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Dark state

In quantum optics and atomic physics, a dark state is a coherent superposition of quantum states—typically ground or long-lived levels—that decouples from the electromagnetic field, preventing absorption or emission of photons due to destructive interference and rendering the state optically inactive or "dark."^[1] This phenomenon arises in multilevel systems, such as Lambda-type three-level atoms, where the dark state is an eigenstate of the interaction Hamiltonian with zero eigenvalue for the light-matter coupling.^[2] First experimentally observed in 1976 by Alzetta et al. during studies of laser-induced resonances in oriented sodium vapor, dark states manifest as narrow, non-absorbing spectral features amid broader absorption lines.^[3] Dark states underpin several key quantum optical processes, including coherent population trapping (CPT), where bichromatic laser fields drive atoms into a stationary, non-radiative superposition, suppressing spontaneous decay and enabling long-lived coherences on the order of seconds.^[2] In stimulated Raman adiabatic passage (STIRAP), dark states facilitate robust, loss-free transfer of population between two ground states via adiabatic following of the instantaneous dark eigenstate, with efficiencies exceeding 99% in dilute atomic gases and applications in state-selective chemistry and quantum state preparation.^[2] These states also play roles in electromagnetically induced transparency (EIT), where they create transparency windows in opaque media, slowing light propagation to group velocities below 1 m/s.^[2] Beyond gaseous atoms, dark states have been realized in solid-state platforms, such as semiconductor quantum dots, where CPT of electron spins achieves coherence times up to microseconds, advancing quantum information processing and spin manipulation.^[4] In superconducting circuits and cavity quantum electrodynamics, collective dark states emerge in multi-atom or multi-mode systems, enabling protected quantum memories and enhanced light-matter interfaces resistant to decoherence.^[5] Recent theoretical extensions describe dark states in non-Hermitian and dissipative environments, revealing their robustness for topological quantum protection and applications in precision magnetometry and atomic clocks with sensitivities below 1 fT/√Hz.

Fundamentals of Dark States

Definition and Characteristics

In quantum mechanics, particularly in the context of atomic and molecular physics, a dark state is defined as a stationary quantum state that does not absorb or emit photons under interaction with specific laser fields, rendering it undetectable by optical spectroscopy due to decoupling from the radiation field.^[1] This decoupling arises from destructive quantum interference among transition pathways or selection rules that prevent coupling to the applied light.^[6] The concept was first experimentally recognized in the 1970s through observations of reduced fluorescence in resonance experiments with atomic vapors, marking the initial identification of such non-radiative states.^[7] Key characteristics of dark states include their orthogonality to bright states, which are the complementary quantum states that actively couple to the light field and exhibit oscillatory behavior under resonant driving. Unlike bright states, dark states demonstrate stability against spontaneous decay in the presence of resonant fields, as they contain no population in optically active excited levels, thereby maintaining coherence without radiative losses.^[1] This leads to non-radiative coherence, where the system remains trapped in a superposition of ground-state sublevels, invisible to the driving radiation.^[7] Dark states are distinct from other quantum states, such as ground states that can absorb light under appropriate conditions or bright states that interact dynamically with fields; in contrast, dark states are effectively "invisible" to the applied radiation, preserving their integrity without photon exchange.^[6] This property underpins phenomena like coherent population trapping, where atomic populations are locked into these decoupled configurations.^[7]

Quantum Interference Mechanism

The formation of dark states relies on destructive quantum interference between transition amplitudes along multiple excitation pathways, which collectively cancel out the coupling to the electromagnetic field and trap atomic population in a coherent superposition of ground states. This interference effect ensures that the net probability amplitude for excitation vanishes, preventing absorption of photons and maintaining the system in a non-radiating configuration.^[8] In this decoupled state, the dark state serves as an eigenstate of the interaction Hamiltonian with a zero eigenvalue relative to the field coupling operator, rendering it orthogonal to the driving fields and immune to their influence. This zero-eigenvalue property isolates the dark state from dissipative processes like spontaneous emission, allowing indefinite persistence under coherent illumination. This mechanism introduces a dichotomy between bright and dark states: bright states, being orthogonal superpositions, exhibit constructive interference that enhances coupling to the field and promotes strong absorption and emission, while dark states remain decoupled due to their destructive interference, effectively "invisible" to the light.^[8] Achieving this interference requires coherent control through multiple phase-locked fields, such as lasers tuned to create the necessary superposition, often under conditions like Raman resonance to align the pathways precisely.^[8]

