Physics Reports 995 (2023) 1–89

Contents lists available at ScienceDirect

Physics Reports
journal homepage: www.elsevier.com/locate/physrep

Nonadiabatic Landau–Zener–Stückelberg–Majoranatransitions, dynamics, and interference∗

Oleh V. Ivakhnenko a,b , Sergey N. Shevchenko a,c,b , , Franco Nori b,da

B. Verkin Institute for Low Temperature Physics and Engineering, Kharkiv 61103, Ukraine
Center for Quantum Computing, Cluster for Pioneering Research, RIKEN, Wako-shi, Saitama 351-0198, Japanc
V. N. Karazin Kharkiv National University, Kharkiv 61022, UkrainedPhysics Department, University of Michigan, Ann Arbor, MI 48109-1040, USAbarticleinfo

Article history:
Received 21 May 2022
Accepted 12 October 2022
Available online xxxx
Editor: David Campbell
Keywords:
Landau–Zener–Stückelberg–Majoranatransition
Adiabatic-impulse model
Coherent destruction of tunneling
Kibble–Zurek mechanism
Photon-assisted tunneling
Rapid adiabatic passage
Quantum control
Classical coherent phenomena
Stückelberg oscillations
Landau-Zener transitiona b s t r a c t
Since the pioneering works by Landau, Zener, Stückelberg, and Majorana (LZSM), it has
been known that driving a quantum two-level system results in tunneling between
its states. Even though the interference between these transitions is known to beimportant, it is only recently that it became both accessible, controllable, and usefulfor manipulating a growing number of quantum systems. Here, we systematicallystudy various aspects of LZSM physics and review the relevant literature, significantly
expanding the review article in Ref. Shevchenko et al. (2010).
© 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC
BY license (http://creativecommons.org/licenses/by/4.0/).

Contents
1.

2.

Introduction............................................................................................................................................................................................... 3
1.1.
Relation to previous work and structure of this paper .......................................................................................................... 3
1.2.
Driven few-level systems............................................................................................................................................................ 4
Linear drive: Landau–Zener–Stückelberg–Majorana (LZSM) transition ............................................................................................. 7
2.1.
Hamiltonian and bases................................................................................................................................................................ 7
2.2.
Brief overview of the original works of Landau, Zener, Stückelberg, and Majorana .......................................................... 9
2.2.1.
Near-adiabatic limit (Landau) ..................................................................................................................................... 9
2.2.2.
Using parabolic cylinder functions (Zener) ............................................................................................................... 9
2.2.3.
Using the WKB approximation (Stückelberg); double-passage solution ............................................................... 10
2.2.4.
Using contour integrals (Majorana) ........................................................................................................................... 10
2.3.
Different properties of the transition........................................................................................................................................ 10
2.3.1.
Adiabatic theorem ........................................................................................................................................................ 10
2.3.2.
Dynamics and times of a transition........................................................................................................................... 11
2.3.3.
Problems with nonlinearities...................................................................................................................................... 14∗ Correspondence to: B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky
Ave., Kharkiv, 61103, Ukraine.
E-mail address: sshevchenko@ilt.kharkov.ua (S.N. Shevchenko).
https://doi.org/10.1016/j.physrep.2022.10.002
0370-1573/© 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/
licenses/by/4.0/).

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

3.

2.4.

Some experimental observations ............................................................................................................................................... 15

2.5.

Transfer matrix (TM) method .................................................................................................................................................... 17

Repetitive passage: interference............................................................................................................................................................. 19
3.1.

3.2.

4.

5.

6.

7.

Physics Reports 995 (2023) 1–89

Adiabatic-impulse model (AIM) ................................................................................................................................................. 19
3.1.1.

Double-passage and multiple-passage cases ............................................................................................................ 19

3.1.2.

Kibble–Zurek mechanism (KZM) ................................................................................................................................ 23

Rotating-wave approximation (RWA) ....................................................................................................................................... 25
3.2.1.

Multi-photon Rabi oscillations ................................................................................................................................... 25

3.2.2.

Multi-photon transitions analyzed by the LZSM theory ......................................................................................... 27

3.3.

Floquet theory .............................................................................................................................................................................. 27

3.4.

Driving fields in the quantum regime ...................................................................................................................................... 29

3.5.

Impact of dissipation and temperature..................................................................................................................................... 30

3.6.

Rate equation ............................................................................................................................................................................... 32

3.7.

Quantum phase tomography ...................................................................................................................................................... 33

3.8.

Comparison of different methods .............................................................................................................................................. 34

Interferometry........................................................................................................................................................................................... 35
4.1.

Superconducting circuits............................................................................................................................................................. 38

4.2.

Semiconductor quantum dots .................................................................................................................................................... 38

4.3.

Donors and impurities (atomic qubits)..................................................................................................................................... 39

4.4.

Graphene....................................................................................................................................................................................... 40

4.5.

Microscopic systems.................................................................................................................................................................... 40

4.6.

Other systems .............................................................................................................................................................................. 41

4.7.

Multilevel systems ....................................................................................................................................................................... 41
4.7.1.

Theory .......................................................................................................................................................................... 42

4.7.2.

Single passage in multilevel systems ....................................................................................................................... 42

4.7.3.

Multiple-passage transitions in multilevel systems ............................................................................................... 42

Quantum control ...................................................................................................................................................................................... 42
5.1.

Coherent control of microscopic and mesoscopic structures................................................................................................. 43

5.2.

Universal single- and two-qubit control................................................................................................................................... 44

5.3.

Shortcuts to adiabaticity ............................................................................................................................................................. 45

5.4.

Adiabatic quantum computation ............................................................................................................................................... 46

Related classical coherent phenomena .................................................................................................................................................. 47
6.1.

From Newton to Schrödinger .................................................................................................................................................... 48

6.2.

Engineered photonic waveguides ............................................................................................................................................ 50

Conclusion ................................................................................................................................................................................................. 50
Declaration of competing interest.......................................................................................................................................................... 51
Acknowledgments .................................................................................................................................................................................... 51
Appendix A. LZSM transition probability .............................................................................................................................................. 51
A.1.

Near-adiabatic limit (Landau)..................................................................................................................................................... 51

A.2.

Using parabolic cylinder functions (Zener) .............................................................................................................................. 52

A.3.

With contour integrals (Majorana) ............................................................................................................................................ 56

A.4.

Using the WKB approximation and the phase integral method (Stückelberg).................................................................... 57

A.5.

Duration of the LZSM transition ................................................................................................................................................ 59

Appendix B. Description of a periodically driven two-level system.................................................................................................. 61
B.1.

Adiabatic-impulse model (AIM) ................................................................................................................................................. 61
B.1.1.

Adiabatic evolution ...................................................................................................................................................... 61

B.1.2.

Multiple-passage evolution ......................................................................................................................................... 61

B.2.

Rotating-wave approximation (RWA) ....................................................................................................................................... 62

B.3.

Floquet theory .............................................................................................................................................................................. 63

B.4.

Rate-equation and white-noise approach ................................................................................................................................. 67
B.4.1.

Transition rate .............................................................................................................................................................. 67

B.4.2.

Rate equation................................................................................................................................................................ 68

B.4.3.

From Bessel to Airy...................................................................................................................................................... 68

B.4.4.

Double-passage regime................................................................................................................................................ 69

References ................................................................................................................................................................................................. 69

2

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Abbreviations and most-often-used symbols
For the readers’ convenience, below we list the main abbreviations used in this work. This list also includes someof the topics covered.
AIM
Adiabatic-impulse model;
CDT
Coherent destruction of tunneling;
JJ
Josephson junction;
KZM
Kibble–Zurek mechanism;
LZSM
Landau–Zener–Stückelberg–Majorana;
PAT
Photon-assisted tunneling;
RAP
Rapid adiabatic passage;
RWA
Rotating-wave approximation;
TLS
Two-level system;
TM
Transfer matrix;
P
Single-passage LZSM transition probability;∆
Minimal energy-level splitting;εEnergy bias;
A, ω, Td
Amplitude, frequency, and period of the driving field;
δAdiabaticity parameter;∆E
Qubit energy-level gap;
Relaxation rate;Γ1 = T1−1
Γ2 = T2−1
Decoherence Rate;T
Temperature.

“Without nonadiabatic transition, this world would have been dead, because no basic chemical and biological processes,
such as electron and proton transfer, could have occurred. Nonadiabatic transition is certainly an origin of mutability ofthis world.”(Zhu et al., 2007)
1. Introduction
The quantum two-level system (TLS) is one of the basic models in quantum physics and describes systems that are
ubiquitous in nature. On the one hand, this is the “simplest nonsimple quantum problem”, quoting (Berry, 1995); and, on
the other hand, this provides the basis for quantum technologies, in which a TLS refers to a qubit.
If a quantum system is excited by a time-dependent drive, it displays a variety of interesting and important effects. Notethat several Nobel Prizes in physics have been awarded to physicists who exploited time-dependent few-level quantum
systems:• 1944: Rabi on molecular beams and nuclear magnetic resonance;• 1952: Bloch and Purcell on magnetic fields in atomic nuclei and nuclear magnetic moments;• 1964: Townes, Basov, and Prochorov on masers, lasers, and quantum optics;• 1966: Kastler on optical pumping;• 1989: Ramsey, Dehmelt, and Paul on atomic spectroscopy, hydrogen maser, and atomic clocks;• 1997: Chu, Cohen-Tannoudji, and Philips on cooling and trapping atoms with laser light;• 2012: Haroche and Wineland on coupled atoms and photons;• 2022: Aspect, Clauser, and Zeilinger on entangled photons and quantum information science.
1.1. Relation to previous work and structure of this paper
We could ask ourselves a question here, quoting Ref. Benderskii et al. (2003):“The title of this paper might sound perplexing at first sight. What else can be said about the Landau–Zener(LZ) problem after the numerous descriptions in both research and textbook literature?”
Below we give several reasons, starting from the fact that this topic should be called LZSM, not only LZ, and ending withthe point that this evergreen topic is nowadays important for many areas of physics and its applications are growing overtime.
3

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

LZSM transitions are ubiquitous and important and have been addressed in several review articles (e.g., Kazantsev et al.
(1985), Shimshoni and Gefen (1991), Grifoni and Hänggi (1998), Zhu et al. (2007), Shevchenko et al. (2010), Dziarmaga(2010), Silveri et al. (2017) and Sen et al. (2021)) and books (Nakamura, 2012; Shevchenko, 2019; Nakamura, 2019). In
particular, the central idea of a previous review Ref. Shevchenko et al. (2010) was a detailed presentation of the theoreticaldescription of periodically driven TLSs. Here, we briefly mention the key aspects where the present work significantly
extends this previous one:

• We show how to derive the LZSM formula by following the original works and not only presenting the readersthese often-cited and difficult-to-access works. We also convincingly demonstrate that what is known as Zeneror Landau–Zener transition/formula should be attributed to the four physicists: Landau, Zener, Stückelberg, andMajorana (LZSM).
• We address different important aspects of the nonadiabatic transition, such as transition time, nonlinearity, anddissipation.
• We relate the LZSM formalism for avoided-level crossing with the Kibble–Zurek mechanism (KZM), which has been
widely used to describe second-order phase transitions.
• We review new results that have appeared in the decade following the previous review article (Shevchenko et al.,2010) and also cover various physical realizations.
• We emphasize that the detailed understanding of LZSM dynamics and its aspects, such as multi-photon transitions,is important not only for spectroscopy or interferometry, but also for quantum control.
For example, the original LZSM problem was covered very briefly in Giacomo and Nikitin (2005). However, we must ask
what was studied in the original works of LZSM. Many (probably, the vast majority of) researchers cite the original works
by LZSM without seeing those papers, which are difficult to access and read. Moreover, out of those five papers (Landau,
1932a,b; Zener, 1932; Stückelberg, 1932; Majorana, 1932), only one (Zener, 1932) was written in English. (See thetranslations in Refs. Haar (1965), Stueckelberg (1970) and Cifarelli (2020a).) One of the tasks in the current review article
is to present a pedagogical summary of these original works of LZSM. We believe that seeing all four approaches together
is both instructive and pedagogical.
The present review paper is organized as follows:1 First, in the rest of Section 1, we present diverse physical systems
that can effectively be described as TLSs. A nonadiabatic transition between energy levels, known as the LZSM transition,
is described in Section 2, with details provided in Appendix A. Various approaches to the description of a periodicallydriven TLS are the subject of Section 3 and Appendix B. We devote Section 4 to the description of experimental studies,in which the LZSM interference is relevant. Quantum control with nonadiabatic transitions and periodic driving is outlined
in Section 5. Related classical coherent phenomena are considered in Section 6. Section 7 presents the conclusions.
1.2. Driven few-level systems
First, we outline the systems in which the phenomena are taking place. These are very different in their physical origins
because the objects can be microscopic (electron or nuclear spins, photons, atoms), mesoscopic (superconducting qubits,
quantum dots, graphene structures), or macroscopic (mechanical or electrical resonators) (Nakamura, 2019; Kjaergaard
et al., 2020). Our aim is to demonstrate that the description of all these can be reduced to a quantum two- or few-levelsystem. Here, the key idea is to show that a basic notion in quantum mechanics—a TLS with avoided-level crossing—
is ubiquitous and that for such systems, LZSM physics is relevant. We have chosen several illustrative examples andpresented them in Fig. 1, with some details given in Table 1 and in the main text below.
Note that neither Fig. 1 nor Table 1 are comprehensive because neither gives a complete picture of the variety of
the respective systems. The aim is to show the diversity of the systems and their characteristic parameters, including
their typical sizes. A goal of this review article is to present different realizations of LZSM phenomena. In particular,
the experimental realizations of the single- and multiple-passage transitions in quantum systems will be presented in
Section 2.4 and Section 4, respectively, while their classical counterparts will be presented in Section 6. After making thesereferences to subsequent sections, we briefly describe the quantum systems. Here, a note is in order. We describe classicalrealizations in a separate section and otherwise consider quantum systems. Interestingly, nonadiabatic LZSM transitions
are largely associated with quantum systems; and it is a rare example in physics when classical related phenomena arestudied later than their quantum counterparts and not vice versa.
It is difficult to give a complete picture of those processes where nonadiabatic transitions between potential energy curves matter because these are ubiquitous in natural sciences. Here, we can briefly consider different physicalrealizations.
* As a recent example of this, in the review article (Köhler et al., 2006), the role of LZSM nonadiabatic transitions
is considered in the production of cold molecules and molecular association and dissociation; single and repeated
nonadiabatic transitions were shown to transfer between molecular and atomic states (Mark et al., 2007b; Lang et al.,2008).
1 Please note that we do not always cite the relevant papers in chronological order. We have mostly aimed to tell the story about LZSM physics,
for which it is sometimes more illustrative to refer to later publications or review articles for the sake of the readers’ convenience.
4

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 1. Various physical systems that can be described by the two-level model. These can be driven by external fields and exhibit Landau–Zener–
Stückelberg–Majorana (LZSM) transitions. (a) Superconducting quantum circuits with Josephson junction (JJ) qubits are illustrated by the flux
qubit, made of a superconducting ring with three tunnel JJs and the qubit states formed by the current direction (e.g., Izmalkov et al. (2004)).
(b) Semiconductor artificial atoms are illustrated here by a gate-defined double quantum dot (Cao et al., 2013), where the charge qubit is formedby electron states localized in either the left (L) or right (R) dot. (c) An impurity-based qubit, formed by phosphorus or arsenium donor atoms in asilicon nanowire transistor (Dupont-Ferrier et al., 2013). (d) Graphene strip contacted with gold electrodes, where the current displays nonadiabatic
transitions between the valence and conduction bands around the Dirac point (Higuchi et al., 2017). (e) Ultracold Caesium Feshbach molecules
in a laser trap, where ramping the magnetic field results in the transfer of molecular states (Mark et al., 2007b). (f) Classical nanomechanicalresonator (Faust et al., 2012), where the coherent superposition of the in-plane and out-of-plane modes behave as a driven TLS. (g) Topologicaldefect formation is schematically represented here by an ion Coulomb crystal; this is a trapped ion chain, where changing the confining potential
results in the chain buckling; this breaking of the axial symmetry is a second-order phase transition, with the density of defects described by the
Kibble–Zurek mechanism (Pyka et al., 2013).

* LZSM transitions become important for describing atoms being scattered by a standing light wave (Kazantsev et al.,1985). The ground and excited states of an atom correspond to two effective potentials and two trajectories of5

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Table 1
Characteristic two-level systems (TLSs) and their parameters, including minimal energy-level splitting ∆ and characteristic driving frequency ω. The
respective systems are described in the main text, while details can be found in the references in the main text. The numbers listed above arecharacteristic values or ranges. The table lists both the size of the core quantum system and the size of the host. For example, for ultracold molecules,
the characteristic size of the atoms is of the order of several Angstroms, while the size of the localized Bose–Einstein condensate (BEC) is typicallya few dozens of micrometers.
System

Size

Basis

Variable

∆/h

ω/2π

Temperature(a)

JJ qubits

1 µm to1 mm

Charge, current

Voltage, flux

10 MHz to 10 GHz

1 GHz

50 mK

(b)

Quantumdots

10 nm to1 µm

charge, spin

Voltage,magnetic field

0.1 to 10 GHz

1 to 10 GHz

50 mK

(c)

Atomicqubits

0.1 GHz

1 MHz to 10 GHz

1 µm

Optical andmicrowave
fieldsGraphene

1 µm

Electric field

100 THz to 1 PHz

100 THzRoom

10 kHz

10 kHz0.01 to 100 µK

1Åelectron charge or spin
nuclear spin

50 mK to 10 K
1 K to room

conduction bands(d)valence bands(e)Ultracold
molecules

1 Å

molecular states

magnetic field

40 µm

lattice bands

lattice tilt(f)Classical
resonators

50 µm

Oscillation modes

Bias voltage

10 kHz

10 kHzRoom(g)

Quantum
phasetransitions

300 µmDefect orientation

Confiningvoltage

100 kHz100 kHz

10 µK

motion. The possibility of nonadiabatic transitions between the two states (beams) results in changing an atom’s
trajectory, which leads to interference of the translational motion states; this is similar to a two-channel opticalinterferometer.
* Superconducting quantum circuits are based on Josephson junctions (JJs) (see, e.g., You and Nori (2005, 2011), Xiang
et al. (2013) and Gu et al. (2017)). Depending on the system parameters, there are three basic types of JJ-based
qubits: charge, phase, and flux ones. The energy-level spacing in these can be controlled by an external parameter:gate voltage, bias current, or magnetic flux, respectively. There are also newer subtypes, including a transmon, whichis the capacitor-shunted charge qubit coupled with a transmission line; such layouts allow for better isolation fromexternal noise, allowing for longer coherence times.
* Quantum dots with controllable parameters are mainly based on electrons that are localized in gate-defined depleted
regions of semiconductor heterostructures (typically a few tens of nanometers in size), such as GaAs/AlGaAs and
Si/SiGe, or in nanowire structures (Zwanenburg et al., 2013). These show Coulomb blockade and display singleelectron physics. Depending on which degree of freedom is relevant, we can have spin or charge qubits, which
involve one or several electrons. The energy levels, including the minimal splitting, can be controlled by an external
magnetic field and gate voltages.
* Atomic impurities, such as nitrogen-vacancy (NV) color centers in diamond and phosphorous impurities in silicon,allow for the manipulation of single electron spins and/or nuclear spins. These can be conveniently coupled witheach other, nicely isolated from the environment, can be controlled by optical and microwave fields, and can be
integrated in solid-state devices. We illustrate this with the device from Ref. Dupont-Ferrier et al. (2013), which
is based on a silicon nanowire. The source–drain current was then defined by the electron transport through twotunnel-coupled donor atoms, of which the electronic-state populations created the charge qubit.
* Energy bands with avoided-level crossings, which are relevant for our consideration, also take place in graphene.
When driven by an external electromagnetic field, the Dirac Hamiltonian for graphene results in LZSM phenomena
near the Dirac points (Higuchi et al., 2017; Heide et al., 2018). It has been shown that thin films of a Weyl semimetal
subjected to a strong AC electromagnetic field should behave similarly to graphene (Rodionov et al., 2016). It has
also been discussed that there is a profound similarity between the effects of spatial and temporal periodicity, which
is one more argument why the avoided-level-crossing structures appear in many different contexts. For a review ofother related materials, the so-called “artificial graphenes”, see Montambaux (2018).
* The theory of LZSM transitions is closely related to the Kibble–Zurek mechanism (KZM) (Damski, 2005) which
will be considered in Section 3.1.2. This describes second-order phase transitions, which occurs when one of thesystem parameters passes through a critical point. The universality of second-order phase transitions makes theirdynamics independent of their microscopic nature. This results in a long list of related realizations, from cosmology
to condensed matter. Leaving this intriguing issue for later, here we illustrate the realization of the KZM with chains6O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

of ions confined in harmonic traps (Pyka et al., 2013). In this situation, weakening the triaxial confining potential in
the transverse direction makes the chain buckle and form a zig-zag shape. This second-order phase transition canlead to the formation of topological defects, which is illustrated by a ‘‘zig’’, followed by another ‘‘zig’’, rather than bya ‘‘zag’’. LZSM theory quantitatively describes the formation of such topological defects. Note that the characteristic
parameters for defect formation in Table 1 are used for this very realization; parameters for other phase transitions
may be completely different.
* Two Majorana works meet when Majorana qubits are described by the LZSM Hamiltonian. These are formed by
the Majorana bound states that reside in topological superconducting systems. A realization of this could be an rf(radio frequency) superconducting quantum interference device (SQUID) with a topological JJ that is formed by a onedimensional nanowire with spin–orbit coupling, quantum spin–Hall edge states, or ferromagnetic atomic chains (Youet al., 2014; Huang et al., 2015; Wang et al., 2018; Feng et al., 2018; Zhang and Liu, 2021; Zhang et al., 2021). Theenergy level structure is controlled by an external magnetic flux. Alternatively, a topological superconductor can
be a weak link between quantum dots (Zazunov et al., 2020). Besides being fundamentally interesting, Majorana
qubits provide the basis of topological quantum computation. For more on engineering gauge fields and triggering
topological order in periodically driven systems, see Goldman and Dalibard (2014).
* Somewhat unexpectedly in this context, some classical systems can also be described as TLSs. This arises because
what is needed for LZSM physics (superposition and transition between discrete states) appears not only in thequantum world, but also in classical physics. To this issue, we devote Section 6; but here, we illustrate this with ananomechanical resonator in the classical regime (Faust et al., 2012). This system is based on the coherent energy
exchange between two strongly coupled high-quality modes of a nanomechanical resonator placed in a vacuum atambient temperature.
* To emphasize the variety of TLSs, for which LZSM physics matters, we kaleidoscopically mention a few otherrealizations: electron spin-polarized 4 He+ ion scattering (Suzuki and Yamauchi, 2010; Suzuki and Sakai, 2016),
low-dimensional conductors (Montambaux and Jérome, 2016; Benito et al., 2016), and charge-density-wave insulators (Shen et al., 2014). Overall, nonadiabatic transitions are relevant in physics, chemistry, biology, economics, andsome other—sometimes unexpected—research fields (Nakamura, 2012). As an exotic example, the LZSM model canbe useful in describing decision making in which there are a few possible outcomes; in Ref. Levi (2013), the author
applies the model to describe free will with afterthoughts.
2. Linear drive: Landau–Zener–Stückelberg–Majorana (LZSM) transition
2.1. Hamiltonian and bases
We now present various approaches to derive the formula for the excitation probability of a TLS. To this end, we
briefly introduce the main steps, while details are presented in Appendix A. Interestingly, this can be done in several ways
within different theoretical formalisms (Giacomo and Nikitin, 2005). We aim to study and compare different techniques
by applying these to the classical LZSM problem with a linear drive to the TLS.
Consider a TLS described by the Hamiltonian
H(t) = −

