks are in order here: (i). Infinite sums require a proper definition of convergence in an appropriate topology. In Section 4, this will be made precise. (ii). Operator products of the form X (µ) HX (µ) are potentially problematic if H is unbounded and therefore only defined on a domain. In our case, however, block structure of operators complementary to X block structure of operators commuting with X I ) with combined atom-cavity transitions matching the block structure of (a) given in red (see Eqs. (10,18)) since commuting with X 1 or X M of Eqs. (12,20), and complementary transitions solely within the atoms given in blue (see Eqs. (15,22)).

(iii) Given the projection F
− and the corresponding subspace At first sight, the definition may look somewhat clumsy, but it allows for proving a controllability result which covers all examples we are going to present in the next section. We will state them here without a proof and postpone the latter to Sect. 5. 3. Consider a strongly continuous representation π : U(1) → U(H) with charge operator X and a family of selfadjoint operators H 1 , . . . , H d on H. Assume that the following conditions hold:

Then the control system (1) with Hamiltonians
Theorem 2.4. The control system (1) is even strongly controllable if in addition to the assumptions of Thm. 2.3 the condition dim H (µ) > 2 holds for at least one µ ∈ N 0 .

Atoms in a cavity
An important class of examples that can be treated along the lines described in the last section are atoms interacting with the light field in a cavity. We will discuss the case of M two-level atoms interacting with one mode in detail and consider three particular scenarios: one atom in Sect. 3.1, individually controlled atoms in Sect. 3.2, and atoms under collective control in Sect. 3

One atom
Let us start with the special case M = 1, i.e. one atom and one mode as discussed in a number of previous publications mostly on pure-state controllability [39,35,34]. Our results go beyond this, in particular because we are considering strong controllability not just pure-state controllability. The Hilbert space of the system is given by and the dynamics is described by the well known Jaynes-Cummings Hamiltonian [26]: where σ α with α ∈ {1, 2, 3} are the Pauli matrices (σ ± = σ 1 ± iσ 2 ), a, a * denote the annihilation and creation operator, and N = a * a is the number operator. The joint domain of all these Hamiltonians is the space with ν ∈ C 2 as canonical basis and |n ∈ L 2 (R) as number basis (Hermite functions). We will assume that the frequencies ω A , ω I and ω C can be controlled independently (or at least two of them) such that we get a control system with control Hamiltonians H JC,j where j ∈ {1, 2, 3} corresponding to the lower half (1 atom) of the energy diagram in Fig. 1 b, where we adopt the widely used convention of forcing the atom (spin) state |↑ to be of 'higher' energy than |↓ to compensate for negative Larmor frequencies, see, e.g., the note in [11, p. 144]. The task is to determine the dynamical group G. To this end, we use the strategy described in Subsection 2.5, which follows in this particular case closely the exact solution of the Jaynes-Cummings model [26]. The charge-type operator X 1 (determining the block structure) then takes the form again with D from (11) as its domain, which in this case turns out to be identical to the space D X 1 of finite-particle vectors. The operator X 1 is diagonalized by the basis |ν ⊗ |n . It is convenient to relabel these vectors in order to get In this basis, we have X 1 |µ, ν = µ|µ, ν and the subspaces H (µ) from (4) become for µ > 0 and H (0) = C|0, 0 for µ = 0. The space D X 1 ⊂ H of finite-particle vectors turns out to be identical with the domain D from (11).
It is easy to see that the operators H JC,j from Eq. (10) commute with X 1 , and therefore we get iH JC,j ∈ u(X 1 ). A more detailed analysis, as will be given in Section 4, shows that iH JC,1 and iH JC,2 generate su(X 1 ), and therefore we get according to Theorem 2.1: To get a fully controllable system, apply Theorem 2.4 to see that one has to add a Hamiltonian which breaks the symmetry. A possible candidate is If we define the spaces H α as H − = span{|µ, 0 | µ ∈ N 0 }, H 0 = {0}, and H + = span{|µ, 1 | µ ∈ N} the operator H JC,4 becomes complementary to X 1 , which can be easily seen since H (µ) Hence, according to Thm. 2.3, the control system with Hamiltonians of Eqs. (9,10).
is pure-state controllable + , and we are recovering a previous result from [39,35,34]. However, with our methods we can go beyond this and prove even strong controllability. Thm. 2.4 cannot be applied since dim H (µ) ≤ 2 for all µ, but the analysis of Sect. 5 will lead to an independent argument. Hence any unitary U on H can be approximated by varying the control amplitudes u 1 = ω A and u 2 = ω I in the Hamiltonian H JC of (9) plus flipping ground and excited state of the atom in terms of H JC,4 (with strength u 3 )-both in an appropriate timedependent manner. The approximation has to be understood in the strong sense as described in Eq. (2).
Finally, note that Theorem 3.2 implies that one can simulate (again in the sense of strong approximations) any unitary V ∈ B(L 2 (R)) operating on the cavity mode alone. One only has to find controls u j such that U(0, T ) φ ⊗ ψ k ≈ φ ⊗ V ψ k for a finite set of states ψ k of the cavity (and an arbitrary auxiliary state φ of the atom).

