Yiğit Perçin and Özgür E. Müstecaplıoğlu at Koc University present a framework combining the Davies master equation with a Morris–Shore decomposition to identify long-lived energy storage states in interacting multilevel quantum batteries. The framework, focusing on two interacting qutrits, analytically constructs dark, bright, and funnel states—states exhibiting directed decay pathways towards protected manifolds—and establishes quantitative key conditions dependent on interaction strength and anharmonicity. The research shows that multilevel systems and exchange interactions enable enhanced energetic storage, offering a principled basis for future protection and control strategies in superconducting platforms.
Multilevel quantum batteries exhibit enhanced storage capacity and sustained energy retention
Stored energy in multilevel quantum batteries now reaches 1.8 times that of comparable qubit systems, a substantial improvement enabled by access to higher excitation levels and the exploitation of funnel states. This surpasses the limitations of prior designs restricted to two-level systems, which could not utilise the full potential of ladder structures for enhanced energy density. This increased storage capacity is maintained even with realistic dissipation, opening avenues for practical quantum energy storage devices.
The framework combines the Davies master equation, a tool for modelling open quantum systems, with a Morris–Shore decomposition, a mathematical technique for analysing energy transfer pathways. The ratio of interaction strength to anharmonicity governs the durability of energy storage states, as revealed by analysis of the energy dynamics. Simulations comparing qutrit batteries to qubits under identical conditions showed that multilevel ladder structures and exchange interactions enable greater energetic storage in qutrits, exhibiting long-lived energy storage under realistic dissipation.
High-energy funnel states, which decay exclusively into protected manifolds, represent a natural design target for multilevel quantum batteries. While dark and subradiant states offer a means of stabilising quantum batteries against dissipation, existing studies have largely focused on qubit ensembles. Perfect dark-state fidelity remains a challenge, with current limitations stemming from the statistical spread of energy distribution; further development is needed to realise practical, high-capacity quantum storage.
Exploiting subradiant states for enhanced quantum battery performance
Quantum batteries—quantum systems designed to store and deliver energy—are increasingly attracting interest as potential applications of quantum coherence, correlations, and collective effects. Early theoretical studies demonstrated that collective charging protocols can enhance charging power relative to classical bounds, while subsequent work emphasised the vital role of dissipation in determining practical battery performance. In realistic settings, quantum batteries operate as open systems and inevitably suffer from energy leakage and self-discharge.
A promising strategy to mitigate these effects is the use of dark or subradiant states, which are protected against dominant decay channels by destructive interference or symmetry. These states are proposed and employed as long-lived storage manifolds in qubit-based batteries, excitonic systems, and open-system charging protocols. However, most existing approaches rely on symmetry arguments and are largely restricted to ensembles of two-level systems.
Many experimentally relevant quantum platforms, most notably superconducting transmon circuits, are intrinsically multilevel systems. Truncating these systems to qubits neglects internal ladder structure that can, in principle, be exploited to enhance both energy density and lifetime. Despite this, a systematic framework for identifying and exploiting protected energy storage states in interacting multilevel quantum batteries remained lacking. This work addresses this gap by introducing a constructive and thermodynamically consistent method to analyse dissipation in multilevel quantum batteries.
Their findings align with recent studies showing that increasing the local Hilbert space dimension sharply enhances the charging energy and power of interacting quantum batteries. The approach combines the Davies master equation, which ensures correct thermalisation behaviour, with a Morris–Shore (MS)-type decomposition applied to Bohr-frequency-resolved dissipative coupling blocks. This framework allows classification of excited states according to their dissipative role and identification of not only dark states, but also a hierarchy of excited states that act as dissipative funnels, decaying exclusively into protected manifolds.
Researchers apply this Davies–MS framework to a minimal yet non-trivial model: two interacting qutrit quantum batteries coupled to a common bath. This model captures essential features of superconducting transmon platforms, including ladder-dependent transition strengths and weak anharmonicity. They analytically construct dark, bright, and funnel states, derive quantitative durability conditions governed by the ratio of interaction strength to anharmonicity, and validate their predictions through numerical simulations.
The condition for weak anharmonicity is given by α ≪ ω, where α is the strength of the anharmonicity and ω is the transition frequency in the absence of anharmonicity. Beyond passive dark-state storage, they propose a battery design strategy in which high-energy funnel states are actively used as storage targets. These states can store more energy than the corresponding dark states while maintaining long lifetimes when combined with selective suppression of specific decay channels.
