Dissipative dynamics in a system in contact with a large environment is described by a quantum master equation. The quantum master equation is microscopically derived from the unitary dynamics under the Hamiltonian of the total system, including the environment. Since dissipation is ubiquitous in nature, it is an important problem of statistical physics to understand (Q1) how the quantum master equation is derived from the microscopic quantum mechanics, and (Q2) what is the property of the quantum master equation that is derived microscopically.
I have thoroughly investigated the both problems (Q1) and (Q2). As for (Q1), I have made efforts to clarify precise conditions that the quantum master equation, especially in the Lindblad form with the complete positivity, is microscopically derived. In [1], I showed that a nontrivial Lindblad quantum master equation is obtained when the system is strongly coupled to a bath and an external periodic field. In [2], we tried to rigorously derive the Lindblad equation for many-body open quantum systems, which are recently paid much attention.
As for the problem (Q2), I and my colleague investigated the spectral property of the Lindblad dynamics and showed that the relaxation of an open quantum many-body system exhibits unexpected delays compared to a naive guess obtained from the spectral gap of the dynamical generator (Lindbladian) [3]. This unexpected delay also happens in a many-body classical stochastic process and gives a new mechanism of metastability [4]. Furthermore, I also clarified that the spectral gap (Liouvillian gap) of the Lindbladian for a many-body quantum system under bulk dissipation shows interesting singularity [5]: if we first take the dissipationless limit and then take the thermodynamic limit, the Liouvillian gap tends to zero, while if we take the dissipationless limit after the thermodynamic limit, the Liouvillian gap converges to a nonzero value. This nontrivial Liouvillian gap that remains even in the dissipationless limit is interpreted as the quantum Ruelle-Pollicott resonance: it gives an intrinsic relaxation timescale of the unitary time evolution without dissipation.
Takashi Mori, "Strong Markov dissipation in driven-dissipative quantum systems", J. Stat. Phys. 192, 1 (2025).
Teruhiro Ikeuchi and Takashi Mori, "Error bounds on the universal Lindblad equation in the thermodynamic limit", Phys. Rev. B 112, 094309 (2025).
Takashi Mori and Tatsuhiko Shirai, "Resolving Discrepancy between Liouvillian Gap and Relaxation Time in Boundary-Dissipated Quantum Many-Body Systems", Phys. Rev. Lett. 125, 230604 (2020).
Takashi Mori, "Metastability associated with many-body explosion of eigenmode expansion coefficients", Phys. Rev. Res. 3, 043137 (2021).
Takashi Mori, “Liouvillian-gap analysis of open quantum many-body systems in the weak dissipation limit”, Phys. Rev. B 109, 064311 (2024).
It is a long-standing fundamental problem to reconcile reversible microscopic laws and irreversible macroscopic phenomena. For example, thermalization, i.e., relaxation to thermal equilibrium, of a macroscopic system is ubiquitous in nature, but it is not at all obvious whether an isolated quantum system can thermalize.
In recent years, this problem has gathered much attention because of the experimental progress in ultra-cold atoms and trapped ions, and our theoretical understanding on this fundamental problem has been refined. Typicality and Eigenstate Thermalization Hypothesis (ETH) are important concepts.
We have derived some rigorous results on the eigenstate thermalization hypothesis [1,2]. I'm also interested in prethermalization [3,4], i.e., the relaxation toward a long-lived quasi-stationary state before reaching thermal equilibrium.
I, with my colleagues, published a review paper, which overviews theoretical development in this field.
Takashi Mori, "Weak eigenstate thermalization with large deviation bound" arXiv:1609.09776
Naoto Shiraishi and Takashi Mori,"Systematic Construction of Counterexamples to the Eigenstate Thermalization Hypothesis", Phys. Rev. Lett. 119, 030601 (2017)
Eriko Kaminishi, Takashi Mori, Tatsuhiko N Ikeda, Masahito Ueda, "Entanglement pre-thermalization in a one-dimensional Bose gas", Nature Phys. 11, 1050 (2015)
Takashi Mori, "Prethermalization in the transverse-field Ising chain with long-range interactions", J. Phys. A 52, 054001 (2019)
When a quantum system is subject to a periodically oscillating external field (e.g. laser light in condensed matter or mechanical oscillation in cold atoms), the Hamiltonian of the system periodically depends on time. Such a system is called a periodically driven system or a Floquet system, which stems from the fact that the Floquet theory is an appropriate theoretical tool to treat such systems.
In recent years, strong (large amplitude) and fast (large frequency) periodic driving has paid much attention. For such driving, we can derive a simple expression of a static effective Hamiltonian by applying the Floquet-Magnus expansion. The simple expression of the effective Hamiltonian enables us to design periodic fields so that they generate an interesting phase of matter that is hard to be realized in a static system.
However, the method of the effective Hamiltonian had not been justified in macroscopic quantum Floquet systems before our works. Indeed, it is known that the convergence radius of the Floquet-Magnus expansion shrinks to zero in the thermodynamic limit, which doubts the applicability of this method. We made clear the condition under which the use of the static effective Hamiltonian is justified by investigating the Floquet-Magnus expansion mathematically [1,2]. It turns out that the timescale of heating due to periodic driving is exponentially slow with respect to the frequency in generic lattice systems. This result implies that genetic lattice systems under a strong and fast oscillating field exhibit Floquet prethermalization, and the method of the effective Hamiltonian is justified in a prethermal regime of the relaxation process. The use of the effective Hamiltonian is also justified for some off-lattice systems, but the timescale of heating is faster than exponential [3].