Many atoms with individual control
Next, consider the case of many atoms interacting with the same mode, and under the assumption that each atom (including the coupling with the cavity) can be controlled + We have omitted the Hamiltonian H JC,3 since it is not needed for the result. However, it can be added as a drift term without changing the result.
individually. Such a scenario is relevant for experiments with ion traps, if the number of ions is not too big as have been studied since [40,41,42]. The Hilbert space of the system is where M denotes the number of atoms. We define the basis |b ⊗ |n ∈ H where n ∈ N 0 , . As before, a and a * denote annihilation and creation operator. The joint domain of all these operators is with the basis |b ⊗ |n as defined above. As depicted by the red parts in Fig 1, all the H IC,k are invariant under the symmetry defined by the charge operator where N = a * a denotes again the number operator and D from (19) is the domain of X M . The eigenvalues of X M are µ ∈ N 0 and the eigenbasis is given by In this basis, X M becomes X M |µ, b = µ|µ, b and the eigenspaces From now on, one may readily proceed as for one atom to arrive at the following analogy to Theorem 3.1: To get strong controllability, one has to add again one Hamiltonian. As before a σ 1 -flip of one atom is sufficient (see the blue parts in Fig 1), and Obviously, all the conditions of Thm. 2.4 are satisfied such that one gets Theorem 3. 4. The control problem (1) with H IC,k and k ∈ {1, . . . , 2M+1} from (18) and (22) is strongly controllable.
As a special case of this theorem, one can approximate any unitary U acting on the atoms alone, i.e. U ∈ U((C 2 ) ⊗M ), by applying Theorem 3.4 to U ⊗ ½. That is, one can simulate U only by operations on one atom and the interactions with the harmonic oscillator. This is used in ion-trap experiments and is known as "phonon bus".

Many atoms under collective control
Now one may modify the setup from the last section by considering again M atoms interacting with one mode, but assuming that one can control the atoms only collectively rather than individually. In other words instead of the Hamiltonians H IC,j and H IC,M +j with j ∈ {1, . . . , M} from Eq. (18) one only has their sums where S α = M j=1 σ α,j and α ∈ {1, 2, 3, ±}, combinded with the free evolution of the cavity. As before, all operators are defined on the domain D from (19). Note that one readily recovers the original setup from Subsection 3.1 with Pauli operators σ α replaced by pseudo-spin operators S α . The multi-atom analogue of the Jaynes-Cummings Hamiltonian, which can be formed from the H TC,j just defined, is called Tavis-Cummings Hamiltonian [27,28].
All the Hamiltonians in Eqs. (23) and (24) are invariant under the U(1)-action generated by X M of Eq. (20). However, this is not the only symmetry, since all these H TC,j are also invariant under the permutation of the atoms. Therefore, one may no longer exhaust the group SU (X M ) as in Theorem 3.3 (since the following operators cannot be reached: those commuting only with X M but not also with permutations of the atoms). A minimal modification is to restrict the states of the atoms to spaces on which permutation-invariant unitaries operate transitively * . The most natural choice is the symmetric tensor product (C 2 ) ⊗M sym ⊂ (C 2 ) ⊗M , i.e. the Bose subspace of (C 2 ) ⊗M . The preferred basis of (C 2 ) ⊗M sym is |ν = Sym M |1 ⊗ν ⊗ |0 ⊗(M −ν) with ν ∈ {0, . . . , M} and the projection Sym M from (C 2 ) ⊗M onto the symmetric subspace (C 2 ) ⊗M sym . In other words |ν is the unique, pure, permutation-invariant state with ν atoms in the excited state |1 and M−ν ones in the ground state |0 . Therefore, (C 2 ) ⊗M sym can be identified with the Hilbert space C M +1 of a (pseudo-)spin-M/2 system. Its basis |ν , with ν ∈ {0, . . . , M} becomes the canonical basis. Combining this with L 2 (R) for the cavity one gets H sym = C M +1 ⊗ L 2 (R) as the new Hilbert space of the system.
All the operators defined above (H TC,1 , H TC,2 , H TC,3 and X M ) can be restricted to H sym (and in slight abuse of notation we will re-use the symbols after restriction) and their domain becomes However, already the restriction to permutation-invariant states will turn out to be difficult enough.
Now one can proceed as the in the previous cases: The operators H TC,1 , H TC,2 , H TC,3 are (as operators on H sym ) invariant under the action generated by X M and therefore elements of u(X M ). However, one still cannot exhaust all of U(X M ) (or SU (X M )). One only gets: Theorem 3. 5. The dynamical group G generated by the operators H TC,1 , H TC,2 , H TC,3 from Eqs. (23) and (24) is a strongly closed subgroup of U(X M ). For each unitary V ∈ U(X M ) and each µ ∈ N 0 we can find an element U ∈ G such that Uψ (µ) = V ψ (µ) holds for all ψ (µ) ∈ H (µ) sym . In other words: As long as the charge µ is fixed, one can still approximate any V ∈ U(X M ), but if one considers superpositions of different charges this is no longer the case, i.e. there are ψ ∈ D X M and V ∈ U(X M ) such Uψ = V ψ for all U ∈ G. We have checked the latter explicitly with the computer algebra system Magma [43] for the case M = 2. To circumvent this problem, one has to add control Hamiltonians. Unfortunately, it seems that one has to add quite a lot. The best result we have got so far is to replace the operators from Eqs. (23) and (24) by (27) The operators H CC,k with k ∈ {1, . . . , M +1} commute with X M and generate (as we will see in Sect. 4  from (27) is strongly controllable.
To be able to control all diagonal traceless operators H CC,k , with k ∈ {1, . . . , M} is a very strong assumption. Unfortunately, a detailed analysis including computer algebra indicates that we cannot recover Theorem 3.7 with fewer resources.

