Decay of transmon qubit in a broadband one-dimensional cavity

This paper investigates the decay dynamics of a transmon qubit coupled to a broadband one-dimensional cavity, using resolvent formalism to identify a transition from Markovian to non-Markovian regimes based on coupling strength and demonstrating how two-photon decay channels significantly influence the system's energy-dependent resonance widths.

The Gist

The Big Picture: A Quantum Swing in a Noisy Room

Imagine a Transmon Qubit (the star of this show) as a very special, three-step staircase.

• Step 0 (Ground): The bottom step, where the qubit likes to rest.
• Step 1 (Middle): The middle step.
• Step 2 (Top): The highest step.

Usually, in quantum physics, we treat these systems like simple two-step ladders (just ground and excited). But this paper looks at what happens when the qubit is actually a three-step ladder, and it's sitting in a "room" filled with invisible sound waves (photons) that can carry energy away. This room is a broadband cavity, which we can think of as a giant, slightly echoey hallway where sound travels in all directions.

The scientists wanted to know: If we kick the qubit to the top step, how fast does it fall down, and what weird things happen on the way?

The Two Main Characters

1. The Qubit: A three-step artificial atom.
2. The Continuum (The Hallway): A sea of electromagnetic waves (like a crowd of people) that can steal energy from the qubit.

The Two "Modes" of Falling

The paper discovers that the qubit behaves differently depending on how "loudly" it talks to the hallway. They identify two distinct regimes:

1. The "Forgetful" Room (Markovian Regime)

• The Analogy: Imagine the qubit is a person trying to whisper a secret to a crowd. If the crowd is huge and the person is whispering very softly, the crowd absorbs the whisper instantly and forgets it immediately. The person doesn't know if the crowd heard them or not; the information is gone forever.
• The Physics: When the qubit's connection to the hallway is weak, it falls down smoothly. The "width" of its energy state (how blurry its position is) stays the same no matter what. It's a standard, predictable decay. The hallway erases the qubit's past faster than the qubit can react to it.

2. The "Echoey" Room (Non-Markovian Regime)

• The Analogy: Now, imagine the person shouts loudly, and the hallway has hard walls that create strong echoes. The person shouts, hears their own voice bounce back, and realizes, "Oh, the crowd heard me!" The crowd hasn't forgotten yet; the information is still bouncing around.
• The Physics: When the connection is strong, the qubit interacts with the hallway so fast that the hallway can't "erase" the information immediately. The qubit feels the "echo" of its own energy. This makes the decay rate depend heavily on the specific energy level. It's messy, complex, and the qubit can even get stuck in a "quasi-stable" state, bouncing back and forth (Rabi oscillations) before finally giving up.

The Plot Twist: The Middle Step Matters

Here is the most surprising part of the paper. In a simple two-step system, the top step only cares about the bottom step. But in this three-step system, the Middle Step (Step 1) changes everything for the Top Step (Step 2).

• The Scenario: The qubit starts at the Top Step. It wants to fall to the Middle Step, and then to the Ground.
• The Problem: The Middle Step is also connected to the Ground. It's like a leaky bucket.
• The Result: Because the Middle Step is "leaking" energy to the ground so fast, it creates a two-photon shortcut. The Top Step can fall to the Ground by emitting two photons at once (or in a rapid sequence) rather than just one.

The "Indistinguishable" Confusion:
Imagine you are watching a magic trick where a rabbit disappears.

• Path A: The rabbit jumps out the front door.
• Path B: The rabbit jumps out the back door.
• If you can't tell which door the rabbit used, the universe gets confused. In quantum mechanics, this confusion causes destructive interference. The two paths cancel each other out.

In this paper, the "rabbit" is the energy. Because the Middle Step is so active, the universe can't tell if the Top Step fell directly or if it fell via the Middle Step first. This confusion destroys the "coherence" (the neat, rhythmic bouncing) that usually happens in strong coupling. Instead of a clean, rhythmic oscillation, the qubit's energy gets scrambled and dissipated much faster.

The "Gaussian" Crowd

To do the math, the authors imagined the "crowd" in the hallway wasn't uniform. They gave it a Gaussian Density of States.

