The studies of the energetic structure of quantum systems – isolated atoms/molecules and solids – aims at determining the allowed stationary states (energy “eigenstates”), that is the states which stay unchanged in time, into which one “projects” the quantum system by means of an energy measurement. The stationary eigenstates are of paramount importance for describing the quantum system since, in principle, all the accessible states for the system are (mathematically/formally) obtained as linear superposition of the stationary ones [Salières01]. In particular, the states which evolve in time – the time-dependent (TD) states - are either coherent superposition (according to Fourier analysis, see Eq 1) or incoherent mixture of steady-states. Establishing the complete energy map of the steady states constitutes the fundamental goal of spectroscopy, which has been worked out for about one century in both isolated and collective systems:

Eq1 : .

Fully describing the time-dependent states, e.g., a coherent superposition of eigenstates as in Eq 1, requires knowledge of the “weight” C_E (complex coefficient) of each eigenstate . This is not provided by conventional (energy resolved) spectroscopy which characterizes only the modulus of the weight but not its complex argument or phase.

To completely characterize the time-dependent state, one has therefore two options: 1) by mean of quantum interferences in phase spectroscopy, measure the spectral phase of the weight coefficient over a large enough energy range, or 2) probe directly the TD state in time in pump-probe experiment.

The spectroscopy option (1) gives the most detailed insight into the TD state. However, it may be practically difficult – or extremely difficult and finally impossible -, e.g., in the case of one or several continua of eigenstates involved in Eq 1, to complete the spectroscopic description.

The “real time” option (2) is more straightforward, for the time-resolved measurement directly probes the time-dependent state as a whole – amplitude and phase – and its time-dependent properties without the need of its projection onto steady eigenstates as in Eq 1.

Spectroscopy and real time pump-probe studies thus appear complementary. Ultrafast studies often combine them.