A Lie algebra of block-diagonal operators
The purpose of this section is to re-discuss abelian symmetries and to provide technical details (in particular proofs) we omitted in Sections 2 and 3. To this end, re-use the notations already introduced in Section 2. 5. In particular, the abelian symmetry induces a block-diagonal decomposition which, in infinite dimensions, allows for defining a block-diagonal Lie algebra and its exponential map onto a block-diagonal Lie group; see Propositions 4.1 and 4.2. We identify the set of all block-diagonal unitaries reachable by block-diagonal time evolutions in Proposition 4.4 as the strong closure of exponentials of block-diagonal Lie algebra elements. A central result is Corollary 4.6, in which the question of controllability for the block-diagonal system of infinite dimensions is reduced to analyzing controllability for all finite-dimensional blocks. Using finite-dimensional commutator calculations one can now establish controllability on the infinite-dimensional but block-diagonal space for each of the three control systems analyzed.

Commuting operators
The first step is a closer look at the Lie algebra u(X) and the corresponding group U(X) introduced in Theorem 2.1 (which we will prove in this context). To this end, let us start with a unitary U commuting with the representatives π(z), i.e. [π(z), U] = 0 for all z ∈ U(1). This is equivalent to Uψ = ∞ µ=0 U (µ) ψ (µ) for all ψ ∈ H with ψ (µ) := X (µ) ψ ∈ H (µ) given a sequence of unitaries U (µ) on the µ-eigenspaces H (µ) of X. Similarly one can consider a selfadjoint H with domain D(H) commuting with X. By definition♯ this means the spectral projections of H commute with the X (µ) , which is equivalent to with a sequence of selfadjoint operators H (µ) on the eigenspaces H (µ) and the ψ (µ) as defined above. The H (µ) are finite-dimensional, and therefore the H (µ) are bounded. Hence the unboundedness of H is inherited only from the unboundedness of the sequence of norms H (µ) . So it is easy to see that all elements of D X are analytic for H and therefore D X becomes a domain of essential selfadjointness for H (i.e. H is uniquely determined by its restriction to D X as a consequence of Nelson's analytic vector theorem [44, Thm. X.39]). Accordingly, we will denote (in slight abuse of notation) the selfadjoint operator H and its restriction to D X by the same symbol. This proves very handy when introducing, on the set u(X) of anti-selfadjoint operators commuting with X, the structure of a Lie algebra by (λQ becomes a Lie algebra with the commutator as its Lie bracket. Since all iH ∈ u(X) are anti-selfadjoint, they admit a well-defined exponential map exp(iH). Boundedness of the H (µ) together with Eq. (28) allows to express exp(iH) very explicitly. More precisely one has and exp(iH (µ) ) = ∞ n=0 (iH (µ) ) n /(n!). This shows that exp : u(X) → U(X) is welldefined and onto as stated in Thm. 2.1, which we are now ready to prove: The exponential map on u(X) is well-defined and given in terms of Equation (31). It maps u(X) onto the strongly closed subgroup Proof. The only statement not yet proven is the closedness of U(X). To this end, we have to show that for any net (U λ ) λ∈I strongly converging to a bounded operator U we have U ∈ U(X). As U λ ∈ U(X) we have [π(z), U λ ] = 0 for all λ. Due to strong continuity of the map A → [π(z), A] and the convergence of the U λ to U it follows that [π(z), U] = 0. Hence U decomposes into a strongly converging series U = µ U (µ) with U (µ) ∈ B(H (µ) ), and for each fixed µ we get lim λ U (µ) λ = U (µ) . Since H (µ) is finitedimensional, the nets (U (µ) λ ) λ∈I converge in norm and therefore U (µ) ∈ U(H (µ) ) which implies U ∈ U(X).
Note that we actually proved more than what we stated. A strongly convergent sequence (or net) of elements of U(X) cannot converge to an isometry which is not unitary as well. Hence U(X) is strongly closed as a subset of B(H)-and not only as a subset of U(H) as generally is the case (cf. corresponding remarks in Sect. 2