Dark States in Atomic Systems

Two-Level Systems

In two-level atomic systems, dark states primarily emerge from selection rules that render certain states uncoupled from the incident light field, preventing absorption or emission of photons. These systems consist of a ground state and an excited state, often with angular momentum quantum numbers that dictate allowed transitions via electric dipole selection rules, such as ΔJ = 0, ±1 (with J ≠ 0 to J = 0 forbidden) and parity change.^[9] In practice, states may be inaccessible due to energy mismatch, for instance, when a probe laser frequency is detuned below resonance for a particular transition, leaving the state dark to excitation. Polarization further enforces darkness; for example, in the hydrogen atom, the Zeeman sublevel of the 2²P_{3/2} state with m_J = -1/2 is dark under σ⁺ polarized light because the selection rule Δm = +1 prohibits coupling from the ground state sublevels.^[9] Metastable states serve as another class of dark states in two-level configurations, characterized by extremely slow decay rates due to forbidden single-photon transitions. A prominent example is the 2S state in hydrogen, which is metastable with a lifetime of approximately 0.12 seconds because the 2S–1S transition violates electric dipole selection rules (both states have even parity and Δl = 0), decaying instead via a two-photon process.^[10] This long-lived nature arises from the absence of allowed single-photon emission pathways, making the state effectively dark to radiative decay in isolation.^[11] True dark states formed through quantum interference, where coherent superpositions decouple from the field, are rare in pure two-level systems without additional control fields or levels, as the simple structure lacks the necessary pathways for destructive interference in coupling amplitudes. Instead, darkness often stems from angular momentum conservation, enforcing selection rules like Δm = 0, ±1 for π and σ polarizations, which leave specific sublevels uncoupled. In experimental contexts, such dark states are observed in resonance fluorescence experiments, where certain Zeeman sublevels of the ground or excited manifold do not scatter light under linearly or circularly polarized illumination, leading to reduced fluorescence from those components.^[9] For instance, in alkali atoms like rubidium, selective pumping with σ-polarized light populates dark sublevels, quenching scattering until a magnetic field or polarization modulation is applied to access them.

Three-Level Systems

In three-level atomic systems, the lambda (Λ)-type configuration consists of two ground states, denoted as |1⟩ and |2⟩, that are coupled to a common excited state |3⟩ through a weak probe field with Rabi frequency Ω_p and a strong coupling field with Rabi frequency Ω_c, respectively, with no direct dipole-allowed transition between |1⟩ and |2⟩. This setup is typically realized in alkali atoms like sodium or cesium, where hyperfine levels serve as the ground states and an optical transition populates the excited state. The dark state in this configuration emerges as a coherent superposition |D⟩ = (Ω_c |1⟩ - Ω_p |2⟩)/Ω, where Ω = √(Ω_p² + Ω_c²) is the effective Rabi frequency, featuring zero amplitude in the excited state |3⟩ and thus decoupling the system from spontaneous decay and light absorption. This superposition arises from the destructive quantum interference between the excitation pathways driven by the two fields, rendering the state "dark" to the applied radiation. Under resonant bichromatic excitation, coherent population trapping (CPT) occurs as the atomic population is fully transferred to the dark state |D⟩, suppressing fluorescence and absorption on the probe transition. First experimentally observed in sodium vapor, CPT manifests as narrow resonances in the absorption spectrum, with linewidths limited by ground-state coherence times rather than excited-state lifetimes. In contrast, V-type configurations, featuring one ground state coupled to two excited states, and cascade (ladder) configurations, with sequential couplings between three levels, can also support dark states through similar interference, but these are generally less stable due to competing decay channels and reduced trapping efficiency compared to the Λ scheme.