The work is organised as follows: Section II discusses the quantum battery model and the Davies master equation, while Section III presents the MS transformation and singular value decomposition. Section IV determines the optimal charging protocol, and Section V concludes the findings. Each quantum battery maintains a weakly anharmonic three-level system (qutrit) with eigenenergies En=nω−α2n(n−1)E_n = n\omega – \frac{\alpha}{2} n(n – 1),
for n=0,1,2n = 0, 1, 2, where ω is the transition frequency in the absence of anharmonicity (i.e., α = 0), and α characterises the strength of the anharmonicity.
As a typical physical system with such anharmonic three-level structure, researchers consider superconducting transmon qutrits, for which α > 0 and α ≪ ω; hence, the upper energy gap E2−E1=ω−αE_2 – E_1 = \omega – \alpha is smaller than the lower one E1−E0=ωE_1 – E_0 = \omega. For a transmon qutrit, the lowering operator truncated to the three lowest levels can be written as a=∣0⟩⟨1∣+2 ∣1⟩⟨2∣a = |0\rangle\langle1| + \sqrt{2}\,|1\rangle\langle2|.
Researchers consider a system of two identical quantum batteries, coupled via an exchange interaction, whose Hamiltonian reads HS=∑j=A,B∑n=02En∣n⟩j⟨n∣+J(aA†aB+aAaB†)H_S = \sum_{j=A,B} \sum_{n=0}^{2} E_n |n\rangle_j \langle n| + J(a_A^\dagger a_B + a_A a_B^\dagger), where J is the coupling strength. Such an interaction is realised by capacitive coupling under the rotating-wave approximation (RWA) for transmon qutrits. Additional indirect coupling could be mediated by using a transmission-line resonator coupled to both quantum batteries, which is not considered here.
The RWA reduces the capacitive coupling term to the exchange interaction term J(aA†aB+aAaB†)J(a_A^\dagger a_B + a_A a_B^\dagger). The batteries are charged using classical drive fields such that the charging Hamiltonian can be written as Hd(t)=iΩR(t)(aA†−aB†)+H.c.H_d(t) = i\Omega_R(t)(a_A^\dagger – a_B^\dagger) + \text{H.c.},
where ΩR(t) is the Rabi frequency associated with the transitions. The choice of an antisymmetric (out-of-phase) drive is not arbitrary but due to optimisation, which will be explained in the next section. The batteries are coupled to a common electromagnetic environment through HSB=[(aA+aB)+(aA†+aB†)]⊗BH_{SB} = [(a_A + a_B) + (a_A^\dagger + a_B^\dagger)] \otimes B, where B stands for bath operators.
For a weakly coupled bath, researchers employ Davies’ construction to write the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation ρ˙=−i[HS+Hd(t),ρ]+DD[ρ]\dot{\rho} = -i[H_S + H_d(t), \rho] + \mathcal{D}_D[\rho], where, assuming a low temperature (T → 0), the dissipator has a contribution only from spontaneous emission, described by DD[ρ]=∑Ω>0Γ(Ω)(A(Ω)ρA†(Ω)−12{A†(Ω)A(Ω),ρ})\mathcal{D}_D[\rho] = \sum_{\Omega > 0} \Gamma(\Omega)\big(A(\Omega)\rho A^\dagger(\Omega) – \frac{1}{2}\{A^\dagger(\Omega)A(\Omega), \rho\}\big).
Here, A(Ω)=∑ε′−ε=ΩΠε(aA+aB)Πε′A(\Omega) = \sum_{\varepsilon’ – \varepsilon = \Omega} \Pi_\varepsilon (a_A + a_B)\Pi_{\varepsilon’}, with Πε\Pi_\varepsilon the spectral projector of HSH_S onto energy ε. The Davies jump operators are constructed from the eigenbasis of the undriven interacting system Hamiltonian HSH_S. The charging field Hd(t)H_d(t) is then added as an external coherent control term in the Schrödinger-picture master equation. This separation allows identification of dark, bright, and funnel states of the interacting open system independently of the specific charging protocol. It is justified provided the drive does not substantially modify the Bohr-frequency structure entering the Davies decomposition.
This separation between coherent driving and dissipative structure is justified provided the drive varies slowly on the reservoir memory timescale. A sufficient condition is ∣ΩR(t)∣τB≪1|\Omega_R(t)|\tau_B \ll 1, ∣Ω˙R(t)∣τB2≪1|\dot{\Omega}_R(t)|\tau_B^2 \ll 1, where τB is the bath correlation time. In practice, this requires the drive modulation timescale τmod(t) ∼ ∣ΩR(t)/Ω˙R(t)∣|\Omega_R(t)/\dot{\Omega}_R(t)| to be long compared with the dissipative timescale Γ⁻¹.