.4).
The remaining statements in this subsection are devoted to the dynamical group G generated by a family of selfadjoint operators H 1 , . . . , H d . Recall that we have introduced it as the smallest strongly closed subgroup of U(H) containing all unitaries of the form exp(itH k ). If the H k are commuting with X, i.e. iH k ∈ u(X), then the group G is a subgroup of U(X), and the simple structure of the latter makes explicit calculations at least feasible. In the following, we show how U(X) is related to the Lie algebra l generated by the iH k . To this end, we need some additional notations. For each K ∈ N, U ∈ U(X), and iH ∈ u(X), let us consider The operators U [K] and H [K] act on the finite-dimensional Hilbert space H [K] . Therefore all operator topologies coincide and we can apply the well-known finite-dimensional theory. The dynamical group G [K] (generated by H with k ∈ {1, . . . , d}) becomes a closed subgroup of the unitary group U(H [K] ), which is a Lie group. Hence G [K] is a Lie group, too, and its Lie algebra l [K] is generated by iH [K] k with k ∈ {1, . . . , d}. Now, the crucial point is that one can approximate the infinite-dimensional objects G and l by the finite-dimensional G [K] and l [K] . To see this, the first step is the following lemma. Lemma 4. 3. Consider the Lie algebras l ⊂ u(X) and l [K] ⊂ B(H [K] ) (with K ∈ N) generated by iH 1 , . . . , iH d and iH [K] 1 , . . . , iH [K] d , respectively. Each elementQ ∈ l [K] can be written asQ = Q [K] for some element Q ∈ l.
Moreover, we now have the tools to prove the relation between the Lie algebra l and the dynamical group G already stated in Thm. 2

.1.
Proposition 4. 4. Consider again iH 1 , . . . , iH d ∈ u(X) and the Lie algebra l generated by them. Then the corresponding dynamical group G coincides with the strong closure of exp(l) ⊂ U(X).

Proof.
Each U ∈ G can be written as the limit of a net (U λ ) λ∈I of operators U λ , which are monomials in exp(it k H k ) with k ∈ {1, . . . , d} with appropriate times t k . This implies in particular that the U λ commute with π(z) for all z, and, by continuity, the same is true for U. Hence U ∈ U(X), and for each K ∈ N we can define U [K] which is the limit of the net (U [K] λ ) λ∈I . The latter converges in norm (since H [K] is finite-dimensional), and therefore U [K] ∈ G [K] . This implies U [K] = exp(Q K ) with Q K ∈ l [K] as G [K] is a Lie group and l [K] its Lie algebra.
For U to be in the strong closure of exp(l), each strong ǫ-neighborhood of U, i.e. the sets N (U; ψ 1 , . . . , ψ f ; ǫ) introduced in Eq. (3), should contain an element of exp(l) for all ψ 1 , . . . , ψ f and all ǫ > 0. However, the unitary group is contained in the unit ball of B(H), and thus it is sufficient to consider only those N (U; ψ 1 , . . . ; ψ f , ǫ) with vectors ψ 1 , . . . , ψ f from a dense subspace of H; cf. [45, I. 3. 1.2]. Hence, in turn, it is sufficient to consider only neighborhoods with ψ j ∈ D X . But then there is a K ∈ N such that ψ j ∈ H [K] for all j ∈ {1, . . . , f }. Now take the operator Q K from the last paragraph andQ K ∈ l withQ [K] K = Q K , which exists due to Lemma 4. 3. By construction we have [U − exp(Q K )]ψ j = [U [K] − exp(Q K ) [K] ]ψ j = [(U [K] − exp(Q [K] K )]ψ j = [(U [K] − exp(Q K )]ψ j = 0 since U [K] = exp(Q K ), as was also seen in the previous paragraph. Hence exp(Q K ) ∈ N (U; ψ 1 , . . . , ψ f ; ǫ) which shows that U is in the strong closure of exp(l). This shows that the dynamical group G is contained in the strong closure of exp(l).
Conversely, consider exp(Q) for Q ∈ l. We have to show that exp(Q) is in the dynamical group G. To this end we observe, for each K ∈ N, that exp(Q [K] ) = exp(Q) [K] , which is obviously in G [K] . Hence there is a U K = exp(iH [K] j 1 ) · · · exp(iH [K] jn ) with j k ∈ {1, . . . , d} which is ǫ-close (in norm) to exp(Q [K] ). As in the last paragraph, this implies thatŨ = exp(iH j 1 ) · · · exp(iH jn ) is in N (exp(Q); ψ 1 , . . . , ψ f ; ǫ) provided ψ j ∈ H [K] for all j ∈ {1, . . . , f }. Hence exp(Q) is in the strong closure of the group of monomials in the exp(iH j ), but this is just the dynamical group G. Since G is strongly closed, the strong closure of exp(l) is contained in G, too. Since we have shown the other inclusion before, the entire proposition is proven.
Moreover, with this proposition the proof of Thm. 2.1 is complete. -The rest of this subsection is devoted to analyzing a related question: If, in finite dimension, two Lie algebras l 1 , l 2 generate the same group, then they are actually identical. However, in infinite dimensions this no longer true. Therefore, the next proposition is meant to decide if dynamical groups generated by two different sets of Hamiltonians do in fact coincide.
Proposition 4. 5. Consider two Lie algebras l 1 , l 2 ⊂ u(X). Assume that for each Q ∈ l 1 and each K ∈ N, there is aQ ∈ l 2 such that Q [K] =Q [K] holds (note that we can have differentQ for the same Q but different K). Then exp(l 1 ) is contained in the strong closure of exp(l 2 ).
Proof. One may readily use the same strategy as in the proof of Prop. 4.4: If the given condition holds, one can find in each neighborhood N (exp(Q); ψ 1 , . . . , ψ f ; ǫ) of exp(Q) with ψ 1 , . . . , ψ f ∈ D X an exp(Q) withQ ∈ l 2 . Hence exp(Q) is in the strong closure of exp(l 2 ).
Inserting su(X) for l 2 provides a useful criterion to check whether the dynamical group G generated by H 1 , . . . , H d ∈ su(X) is as large as possible in the sense that G = SU (X). To this end, let us introduce the truncated versions su [K] (X) = {Q [K] | Q ∈ su(X)} = ⊕ K µ=0 su(H (µ) ), SU [K] where we have used for any finite-dimensional subspace K of H the notations su(K) for the Lie algebra of traceless operators on K and similarly SU (K) for the Lie group of unitaries on K with determinant 1. Note that elements of su(K) and SU (K) have-as operators on H-a finite rank and their support and range are both contained in K.
Proof. Simple application of Props. 4.4 and 4.5.

