As we know, annihilation operator is a non-hermitian operator which can take complex eigenvalues. I would like to know if the eigenvalue of this operator was ever observed in the laboratory or at least a proposal? If so, please provide me a reference.
Whilst Stam is quite correct in coherent states being the eigenstates of the annihilation operator, one needs to be careful about the concept of the annihilation operator as an observable in the conventional quantum mechanical sense.
As Stam also points out, the annihilation operator is non-Hermitian. This means that in the conventional von-Neumann sense, it cannot be an observable and cannot give rise to measurement outcomes for strong projective measurements (such measurements need to yield real number outputs).
However it is possible to measure the annihilation operator directly as a weak operator. This has been done and is found in a few places, but for convenience I am just going to point you to Lundeen and Resch, "Practical measurement of joint weak values and their connection to the annihilation operator" Physics Letters A 334 (2005) 337-344, DOI: 10.1016/j.physleta.2004.11.037.
Hope this helps
To measure the eigenvalue(s) of any operator means constructing its eigenstates. The eigenstates of the annihilation operator are the coherent states. Read, for example, http://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_5.pdf`
http://www.scholarpedia.org/article/Coherent_state_%28Quantum_mechanics%29
for the theory and
http://arxiv.org/ftp/quant-ph/papers/9710/9710025.pdf
for the experiments.
The annihilation operator is a quite particular non-hermitian operator.
Whilst Stam is quite correct in coherent states being the eigenstates of the annihilation operator, one needs to be careful about the concept of the annihilation operator as an observable in the conventional quantum mechanical sense.
As Stam also points out, the annihilation operator is non-Hermitian. This means that in the conventional von-Neumann sense, it cannot be an observable and cannot give rise to measurement outcomes for strong projective measurements (such measurements need to yield real number outputs).
However it is possible to measure the annihilation operator directly as a weak operator. This has been done and is found in a few places, but for convenience I am just going to point you to Lundeen and Resch, "Practical measurement of joint weak values and their connection to the annihilation operator" Physics Letters A 334 (2005) 337-344, DOI: 10.1016/j.physleta.2004.11.037.
Hope this helps
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