• Analogy: Imagine the hallway isn't filled with people evenly. Instead, there is a dense crowd right in the middle of the room, and the crowd thins out as you move to the edges.
• This shape allowed them to smoothly transition between the "Forgetful" and "Echoey" regimes and see exactly how the qubit's behavior changes as the crowd gets denser.

The Takeaway

1. Three steps are different from two: You can't just ignore the middle step of a transmon qubit. Its connection to the ground state drastically changes how the top step decays.
2. Strong coupling creates echoes: When the qubit talks loudly to its environment, it creates complex, energy-dependent behaviors (non-Markovian dynamics).
3. Coherence gets destroyed: While strong coupling usually creates beautiful, long-lasting quantum oscillations (like a bell ringing), the specific "leakiness" of the middle step in a transmon acts like a dampener, destroying that rhythm and causing the energy to vanish faster.

In short: The paper explains why a three-step quantum ladder behaves very differently than a simple two-step one, especially when the environment is noisy and the connections are strong. It's a warning to engineers building quantum computers: Don't ignore the middle step; it's the one that ruins the party.

Technical Summary

1. Problem Statement

The paper investigates the decay dynamics of a superconducting transmon qubit (modeled as a three-level artificial atom with levels $|g\rangle$, $|e\rangle$, and $|f\rangle$) weakly coupled to a continuum of modes in a broadband, one-dimensional cavity (open waveguide).

While the decay of two-level systems in open waveguides is well-understood, the behavior of multi-level transmons is more complex due to:

• Anharmonicity: The energy spacing between levels is not uniform ($E_f - E_e \neq E_e - E_g$).
• Multi-photon pathways: The coupling between the second level ($|e\rangle$) and the ground state ($|g\rangle$) creates a two-photon decay channel that influences the decay of the third level ($|f\rangle$).
• Coupling Regimes: The transition between Markovian (memoryless) and non-Markovian (memory-retaining) dynamics depends on the ratio of the qubit's coupling strength to the bandwidth of the continuum.

The authors aim to derive analytical expressions for resonance frequency shifts and linewidths, specifically analyzing how the interaction between the lower levels affects the upper level's decay.

2. Methodology

The authors employ a rigorous theoretical framework combining resolvent formalism and projection operator techniques.

• Hamiltonian Formulation: The system is described by a multimode Jaynes-Cummings Hamiltonian under the Rotating Wave Approximation (RWA). The interaction terms $V_1$ (coupling $|g\rangle \leftrightarrow |e\rangle$) and $V_2$ (coupling $|e\rangle \leftrightarrow |f\rangle$) are treated perturbatively.
• Projection Operators: The Hilbert space is divided into a subspace $P$ (discrete states with zero photons: $|e_0\rangle, |f_0\rangle$) and a subspace $Q$ (continuum states with one or two photons).
• Resolvent Operator ($G(z)$): The authors calculate the matrix elements of the resolvent operator $G(z) = (z - H)^{-1}$ to determine the survival probability of the excited state. This involves calculating the shift operator $R(z)$, which accounts for the self-energy corrections due to the continuum.
• Analytical Derivation:
  • They derive exact expressions for the frequency shift ($\Delta$) and decay width ($\Gamma$) of the third level ($|f\rangle$).
  • Crucially, they demonstrate that the decay of $|f\rangle$ depends on both $V_2$ (direct coupling) and $V_1$ (indirect coupling via the two-photon channel).
• Continuum Limit & Density of States (DOS):
  • The discrete mode sums are transformed into integrals over a continuous frequency spectrum.
  • A Gaussian Density of States is assumed to model the broadband nature of the waveguide (approximating a low-quality factor cavity), allowing for the exploration of both weak and strong coupling regimes.
• Numerical Simulation: The derived analytical expressions are evaluated numerically for specific transmon parameters (typical frequencies of 5 GHz and 9.85 GHz) to visualize the transition between dynamical regimes.

