Double-weak-link interferometer of hard-core bosons in one dimension

This paper demonstrates that the expansion dynamics of a one-dimensional hard-core boson gas released from a domain wall in the presence of two weak links exhibit genuine quantum interference fringes and coherent patterns that deviate from standard hydrodynamic predictions, providing an exact analytic description of these effects through fermionic propagators.

The Gist

Imagine you are watching a crowd of people (the bosons) trying to walk through a long, narrow hallway (the one-dimensional lattice).

In a perfect, empty hallway, if you suddenly open a gate letting people rush from the left side to the right, they would spread out smoothly and predictably. Physicists call this "hydrodynamics." It's like water flowing in a pipe; you can predict exactly where the water will be at any time just by knowing the speed of the flow.

The Twist: The Obstacles
Now, imagine you put two small, tricky obstacles (the defects) in the hallway.

• Obstacle 1 is a turnstile that lets some people through but bounces some back.
• Obstacle 2 is a similar turnstile further down the hall.

In the world of everyday objects (like water or cars), if you put two turnstiles in a line, the flow eventually just looks like it's passing through one big, slightly slower turnstile. The water doesn't "know" there are two separate ones; it just averages them out. This is what the standard "Generalized Hydrodynamics" theory predicts.

The Quantum Surprise
But these aren't normal people; they are quantum particles. In the quantum world, particles act like waves. When a wave hits an obstacle, it doesn't just bounce or pass; it splits.

Here is the magic (and the problem for standard physics):

1. A particle hits the first turnstile. Some pass, some bounce back.
2. The ones that pass hit the second turnstile. Some pass, some bounce back.
3. The Bounce-Back: The particles bouncing off the second turnstile travel back to the first one. They hit it again, bounce off, go back to the second, and so on.

Because they are waves, these particles are constantly overlapping with themselves. They interfere with each other like ripples in a pond. Sometimes the ripples add up (making a big splash), and sometimes they cancel out (making a flat spot).

What the Paper Found
The authors of this paper studied exactly what happens when these "quantum waves" bounce back and forth between two obstacles.

• The Failure of the Old Map: They found that the standard "hydrodynamic" map (the one that predicts smooth water flow) fails completely. It predicts a smooth, boring density profile. But the real quantum system creates a complex, wiggly pattern of high and low density—like a barcode or a musical equalizer. These are interference fringes.
• The New Map: The authors derived a new, complex mathematical formula (a "propagator") that accounts for every single bounce and every single wave interference.
• The Result: When they compared their new formula to computer simulations, it matched perfectly. The "wiggles" in the density were real and caused by the particles playing a game of quantum ping-pong between the two obstacles.

The Long-Term View
Interestingly, if you wait long enough (much longer than the time it takes to bounce back and forth a few times), the system starts to look "classical" again. The rapid bouncing averages out, and the two obstacles start to look like a single, composite obstacle. The quantum "wiggles" smooth out, and the old hydrodynamic rules start to work again, but only after a very long time.

The Analogy Summary

• Standard Physics (Hydrodynamics): Imagine a stream of cars hitting two toll booths. The traffic slows down, but the line of cars looks smooth and predictable.
• This Paper (Quantum Interference): Imagine the cars are actually ghostly waves. When they hit the first booth, they split. The ones that bounce back hit the second booth, bounce again, and crash into the incoming waves. This creates a chaotic, beautiful pattern of "traffic jams" and "open lanes" that no smooth-flow theory could ever predict.

Why It Matters
This research is important because it shows that Generalized Hydrodynamics, a powerful tool used to understand quantum gases, has a blind spot. It works great for single obstacles or no obstacles, but it breaks down when you have multiple obstacles close together because it misses the "quantum echo" effects.

The authors have provided the first complete recipe to calculate exactly what happens in these tricky situations, proving that even in a simple line of particles, quantum mechanics can create surprisingly complex and beautiful patterns.

Technical Summary

1. Problem Statement

The paper investigates the non-equilibrium dynamics of a one-dimensional lattice gas of hard-core bosons released from a domain-wall initial state in the presence of two conformal defects (weak links).

• Context: Generalized Hydrodynamics (GHD) has successfully described the ballistic expansion of integrable 1D quantum gases, treating quasi-particles as classical entities propagating ballistically. Previous work (e.g., Ref. [33]) showed that GHD works well for a single conformal defect, where the defect acts as a boundary condition modifying transmission and reflection probabilities.
• The Gap: It was unclear how GHD behaves when multiple defects are present. Specifically, does the standard hydrodynamic description (which assumes incoherent scattering) hold, or do quantum interference effects between multiple defects lead to deviations?
• Goal: To determine if the density profile of a gas expanding through two defects can be captured by standard hydrodynamics or if genuine quantum interference effects dominate, and to derive an exact analytic description of this dynamics.

2. Methodology

The authors employ an exact analytical approach based on the mapping of hard-core bosons to free fermions via the Jordan-Wigner transformation.

