Projection Postulate of Quantum Mechanics. Where does the Projection occur in the detection process?

Question

This question is about a specific instance of the measurement problem.

Let's say we excite a single emitter producing a single photon (described by state $ |\psi_{i}>$), which then "travels" through the experimental setup until it can interact with a photon counting detector.

In a simplified picture, the detector can be described by an array of atoms with a ground state and a continuum of excited states. The incident radiation can interact with these atoms and excite electrons to the continuum, which are then used to generate an electronic signal. If its amplitude is above a certain threshold, it will result in a digital signal being sent from the detector to a computer, where the signal can be processed, visualized and seen by a human. (The threshold corresponds to a minimum energy level of the incident radiation.)

With this model, one can calculate the probability to detect a photo-ionization process per unit time: $$w(r,t) = \frac{dP_{e}}{dt} = \xi < \psi_{i}|E^{-}(r,t) E^{+}(r,t)|\psi_{i} >$$

Where $\xi$ is a parameter describing the efficiency of the detector.

With this detection probability density and using the mathematical trick of inserting the identity as a complete set of basis states, one can now define a spatio-temporal wavefunction. (Defining a meaningful wavefunction for the free photon before detection is not straight forward because of its infinite de Broglie wavelength). (The calculations can be found in standard Quantum Optics books)

$$\psi(z,t) = <0|E^{+}(z,t)|\psi_{i}>$$

Going to a single positional variable for simplicity (experimentally for example through the use of a single mode fiber right in front of the detector).

The probability associated with this wavefunction $w(r_{0},t)$ defined above, can be sampled in a typical TCSPC (time correlated single photon counting) measurement.

With each click/measurement of the detector, the state is projected (the wavefunction collapses in the Copenhagen interpretation) and the times between excitation of the emitter and detection of the photons can be plotted. A histogram can be obtained, approximating the probability density $w(r_{0},t)$ (fixed detector position, variable detection time).

The question is now, where does the quantum mechanical "measurement" occur in the detection process. In the sense of projection onto the measured state. Does it happen when the photon ionizes an atom in the detector? This process can be described purely quantum mechanically and projection is not necessary here. The postulated projection could just be an effective model to explain a more complete quantum field theory that has not been discovered yet. where the final state is still probabilistic, however the relevant information about the detector producing a click at time t is determined with unit probability.

Analogous to the process that could happen in a detector is the process of decoherence, which leads to an evolution of the relevant subsystem that is not unitary. (Here the whole system still evolves according to the full Hamiltonian producing unitary evolution.)

The uncertainty is in the exact atomic configuration of the detector about which the physicist does not care. This complicated configuration of the atomic states does however allow for sampling of $w(r_{0},t)$ at different times, otherwise it wouldn't be a detector. If this last part were true, then the measurement problem is still not solved though, because a probability that is almost one does not represent the system going from the "possible" to the "actual" physical realization. For this, the probability has to be exactly one for the realized state. Maybe with a further extension to quantum field theory, it will be possible to achieve this. Then no special interpretation of quantum mechanics will be necessary, because the projection axiom can be dropped completely.

Or does it occur when the experimental physicist looks at his computer screen? This would mean that before he looked at the screen, some electronic bits in the computer representing the arrival time of the photon were neither 0 or 1. The last explanation that only a "conscious being" can project a system onto a quantum mechanical state would imply that quantum mechanics is only a valid model of a reality where these conscious beings exist right now.

Answer 1

This question is asking about the location of theHeisenberg cut in an experiment. Different theories* of quantum mechanics have different answer to this question.

• The Copenhagen theories (or textbook version of Quantum mechanics) insists that a Heisenberg cut does exist somewhere between the microscopic regime and macroscopic regime but it does not specify in a quantitative or scientific way where the cut happens. Because of this the Copenhagen interpretation results in an incomplete scientific theory. For the experiment in question, my guess is that most proponents of this interpretation would suggest that the cut occurs either when the single emitter decays (i.e. the interaction between the emitter and the EM field) or when the photon is detected by the "macroscopic" sensor (i.e. the interaction between the EM field and the sensor).
• Objective collapse theories attempt to remedy the wrongs of the Copenhagen interpretation by providing a quantitative scientific theory of where the Heisenberg cut should be. The gist of theories is that they say that once a quantum state involves a "sufficiently large" superposition it spontaneously collapse. The collapse probability might be a function of the spatial extant of a quantum superposition state, the total mass of a superposition state, or the number of particles included in an entangled superposition state. These theories might involve explicit non-linear extensions to the Schrodinger equation. These theories are experimentally testable and as we generate quantum superpositions of larger and larger quantum systems we put bounds on the parameter space of these theories.
• The Everettian theories (my preferred name for the many worlds theory) insists that a Heisenberg cut never happens. There is no collapse of the wavefunction ever, only unitary evolution. As the OP rightly points out, a sufficiently complex quantum theory can describe the interaction between the photons and the sensor, the sensor electrons and the downstream electronics, the digital bits, the photons sent to the human's eye, the ions carrying electric signals from the human's eye to their brain etc. All of these can be described with quantum mechanics without collapse. The Everettian interpretation takes all of this seriously and as sufficient. In this interpretation the human and their organs (i.e. their brain) will be in entangled states with the emission source throughout and at the end of the experiment.

The upside to the Copenhagen and Objective Collapse Theories compared to the Everettian theory is that they don't ever have human brains in superposition states. This means that one can maintain a naive dualist notion of consciousness which purports that physical states of the brain are 1:1 with mental states. The Everettian interpretation, on the other hand, allows humans and brains to be in superpositions but does not provide any instruction to the user as to how to relate these superposition physical states of the brain with conscious mental states.

The upside of the Everettian interpretation is of course that it is the most mathematically simple version of quantum mechanics in that it only had unitary evolution and does not need to try to explain where and how the non-linear of quantum collapse/Heisenberg cut happens.

All of these theories are consistent with all experiments to date, but it is possible experimental evidence will arise to cause us to reject one or more of these. Also, there are other interpretations/theories of quantum mechanics flying around that aren't addressed in this answer. All of this is to say: the quantum measurement problem is NOT solved despite what anyone might tell you. Work still needs to be done in terms of theoretical and experimental physics as well as philosophy and possibly neuroscience to find solutions.

*I use the word theory here instead of interpretation. Some people insist that two different interpretations must make the same predictions because that is the definition of an interpretation. I disagree with that constraint on the definition of itnerpretation. Nonetheless, whatever your definitions this much is true: The Copenhagen, objective collapse, and Everettian "versions" of quantum mechanics make different physical predictions so use whatever word you think appropriate for those.