One atom
The material just introduced readily applies to the systems studied in Sect. 3. This includes in particular the proofs of Thms. 3.1, 3.3, 3.5 and 3. 6. The first step is again one atom interacting with a cavity (Sect. 3.1). Hence the Hilbert space is H = C 2 ⊗L 2 (R) and the U(1)-symmetry under consideration is generated by the operator X 1 = σ 3 ⊗½+½⊗N already defined in (12). The domain of X 1 is D from Eq. (11), which is identical to D X 1 introduced in (5). The next step is to characterize the Lie algebra l generated by the control Hamiltonians H JC,1 and H JC,2 as defined in (10). They admit D = D X 1 as a joint common domain, and it is easy to see that they commute with X 1 (in the sense introduced in the previous subsection). Hence l ⊂ u(X 1 ), and all the machinery from Subsection 4.1 applies. This includes in particular the block-diagonal decomposition of operators A ∈ u(X 1 ) given in Eq. (28). In our case the subspaces H (µ) with µ ∈ N are given by (cf. Eq. (14)) H (µ) = span{|µ, 0 , |µ, 1 } using the basis |µ, ν ∈ H introduced in (13). For µ = 0, we get the one-dimensional space H (0) = C|0, 0 . The restrictions H where we have introduced the operators ς α = µ ς α is just the corresponding Pauli operator on H (µ) given in the basis |µ, 0 , |µ, 1 . We have used the core symbol ς rather than σ in order to avoid confusion with the operators σ α ⊗ ½ acting only on the atom. In addition we introduce the operators A α,k ∈ u(X 1 ) with α ∈ {0, . . . , 3} and k ∈ N 0 by In terms of the A α,k , now the H JC,j can readily be re-expressed as The next lemma shows that the Lie algebra l generated by the H JC,j is spanned as a vector space by a subset of the A α,k .
Proof. Obviously the operators iA α,k are in su(X 1 ). Hence, they span a subspacel ⊂ su(X 1 ). To prove thatl is a Lie subalgebra of su(X 1 ) one only has to check that [A α,k , A β,j ] ∈l for all α, β ∈ {1, 2, 3} and j, k ∈ N 0 . This follows easily, because the A α,k are just products of powers of X 1 and the ς α . But the latter are representatives of the Pauli operators. Hence All operators vanish in the case of µ = 0. Hencel is a Lie algebra and Eq. (38) proves that l ⊂l. For provingl = l, one has to express the A α,k for α ∈ {1, 2, 3} and k ∈ N 0 in terms of repeated commutators of the H JC,2 and H JC, 3 . By the commutation relations in Equation (39) it is obvious thatl is generated (as a Lie algebra) by A α,0 with α ∈ {1, 2, 3}. Therefore, the statement follows from Eq. (38), which in turn shows that With this Lemma and the material developed in the last subsection, one can proceed to determine the structure of the dynamical group generated by H JC,1 and H JC,2 . This is the content of Thm. 3.1, which is restated (and proven) here as a proposition. Proof. According to Prop. 4.4 the dynamical group G is the strong closure of exp(H) with H ∈ l, i.e. the Lie algebra generated by H JC,1 and H JC,2 , while SU (X) is the strong closure of exp(su(X)). Hence, by Cor. 4.6 we have to show that the truncated algebras l [K] and su [K] (X) are identical. The inclusion l [K] ⊂ su [K] (X) is trivial, since all the blocks H Hence it is sufficient to check that for each fixed 0 < µ 0 ≤ K and each iH ∈ su [K] (X) with H (µ) = 0 for µ = µ 0 there is an iA ∈ l such that iA (µ 0 ) = iH (µ 0 ) and A (µ) = 0 for all 0 < µ ≤ K with µ = µ 0 . The rest follows by linearity.
For constructing such an A, recall from Lemma 4.7 that l is spanned (as a vector space) by the A α,k with α ∈ {1, 2, 3} and k ∈ N 0 . Now consider a polynomial f in one real variable satisfying f (µ) = 0 for all 0 < µ ≤ K with µ = µ 0 and f (µ 0 ) = 1. The operators B α,f = f (X) √ Xς α with α ∈ {1, 2} and B 3,f = f (X)ς 3 are linear combinations of the A α,k , and they satisfy the condition B Before proceeding to the next subsection, consider the free Hamiltonian of the cavity H JC, 3 . We have omitted it from the discussion of the dynamical group, and the reason can be seen easily from (39): H JC,2 differs from H JC,1 only by X 1 + ½/2 which commutes with all elements of su(X). Hence adding H JC,3 as a control Hamiltonian would just add a one-dimensional center to the dynamical group G = SU (X). For the same reason, H JC,3 could be easily added as a drift term. Any effect it may have can be undone by evolving the system with H JC,1 , and the remaining relative phase between sectors of different charge µ does not affect the discussion of strong controllability in Sect. 5. Finally, let us remark that-due to the same reasons just discussed-we could exchange H JC,1 and H JC,3 almost without changes to the results of this subsection.

