Energy and Identity in Atomic Photon Absorption

Paper idea

The paper asks a localization question

A photon is absorbed by an atom. Its energy becomes atomic excitation, with only a tiny recoil correction. But the photon's identity means something different: which photon alternative was absorbed?

The paper tracks that identity with a reference system \(R\). The full output \(A'D\) can still preserve the reference correlation. The atom \(A'\) alone may not.

\[ A\gamma \longrightarrow A'D \]

\(A'\) is the final atom. \(D\) is everything not inspected: recoil, motional modes, fields, environment, or other records.

The point of writing the process this way is to keep the missing degrees of freedom visible. If identity is not found in the atom, the framework asks whether it moved into \(D\), rather than treating it as simply destroyed.

Translation key

Symbols in plain language

Paper symbols
\(R\)	Reference labeling the photon alternatives.
\(I(R{:}S)\)	Reference bits recoverable from system \(S\).
\(A\gamma\)	Initial atom plus incoming photon.
\(A'\)	Final atom after absorption.
\(D\)	Unread recoil, motion, fields, environment, or records.
\(\rho^-,\rho^+\)	States before and after absorption.
\(\eta_{\rm abs}\)	Fraction of photon identity retained by \(A'\).

Budget and experiment symbols
\(\mathcal J_{\rm abs}\)	Identity budget before absorption.
\(\mathcal I_{\rm ret}\)	Identity retained in the atom.
\(\mathcal L_{\rm abs}\)	Identity leaked into \(D\).
\(\eta_{\rm int}\)	Eta-int: fraction retained in the internal manifold.
\(d_e\)	Dimension of the internal excited carrier.
\(d_A,d_D\)	Atom-space and discarded-space dimensions in simulations.
\(N\)	Number of possible photon labels.

Core asymmetry

Energy and identity obey the same global accounting, but not the same local behavior

Energy is almost forced into the atom. For an optical photon on hydrogen, the recoil fraction is roughly:

\[ \frac{\hbar\omega}{2Mc^2}\sim 10^{-9} \]

Here \(\hbar\omega\) is the photon energy, \(M\) is the atomic mass, and \(c\) is the speed of light. The atom keeps essentially all of the photon's energy.

Identity is measured by retained correlation with \(R\). Its localization fraction \(\eta_{\rm abs}\), pronounced eta-abs, is:

\[ \eta_{\rm abs} = \frac{I(R{:}A')_{\rho^+}}{I(R{:}A\gamma)_{\rho^-}} \]

This can range from \(0\) to \(1\). It is dynamical, not recoil-pinned.

The claim is not that mutual information is new. The claim is the contrast: energy is locally pinned near one, while identity can localize, leak, or disappear from the atom's accessible state.

Paper results

The two main mathematical handles

The paper builds a conserved identity budget, then studies how much of that budget is locally readable from the atom.

Accounting identity

The full absorption map is reversible on \(A'D\). Globally, the reference correlation is not destroyed.

\[ \mathcal J_{\rm abs} = \mathcal I_{\rm ret} + \mathcal L_{\rm abs} \]

The atom keeps \(\mathcal I_{\rm ret}\). The rest lives in \(D\).

This is the bookkeeping version of conservation: identity is not erased by the full map, but it may leave the subsystem we actually measure.

That is why the paper separates retained identity from leaked identity. The total is the stable object; the split tells us what an atom-only observer can actually recover.

Two-state curve

For two possible photon labels, the atom's retained identity depends only on the overlap of the two final atomic states.

Here \(|e_0\rangle\) and \(|e_1\rangle\) are the two possible final atomic states, \(c_A\) is their overlap, and \(h_2\) is the binary entropy function.

\[ \eta_{\rm abs} = h_2\!\left(\frac{1+c_A}{2}\right), \qquad c_A=|\langle e_0|e_1\rangle| \]

Orthogonal outputs retain the bit. Identical outputs retain none of it.

This curve is the cleanest falsifiable shape in the paper: if the overlap is swept experimentally, the retained fraction should follow the binary-entropy curve.

Finite internal dimension no-go

If only an internal excited manifold of dimension \(d_e\) is read out, it can hold at most \(\log_2 d_e\) identity bits.

The symbol \(\eta_{\rm int}\), eta-int, is the internal localization fraction: retained internal identity divided by the starting identity budget.

\[ \eta_{\rm int} \le \frac{\log_2 d_e}{\mathcal J_{\rm abs}} \]

For \(N\) balanced, distinguishable photon labels, \(\mathcal J_{\rm abs}=\log_2 N\).

The result is a no-go, not a typical-case statement. Even a perfect internal memory cannot store more independent label bits than its internal dimension allows.

Experimental reanalysis workflow for retained internal identity — Existing tomography can check endpoints and carrier ceilings. A controlled sweep of overlap \(c_A\) tests the functional curve.