∆
2

σx −

ε (t)
2

σz = −

ε
2 ∆

1

(

)
∆
,
−ε

(1)

with the linear bias

ε (t) = v t ,(2)∆ being a time-independent quantity and time(1) t ∈ (−∞,(0∞) ). Let us now define the diabatic states, which are the
Hamiltonian eigenfunctions at ∆ = 0: |0⟩ = 0 and |1⟩ = 1 . The respective (diabatic) energy levels are E0,1 = ∓ε/2.
These are plotted by the dashed lines in Fig. 2. In general, the wave function is a superposition state|ψ (t)⟩ = α |0⟩ + β |1⟩ =

( )
α
β

(3)

with α and β being the time-dependent coefficients.
The adiabatic eigenvalues E± (t) and eigenstates |E± (t)⟩ are given by the Schrödinger equation, where time is aparameter, H(t) |E± (t)⟩ = E± (t) |E± (t)⟩. We obtain the adiabatic energy levels
E± (t) = ±

1√
2

1

∆2 + ε (t)2 = ± ∆E(t).

(4)

2Here, ∆E = E+ − E− is the distance between the energy levels, which are presented by the solid curves in Fig. 2. Now we
can see the meaning of the parameter ∆ (the minimal energy spacing, or gap), while the parameter ε is the energy bias.
The energy gap is smallest at ε = 0; accordingly, we say that at this point, we have an avoided-level crossing.
7

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 2. Schematic of a driven two-level system (TLS). Here, E± are the upper and lower energy levels, ∆ is the minimal distance between energy
levels, and ε describes the energy bias. The avoided-crossing region corresponds to the vicinity of ε = 0. Three problems are considered in thiswork: (a) single passage of the avoided-crossing region, which is described by the probability P ; (b) double-passage problem for Pdouble ; and (c)multiple-passage problem with the solution for the time-averaged probability P + .

The adiabatic energy eigenstates are|E± (t)⟩ = γ∓ |0⟩ ∓ γ± |1⟩ ,
√
1
ε (t)
γ± = √ 1 ±
.
∆E(t)
2

(5)
(6)

In particular, at the point of the avoided-level crossing, ε = 0, |E± (0)⟩ = √1 (|0⟩ ∓ |1⟩). For ε ≫ ∆, the adiabatic energy
2levels approach the diabatic ones.
Now the problem is in finding the probability of a TLS to be in the upper state after passing the avoided-level crossing.
Let us assume that we start from the ground state |E− ⟩ on the left-hand side of Fig. 2(a), that is, at t → −∞. We areinterested in the probability P of finding the system in its excited state |E+ ⟩ after passing the avoided-crossing region,that is, at t → +∞. Alternatively, the problem can be formulated in terms of diabatic states: what is the probability P
to stay in the same diabatic state |0⟩, or what is the probability (1 − P ) of changing the state from |0⟩ to |1⟩?
The time-dependent Schrödinger equation gives us the solution, which is known as the LZSM formula:P = exp {−2π δ},

(7)

where

δ=

∆2
4h̄v

(8)

is the adiabaticity parameter. For slow changes, δ ≫ 1 (i.e., v ≪ ∆2 /h̄), we have an adiabatic evolution, where thetwo-level system (TLS) mostly stays in the ground state, P ≈ 0.
For fast changes, δ ≪ 1 , we have the diabatic evolution, where the system dominantly follows the diabatic state,(1−P ) ≈ 0; this means that by starting from the |0⟩ state at t = −∞ in Fig. 2(a), we end up with an almost unit probability
in the same |0⟩ state at t = ∞. We emphasize that the LZSM formula, Eq. (7), describes the transition probability if startingfrom an eigenstate; the case when the system starts in a superposition state will be considered later.
Besides the absolute value of the wave function, the phase obtained during the LZSM transition becomes crucial forinterferometry and quantum control. This phase is known as the Stokes phase,φS (δ ) =π4

+ δ (ln δ − 1) + Arg [Γ (1 − iδ )] ,(9)

where Γ here refers to the Gamma function.
In what follows, we present the derivation of the formula (7) as used in four different methods.
8

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Table 2The works of Landau, Zener, Stückelberg, and Majorana at a glance.
Article bySubmission

Publication

SystemMethodPhase

Majorana (1932)
Landau (1932b)
Zener (1932)
Stückelberg (1932)?-1931
12-1931
07-1932?-193202-1932
06-1932
09-1932
11–1932

Spin 1/2 in a magnetic field
Inelastic adiabatic atomic collisions
Crossing polar and homopolar states in molecule
Inelastic adiabatic collision

Laplace transform
Quasiclassical approach
Parabolic cylinder function
WKB approximation

Yesa
No
Yesb
No

a

Note that Majorana in his work obtained only the probability and did not pay attention to the phase change. Also note that he published Eq. (7)before others, which is not known to many. For detailed derivations of the full wave function, including the phase change, within Majorana’s
approach see Appendix A.3 as well as Rodionov et al. (2016) and Kofman et al. (2022).
b
Zener obtained the full wave function in terms of the parabolic-cylinder functions. However, in his work, the author discussed only the absolutevalue, that is the probability, Eq. (7). For detailed discussion of the solution, including the phase, see Appendix A.2 and Child (1974), Kayanuma(1997).

2.2. Brief overview of the original works of Landau, Zener, Stückelberg, and Majorana
Let us briefly consider the approaches of LZSM, which are summarized in Table 2 and of which the details are presented
in Appendix A. Importantly, all four of them published the very same year papers where one of the key results was exactly
Eq. (7).
From the dates in Table 2 we can see that the correct ordering would be MLZS. Concerning Majorana’s contribution,
see also Wilczek (2014), Kofman et al. (2022).
2.2.1. Near-adiabatic limit (Landau)
In his first work concerning the nonadiabatic transitions (Landau, 1932a), L.D. Landau studied adiabatic nonelastic
atomic collisions. He derived a general expression for the probability of nonadiabatic transitions within perturbation theory, which was applied for the near-sudden limit, with δ ≪ 1. The resulting excitation probability for the double-passage
process was(L)

Pdouble = 8π δ sin2 ΦL ,(10)

which is a rapidly oscillating function. Being averaged over a large dynamical phase ΦL , accumulated during double(L)passage evolution [see Fig. 2(b)], this would give P double = 4πδ . Indeed, from Eq. (7) at δ ≪ 1, we have P ≈ 1 − 2πδ .
These are consistent results, if we note that the latter gives the probability P0→1 ≈ 2πδ of staying in the ground state
after the first passage; then, there are two possibilities to be excited during the second passage: P0→1→1 ≈ 2πδ andP0→0→1 ≈ 2π δ , which would add up to Landau’s value of 4πδ .
In his second related paper (Landau, 1932b), the author applied the general formula of the transition to a generic caseof almost-crossing potential curves in the near-adiabatic limit, that is, for δ ≫ 1 and the obtained excitation probability
in the form of Eq. (7), but with the prefactor C being presumably of the order of unity. If analyzed by the other (moreprecise) methods, which are presented below, this constant becomes exactly equal to 1.
2.2.2. Using parabolic cylinder functions (Zener)
The second relevant approach is by Clarence Zener (Zener, 1932). The author studied the crossing of the polar andhomopolar states of a molecule. The energy bias for the electronic states was the slow variable of the nuclei position.
The task was reduced to the very same problem formulated in Fig. 2(a): What is the probability of excitation if startingfrom the ground state to the left and linearly driving to the right when passing the avoided crossing? The respective
Schrödinger equation was transformed into a second-order differential equation, of which the solution was the parabolic
cylinder Weber functions. This exact solution, after taking the asymptotes, resulted in Eq. (7).
Here, a note about Zener tunneling/effect/diode is in order. Two years later, Zener published another paper (Zener,
1934), in which he studied the dielectric breakdown, which is the electrical breakdown in solid insulators when applying
a strong constant electric field. The breakdown occurs because of the tunneling between the conduction bands througha forbidden band. Later, such sort of electric breakdown was studied for semiconductors (Kane, 1960) and is the basis of
the Zener diode (stabilitron).
Some authors have analyzed the analogy between Zener tunneling and LZSM transitions, for example, Romanova et al.
(2011); however, we differentiate Zener tunneling from LZSM transitions because the former does not involve an avoidedlevel crossing but instead needs strong fields, while the avoided-level crossing is the origin for LZSM interferometry. Basedon this, the two cases can also be called nonresonant and resonant (Zener) tunneling, respectively (Glutsch, 2004). As a
special case, one can mention here the so-called Bloch(-Landau)-Zener dynamics (Rotvig et al., 1995; Holthaus, 2000; Wu
and Niu, 2003; Ke et al., 2015; Khomeriki and Flach, 2016; Xia et al., 2021), which involves LZSM transitions between
energy bands when these display Bloch oscillations with avoided-level crossing.
9

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

2.2.3. Using the WKB approximation (Stückelberg); double-passage solution
Much like the above, E.C.G. Stückelberg also considered atomic collisions, for which he used the Wentzel–Kramers–
Brillouin (WKB) approximation and the phase integral method (Stückelberg, 1932). As a result, Stückelberg obtained theformula for the double-passage problem
Pdouble = 4P (1 − P ) sin2 ΦSt ,(11)

where the single-passage probability P is again given by Eq. (7), and ΦSt is the phase, that is, the so-called Stückelberg
phase, which is accumulated by the wave function during evolution. We will see that this consists of two parts: the one
accumulated during the adiabatic motion and the other (called dynamical or Stokes phase) φS acquired during the single
passage of the avoided-crossing region (Nikitin, 1999). Interestingly, Stückelberg pointed out that, particularly for δ ≪ 1,
his result gives what Landau obtained in the work (Landau, 1932a) with ΦSt = ΦL and 2P (1 − P ) = 4πδ . In Appendix A.4,
we present some details about the Stückelberg approach. In particular, we see that even with all the complications andgeneralities of this approach, the expression for the dynamical part of the Stückelberg phase cannot be obtained within
this formalism (Child, 1974).
2.2.4. Using contour integrals (Majorana)
In the fourth approach, Ettore Majorana considered an oriented atomic beam passing a point of a vanishing magnetic
field (Majorana, 1932). The problem was reduced by the author to a spin-1/2 particle in a linearly time-dependent
magnetic field, exactly as described by the Hamiltonian (1) with the bias (2). Much like the approach by Zener, Majorana
reduced the problem to a mathematical treatment of a second-order differential equation. This time, the author solved
the equation using the direct and inverse Laplace transform by calculating the respective contour integrals in the limitsof t → ±∞, resulting again in Eq. (7). Expectedly, that integral is similar to the integral representation of the parabolic
cylinder function.
We note that, previously, most of the papers on the subject of nonadiabatic transitions called these either LZ or LZS
transitions. Paradoxically enough, to some extent, the paper by Majorana is even more relevant and better suited for the
problem:• Majorana’s derivation does not contain undefined exponential prefactors or limitations for the value of the adiabaticity parameter δ , as in the derivation by Landau.
• It does not refer to special functions that require using asymptotics from books or numerics, as in Zener’s approach.
• Majorana’s derivation is less complicated than the one by Stückelberg.
The work of Majorana was both stimulated and verified by experimental observation (Frisch and Segre, 1933).
For the history of this, see Esposito (2014, 2017) and Cifarelli (2020b). With similar arguments, F. Di Giacomo andE.E. Nikitin (Giacomo and Nikitin, 2005) proposed, first, to make Majorana’s approach a central problem for textbooks
on quantum mechanics and, second, to denote the problem and formula, Eq. (7), using all four names: LZSM problem and
LZSM formula, respectively. From the dates in Table 2 we can see that if one follows the dates of publication, the correct
ordering would be MLZS. However, to avoid introducing confusion, we will call this LZSM, as almost all other authors
who acknowledge Majorana’s role. Concerning Majorana’s contribution, see also Wilczek (2014), Kofman et al. (2022).
As an additional advantage of Majorana’s formulation, we note that he (in contrast from LZS) formulated the problem
in terms of the spin-1/2 Hamiltonian, exactly in the form employed in quantum information nowadays.
Finally, Majorana’s approach allows for explicitly obtaining the phase acquired during the transition, like in Zener’s
approach, while this cannot be done in the semiclassical calculations by Landau and Stückelberg.
See Appendix A for further details, where we present the approaches developed by LZSM. Among other approaches,
we can mention the one by Wittig (2005), which was also presented in §1.5.2 of the textbook (Zagoskin, 2011), and the
Zhu–Nakamura theory (Nakamura, 2012, 2019). See also Hagedorn (1991), Chichinin (2013), Ho and Chibotaru (2014), Liuet al. (2019), Rodriguez-Vega et al. (2021) and Wang (2022).
Hence, there are different ways to find the LZSM transition probability, including shortcuts to finding the solutions
without solving the differential Schrödinger equation. However, being interested in the complete wave function—not onlyin the transition probability—we emphasize that this can be done only by one of the differential equation methods (Child,
1974; Nikitin, 1999). We illustrate this in the last column of Table 2, which responds to the following question: Can themethod be directly applied to derive the phase factor acquired after the transition? Only two answers are positive, and weaddress these in Appendices A.2 and A.3. Namely, we examine the approaches by Zener and Majorana, where the former
is quite known and the latter much less so. For these reasons, we present Majorana’s approach briefly in Appendix A, forthe readers’ convenience, with details given elsewhere (Kofman et al., 2022).
2.3. Different properties of the transition
2.3.1. Adiabatic theorem
The adiabatic theorem is one of the oldest and most important theorems in quantum mechanics. It provides thefoundation for various techniques (such as the adiabatic-impulse method described below) and for emergent devices
10

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

(such as adiabatic quantum computers, also discussed below). The adiabatic theorem is limited by nonadiabatic transitions,making this natural to be discussed here.
The adiabatic theorem states that in a system with a discrete energy spectrum under certain conditions, an infinitely
slow—or adiabatic—change of the Hamiltonian does not change the level populations; for example, see Chapter 1.5
in Zagoskin (2011). Let us now discuss this formulation and clarify those conditions.
First, we note that it is not enough to formulate the adiabatic theorem as it is often formulated: a physical systemremains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between
the eigenvalues (Albash and Lidar, 2018). Even under slow perturbation, resonant and interference effects may resultin significant changes in the energy-level populations. Below, in Section 3.1, we show that even with a small LZSM
probability of excitation during a single passage (P ≪ 1), under a condition of constructive interference, the upperlevel occupation probability would increase in a step-like manner. Then, during many driving periods, the occupation
probability could reach significant values, up to unity, displaying as a result slow Rabi-like oscillations. This could
be termed the “inconsistency” of the adiabatic theorem (Marzlin and Sanders, 2004). Hence, we should add “certain
conditions” (Amin, 2009; Tagliaferri et al., 2018; Hatomura and Kato, 2020) that can be formulated as either absenceof resonance or as a limitation on the time duration of the process, which, in our example, means that the time spanshould be much less than the period of the Rabi-like oscillations.
In the general case of a multilevel system, the eigenstates are defined by
H(t) |En (t)⟩ = En (t) |En (t)⟩ .

(12)Then, the adiabatic condition is usually quantified in either one of the two equivalent forms (e.g., Silveri et al. (2017)):⏐⟨⟩⏐⏐ Em (t)| Ėn (t) ⏐ ≪ |ωnm (t)|

(13)

or
max

t ∈[t0 ,t0 +∆t]

⟨Em (t)| Ḣ(t) |En (t)⟩≪ 1,
ωnm (t)2(14)

where h̄ωnm (t) = En (t) − Em (t) and m ̸ = n; the evolution is considered from t = t0 until t0 + ∆t. One can derive thatEq. (14) follows from Eq. (13) by differentiating Eq. (12). The interpretation of the adiabaticity condition (13) is that for all
pairs of energy levels, the expectation value of the time rate of change of the Hamiltonian must be small compared withthe gap (Sarandy et al., 2004). To be more precise, we could add the max and min, with respect to time, to the two sidesof this inequality, respectively. The value standing on the left-hand-side of Eq. (14) can be considered as the quantitative
measure of adiabaticity (Skelt et al., 2018).
In particular, for a TLS, from Eq. (14) with Eqs. (4), (5), we obtain h̄v/∆2 ≪ 1. This means δ ≫ 1, and explains
why δ is called the adiabaticity parameter. Indeed, in this adiabatic limit, the nonadiabatic transitions are suppressed:
P = exp {−2π δ} → 0 when δ → ∞. In general, the adiabaticity parameter δ changes from zero with the diabatic
transition, P = 1, and to infinity when the evolution is adiabatic and without nonadiabatic transitions, P = 0. However,recall that for the adiabatic theorem to be fulfilled, the time step ∆t should be shorter than any possible resonance time,such as the Rabi period.
2.3.2. Dynamics and times of a transition
We stated above that for linear driving, ε = v t, a TLS starting from the ground state with the LZSM probability P
can then be found in the excited state. For a graphical representation of the problem with linear drive, see Fig. 2(a). We
consider this now in more detail by addressing the following questions: What is the system’s dynamics P+ (t)? How doesit change when not starting from the ground state? What are the characteristic times describing how P+ (t) tends to P ?
What changes if the driving is nonlinear?
To start with, the dynamics depends on the representation. Both theoretically and experimentally, we can study
evolution in various bases (Tayebirad et al., 2010). The most important bases are the diabatic and adiabatic ones, which
we have introduced above. Given the relevance of these two bases, both for theory and measurements, we consider the
dynamics and its characteristic features for the two cases. For further studies, see Ref. Sun et al. (2015) on the experimental
visualization of the single-passage dynamics; Refs. Barra and Esposito (2016) and Thingna et al. (2017, 2019) on manylevel crossing in open quantum systems; and Refs. Vitanov and Garraway (1996) and Ribeiro et al. (2013b) on the finite
coupling solution where ∆ is a step function.
In the simplest approach, the dynamics is described by what we call the adiabatic-impulse model (AIM). Given its
importance, we consider this in much more detail in the next section when we examine periodic driving. For the singlepassage problem, the model consists of the adiabatic state following the ground state, then resonant (impulse-type)excitation to the upper level at the quasicrossing point, and then again the adiabatic state, now with a certain probability at
occupying the excited state. These dynamics are shown in Fig. 3(a) with dashed lines. Mathematically, such step-function
behavior is conveniently described by the transfer matrix (TM) method, where each type of evolution is attributed to therespective matrices.
Let us now clarify how accurate the TM approach is and what are the limitations on the application of the AIM (Mullen
et al., 1989). Accordingly, we will solve the Schrödinger equation exactly and describe the transient behavior by
introducing relaxation times.
11

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 3. Single-passage Landau–Zener–Stückelberg–Majorana (LZSM) transition with different probabilities (a) and with different transition times
for a fixed value of the transition probability P (b). In (a), we plot the time dependence of the upper-level occupation probability P+ (t), hence
demonstrating dynamics for a given value of the final LZSM probability P . Black dashed curves are given by the transfer matrix (TM) method, while
the solid colored curves show the numerical solution of the Schrödinger equation. In (b), all the curves are for a fixed LZSM probability P = 0.1,
while both ∆ and v are varied; this demonstrates the impact of these parameters on the transient dynamics. Here, it is convenient to normalize
time with a fixed value ∆0 , to which also the ∆ in the legend are related. The evolution is characterized by the two transition times: the jump
time tjump and the relaxation time trelax , which are shown in the figure for the thick magenta curve.