Theoretical Frameworks

Mathematical Formulation for Few-Level Systems

In few-level atomic systems, the mathematical description of dark states relies on the quantum mechanical treatment of the atom-light interaction via the Hamiltonian and the resulting time evolution of the state amplitudes. For a two-level system consisting of a ground state |g⟩ and an excited state |e⟩ coupled by a resonant laser field with Rabi frequency Ω, the interaction is governed by the simplified Hamiltonian in the rotating frame under the rotating wave approximation (RWA),

H = \frac{\hbar \Omega}{2} \left( |g\rangle\langle e| + |e\rangle\langle g| \right),

leading to coherent Rabi oscillations between the states with period 2π/Ω. However, true dark states in such systems arise not from the coherent dynamics but from selection rules, such as Δm = ±1 for σ-polarized light, which leave certain magnetic sublevels (e.g., the stretched sublevel m = J in a transition from a ground state with angular momentum J under circularly polarized (σ) light, which remains uncoupled as there is no excited sublevel with m = J + 1 accessible under the Δm = +1 selection rule) uncoupled and thus optically dark, preventing absorption and enabling accumulation of population in these non-interacting states during optical pumping. For three-level systems in the Λ configuration, where two ground states |1⟩ and |2⟩ couple to a common excited state |3⟩ via probe (Ω_p) and control (Ω_c) fields with frequencies ω_p and ω_c, the full Hamiltonian is H = H_0 + H_1. The free atomic part is

H_0 = \hbar \omega_1 |1\rangle\langle 1| + \hbar \omega_2 |2\rangle\langle 2| + \hbar \omega_3 |3\rangle\langle 3|,

and the interaction Hamiltonian in the dipole approximation and semiclassical limit is

H_1 = -\frac{\hbar}{2} \left( \Omega_p e^{i \omega_p t} |1\rangle\langle 3| + \Omega_c e^{i \omega_c t} |2\rangle\langle 3| + \text{h.c.} \right),

assuming resonant coupling (ω_p ≈ ω_3 - ω_1, ω_c ≈ ω_3 - ω_2). Under the RWA, which neglects rapidly oscillating counter-rotating terms, and transforming to a rotating frame where the state vector is |ψ(t)⟩ = c_1(t) e^{-i \delta_p t} |1⟩ + c_2(t) e^{-i \delta_c t} |2⟩ + c_3(t) |3⟩ (with detunings δ_p = ω_p - (ω_3 - ω_1), δ_c = ω_c - (ω_3 - ω_2)), the time-dependent Schrödinger equation yields the coupled differential equations for the amplitudes:

i \dot{c_1} = \frac{\Omega_p}{2} c_3, \quad i \dot{c_2} = \frac{\Omega_c}{2} c_3, \quad i \dot{c_3} = \frac{\Omega_p^*}{2} c_1 e^{i \delta_p t} + \frac{\Omega_c^*}{2} c_2 e^{i \delta_c t} + \delta_3 c_3,

where δ_3 is the effective detuning for |3⟩; for simplicity, assume δ_p = δ_c = 0 and real Rabi frequencies. In the resonant case, the steady-state solutions reveal a dark state |D⟩ as the zero-eigenvalue eigenstate of the effective non-Hermitian Hamiltonian (incorporating decay if needed, but ideally zero coupling), given by the coherent superposition |D⟩ = (Ω_c |1⟩ - Ω_p |2⟩)/√(Ω_p² + Ω_c²), or equivalently |D⟩ = cos θ |1⟩ - sin θ |2⟩, where the mixing angle θ satisfies tan θ = Ω_p / Ω_c. This state is orthogonal to the bright state |B⟩ = sin θ |1⟩ + cos θ |2⟩, which couples to |3⟩, and experiences no light-induced transitions due to destructive quantum interference between the probe and control paths. For coherent population trapping (CPT), starting from an initial condition such as population in |1⟩ (c_1(0) = 1, c_2(0) = c_3(0) = 0), the dynamics under continuous illumination drive the system toward the dark state: the excited-state amplitude c_3(t) decays to zero as spontaneous emission populates the ground manifold, while the population transfers to |D⟩, with |c_1(t)|² → cos² θ and |c_2(t)|² → sin² θ, trapping the population indefinitely in the non-absorbing superposition without further excitation. This trapping efficiency depends on the ratio of Rabi frequencies and initial conditions, highlighting the role of the dark eigenvalue in stabilizing the system against dissipation.

