Weak covariance and the correlation of an observable with pre-selected and post-selected state energies during its time-dependent weak value measurement
Tóm tắt
The peculiar weak energy of evolution appears as a factor in the equation of motion
$$ \dot{A}_{w}$$
for a time-dependent weak value of an observable
$$\hat{A}$$
. This energy has the mathematical form of the weak value of the difference between the two Hamiltonian operators
$$\hat{H}_i $$
and
$$\hat{H}_f $$
that describe the evolution of the associated pre- and post-selected states, respectively. Here, the weak covariance
$$\mathrm{cov}_w \left( {\hat{X},\hat{Y}} \right) $$
for operators
$$\hat{X}$$
and
$$\hat{Y}$$
is introduced and it is shown that
$$ \left| {\dot{A}_{w}} \right| $$
can be expressed entirely in terms of
$$\mathrm{cov}_w \left( {\hat{H}_f ,\hat{A}} \right) $$
,
$$\mathrm{cov}_w \left( {\hat{A},\hat{H}_i } \right) $$
, and an angle
$$\theta $$
that is governed by the complex valued nature of the terms defining each covariance. Several cases are briefly discussed and an experiment is used to illustrate the
$$\hat{H}_i =\hat{0}\ne \hat{H}_f $$
case. It is shown that
$$cov_w \left( {\hat{H}_f ,\hat{A}} \right) $$
is observed in the associated experimental data.
Tài liệu tham khảo
Aharonov, Y., Albert, D., Casher, A., Vaidman, L.: Novel properties of preselected and postselected ensembles. In: Greenberger, D. (ed.) New techniques and ideas in quantum measurement theory, pp. 417–421. New York Academy of Sciences, New York (1986)
Aharonov, Y., Albert, D., Vaidman, L.: How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351–1354 (1988). https://doi.org/10.1103/PhysRevLett.60.1351
Aharonov, Y., Vaidman, L.: Properties of a quantum system during the time interval between two measurements. Phys. Rev. A 41, 11–20 (1990). https://doi.org/10.1103/PhysRevA.41.11
Ritchie, N., Story, J., Hulet, R.: Realization of a measurement of a ‘weak value’. Phys. Rev. Lett. 66, 1107–1110 (1991). https://doi.org/10.1103/PhysRevLett.66.1107
Parks, A., Cullin, D., Stoudt, D.: Observation and measurement of an optical Aharonov-Albert-Vaidman effect. Proc. R. Soc. 454, 2997–3008 (1998). https://doi.org/10.1098/rspa.1998.0288
Resch, K., Lundeen, J., Steinberg, A.: Experimental realization of the quantum box problem. Phys. Lett. A 324, 125–131 (2004). https://doi.org/10.1016/j.physleta.2004.02.042
Wang, Q., Sun, F., Zhang, Y., Li, J., Huang, Y., Guo, G.: Experimental demonstration of a method to realize weak measurements of the arrival time of a single photon. Phys. Rev. A 73, 023814 (2006). https://doi.org/10.1103/PhysRevA.73.023814
Hosten, O., Kwiat, P.: Observation of the spin Hall effect of light via weak measurements. Science 319, 787–790 (2008). https://doi.org/10.1126/science.1152697
Parks, A.: The geometry and significance of weak energy. J. Phys. A: Math. Gen. 33, 2555–2567 (2000)
Parks, A.: A weak energy stationary action principle for quantum state evolution. J. Phys. A: Math. Gen. 36, 7185–7192 (2003)
Parks, A.: Weak energy: form and function. In Struppa, D., Tollaksen, J. (eds.) Quantum Theory: A Two-Time Success Story, pp. 291–302 . Springer, Milano, (2014). https://doi.org/10.1007/978-88-470-5217-818
Parks, A.: Time-dependent weak values and their intrinsic phases of evolution. J. Phys. A: Math. Theor. 41, 335305 (16pp) (2008). https://doi.org/10.1088/1751-8113/41/33/335305