There are two different ways to obtain the exact solution. The first one consists of the numerical solution for the
Schrödinger equation
ih̄

d
dt

|ψ (t)⟩ = H(t) |ψ (t)⟩ ,(15)which can be used for all cases, including different nonlinear excitation signals and different initial conditions.
The second approach involves solving the Schrödinger equation with a linear excitation in terms of the parabolic
cylinder functions Dν (x) (see Appendix A.2). This approach gives a simple expression for the probability in the diabatic
basis {|0⟩ , |1⟩}. When solving the Schrödinger equation following Zener, it is natural to first introduce the dimensionlesstime, τ , and then the related complex value, z, which can be called the “Zener” variable:

τ =t

√

v
2h̄

,

z = τ eiπ /4√2eiπ /4 = t

√

v
h̄

.

(16)

Then, starting the evolution from the ground level, |E− ⟩ to the left in Fig. 2(a), we can obtain the time-dependent solution
for the upper level occupation probability in diabatic basis (see Appendix A.2)
Pd (z) = δ exp (−πδ/2) |D−1−iδ (−z)|2 .

(17)

To obtain transition dynamics for the upper level occupation probability in the adiabatic basis {|E− (t)⟩ , |E+ (t)⟩}, we use
formulas Eqs. (5), (6),⏐⏐Pa (z) = exp (−π δ/2) ⏐D−iδ (−z)γ+ −⏐2√ − iπ⏐δ e 4 D−1−iδ (−z)γ− ⏐ .

(18)An analytical solution like this has the advantage that one does not need to find all the values of the wave function fromthe initial time to the desired moment of time; this is in contrast to the numerical solution, where we need to calculate
all the previous values of the wave function between the current and initial times.
In Fig. 3, we illustrate the evolution of the upper-level occupation probability, emphasizing several different aspects.
In Fig. 3(a), we first fix several values of the final LZSM transition probability P . These are defined by the adiabaticity
parameter δ , Eq. (8). Inverting the relation for P , Eq. (7), we obtain the expression
h̄v
π
=−
,
∆2
2 ln (P )(19)which defines the ratio between v and ∆ for a given value of P . With the defined values of v and ∆, we plot in Fig. 3(a)both the analytical solution (dashed lines), which is the step function from 0 to P , and the numerical solution (shownwith the solid lines). We emphasize that, for the numerical approach, we can equally use either the direct solution for the12

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 4. Dynamics of a single LZSM transition in the adiabatic basis (a,c) and in the diabatic basis (b,d). We illustrate two limit cases of the adiabaticity
parameter: δ = 2 ≳ 1 in P+ (τ ) (a,b) and δ = 0.05 ≪ 1 in P0 (τ ) (c,d). Horizontal black double arrows show tjump and red double arrows trelax . Inclined
dashed green lines show the derivative at the avoided crossing point, τ = 0. Recall that the dimensionless time is τ = t

√

v
2h̄

.

Schrödinger equation or the formulas above, that is, Eqs. (17), (18). Note that with an increasing P , the evolution becomes
more similar to the analytical solution: the step function. Importantly, Fig. 3(a) vividly shows that the LZSM formula isrobust and is valid in the whole range of TLS parameters, which was theoretically grounded in Refs. Hagedorn (1991), Joe(1994), Vitanov and Suominen (1999) and Nakamura (2019).
In Fig. 3(b), we take the fixed value of the LZSM transition probability, P = 0.1. Then, for different curves, we
simultaneously vary both ∆ and v to keep this P constant; the values of ∆ are displayed. Fig. 3(b) demonstrates that fora given P , the other parameters drastically influence the dynamics. We characterize this using two transition times: tjump
and trelax . Consider now the definition and calculation of these important values; the details are presented in Appendix A.5.
Importantly, the time scales of the transition processes are very different in the adiabatic and diabatic bases; thus, we
describe the duration of the LZSM transition in both bases, following Vitanov (1999).
Therefore, the transition process has two subsequent phases. The first one is when the probability jumps from theinitial value P(−∞) to the vicinity of the final value P(∞). For the adiabatic basis, we have P(∞) = P , while for the
diabatic basis, we have P(∞) = 1 − P . To quantify this time span tjump , we note that the slope at zero is approximatedby P ′ (0) ∼ ∆P /∆t; replacing ∆P with P(∞) and ∆t with tjump , we come to the definition
tjump =P(∞)
P ′ (0).

(20)

However, this definition is not always appropriate, as we discuss in Appendix A.5.
13

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89ddFig. 5. Transition times versus the adiabaticity parameter δ . We plot both τrelaxand τjumpfor the transition in the diabatic basis and the relaxationaaatime τrelax
in the adiabatic basis. We do not plot the jump time in the adiabatic basis because this is very close to the relaxation time, τjump≈ τrelax.

Here, the dimensionless time is τ = t(v/2h̄)1/2 . See Appendix A.5 for more details.

The second phase of evolution is the time when the probability exhibits damping oscillations around its final value
P(∞); the duration of this process is denoted as trelax . This relaxation time can be quantified by introducing the small
parameter η ≪ 1, which describes that, after trelax , the amplitude of the oscillations becomes less than ηP(∞).
We must distinguish the upper-level occupation probability in the adiabatic basis P+ from the one in the diabatic basis
P0 . The dynamics of the probabilities in different bases was theoretically investigated in Wubs et al. (2005) and Dangaet al. (2016) and experimentally in Zenesini et al. (2009) and Tayebirad et al. (2010). We demonstrate the dynamics inthese two bases in Fig. 4. For the adiabaticity parameter δ , which describes the dynamics, we take two opposite limits:δ ≪ 1 (diabatic limit) and δ ≫ 1 (adiabatic limit). To be more precise, in the adiabatic limit, we take δ ≳ 1 because withδ ≫ 1, the LZSM probability P = exp (−2πδ ) becomes too small. With these four possibilities, in Fig. 4(a–d), we canobserve quantitatively different types of dynamics.
Note( that the relaxation) times are also very different in the adiabatic and diabatic bases. Namely, the total transition
time tLZSM = tjump + trelax in the adiabatic limit (δ ≳ 1) is much longer in the adiabatic basis than in the diabatic one.
The opposite is true, as well, where the transition time in the diabatic limit (δ ≪ 1) is much longer in the diabatic limit
than in the adiabatic one. In particular, for δ ≳ 1, in the diabatic basis, there are no oscillations, which means there isd= 0.
zero relaxation time, trelax
The jump and relaxation times can be obtained analytically from the exact solution that describes a single LZSM
transition with linear excitation; the details are presented in Appendix A.5. Here, we define the simplified transition
times tLZSM , which allow us to check the validity of the TM method. The characteristic time for a single-passage process
is the sweeping time from the initial to the final state; if the driving is periodic, then the sweeping time equals half theperiod, Td /2. Hence, this characteristic time should be much larger than the LZSM transition time, which can be compactly
written as follows (see Appendix A.5):

√

{ √ }
δ ,
v
{( )
√
)1/6 }
1/3 (
h̄
δ
δ
a
tLZSM
∼π
max
,
P
,
v
η
η

d
tLZSM

∼4

h̄

max 1,

(21a)
(21b)

in the diabatic and adiabatic bases, respectively. In Eq. (21b), η ≪ 1 is the small parameter that describes the magnitudeof the vicinity near the initial and final probabilities. These formulas are illustrated in Fig. 5, using τ = t(v/2h̄)1/2 .
2.3.3. Problems with nonlinearities
In general, the bias ε (t) is not a linear function of time. To obtain the linear model, Eq. (2), which we considered before,
we need to linearize the otherwise nonlinear bias around the point t0 , where ε (t0 ) = 0:

ε (t) = (t − t0 )

(

dε ⏐

)⏐
⏐
dt ⏐

+ o (t − t0 ) .

(22)

t = t014

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

The linearization is appropriate if both the first derivative is nonzero and other terms are negligible. Then, the probabilityof the nonadiabatic transition is given by Eq. (7), withδ=∆2( dε )⏐
⏐
4h̄
dt

.

(23)

t =t0

Alternatively, instead of ε (t), we can write the distance between the diabatic states: ε (t) → (E0 − E1 ). Even more generally,
instead of ∆/2, we can write the off-diagonal part of the Hamiltonian as the corresponding matrix element V10 ; then, theadiabaticity parameter is written, as, for example, in Ref. Bendersky et al. (2013),

⏐
⏐
2
V10
⏐
⏐
δ=⏐ d
⏐ h̄ dt (E0 − E1 )⏐

t =t0

⏐
⏐
⏐
⏐.
⏐

(24)

Let us now consider the cases where the bias cannot be linearized or nonlinear corrections are relevant. Different
nonlinear-level-crossing models were analyzed in Refs. Suominen et al. (1991), Vitanov and Suominen (1999) and Ashhab
et al. (2022). In that study, the authors used the Dykhne–Davis–Pechukas formula [this appears in Appendix A.1 as
Eq. (A.4)] to calculate the nonadiabatic transition probability when driven by different nonlinear biases ε (t).
The biases can be grouped in two types: perturbative and essential nonlinearities. The former relates to the case when
the linear term in Eq. (22) is dominant and the nonlinear corrections result in perturbative changes of the LZSM formula,while the latter relates to the case where there is no linear term and the essentially nonlinear bias is analyzed for thecases ε (t) ∝ t N with N = 2, 3, . . . (Shimshoni and Gefen, 1991; Vitanov and Suominen, 1999; Lehto and Suominen, 2012,2015; Kaprálová-Žďánská, 2022).
A special case of nonlinear LZSM tunneling relates to a TLS where the energy levels depend on the occupation of theselevels. This may arise in a mean-field treatment of many-body systems where the particles predominantly occupy twoenergy levels (Liu et al., 2002). Then, the system is described by the Hamiltonian

ε (t)
C
σx −
σz − (P+ − P− ) σz ,
(25)
2
2
2
where C is the parameter of nonlinearity, which describes the dependence of the energy levels on the state populations
P± . This parameter is not necessarily small and can be tunable. This form of the Hamiltonian is characteristic for thesystems described by the Gross–Pitaevskii equation (Ishkhanyan et al., 2010; Liu et al., 2016). This Hamiltonian—bothwith the linear driving, ε (t) ∝ t, and periodic driving, ε (t) ∝ sin ωt—may be experimentally realized in several ways with
BECs (Zhang et al., 2008). If taking the nonlinearity into account for such systems, all the features of the nonadiabatic
transitions should be revisited, including the quantum adiabatic theorem (Wu and Niu, 2003). When ε (t) is a sinusoidal
driving field, we should focus on nonlinear LZSM interferometry (Li et al., 2018).
One more development of the linear LZSM problem can be obtained using an asymmetric linear bias, where the slopes
are different on the left and right of the avoided-level crossing (Damski and Zurek, 2006). When the bias is a nonanalytic
function, similar cases were studied in Ref. Garanin and Schilling (2002a). Various aspects of the nonlinearity of
nonadiabatic transitions have been studied recently for realizations in such systems as a periodic lattice system (Takahashi
and Sugimoto, 2018), spin qubit in a quantum wire (Tchouobiap et al., 2018), superconducting qubits (Wu et al., 2019),
and topological systems (Kam and Chen, 2020).
Therefore, one can consider versatile nonlinear biases in the contexts of LZSM-like problems. At the end of thissubsection, we illustrate this also with the idea of reverse engineering (Kang et al., 2022). This is formulated as finding a
Hamiltonian H̃(t) generating a given dynamics, here evolving in states that are the instantaneous eigenstates of a given
Hamiltonian H(t) (Berry, 2009). One formulation of this is the inverse LZSM problem, which is formulated as finding thebias ε (t) resulting in any required time dependence of the level populations. This problem was formulated and solved inRef. Garanin and Schilling (2002b). Then, in Ref. Shevchenko et al. (2012a), a similar problem was studied for a qubitresonator system as restoring a driven qubit Hamiltonian, provided its stationary behavior is known; see also Barnes(2013). Another problem related to the reverse engineering approach is transitionless quantum driving, which is analogous
to the explanation of reflectionless potentials. This was studied both theoretically (Berry, 2009; Villazon et al., 2021) andexperimentally (Bason et al., 2012; Xu et al., 2018). We address this in more detail in Section 5.3, where we demonstratethat a Hamiltonian with appropriate nonlinear driving results in a transitionlessness system dynamics.
HC (t) = −∆2.4. Some experimental observations
Driven TLSs are ubiquitous, which is true for the observations of the LZSM transition. In this section, we present
illustrative examples of observations of the single passage LZSM transition.
* Historically, the first works by Landau, Zener, and Stückelberg related to inelastic atomic and molecular collisions.
These described energy and charge exchange, as well as predissociation and associative recombination. The patterns
observed in the scattering form the subject of collision spectroscopy. This was demonstrated for the inelastic
scattering of He+ by Ne (Coffey et al., 1969). In that work of more than half-century ago, the authors demonstrated
both LZSM transitions and Stückelberg oscillations.
15

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

* As one realization of Majorana’s problem, a spin moves through the point of a vanishing magnetic field, also
considered experimentally in Ref. Betthausen et al. (2012). In this case, a spin moved in a spin transistor over a
distance of 50 micrometers while experiencing an adiabatically variable magnetic field. Alternation of adiabatic
evolution and nonadiabatic transitions allows for accurate transistor control, making the spin transistor tolerant
against disorder.
* Nonadiabatic transitions were experimentally shown in a strong electric field between the Stark states of highly
excited (Rydberg) states in lithium (Rubbmark et al., 1981). In contrast to molecular collisions, all the parameters
there can be controlled accurately, which allowed obtaining quantitative agreement with theory in the two-levelapproximation, regarding their multilevel energy diagram. This allowed to better understand the dynamics ofRydberg atoms in rapidly rising electric fields.
* The tunneling dynamics of a BEC of ultracold atoms in a tilted periodic potential is realized by accelerating thelattice (Zenesini et al., 2009; Tayebirad et al., 2010). Researchers studied LZSM tunneling both in the diabatic andadiabatic bases. Another study of an ultracold Fermi gas in a tunable honeycomb lattice was presented in Ref.
Uehlinger et al. (2013). The authors realized two successive LZSM transitions without interference after sequentiallypassing through two Dirac points. The authors of Ref. Thalhammer et al. (2006) experimentally demonstrated theassociation and dissociation of the so-called Feshbach molecules, which are the molecules of a BEC formed by meansof a Feshbach resonance.
* In quantum systems, it is often the case that many levels are relevant, and this will be a subject of one of the following
sections. However, sometimes, only the dynamics of two close levels is relevant. We find this in the experimentof Zhao et al. (2017) on a large-spin system, S = 7/2. The authors observed the nonadiabatic dynamics around oneof the avoided-level crossings controlled with an external magnetic field in a Gd3+ impurity ion ensemble, whichmakes a qubit system with “a virtually unlimited relaxation time”.
* Qubit states in an NV center in diamond also have the advantage of good isolation. A single NV center electronic spin
was coupled with a single nitrogen nuclear spin, creating a hybridized electronic-nuclear state (Fuchs et al., 2011).
In this case, the LZSM transition can transfer the excitation between the two subsystems, which was proposed as abasis for a room temperature quantum memory. To create the avoided-level crossing with the NV centers, the authorsof Ref. Xu et al. (2019b) first applied a resonant microwave and considered the RWA, and then they explored theadiabatic evolution and nonadiabatic transitions.
* Micrometer-size superconducting qubits allow the realization of macroscopically distinct quantum states. The nonadiabatic transitions between them were demonstrated for a variety of JJ-based qubits, including the flux (Izmalkov
et al., 2004), quantronium (Ithier et al., 2005), charge (Cooper pair sluice) (Gasparinetti et al., 2012), and phase (Tanet al., 2015) qubits. In these works, it was demonstrated that such measurements are useful for probing andcontrolling both the qubits themselves and the coupled microscopic systems, hence providing a fast and sensitive
tool to study and control qubits.
* Another mesoscopic-size platform is provided by (double-)quantum dots. LZSM transitions were studied in singlettriplet qubits in silicon in Harvey-Collard et al. (2019) and Khomitsky and Studenikin (2022). In this case, tunnelingwas demonstrated to be useful to extract the spin–orbit coupling. Besides the electronic degrees of freedom, quantum
dots can be used to manipulate the nuclear spin ensemble with chirped magnetic resonance pulses (Munsch et al.,2014).
* Somewhat unexpectedly, LZSM physics is related to second-order phase transitions, of which the dynamics is
described by the KZM. We discuss this in the SubSection 3.1.2, but here, to better describe the second-orderphase transitions dynamics, we present a variety of systems in which the KZM was experimentally studied inlaboratories. The KZM correctly predicts the creation of topological defects during a single passage through asymmetry-breaking transition. Following the original proposal by Zurek (1985), defects (vortices) in superfluid 4 He
were created during the phase transition induced by fast expansion through the critical density, crossing the λ-lineon the pressure–temperature phase diagram (Hendry et al., 1994). The vortices in superfluid 3 He were created using
neutron-induced nuclear reaction (n+ 3 He→ 3 H+p) to heat small regions of superfluid 3 He above the superfluid
transition temperature (Bäuerle et al., 1996; Ruutu et al., 1996). The probability to trap a single flux line in annular
JJs was shown to work because of a causal KZM rather than because of thermal activation (Monaco et al., 2009).
Other examples include zig-zag defects in buckled chains of ions (introduced above, in Fig. 1), defect textures inliquid crystals, flux lines in superconductors quenched through the critical temperature, and spin domains in Bose
condensates realized in a shaken optical lattice. For a review, see Pyka et al. (2013), Hedvall and Larson (2017) andDziarmaga (2010).
* The analogy between nonadiabatic transitions in classical and quantum mechanics has been known for quite a longtime (Maris and Xiong, 1988). However, this idea was realized only recently on mechanical resonators (Faust et al.,2012), in which the authors demonstrated that the energy transfer between the two modes of a nanomechanicalresonator obeys LZSM behavior. We explore analogies and studies in a separate Section 6.
For a description of additional experimental observations, see Section 4. We would like to note that in many cases it
is possible to observe either single-passage LZSM transitions or multiple-passage LZSM interference, which are describedhere and in Section 4, respectively. For this reason, we describe the related experiments in these two places. Before moving
to the latter observations, let us consider the underlying physics first.
16

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

2.5. Transfer matrix (TM) method
Consider now the dynamics during a single-passage transition as a sequence of three stages:• (i) adiabatic evolution starting from the time ti < 0 until an avoided-level crossing is reached,• (ii) a nonadiabatic transition very near t = 0, and• (iii) adiabatic evolution starting from an avoided-level crossing passed until the time tf = −ti > 0.
The wave function can be expanded in the basis of the adiabatic eigenstates |E± (t)⟩:)α (t)|ψ (t)⟩ = α (t) |E− (t)⟩ + β (t) |E+ (t)⟩ =
.
β (t)
(

(26)

The normalization condition results in all the transfer matrices being unitary ones. Consider this for both (i,iii) adiabaticand (ii) nonadiabatic evolutions. For the former, from a nonstationary Schrödinger equation, Eq. (15), we obtain the
adiabatic time-evolution operator(

U(ζ (t , ti )) =exp [−iζ (t , ti )]0)0
exp [iζ (t , ti )]= exp [−iζ (t , ti )σz ],(27)

where ζ (t , ti ) is the phase accumulated during the adiabatic evolution from the time ti until the time moment t

ζ (t , ti ) =

1

∫ t

2h̄

ti

∆E(t) dt .