Generalization to Multilevel Systems

In multilevel quantum systems, dark states generalize the concept observed in three-level Λ configurations, where coherent superpositions of multiple ground states become decoupled from the applied laser fields due to destructive quantum interference. These states are particularly relevant in N-level atoms, often realized in ladder schemes or more complex architectures, allowing for the storage of population in subspaces immune to optical excitation. Unlike the single dark state in a basic Λ system, multilevel dark states form a degenerate manifold that can span multiple orthogonal combinations of the ground-state basis.^[12] The general theoretical framework describes dark states as the null space, or kernel, of the coupling matrix that connects ground states to excited states via the driving fields. For a system with N total levels driven by M coherent fields coupling to one or more excited states, the dimension of this kernel yields up to N - M dark states, providing a systematic way to identify the decoupled subspace without solving the full time-dependent Schrödinger equation. This kernel-based approach applies to arbitrary coupling configurations and highlights the role of field polarizations and detunings in shaping the dark manifold. Coherent population trapping emerges as a special instance in such frameworks when the system relaxes into these steady states under continuous illumination.^[12]^[1] Specific configurations illustrate the utility of multiple dark states. In V-type systems, where a single ground state couples to two excited states, dark states arise from interference between the excited levels, enabling selective addressing. Double-Λ schemes, involving two interleaved Λ systems, support paired dark states for enhanced control over population dynamics. Tripod configurations, with three ground states coupling to a common excited state, feature a two-dimensional dark subspace that facilitates selective population transfer between specific ground-state superpositions by tuning field amplitudes. These setups underscore the scalability of dark states for manipulating high-dimensional quantum information.^[12]^[13] Recent theoretical advances have formalized a comprehensive dark-state theory for arbitrary multilevel systems, leveraging destructive interference to predict and engineer these states across diverse coupling topologies. This framework, developed in 2025, emphasizes scalability for applications in complex quantum networks by providing analytical conditions for dark-state existence, such as required degeneracies in the ground manifold. It extends beyond perturbative treatments, offering tools to diagonalize effective Hamiltonians within subspaces and predict transitions in N-level atoms.^[12] Decoherence introduces significant challenges in higher-dimensional systems, where environmental noise can lift the degeneracy among multiple dark states, causing mixing between them and gradual population leakage to bright states. In particular, amplitude damping and phase noise lead to dephasing rates that scale with the system's dimensionality, reducing the effective coherence time of the dark manifold and necessitating robust preparation protocols. Such effects are pronounced in configurations with closely spaced dark states, where even weak interactions with phonons or stray fields destabilize the interference conditions.^[14]^[15]

Applications

Coherent Population Trapping and STIRAP

Coherent population trapping (CPT) is a quantum interference phenomenon observed in three-level Λ-type atomic systems, where two coherent laser fields, resonant with the transitions from two ground states to a common excited state, drive the system into a coherent superposition known as the dark state. This superposition is orthogonal to the excited state, effectively decoupling the atoms from the optical fields and trapping the population in the ground-state manifold, thereby suppressing spontaneous emission and absorption. The effect was first experimentally observed in sodium vapor by Alzetta et al. in 1976, who noted a reduction in fluorescence due to the formation of non-absorbing states under bichromatic excitation. CPT has become a cornerstone for high-resolution spectroscopy, enabling narrow-linewidth resonances for atomic clocks and magnetometers with linewidths below 1 Hz in alkali vapors. Stimulated Raman adiabatic passage (STIRAP) extends the CPT concept to achieve efficient, coherent population transfer between the two ground states in a Λ system without populating the intermediate excited state, thus minimizing losses from spontaneous decay. In STIRAP, the dark state is adiabatically followed by applying the fields in a counter-intuitive sequence: the Stokes (coupling) field precedes the pump (probe) field, with their intensities varying such that the mixing angle θ—defined by \tan \theta = \Omega_p / \Omega_s, where \Omega_p and \Omega_s are the Rabi frequencies—slowly rotates from 0 to \pi/2. This process, proposed by Gaubatz et al. in 1990 for selective transfer between molecular vibrational levels, relies on the adiabatic theorem to ensure the system remains in the instantaneous dark eigenstate. Under ideal conditions, STIRAP achieves near 100% population transfer efficiency in Λ systems, as demonstrated in atomic and molecular experiments with transfer fidelities exceeding 99% in rubidium and sodium atoms. The technique's robustness against decoherence stems from the adiabaticity condition \left| \frac{d\theta}{dt} \right| \ll |\Omega|, where |\Omega| = \sqrt{|\Omega_p|^2 + |\Omega_s|^2} is the effective Rabi frequency, ensuring minimal nonadiabatic excursions to lossy bright states even in the presence of moderate detunings or fluctuations. This condition allows STIRAP to operate over pulse durations from nanoseconds to milliseconds, depending on the system coherence time.

