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arXiv:2605.30032

A comparison of different master equations for driven-dissipative dynamics in composite quantum systems: Dispersive readout in structured electromagnetic environments

Prakritish Gogoi, Angela Riva, Émile Cochin, Alex Chin

quant-phSubmitted May 28, 2026• Announced May 29, 2026• 0 citations
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Abstract

Driven-dissipative qubit-resonator dynamics, which are the basis of most dispersive superconducting qubit measurement schemes, are often modeled with Lindblad master equations built from subsystem local jump operators, even when the qubit and resonator are appreciably hybridized. In this work we revisit this setting using a microscopic Bloch-Redfield approach, where dissipation is constructed in the eigenbasis of the coupled qubit-resonator Hamiltonian with a complete, frequency dependent, open system description of the transmission line environment. Here, we show that the Lindblad and Bloch-Redfield decay rates can be quantitatively different in the absence of driving, while in the driven case we demonstrate that the time-independent Redfield dissipator and its time-dependent generalization can show qualitatively different behaviors as a function of driving strength. Finally, we investigate the effects of driving in structured spectral densities, recovering the suppression of measurement-induced relaxation in the presence of a so called Purcell filter.

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First-principles study of dispersive readout in circuit QED

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The speed and fidelity of dispersive readout of superconducting qubits should improve by increasing the amplitude of the measurement drive. Experiments show, however, that beyond some drive amplitude there is always a saturation or drop in fidelity, often associated with a decrease in qubit energy relaxation time $T_1$. A simple Lindblad master equation does not capture the latter effect. More involved approaches based on effective master equations rely on strong assumptions about the spectra of the system and the bath and only partially agree with observations. Here, we perform a first-principles simulation of the full unitary dynamics of dispersive readout by considering the circuit QED Hamiltonian coupled to a microscopic model for the measurement transmission line, allowing for its arbitrary spectrum, including filters. Our access to the dynamics of the bath degrees of freedom allows us to investigate the emission spectrum of the system as a function of drive power. We show how the dependence of qubit $T_1$ on readout drive amplitude is sensitive to the details of the bath spectrum. In particular, we find that $T_1$ drops with increasing drive amplitude when a Purcell notch filter is placed at the qubit frequency, and that the Lindblad master equation shows general qualitative defects compared to the first-principles model.

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Generalized master equation for driven quantum oscillators: microscopic origin of nonlinear dissipation and asymmetric resonances

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Driven nonlinear quantum oscillators are a central platform for quantum technologies, yet their dissipative dynamics are typically described using Lindblad or Caldeira-Leggett master equations derived under assumptions that exclude nonlinearities and driving. Here, we derive a generalized Caldeira-Leggett master equation for driven nonlinear oscillators by retaining the full nonlinear and time-dependent system dynamics in the construction of the dissipator. For position- and momentum-dependent system-bath coupling, the dissipator itself becomes dynamically dressed, generating nonlinear and drive-dependent dissipative channels beyond conventional fixed-dissipator approaches. This produces nonlinear damping together with dissipation-induced corrections to the effective drive. The resulting dissipative dynamics suppress large-amplitude excitations and reduce phase-space fluctuations. For a driven Kerr oscillator, this leads to the suppression of bistability, asymmetric resonance responses, and strongly modified fluctuation distributions. More broadly, our results establish a microscopic framework in which nonlinear dynamics and driving directly reshape the dissipative sector of driven open quantum systems.

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Dynamics of dispersive single-qubit readout in circuit quantum electrodynamics

R. Bianchetti, Stefan Filipp, Matthias Baur +6 more

The quantum state of a superconducting qubit nonresonantly coupled to a transmission line resonator can be determined by measuring the quadrature amplitudes of an electromagnetic field transmitted through the resonator. We present experiments in which we analyze in detail the dynamics of the transmitted field as a function of the measurement frequency for both weak continuous and pulsed measurements. We find excellent agreement between our data and calculations based on a set of Bloch-type differential equations for the cavity field derived from the dispersive Jaynes-Cummings Hamiltonian including dissipation. We show that the measured system response can be used to construct a measurement operator from which the qubit population can be inferred accurately. Such a measurement operator can be used in tomographic methods to reconstruct single and multiqubit states in ensemble-averaged measurements.