Experiments PDF

What the numerical experiments are doing

These are simulations, not laboratory data. They make the paper's bookkeeping visible by sending photon labels through idealized absorption maps and measuring how much label information remains in the atom.

In this section, \(x\) is one photon label, \(|\Psi_x\rangle\) is the final joint state created by absorbing that label, and \(\rho_A^x\) is what the atom alone looks like after we ignore \(D\). The operation \(\operatorname{Tr}_D\) means "trace out \(D\)," or discard the unobserved degrees of freedom mathematically.

\[ |x\rangle \mapsto |\Psi_x\rangle_{A'D}, \qquad \rho_A^x=\operatorname{Tr}_D|\Psi_x\rangle\!\langle\Psi_x| \]

The readable atomic identity is the Holevo information. Here \(S(\rho)\) is von Neumann entropy in bits, \(\chi_A\) is the number of label bits readable from the atom, and \(\eta_A\) is the retained fraction:

\[ \chi_A = S\!\left(\frac1N\sum_x \rho_A^x\right) - \frac1N\sum_x S(\rho_A^x), \qquad \eta_A=\frac{\chi_A}{\log_2N} \]

Figure 1

One bit, one overlap

Two photon labels lead to two final atomic states. As those states become more alike, the atom has less identity information.

The horizontal axis is the overlap between those two atomic states. Moving right means the atom's two possible outputs are harder to tell apart.

The vertical axis is the fraction of the original one-bit label that remains readable from the atom. Near the left edge the atom acts like a good record; near the right edge it has effectively forgotten which label arrived.

This is distinguishability made visible.

Two-state identity localization curve — Orthogonal atomic outputs retain one bit; identical outputs retain none.

Figure 2

The best possible finite memory

This is the ideal capacity limit. A finite atom can store only so many label bits internally.

Here \(d_A\) is the dimension of the atomic space used in the simulation, and \(N\) is the number of possible photon labels.

Each curve asks a best-case question: if the atom were used as an ideal memory, what fraction of the label could fit inside it?

\[ \eta_A\le\min\!\left(1,\frac{\log_2d_A}{\log_2N}\right) \]

If the labels do not fit, identity must be outside the atom.

Finite carrier envelope — The dimension envelope for an ideal internal carrier.

Figure 3

Generic absorption is worse than ideal memory

Random reversible maps usually spread the label into \(D\). The atom retains much less than the theoretical ceiling.

Each sample is a different generic absorption map. The plot asks what happens without carefully engineering the atom to act as a memory.

The dashed curves show what is allowed in principle. The solid curves show what random maps usually achieve, which is far lower.

A good memory is structured. A generic coupling is not.

Random absorption channels below the dimension envelope — Solid curves are random medians; dashed curves are dimension ceilings.

Figure 4

Dimension pressure

Rows vary atom dimension \(d_A\). Columns vary the number of photon labels \(N\). Brighter cells mean more retained identity.

Read it like a map: moving downward gives the atom more room, while moving right gives it more labels to remember.

The dark upper-right region is the hard regime: small atomic carrier, many possible labels, and most identity outside the atom.

Bigger atoms help, but generic maps still leak most high-\(N\) identity.

Heatmap of retained identity by atom dimension and number of labels — A compact view of retained fraction across \(d_A\) and \(N\).

Figure 5

The bookkeeping line

For \(N=16\), the incoming identity budget is four bits. Each random map splits those bits between retained identity and leakage.

Here \(L_A\) means leaked identity: the part not readable from the atom alone.

Moving right means the atom remembers more. Moving up means more of the same fixed budget has gone into discarded degrees of freedom.

\[ L_A=\log_2N-\chi_A \]

Retained plus leaked identity equals the starting budget.

Identity budget split between atom and discarded modes — Every point respects the fixed identity budget.

Figure 6

More unobserved space means less atomic memory

Holding \(N=16\) fixed, increasing \(d_D\) gives the ignored space more room to carry the label.

Here \(d_D\) is the dimension of the discarded space \(D\).

This isolates the role of the unobserved sector. More available hidden space makes it easier for identity to leave the atom.

The curves should be read as a generic trend, not a lab claim: once \(D\) is larger, random maps have more room to make the atom's reduced states look alike.

The full output still remembers; the atom alone remembers less.

Retained identity decreases as discarded dimension grows — Generic coupling decouples the atom as the discarded space grows.

Takeaway

What the project says

The paper's thesis is the asymmetry: energy and identity both obey global accounting across \(A'D\), but energy is pinned near the atom while identity can be distributed.

The simulations show the geometry behind that statement. A finite atom can absorb the photon's energy while much of the photon's label lives in discarded degrees of freedom.

The numerical plots are explanatory simulations. They do not replace the proposed experimental test: a controlled sweep of the atomic-state overlap \(c_A\), plus tomography to reconstruct retained identity.