Laser Cooling and Quantum Information Processing

In laser cooling, dark states play a crucial role in achieving temperatures below the Doppler limit through mechanisms such as polarization-gradient cooling in optical molasses and Sisyphus cooling. In optical molasses, atoms are optically pumped into dark states—non-absorbing superpositions decoupled from the laser light—near the intensity maxima of the light field, where they experience minimal momentum kicks from photon absorption and re-emission. As atoms move toward intensity minima, they are repumped into bright states, undergoing stimulated emission that preferentially imparts momentum opposite to their velocity, thereby damping motion and reducing recoil heating. This process enables sub-Doppler temperatures on the order of the recoil temperature, as demonstrated in seminal theoretical models for alkali atoms. Similarly, in Sisyphus cooling, atoms climb potential hills formed by light-shift-induced adiabatic potentials, losing kinetic energy upon reaching the top where they are pumped to dark states at the bottom of adjacent hills, effectively converting kinetic energy into internal energy that is dissipated via spontaneous emission. These techniques have been extended to molecules, achieving temperatures as low as 50 μK in three-dimensional optical molasses by leveraging dark-state pumping to minimize off-resonant scattering. In quantum information processing, dark states form the basis of decoherence-free subspaces (DFS), which encode logical qubits in superpositions immune to collective noise, such as phase damping from environmental fluctuations. For instance, in spin ensembles, the singlet state |↑↓⟩ - |↓↑⟩ serves as a dark state orthogonal to the total spin operator, protecting it from collective dephasing while allowing single-qubit operations via local addressing. This approach enhances coherence times in solid-state systems and is integral to quantum repeaters, where dark states in atomic ensembles store photonic entanglement without decoherence from shared noise channels, enabling long-distance quantum networks. Coherent population trapping underlies the formation of these dark states in multilevel systems, providing a foundational mechanism for noise-resistant encoding. Electromagnetically induced transparency (EIT) exploits dark states to create a transparency window in an otherwise absorbing medium, enabling slow light propagation via dark-state polaritons—hybrid quasiparticles combining photonic and atomic excitations. In a three-level Λ system, the dark-state polariton is a coherent superposition Ψ(z,t) = cosθ(z) E(z,t) - sinθ(z) √N S(z,t), where E is the probe field, S the collective spin coherence, N the atomic density, and θ the mixing angle determined by the control field Rabi frequency; as θ varies from 0 to π/2, the polariton transitions from pure light to pure matter, slowing the group velocity to c cos²θ and reducing it to millimeters per second in dense vapors. This has been experimentally realized with pulse velocities as low as 17 m/s in ultracold sodium atoms.^[16] opening applications in quantum memories and nonlinear optics at low intensities. Specific examples highlight dark states' role in protecting coherence for quantum gates. In nitrogen-vacancy (NV) centers in diamond, collective dark states emerge in hybrid systems coupled to superconducting qubits, exhibiting narrow linewidths (∼1 MHz) and long lifetimes due to subradiant isolation from inhomogeneous broadening, facilitating high-fidelity storage and transfer of quantum information. In Rydberg atom arrays, dark states minimize scattering during entangling gates by maximizing population in the non-fluorescing superposition |D⟩ ∝ |g⟩ - |r⟩ (where |g⟩ is ground and |r⟩ Rydberg), enabling 99.5% fidelity controlled-phase gates on up to 60 qubits through optimized pulse shaping that suppresses bright-state leakage. Multilevel dark states in such systems support scalable qubit encoding by extending DFS dimensionality.