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Control of open quantum systems: Manipulation of a qubit coupled to a thermal bath by an external driving field

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Fast and reliable manipulation with qubits is fundamental for any quantum technology. The implementation of these manipulations in physical systems is the focus of studies involving optimal control theory. Realistic physical devices are open quantum systems. So far, studies in optimal control theory have primarily utilized the Redfield/Lindblad quantum master equation to simulate the dynamics of such systems. However, this Markov description is not always sufficient. Here, we present a study of qubit control utilizing the nonequilibrium Green's function method. We compare the traditional master equation with more general Green's function results and demonstrate that even in the parameter regime suitable for the application of the Redfield/Lindblad approach, the two methods yield drastically different results when addressing evolution involving mixed states. In particular, we find that, in addition to predicting different optimal driving profiles, a more accurate description of system evolution enables the system to reach the desired final state much more quickly. We argue that the primary reason for this is the significance of the non-Markov description of driven system dynamics due to the effect of time-dependent driving on dissipation.

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Structured Environments and Engineered Dissipation in Superconducting Circuits: Bound States, Dissipative Phase Transitions, Purcell Filtering and Power sensors

Claudia Castillo-Moreno

Superconducting circuits are a powerful platform for studying and controlling the interaction between light and matter at the quantum level. In these systems, superconducting qubits act as artificial atoms (with matter-like degrees of freedom), whereas microwave resonators play the role of oscillators hosting photons (with photonic-like degrees of freedom). A major advantage of this platform is the ability to tailor the electromagnetic environment to which the qubits are coupled. This degree of control makes it possible to realize well-known quantum optics models, like the Jaynes-Cummings model, as well as more complex systems, like quantum electrodynamics close to a bandgap. In this thesis, we engineer structured electromagnetic environments and couple superconducting qubits to them to experimentally study their interaction. By coupling multiple microwave resonators together, we create a metamaterial. When an atom interacts with the metamaterial, an atom-photon bound state emerges. We observe the dynamics of the creation and dissolution of the atom-photon bound state, as well as the spectral content of its photonic component, for the first time to our knowledge.Our metamaterial is intrinsically nonlinear because each resonator is formed by an array of Josephson junctions to serve as an inductor. When the system is pumped, its phase diagram exhibits multimode dissipative phase transitions with timescales on the order of hundred seconds. There is ongoing theoretical debate about whether such phase transitions can occur in one-dimensional systems, a question we address and resolve in this work.Furthermore, we engineer qubit decay using two different approaches. In the first one, we engineer the decay of two coupled atoms to two waveguides by using a symmetric–antisymmetric coupling configuration. This approach enables selective coupling to bright and dark states of the system. In the second one, we suppress the Purcell decay of a qubit using a compact lumped-element Purcell filter. In addition, we characterize the Purcell decay by employing an auxiliary superconducting qubit as a sensitive power detector directly coupled to the feedline waveguide used for the qubit readout.In summary, this thesis presents three different methods for controlling light–matter interactions in structured quantum systems, including decay suppression with both Purcell filters and symmetry-selective couplings. The thesis also introduces new experimental techniques for probing atom-photon bound states and provides direct evidence of collective phenomena in driven-dissipative quantum metamaterials.

•Oct 20, 2025•0 citations

Quantum master equation with dissipators regularized by thermal fluctuations

Arnab Chakrabarti, Rangeet Bhattacharyya

We report an alternate formulation of the quantum master equation (QME) to describe the dynamics of a quantum system weakly coupled to a heat bath, in the presence of weak external driving. A key feature of this approach is the introduction of an explicit Hamiltonian to model the thermal fluctuations in the heat bath. We show that the resulting time coarse-grained dynamical equation for the quantum system has dissipators with a natural regulator, which emerges from an ensemble average over the fluctuations. Importantly, such regularized dissipators arise from the second-order contributions of both the external drive as well as the system-environment coupling. We show that the second-order drive terms, regularized to time-scales set by the fluctuations, result in dynamic drive-induced frequency shifts as well as drive-dependent relaxation phenomena. Considering the specific case of an ensemble of two-level systems, subjected to a linearly polarized external drive, we derive the modified Bloch Equations with such drive-dependent shift and damping terms. The resulting drive-induced frequency shifts converge to the known forms of dynamic frequency shifts, such as the Bloch-Siegert shift or the dynamic Stark shift, in appropriate limits. The Kramers-Kronig pairs of these frequency shifts - manifest as drive-dependent damping terms in the modified Bloch Equations and help explain the Redfield limit of free-induction-decay (FID) rates as well as the non-Bloch decay of Rabi oscillations in isotropic medium. Our method predicts correct orders of magnitudes of non-Bloch decay rates in isotropic medium as well as their observed temperature dependence. The QME reported here, correctly describes all known aspects of the driven-dissipative dynamics up to second-order of an open quantum system with appropriate thermal signatures and as such is expected to provide deeper insights into the study of quantum information processing on real systems.

