Spontaneous emission of a three-level artificial atom in a one-dimensional open waveguide

1. Problem Statement

The paper investigates the spontaneous emission dynamics of an initially excited three-level ladder-type system (specifically a superconducting transmon qubit) coupled to a one-dimensional (1D) open waveguide.

While traditional studies in waveguide QED often focus on:

• Single-photon limits (scattering of monochromatic waves or pulses).
• Two-level systems (qubits).
• Cavity QED (discrete modes).

This work addresses the two-excitation subspace of a three-level system (3LE) interacting with a continuum of photon modes. The specific challenge is to derive an analytical solution for the non-stationary Schrödinger equation to understand how a fully excited 3LE ($|f\rangle \to |e\rangle \to |g\rangle$) decays into the waveguide, emitting two photons. The authors aim to determine the probability of detecting specific atomic states and the spectral density of the emitted photon pairs, particularly focusing on the emergence of photon-photon correlations in the strong coupling regime.

2. Methodology

The authors employ a real-space description of the photon field combined with the Rotating Wave Approximation (RWA).

• System Model:

  • A ladder-type 3LE (states $|g\rangle, |e\rangle, |f\rangle$) located at $x=0$ in a 1D waveguide.
  • The system is modeled using a Hamiltonian that includes the free atom, the free field, and the interaction terms coupling the atom to the field via coupling constants $V_1$ (transition $|e\rangle \leftrightarrow |g\rangle$) and $V_2$ (transition $|f\rangle \leftrightarrow |e\rangle$).
  • A chiral approximation is used, assuming photons propagate only in the forward direction (right-moving), simplifying the boundary conditions.
  • Non-radiative decay and decay to non-guided modes are neglected, assuming a coupling efficiency $\beta \approx 1$ (valid for superconducting circuits).

• Wavefunction Formulation:
  The total wavefunction $|\Psi(t)\rangle$ is expanded in the two-excitation subspace:
  $$|\Psi(t)\rangle = \alpha(t)|f, 0\rangle + \int dx \beta(x,t) a^\dagger(x)|e, 0\rangle + \frac{1}{\sqrt{2}} \iint dx_1 dx_2 \gamma(x_1, x_2, t) a^\dagger(x_1)a^\dagger(x_2)|g, 0\rangle$$
  where $\alpha$, $\beta$, and $\gamma$ are probability amplitudes for the atom being in states $|f\rangle$, $|e\rangle$ (with 1 photon), and $|g\rangle$ (with 2 photons), respectively.

• Analytical Derivation:

  • The authors derive a set of coupled differential equations for the amplitudes $\alpha(t)$, $\beta(x,t)$, and $\gamma(x_1, x_2, t)$.
  • They utilize a Fourier transform to switch between real space ($x$) and momentum/frequency space ($k$ or $\omega$).
  • By solving the equations iteratively and applying the initial condition of a fully excited atom ($\alpha(0)=1$, no incident photons), they obtain exact analytical expressions for all probability amplitudes in both real space and frequency space.

3. Key Contributions

1. Exact Analytical Solution: The paper provides a complete non-stationary analytical solution for the spontaneous decay of a 3LE in a 1D waveguide, covering the full time evolution from the excited state to the ground state with two emitted photons.
2. Real-Space to Frequency-Space Mapping: The derivation explicitly connects the time-domain dynamics of the wavefunction amplitudes to the spectral density of the output field, allowing for the calculation of state-detection probabilities and photon spectra.
3. Discovery of Frequency Correlation: The work identifies a regime where strong coupling leads to frequency correlations between the two emitted photons, potentially resulting in the emission of two photons with identical energies, even if the atomic levels are anharmonic.

4. Key Results

A. State Detection Probabilities

The probability of finding the system in specific states ($|f,0\rangle$, $|e,1\rangle$, $|g,2\rangle$) depends primarily on the ratio of the decay rates $\Gamma_2/\Gamma_1$ (where $\Gamma_2$ is the decay rate of $|f\rangle \to |e\rangle$ and $\Gamma_1$ is $|e\rangle \to |g\rangle$).

• Fast Upper Decay ($\Gamma_2 \gg \Gamma_1$): The atom decays rapidly to $|e\rangle$ and then slowly to $|g\rangle$. The system behaves like a sequential cascade.
• Slow Upper Decay ($\Gamma_2 \ll \Gamma_1$): The atom decays to $|e\rangle$ and immediately emits the second photon. The two photons are emitted nearly simultaneously.
• Typical Transmon ($\Gamma_2/\Gamma_1 \approx 1.5$): The probabilities show a distinct intermediate behavior where the population of the intermediate state $|e\rangle$ is significant for a measurable duration.

B. Two-Photon Spectral Density

The spectral density $S_{sp}(\omega_1, \omega_2)$ reveals the energy distribution of the emitted photon pairs.

• Weak Coupling Regime: Two distinct peaks appear at frequencies $\omega = \Omega_1$ (transition $|e\rangle \to |g\rangle$) and $\omega = \Omega_2 - \Omega_1$ (transition $|f\rangle \to |e\rangle$). This corresponds to the standard cascaded emission where photons have distinct frequencies.
• Strong Coupling Regime: A third peak emerges at the resonant frequency $\omega = \Omega_1$. This indicates that the two photons can have identical frequencies ($\omega_1 = \omega_2 = \Omega_1$).
  • This effect arises from the interference between the first emitted photon and the atom remaining in the excited state $|e\rangle$ while the second photon is being emitted.
  • It is most pronounced when the anharmonicity is low (levels are nearly equidistant) and the coupling to the waveguide is strong.

C. Spectral Density for Identical Photons

Analyzing the diagonal $\omega_1 = \omega_2$, the authors find that the spectral density has two maxima:

1. Resonant Peak: At $\omega = \Omega_1$. This dominates when $\Gamma_2/\Gamma_1$ is high (e.g., $\approx 3$).
2. Anharmonicity-Shifted Peak: At $\omega \approx \Omega_1(1 + \alpha_r/2)$, where $\alpha_r$ is the relative anharmonicity. This dominates for typical transmon ratios ($\Gamma_2/\Gamma_1 \approx 1.5$).

5. Significance and Implications

• Source of Identical Photons: The findings suggest that by tuning the decay rates (via coupling strength) and the anharmonicity of superconducting transmons, one can engineer a source of correlated, identical photons in a waveguide QED setup. This is crucial for quantum information processing where indistinguishable photons are required for interference-based protocols (e.g., Boson sampling, quantum gates).
• Beyond the Cascaded Approximation: The work challenges the standard view that 3LE spontaneous emission is strictly a sequential, uncorrelated cascade. It demonstrates that strong coupling induces quantum interference that alters the spectral properties of the emitted light.
• Universality: While the model uses superconducting transmons, the analytical framework is universal and applicable to other 3-level systems in 1D waveguides, such as Rydberg atoms or quantum dots in nanoplasmonic waveguides.
• Future Applications: The derived solutions can be extended to study stimulated emission and the scattering of arbitrary photon pulses on 3LEs, opening avenues for designing single-photon routers and quantum logic gates in open waveguides.

In conclusion, this paper provides a rigorous theoretical foundation for understanding and controlling the emission statistics of multi-level artificial atoms, highlighting the potential to generate tunable, correlated photon pairs through the manipulation of system anharmonicity and coupling strength.