Collapse and transition of a superposition of states under a delta-function pulse in a two-level system

This paper derives exact analytical expressions for the transition of a two-level quantum system from an initial linear superposition of eigenstates to a definite eigenstate under a delta-function pulse, demonstrating that specific interaction strengths can induce an abrupt, measurement-like "collapse" of the wavefunction that is independent of the system's energy gap.

The Gist

Imagine you are holding a spinning coin. While it's spinning, it's not just "Heads" and it's not just "Tails." It is a magical blur of both at the same time. In the quantum world, we call this a superposition.

Now, imagine you have a special, invisible hammer that can hit this spinning coin. If you hit it just right, the coin doesn't just land; it instantly snaps into a definite state—either 100% Heads or 100% Tails.

This paper by Ariel Edery is about exactly that kind of "magic hammer," but for atoms. Here is the story of what happens, explained simply.

1. The Setup: The Quantum Coin

In the world of atoms, particles have energy levels. Think of these as floors in a building.

• Floor 1 (E1): The ground floor.
• Floor 2 (E2): The second floor.

Usually, an atom sits on one floor. But in this experiment, the atom starts out in a superposition. It's like the atom is simultaneously standing on Floor 1 and Floor 2, spinning between them. We describe this spinning state with two numbers (coefficients), let's call them Alpha-1 and Alpha-2. These numbers tell us how much "Heads" and how much "Tails" are in the mix.

2. The Hammer: The Delta-Function Pulse

Usually, if you want to knock an atom from one floor to another, you use a gentle, rhythmic push (like a radio wave) that matches the energy difference between the floors. It takes time, and the atom "feels" the gap between the floors.

But this paper uses a Delta-Function Pulse.

• The Analogy: Imagine a hammer strike that is infinitely thin but infinitely heavy. It happens in zero time. It's a "snap" rather than a "push."
• The Magic: Because this hammer strikes so instantly, the atom doesn't have time to notice the distance between Floor 1 and Floor 2. The "gap" between the floors disappears from the equation. The transition happens so fast that the relative "phase" (the timing of the spin) is wiped out.

3. The Result: The "Collapse"

When this instant hammer hits the spinning coin (the superposition), something fascinating happens. The paper calculates exactly what the coin looks like after the hit.

Usually, when we measure a quantum system, it "collapses" randomly. If you have a 30% chance of Heads and 70% of Tails, a measurement gives you Heads 30% of the time and Tails 70% of the time. This is the famous Born Rule.

But this paper shows a different kind of collapse.
By adjusting the "strength" of the hammer (how hard you hit it), you can force the coin to land on 100% Heads or 100% Tails with absolute certainty, no matter what the starting mix was.

• If you hit it with a specific strength, the atom that was spinning between floors is suddenly forced to sit firmly on Floor 1.
• If you hit it with a slightly different strength, it snaps to Floor 2.

This is a "controlled collapse." It's not a random guess; it's a precise engineering trick using a mathematical hammer to force the universe to pick a side.

4. The Twist: It's Reversible!

Here is the most mind-bending part.
In the real world, if you measure a quantum coin and it lands on Heads, that's it. You can't un-measure it. That process is irreversible.

However, because this "collapse" was caused by a mathematical equation (Schrödinger's equation) and not a messy real-world measurement, it is reversible.

• The Analogy: Imagine you have a spinning top that you slam with a hammer to make it stop and stand still. In this quantum world, if you hit it again with a hammer of the exact opposite strength, the top will start spinning again, returning to its original superposition state.
• You can "un-collapse" the wave. You can take a definite state and turn it back into a blur of possibilities.

5. Why Does This Matter?

• Speed: It shows that transitions can happen instantly, without caring about the energy gap between states.
• Control: It proves we can design interactions that force a quantum system into a specific state with 100% certainty, rather than leaving it to chance.
• The Connection to Reality: The author notes that this "instant loss of phase" (the spinning stopping) looks a lot like decoherence, which is what happens when a quantum system interacts with the noisy environment (like air molecules or heat). In decoherence, the environment "measures" the system and destroys the superposition. This paper suggests that a single, perfect "kick" can mimic that messy environmental effect in a clean, controlled way.

Summary

Think of this paper as a manual for a Quantum Switch.

1. Start: A coin spinning (Superposition).
2. Action: A super-fast, super-strong hammer strike (Delta-function pulse).
3. Result: The coin snaps to a definite side (Collapse).
4. Special Feature: If you hit it with the opposite force, you can make it spin again (Reversibility).

The author shows us that by tuning the strength of the hammer, we can decide exactly which side the coin lands on, turning a quantum mystery into a predictable, reversible switch.

Technical Summary

Here is a detailed technical summary of the paper "Collapse and transition of a superposition of states under a delta-function pulse in a two-level system" by Ariel Edery.

1. Problem Statement

The paper addresses the dynamics of a quantum two-level system subjected to a time-dependent perturbation. While standard time-dependent perturbation theory typically calculates transition probabilities between two specific eigenstates (e.g., ground to excited state), this work investigates a more general scenario:

• Initial State: A linear superposition of two energy eigenstates ($\Psi_{initial} = \alpha_1 \psi_1 + \alpha_2 \psi_2$), rather than a single eigenstate.
• Perturbation: A delta-function pulse ($\delta(t)$) acting at $t=0$, representing an abrupt, instantaneous interaction.
• Objective: To derive exact analytical expressions for the final state coefficients ($c_1, c_2$) and transition probabilities, and to explore the possibility of a "collapse" where the superposition transitions abruptly to a single definite eigenstate with unit probability.

2. Methodology

The author employs exact analytical methods to solve the time-dependent Schrödinger equation for a two-level system with Hamiltonian $\hat{H} = \hat{H}_0 + \hat{H}'(t)$.