Physical Review A•Jun 18, 2018•15 citations
Driven-dissipative qubit-resonator dynamics, which are the basis of most dispersive superconducting qubit measurement schemes, are often modeled with Lindblad master equations built from subsystem local jump operators, even when the qubit and resonator are appreciably hybridized. In this work we revisit this setting using a microscopic Bloch-Redfield approach, where dissipation is constructed in the eigenbasis of the coupled qubit-resonator Hamiltonian with a complete, frequency dependent, open system description of the transmission line environment. Here, we show that the Lindblad and Bloch-Redfield decay rates can be quantitatively different in the absence of driving, while in the driven case we demonstrate that the time-independent Redfield dissipator and its time-dependent generalization can show qualitatively different behaviors as a function of driving strength. Finally, we investigate the effects of driving in structured spectral densities, recovering the suppression of measurement-induced relaxation in the presence of a so called Purcell filter.
The speed and fidelity of dispersive readout of superconducting qubits should improve by increasing the amplitude of the measurement drive. Experiments show, however, that beyond some drive amplitude there is always a saturation or drop in fidelity, often associated with a decrease in qubit energy relaxation time T1T_1T1​. A simple Lindblad master equation does not capture the latter effect. More involved approaches based on effective master equations rely on strong assumptions about the spectra of the system and the bath and only partially agree with observations. Here, we perform a first-principles simulation of the full unitary dynamics of dispersive readout by considering the circuit QED Hamiltonian coupled to a microscopic model for the measurement transmission line, allowing for its arbitrary spectrum, including filters. Our access to the dynamics of the bath degrees of freedom allows us to investigate the emission spectrum of the system as a function of drive power. We show how the dependence of qubit T1T_1T1​ on readout drive amplitude is sensitive to the details of the bath spectrum. In particular, we find that T1T_1T1​ drops with increasing drive amplitude when a Purcell notch filter is placed at the qubit frequency, and that the Lindblad master equation shows general qualitative defects compared to the first-principles model.
Driven nonlinear quantum oscillators are a central platform for quantum technologies, yet their dissipative dynamics are typically described using Lindblad or Caldeira-Leggett master equations derived under assumptions that exclude nonlinearities and driving. Here, we derive a generalized Caldeira-Leggett master equation for driven nonlinear oscillators by retaining the full nonlinear and time-dependent system dynamics in the construction of the dissipator. For position- and momentum-dependent system-bath coupling, the dissipator itself becomes dynamically dressed, generating nonlinear and drive-dependent dissipative channels beyond conventional fixed-dissipator approaches. This produces nonlinear damping together with dissipation-induced corrections to the effective drive. The resulting dissipative dynamics suppress large-amplitude excitations and reduce phase-space fluctuations. For a driven Kerr oscillator, this leads to the suppression of bistability, asymmetric resonance responses, and strongly modified fluctuation distributions. More broadly, our results establish a microscopic framework in which nonlinear dynamics and driving directly reshape the dissipative sector of driven open quantum systems.
The quantum state of a superconducting qubit nonresonantly coupled to a transmission line resonator can be determined by measuring the quadrature amplitudes of an electromagnetic field transmitted through the resonator. We present experiments in which we analyze in detail the dynamics of the transmitted field as a function of the measurement frequency for both weak continuous and pulsed measurements. We find excellent agreement between our data and calculations based on a set of Bloch-type differential equations for the cavity field derived from the dispersive Jaynes-Cummings Hamiltonian including dissipation. We show that the measured system response can be used to construct a measurement operator from which the qubit population can be inferred accurately. Such a measurement operator can be used in tomographic methods to reconstruct single and multiqubit states in ensemble-averaged measurements.