(28)

Hence, the adiabatic evolution is described by the relation|ψ (t)⟩ = U(ζ (t , ti )) |ψ (ti )⟩ ,(29)

which corresponds to no transitions between the adiabatic states, with only the phase difference accumulated.
Next, consider an impulse-type transition at t = 0. In Appendix A, we obtain the transition matrix in the diabatic
basis; after transferring from the diabatic basis to the adiabatic one, using Eq. (6) and assuming ε (ti,f ) ≫ ∆, we obtain thetransition matrix(
N =Re−iφS
T

−T

ReiφS

)

,

(30)

where

√
P = Transition coefficient,√
R = 1 − P = Reflection coefficient;
T =

(31a)
(31b)

φS is the Stokes phase (9), which appears in the theory of second-order differential equations. For more details, see Nikitin
and Reznikov (1972), Child (1996), Kayanuma (1997) and Gasparinetti et al. (2011), particularly for how this phase appears
in terms of the Bloch vector evolution and Berry phase accumulation. Hence, the impulse-type nonadiabatic transition ataround t = 0 is described by|ψ (+0)⟩ = N |ψ (−0)⟩ .

(32)

Note that given the asymptotics of the gamma function, the monotonous function φS (δ ) changes from 0 in the adiabatic
limit (δ ≫ 1) to π /4 in the diabatic limit (δ ≪ 1).
Now, we can define the total single-transition evolution matrix in the general case ζi ≡ ζ (0, ti ) ̸ = ζ (tf , 0) ≡ ζf
U(ζf )NU(ζi ) =(R exp [−i(φS + ζi + ζf )]T exp [i(ζi − ζf )])−T exp [−i(ζi − ζf )].
R exp [i(φS + ζi + ζf )](33)

We also can find the total single-transition evolution matrix in the case of symmetric adiabatic evolution ζ (0, ti ) =ζ (tf , 0) ≡ ζ ,U(ζ )NU(ζ ) =(R exp [−i(φS + 2ζ )]
T

)
−T
.
R exp [i(φS + 2ζ )](34)

Note that when we consider the inverse transition we should replace the direct-transition matrix N with the inverse
transition matrix
Ninverse ≡ N ⊤ ,(35)which is the transposed matrix, see Eq. (A.29a), and Ref. Teranishi and Nakamura (1998).
17

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 6. Nonadiabatic transition with a superposition state. Comparison of the numerical,
analytical, andmatrix (TM) solutions for a single√√ transfer
LZSM transition starting from a superposition state. For this, we took α (ti ) =0.2 and β (ti ) =0.8eiφi , here with two cases for the phasedifference: one for a destructive interference case, φi = 5π/3, and another case for constructive interference, φi = 2π/3 in (a), which can realize
the maximum and minimum possible values of the final probability, Eqs. (38) and (39), and the probability-conserving case in (b), here with theinitial phase difference defined in Eq. (41) and a transition without changes in the probability. The black double-arrow shows the range of thepossible final values of the probability. This demonstrates the dramatic dependence of the evolution on the initial phase difference and the role ofinterference.

As a generic initial condition at t = ti , we take a superposition state|ψ (ti )⟩ = α (ti ) |E_ (ti )⟩ + β (ti ) |E+ (ti )⟩ ,(36)

where |E± (t)⟩ are the instantaneous eigenstates of the time-dependent Hamiltonian H(t) and{√α (ti ) = P− (ti ),√β (ti ) = P+ (ti )eiφi ,(37)where P± (ti ) are the occupation probabilities of the respective states and φi describes the initial phase difference.
Importantly, for a superposition state, the phase difference significantly influences the dynamics (Emmanouilidou et al.,
2000; Wubs et al., 2005). Now, with the evolution matrix UNU, we can obtain the final upper-level occupation probability
P+ (tf ) = T 2 P− (ti ) + R2 P+ (ti ) − 2RT√P− (ti )P+ (ti ) cos(−φS − 2ζ + φi ).

(38)This formula describes several important aspects. First, when the cosine equals +1 or −1, we have maximal and minimal
max / minexcitation probabilities P+(tf ), respectively. These correspond to the constructive and destructive interference of theincoming states. The respective conditions are{max−φS − 2ζ + φi = 2π nfor P+,
1−φS − 2ζ + φi = 2π (n + 2 ) for P+min ,

(39)

min
max
where n is an integer. Second, note that the range between the extremal values P+
and P+
includes the initialprobability P+ (ti ). This means that we can select the value of the initial phase difference φi , which gives us the transition
without any change of the probability so that

P+ (tf ) = P+ (ti ).

(40)

This process can be named occupation-conserving transition (OCT). This takes place for the phase difference φi = φ

φiOCT = −φS − 2ζ + arccos[

T (P+ (ti ) − 1/2)√

R P+ (ti )P− (ti )

]

.

OCT
i

(41)

Note that this is possible only for a superposition initial state; when starting from a ground state, there is no effect on
the phase difference.
In the case of starting from a superposition initial state, all these dynamical features are illustrated in Fig. 6. In thisfigure, we compare the numerical solution with the analytical one, which is given by Eq. (A.15a), with a perfect agreement
18

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

between the two. In addition, we show the asymptotic solution described by the transfer matrix method with a step-like
transition, as described by the equations above. For the calculations, we take the adiabaticity parameter δ = 0.4, whichcorresponds to the LZSM probability P = 0.08. In Fig. 6, we present the solutions for three different cases: for constructive
interference with initial phase difference φi = φ1 = 2π/3, for destructive interference with φi = φ2 = 5π/3, and for the
probability-conserving case with φi = φiOCT , Eq. (41). Note that P is the probability of excitation if starting from the groundstate, while now, we can demonstrate a more general case of starting from the superposition state. This demonstrates
that the upper-level occupation probability P+ (∞) is essentially different from P and that this is defined by not only δ ,but also by the initial condition.
3. Repetitive passage: interference
In the previous section, we considered a TLS when driven by a linear drive ε (t) = v t. From now on, we consider theevolution for a generic periodic bias with an offset ε0 ,ε (t) = ε0 − A cos ωt .

(42)

We now consider several approaches.
3.1. Adiabatic-impulse model (AIM)
3.1.1. Double-passage and multiple-passage cases
The adiabatic-impulse model is possibly the most intuitive model for describing the repetitive LZSM passage (Damski
and Zurek, 2006; Tomka et al., 2018). In this model, we consider the evolution of the TLS beyond the avoided-crossing
region as adiabatic evolution, and in the avoided-crossing region, we consider the diabatic evolution of the TLS. Also, we
approximate the avoided-crossing region by the point of minimum distance between energy levels. Therefore, we havethe adiabatic evolution everywhere save for the points of the minimal distance between energy levels, where nonadiabatic
LZSM transitions occur. Thus, the AIM consists in that the evolution is modeled (approximated) as adiabatic one besides
the non-adiabatic transitions in the avoided-level-crossing points (where the driving is approximated as a linear one).
Given these approximations, instead of AIM this technique can alternatively be called adiabatic-impulse approximation.
Essentially, the AIM is described by using the TM method. The latter was developed by Bychkov and Dykhne (1970),
Averbukh and Perel’man (1985), Kayanuma (1993), Vitanov and Garraway (1996), Garraway and Vitanov (1997), Teranishiand Nakamura (1998) and Delone and Krainov (2012).
In the section “Transfer Matrix Method” we introduced the matrices U and N for the adiabatic evolution andnonadiabatic impulse-type transition, respectively. This describes the evolution during half the period. When speaking
about driven systems, it is illustrative to sequentially consider three cases: single-passage transition probability, doublepassage case, and the multiple-passage transition probabilities (Nikitin, 2006). We now consider the evolution during one
full period and then during many periods, to which we refer to as double-passage and multiple-passage cases, respectively.
The adiabatic energy levels

∆E(t)

1√ 2
=±
∆ + ε (t)2(43)
2
2have a minimum distance (equal to ∆) at times t1,2 + nTd , where ωt1 = arccos(ε0 /A) and ωt2 = π + ωt1 , see Fig. 7(a).
The adiabatic evolution is described by
E± (t) = ±U1,2 = exp(−iζ1,2 σz )(44)

with the phase differences

ζ1 =

1

∫ t2

2h̄

t1

∆E(t)dt ,ζ2 =1∫ t1 +Td

2h̄

∆E(t)dt .

(45)

t2

The nonadiabatic transitions between the states |E± (t)⟩ are described by the transfer matrix N, which is true for bothtransitions and corresponds to the sweeping occurring both to the right and left in Fig. 2, respectively. Note that theevolution in the diabatic basis should be described differently (Ashhab et al., 2007; Shevchenko et al., 2010). In some
papers (Shevchenko et al., 2010) double-passage evolution is described starting at the quasicrossing point and finishingalso at the quasicrossing point [see Vitanov and Garraway (1996) and Cucchietti et al. (2007)].
And that() way gives a correct
result for averaged level occupations under the periodic driving Ξ = Ninverse U2 NU1 =

Ξ11
∗
−Ξ12

Ξ12
∗ .
Ξ11

It is more convenient to describe the evolution starting far from the quasicrossing point and finishing also far fromit. Then the double-passage evolution takes place after the full period and is described by the double-passage transfer
matrixΞ≡√

U2 Ninverse U1 N

√

(
U2 =

Ξ11
Ξ12

)
Ξ12
,
∗
Ξ11

(46)
19

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 7. Evolution of energy levels in the adiabatic-impulse model (AIM). (a) The time-dependent adiabatic energy levels E± (t) define two stagesof evolution: first, the adiabatic evolution with the transfer matrices U1 and U2 and, second, the transition in the region of avoided level crossing
as determined by the matrix N. Here we consider a single-period
the two vertical√ evolution from√ t = 0 to t = Td , which is the time between √dashed black lines in panel (a). This evolution is described by U2 Ninverse U1 N U2 , in terms of time-evolution matrices, where U2 = U(ζ2 /2).
(b) Equivalent optical scheme based on Mach–Zehnder interferometer, where the optical beam splitters are analogous to the transition pointsin (a). The trajectories can have different lengths (e.g. due to the moving upper mirror) resulting in the relative phases of the two partial rays, inanalogy with the wave-function phase difference accumulated during the LZSM transitions in (a). (c) Typical scheme of the optical Mach–Zehnderinterferometer, analogous to the double-passage LZSM problem.

whereΞ11 = −R2 e−iζ+ − T 2 e−iζ−∗Ξ12 = −2iRT sin(ΦSt ) = −Ξ12
,

(47b)

ζ+ = ζ1 + ζ2 + 2φS , ζ− = ζ1 − ζ2 , ΦSt = φS + ζ1 ,

(47c)

(47a)

for the inverse transition matrix Ninverse see Eq. (35). We obtain the same Ξ11 as in Ref. Shevchenko et al. (2010) for the
double transition, but Ξ12 is different due to using a shifted driving signal. From Eq. (47b), one can see that the upper-leveloccupation probability, if starting from the ground state, becomesdoubleP+= |Ξ12 |2 = 4P (1 − P ) sin2 ΦSt .

(48)

In this way, following Zener’s approach we confirmed the Stückelberg formula, Eq. (11).
20

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 8. Constructive versus destructive interference. Comparison of the upper-level occupation probability obtained numerically and from the AIM
in the regime of constructive interference (blue and light blue curves: P+ (t)c calculated numerically and in the AIM, respectively) and destructive
interference (red and green curves: P+ (t)d calculated numerically and in the AIM, respectively). The light blue and green curves present the analyticalsolutions, while the blue and red curves show numerical solutions. We take the following parameters: ε0 = 0 (then ζ1 = ζ2 ) and P = 0.1; and theconditions for the constructive and destructive interference, ζ1 + φS = π k and = π k + π2 , respectively; these conditions define ω and A. The brownline is plotted with Rabi frequency obtained in Eq. (53).

Here, an instructive analogy with the Mach–Zehnder interferometer is appropriate (Oliver et al., 2005; Petta et al., 2010;
Burkard, 2010; Ma et al., 2011). We illustrate this analogy graphically: Fig. 7(b) shows that our dynamics in Fig. 7(a) is
analogous to multiple periodic Mach–Zehnder interferometers (Oliver et al., 2005), and Fig. 7(c) shows that our doublepassage LZSM problem is analogous to an optical Mach–Zehnder interferometer (Burkard, 2010). Namely, passing the
avoided-level crossing is analogous to light passing a partly transparent mirror functioning as a beam splitter with thecoefficients R and T . After the two beams meet, the outcome is the result of the interference, which depends on the phasedifference ΦSt . For more about double-passage regime theory, see, e.g., Saxon and Olson (1975), Garraway and Stenholm(1992b), Garraway and Suominen (1995), Nagaya et al. (2007) and Gasparinetti et al. (2011).
For the multiple-passage case, after n full periods, the time evolution is described by the following evolution matrices:
U (t , t1 + nTd ) Ξ n

for t − nTd ∈ (t1 , t2 ) ,U (t , t2 + nTd ) NU1 Ξ n(49)

for t − nTd ∈ (t2 , t1 + Td ) .

(50)

Hence, the system state after n full periods of evolution is given by Ξ , which reads as Bychkov and Dykhne (1970)
n

Ξn =

Ξn11
∗
−Ξn12

(

)
Ξn12
,
∗
Ξn11

Ξn11 = cos nφ + iImΞ11

sin nφ
sin φ

(51a)

,

Ξn21 = Ξ21

sin nφsin φ, φ = arccos ReΞ11 .

(51b)Then, for the respective upper-level occupation probability, if starting from the ground state, we obtain P+ (n) = |Ξn12 |2 ,which gives
P+ (n) = |Ξ12 |2

sin2 nφsin2 nφ= 4P (1 − P ) sin2 ΦSt.
 sin2 φ
sin φ
2

(52)

P double+This describes the time evolution, with n denoting the integer number of periods passed, as shown in Fig. 8. Note theimpressive agreement between the results of the AIM and numerics; see also Mukherjee et al. (2018), Kuno (2019). Fora similar description of the multilevel systems, see Qin (2016), Neilinger et al. (2016), Niranjan et al. (2020), Suzuki and
Nakazato (2022).
Rabi oscillations from LZSM transitions: Adiabatic dynamics is characterized by small steps which, under resonancecondition, result in Rabi-like oscillations (Garraway and Stenholm, 1992a; Pu et al., 2000; Shevchenko et al., 2005). This
21

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

2
can be( seen) in Fig. 8. The frequency of these Rabi-like oscillations can be found from Eq. (52) if we identify sin (nφ ) with2 ΩR
sint (Ashhab et al., 2007; Neilinger et al., 2016; Liu et al., 2021). Then, we note that during one driving period, the2integer n changes by unity, and this corresponds to changing the time t by 2π/ω. With this, we obtain the relation forthe coarse-grained oscillations

ΩR =

ω
ω
|φ| = arccos [(1 − P ) cos ζ+ − P cos ζ− ] .
π
π

(53)

This formula correctly describes the Rabi oscillations induced by strong driving, as studied in Refs. Zhou et al. (2014) and
Neilinger et al. (2016). Indeed, we note that for a small offset ε0 /A ≪ 1, the expressions for ζ± can be simplified:ζ1 + ζ2 ≈ 2A/h̄ωandζ1 − ζ2 ≈ πε0 /h̄ω.

(54)With this, in the adiabatic limit (P ≪ 1), we obtain ΩR ∼ A/h̄. This correctly describes the resonant Rabi frequency andjustifies the term “Rabi oscillations”, which we used above.
The long-time averaged occupation probability, is given by averaging over large n,P+ =|Ξ21 |2|Ξ21 |2
1=.
2 |Ξ21 |2 + (ImΞ11 )2
2 sin2 φ(55)

It follows that the upper-level occupation probability P+ is maximal at ImΞ11 = 0. This results in the resonance condition:(1 − P ) sin ζ+ − P sin ζ− = 0.

(56)

In particular, in the adiabatic (slow) and diabatic (fast) limits, the resonance condition takes the following forms:

ζ1 + ζ2 = kπ
ζ1 − ζ2 = kπ

for δ ≫ 1

(adiabatic),(57a)

for δ ≪ 1(diabatic).

(57b)

With the limiting expressions (51), the resonance condition for the adiabatic limit reads 2A ≈ kπ h̄ω, and for the
diabatic limit, this gives ε0 ≈ kh̄ω. Because in the diabatic limit ∆ is relatively small (∆E ≈ |ε0 |), the latter condition canbe interpreted as an exchange of k photons between the driving field and our two-level system.
In the slow passage limit, where δ ≫ 1 and P ≪ 1, we directly obtain the time-averaged occupation probability of theupper state:P+ =P (1 + cos ζ+ cos ζ− )sin ζ+ + 2P (1 + cos ζ+ cos ζ− )
2

,

(58)

which describes the dependence on the variable and controllable parameters ε0 , A, and ω.
In the case of fast passage, where (1 − P ) ≈ 2πδ ≪ 1, there is a large probability (P ∼ 1) for transitions between the
adiabatic states in a single passage, while the transition probability between diabatic states is small, (1 − P ) ≪ 1. Hence,
we consider the time-averaged probability of the upper diabatic state Pup . Then, one can obtain:
P up =

1

4π δ cos2 (ζ2 − π/4)

2 sin2 ζ− + 4πδ cos2 (ζ2 − π/4)

.

(59)

On resonance, we have ζ− = kπ and P up = 1/2. Then, in particular, for a small offset, with ζ− ≈ πε0 /h̄ω, we obtain theresonance frequency h̄ω(k) = ε0 /k, meaning that the resonance transitions are described by their multi-photon relation.
In the vicinity of the kth resonance, for ω ∼ ω(k) , to first approximation in ε0 /A, we obtain
(k)

1

∆2k

,
+ (kh̄ω − ε0 )2
√
(
)
2h̄ω
A
π
cos
− (2k + 1) .
∆k = ∆πAh̄ω4
P up =

(60)

2 ∆2k

(k)

The total probability P up is obtained as the sum of the partial contributions P up . Note that the above derivation within
the AIM assumes that the excitation probability may become nonzero when the energy quasicrossing is reached, that is,
at |ε0 | < A; otherwise, at |ε0 | > A, this model gives a zero transition probability.
Coherent destruction of tunneling (CDT): From the formulas above, we can see that there are conditions under which
the driven system can stay unexcited, with no tunneling between the states, even under the impact of a strong resonant
drive. This phenomenon is known as coherent destruction of tunneling or CDT (Grossmann et al., 1991) and can be applied
for controlling tunneling in TLSs (Llorente and Plata, 1992; Hu et al., 2022). It can be understood and described as either
22

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 9. LZSM interferograms calculated within the adiabatic-impulse model (AIM). (a) The low-frequency driving corresponding to the slow-passage
regime, Eq. (58), with h̄ω/∆ = 0.8 < 1. (b) The high-frequency driving corresponds to the fast-passage limit, Eq. (60), with h̄ω/∆ = 1.4 > 1.

a degeneracy of the Floquet quasienergies (Hijii and Miyashita, 2010) or, equivalently, as a result of destructive LZSM
interference (Kayanuma, 1994; Kayanuma and Saito, 2008). Indeed, from Eq. (52), looking at a general case, there areno transitions between the states under the antiresonant condition, Ξ21 = 0. Another case is Eq. (60), which gives zeroexcited-state populations if driven with the amplitude A = Ak ≡ (π/4)(2k − 1)h̄ω.
Complete CDT is only possible for isolated systems, such as the ones we consider here; taking into account dissipation
spoils the effect (Miao and Zheng, 2016a). CDT happens to be important for the description of various systems: transitions
in graphene (Gagnon et al., 2016, 2017) or in biomolecular protein—solvent reservoirs in photosynthetic light harvesting
complexes (Eckel et al., 2009); for a review of CDT in different systems, see Wubs (2010).
Latching modulation: Our derivations in this section are mainly developed for sinusoidal driving. These can be adapted
to a description of any other periodic driving, ε (t + Td ) = ε (t). Particularly, consider now the situation when a TLS isdriven so that the level separation is switched abruptly between two values and is kept constant otherwise (Silveri et al.,2015). In this case, we have the bias

ε (t) = ε0 + A sgn [cos(ωt)] ,(61)

which results in the periodic latching modulation of the qubit energy levels. Direct application of the LZSM approach would
give infinite speed v , resulting in exact unit probability P = 1 with no interference. In this case, with
√ v = ∞, the AIM isd
not directly applicable because then the width of the transition region becomes infinite: v tLZSM∼ h̄v , see Eq. (21a). Thishas been analyzed in Ref. Silveri et al. (2015), where the generalization of the AIM is presented. Interestingly, most of theformulas above, which describe the upper-level occupation probability, remain valid with only one important substitution:(l)(r)
now, instead of the LZSM probability P , we write⏐ the⟩ sudden-switch transition probability ps = ⟨ψ+⏐ |ψ−⟩ ⟩, which occurs(r)

between the lower state in one latch position ⏐ψ−⏐(l)and the upper state in the other latch position ⏐ψ+ . The successful⏐

application of the AIM with this substitution P → ps in Ref. Silveri et al. (2015) was not only compared with the numericalsolution and experiment, but also with the rotating-wave approximation (RWA). We return to this later when discussing
the RWA.
Interferograms: With the formulas above, Eq. (55), as well as its limiting expressions, Eqs. (58) and (60), we cangraphically visualize the interference. These dependencies, say, on ε0 and ω, or on ε0 and A, can be called interferograms.
These are shown in Fig. 9(a) and (b) for h̄ω/∆ = 0.8 and 1.4, respectively.
3.1.2. Kibble–Zurek mechanism (KZM)
Making analogies can bring us very far from where we started. The KZM is much like this, bringing us from qubits
to the Big Bang. The KZM started from the proposition by T.W.B. Kibble to model the physics of the early universe as23

