Steady-State Intracavity Photon Number

• Steady-State Intracavity Photon Number is a metric that measures the average photon count in a cavity under continuous drive and loss, highlighting key quantum and nonlinear dynamics.
• It plays a critical role in systems like quantum optics, cavity QED, and optomechanics by linking coherent pumping, photon leakage, and system-environment interactions.
• Recent methods, including rate-equation analysis and QND measurement techniques, have enabled precise determination and control of these photon populations in various experimental setups.

The steady-state intracavity photon number, commonly denoted $\langle n \rangle_{\mathrm{ss}}$, quantifies the mean number of photons present in a cavity mode in the stationary regime of a driven-dissipative photonic system. This macroscopic observable encodes the interplay of coherent drive or pumping, photon loss mechanisms, nonlinearities, and system-environment couplings. Its theoretical determination and experimental measurement are central to quantum optics, cavity QED, optomechanics, and mesoscopic transport in hybrid electron-photon systems.

1. Operator Definition and General Formulation

The intracavity photon number operator for a single bosonic mode is defined as

$\hat n = a^{\dagger} a,$

where $a^\dagger$ and $a$ are the creation and annihilation operators, satisfying $[a,a^\dagger]=1$. For a quantum system described by a density operator $\rho$, the expectation value at time $t$ is

$$n(t) = \langle \hat n \rangle (t) = \mathrm{Tr}\{a^{\dagger} a\, \rho(t)\}.$$

In the steady state (if it exists), $$n_{\mathrm{ss}} = \lim_{t\to\infty} n(t) = \mathrm{Tr}\{a^{\dagger}a\,\rho_{\mathrm{ss}}\}$$, where $\rho_{\mathrm{ss}}$ is time-independent. In multi-mode or multi-cavity contexts, analogous definitions apply for each mode.

The evolution towards steady state is generally governed by a Markovian or non-Markovian master equation, often of the Lindblad type or with additional nonlocal dissipators. Dissipative terms account for photon leakage, absorption, and pumping, while coherent terms encompass driving, electron-photon coupling, or Kerr-type interactions. Setting $d\langle n\rangle/dt = 0$ delivers the steady-state condition.

2. Rate-Equation and Master Equation Approaches

A wide range of physical scenarios admit a rate-equation description for the photon number, derived from the master equation. In the presence of incoherent processes (e.g., gain at rate $\kappa_+$, loss at $\kappa_-$), the population dynamics for mode occupation probabilities $p_n$ follow a birth-death process: $$\frac{d}{dt} p_n = \kappa_- (n+1) p_{n+1} + \kappa_+ n p_{n-1} - [\kappa_- n + \kappa_+ (n+1)] p_n.$$ The mean photon number then evolves as

$$\frac{d}{dt} \langle n \rangle = \kappa_+ - (\kappa_- + \kappa_+) \langle n \rangle,$$

leading in steady state to

$$\langle n \rangle_{\mathrm{ss}} = \frac{\kappa_+}{\kappa_- + \kappa_+}.$$

This simple form generalizes to multi-reservoir settings; for example, two thermal reservoirs at temperatures $T_1, T_2$ give (Rüting et al., 2014): $$\langle n \rangle_{\rm NEST} = \frac{\kappa_1 F(\beta_1) + \kappa_2 F(\beta_2)}{\kappa_1 + \kappa_2},$$ with $F(\beta_j) = ( e^{\hbar\omega\beta_j} - 1 )^{-1}$.