3. Key Contributions

A. Identification of Two Dynamical Regimes

The paper defines two distinct regimes based on the dimensionless coupling parameter $\Lambda/\delta^2$ (where $\Lambda$ is the collective coupling strength and $\delta$ is the continuum bandwidth):

1. Markovian Regime (Weak Coupling, $\Lambda/\delta^2 \ll 1$):
   • The resonance width is practically independent of energy within the continuum band.
   • The continuum erases information about the qubit's past faster than the qubit interacts with it.
2. Non-Markovian Regime (Strong Coupling, $\Lambda/\delta^2 \geq 1$):
   • The resonance width becomes strongly energy-dependent.
   • The qubit interacts with the continuum faster than the continuum can "forget" the interaction, leading to memory effects.

B. The "Two-Photon" Decay Channel and Coherence Destruction

A central finding is that the decay of the third level ($|f\rangle$) is significantly altered by the coupling of the second level ($|e\rangle$) to the ground state ($|g\rangle$).

• Mechanism: The interaction $V_1$ opens a fast two-photon decay channel ($|f\rangle \to |e\rangle \to |g\rangle$).
• Incoherence: Unlike a ladder system where two coherent photons might be emitted, the indistinguishability of the photon emission paths ($f \to e \to g$) leads to destructive interference.
• Result: This interaction broadens the second level and effectively destroys the coherence of the decay of the third level. In the strong coupling regime, this prevents the formation of long-lived quasistable states that would otherwise support Rabi oscillations.

C. Analytical Expressions for Shift and Width

The authors provide closed-form analytical expressions (Eqs. 61, 62, 70, 71) for the frequency shift $\Delta_2(E)$ and width $\Gamma_2(E)$ of the third level. These expressions explicitly include terms describing the influence of the lower-level coupling ($\Delta_1$ and $\Gamma_1$), distinguishing the transmon decay from a simple two-level system.

4. Key Results

• Spectral Function Evolution:

  • Weak Coupling ($L_2 = 0.1$): The spectral function $U_{ff}(E)$ resembles a standard Lorentzian. The presence of $V_1$ causes minor broadening but does not qualitatively change the decay profile.
  • Intermediate Coupling ($L_2 = 0.3$): The system approaches the crossover point. The frequency shift curve develops non-linearities.
  • Strong Coupling ($L_2 = 1$ to $6$):
    • Two-Level Case ($V_1=0$): The spectral function splits into multiple narrow peaks. This indicates the formation of quasi-stable states within the continuum, leading to weakly damped Rabi oscillations in the time domain.
    • Three-Level Case ($V_1 \neq 0$): The interaction with the ground state suppresses these narrow peaks. The spectral function remains broad, and the Rabi oscillations are heavily damped or destroyed. The peaks that do appear are associated with regions where the decay width is minimal, but their heights are significantly reduced compared to the two-level case.

• Resonance Splitting: In the strong coupling regime without the lower-level interaction, the equation for resonance points ($E - E_f - \Delta_2(E) = 0$) yields multiple roots, corresponding to split energy levels. When $V_1 \neq 0$, the number of observable peaks changes, and the central peak (associated with the original transition) is suppressed due to the increased width caused by the two-photon channel.

• Rabi Oscillations:

  • In the two-level strong coupling scenario, Rabi oscillations are well-resolved ($T_R \ll \tau$, where $T_R$ is the oscillation period and $\tau$ is the decay time).
  • In the three-level scenario, the destructive interference causes $T_R \approx \tau$, meaning the oscillations are not well-resolved and the coherence is lost.

5. Significance

• Fundamental Physics: The paper clarifies the role of multi-level structure and anharmonicity in open quantum systems. It demonstrates that the "effective" decay of an upper level is not isolated but is intrinsically linked to the decay properties of lower levels via multi-photon channels.
• Quantum Computing: For superconducting quantum processors, understanding these decay dynamics is crucial for error correction and qubit design. The results suggest that in open waveguide architectures (used for high-fidelity readout and coupling), the coupling between qubit levels can fundamentally alter coherence times and the nature of the emitted radiation.
• Non-Markovian Dynamics: The work provides a concrete theoretical model for transitioning from Markovian to non-Markovian behavior in a realistic physical system (transmon in a waveguide), offering a way to engineer or mitigate memory effects in quantum devices.
• Generalizability: While focused on transmons, the derived framework applies to any ladder-type artificial atom interacting with a structured continuum, making it relevant for broader research in open quantum systems.