Complete controllability of finite-level quantum systems (Q2729404)

The authors consider the single input bilinear control system on \(U(N)\): NEWLINE\[NEWLINEi\hbar\dot U= (H_0+ vH_1)U,NEWLINE\]NEWLINE where \(v\) is the input, \(H_0= \sum^{n= N}_{n=1} E_n e_{nn}\) and \(H_1= \sum^{n= N}_{n=1} d_n(e_{n,n+1}+ e_{n+1,n})\); \(e_{ij}\) is the standard basis of \(M_N(\mathbb{R})\).NEWLINENEWLINENEWLINEThey first invoke previous results which say that complete controllability for the above system means that \({\mathcal L}= \langle iH_0, iH_1\rangle_{LA}= {\mathfrak u}(n)\). One uses the basis of \({\mathfrak s}{\mathfrak u}(N): x_{nn'}= e_{nn'}- e_{n'n}\), \(y_{nn'}= i(e_{nn'}+ e_{n'n})\) and \(h_n= i(e_{nn}+ e_{n+1,n+1})\) and some of its Lie bracket properties (involving in particular \(iH_1\)) allow to establish lemmas which guarantee that if \(x_{p,p+1}\), \(y_{p,p+1}\) are in \({\mathcal L}\) for some \(p\) then this will also be the case of all elements of the basis. (The gap between \({\mathfrak s}{\mathfrak u}(N)\) and \({\mathfrak u}(N)\) is filled when \(\text{tr}(H_0)\neq 0\).) The main result which states that the above sufficient conditions are fulfilled if some properties of the spectrum of \(H_0\) hold, is proved. As before the proof is computational and involves here iterated Lie brackets using \(iH_0\) and the dependence on \(iH_1\) is linear. One potential bad case is when \(E_i= {E_{i-1}+ E_{i+1}\over 2}\) but it is handled by computing successively Lie brackets of a quantity of the type \([iH_0,[iH_0, iH_1]]\) (which involves the \(e_{nn}\) and is modified in the succession) with alternatively quantities of the type \(iH_1\) (which involves the \(y_{n,n+1}\) with successive modifications) or \([iH_0,iH_1]\) (which involves the \(x_{n,n+1}\) with successive modifications); here one supposes that the \(d_i\) are well behaved. A bad case is when moreover \(2d^2_n- d^2_{n-1}- d^2_{n+1}\) is constant and this leads to situations with a lack of controllability but this does not cover all of them. An example shows that controllability holds for all N-level harmonic oscillators of a certain type but fails in the infinite-dimensional case. A last section treats the case \(N=4\).NEWLINENEWLINENEWLINEThe lesson is that generically one has complete controllability.