With additional structured coupling (electron-photon interactions, nonlinearities, multi-level transitions), the steady-state condition involves generalized rates. For a quantum dot in a cavity coupled to electronic leads, the dynamics yield (Gudmundsson et al., 2016): $$0 = \Gamma_\mathrm{em}^{(+)} (1 + n_{\mathrm{ss}}) - \Gamma_\mathrm{em}^{(-)} n_{\mathrm{ss}} - \kappa n_{\mathrm{ss}}$$ which admits the solution

$$n_{\mathrm{ss}} = \frac{\Gamma_\mathrm{em}^{(+)} }{ \Gamma_\mathrm{em}^{(-)} + \kappa - \Gamma_\mathrm{em}^{(+)} }.$$

For $$\Gamma_\mathrm{em}^{(+)} \ll \Gamma_\mathrm{em}^{(-)} + \kappa$$, this reduces to $$n_{\mathrm{ss}} \approx \Gamma_\mathrm{em}^{(+)}/(\Gamma_\mathrm{em}^{(-)} + \kappa)$$.

3. Parameter Dependencies and Regimes

The steady-state photon number is acutely sensitive to system parameters, including:

• Emission/Absorption Rates: $\Gamma_\mathrm{em}^{(+/-)}$ typically scale as $g^2$ (electron-photon coupling), tunnel rates ($\Gamma_{L,R}$), Fermi factors, temperature $T$, and bias window alignment (resonance condition).
• External Coupling: Photon loss ($\kappa$), coherent/incoherent external drive, and reservoir thermal population ($\bar n_\mathrm{res}$).
• Nonlinearities: Kerr interactions, optomechanical coupling, parametric drive—captured via higher-order nonlinear equations for $n_{\mathrm{ss}}$ (e.g., cubic equations (Shahidani et al., 2013), quartics in triplet down-conversion (Denys et al., 2019)).
• Bistability and Multistability: In regimes with strong nonlinearity or coupled degrees of freedom (e.g., optomechanics (Wang et al., 2021), Kerr-OPA cavities (Shahidani et al., 2013)), $n_{\mathrm{ss}}$ can exhibit multiple stable branches and hysteresis, associated with saddle-node bifurcations of the steady-state equation.
• Non-equilibrium Constraints: Inhomogeneous pumping, frequency-dependent gain, or multi-reservoir coupling leads to steady states not describable by an equilibrium Bose-Einstein distribution (Lebreuilly et al., 2015, Marelic et al., 2014).

Table 1: Schematic Parameter Dependence

Regime	Main Control Variables	$n_{\mathrm{ss}}$ scaling/trend
Nonradiative transport	bias window outside photon-dressed	$n_{\mathrm{ss}} \approx 0$
Resonant radiative	bias aligns with photon-dressed	$n_{\mathrm{ss}} \sim g^2, \Gamma_{L,R}$
Strong nonlinearity	Kerr, OPA, triplet coupling	Discrete/quantized, multistable
Multi-thermal NEST	reservoirs $T_1, T_2$	Arithmetic mean of thermal occupations

4. Representative Physical Realizations

Hybrid Electron-Photon Systems

In a cavity-coupled quantum dot, steady-state $n_{\mathrm{ss}}$ discriminates between radiative and nonradiative transport. When the bias window encompasses only bare electronic states, $n_{\mathrm{ss}} \simeq 0$; when photon-dressed states mediate transport, $n_{\mathrm{ss}}$ becomes finite and highly tunable by $g$, $V_g$, and bias window (Gudmundsson et al., 2016).

Optomechanical and Hybrid Cavity Systems

In optomechanical arrays or coupled cavity-mechanical oscillator networks, the steady-state photon number in a chosen mode is extracted from coupled nonlinear algebraic equations involving shifted detunings, tunnelings, and optomechanical backaction (Wang et al., 2021). Multiple roots yield optical bistability or even double/tristability depending on detuning and pump power.

Driven-Dissipative and Nonlinear Optical Cavities

Kerr, OPA, and photon-downconversion processes are realized in strongly nonlinear optical microresonators and superconducting circuits. The photon number in these systems is determined by roots of cubic or quartic polynomials that encode the system's multi-stable landscape and yield pronounced S-shaped or multi-humped response curves as a function of drive (Shahidani et al., 2013, Denys et al., 2019).