I have also investigated the steady state of a many-body Floquet system under dissipation by using the quantum master equation. This problem is important for Floquet physics in solids since dissipation is inevitable there. We made clear the condition under which the notion of the Gibbs state in terms of the effective Hamiltonian is valid in the steady state [4, 5].
Tomotaka Kuwahara, Takashi Mori, and Keiji Saito, "Floquet–Magnus theory and generic transient dynamics in periodically driven many-body quantum systems" Ann. Phys. 367, 96 (2016)
Takashi Mori, Tomotaka Kuwahara, and Keiji Saito, "Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems" Phys. Rev. Lett. 116, 120401 (2016)
Takashi Mori, "Floquet resonant states and validity of the Floquet-Magnus expansion in the periodically driven Friedrichs models", Phys. Rev. A 91, 020101 (2015)
Tatsuhiko Shirai, Takashi Mori, and Seiji Miyashita, "Condition for emergence of the Floquet-Gibbs state in periodically driven open systems", Phys. Rev. E 91, 030101 (2015).
Tatsuhiko Shirai, Juzar Thingna, Takashi Mori, Sergey Denisov, Peter Hänggi, and Seiji Miyashita, "Effective Floquet–Gibbs states for dissipative quantum systems", New J. Phys. 18, 053008 (2016).
Equilibrium statistical mechanics of long-range interacting systems shows various peculiar properties such as the violation of ensemble equivalence and negative specific heat. In an extreme case where the interaction potential is independent of the distance, we can investigate the problem analytically since it is known that the mean-field theory is exact for such fully connected models in the thermodynamic limit. However, long-range interaction potentials in nature decay with distance slowly, and not many rigorous results have been known.
For long-range interacting spin systems in which the pair potential decays with distance r slower than 1/r^d, where d is the spatial dimension, I rigorously proved that the mean-field theory is still exact in the canonical ensemble [1,2]. On the other hand, the mean-field theory can break down in the canonical ensemble with a fixed value of the magnetization [1,2] or in the microcanonical ensemble [3]. These results also hold in quantum spin systems [4]. See also [5] for a brief review.
I also explored the possibility that a short-range interacting system exhibits peculiar properties of equilibrium statistical mechanics of long-range interacting systems, such as the negative specific heat. Rigorous results of equilibrium statistical mechanics rule out this possibility in thermal equilibrium. I found that a metastable state of a certain short-range interacting system resembles an equilibrium state of a certain long-range interacting model [6,7]. As a result, this short-range model exhibits the violation of the ensemble equivalence, the negative specific heat, and the negative susceptibility in its metastable state. This result indicates that considering equilibrium statistical mechanics of long-range interacting systems may be much more important than considered previously.
Takashi Mori, "Analysis of the exactness of mean-field theory in long-range interacting systems", Phys. Rev. E 82, 060103(R) (2010)
Takashi Mori, "Instability of the mean-field states and generalization of phase separation in long-range interacting systems", Phys. Rev. E 84, 031128 (2011)
Takashi Mori, "Microcanonical Analysis of Exactness of the Mean-Field Theory in Long-Range Interacting Systems", J. Stat. Phys. 147, 1020 (2012)
Takashi Mori, "Equilibrium properties of quantum spin systems with nonadditive long-range interactions", Phys. Rev. E 86, 021132 (2012)
Takashi Mori, "Phase transitions in systems with non-additive long-range interactions", J. Stat. Mech. P10003 (2013)
Takashi Mori, "Nonadditivity in Quasiequilibrium States of Spin Systems with Lattice Distortion", Phys. Rev. Lett. 111, 020601 (2013)
Takashi Mori, "Quasi-Equilibrium Nonadditivity", J. Stat. Phys. 159, 172 (2015)
Modern machine learning achieves unparalleled successes in the overparameterized regime, in which the number of parameters in the model greatly exceeds that of the training data samples. This is surprising because traditional learning theory suggests that such an overparameterized model suffers from serious overfitting and does not show good generalization ability.
It is a theoretical challenge to understand why overparameterized deep neural networks exhibit astonishing generalization ability without serious overfitting. There are several perspectives to explain it: the importance of data structure, some special properties of network architecture, and implicit biases of the optimization algorithm like stochastic gradient descent (SGD).
In our recent works, I investigated the efficiency of SGD [1,2]. In [2], I focused on the property of dynamical noise in SGD, and showed that noise strength is proportional to the loss function as well as its Hessian near a local minimum. This property of SGD noise suggests that the escape rate from a local minimum obeys an unconventional law. The new escape rate formula implies that SGD prefers minima with a low effective dimension. This result will trigger further studies on SGD and its implicit biases.
Liu Ziyin, Kangqiao Liu, Takashi Mori, and Masahito Ueda, "Strength of Minibatch Noise in SGD", In the Proceedings of the 10th International Conference on Learning Representations (ICLR2022)
Takashi Mori, Liu Ziyin, Kangqiao Liu, and Masahito Ueda, "Power-Law Escape Rate of SGD", In Proceedings of the 39th International Conference on Machine Learning (ICML2022), PMLR 162, 15959 (2022)