Fast and reliable manipulation with qubits is fundamental for any quantum technology. The implementation of these manipulations in physical systems is the focus of studies involving optimal control theory. Realistic physical devices are open quantum systems. So far, studies in optimal control theory have primarily utilized the Redfield/Lindblad quantum master equation to simulate the dynamics of such systems. However, this Markov description is not always sufficient. Here, we present a study of qubit control utilizing the nonequilibrium Green's function method. We compare the traditional master equation with more general Green's function results and demonstrate that even in the parameter regime suitable for the application of the Redfield/Lindblad approach, the two methods yield drastically different results when addressing evolution involving mixed states. In particular, we find that, in addition to predicting different optimal driving profiles, a more accurate description of system evolution enables the system to reach the desired final state much more quickly. We argue that the primary reason for this is the significance of the non-Markov description of driven system dynamics due to the effect of time-dependent driving on dissipation.
Superconducting circuits are a powerful platform for studying and controlling the interaction between light and matter at the quantum level. In these systems, superconducting qubits act as artificial atoms (with matter-like degrees of freedom), whereas microwave resonators play the role of oscillators hosting photons (with photonic-like degrees of freedom). A major advantage of this platform is the ability to tailor the electromagnetic environment to which the qubits are coupled. This degree of control makes it possible to realize well-known quantum optics models, like the Jaynes-Cummings model, as well as more complex systems, like quantum electrodynamics close to a bandgap. In this thesis, we engineer structured electromagnetic environments and couple superconducting qubits to them to experimentally study their interaction. By coupling multiple microwave resonators together, we create a metamaterial. When an atom interacts with the metamaterial, an atom-photon bound state emerges. We observe the dynamics of the creation and dissolution of the atom-photon bound state, as well as the spectral content of its photonic component, for the first time to our knowledge.Our metamaterial is intrinsically nonlinear because each resonator is formed by an array of Josephson junctions to serve as an inductor. When the system is pumped, its phase diagram exhibits multimode dissipative phase transitions with timescales on the order of hundred seconds. There is ongoing theoretical debate about whether such phase transitions can occur in one-dimensional systems, a question we address and resolve in this work.Furthermore, we engineer qubit decay using two different approaches. In the first one, we engineer the decay of two coupled atoms to two waveguides by using a symmetric–antisymmetric coupling configuration. This approach enables selective coupling to bright and dark states of the system. In the second one, we suppress the Purcell decay of a qubit using a compact lumped-element Purcell filter. In addition, we characterize the Purcell decay by employing an auxiliary superconducting qubit as a sensitive power detector directly coupled to the feedline waveguide used for the qubit readout.In summary, this thesis presents three different methods for controlling light–matter interactions in structured quantum systems, including decay suppression with both Purcell filters and symmetry-selective couplings. The thesis also introduces new experimental techniques for probing atom-photon bound states and provides direct evidence of collective phenomena in driven-dissipative quantum metamaterials.
We report an alternate formulation of the quantum master equation (QME) to describe the dynamics of a quantum system weakly coupled to a heat bath, in the presence of weak external driving. A key feature of this approach is the introduction of an explicit Hamiltonian to model the thermal fluctuations in the heat bath. We show that the resulting time coarse-grained dynamical equation for the quantum system has dissipators with a natural regulator, which emerges from an ensemble average over the fluctuations. Importantly, such regularized dissipators arise from the second-order contributions of both the external drive as well as the system-environment coupling. We show that the second-order drive terms, regularized to time-scales set by the fluctuations, result in dynamic drive-induced frequency shifts as well as drive-dependent relaxation phenomena. Considering the specific case of an ensemble of two-level systems, subjected to a linearly polarized external drive, we derive the modified Bloch Equations with such drive-dependent shift and damping terms. The resulting drive-induced frequency shifts converge to the known forms of dynamic frequency shifts, such as the Bloch-Siegert shift or the dynamic Stark shift, in appropriate limits. The Kramers-Kronig pairs of these frequency shifts - manifest as drive-dependent damping terms in the modified Bloch Equations and help explain the Redfield limit of free-induction-decay (FID) rates as well as the non-Bloch decay of Rabi oscillations in isotropic medium. Our method predicts correct orders of magnitudes of non-Bloch decay rates in isotropic medium as well as their observed temperature dependence. The QME reported here, correctly describes all known aspects of the driven-dissipative dynamics up to second-order of an open quantum system with appropriate thermal signatures and as such is expected to provide deeper insights into the study of quantum information processing on real systems.