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

cosmological phase transitions that result in the formation of topological defects in the form of monopoles or cosmic
strings (Kibble, 1976). This was shown by W.H. Zurek to be a universal feature for second-order phase transitions andrelated to the adiabatic-impulse approximation (Zurek, 1985). The latter can be equally applicable to two-level systems,
hence relating the KZM and LZSM transitions (Damski, 2005).
It was shown that for what we call a single-passage process, LZSM theory can be used to accurately describe theKZM (Damski, 2005; Zurek et al., 2005; Dziarmaga, 2005). Namely, the AIM for an avoided-level crossing is a generalmodel that describes both qubit dynamics and symmetry-breaking second-order phase transitions. This allows us to passfrom the LZSM evolution to phase transition dynamics and back again (Damski and Zurek, 2006).
Following Damski (2005), consider this for a pressure quench that drives liquid 4 He from a normal phase to a superfluid
one. The process is described by the distance λ from the critical point. The quench comes with the linear increase

λ=

t

τQ

,

(62)

with the rate τQ−1 and time taken such that the transition point is crossed at t = 0. In the case of quenching liquid helium,
λ is the relative temperature, such that λ(t = 0) = 0. Changes in pressure translate into changes of the dimensionlessparameter λ. So, we start at t → −∞, here with helium in the liquid phase. The dynamics is described by the relaxation
time τr , which is the time needed for the system to adjust to new thermodynamic conditions. Far from the critical point,τr is small and the evolution is adiabatic.
Moving closer to the transition point, critical slowing down (longer relaxation) occurs, which is the divergence of therelaxation time, τr = τ0 /|λ|, with the characteristic time value τ0 . This dynamics is shown in Fig. 10(a); the adiabaticand impulse regions are separated by tKZ , which is called the freeze-out time. This characteristic time is defined by theconditionτr (tKZ ) = α tKZ ,(63)

as graphically shown by the inclined lines in Fig. 10(a).
√ Here, α = O(1) is the system-specific parameter, which is taken as
unity in the figure. From Eq. (63), we obtain tKZ = τ0 τQ /α . Knowledge of tKZ allows us to find the density of topologicaldefects, which appear as a result of the nonequilibrium phase transition, interestingly without solving those equations
describing the dynamics of the system. For this, the analogy with the LZSM model is beneficial.
Considering
the analogy with a TLS, in Fig. 10(b), we plot the inverse distance between the energy levels of a TLS, which√is ∆E = ∆2 + (v t )2 . This analogy is based on the adiabatic theorem, which states that a system stays in the ground state
as long as the inverse gap ∆E −1 is small enough. Hence, the inverse of the gap can be considered a quantum-mechanicalequivalent of the relaxation time, τr = h̄∆E −1 . Then, the equivalent of the quench time τQ is ∆/v , while ∆ is identified
with τ0−1 . There, the LZSM transition is analogous to a phase transition. Indeed, by solving Eq. (62) with τr = h̄∆E −1 ,
τQ = ∆/v , and h̄∆ = τ0−1 , one exactly reproduces the above-mentioned 4 He result for tKZ in the fast quench/transition
limit.
Analogously to the above-considered quenched 4 He, a quantum Ising model can be used to describe the paramagnet–
ferromagnet quantum phase transition (Zurek et al., 2005; Dziarmaga, 2005; Polkovnikov, 2005; Dutta et al., 2015a). In a
more general context, the characteristic transition time tKZ and length ξKZ (size of regions in which the order parameter
1/(1+ν z )ν/(1+ν z )is smooth) are defined by the universal critical exponents z and ν : tKZ ∼ τQ
and ξKZ ∼ τQ
(Zurek, 1985;
Dziarmaga, 2010). Hence, this shows that equilibrium critical exponents can be used to predict the nonequilibrium
dynamics and that the KZM correctly describes the results of this dynamics by giving the density of residual topologicaldefects. We emphasize that the deep analogy between the LZSM and KZM is in the adiabatic-impulse approximation, which
has been shown to quantitatively well describe both the dynamics of quantum TLSs (which is the subject of the present
review article) and those of quantum phase transitions.
There are some difficulties in observing the time evolution of second-order phase transitions, and these are related
to their rapid speed (sufficient range of quench time scales) or controlling and counting the defects. Then the quantum
simulation can effectively be used by means of a convenient controllable quantum system.
Making use of the interrelation between the LZSM and KZM, this simulation was done recently with such diverse
systems as an optical interferometer (Xu et al., 2014b), a semiconductor charge qubit in a double quantum dot (Wanget al., 2014), superconducting phase and transmon qubits (Gong et al., 2016a), a single trapped 171 Yb+ ion (Cui et al., 2016,2020), spin-1 Bose–Einstein condensate (Damski and Zurek, 2009; Anquez et al., 2016), and NMR based studies (Zhang
et al., 2017). These simulations correctly reproduced the main KZM results: the boundary between the adiabatic andimpulse regions, the freeze-out phenomenon, and the dependence of the topological defect density on the quench rate.
Nowadays, the KZM is a general model that provides a description of the nonequilibrium dynamics and creationof topological defects such as strings, vortices, and domain walls. Here, using an analogy with the LZSM theory, theseappear during symmetry-breaking phase transitions in the following systems: Ising chains (Quan and Zurek, 2010; Das,
2010; Henriet and Hur, 2016) and spin-1/2 XX and XY chains when the transverse field or anisotropic interaction isquenched (Mukherjee et al., 2007; Divakaran et al., 2009; Roósz et al., 2014), graphene in a time-dependent electricfield (Fillion-Gourdeau et al., 2016), and biaxial paramagnet in an external magnetic field (Zvyagin, 2018). For deviationsof realistic systems from the paradigmatic KZM, see for example Gao et al. (2017).
24

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 10. Adiabatic-impulse model (AIM) which relates the Landau–Zener–Stückelberg–Majorana (LZSM) with the Kibble–Zurek mechanism (KZM).
In (a) we show the time dependence of the relaxation time τr (t) of second-order phase transitions. The inclined blue lines at τr = |t | give themoments of time tKZ , which separate the adiabatic (A) and impulse (I) stages of evolution. In (b), we plot the inverse energy-level gap ∆E −1 for a
two-level system.

Even though there are some studies on periodic driving, for example, Mukherjee and Dutta (2009), Setiawan et al.
(2015), Dutta et al. (2015b), Kar et al. (2016) and Higuera-Quintero et al. (2022), we note that in the context of phasetransitions, the vast majority of research is devoted to the single-passage transition/quench. In quantum simulations of the
KZM, as mentioned above, it is straightforward to realize double or even multiple passages. This introduces interference, in
addition to the possibility of nonadiabatic transitions. This may become one of the new twists in the interrelation between
LZSM physics and symmetry-breaking second-order phase transitions. Much like how the KZM for a single passage wasused to describe the Big Bang, the respective development may be useful for speculating about the Big Bounce theory.
3.2. Rotating-wave approximation (RWA)
3.2.1. Multi-photon Rabi oscillations
Consider now the situation of resonant driving, with those parameters close to where the energy distance ∆E equals
the photon energy h̄ω or, more generally, close to the energy of k photons, ∆E ∼ kh̄ω. The former (the single-photon
resonant excitation) is critical for microscopic systems where there are electron paramagnetic (spin) resonance andnuclear magnetic resonance (Rabi, 1937). In this case, the amplitude A is usually small, and the k-photon resonances
appear within perturbation theory (Shirley, 1965; Krainov, 1976; Krainov and Yakovlev, 1980).
In contrast, for mesoscopic systems, the strong-driving regime is both accessible and important. Hence, we will firstconsider the solution of the Schrödinger equation for a weak driving, afterwards for a strong driving, and then finally thesolution of the Bloch equation. These solutions are based on the RWA (also called secular approximation), where the terms
that are quickly varying in time are neglected.
For weak resonant driving, with A ≪ ∆ and δω ≡ ω − ∆E /h̄ ≪ ω, the upper-level occupation probability exhibits
Rabi oscillations (see e.g. Neilinger et al. (2016) and Shevchenko (2019)):
Pup (t) = Pup (1 − cos ΩR t),

ΩR =

√

2
ΩR0
+ δω2 ,

ΩR0 =

(64a)
A∆
2h̄∆E

,

2
ΩR0
Pup =
.
2
2 ΩR0 + δω2

1

(64b)

Here, Pup describes the time-averaged occupation probability; it is maximal in resonance at δω = 0.
For stronger driving, one usually assumes A ≫ ∆2 /h̄ω, which corresponds to what we call the diabatic limit, withδ ≪ 1. This condition also means that the minimal energy distance ∆ is small; we then, have ∆E ≈ |ε0 |. The solution of
the Schrödinger equation can be obtained close to the kth resonance, withδω(k) ≡ kω −

|ε0 |
h̄

≪ ω,

(65)

as in Refs. Henry and Lang (1977), Kmetic and Meath (1985) and Lopez-Castillo et al. (1992),
(k)

(k)

(k)
Pup
(t) = Pup (1 − cos ΩR t),

(66a)
25

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori(k)√

Physics Reports 995 (2023) 1–89(k)2ΩR0 + δω(k)2 ,
( )
∆
A
(k)
ΩR0 = Jk
,
h̄
h̄ω
ΩR =

(66c)

(k)2

ΩR0

1

(k)

Pup =

(66b)

2 Ω (k)2 + δω(k)2

.

(66d)

R0

These describe multi-photon Rabi oscillations. Here, the Rabi frequency is modulated by the Bessel function of the first kindJk . If the system’s parameters vary, we must take into account all the resonances:
Pup =

∑

(k)

Pup .

(67)

k

We could study the moment of time tmax , when the upper-level occupation reaches its maximum for the first time and the(k)probability Pup (Td ) after the double passage of the avoided crossing (i.e., after a full period) in resonance (Lopez-Castillo(k)
et al., 1992). From Eq. (66a), these are given by tmax = π/ΩR and

(

(k)
Pup
(Td ) = 1 − cos 2π

∆
Jk
h̄ω

(

))

A
h̄ω

.

(68)

In the next approximation, in the small parameter ∆ (to be more precise ∆2 /Ah̄ω ≪ 1 here), we can obtain the shiftof the resonance frequency, which is known as the Bloch–Siegert shift (Lopez-Castillo et al., 1992). Although for small ∆the resonance frequency for √the first resonance (k = 1) from Eq. (66d) is ω = |ε0 | /h̄, the resonance for larger ∆ is given
by Eq. (64a): ω = ∆E /h̄ =ε02 + ∆2 /h̄. Hence, when either ∆ is large or the frequency ω is small, the results of theRWA can be improved by adding the Bloch–Siegert shift and higher corrections (generalized Bloch–Siegert shift) (Tuorila
et al., 2010). See Refs. Abovyan and Kryuchkyan (2016), Sun et al. (2016), Wang et al. (2017), Huang et al. (2017), Saikoet al. (2018) and Kohler (2018) for those cases beyond the RWA.
More generally, taking into account relaxation, the RWA for a periodically driven system is presented in Appendix B.2.
This is considered for a generic periodic drive ε (t) = ε0 + ε̃ (t), with ε̃ (t) = A cos ωt as a particular case. The TLS is considered
as being coupled to a dissipative environment, the impact of which is taken into account by introducing the relaxation anddecoherence rates Γ1,2 = T1−,21 (Silveri et al., 2012, 2015). Then, the solution of the Bloch equations gives the upper-leveloccupation probability Pup , which for the stationary case reads
∞

Pup =

(k)2

ΩR0

1∑
2

ΩR0 + ΓΓ1 (kω − |ε0 | /h̄)2 + Γ1 Γ2
(k)2

k=0

,

(69)

2

where
(k)

ΩR0 =

∆ ω
h̄ 2π

∫ 2π /ω[ ∫ t
dt exp

0

i

h̄

]

dt ′ ε̃ t ′ − ikωt .

( )

(70)

0

These are quite general results that can be applicable to any driving ε̃ (t ). In particular, for ε̃ (t ) = A cos ωt, we obtain the
Rabi frequency (66c).
Role of the driving shape: Although most of this review is devoted to sinusoidal driving, we consider here the role ofdriving shape on the LZSM interference (Blattmann et al., 2015). In general, equations like (69), (70) can be useful for
describing any periodic perturbation ε̃ (t ), including multiharmonic drivings. Although the qualitative picture is similar
for any periodic signal, the overall interference fringes can differ. This was studied for such pulses as triangular (Xu et al.,2010), hyperbolic tangent and Gaussian (Cao et al., 2010), rectangular (Silveri et al., 2015; Shi et al., 2021), secant (Zhaoet al., 2018), and other ones (Mukherjee et al., 2016; Xie, 2018).
As a special and illustrative case, we now consider biharmonic driving (Blattmann et al., 2015). This allows to study theeffect of commensurate versus incommensurate driving frequencies (Forster et al., 2015). It has been demonstrated that
depending on the phase difference between its two components, bichromatic driving with commensurable frequencies
may break time-reversal symmetry, which is visible in the Fourier transform of the LZSM interference pattern and which isuseful for quantum simulation. Depending on the desired properties of the transitions, we can tailor the requested driving
signal. We will return to this in the section on quantum control. Here, we illustrate this by referring to the developmentof the Lyapunov-based control method. In Ref. Cong et al. (2015), it was demonstrated that an ultrafast excitation can begained by using a signal based on the Lyapunov control method, which is a design control method based on the Lyapunov
indirect stability theorem.
Now, with the equations above, Eqs. (69) and (66c), we can plot the interferograms for sinusoidal driving, as shownin Fig. 11. Regarding the dependence of the time-averaged occupation of the upper diabatic level Pup on the bias ε0 anddriving amplitude A, we can see that the resonances are along the lines |ε0 | = kh̄ω, and these are interrupted by the26

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 11. LZSM interferogram for different frequencies. For this, we took h̄ω/∆ = 10, 1.14, 0.32, for (a–c) and Γ1 = 0.1Td−1 , Γ2 = Γ1 /2.

zeros of the Bessel functions, resulting in the CDT. The width of the resonances, as defined by Eq. (69), depends on theinterrelation of the parameters; this is illustrated in the figure by varying the driving frequency ω.
For strong driving, A > h̄ω, from Eqs. (69), (70), we arrive at Eq. (60). For this, we use the asymptotics of the Bessel
functions and neglect the dissipation by considering the case when T1 = T2 = ∞. Importantly, we now see that two
different theoretical approaches, the AIM and RWA, which we redeveloped in different parameter regimes, lead to thevery same result: Eq. (60). Moreover, in the following subsections, we arrive at the same result while presenting twoother approaches.
3.2.2. Multi-photon transitions analyzed by the LZSM theory
It is well known that “an atom submitted to sufficiently intense radiation can absorb several incident photons andgo from a discrete level a to another discrete level b located at a higher energy, at a distance equal to the sum of theenergies of the absorbed photons. Such a process is called multi-photon absorption, the reverse process being multiphoton stimulated emission” (Cohen-Tannoudji et al., 1998). The observation in the previous subsection demonstrates
that multi-photon transitions are essential and appear consistently in different pictures, both within resonant RWA andwithin LZSM theory. In particular, from Eq. (69), we can see that the resonances take place at kh̄ω = |ε0 | ≈ ∆E.
Multi-photon processes have been studied in various systems: superconducting qubits (Wallraff et al., 2003; Saito
et al., 2004; Tornes and Stroud, 2008; Shevchenko et al., 2012b), two-dimensional electron systems (Zudov et al., 2006),
atomic and molecular structures (Arbó et al., 2010), Rydberg atoms (Førre, 2004; van Ditzhuijzen et al., 2009), TLSs
in superconducting circuits (Burin et al., 2014), quantum dots (Giavaras and Tokura, 2019a,b), and electromechanicalsystems (Heinrich et al., 2010).
A TLS can be subjected to resonant driving with the energy of k photons matching the distance between the energy
levels kh̄ω ≈ ∆E; this induces a transition from one level to another. These can be described as the exchange ofenergy quanta (here, photons) with the oscillating driving field, which is known as photon-assisted tunneling (PAT); see,for instance, Ref. Li et al. (2013), where this was studied for a superconducting qubit. PAT was studied extensively forother various quantum systems with tunneling, especially for semiconductor nanostructures (Platero and Aguado, 2004),
including driven quantum dots (Mavalankar et al., 2016; Osika and Szafran, 2017) and qubit-resonator dimers (Zheng
et al., 2021). Through a double quantum dot, PAT is described by the tunneling current defined by a formula analogous
to Eq. (69) (Stoof and Nazarov, 1996; Gallego-Marcos et al., 2015).
Historically, the first experiments with PAT were performed on AC-driven Josephson contacts (Tien and Gordon, 1963).
In this context, we can note the similarity and difference of the PAT (or, equivalently, multi-photon Rabi oscillations) and
AC Josephson effect with the Shapiro steps, both of which appear for an irradiated superconducting tunnel junction. The
former relates to the tunneling of electrons (quasiparticles) and results in steps in the current–voltage curve displacedin voltage by kh̄ω/e; the latter assumes the supercurrent response, which occurs as the Shapiro steps at voltageskh̄ω/2e (Nakamura et al., 2001). Both sets of resonances were observed experimentally and described theoretically in Borisand Krasnov (2015) and Shaikhaidarov et al. (2022). In that study, it was demonstrated that the Shapiro steps and PAT
states originate from Cooper-pair and quasiparticle currents; and if a JJ is subjected to an intense microwave signal, bothsets of resonances can be observed in the current, changing both the voltage and microwave power (Boris and Krasnov,
2015; Snyder et al., 2018).
3.3. Floquet theory
It is natural to use the Floquet theorem for solving the Schrödinger equation for a periodically driven system. The
theory based on this— Floquet theory—has been covered in many reviews (Chu, 1989; Grifoni and Hänggi, 1998; Chu and27

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Telnov, 2004; Son et al., 2009; Shevchenko et al., 2010; Eckardt, 2017; Rodriguez-Vega et al., 2021; Sen et al., 2021). We
now consider this for our problem of describing the dynamics and stationary state of a driven TLS; the details are given
in Appendix B.3.
For a time-dependent Schrödinger equation, Eq. (15), for the wave function |ψ (t)⟩ with a periodic Hamiltonian
H(t) = H(t + Td ), we can use the Floquet theorem. Accordingly to this, the solution (so-called Floquet-state solution)
is

(
)⏐
⟩
|ψ (t)⟩ = Σj=1,2 Cj exp −iϵj t /h̄ ⏐Φj (t) ,(71)⏐⟩
where ⏐Φj (t) is a periodic state, called the Floquet mode. The index j = 1, 2 appears because we are dealing witha two-dimensional Hilbert space. The real-valued ϵj are called quasienergies, to reflect the formal analogy with aquasimomentum k, which characterizes electron Bloch eigenstates in a periodic solid state (Zel’dovich, 1967, 1973;
Averbukh and Perel’man, 1985). The quasienergies are unique up to multiples of 2π n/Td . It follows that these are theeigenstates in the problem(
H(t) − ih̄

d

)

dt

⏐
⏐
⏐
⟩
⟩
⟩
⏐Φj (t) ≡ H(t) ⏐Φj (t) = ϵj ⏐Φj (t) .