Many atoms with individual control
First, recall some notations from Sect. 3.2. The Hilbert space is H M = (C 2 ) ⊗M ⊗ L 2 (R) using the distinguished basis |µ; b with b ∈ Z M 2 from Eq. (21). The charge operator is Eq. (20), with domain D M from Eq. (19). In addition, let us introduce the re-ordered tensor product (where |µ, b 1 , . The key result of this section is split into the following three lemmas, which eventually will lead to a proof of Thm. 3. 3.   The elements of su C (H (µ) )⊗ k ½ are of the form A = a⊗ k |0 0| + a⊗ k |1 1| with a ∈ su C (H (µ) ). We will show that both summands are elements of g C , i.e. a⊗ k |b b| ∈ g C for b ∈ {0, 1}. The same holds for µ−1. The statement then follows from Lemma 4. 10.   (18). We will use Lemma 4.11 and an induction in M to prove Thm. 3.3, which we restate here as a proposition. Proof. According to Corollary 4.6 we have to show that for each K, we find that l for all K. As operators on H M +1 , they generate the Lie algebra l M⊗k ½ ⊂ l M +1 and according to (43) one finds that su(H (µ) M +1 holds for all µ ≤ K and k ∈ {1, . . . , M+1}. Thus, we can apply Lemma 4.11 and su(H M +1 for all µ ≤ K. But since l (K) M +1 ⊂ su [K] (X M +1 ) = su(H (K) M +1 ), one even gets su [K] (X M +1 ) ⊂ l [K] M +1 , just as was to be shown.

Many atoms under collective control
As a last topic in this section, we provide proofs for Thms. 3.5 and 3.6. To this end, recall the notation from Sect. 3

.3. The Hilbert space is
The charge operator is again X M = S 3 ⊗ ½ + ½ ⊗ N from Eq. (20) but now as an operator on H sym with domain D sym defined in (25) and the µ-eigenspaces H  (23) and (24). In addition let us introduce the operators Y 3 , Y ± ∈ su C (X M ) (which denotes again the complexification of su(X M )) given by They are related to the H TC,j by where f is a function in two variables x, y given by and f (X M , Y 3 ) has to be understood in the sense of functional caculus (both operators commute). As operators on H (µ) sym for fixed µ, the Y ± satisfy and for any function g(y) which is continuous on the spectrum of Y 3 , one finds We are now prepared for the first lemma.
we cannot exhaust all of SU (X M ). In other words: After U (µ 0 ) is fixed, we loose the possibility to choose an arbitrary U (µ) ∈ SU (H (µ) sym ) for another µ. Our analysis for two atoms suggests that the Lie algebra generated by the H TC,j is almost as big as su(X 2 ), but does not contain operators of the form A ⊗ ½ with a diagonal traceless operator A (except H TC,1 ). This observation suggests the choice of the Hamiltonians H CC,k with k ∈ {1, . . . , M+1} in Eq. (27), which lead to a dynamical group exhausting SU (X M ). This is shown in the next proposition, which completes the proof of Thm. 3. 6. Proof. Let us introduce the operators κ(k, j) ∈ u C (X M ) (the complexification of u(X M )) given by κ(k, j) (µ) = |µ; k µ; j| with k, j ∈ {0, . . . , M} and κ(k, j) = 0 if k ≥ d µ and j ≤ d µ , where d µ = min(µ, M+1); cf. Eq. (44). We can re-express Y ± in terms of κ(k, j) as Compare this to the definition of Y ± in (45). The truncation of the sums occuring for µ < M is now built into the definition of the κ(k, j). Similarly we can write the H CC,j for j ∈ {1, . . . , M} as H CC,j = κ(k, k) − κ(k−1, k−1). The κ(k, j) are particularly useful because their commutator has the following simple form: [κ(k, j), κ(p, q)] = δ jp κ(k, q) − δ kq κ(p, j). Note that all truncations for small µ are automatically respected. This can be used to calculate the commutator of H CC,k and Y ± . To this end we introduce the M × M matrix (A jk ) with A jj = 2, A j,k = −1 if |j − k| = 1 and A jk = 0 otherwise. Using (A jk ) we can write [H CC,j , Y + ] = k A jk κ(k, k−1). The matrix (A jk ) is tridiagonal, and therefore its determinant can be easily calculated and it equals M+1. Hence (A jk ) is invertible, and we can express κ(j, j−1) for j ∈ {1, . . . , M} as linear combination of the commutators [H CC,k , Y + ].
Now consider the Lie algebra l CC generated by H CC,k with k ∈ {1, . . . , M+1} and its complexification l CC,C . We have H TC,1 ∈ l CC since it can be written as a linear combination of the H CC,j . In addition H TC,3 = H CC,M +1 ∈ l CC and since S + ⊗ a, S − ⊗ a * can be written as (complex) linear combinations of H TC,3 and its commutator with H TC,1 we get S + ⊗ a, S − ⊗ a * ∈ l CC,C . To calculate the commutators [H CC,j , S + ⊗ a] note that according to (45) . Using the reasoning from the last paragraph, we see that f (X M , Y 3 )κ(k, k−1) ∈ l CC . Similarly we can show by using commutators with S − ⊗ a * that all κ(k, k+1)f (X M , Y 3 ) are in l CC , too. By expanding the function f we see in this way that for k ∈ {1, ..., M} the operators with P (k) := X M + (1−k)½ are elements of l CC,C .
To conclude the proof, we apply again Corollary 4. 6. Hence we have to consider the truncated algebra l [K] CC . To this end, look at the subalgebra l CC,k of l CC generated by the operators in (53). They are acting on the subspace generated by basis vectors |µ; k , |µ; k−1 and if we write A 1 = A + + A − , A 2 = i(A + − A − ) we get (up to an additive shift in the operator X M ) the same structure already analyzed in Lemma 4.7 (cf. also the operators A α,k in Eq. (37)). Hence we can apply the method from Sect. 4.2 to see that for all µ ∈ {0, . . . , K} the operators |µ; k µ; k| − |µ, k−1 µ, k−1|, |µ; k µ; k−1| and |µ; k−1 µ; k| are elements of l [K] CC,C (provided k ≤ d µ ). Now we can generate all operators |µ; p µ, j| with p, j ≤ d µ by repeated commutators of |k k−1| and |k−1 k| for different values of k. This shows that su C (H (µ) sym ) ⊂ l [K] CC,C for all µ ≤ K. By passing to anti-selfadjoint elements we conclude that l [K] CC = su(X M ) [K] holds for all K. Hence the statement follows from Corollary 4.6.