• System Definition:
  • Unperturbed Hamiltonian $\hat{H}_0$ with eigenstates $\psi_1, \psi_2$ and energies $E_1, E_2$.
  • Perturbation $\hat{H}'(t)$ has off-diagonal elements H'_{12} = H'_{21}^* = \beta q_n(t), where $\beta$ is the interaction strength and $q_n(t)$ is a sequence of functions (specifically Gaussians) converging to a Dirac delta function as $n \to \infty$. Diagonal elements are set to zero.
• Mathematical Approach:
  1. Coupled Equations: The problem is formulated as a set of coupled first-order differential equations for the time-dependent coefficients $c_1(t)$ and $c_2(t)$.
  2. Exact Solution (Method 1): The coupled first-order equations are converted into a single second-order differential equation for $c_1(t)$. This equation is solved exactly using the properties of the Gaussian sequence $q_n(t)$ and the error function ($\text{erf}$). The limit $n \to \infty$ is taken to obtain the final coefficients.
  3. Verification (Method 2): A second, independent method is presented in Appendix A. Instead of converting to a second-order equation, the author solves the coupled first-order equations directly by dividing them to find a relationship between $c_1$ and $c_2$, integrating, and applying boundary conditions. This confirms the results of Method 1.
• Key Approximation: In the limit of the delta-function pulse ($n \to \infty$), the oscillatory phase terms $e^{\pm i \omega_0 t}$ (where $\omega_0 = (E_2-E_1)/\hbar$) are treated as unity because the pulse acts instantaneously at $t=0$, effectively "freezing" the relative phase evolution.

3. Key Contributions and Results

A. General Analytical Expressions

The paper derives exact expressions for the final state coefficients $c_1(t>0)$ and $c_2(t>0)$ in terms of the initial coefficients ($\alpha_1, \alpha_2$) and the interaction strength $\beta$:
$$c_1(t>0) = \alpha_1 \cos\left(\frac{|\beta|}{\hbar}\right) - i \alpha_2 \frac{|\beta|}{\beta^*} \sin\left(\frac{|\beta|}{\hbar}\right)$$
$$c_2(t>0) = \alpha_2 \cos\left(\frac{|\beta|}{\hbar}\right) - i \alpha_1 \frac{|\beta|}{\beta} \sin\left(\frac{|\beta|}{\hbar}\right)$$

B. Independence from Energy Gap

A critical finding is that the final coefficients and transition probabilities do not depend on the energy gap ($E_2 - E_1$) or the relative phase between the eigenstates.

• In finite-width pulses, the transition probability depends on the energy gap (via the Rabi frequency and detuning).
• In the delta-function limit, the interaction is so rapid that the system does not "feel" the energy difference; the relative phase $e^{i\omega_0 t}$ is lost during the abrupt transition. This mimics the loss of coherence seen in decoherence theory but occurs via a deterministic unitary evolution in the Schrödinger equation.

C. Transition Probability Formula

The general transition probability from an initial superposition to the second eigenstate ($P_{\alpha_1, \alpha_2 \to 2}$) is derived as:
$$P = |\alpha_2|^2 \cos^2\left(\frac{|\beta|}{\hbar}\right) + |\alpha_1|^2 \sin^2\left(\frac{|\beta|}{\hbar}\right) + \text{interference terms}$$
The paper shows that for specific values of $\beta$ (multiples of $\pi\hbar/2$), the interference terms vanish or simplify, leading to specific outcomes like state swapping or no change.

D. The "Collapse" Scenario

The most significant contribution is the demonstration of a deterministic "collapse" of the wavefunction without measurement.

• Condition: For specific values of the interaction strength $\beta$ (dependent on the initial superposition ratio $|\alpha_1|$), the system transitions from a superposition to a definite eigenstate with probability 1 (e.g., $|c_2|^2 = 0$ and $|c_1|^2 = 1$).
• Mechanism: The required interaction strength $|\beta|$ is a function of $|\alpha_1|$ only, given by $|\beta|/\hbar = \cos^{-1}(\pm |\alpha_1|)$.
• Reversibility: Unlike the collapse induced by a quantum measurement (which is irreversible and probabilistic according to the Born rule), this pulse-induced collapse is reversible. By applying a pulse with the negative interaction strength ($-\beta$), the system returns exactly to the original superposition state.

4. Significance and Implications

1. Redefining "Collapse": The paper distinguishes between "measurement collapse" (probabilistic, irreversible, governed by the Born rule) and "pulse-induced collapse" (deterministic, reversible, governed by unitary Schrödinger evolution). It demonstrates that a superposition can transition to a single eigenstate via a specific physical interaction without an external observer or apparatus.
2. Exact Solvability: It reinforces the utility of delta-function pulses in quantum dynamics, showing they allow for exact analytical solutions in systems that are otherwise difficult to solve, particularly for arbitrary initial superpositions.
3. Decoherence Analogy: The loss of the energy-gap dependence (and thus the relative phase) in the delta-function limit provides a theoretical parallel to environmental decoherence, where rapid interactions destroy phase relationships. However, here the process is coherent and reversible, unlike true environmental decoherence.
4. Control of Quantum States: The results suggest a method for controlling quantum states. By tuning the interaction strength $\beta$ based on the initial state's composition, one can deterministically steer a superposition into a specific eigenstate or swap the probabilities of the energy levels.

Conclusion

Edery's work provides a rigorous mathematical framework for understanding how a two-level system evolves under an instantaneous delta-function pulse. It challenges the conventional view that "collapse" is exclusively a measurement phenomenon by showing that specific unitary interactions can achieve the same result (transition to a definite eigenstate) while remaining fully reversible and deterministic. The independence of the transition from the energy gap highlights a unique feature of instantaneous interactions in quantum mechanics.