(72)

The quasienergy states for time-dependent problems play a role analogous to the stationary states for time-independentproblems.
Using the Floquet formalism allows us to reduce the problem of periodic perturbation to the stationary problem (Shirley, 1965; Barone et al., 1977; Ikeda et al., 2022), which is known as the Floquet Hamiltonian(Chu, ⏐1989;⏐ method⟩⟩⏐Φj (t) = Σn einωt ⏐Φj,n .
Chu and Telnov, 2004). For this, one needs to expand the quasienergy function into a Fourier ⏐series,⟩
Hence, the quasienergy states become expressed as a superposition of stationary states ⏐Φj,n , with energies equal toϵj + nh̄ω. Using the Fourier series expansion, from the Schrödinger equation, we obtain the relation

(
)
⏐
⏐
⟩
⟩)
⟩ ⏐
⟩ A (⏐∆ε0ϵj ⏐Φj,n = − σx − σz + nω ⏐Φj,n − σz ⏐Φj,n−1 + ⏐Φj,n+1 .
2

2

(73)

2

Multiplying this to the left by Φi,m ⏐, we obtain the matrix equation for the eigenvalues ϵj and eigenfunctions ⏐Φj,n . Then,
we truncate the matrix to solve the equation numerically, say, with n ranging from −50 to 50 (Deng et al., 2016).
Close to the resonance, δω(k) ≡ kω − |ε0 | /h̄ ≪ ω, the contribution of the nonresonant states can be neglected. Thismeans using the RWA, for example, as in Autler and Townes (1955), Aravind and Hirschfelder (1984) and Silveri et al.
(2013). Then, the Floquet Hamiltonian consists just of the copies of

⏐

⟨

(

(k)

HRWA =−ε0 /2(k)−h̄ΩR0/2⏐)(k)−h̄ΩR0/2,
ε0 /2 + kh̄ω

⟩

(74)

(k)

where the Rabi frequency ΩR0 is the one defined in Eq. (66c). Diagonalization of this Hamiltonian produces the
(k)
(k)′
(k)
quasienergy difference, which we denote as h̄ΩR : HRWA = σz h̄ΩR /2. This brings us to the same result as RWA; see
Eq. (66b). Hence, Floquet theory in the secular approximation (i.e., in the RWA) gives us again the upper-level occupation
probability in Eq. (66d) (Son et al., 2009).
These RWA results are accurate, provided ∆/h̄ω ≪ 1. The higher-order terms can be obtained within the generalized
van Vleck perturbation theory (Hausinger and Grifoni, 2010), resulting in a shift of the resonance frequency by

δk =

1 ∑
2

l̸ =−k(l)2ΩR0.
ε0 /h̄ + lω(75)This means√ that the resonances are situated at ε0 = kh̄ω − δk , and that the Rabi frequency becomes δk -shifted,

(
)2
(k)2
ΩR0 + δω(k) − δk . Note that these formulas are calculated in the diabatic basis, while for the adiabatic basis,

(k)

ΩR =

some modifications must be made (Silveri et al., 2013).
In particular, consider the first-order correction from Eq. (75), which is δ1 = ∆2 /2ε0 . Then, we expect the firstresonance at ε0 = h̄ω − ∆2 /2ε0 (where we can apply ε0 = h̄ω on the√right-hand-side). This matches the original
Rabi approach. Indeed, from Eq. (64a), we have the resonances at h̄ω =∆2 + ε02 , where the expansion gives ε0 =h̄ω − ∆ /2ε0 . Hence, generalized van Vleck perturbation theory gives the correction exactly consistent with the Rabi
RWA, which is valid for small driving amplitudes (Sambe, 1973). For more about the resonance shift, termed AC Stark
shift or Bloch–Siegert shift, see Autler and Townes (1955), Aravind and Hirschfelder (1984), Yan et al. (2017).
Because they are related to the Rabi frequency, the Floquet quasienergies can be visualized in experiments, known
as Floquet spectroscopy. This was realized recently with qubits in cavities. In this case, the cavity is a superconducting
microwave resonator and qubits are based either on superconducting circuits (Silveri et al., 2013; Deng et al., 2015) or on
double quantum dots (Koski et al., 2018; Chen et al., 2021a). Other possible realizations include such systems as a strongly
driven Anderson insulator (Agarwal et al., 2017) or pumping in a Cooper pair sluice (Russomanno et al., 2011). As a further
228

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

development of the theory, this approach can be used to study the low-frequency limit (Rodriguez-Vega et al., 2018),
include the dissipation (Henriet et al., 2014; Restrepo et al., 2016; Kohler, 2017; Mori, 2022) (Floquet–Markov theory),and considering multilevel systems (Denisenko et al., 2010; Ganeshan et al., 2013; Satanin et al., 2014; Han et al., 2019,2020; Munyaev and Bastrakova, 2021; Zhou et al., 2021).
3.4. Driving fields in the quantum regime
Our approach above, in which the driving field is treated as a classical field and enters as ε (t), is essentially thesemiclassical approach. This takes place when the system of interest is described by the Schrödinger equation anda field satisfies classical Maxwell equations. In the other, fully quantum-mechanical approach, both the system andelectromagnetic field are treated quantum mechanically. Importantly, semiclassical theory gives rise to results that areequivalent to those obtained from fully quantized theory in intense fields (Aravind and Hirschfelder, 1984; Grifoni and
Hänggi, 1998; Chu, 1989). We discuss this below. For experimental realizations, see Nakamura et al. (2001), Saito et al.
(2004), Wilson et al. (2007), Kervinen et al. (2019).
A TLS coupled to a quantum field is described by the Janes–Cummings Hamiltonian with driving:
HQ (t) = −

∆
2

σx −

ε0
2

)
)
(
(
σz + h̄ωr a† a − g σz a + a† + ξ a† e−iωt + aeiωt ,(76)

where a and a† are the annihilation and creation operators for photons in the electromagnetic field, g is the coupling
constant, ξ describes the amplitude of the driving field, and the driving frequency ω is close to the resonator frequency
ωr . For example, for a flux qubit coupled to a transmission line resonator, the coupling constant is proportional to thepersistent current in a flux qubit Ip , which is the current constant in the resonator Ir0 , and the mutual inductance M:
g = MIp Ir0 ; the driving amplitude is defined by the amplitude of the applied microwave voltage VA , the voltage constantin the resonator Vr0 , and a coupling capacitance C0 : ξ = C0 VA Vr0 (Greenberg, 2007; Shevchenko, 2019). Note that thelast term in Eq. (76) describes driving through the resonator; if the driving is through the qubit, this term would be
proportional to σz cos ωt (Zhao et al., 2015).
()
Next, we need to get rid of the temporal dependence in the driving term with the transformation U = exp iωta† a ,resulting in HQ → H̃Q , and to average this over the coherent states |α⟩ (Aravind and Hirschfelder, 1984; Sun et al., 2012).
One way to define the coherent states is to assign them as the eigenstates of the annihilation operator, a |α⟩ = α |α⟩; itfollows that the value α is given by the mean number of photons in the resonator: ⟨n⟩ = |α|2 . The resulting HamiltonianH = ⟨α| H̃Q |α⟩ becomes our quasiclassical qubit Hamiltonian H(t) in Eq. (1), with the cosine bias with the amplitude
A = 4α g in Eq. (42), quod erat demonstrandum, which is one of the cornerstones of circuit quantum electrodynamics(QED).
Alternatively, one can show that HQ (t) in matrix form is exactly reduced to the Floquet Hamiltonian HF for an intense
driving field (Aravind1984). For√this, the annihilation and creation operators satisfy the following√ and Hirschfelder,
relations: a |n⟩ =n |n − 1⟩ and a† |n − 1⟩ =n |n⟩. Then, with ∆ = 0, the Hamiltonian can be diagonalized
exactly (Cohen-Tannoudji et al., 1998; Nakamura et al., 2001; Saito et al., 2004; Wilson et al., 2007, 2010) with the
eigenstates( ())
g|0, n⟩ = |0⟩ ⊗ D (g̃ ) |n⟩ , |1, n⟩ = |1⟩ ⊗ D (−g̃ ) |n⟩ , where D(g̃) = exp g̃ a† − aand g̃ =.
(77)h̄ω
These states are called the dressed-state basis; their eigenenergies are
Ej,n = nh̄ω −g2h̄ω+ (−1)j ε0 .

(78)Here, D is the displacement operator, the parameter g̃ defines the strength of the coupling, while j = 0, 1 is attributed to
the TLS, and n to the number of photons in the cavity. Then, in this dressed-state representation, the matrix elements of thefull Hamiltonian (with nonzero ∆) coincide with the ones of the Floquet Hamiltonian HF , Eq. (B.30). When E0,n+k ≈ E1,n ,that is, when the k-photon resonance condition is met, kh̄ω = ε0 , the two dressed-states experience avoided-level crossing(k)with the coupling (off-diagonal element) 21 ∆Jk (α ), resulting in k-photon Rabi oscillations with frequency ΩR = ∆Jk (α )/h̄,
as above in the RWA.
Ultra-strong coupling regime: The picture above is in agreement with the semiclassical approach. However, there is aparticular regime with very strong coupling and small photon numbers where some differences may appear (Saiko et al.,2016; Ashhab, 2017). To understand this, note that with the driving field in the quantum regime, when we consider strong
driving, we mean large A = 4α g; but this can be reached with either a large coupling g or a large number of photons ⟨n⟩.
Accordingly, we can consider several limiting cases of the strong driving (Li et al., 2013; Ashhab, 2017; Bonifacio et al.,2020):⎧⎨g̃ ≪ 1, weak coupling with ⟨n⟩ ≫ 1,
A = 4 ⟨n⟩g ≫ h̄ω ↔
g̃ ∼ 1, (deep-)strong coupling with ⟨n⟩ ≫ 1,⎩g̃ ≫ 1, ultra-strong coupling with ⟨n⟩ ∼ 1.
√

29

(79)

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

For all these regimes, the Rabi frequency in the quantum case, around the k -photon resonance, kh̄ω = ε0 , reads (Ashhab,
2017)
(k)
ΩR,Q

=

∆
h̄

√
]
[
(2g̃ ) exp −2g̃ 2n!

k(n + k)!

Lkn 2g̃ 2 ,

(

)

(80)

where Lkn are the associated Laguerre polynomials. The numerical calculations in Ref. Ashhab (2017) demonstrate that thesemiclassical dynamics is fully consistent with the quantum dynamics with the Rabi frequency in Eq. (80), but only forsmall coupling g̃ and large photon number n. In the ultra-strong coupling regime, from Eq. (80), where the fully quantum
approach provides decaying oscillations because of quantum fluctuations and where the Rabi frequency deviates from theresult of the semiclassical approach.
Dressed-states and interferometry for systems which are either doubly driven or driven by a chirped microwave:
The notion of dressed-states, which we just considered for the quantum driving fields, appears when describing driving
systems in a different context. Here, we mean the case when both the Hamiltonian of a TLS and driving can be rewritten as
an effective TLS; hence describing entangled light-matter states, which are also known as dressed-states (Magazzù et al.,2018; Chang et al., 2020). This notion appears because the qubit is now dressed by the driving field. This approach is usefulfor systems that do not have avoided energy-level crossing, or where their energy-level spacing does not depend on theexternal bias parameters (Gong et al., 2016b). Here, a typical case with these two properties is a transmon qubit (Fink
et al., 2009); to make its effective energy levels controllable and, in particular, for realizing the LZSM regime, this can beachieved by using either a chirping field or a second signal (Garraway and Stenholm, 1992a; Gauthey et al., 1997; Vitanov
et al., 2001; Sun et al., 2011; Silveri et al., 2015; Garraway and Perrin, 2016; Gong et al., 2016b).
To illustrate dressed-states and interferometry, consider now a doubly driven TLS, described by the “pump–probe”
Hamiltonian, for example (Wen et al., 2020),
Hpp = −σx ∆ cos ωprobe t +σz (2h̄ω10 + δ cos ωpump t .

)

(81)

This corresponds to a TLS with distance between the energy levels equal to h̄ω10 and driven by( two signals, )with
amplitudes proportional to ∆ and δ , assuming ωpump ≪ ωprobe . Using the transformation U = exp −iωprobe σz t /2 andthe RWA lead to the Hamiltonian (1) with)(
ε (t) = h̄ ω10 − ωprobe + δ cos ωpump t .

(82)

Now, after arriving at the pseudo-spin Hamiltonian for our driven system (which we can now call the dressed Hamiltonian), we can understand the two options above.
two signals, the pump (dressing) signal and probe signal, obtaining the bias (42) with ε0 =)( First, we applyh̄ ω10 − ωprobe (for a transmon, this is controlled by the DC magnetic flux, changing ω10 ), A = δ , and ω = ωpump (Silveri
et al., 2015).
Second, we do not have a second signal, δ = 0, while the “probe” frequency is time dependent, h̄ωprobe (t) →ε0 + A cos ωt; see Ref. Gong et al. (2016b) for chirping with triangular pulses, Refs. Childress and McIntyre (2010),
Blattmann et al. (2014) and Ono et al. (2019) for sinusoidal frequency modulation, and Refs. Saiko et al. (2007, 2014,2019) and Wang et al. (2021) for the impact of bichromatical driving.
LZSM spectroscopy of multilevel systems: For a multilevel system, a similar dressing can also be applied to create
an avoided-level structure of energy levels (Mi et al., 2018; Shevchenko et al., 2018). For this, we apply the driving
signal with frequency ωd close to the energy-level separation ∆0 at ε = 0, to obtain the dressed avoided-level gap∆ ≈ (∆0 − h̄ωd ) ≪ ∆0 . Then, the dressed energy levels are related to the bare ones as Ẽi ≈ Ei ± h̄ωd . As a result,in addition, this dressing can significantly increase the distance between the lower two energy levels and upper ones,
providing an instrument for creating controllable TLSs out of multilevel ones. In such case, note that features of the
TLS’s spectrum, such as the curvature of its energy levels, essentially depend on the entire energy spectrum of the baremultilevel system (Mi et al., 2018; Shevchenko et al., 2018). Hence, the LZSM interferogram of the effective dressed TLS
is a convenient tool for the spectroscopy of multilevel systems.
3.5. Impact of dissipation and temperature
Any real quantum system is coupled with the environment, and for mesoscopic systems, it is particularly important
to include dissipation. Many authors study the impact of temperature, relaxation, and decoherence on the dynamics
of TLSs (Leggett et al., 1987; Grifoni and Hänggi, 1998). The general approach is to start from the Liouville–von
Neumann equation for the density matrix ρtot of the system, here comprising of our quantum system and the dissipativeenvironmentddt

iρtot (t) = − [Htot (t), ρtot (t)] .

(83)

h̄

30

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

The total Hamiltonian, Htot = H + Henv + Hint , consists of our system’s part H, the environment Hamiltonian Henv ,and the interaction between them Hint , for example, Nalbach and Thorwart (2009). Provided the coupling is weak, theenvironment can be represented as a set of harmonic oscillators with a coupling linear in∑
the oscillator coordinates. Within†
this “spin-boson” model, the environment is characterized by the Hamiltonian Henv =h̄ωi bi bi with the frequencies ωiand annihilation operators bi ; a bosonic reservoir can represent
phononsif
oursystemiscoupledwith a crystal lattice.
∑†
The general form of the interaction is given by Hint = − 12 S
h̄λi (bi + bi ), with S representing a spin operator; usually,S = σz (Kayanuma and Nakayama, 1998; Saito and Kayanuma, 2002; Chen et al., 2015; Wertnik et al., 2018; Lambert
et al., 2019, 2020; Funo et al., 2021).
∑ 2
The environment is described by the spectral density function J(ω) = πλi δ (ω − ωi ) (Leggett et al., 1987). At
small energies, this is characterized by a power-law dependence and can be further approximated by introducing thecutoff frequency ωc : J(ω) = αω(ω/ωc )s−1 exp (−ω/ωc ), where the dimensionless parameter α defines the strength of thecoupling and the power law is described by s, with s = 1 for the ohmic environment (Ao and Rammer, 1989, 1991). For
nonohmic cases, see Leggett et al. (1987). As a simple approach, we can consider the environment being modeled as only
one (finite-temperature) harmonic oscillator (Saito et al., 2007; Ashhab, 2014; Malla and Raikh, 2018).
Tracing out the environment’s degrees of freedom, we can obtain a master equation for the reduced density matrix ρ (t)of our system, which can be written as the Lindblad equation (also known as the Lindblad–Gorini–Kossakowski–Sudarshan
equation, Chruściński and Pascazio (2017))iρ̇ = − [H(t), ρ ] +
h̄

∑(

†

Lk ρ Lk −

k

1 †
1 †
Lk Lk ρ − ρ Lk Lk
2
2

)

.

(84)

Different channels of relaxation are described by the Lindblad operators Lk . Note that Lindblad equation is correct undercertain conditions such as:• separability (no correlations between the system and its environment at t = 0),• Born approximation (environment is static despite interaction with quantum system, and weak coupling between
quantum system and environment),• Markov approximation (time scale of decay is much shorter than the smallest characteristic time of the system),• secular approximation (all fast rotating terms in the interaction picture can be neglected).
For the limitations of the Lindblad equation, see Teixeira et al. (2021).
√ In particular, for a TLS, the√impact of the environment results in relaxation and dephasing, as described by Lrelax −=1
Γ1 (σx − iσy )/2 and Lφ =
Γφ /2σz . Instead of the relaxation rates, we can use the respective times: T1 = Γ1
and Tφ = Γφ−1 . Then, the Lindblad equation is reduced to the Bloch equations for the components of the Bloch vector
s = Tr(σ ρ ); writing the Hamiltonian in the form H = −B · σ /2, with the effective “magnetic” field B, these read as
d

) 1
1 (
eq
s∥ − s∥ −
s⊥ ,
T1
T2

s = −B × s −

(85)
dt
1
seq
(86)
T2−1 = T1−1 + Tφ−1 ,z = tanh(∆E /2kB T ).
2
Importantly, the Bloch equations are obtained based on the qubit eigenstates (Krzywda and Cywiński, 2021) (see also Xu
et al. (2014a)). In particular, this assumes a relaxation to the ground state at low temperatures. Thus, the Hamiltonian H(t)in the Lindblad and Bloch equations should be written in the representation of the adiabatic basis. The diagonalizationof the Hamiltonian is done by a rotation on the angle η. The relaxation rates are related to the spectral density (Makhlin
et al., 2001) and depend on the energy offset ε0 and temperature TΓ1 = π α sin2 η

∆E
h̄

coth

∆E
2kB T

Γφ = πα cos2 η

,

2kB T
h̄

.

(87)

The Bloch equations can be extended to a multilevel system (Shimshoni and Gefen, 1991; Gefen and Thouless, 1987)i

ρii (t) − ρiieq (t)h̄iρij (t)

T1,i

h̄

T2,ij

ρ̇ii (t) = − [H(t), ρ ]ii −ρ̇ij (t) = − [H(t), ρ ]ij −

,

,

(88a)

i ̸ = j.

(88b)

Therefore, the most straightforward approach to demonstrate the impact of both dissipation and temperature is tosolve the Bloch equations; see this also in Refs. Kayanuma (1993), Zueco et al. (2008), Nalbach and Thorwart (2009), Scalaet al. (2011), Whitney et al. (2011), Orth et al. (2013), Xu et al. (2014a) and Dai et al. (2022). In Fig. 12, we present theinterferograms for different relaxation and dephasing rates and their Fourier images. In panel (a), we show the case of
low dissipation and relaxation rates, in which all the resonances are distinguishable. In panel (b), we show the case of
the average dephasing regime, which we also refer to as the double-passage regime, when several passages occur before
dephasing dominates, which leads to merging resonances into lines.
31

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 12. Impact of dissipation and dephasing on interferograms and tomograms. The upper panels present the LZSM interferograms and the lower
panels their respective 2D Fourier transforms also known as tomograms. These interferograms and their tomograms are demonstrated in several
cases: (a) and (e) low relaxation and dephasing, T1 = 10Td , T2 = 20Td ; (b) and (f) low relaxation and the dephasing time comparable with thedriving period, T1 = 10Td , T2 = 0.5Td ; (c) and (g) weak relaxation and strong dephasing, T1 = 10Td , T2 = 0.1Td ; and (d) and (h) both the relaxation
time and the dephasing time comparable with the driving period, T1 = 0.25Td , T2 = 0.5Td . For all panels, the driving frequency is ω = 1.14∆/h̄.
The inset (i) shows that fitting the linear part of the dependence can be used for defining the decoherence time T2 .

The next regime is with large dephasing, which is shown in panel (c), where dephasing dominates over excitation andresonances merge into one region A > |ε|. Panel (d) presents the strong relaxation case when relaxation dominates overexcitations. The characteristic pattern for this case is the peak of probability near (ε0 = 0, A = 0), where an increasing in
the driving amplitude results in the resonances disappearing.
Dissipative LZSM transitions were studied for quantum dots (Fouokeng et al., 2014; Ota et al., 2017; Huang and Hu,
2021), superconducting qubits (Gramajo et al., 2019), and molecular nanomagnets that include the high-spin case (Calero
et al., 2005; Vogelsberger and Garanin, 2006; Földi et al., 2008). The transition in a system subject to continuous
measurement was considered in Ref. Haikka and Mølmer (2014). When the environment can be treated as a continuum ofstates, LZSM transitions can be formulated within the open-multistate model (Dodin et al., 2014). The impact of dissipation
on CDT is the suppression of those processes (Hausinger and Grifoni, 2010).
3.6. Rate equation
The master equation which describes a TLS can be written in the form of a rate equation. (This was not consideredin Shevchenko et al. (2010), but this approach is also useful and important.) Consider here the microscopic analysis of
the dynamics of a TLS based on the rate equation and adding classical noise to model decoherence (Berns et al., 2006).
Details are presented in Appendix B.4.
First, we calculate the transition rate W for a bias ε (t) = ε0 + A cos ωt + δε (t), in the presence of classical noise δε (t).
For concreteness, we present the derivations and results for harmonic driving. This approach can be used for any otherperiodic driving, such as sawtooth-like and biharmonic ones (Rudner et al., 2008; Berns et al., 2008; Cao et al., 2013).
Then, the averaging over δε (t) using the white-noise model gives the rate of transitions between the TLS states
W =

∞
1 ∑

2

n=−∞

Γ2 ∆2n(ε0 /h̄ − nω)2 + Γ22

,

(89)

where the splitting is modulated with the Bessel function, ∆n = ∆Jn (A/h̄ω) and Γ2 = T2−1 is the decoherence rate (Duet al., 2010, 2013a).
32

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Next, we construct the rate equation, which includes possible transitions between the states, with rate W , andrelaxation, with rate Γ1 = T1−1 . From the stationary solution of the rate equation, we obtain
P+ =W
2W + Γ1.