Strong controllability
The purpose of this section is to show how one can complement the block-diagonal dynamical groups from the last section to get strong controllability. We add one generator which breaks the abelian symmetry of the block-diagonal decomposition. The proofs for pure-state controllability and strong controllability are given in Proposition 5.2 and Proposition 5.6, respectively. This completes the proof of Theorems 3.2, 3.4 and 3.7.

Pure-state controllability
Consider a family H 1 , . . . , H n of control Hamiltonians on the Hilbert space H with joint domain D ⊂ H admitting a U(1)-symmetry defined by a charge operator X with the same domain. Since all the subspaces H (µ) are invariant under all time evolutions, which can be constructed from the H k , pure-state controllability cannot be achieved. For rectifying this problem, we have to add a Hamiltonian that breaks this symmetry in a specific way. We will do so by using complementary operators as in Definition 2. 2. Hence in addition to the projections X (µ) , µ ∈ N 0 we have the mutually orthogonal projections E α , α ∈ {+, 0, −} introduced in Sect. 3 and the corresponding derived structures. This includes in particular the subprojections X (µ) α ≤ X (µ) , µ ∈ N 0 and the Hilbert spaces H (µ) α onto which they project. Recall, that they satisfy X ± are required to be non-zero. For the following discussion we need in addition the Hilbert spaces H [K] = H [K] ⊕H (K+1) − , the projections F [K] onto them and the group SU (X, F [K] ) of U ∈ SU (X) commuting with F [K] . Furthermore we will indicate restrictions to the subspaces H [K] by a subscript [K], e.g. SU [K] (X, F [K] ) denotes the corresponding restriction of SU (X, F [K] ) which has the form SU [K] (X, F [K] ) = SU [K] (X) ⊕ SU (X (K+1) − ). Now one can prove the following lemma, which will be of importance in the subsequent subsections.
Lemma 5. 1. Consider a strongly continuous representation π : U(1) → U(H) with charge operator X, an operator H complementary to X, and the objects just introduced. For all K ∈ N, introduce the Lie group G X,F,K generated by SU [K] (X, F [K] ), exp(itH), t ∈ R and global phases exp(iα)½, α ∈ [0, 2π). Then the group G X,F,K acts transitively on the unit sphere of H [K] .
Proof. Consider φ ∈ H [K] and chooseŨ 1 ∈ SU [K] (X, F [K] ) such that X (µ) +Ũ1 φ = 0 for all µ > 0. This is possible, since SU (H (µ) ) acts transitively (up to a phase) on the unit vectors of H (µ) = H − holds. Hence exp(itH)φ ∈ H [K] and we can find aŨ 2 ∈ SU [K] (X, F [K] ) with φ 1 =Ũ 2 exp(itH)Ũ 1 φ ∈ H [K−1] . Applying this procedure K times we get φ K = U K · · · U 1 φ ∈ H [0] with U j ∈ G X,F,k . Similarly we can find V 1 , . . . , V K ∈ G X,F,k with ψ K = V k · · · V 1 ψ ∈ H [0] . Now note that the group G X,F,0 can be regarded as a subgroup of G X,F,k (which acts trivially on the orthocomplement of H [0] in H [K] ). Hence, the statement of the lemma follows from the fact that, due to condition (iii) of Def. 2.2, the group G X,F,0 acts transitively on the unit vectors in The first easy consequence of this lemma is the following result which is a proof of Thm. 2.3 which we restate here as a proposition.   (iv) The operator H d is complementary to X.
Then the system (1) with Hamiltonians H 0 = ½, H 1 , . . . , H d is pure-state controllable. Proof. We have to show that for each pair of pure states ψ, φ ∈ H and each ǫ > 0 there is a finite sequence U k ∈ U(H) with k ∈ {1, . . . , N} and either U k ∈ SU (X), U k = exp(itH d ), or U k = exp(iα)½ such that ψ − U N · · · U 1 φ < ǫ. To this end, first note that we can find K ∈ N such that ψ − F [K] ψ < ǫ/3 and φ − F [K] ψ < ǫ/3, where F [K] is the projection defined in the first paragraph of this subsection. Therefore ψ − U N · · · U 1 φ ≤ ψ−F [K] ψ + F [K] ψ−U N · · · U 1 F [K] φ + U N · · · U 1 F [K] φ−U N · · · U 1 φ < ǫ provided F [K] ψ − U N · · · U 1 F [K] φ < ǫ/3. Hence we can assume that ψ, φ ∈ H [K] and apply Lemma 5. 