The Davies generator decomposes the system–bath coupling into Bohr-frequency-resolved jump operators A(Ω), each connecting eigenstates of HSH_S whose energies differ by Ω. For a fixed Ω, the operator A(Ω) therefore defines a linear map between two excitation manifolds: an upper manifold HAH_A (e.g., N = 2) and a lower manifold HBH_B (e.g., N = 1). Choosing orthonormal bases {ψj(A)}\{\psi^{(A)}_j\} in the upper manifold and {ϕi(B)}\{\phi^{(B)}_i\} in the lower manifold, the jump operator can be written as A(Ω)=∑i,jMij∣ϕi(B)⟩⟨ψj(A)∣A(\Omega) = \sum_{i,j} M_{ij} |\phi^{(B)}_i\rangle \langle \psi^{(A)}_j|, where the matrix elements are obtained directly from the jump operator as Mij=⟨ϕi(B)∣A(Ω)∣ψj(A)⟩M_{ij} = \langle \phi^{(B)}_i | A(\Omega) | \psi^{(A)}_j \rangle.
Thus, the matrix M is simply the representation of the Davies jump operator between two excitation manifolds. It contains the dissipative transition amplitudes from the upper manifold to the lower one.
The mathematical mechanism underlying the MS transformation is the singular value decomposition (SVD) in linear algebra. Traditionally used in closed quantum systems to reduce complex coherent drives to independent two-level subsystems, this method has been extended to open quantum systems in the framework presented and applied to dissipative coupling blocks decomposed into Bohr frequencies.
When SVD is applied to the jump matrix M of dimensions NB×NAN_B \times N_A, it is factorised as M=UΣV†M = U \Sigma V^\dagger. In this separation, the physical meanings of the matrices match as follows: the columns of the unitary matrix V define orthonormal linear combinations of states within the upper manifold, representing the effective source states that participate in dissipative transitions. Correspondingly, the columns of the matrix U characterise the orthonormal basis of the lower manifold, acting as the designated target states for these dissipative channels. Finally, knowledge of these pathways offers a principled basis for developing future protection and control strategies in superconducting multilevel platforms.
Dark and subradiant states have emerged as a promising resource for stabilising open quantum batteries against dissipation, but existing studies are largely limited to qubit ensembles and symmetry-based constructions. A systematic framework for identifying long-lived energy storage states in interacting multilevel quantum batteries combines the Davies master equation with a Morris–Shore (MS)-type decomposition of dissipative coupling blocks. Focusing on a minimal model of two interacting qutrits coupled to a common bath, analytical construction of dark, bright, and funnel states—excited states that decay exclusively into protected manifolds—is possible.
Quantitative durability conditions are governed by the ratio of interaction strength to anharmonicity, and multilevel ladder structure and exchange interactions enable energetic storage states beyond the qubit case. Numerical simulations confirm that these states exhibit long-lived energy storage under realistic dissipation. These null vectors define the spectator states of the MS transformation. When such states are also eigenstates of HSH_S, they become exact dark states of the open-system dynamics.
Conversely, singular vectors with nonzero singular values define dissipative channels. Depending on their decay targets, these states may act as funnel or bright states, as classified below. Researchers classify eigenstates based on the structure of the jump operator as follows: dark states, which are states annihilated by all relevant jump operators and invariant under the coherent Hamiltonian; funnel states, which are excited states corresponding to nonzero singular values whose decay targets lie entirely inside the dark subspace of the lower manifold; bright states, which are states whose decay channels connect to radiative sectors that eventually lead to the ground state; and spectator states, which are null vectors of a dissipative block that are not eigenstates of the system Hamiltonian.
Dark and subradiant states have emerged as a promising resource for stabilising open quantum batteries against dissipation, but existing studies are largely limited to qubit ensembles and symmetry-based constructions. A systematic, thermodynamically consistent framework for identifying long-lived energy storage states in interacting multilevel quantum batteries is introduced, combining the Davies master equation with a Morris–Shore (MS)-type decomposition of dissipative coupling blocks. Focusing on a minimal model of two interacting qutrits coupled to a common bath, dark, bright, and funnel states—excited states that decay exclusively into protected manifolds—are analytically constructed. Quantitative durability conditions are derived, governed by the ratio of interaction strength to anharmonicity.
The research successfully identified long-lived energy storage states within interacting quantum batteries composed of two qutrits. This is important because it demonstrates a method for stabilising these batteries against energy loss, moving beyond previous limitations focused on simpler qubit systems. Researchers analytically constructed dark, bright, and funnel states, revealing how energy can be stored in protected manifolds within the system. The study establishes quantitative conditions relating interaction strength to energy level spacing, offering a basis for future protection strategies in superconducting platforms.
👉 More information
🗞 Davies-Morris-Shore Framework for Multilevel Quantum Batteries: Dark and Funnel States in Interacting Qutrit Systems
🧠 ArXiv: https://arxiv.org/abs/2604.18212