(90)

The summation of all possible resonances gives us exactly the same solution as the one above: Eq. (69). This derivationof the formula (69) reveals its robustness.
The formalism presented here has the advantage that it is straightforward to be generalized for multilevel systems (Wen et al., 2009). For a system with N energy levels, we must solve a system of N rate equations (like we dofor a TLS in Appendix B.4) to describe the transitions between levels i and j with the rates Wij , given by Eq. (89), replacing
the splitting ∆ → ∆ij and offsets from them ε0 → εij . This approach was developed and applied to multilevel systems,
including the ones based on superconducting qubits (Wen et al., 2010; Du and Yu, 2010; Wang et al., 2010; Chen et al.,2011) and double quantum dots (DQD) (Chatterjee et al., 2018; Liul and Shevchenko, 2022).
3.7. Quantum phase tomography
The Fourier transform is a useful tool for analyzing different periodic patterns. In the case of LZSM interferograms,
the Fourier transform results in a highly ordered structure of one-dimensional arcs (Berns et al., 2008). Because theFourier image gives the structured mapping of the interferograms, it presents the tomographic imaging and the respective
graphs give rise to a technique called quantum phase tomography (Rudner et al., 2008). This approach gives additional
information about the physical processes in a driven quantum system and is useful for defining system parameters,
including for probing the dephasing mechanisms (Rudner et al., 2008; Cao et al., 2013).
Having an analytical expression for Pup , we can use the continuous 2D Fourier transformation
PFT (kε , kA ) =∫∫ ∞

Pup (ε0 , A) exp [(−ikε ε0 − ikA A)] dε0 dA,

(91)

−∞

where kε and kA are the reciprocal-space variables corresponding to the variables ε0 and A, respectively. In the case whenthe matrix of the data is given instead by a formula for Pup , we can use the discrete 2D Fourier transform. In the lower
panels of Fig. 12, we present the tomograms related to the interferograms in the upper panels; these were calculatedby the discrete Fourier transform of the data for the interferograms. We can see that the relaxation suppresses the lowfrequency resonance curves; when increasing the relaxation rate, only the main curve (with l = 1) survives, as can beseen in Fig. 12(h). Note that the vertical and horizontal lines of increased PFT along the axes, especially at kε = 0 andkA = 0, are artifacts because of the finite size of the interferograms.
The overall tomograms in Fig. 12(e–h) are lemon-shaped structures formed by the 1D sinusoids of the formkA (kε ) = ±

2l

ω

(
sin

ωkε + 2π l′
2l

)

,

(92)

with l = 1, 2, 3, . . ., l′ = 0, 1, 2, . . ., and l′ < l. This can be seen by either analysis of the phase acquired during theperiodic evolution (Rudner et al., 2008) or by directly transforming Eq. (69) (Forster et al., 2014; Blattmann et al., 2015).
From the lower panels in Fig. 12, we can see how relaxation and dephasing impact the Fourier images. With low relaxationand decoherence rates, there are many resonances in panel (e). From panels (f) and (g), we can see that increasing thedephasing rate leads to increasing the slope rates along the |kε | axis. Then, when we compare Fourier images on panels(f) and (h), we can see that for the same dephasing rate, increasing the relaxation rate leads to the disappearance of thehigher resonances. In panel (i), we show the cross-sections of panels (f,g,h) along kA = 0.85; the slopes are given by
the dephasing rates, as shown by the dashed lines. Note that in this panel, this is very noisy because of resonances andartifacts; to better see the slopes, one can consider the response along the resonance lines, as in Forster et al. (2014).
From Eqs. (89)–(90), we can find that the slope of the Fourier image is proportional to the decoherence rate:ln PFT ∝ −Γ2 |kε | (Rudner et al., 2008). This is demonstrated as the inset, Fig. 12(i). This observation provides a convenient
tool for obtaining the decoherence rate without additional measurement or fitting (Cao et al., 2013; Gonzalez-Zalba et al.,2016). If this theory accounts for inhomogeneous broadening Γ2∗ , this results in
ln PFT (kε , kA (ε )) ∝ −Γ2 |kε | − Γ2∗2 k2ε /2,(93)

where kA (kε ) is defined by the relation Eq. (92) (Forster et al., 2014). In addition to Γ2 , this provides a convenient tool for
defining Γ2∗ , most importantly without any additional measurements, such as spin-echo experiments (Cao et al., 2013).
Hence, the Fourier transform provides a useful tool, quantum-phase tomography, providing an additional visualizationof the interference and the system parameters, such as the decoherence rate. This methodology can be further developed
and applied to other systems, specifically for bichromatic driving (Forster et al., 2015) and multilevel systems (Berns et al.,2006; Liul and Shevchenko, 2022).
33

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Table 3
Relative efficiency of different methods. This is quantified with the time (in seconds) needed to calculate the respective interferograms shown in
Fig. 13. The first line relates to the interferograms in the coordinates (ε0 , A) and the second line to the coordinates (ε0 , ω). The numbers in the tablewere obtained using an eight-core CPU Intel 10875H using eight threads.
(ε0 , A)(ε0 , ω)Numerical ODE

Numerical QuTiP

Floquet method

RWA

Optimized RWA

AIM

3144
1400

2050
1241

1300
774

16.4
9.3

4.7
4

9.4
8.6

3.8. Comparison of different methods
Above we have considered in detail different methods for describing driven quantum systems. Now, we would like tocompare these different approaches; see also Ashhab et al. (2007). We summarize their key aspects and then comparethem by computing the interferograms.
Numerically solving the Schrödinger or Lindblad equation: This method is the most commonly used because it isbased on the integration of a system of differential equations. This gives the best-quality results and can be used in all
range of parameters. We compare all other methods with this one, because the numerical integration can be performed
accurately. However, the numerical integration has one significant disadvantage: it needs considerable computing power,
so it requires considerable CPU time to compute interferograms. We use the numerical solution based on the modified
Runge–Kutta method, which is a standard approach for solving systems of ordinary differential equations (ODE) in Python.
Numerically solving using QuTiP: The QuTiP (Quantum Toolbox in Python) framework (Johansson et al., 2012, 2013;
Shammah et al., 2018) is a very useful tool to calculate the dynamics of quantum systems. We need only to introduce a
Hamiltonian and set of parameters; as a result, we have the full evolution of the system (all terms of the density matrix,in all the defined moments of time) or, alternatively, the averaged value of population of a certain level. This library also
has a detailed manual and many different solvers.
Floquet method: This approach can be used only for periodic Hamiltonians and is based on the Floquet theorem. The
Floquet method is faster than the direct numerical solution, but slower than other analytical methods. Importantly,
relaxation and dephasing can be included. This method is analogous to a decomposition on eigenfunctions in methods ofmathematical physics.
Rotating-wave approximation (RWA): The RWA corresponds to the first term in a Floquet formalism by the small
distance between levels ∆ ≪ ω, ε . It is the most commonly used analytical method because it combines adequate accuracy
with a moderate need in computing power. Here, for calculating interferograms, the range of application is limited by
the overlap of resonances, which occurs at increasing relaxation and dephasing or decreasing driving frequency. In thesimplest approach, we can use the formula Eq. (69), which assumes calculating a number of terms in each point. However,
if we take a few close resonances only, that is, with k close to the resonance condition ε0 = kh̄ω, this significantly speed-up
the calculation; this approach can be called optimized RWA.
Adiabatic-impulse model (AIM): The Adiabatic-impulse model can provide interferograms in different bases. This is
generally done neglecting relaxation and decoherence; thus the time-averaged result obtained by this method depends on
the initial condition. This correctly gives the location of the resonances; but between them this method may give different
values from those obtained using more precise numerical calculations, which is a result of not including relaxation anddephasing. This means that the time-averaged result depends on the initial condition; but when we use the ground-state
initial condition in this AIM and small dissipation in other methods, then the AIM gives the correct result. The range ofusability of this method is defined by a high amplitude of excitation A > ∆, which is needed for the adiabatic evolution
between transitions. The second limitation is the times of the transition process: Td /2 > (τrelax + τjump ). See the transition
times described in paragraph 2.3.2.
Interferograms and time of calculation: We illustrate our conclusions in Fig. 13 and Table 3. First, in Fig. 13: (a,b,c,d),
we present the interferograms using the coordinates (ε0 , A); and in (e) using (ε0 , ω). Even the former shows not somany differences, and these become even less for the latter in (e). For the calculations, we chose the driving frequencyh̄ω/∆ = 1.14 and the relaxation rates for (b–d), Γ1 = 0.1Td−1 and Γ2 = Γ1 /2. Note that the adiabatic-impulse modelin (a) is valid under the condition A > ε0 , with no relaxation and decoherence included. The RWA in (b) provides a result
more similar to the numerical solution in (c), except the shifting of the resonances when increasing the driving amplitude,
here in the region A < ε0 .
In Table 3, as a quantitative measure of the efficiency, we have chosen the time needed to compute the interferograms.
For this, we used an eight-core CPU Intel 10875H using eight threads. Calculations were performed as in Fig. 13, with 401
points in each axis. Note that the most significant impact on calculation time is given by the number of points, while thedependence on other parameters vary for different methods. As we see from Table 3, the analytical methods are much
faster than the numerical calculations, with a difference of about two orders of magnitude in calculation time. However,the numerical calculations are more flexible and can be easily used for more complicated systems, including many levels,
34

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 13. LZSM interferograms calculated by different methods. (a–d) and (e) present the time-averaged upper-level occupation probability as afunction of the energy offset ε0 , the driving amplitude A for (a–d) and the driving frequency ω for (e). Comparison between the regions of validity
for different methods with small relaxation and dephasing rates (f). The methods used are the adiabatic-impulse model (AIM) (blue circles in panel(f)), rotating-wave approximation (RWA), and the second approximation by the Floquet method given by the generalized Van-Vleck perturbation
theory (GVVPT), which has the same region of validity (red ovals in panel (f)). Numerical solution of the ordinary differential equations (ODE), andthe Floquet approach can be applicable everywhere. Also, panel (f) demonstrates the region of validity for the Rabi oscillations method (green ovals).

adding noise, and so forth. For describing the time-averaged occupation probability, the most convenient method is the
RWA. For describing the dynamics, one of the most illustrative methods is the adiabatic-impulse model, but it is limitedby the time of the transition process; for more related information, see Section 2.3.2.
4. Interferometry
The interference in a driven two- or multi-level system is essentially defined by the problem parameters. It is important
for applications to demonstrate this through calculations and to formulate recipes of how to extract not only the quantum
system parameters from the interferograms, but also the information about systems coupled with them, as well as about
the dissipative environment (Shytov et al., 2003; Yang et al., 2017).
Although we have given a detailed description of various systems with LZSM interference, we display the diversityof these, presenting several systems in Fig. 14. (Please note that the choice of works and figures for Fig. 14 are used
only for illustrative purposes, while many other works are analyzed below and throughout this review paper.) A detailed
description of Fig. 14 follows.
Extended explanation of Fig. 14:(a) Superconducting circuits: Although JJ-based quantum superconducting circuits are diverse, we would like to illustrate
the work on these systems with Ref. Berns et al. (2006). A superconducting flux qubit, also called a persistent-current
qubit, was realized as a niobium (but it could also be aluminum) superconducting loop with three JJs. The basis states
are created by a persistent current flowing either clockwise or in the counterwise direction. Strong driving by an AC
magnetic flux induces transitions between the two states. In Ref. Oliver et al. (2005), these qubit states were read out
with an outer DC-SQUID, a sensitive magnetometer that distinguishes the flux generated by the persistent currents. Up to
20-photon transitions were observed, with an impressive increase up to 45-photon transitions in Ref. Berns et al. (2006).
35

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 14. LZSM interferograms in different systems. See the extended explanation in the text for a detailed description. Figure (a) is reprinted from
Berns et al. (2006) with permission; copyright (2006) by APS. Figure (b) is reprinted from Forster et al. (2014) with permission; copyright (2014) by
APS. Figure (c) is reprinted from Childress and McIntyre (2010) with permission; copyright (2010) by APS. Figure (d) is reprinted from Heide et al.
(2018) with permission; copyright (2018) by APS. Figure (e) is reprinted from Mark et al. (2007b) with permission; copyright (2007) by APS. Figure(f) is reprinted from Seitner et al. (2016) with permission; copyright (2016) by APS.

36

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

The energy-level distance was controlled by a DC magnetic flux; the tunneling amplitude was rather small, ∆ ≃ 13 MHzh, and the driving frequency ω/2π = 0.27 GHz ≫ ∆/h.
(b) Quantum dots: These are based on electrons localized in semiconductor low-dimensional systems. Specifically, atwo-electron charge qubit, which is defined in a lateral double quantum dot, is considered in Ref. Forster et al. (2014).
The source and drain leads, tunnel-coupled with dots, allow current flow by single-electron tunneling. This current isused to detect the properties of the driven double quantum dot (DQD) device. The charge qubit is formed by the singlet
states S11 (one electron in each dot) and S20 (two electrons in the left dot). The source–drain current is proportional tothe occupation probability of S20 . The three triplets T11 are hindered by a Pauli-spin blockade; however, blockade is liftedusing an on-chip nanomagnet to quickly initialize the qubit. In the energy diagram, the singlets (the qubit states) arerepresented as black and red lines; the triplets, which are Zeeman split, are shown as gray lines. The interdot coupling is∆ ≃ 3.1 GHz h, and the energy detuning ε is defined by a DC+AC gate voltages ε (t) = ε + A cos ωt. The driving frequency
for the interferogram shown is ω/2π = 2.5 GHz, which is smaller than ∆/h.
(c) Atomic qubits (impurity or donor based): These may involve electrons, holes, and/or nuclear spins in impurities
within silicon, diamond, or other materials. For illustration purposes, we now consider an NV center in diamond, which
consists of a substitutional nitrogen atom replacing a carbon atom and neighboring one vacancy. Its electronic ground state
has a spin S = 1, which, in the absence of a magnetic field, has the spin singlet sublevel ms = 0 situated 2.87 GHz below
the ms = ±1 two-fold-degenerate triplet sublevels. The optically excited state has the zero-field splitting 1.42 GHz. Thesecan be further changed by the magnetic field. We illustrate this with an energy diagram displaying the excited-state level
anticrossing (ESLAC) and the interferogram from Ref. Childress and McIntyre (2010). The system was doubly driven by amicrowave and rf signals, resulting in characteristic dressed-state interference fringes. This was displayed by measuring
a spin-dependent fluorescence intensity, which is proportional to the excitation probability in the TLS formed by the
ms = 0 and ms = −1 states; on the color scale, darker indicates lower fluorescence. Versatile analysis, which included
impressively elaborated experimental, numerical, and analytical aspects, allowed the authors (Childress and McIntyre,2010) to describe the dressed-state interferometric picture with aspects such as multi-photon transitions and coherent
destruction of tunneling (CDT).
(d) Graphene structures: Electrons in graphene have a Dirac-cone dispersion, with a slope vF around the K and K′ points.
If electrons are driven by an electric field E = E0 ex of linearly x-polarized light with an amplitude of E0 , only thekx (t) component of the wave vector is affected, meaning that the relevant electron dispersion is a hyperbola (Higuchi
et al., 2017; Heide et al., 2018). This dispersion can be described with the TLS Hamiltonian: H = − [∆σx + ε (t)σz ] /2,
with ε (t) = 2h̄vF kx (t) = 2vF eω−1 E0 (t) (Heide et al., 2021b). The avoided-level spacing ∆ = 2h̄vF ky is continuousand can be swept with E0 for optimum splitting of the occupation probability. The central driving photon energy wash̄ω ≈ 1.55 eV, which is ω/2π ≈ 375 THz; vF ≈ 1 nm fs−1 and E0 from 0 to 3 V nm−1 (Heide et al., 2020, 2021a).
Hence, in a strong-field regime, the electron dynamics is governed by sub-optical-cycle LZSM interference, here on a
femtosecond timescale (Ishikawa, 2013). For this, a graphene strip on a SiC substrate was coupled with Au/Ti leads andilluminated by two-cycle laser pulses with a controlled carrier-envelope phase, in a vacuum chamber. Since the LZSM
interference takes place on a time scale where the coherence of the electron wave function is preserved, this experimentcan even be performed at room temperature. Adiabatic evolution and nonadiabatic transitions correspond to intrabandand interband processes for the electron wavefunction. The excitation probability of the upper (conduction) band resultsin the generation of a residual persistent current after the laser pulse ended. The right figure shows the conduction band
population imbalance ∆ρ with respect to kx = 0 (dashed line) generated with E0 = 2.5 V nm−1 , resulting in a net electron
flow to kx > 0. An increase of E0 leads to an inversion of the imbalance and hence, the current direction reverses, which
is a direct experimental indication for LZSM interference. The curved lines in the kx –ky plot indicate the multi-photon
resonances where the energy difference between the two bands is n · h̄ω, with n = 1 on the innermost circle and thesubsequent ring with n = 2.
(e) Ultracold atoms and molecules: When a bound molecular dimer state is magnetically tuned near a two-atom
scattering state, the Feshbach resonance leads to a resonant atom–molecule transition. The internal state structure ofultracold Feshbach molecules has several avoided crossings, which is considered in the⏐example
of weakly bound Cs2⟩
dimers (Mark et al., 2007b). The crossing used for the interferometer is the one between ⏐g ′ and |l⟩ states, which can be
determined upon molecular dissociation. Two subsequent passages through a weak avoided crossing between these twodifferent orbital angular momentum states result in LZSM interference, which is defined by the variable hold time τ . The
driving is realized by ramping the magnetic field B around the point of the avoided crossing, here at B = Bc . The acquired
phase difference is defined by the product of ∆E and τ . The interference is visualized in the g-wave molecular fractionof the state |g ⟩ as a function of the hold time τ . The oscillation frequency of the sinusoidal fit is about 100 kHz; and up
to 100 oscillations were observed.
(f) Mechanical oscillators: One classical system that behaves as a two-level system is a two-mode mechanical oscillator.
This system could be either two coupled mechanical oscillators or two modes of one mechanical oscillator. The lattercase is realized by Seitner et al. (2016) as two flexural modes of a string resonator. The modes represent in-plane andout-of-plane oscillations with an eigenfrequency of about 7 MHz. The resonator is made of a high-stress silicon nitride37