1. This leads to a sequence V 1 , . . . , V N ∈ G X,F,K with V N · · · V 1 φ = ψ. Now note that the dynamical group G generated by H 0 , . . . , H d contains by assumption the group SU (X), the unitaries exp(itH d ) and the global phases exp(iα)½. Hence with the definition of G X,F,K , we get for j ∈ {1, . . . , N} a W j ∈ G with [W j , F [K] ] = 0 and F [K] W j = V j , and therefore ψ = W N · · · W 1 φ. But by definition the dynamical group is the strong closure of monomials U N · · · U 1 with U j = exp(it j H k j ) for some t j ∈ R and k j ∈ {0, . . . , d + 1}. In other words for all U ∈ G, ξ ∈ H and ǫ > 0 we can find such a monomial satisfying U N · · · U 1 ξ − Uξ < ǫ. Applying this statement to the operators W j and the vectors W j−1 · · · W 1 φ concludes the proof. cf. [46,47,48]. Assume SG X,F,K = USp(K [K] ) holds. This would imply that SG X,F,K is self-conjugate (or more precisely the representation given by the identity map on SG X,F,K ⊂ B(H [K] ) is self-conjugate). In other words, there would be a unitary V ∈ U(H [K] ) with V UV * =Ū for all U ∈ SG X,F, K . HereŪ denotes complex conjugation in an arbitrary but fixed basis (cf. footnote 9). Now consider SU (H (µ) ) with d (µ) > 2. It can be identified with SU(d) in its first fundamental representation λ 1 (i.e. the "defining" representation). At the same time it is a subgroup of SG X,F,K (one which acts nontrivially only on H (µ) ⊂ H [K] ). Existence of a V as in the last paragraph would imply that λ 1 is unitarily equivalent to its conjugate representation, which is the d − 1 st fundamental representation. This is impossible if d (µ) > 2 holds. Hence V with the described properties does not exist and SG X,F,k has to coincide with SU (H [K] ) and therefore G X,F,K = U(H [K] ) as stated.
Finally we can conclude the proof of Thm. 3.4 which we restate here as the following proposition: Proof. Consider an arbitrary unitary U ∈ U(H). By Lemma 5.3, there is a sequence of partial isometries U [K] converging strongly to U, and by Lemma 5.5 we can assume that U [K] ∈ G X,F,K . Now considering the dynamical group G generated by the H j , define the subgroup G(F [K] ) of U ∈ G commuting with F [K] , and the restriction G [K] of G(F [K] ) to H [K] . The assumptions on the H j imply that G [K] = G X,F,K = U(H [K] ). Hence there is a sequence W K , K ∈ N of unitaries with W [K] ∈ G(F [K] ) ⊂ G and F [K] W [K] = U [K] . Since U [K] converges to U strongly, Lemma 5.4 implies that the strong limit of the W [K] is U, which was to show. This proposition shows strong controllability for all the systems studied in Sect. 3. The only exception is one atom interacting with one harmonic oscillator (Sect. 3.1). Here we have d (µ) = dim H (µ) ≤ 2 and we can actually find a unitary V with V UV =Ū for all U ∈ SU [K] (X 1 , F [K] ). However, the elements U of SU (X 1 ) are block diagonal where the blocks U (µ) ∈ SU (H (µ) ) can be chosen independently. This implies V ∈ SU K] (X 1 , F [K] ), which is incompatible with V H JC,4 V * = −H JC,4 (cf. Eq. (15) for the definition of H JC,4 ) which would be necessary for the group G X 1 ,F,K to be self-conjugate. Hence we can proceed as in the proof of Prop. 5.6 to prove Thm. 3.2.

Conclusions and Outlook
Many of the difficulties of quantum control theory in infinite dimensions arise from the fact that, due to unbounded operators, the group U(H) of all unitaries on an infinitedimensional separable Hilbert space H is in fact no Lie group as long as it is equipped fairly small set of control Hamiltonians for guaranteeing strong controllability, i.e. simulability. -Thus we anticipate the methods introduced here will find wide application to systematically characterize experimental set-ups of cavity QED and ion-traps in terms of pure-state controllability and simulability.