Bose–Einstein Condensation of Photons

In dye-filled microcavities, the equilibrium prediction is modified by inhomogeneous pumping and energy-dependent losses: the observed $N_{\mathrm{ph}}$ and condensation threshold power depend nontrivially on spatial pump overlap, cavity geometry, and spectral losses (Marelic et al., 2014).

Non-equilibrium Photon States

When a single mode is coupled to multiple non-identical reservoirs, the steady-state photon number deviates substantially from an equilibrium Planckian expression. In general, $n_{\mathrm{ss}}$ interpolates between the occupations set by each reservoir's temperature and coupling, and deviation from fitting by a single effective temperature can exceed 10%, impacting precision measurement (Rüting et al., 2014).

5. Experimental Determination and Quantum Measurement Techniques

Steady-state intracavity photon numbers are routinely probed via output port leakage (direct intensity measurement), photon correlation statistics, or quantum non-demolition (QND) protocols. In superconducting circuit QED, CPMG pulse sequences applied to a dispersively coupled qubit enable extraction of $\bar n$ with sensitivity $\sim 10^{-4}$ via photon-induced qubit dephasing. The dephasing rate, exactly calculated in both Gaussian and non-Gaussian regimes (arbitrary $2\chi/\kappa$), functions as a quantitative probe of steady-state photon populations for both thermal and coherent excitations (Atalaya et al., 2023).

Protocols for precise QND measurement using intracavity Kerr nonlinearity, pre-squeezing, and parametric amplification have been shown capable of achieving single-photon sensitivity for $\bar n$ in ultra-high-Q microresonators, provided SPM noise is cancelled (e.g., via intracavity squeezing) (Salykina et al., 16 Sep 2024).

6. Quantum and Nonlinear Fluctuations, Blockade, and Photonic Correlations

The steady-state photon number encodes not only the mean intensity but also signatures of quantum correlations, statistical blockade, and parity effects. In quantum-reservoir-engineered environments allowing two-photon driving and loss, the final $\langle n \rangle_{\mathrm{ss}}$ demonstrates sensitively parity-dependent population structure—i.e., a linear combination of different Fock sectors determined entirely by the initial even/odd photon-number content, which is a marked departure from ordinary single-photon blockade (Miranowicz et al., 2014). Multi-stability tends to be reflected in the photon-number histograms and critical slowing down near transition points (Denys et al., 2019, Shahidani et al., 2013).

7. Multimode, Array, and Non-Markovian Generalizations

In photonic arrays, nonlinearities, frequency-dependent gain, and nonlocal pumping mediate complex steady-state photon distributions. For arrays of dissipative nonlinear cavities under a frequency-dependent incoherent pump, the diagonal steady state $$\rho_{\mathrm{ss}} = \sum_N \pi_N |N\rangle\langle N|$$ yields

$$\langle n \rangle_{\mathrm{ss}} = \sum_{N=0}^\infty N \pi_N,$$

with $\pi_N$ constructed recursively using the detailed-balance ratio reflecting both Markovian loss and non-Markovian pump. In the "grand-canonical" limit, the steady state maps onto a Bose–Einstein like distribution with an effective chemical potential set by the pump linewidth and emission/loss rates, marking a formal analogy to equilibrium thermodynamics in a fundamentally non-equilibrium setting (Lebreuilly et al., 2015).

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In summary, the steady-state intracavity photon number provides a unifying metric for quantifying photon populations in cavities and resonators subject to complex interplay between pumping, dissipation, coupling, and nonlinearity. Its analytical determination requires identifying and solving the relevant set of nonlinear algebraic equations or rate equations set by the system dynamics, boundary conditions, and noise characteristics. Measurement methodologies span classical output detection, qubit-based QND schemes, and full quantum tomography, with regimes of strong nonlinearity and multi-stability revealing the intricacy and richness of quantum optical steady states.