Complementarity and entanglement in a simple model of inelastic scattering

An understanding of quantum concepts is often built by overlapping classical analogies, analytical models, and numerical illustrations. After learning about a particle’s classical momentum, students are shown that a particle’s de Broglie wavelength depends on the inverse of that momentum. Students may then analytically model the one-dimensional (ID) reflection and transmission of de Broglie waves from a potential barrier, whose analogous classical counterparts would all have been stopped by that same barrier. Further insights can be gained by numerically modeling such reflection and transmission events using wave packets. Each of these approaches teases out new qualitative and quantitative connections ¹.

Recent years have seen an increasing consensus that the concept of entanglement—the inability of some quantum states to be written as the product of individual particle states—should be a part of every student’s quantum tookit. As Daniel V. Schroeder pointed out in “Entanglement isn’t just for spins,” entanglement generically arises when quantum particles interact with each other ². In that paper, Schroeder presented two dynamical models showing how entanglement emerges, but lamented that such examples are rarely included in quantum mechanics textbooks. “The reason,” he conceded, “is probably that despite their conceptual simplicity, a quantitative treatment of either scenario requires numerical methods.”

This paper presents a conceptually simple model that can model entanglement without resorting to numerical methods. The level of mathematical difficulty in this treatment is similar to that of the commonly taught models involving potential barriers. The model is a simplified 1D treatment of inelastic scattering. It is well-known to the electron spectroscopy community ³, and is similar to the model of a 1D atom scattering off a 1D harmonic oscillator presented in this journal several decades ago by Knudson ⁴, though this treatment differs in its attention to the time-evolution of the scattering process.

In the model under review, a beam particle is coupled to a harmonic oscilator, as illustrated in Fig. 1. The model Hamiltonian sums the contributions of a 1D free particle of mass $m$ (position variable $x$, conjugate momentum $p_{x}$), a 1D harmonic oscillator of reduced mass $\mu$ and resonance frequency $\omega_{0}$ (position variable $y$, conjugate momentum $p_{y}$), and an interaction term $H^{1}$ that couples the two systems:

The coupling constant $\alpha$ rationalizes units and tunes the strength of the interaction, and the spatially-varying “window function” $f(x)$ is taken to have units of length^-2, and to die off as $x$ approaches $\pm\infty$. The negative sign in $H^{1}$ means that the oscillator is attracted toward the beam when $\alpha$ and $f(x)$ are positive, but changing this sign would not significantly alter any results.

Though its form is simple, this model can be used to capture real physics. For instance, in “Characterizing Localized Surface Plasmons Using Electron Energy-Loss Spectroscopy,” Cherqui et al.⁵ derive “classical” and “quantum” Hamiltonians for the electron-plasmon interaction (their Eqs. 14 and 15) that map respectively onto our Eq. I and Eq. 30, but for the fact that an electron couples to infinitely many plasmonic modes, while our beam couples to just one oscillator. For the plasmonic case, as an electron nears a nanoparticle, a surface charge is induced, which in turn interacts with the electron via the Coulomb potential. The physics in this model is analogous, with the passing beam tugging on the oscillator, and the oscillator in turn tugging back on the beam.

In addition to its discussion of entanglement, this presentation also highlights conceptual issues surrounding classical/quantum complementarity.⁶ Complementarity is explored implicitly as different approaches to the model are presented—a classical approach (Sec. II), and, after a review of perturbation theory (Sec. III), partially and fully quantum approaches (Secs. IV and V). Complementarity then is addressed explicitly in Sec. VI, where the transition from one approach to the next is discussed, and Sec. VII explores post-interaction entanglement, showing how predictions may change in entangled systems following a measurement. Sec. VIII is a short summary.

Solving the model in the classical approach requires only standard tools. Applying Hamilton’s equations

to Eq. I yields the classical equations of motion for the beam position $x$ and oscillator displacement $y$

which can be solved either by analytical or numerical methods.

But there is a problem. The classical approach is empirically inadequate for micro- or mesoscopic systems, since the beam particle in fact will not lose energy each time it passes the (generalized) oscillator. Sometimes the beam particle will be observed to lose energy, but most of the time it will pass by without any energy loss at all.

As a result, we will want to develop a quantum version of the model. We are interested in both analytical and numerical solutions, and when we want analytical solutions in quantum mechanics, we often turn to perturbation theory. So let’s briefly remind ourselves, now, about time-dependent perturbation theory ⁹.