Experimental Realizations and Extensions

In Gaseous Atomic Ensembles

Experimental demonstrations of dark states in gaseous atomic ensembles typically involve dilute vapors of alkali atoms, such as rubidium-87 (^87Rb) or sodium, confined in magneto-optical traps (MOTs) or thermal cells. In MOT setups, atoms are laser-cooled to microkelvin temperatures, and counter-propagating laser beams with appropriate polarizations and frequencies excite a Λ-type three-level configuration, enabling the observation of coherent population trapping (CPT). This setup traps atoms in a non-absorbing dark state, reducing spontaneous emission and allowing coherent manipulation.^[17] Key experiments in the 1970s and 1990s established these phenomena in gaseous systems. The first observation of CPT occurred in 1976 using a sodium vapor cell, where two-frequency laser excitation on the D1 line led to a narrow resonance in the fluorescence spectrum due to population trapping in a coherent superposition of ground states. In the 1990s, electromagnetically induced transparency (EIT) was demonstrated in warm rubidium vapor cells, where a coupling laser created a transparency window for a weak probe beam, enabling slow light propagation with group velocities reduced to tens of meters per second. These experiments highlighted dark states as Λ-system superpositions decoupled from the light fields, with applications briefly extended to stimulated Raman adiabatic passage (STIRAP) for population transfer in cold atomic clouds. Diagnostics for dark states in these ensembles rely on optical probing. Reduced fluorescence intensity signals CPT, as atoms in the dark state do not emit light, while absorption spectra of the probe beam exhibit transparency dips at the EIT resonance frequency, confirming the coherent interference. In Bose-Einstein condensates (BECs) of ^87Rb, dark states have been realized through similar Λ configurations, showing enhanced coherence times due to suppressed thermal motion. A primary challenge in hot vapor experiments is Doppler broadening, which shifts transition frequencies based on atomic velocity, potentially washing out the narrow dark resonances. This is mitigated by velocity-selective CPT (VSCPT), where co-propagating or appropriately detuned beams selectively trap atoms with near-zero velocity components along the light propagation direction, achieving sub-Doppler cooling and sharper resonances.^[18]

In Solid-State and Hybrid Systems

In solid-state systems, dark states manifest prominently in semiconductor quantum dots (QDs), where spin-forbidden dark excitons—characterized by parallel electron and hole spins—exhibit exceptionally long coherence times due to their optical inactivity. These states, observed in materials like CdSe/ZnS core-shell QDs, suppress radiative recombination, enabling lifetimes exceeding microseconds at cryogenic temperatures and making them ideal for quantum memory applications in repeaters. For instance, the dark exciton ground state in such QDs promotes efficient photon-pair emission and entanglement generation, crucial for scalable quantum networks.^[19] Recent advances have focused on accessing these dark states optically, overcoming their forbidden nature. In July 2025, researchers demonstrated all-optical control of dark excitons in semiconductor QDs using chirped laser pulses combined with magnetic fields, achieving storage and retrieval efficiencies up to 70% while incorporating phonon coupling in theoretical models to account for decoherence. This phonon-mediated access, modeled via tensor-network methods and Lindblad dynamics, allows manipulation without spin flips, enhancing prospects for on-demand quantum emitters in hybrid quantum technologies.^[20]^[21] In superconducting circuits, dark states emerge in multi-level transmon qubits, where specific superpositions decouple from photonic environments, rendering them insensitive to cavity-mediated decay. A seminal 2015 experiment realized such a dark state in a three-dimensional transmon qutrit with cascading energy levels (|0⟩, |1⟩, |2⟩), achieving population trapping via microwave drives and demonstrating coherence times limited primarily by qubit relaxation rather than photon interactions. These photon-insensitive modes facilitate robust quantum state preservation in circuit quantum electrodynamics (cQED) platforms, with applications in error-corrected quantum computing. Hybrid systems combining solid-state elements with optical cavities further extend dark state realizations through collective excitations. In cavity QED setups with atomic ensembles coupled to high-finesse resonators, dark-state polaritons—superpositions of atomic coherence and photonic fields—form via electromagnetically induced transparency (EIT), propagating without dissipation in the cavity. These collective modes, tunable by cavity-atom detuning, enable slow-light storage and retrieval with minimal losses, bridging atomic and solid-state quantum interfaces for integrated photonics. Extensions in 2025 have explored photon dark states in engineered interference setups at the Max Planck Institute of Quantum Optics (MPQ). Experiments demonstrated single photons in dark states within double-slit interference screens, where the photonic wavefunction occupies "shadow" regions invisible to atomic detectors, preventing interaction and scattering. This invisibility arises from quantum interference in the photon's collective state, offering new insights into classical interference patterns and potential for stealth quantum communication protocols.^[22]

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