The quote from E.T. Jaynes is perhaps not as well known as
Schrödinger's "In the first place it is fair to state that we are not
experimenting with single particles, any more than we can raise Ichthyosauria in
the zoo." (Are There Quantum Jumps?
Part I, The British Journal for the Philosophy of Science, 3, (1952), 109-123 [B
12]), and of course Schrödinger's cat is even better known. The problem is
always the same: does quantum theory (QT) describe the behavior of an ensemble
or individual particles? And the question arises whether there is an "objective
randomess of the QT" (quantum indeterminism), or a deterministic "development of
the individual" (cat, dinosaur or atom) runs continuously, or only "jumps" to a
value upon observation. Not to mention the multitude of interpretations of the
QT with which one tries to sweep these contradictions under the rug. However, another problem arises from this: the linear Schrödinger equation always leads to linear differential equations, i.e. to (a sum of) exponential functions as a solution (see Weisskopf-Wigner). While the temporal evolution of the transition measured by Minev et al. is described by a tanh-function (curve fit), which is typical for a non-linear process. We therefore want to take a closer look at whether it is really possible to derive a non-linear result from a linear theory (QTT).From arXiv:1803.00545v3 [quant-ph] 12 Feb 2019, Supplement page 4: Supplement page 5:
From a purely mathematical point of view, the procedure for deriving (5) can be divided into an "exact solution (3x3-system)" and an "approximate solution (2x2-system)":
With the definition
the "usual notation of inversion" (squares of the amplitudes = densities)
becomes
One wonders what this definition is good for (also for the X and Y components of the Bloch vector).
State of the atom with G = ground, B = bright, D = dark:
So a Equation (5) (Supplement page 4) leads to the matrix
respectively the
or
Note:
So there is a closed solution (there is always a numerical one), but with Maple's solution algorithm it is so extensive that it has "no simple form" (the output comprises more than 1 million characters :). But you can represent them graphically, e.g. the
Red: ground state (G), green: dark state (D), blue: bright state (B), values of the four parameters in MHz, animations with 1 frame per second. 1. Order of magnitude of the numbers as in Minev et al., the decay rate of the dark state is changed.
The amplitude of the bright state is practically zero! 2. As 1., but with a small decay rate of the bright state, which is now visible.
3. The Rabi frequency of the DG transition is changed. "Damped oscillation" furthermore...
4. The inversion Z (light blue) is also shown. The Rabi frequency of the BG transition is changed. Transition from Z periodic, to Z aperiodic.
5. Amplitude
squares of G and D increased by a factor of 100. Ω
But you can also just insert numbers to see the simple form of the solution, e.g .
results in
thus the mentioned linear combination of exponential functions (here with real eigenvalues).
The solution of the 3x3 system, which is known in principle, is
not used. Instead, an approximation is introduced (which can only be found in
the dissertation): The change in the bright state (B) over time is set to zero,
which means that the bright state can be eliminated from the 3x3 system, and a
The solution of this DE-system is in a closed form:
In other words, an exact and manageable solution that can of course also be calculated without CAS. Instead of animations, here are just a few examples of how you
can set the parameters with Maple with sliders, e.g. for the
C (red and blue)_{D}
With the exact solution of the 2x2 system, it is
easy to investigate how the change in the four parameters affects the Rabi-oscillations
of a 2-level system. But even this simplified solution (
The 2x2 system (see above, (3.1)) is
With the definition
the equation of motion for W is obtained:
Insertion of the time derivatives of the
amplitudes
The solution W(t) of this DE is a tan-function, which does not become a tanh-function even for negative radicands of the root:
But this exact solution is not used either (for good reason: it is "unphysical"). Instead, the equation of motion for W is replaced by this approximation
in which, however, beyond the approximation mentioned above, the quadratic term in W is also missing, which would only make sense for W < 1. With these simplifications (and the neglect of a summand -1) one then obtains an exponential function for W, and finally
These equations are then used to
Quoting from Nature 570, 200–204 (2019). https://doi.org/10.1038/s41586-019-1287-z
Note 1: " The phrase "probability of success to reverse the
transition (or not)" probably means the density (population) of one or the other
state (G or D). This means
that P Note 2: With
C i.e. non-linear differential equations that cannot be represented with the QTT (and the linear Schrödinger equation used there).
1. With the QTT, a 3x3 DE-system is set up for
the amplitudes, which has an exact but unwieldy solution (cubic equation). 8. A consistent treatment of atomic transitions can be found in Ensemble - Individuum, or in Über die spontane Emission von Photonen (quote with the symbols from there):
"The
change
of the
ground state ρ Putting
this physical fact in the foreground instead of assuming
an "exponential decay" as the solution, one obtains the differential equation
with the proportionality constant
also known as the = also known as the The question now is how, of all things, a series of approximations can be used to arrive at a tanh-solution that was already known to the neoclassic, without citing the neoclassic (e.g. E.T. Jaynes). 9. The decay constants are not determined directly in the experiment, but serve as "free parameters" that are calculated / adapted. 10. In the transition to the dark state, only the bright state occurs in the time constant. (Only the amplitude of W is co-determined by the rabi frequency of the dark state.) 11. A value for the dark state decay constant is
nowhere to be found in the article. It is eliminated by the assumption
γ
Minev et al.: Porrati et al.: © Juni 2021, Dr. Michael Komma (VGWORT)
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