Suppose we have an unperturbed Hamiltonian operator $\hat{H_{0}}$ whose eigenvectors $\ket{n^{0}}$ are known:

Solutions of the Schrodinger equation

can be written, without loss of generality, in terms of coefficients $c_{n}(t)$:

When the Hamiltonian in question is the unperturbed Hamiltonian $\hat{H}=\hat{H^{0}}$, one can substitute Eq. 21 into the Schrodinger equation to confirm that the coefficients $c_{n}$ are constant in time. When the Hamiltonian is perturbed by a weak (and possibly time-dependent) term $\hat{H^{1}}(t)$ such that $\hat{H}=\hat{H^{0}}+\hat{H^{1}}(t)$, the $c_{n}$ coefficients will still be useful, since they vary more slowly in time than state coefficients would without factoring out $\exp(-iE_{n}^{0}t/\hbar)$.

Applying the operator $\bra{f^{0}}\exp(+iE_{f}^{0}t/\hbar)$ to both sides of Eq. 20 and inserting the expansion of Eq. 21, we can find the time-dependence of any state coefficient $c_{f}$ as

where

For a system that begins in initial state $\ket{i^{0}}$ with energy $E_{i}$ at time $t=-\infty$, Eq. 22 leads to a first-order perturbation coefficient $c_{f}$ for exciting the system to some final state $\ket{f^{0}}$ with energy $E_{f}$ as

which means the first-order probability of transition to state $\ket{f^{0}}$ is found by allowing $t\rightarrow\infty$ and calculating

from the Born rule. When the initial and final states are distinct, inserting Eq. 24 into Eq. 25 yields

which will be sufficient to let us calculate excitation probabilities analytically for both of the common quantum-mechanical approaches to the model being reviewed.

The partially quantized approach has us treat the oscillator as quantized while treating the beam only as the source of a time-dependent perturbation. If the beam is still modeled as a classical particle whose path follows $x\approx vt$, the Hamiltonian that acts on the oscillator wavefunction is

This form will allow us to use perturbation theory, since the energy eigenstates of the quantum harmonic oscillator $\ket{n}$, with $E_{n}=(n+1/2)\hbar\omega_{0}$, are well-known.

If we want to treat our model in a fully quantum-mechanical way, the simplest way is to include the beam and the oscillator in a combined quantum state.

To make the problem analytically tractable, it will be useful to introduce a box length $L$ over which our wavefunction runs in $x$, so as to make it normalizable. This will allow us to smuggle in classical notions like the velocity of the particle, since the time integral as the beam goes from $-L/2$ to $+L/2$ can be written, if we consider that the beam travels at a speed of approximately $v$ throughout, as going from $t_{-}=-L/2v$ to $t_{+}=+L/2v$.

Plane waves constitute energy eigenfunctions of the free particle, which we can write in terms of $k$-vectors with $k=mv/\hbar$. We can also reuse the energy eigenstates of the harmonic oscillator, which we will now express as $\ket{n}_{y}$. So we can write a general quantum-mechanical state for the beam-oscillator system as

where the $\sqrt{L}$ in the denominator is for plane-wave normalization, and the energies are just

How might one relate the fully quantum approach to the partially quantum approach? The fully quantum approach of Sec. V uses maximally dispersed plane waves states to describe the beam, while the partially quantum approach of Sec. IV treats the beam as a classical particle whose position is sharply defined at all times. Yet the reduced wavefunction $\psi(y)$ should follow

What assumptions might allow this to be true?

Thinking it through more carefully, we can integrate out $x$ on both sides of the two-particle Schrodinger equation (Eq. 56) to obtain a correct expression for the dynamics of $\psi(y)$, assuming that the wavefunction and its derivatives vanish at infinity. In this case, we find that

Comparing this to the Hamiltonian of the partially quantum approach (Eq. 27), we can see that the only difference will occur in the interaction terms of the two approaches. Setting these terms equal, we find

In other words, for the partially and fully quantum approaches to agree, $f(x)$ must both be moved outside the integral, which is reasonable only if $f(x)$ varies much more slowly in $x$ than $\psi(x,y)$, and must also follow $f(x)\approx f(vt)$, which is reasonable only when $x$ is near $\langle x\rangle$, presuming that $\langle x\rangle\approx vt$. Since the region near $\langle x\rangle$ is also where the interaction term will be most significant to the dynamics of the combined wavefunction, such an approximation may be less egregious than it first seems.

On a more limited level, how might we recover the time-dependence of $\psi^{*}(y)\psi(y)$ when our approximate $\braket{\psi_{f}}{\psi_{f}}$ appears to be stationary in time? Here we need to integrate out the $\ket{k}_{x}$ parts of $\ket{\psi(k_{x},y,t)}$, including their time dependence. We can do this reduction explicitly for our approximate final-state wavefunction, Eq. 59:

An easy way to check that this reproduces the dynamics of $\psi(y)$ as calculated in Sec. IV is to obtain the expected value of $y$. For our reduced approximate wavefunction (Eq. 64), we can calculate $\langle y\rangle$ as

The amplitude of the oscillation for this expected value exactly matches the amplitude of the classical displacement for the oscillator (Eq. 34), just as it should.

