論文 / 著書情報
Article / Book Information
題目(和文)

開放量子系におけるパリティ・時間対称性由来 の散逸連続時間結晶

Title(English)

Dissipative continuous time crystals originating from parity-timesymmetry in open quantum systems

著者(和文)

中西優馬

Author(English)

Yuma Nakanishi

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第12640号,
授与年月日:2024年3月26日,
学位の種別:課程博士,
審査員:笹本 智弘,村上 修一,西田 祐介,石塚 大晃,納富 雅也

Citation(English)

Degree:Doctor (Science),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第12640号,
Conferred date:2024/3/26,
Degree Type:Course doctor,
Examiner:,,,,

学位種別(和文)博士論文Category(English)

Doctoral Thesis

種別(和文)

要約

Type(English)

OutlinePowered by T2R2 (Tokyo Institute Research Repository)

（博士課程）Doctoral Program論 文 要 約ＴＨＥＳＩＳ OUTLINE

系・コース：
Department of, Graduate major in

学生氏名：
Student’s Name

物理学
物理学
中西 優馬系コース

申請学位（専攻分野）：

博士

Academic Degree Requested

Doctor of

審査員主査：
Chief Examiner（ 理学 ）
笹本 智弘

要約
Exploration of novel non-equilibrium phases of matter or phase transitions and clarification of their mechanisms areimportant problems for non-equilibrium statistical physics. In particular, (continuous) time crystals that spontaneously breakcontinuous time-translation symmetry into discrete time-translation symmetry are a typical example, and these have been
observed in open quantum systems but proved to be impossible to exist in closed quantum systems. Here, continuous time
crystals in open quantum systems are called dissipative continuous time crystals. In this thesis, we provide novel nonequilibrium phase transitions with anti-unitary symmetry breaking in Markovian open quantum systems and show that theycan explain the mechanism of a class of dissipative continuous time crystals in open collective spin systems.
First, we give a novel phase transition similar to the parity-time (PT) phase transitions, which are studied in non-Hermitiansystems well, for investigating a two-collective spin system with balanced gain and loss described by the GKSL (GoriniKossakowski-Sudarshan-Lindblad) equation. Here, a generator of the GKSL equation is called the Lindbladian. In the nonHermitian PT phase transitions, for the Hamiltonian with the PT symmetry, the eigenvalues change from real to complex
eigenvalues, thereby changing the dynamics from periodic oscillations to diverging or decaying dynamics, and the PTsymmetry of the eigenstate is broken at the same parameter point. On the other hand, for the Lindbladian case, we show thatwhen the Lindbladian has a PT symmetry, the Lindbladian spectrum transition, dynamics transition, and PT symmetry
breaking of the stationary state occur at the same parameter point. Here, we use the Holstein-Promakoff approximation,
which linearly maps the collective spin operators to bosonic operators, and the third quantization, which is the prescription
to diagonalize the Lindbladian with quadratic Hamiltonian and linear Lindblad operators. Also, in the non-Hermitian PT
phase transition, the transition point is known as the exceptional point where more than two eigenvectors coalesce. We will
explain that the transition point is expected to be an exceptional point in the Lindbladian case as well. These properties are
similar to those of non-Hermitian PT phase transitions. This dissipative phase transition is called the Lindbladian PT phasetransition. Then, we can define the Lindbladian PT phase as satisfying the following three properties: (i) the model has astrong Lindbladian PT symmetry, (ii) purely imaginary eigenvalues exist in the thermodynamic limit, and (iii) the PT
symmetry of a stationary state holds in the thermodynamic limit.
Next, we show the physical origin of the boundary time crystal (BTC) for specific models, which is a dissipative time
crystal only in the thermodynamic limit. In particular, we focus on the dissipative one-collective spin model with transverse
magnetic field and excitation loss (hereafter, we refer to this model as the Driven Dicke model). In the Driven Dicke model,
the exact solution of the stationary state had already been found in 1979, and the static dissipative continuous phase transition
was known to occur. However, the mechanism of this phase transition is unclear. Notably, this continuous phase transition
has been recently observed in a cold atom gas system. Also, the persistent oscillation appearing in this model was reaffirmed
as the BTC in 2018, and this mechanism is also unclear.
In this thesis, we analytically show that the Driven Dicke model has the Lindbladian PT symmetry when we choose theparity operator to reflect the basis of Sz. Physically speaking, it means that the Driven Dicke model returns to itself aftertransforming the parity and time reversal operations. Moreover, we analytically show that the PT symmetry breaking of thestationary state occurs at the point where BTC breaks down. Also, we numerically show a similar statement for another
BTC (generalized Driven Dicke) model. Here, we introduce the PT symmetry parameter that characterizes the PT symmetryof the density matrix. Finally, using perturbation theory with degeneracy, we prove that BTC arises under weak dissipation
for a class of one-collective spin systems only when gain and loss are balanced. These results indicate that there exist BTCs
in collective spin systems originating from parity-time symmetry.
Furthermore, we give a simple and clear description of BTCs and the Lindbadian PT phase transitions. Our strategy isthat the Schwinger boson transformation makes it easier to discuss the effect of PT symmetry, even at the mean-field level.
In particular, we show that when the Lindbladian has PT symmetry at the microscopic level, the Schwinger boson meanfield equation becomes a nonlinear Schrodinger equation with U (1) symmetry. Moreover, we prove that if a Lindbladian
has PT symmetry at the microscopic level, the effective nonlinear Hamiltonian also has PT symmetry for the general

collective spin system. Also, the continuous and discontinuous phase transitions can occur with spontaneous non-linear PT
symmetry breaking. Then, we demonstrate that all collective excitation modes are shown to be destabilized at the transition
point using the linear stability analysis. As a result, the time-dependent phases appear, such as the swap or chiral phases, inwhich the amplitude or phase periodically oscillates respectively. In particular, the phase where amplitudes are timedependent (e.g., swap phase) corresponds to the time-crystalline (Lindbladian PT) phase in the notation of collective spin.
Moreover, we show that critical exceptional points (CEPs), in which two excitation modes coalesce, emerge in the case ofcontinuous phase transitions. Finally, we confirm the validity of these analyses by comparing them with numericalcalculations. As a result, we can expect that the exceptional point also emerges for the Lindbladian.
Lastly, we summarize and mention the outlook on the BTCs and Lindbladian PT phase transitions. In particular, wemention the application of Lindbladian exceptional points. The PT phase transitions in non-Hermitian systems have beenused for many applications, such as sensors and lasers, by exploiting the properties of exceptional points. In particular, inorder to be used as a sensor, both (i) the response coefficient of the output quantity to the input quantity of interest and (ii)the signal-to-noise ratio must be high. Around exceptional points, the property (i) satisfies. However, it has been shown thatthe signal-to-noise ratio for classical and quantum noise cannot improve even around exceptional points. On the other hand,by including nonlinearity to the Hamiltonian, it has recently been shown that the signal-to-noise ratio for classical noise canbe improved around the exceptional point. Since the Driven Dicke model has nonlinearity at the mean-field level, it can beexpected to improve the signal-to-noise ratio for classical noise around the transition point. In addition, since the Driven
Dicke model shows exceptional points at the Lindbladian level, it would be very interesting to investigate whether thesignal-to-noise ratio for quantum noise is also improved around the transition point. My thesis will not only further ourunderstanding of non-equilibrium phase transitions or phases of matter, but also help us develop applied techniques using
exceptional points in open quantum systems.