• Model: The system is defined by a Hamiltonian with nearest-neighbor hopping and two localized defects at positions $x=0,1$ and $x=N, N+1$. The defects are characterized by parameters $\lambda_1, \lambda_2 \in [0,1]$, representing the strength of the weak links (transmission coefficients).
• Exact Diagonalization:
  • The authors solve the single-particle Schrödinger equation for the system with periodic boundary conditions to find the exact eigenstates $\Psi_k(x)$ and eigenvalues $\epsilon_k$.
  • They derive a quantization condition for the quasi-momenta $k$ in the presence of two defects, which involves a transcendental equation coupling the defect strengths and the separation distance $N$.
• Propagator Derivation:
  • Using the exact eigenstates, they construct the time-dependent fermionic propagator $K_t(x, y) = \langle \Psi(t) | \hat{c}_x^\dagger \hat{c}_y | \Psi(t) \rangle$.
  • By taking the thermodynamic limit ($L \to \infty$), the discrete sum over quasi-momenta is converted into integrals. The authors show that the propagator can be expressed as an infinite series of Bessel functions ($J_n(t)$).
  • This series represents the sum over all possible classical paths involving multiple reflections and transmissions between the two defects.
• Density Calculation:
  • The particle density $\rho(x, t)$ is calculated from the propagator as $\rho(x, t) = \sum_{y \le 0} |K_t(x, y)|^2$, corresponding to the domain-wall initial condition where sites $x \le 0$ are filled.
  • The authors separate the result into diagonal terms (which correspond to the hydrodynamic limit) and off-diagonal terms (which encode quantum interference).

3. Key Contributions

1. Exact Analytic Propagator: The paper provides the first closed-form analytic expression for the fermionic propagator in a system with two conformal defects (Eq. 62). This expression is a sum of Bessel functions representing an infinite series of scattering paths.
2. Breakdown of Hydrodynamics: The authors demonstrate that the standard Euler-scale hydrodynamic description (GHD) fails to capture the density profile in the presence of two defects at finite times. While GHD predicts a smooth profile based on transmission probabilities, the exact solution reveals interference fringes and coherent patterns.
3. Semiclassical Interpretation: Despite the failure of standard GHD, the authors recover a semiclassical picture where particles follow classical trajectories but accumulate quantum phases due to multiple reflections. The interference arises from the coherent superposition of paths that bounce back and forth between the two defects.
4. Large-Time Limit: The paper analyzes the long-time limit ($t \gg N$). In this regime, the interference effects are "washed out" due to the summation over many paths, and the system effectively behaves as if it has a single composite defect with a momentum-dependent transmission amplitude $T(k)$.

4. Key Results

• Density Profile Formulas: The authors derive exact expressions for the density profile $\rho(x, t)$ in three regions: left of the defects ($x \le 0$), between the defects ($0 < x \le N$), and right of the defects ($x > N$).
  • These formulas (Eqs. 73a–73c) consist of diagonal terms (recovering hydrodynamics) and off-diagonal terms (interference corrections).
  • Figure 6 illustrates that the hydrodynamic prediction (blue dashed line) deviates significantly from the exact numerical simulation (red solid line), showing clear oscillations. The analytic formula including interference (green dashed line) matches the numerics perfectly.
• Interference Mechanism: The interference fringes are caused by the coherent superposition of particles that have undergone different numbers of reflections between the two defects. The phase difference depends on the distance $N$ and the quasi-momentum $k$.
• Composite Defect Behavior: In the limit $t \to \infty$ (with $N/t \to 0$), the two defects merge into a single effective scatterer. The transmission amplitude becomes:
  $$T_\lambda(k) = \frac{1}{4(1-\lambda^2)} \frac{\lambda^4}{\cos^2(Nk) + 1}$$
  (assuming $\lambda_1 = \lambda_2 = \lambda$). This amplitude is momentum-dependent, unlike the constant transmission of a single conformal defect.
• Convergence: Figure 8 shows that as time increases, the exact density profile converges to the hydrodynamic prediction derived from the composite transmission amplitude, confirming that interference effects are transient but dominant at intermediate times.

5. Significance

• Limit of GHD: This work provides a concrete counter-example to the universality of Generalized Hydrodynamics. It proves that GHD, which relies on the assumption of incoherent quasi-particle propagation, cannot describe systems with multiple localized impurities where quantum coherence is preserved over macroscopic distances.
• Quantum Interferometry: The system acts as a "double-weak-link interferometer" for quantum particles. The results highlight how quantum interference can be engineered and observed in 1D quantum gases, even in the presence of strong interactions (mapped to free fermions).
• Theoretical Framework: The methodology of deriving exact propagators and interpreting them as sums over classical paths with quantum phases offers a powerful tool for studying non-equilibrium dynamics in integrable systems with multiple defects.
• Future Directions: The authors suggest that this approach could be extended to study entanglement entropy and other correlation functions, and potentially generalized to non-conformal defects or higher numbers of impurities. It opens the door to developing a "Generalized Hydrodynamics with Interference" that accounts for coherent scattering.

In summary, the paper rigorously demonstrates that multiple defects induce genuine quantum interference effects that invalidate standard hydrodynamic descriptions of 1D quantum gases, and it provides the exact mathematical machinery to predict and understand these phenomena.