The single-particle wavefunction $\psi(y)$ can also be linked to the classical oscillator displacement $y$ using Ehrenfest’s theorem ¹⁴. For any quantum operator $\hat{A}$, Ehrenfest’s theorem predicts

We may apply this machinery to connect the partially quantum approach to the classical approach. Using the Hamiltonian Eq. 27 and the position-momentum commutator Eq. 28, we can easily calculate

which can be inserted into Eq. 66 to yield

This can be compared with the classical equation

demonstrating that $\langle\hat{p}_{y}\rangle$ and $p_{y}$ follow the same dynamics. The same exercise can be carried out for $\langle y\rangle$ and $y$, and the first-order equations in time can then be combined to yield second-order equations a la Eq. 4 above.

Schroeder has pointed out that interactions in quantum systems generically introduce entanglement ². Returning to the approximate final-state wavefunction of Eq. 59, we can easily confirm that the state is indeed entangled, since it cannot be written as the product of single-particle wavefunctions. How, then, would our expectations about measurements be altered by the order in which we measure the oscillator and the beam?

In the usual way of discussing quantum mechanics, the wavefunction $\ket{\psi}$ “collapses” upon measurement ¹⁵. The modified wavefunction post-measurement $\ket{\psi^{\prime}}$ can be written in terms of a projection operator $\hat{\Pi}$ as

The projection operator, here, collapses the wavefunction into a determinate state for whatever relevant quantity has just been measured, and the denominator serves to reinforce normalization on the collapsed wavefunction.

Measuring one particle in a system but not the other requires us to alter the final state wavefunction using a partial projection operator ¹⁶. For instance, were we first to measure the final momentum of our beam particle with a value $k_{1}$, the projection operator

could be applied to our wavefunction approximation (Eq. 59) to produce a post-collapse wavefunction of

where we have omitted the factor of $d_{1}e^{-iE_{i}t/\hbar}$, since this is a phase factor of unit magnitude. Notice, then, that any subsequent predictions for the oscillator would simply match those of its first energy eigenstate.

Likewise, were we first to measure the oscillator’s displacement as $y^{\prime}$, we could update our wavefunction using the projection operator

which, realizing that $\braket{y^{\prime}}{n}_{y}=\psi_{n}(y^{\prime})$, would lead to the updated wavefunction (again, omitting a phase factor) of

From this, we find that a measurement of $y^{\prime}$ far from the origin increases the probability that the beam has reduced its momentum from $k_{0}$ to $k_{1}$, since the oscillator’s ground-state wavefunction $\psi_{0}(y)$ (Eq. 35) has thinner tails than the excited-state wavefunction $\psi_{1}(y)$ (Eq. 36).

These results are summarized in Fig. 5, which uses all the same model parameters as in Figs. 3 and 4. If the beam momentum is measured first, this changes our predictions for the oscillator displacement probabilities. If the oscillator displacement is measured first, this changes our predictions for the beam momentum probabilities.

While such claims are undoubtedly of academic interest, we should concede how minimally they bear on most real experiments. To make an analogy with real experiments, the “oscillator” is just the sample being studied, and the “beam” is just the probe being used. Given the results of Fig. 5, why is the order of quantum measurements not a critical part of all experimental protocols?

It is because any “measurement” typically involves further interactions, and thus further branching and entanglement. The entanglement of the sample with the probe allows experimental measurements of the probe to tell us something about the sample ¹⁷, but it typically is difficult to make further measurements on the sample that preserve quantum coherence. After all, most quantum systems whose positions can be pinned down (e.g., an atomic defect in a crystal) interact with their substrates and the beam alike, and the intrusions of environmental decoherence conspire to obviate the need for such worries.¹⁸

What, then, has been learned? Three complementary approaches to a simple model Hamiltonian yield results that conceptually differ, but quantitatively match. In the classical approach, the beam particle transfers a predictable quantity of energy to the oscillator as it passes. When the classical oscillator is replaced by a quantum oscillator, the quantum oscillator’s displacement expectation value oscillates with an amplitude matching that of the classical oscillator, and the quantum amplitude of the first excited state has a magnitude that is directly proportional to the classical amplitude. When both the beam particle and the harmonic oscillator are treated as quantum objects, however, the interaction between the two objects induces entanglement. The conditions allowing these three approaches to match were explored, and the consequences of partial wavefunction “collapse” on measurements of entangled systems were demonstrated.

The anonymous reviewers and the journal editor offered helpful suggestions that improved this manuscript.