  We propose a method to reconstruct the vibrational quantum state of molecules excited by a general excitation laser pulse. Unlike existing methods, we do not require the molecules before excitation to be in a pure state, allowing us to treat the important case of initially thermally excited molecules. Even if only a single initial level is appreciably populated, initial levels with small populations can still give major contributions to the unknown vibrational state, making it essential to take them into account. In addition to the excitation pulse, the method uses two incident, short laser pulses in a non-co-linear geometry to create four-wave mixing in the molecules. The measurements used in the reconstruction are spectra of the outgoing four-wave mixing pulse at different time delays of the excitation laser pulse. An important point is that the method does not require detailed knowledge of molecular transition moments between excited states nor of any of the incoming laser pulses, but circumvents this requirement by using one or more calibration laser pulses in a separate experiment either before or after the main data are recorded. The only requirements for the calibration laser pulses are that the constant parts of their spectrums should together cover the spectral range of the excitation laser pulse, and the constant part of each should have sufficient spectral overlap with one other calibration pulse to populate two of the same levels. Finally, we discuss the extension of the reconstruction method in this paper to more general situations, hereby presenting the new idea of quantum state reconstruction through perturbations with calibration. 
Reconstructing vibrational states in warm molecules using four-wave   mixing with femtosecond laser pulses