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

and has the cross-section of 100 × 270 nm, here with the length being 50 µm. The sweep voltage controls the distance
between the eigenfrequencies, with the minimal splitting ∆/2π of the order of 10 kHz. Triangular changes of the sweep
voltage result in transgressing the avoided crossing twice. Then, the interference results in the fringes shown as amplitudeof the transitions between the modes versus the sweep rate β −1 and amplitude Up .
4.1. Superconducting circuits
Flux qubits: This is presented above in relation to Fig. 14, with references to Oliver et al. (2005) and Berns et al. (2006).
Based on such interferometry, these authors also developed several techniques: amplitude spectroscopy (Berns et al.,2008), quantum phase tomography (Rudner et al., 2008), and pulse imaging (Bylander et al., 2009). These works weresummarized in Ref. Oliver and Valenzuela (2009) and scrutinized theoretically in Refs. Ferrón et al. (2010, 2012, 2016).
Other close observations were realized: an rf-SQUID-based flux qubit with the persistent-current states probed by the
nearby dc-SQUID magnetometer (Sun et al., 2009) and three-junction aluminum flux qubit probed by a niobium LC (tank)circuit (Il’ichev et al., 2004; Izmalkov et al., 2008). In this latter case, the sign-changing response was explained in termsof the impedance measurement technique (Il’ichev and Greenberg, 2007; Shevchenko et al., 2008; Shevchenko, 2008),
where the quantum inductance is related to the qubit occupation probability and its derivative. In a flux qubit, the roleof a weak link can be played not only by a conventional JJ, but also by a phase slip based on nanowires made from thinfilms of niobium nitride, for which the interferometry was observed in Ref. Neilinger et al. (2016).
Charge qubits: A superconducting charge qubit is based on a Cooper pair box that is created using JJs. In Ref. Sillanpää
et al. (2006), this was controlled by the magnetic flux piercing the loop and biased by the gate voltage; the state was readout by an LC circuit. The latter feels changes in the (quantum) capacitance, which probes the charge qubit state (Sillanpää
et al., 2005, 2007).
Alternatively, the qubit was probed via an inductive coupling to the resonant circuit (Tuorila et al., 2013). Two typesof multi-photon transitions in this system were further studied by Paila et al. (2009). A similar aluminum Cooper pairbox was probed via the reflection coefficient in a coupled electric resonator in Refs. Wilson et al. (2007) and Wilson et al.
(2010), where an emphasis was made on the dressed-state description, and in Refs. de Graaf et al. (2013) and Leppäkangas
et al. (2013), where the emphasis was made on quasiparticle tunneling.
An analogous qubit was also probed by a nanomechanical resonator (LaHaye et al., 2009), where a charge qubit state
influenced the displacement of the resonator, of which the resonant frequency was probed. The sign-changing behaviorof the response in that case was explained in Ref. Shevchenko et al. (2012a) in terms of the quantum capacitance.
Transmon qubits: This refers to a charge qubit shunted by a capacitance (Xiang et al., 2013; Krantz et al., 2019). This
results in a small sensitivity to the charge noise because of the alignment of the energy levels as a function of a gate
voltage, meaning there is no intrinsic avoided-level crossing. Because this is needed for LZSM interferometry, the solution
is to create dressed-states by applying a microwave signal; then, the slow driving would result in traversing these dressedstates around the avoided-level crossing. LZSM interference was realized by several groups with the following features:
driven by sinusoidal and noisy signals (Li et al., 2013), performing a time-resolved state tomography measurement (Gonget al., 2016b), resolving multi-photon processes (Chang et al., 2020) with variable coupling between the qubit and thetransmission line (Wen et al., 2020), and coupling to a mechanical resonator (Bera et al., 2021).
4.2. Semiconductor quantum dots
Quantum dots are based on an electron charge, spin, or even valley degree of freedom (for brevity, we refer to electrons;
however, sometimes, these could be holes). LZSM interferometry was realized and studied on different types of quantum
dots, as we describe below. Mainly, these are DQDs exploiting either an electron spin (namely, a singlet–triplet transition)or an electron charge (being placed in one of the quantum dots).
Spin qubits: A system of two electrons in a DQD can be in a singlet or triplet state, which form a singlet–triplet or spin
qubit (Jirovec et al., 2022; Burkard et al., 2021). The states are coupled via the hyperfine interaction between the trappedelectron spins and surrounding nuclear spins. The singlet and triplet states form an avoided-level crossing controlled bya detuning gate voltage and external magnetic field (which splits a triplet state into three states). This was realized inRefs. Petta et al. (2010) and Stehlik et al. (2012) in a triple quantum dot geometry, where one of the dots was used as ahigh-sensitivity quantum-point-contact charge sensor.
Related studies were performed in a linear triple-dot device in Refs. Gaudreau et al. (2011), Studenikin et al. (2012)
and Granger et al. (2015), realizing a spin qubit on either two three-spin states or three two-spin states; in a three-qubit
chain, the transitions in the avoided-crossing regions were used to shuttle the entanglement (Nakajima et al., 2018); ina DQD fabricated in a Si/SiGe heterostructure, here with an integrated micromagnet (Wu et al., 2014); and in a silicon
metal–oxide–semiconductor DQD with both single-spin addressability and single-shot readout (Fogarty et al., 2018). For atheoretical analysis of the dynamics in singlet–triplet transitions in DQDs, see Brataas and Rashba (2011), Särkkä and Harju(2011), Ribeiro et al. (2013b), Mehl and DiVincenzo (2013) and Zhao and Hu (2018b). For extreme harmonic generation ina multielectron double quantum dot, displaying multi-level interference resonances, see Stehlik et al. (2014) and Danonand Rudner (2014).
38

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Charge qubits: These can be based on a DQD that is laterally defined from a GaAs/AlGaAs-heterostructure double
quantum dot with the voltage pulses applied to depletion gates (Petersson et al., 2010; Nalbach et al., 2013). There,
the two states of a charge qubit correspond to one electron in the left and right dots; the charge occupation of thedots can be continuously detected via the electric current through a capacitively coupled quantum point contact. With asimilar layout, the authors of Ref. Cao et al. (2013) used LZSM interference for ultrafast universal quantum control at thepicosecond scale, which is important given that the decoherence times in semiconductor quantum dots are typically lessthan a few nanoseconds.
A similarly driven DQD was realized in Ref. Braakman et al. (2013) in an array of three dots to demonstrate coherent
coupling via long distance, which involved two three-dot states. The charge qubit in Ref. Forster et al. (2014) was probed
through the DC source–drain current, as discussed above. The effect of the pulse shape and distortion was studied inRef. Ota et al. (2018). Further development is a coupling of charge qubits: two DQDs were strongly coupled capacitively
by the authors of Ref. Ward et al. (2016), where the LZSM interference in one DQD (right one) was shown to be defined by
the charge state of the other DQD (left one), which was called charge-state-conditional coherent quantum interference.
Charge qubits can be defined not only on a two-dimensional gas, but also based on a nanowire. In Ref. Stehlik et al.
(2014), this was an InAs nanowire that was placed on top of prepatterned depletion gates; the authors studied strong
harmonic generation involving up to eight photons for qubit excitation. In the circuit quantum electrodynamics (cQED)
architecture, such DQD can be placed in a microwave cavity (with a resonant frequency in a GHz domain), where driving
the qubit results in interferometric cavity power gain (Stehlik et al., 2016a).
Using silicon allows for exploiting the advances of complementary metal–oxide–semiconductor (CMOS) technology, as
in Ref. Gonzalez-Zalba et al. (2016), in which the DQD was realized on the corner states in a silicon nanowire transistor.
There, an electrical resonator response probed the differential capacitance of a DQD, which contained two competing
terms: one, called the quantum capacitance, is proportional to the occupation probability, and the other one, which is thetunneling capacitance, is proportional to the derivative of the occupation probability; in resonance, the former gives theLorentzian peak and the latter the alternation of a peak and a dip. Because of the quantum capacitance, similar peak-anddip features were observed on a DQD charge qubit based on a carbon nanotube (Penfold-Fitch et al., 2017). The authorsof Ref. Khivrich and Ilani (2020) constructed a TLS in a carbon nanotube single quantum dot from electron wavefunctions
with different magnetic moments and spatial charge distributions.
4.3. Donors and impurities (atomic qubits)
The electrically controlled electronic states in some donors and impurities have much in common with quantum dots,
both conceptually and in terms of qubit specifications and hardware requirements. Similar to quantum dots, a donor
atom has one excess electron, here compared with the atoms in the surrounding lattice (less frequently in this context,
these could be acceptors with single excess holes). It is the electron spin states that form the spin qubit basis. These spin
qubits have many key assets that are well separated from the noisy environment and that use nuclear spins in addition
to electronic ones (Vandersypen et al., 2017). Good isolation from the environment is characterized by long coherence
times (∼1 ms for electron spins and ∼1 s for nuclear spins). Similar to the gate-defined quantum dots above, quantum
information can be encoded in the charge states of an electron in a double-well potential of two nearby impurities (acharge qubit) or in a spin state of an electron or nuclear spin in a single impurity (spin qubit). We consider several works,
where LZSM interference was realized.
Our first example is phosphorous (P) or arsenium (As) donor atoms in silicon (Si). The authors of Refs. Dupont-Ferrier
et al. (2013) and Jehl et al. (2015) studied a system of two tunnel-coupled donor atoms in silicon. A TLS was formed by asingle electron located on either of the two donors. The donors were implanted in a silicon nanowire, and the coherent
interference pattern was observed by measuring the source–drain current through the double-donor system. In Ref. Ono
et al. (2019), a combination of a shallow impurity and a deep impurity in a silicon field-effect transistor created a DQD-likestructure controlled by a magnetic field and rf and mw signals, of which the layout allows quantum coherent dynamics
at temperatures above one Kelvin.
In Ref. Miao et al. (2019), the authors demonstrated electrically driven coherent quantum interference in the opticaltransition of a divacancy in commercially available 4H silicon carbide (SiC). The divacancy consisted of a carbon vacancy
adjacent to a silicon vacancy; the TLS was formed by the ground and excited states of electronic orbital levels in thedivacancy; the monochromatic and bichromatic LZSM interference patterns involved up to 15-photon excitations.
The NV centers in diamond have a nearly spinless environment, leading to extremely long electronic and nuclear spin
coherence times. In Ref. Childress and McIntyre (2010), the authors simultaneously excited both the electronic transition
by weak microwaves and nuclear spin resonance by strong rfs, while the experimentally observable fluorescence wasdominated by the response of the electronic spin qubit. In Ref. Scheuer et al. (2017), the electron spin polarization wastransferred from a single NV center to the surrounding carbon nuclear spins; the polarization dynamics of the nuclear
spin ensemble displayed the LZSM oscillations.
In Ref. Huang et al. (2011), the authors considered the doubly driven system of the two states, ms = 0 and ms = +1,
coupled with a nearby 14 N nuclear spin. First, a microwave field matched those two bare energy levels and created the
dressed-states with the avoided-level distance defined by the microwave amplitude. Second, the repeated passage aroundthe avoided-level crossing was made by applying rectangular rf pulses. Hence, these demonstrated the quantum coherent
control of electronic single spins at room temperature.
39

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

4.4. Graphene
Nonadiabatic transitions between Dirac cones in the vicinity of the Dirac points take place if these display avoided-levelcrossing and are driven appropriately. This was realized in graphene by illuminating it with a strong linearly polarized
light field (Higuchi et al., 2017; Heide et al., 2018; Li et al., 2021). This was described above in relation to Fig. 14, here
in terms of the one-dimensional trajectories of electrons in reciprocal space; see also Refs. Syzranov et al. (2013) and
Lefebvre et al. (2018). If the light becomes circularly polarized, the intraoptical-cycle LZSM interference cannot occur andthere is no respective change in the current. The intermediate situation takes place for laser pulses with various degrees
of ellipticity (Heide et al., 2018, 2019, 2020; Boolakee et al., 2020).
The theory of nonadiabatic transitions in the vicinity of a conical intersection and related interference were addressedin several works for graphene (Rozhkov et al., 2016) and other related materials (Montambaux, 2018; Li et al., 2021; Wu
et al., 2020; Huang et al., 2021; Heide et al., 2021a). Here, we note that the Dirac cones and Dirac points are important
to many objects, including massless electrons in graphene and in conducting edge states in topological insulators. In
particular, transitions that happen between two massive Dirac cones were shown to create an interferometer, revealing
information on the band eigenstates, such as the chirality and mass sign (Fuchs et al., 2012; Lim et al., 2014). Similar
descriptions of the driving near the Dirac points are applicable to various systems, such as Bloch oscillations of ultracold
atoms in a honeycomb optical potential (Xu, 2014), surface states of 3D topological insulators (Lim et al., 2012), ultrafast
transfer of excitation energy in photoactive molecules (Nalbach et al., 2016), and electron–hole pair production in anelectric field (Fillion-Gourdeau et al., 2016; Taya et al., 2021).

4.5. Microscopic systems
In microscopic systems, the energy levels often experience avoided crossing, which is relevant here. In particular,
ultracold atomic and molecular quantum gases have an extraordinary degree of control (Chin et al., 2010). Ultracold gases,
which are produced by laser cooling and trapping techniques, have characteristic temperatures in the (sub-)µK range
and typical densities below 1015 cm−3 . Under these conditions, the atomic de Broglie wavelength exceeds the averageinteratomic distance, resulting in the quantum-degenerate state of a BEC. In such systems, when the bound molecular stateenergetically approaches the scattering state, two colliding atoms couple with a molecular bound state, a phenomenon
known as Feshbach resonance.
Several phenomena associated with the Feshbach resonance, such as molecular association and dissociation, areconveniently described in terms of the LZSM model (Köhler et al., 2006). The energy levels of the molecular states(produced by atoms in the BEC state) around the Feshbach resonance are often manipulated using time-dependent
magnetic fields. Tunneling at the avoided-level crossing leads to the formation of stable dimer molecules (Mies et al.,
2000; Góral et al., 2004). Using the repetitive passage of the avoided-level crossing allows for a high degree of control
over the interferometric dynamics, which was demonstrated on caesium ultracold Feshbach molecules confined in a CO2 laser trap (Mark et al., 2007b,a) and presented above, here in relation to Fig. 14. Such a BEC-based tunable “beam splitter”
allows for controllably transferring the molecular states. The driving of ultracold atoms was studied theoretically (Li andFu, 2020; He and Li, 2020).
The dynamics of ultracold atoms in an optical lattice can be described in terms of LZSM transitions between Bloch bandsof the optical lattice (Kling et al., 2010). The authors loaded an atomic rubidium BEC into a regular optical lattice so thatthe atoms were transferred into the lowest energy band; then, the lattice was accelerated, which was equivalent to theapplication of an external force, resulting in interferometric tunneling between the two bands. Analogous Stückelberg
oscillations were studied in Ref. Zenesini et al. (2010), where the respective double-passage model was used for a
description of the experiment. These systems can also be used for multipassage interferometry (Plötz and Wimberger,
2011; Liang et al., 2020; Béguin et al., 2022).
Among other microscopic systems, Rydberg atoms play a special role because they allow for early observation of
the Bessel-function modulated multi-photon transitions and Stückelberg oscillations. This was studied by Baruch andGallagher (1992), where a thermal beam of K atoms passed through a microwave cavity, to which a static electric field
was applied for tuning and a pulse for field ionization by two laser beams. In the same year, Yoakum et al. (1992) studied
a monoenergetic beam of helium atoms in the Rydberg state and drove them using a microwave electric field, with
nice agreement between experimental Stückelberg oscillations and a numerical solution of the Schrödinger equation.
The Stückelberg oscillations were observed in the dipole–dipole interactions between Rydberg atoms and describedtheoretically within the Floquet approach in Ref. van Ditzhuijzen et al. (2009).
LZSM interferometry was also realized with single ions and free radicals. A single 171 Yb+ ion was trapped in a fourrod radio-frequency trap and driven by microwaves by Zhang et al. (2014). The molecular free radical 138 BaF, with one
unpaired electron in the ground state, was Zeeman tuned and electrically driven by Cahn et al. (2014).
40

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 15. Three-level systems. Diabatic energy levels of states |i⟩ (thin black lines) intersect in one (a), two (b), or three (c) points. Eigenenergies ofthe full Hamiltonian H(t), as shown with thick color lines, exhibit avoided-level crossing. Note that in (c) there are two possible trajectories to getfrom |1⟩ to the left into |3⟩ to the right, which implies interference during the single-passage process.

4.6. Other systems
For classical systems, mechanical resonators included, we devote a separate section. This is because to the best of ourknowledge, no review on this important subject exists. Here, we would rather mention several other possible realizationsof LZSM interferograms.
In principle, any TLS subjected to a periodic driving can display LZSM interference. However, there are limitations;in particular, the system needs to preserve coherence long enough. This may take place in superconducting weak links,
where the Andreev levels as a function of time display avoided-level crossing (Gorelik et al., 1998; Mélin et al., 2019; Mélin
and Douçot, 2020; Douçot et al., 2020; Oriekhov et al., 2021); there, it was pointed out that the microwave-induced LZSM
interference should be revealed as a significant source of energy absorption; for a mini review, see Roy (2013). Similar
studies of coherent interference effects were done for Schwinger vacuum pair production (Akkermans and Dunne, 2012),
many body localized systems (Gopalakrishnan et al., 2016), and photoisomerization reaction (Duan et al., 2016).
If in place of energy levels we have energy bands, that is, Bloch bands, we can call these Bloch–LZSM transitions orinterference. As in the other cases, in the literature we can meet shorter names such as Bloch–(Landau–)Zener transitions /oscillations / interference. In the case of the Bloch bands, these effects also include Bloch oscillations, which are theperiodic motion of particles in a superlattice subjected to a constant external field. These take place in Bose–Einsteincondensates in tilted and driven optical superlattices (Witthaut et al., 2011; Reid et al., 2016), in graphene and topologicalinsulators (Krueckl and Richter, 2012; Sun et al., 2018), in photonic Lieb lattices (Long and Ren, 2017), for trappedions (Gagge and Larson, 2018), and for the high-order harmonic generation in solids (Jin et al., 2018), and in moiré systems
under a uniform magnetic field (Paul et al., 2022).
4.7. Multilevel systems
Having considered two-level systems, it is important to extend the study to possible multilevel systems, with multiple
avoided-level crossings. These may relate to diverse systems such as a chain of interacting spins-1/2 (Ostrovsky andVolkov, 2006; Larson, 2014; Grimaudo et al., 2022), a spin or multiple spins coupled to a resonator (O’Keeffe et al., 2013;
Singh et al., 2020), molecular nanomagnets (Földi et al., 2007; Garanin et al., 2008; Pavlyukh, 2020), interacting bosons
in optical lattices (Tschischik et al., 2012), broad periodic photonic waveguide (Benisty et al., 2011), or NV center coupledwith a substitutional nitrogen center in diamond (Zangara et al., 2019; Band and Japha, 2022), to name a few.
Let us start here from the simplest realization of a multilevel system: the three-level one. If the system diabatic levels|i⟩ depend linearly on the energy bias ε (t) with possible offset between levels h, then the respective diabatic energy levels
can be drawn as straight lines, as in Fig. 15, with one, two, or three intersections; here, h = 0 is used for the case witha single intersection and h ̸ = 0 for the other cases. Taking into account possible tunneling, proportional to ∆, lifts thedegeneracies and results in avoided-level crossings. We can describe this with the Hamiltonian
H(t) = −

ε (t)
2

Â −

h
2

B̂ −

∆
2

Ĉ ,

(94)

with Â, B̂, and Ĉ being 3 × 3 matrices; these can be expanded with Gell–Mann matrices (Kiselev et al., 2013). Then, taking
these matrices with the parameters, say, close to the ones of Ref. Ashhab (2016), we display in Fig. 15 several situations: (a)the “bow-tie” model, when all three levels (quasi-)intersect at one point, (b) the “equal-slope” model with the (avoided)crossing of the three levels in two points, and (c) the “triangle” model with three pairwise (quasi-) intersections. The mostimportant aspect that distinguishes (c) from (a) and (b) is the possibility of interference during a single-passage dynamics.
For example, there are two possible trajectories that lead from |1⟩ to the left into |3⟩ (or alternatively to |2⟩) to the right41

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

in Fig. 15(c). This means that the final occupation probability is essentially defined by the phase difference between the
two trajectories. Note that a three-level system can be described with a spin-1 Hamiltonian, so all theory for both singleand multiple-passage problems can be extended from two-level systems to three-level ones (Zhang et al., 2011).
Let us now consider previous work done for driven multilevel systems, here separating the subject into three topics.
4.7.1. Theory
From the above example of a three-level system, we can formulate a generalized LZSM problem for a multilevel system,
for example, as follows: what is the occupation probability of a state |j⟩ if starting from a state |i⟩? More generally, theevolution can be described by the matrix S linking the final vector-state to an initial one (Shytov, 2004; Suzuki and
Nakazato, 2022). Here, as for the evolution of a TLS, it is instructive and most common to consider either a single-passage
problem or periodic evolution with multiple passages. The “bow-tie” model describes a quasi-intersection of N energy
levels (Ostrovsky and Nakamura, 1997), while the “equal-slope” model is the (quasi-) intersection of the energy band, herewith N − 1 levels, by one more level (Demkov and Osherov, 1968). The single passage for a “bow-tie” and “equal-slope”
models is described by sequential pairwise level crossings (Brundobler and Elser, 1993). Magnetic sub-levels of two atomic
levels with nonzero total angular momenta can be described by degenerate LZSM model, which involves two crossing setsof degenerate energy levels with quasi-crossing between them (Vasilev et al., 2007).
As for a three-level case, for an N-level system, we can also have an extended “triangular” model, with several
sequential crossings making trajectories meet (Demkov and Ostrovsky, 1995; Fai et al., 2015). In this case, even for asingle-passage evolution, we can take possible interference into account (Ostrovsky et al., 2007; Sinitsyn, 2015; Malla
et al., 2021).
The description of the S-matrix approach for single-passage problems is reviewed in Sinitsyn and Chernyak (2017). As
an extension of this, we can interpret the method of solving rate equations, which we consider in a separate subsection.
This method is applicable to periodically driven systems, as shown by Wen et al. (2010), for the example of multilevel
devices based on superconducting circuits (Berns et al., 2008; Oliver and Valenzuela, 2009). Importantly, this latterapproach takes a dissipative environment into account. An intuitive picture of open multilevel systems can be obtainedby modeling the external environment by a harmonic oscillator (Ashhab, 2016). In the case of a weak interaction between
N spins- 12 , the LZSM transitions can be described by a mean-field approach (Garanin, 2003).
4.7.2. Single passage in multilevel systems
Although a three-level system is useful as a prototypical model for introducing the features of multilevel systems, more
realistic and widely used are those models with four levels. These describe coupled two-level systems or genuine fourlevel systems, such as double quantum dots with two electrons. The single passage problem with the four-level model wasaddressed, for instance, for two antiferromagnetically coupled tunneling systems, such as Mn4 dimers (Garanin, 2004),
Rydberg atoms (Mallavarapu et al., 2021; Niranjan et al., 2020), and quantum dots (Sinitsyn, 2002; Reynoso and Flensberg,
2012; Krzywda and Cywiński, 2020). Not only qubits, but also qutrits, can be coupled, as in Refs. Grimaudo et al. (2019,2020), which demonstrates that such coupling is useful for entanglement control.
4.7.3. Multiple-passage transitions in multilevel systems
Periodically driven three-level systems may describe a qubit coupled with two microscopic systems (Sun et al., 2010)or a superconducting charge qubit that is taking a higher energy level into consideration (Parafilo and Kiselev, 2018a). A
three-level model for a driven system is convenient for describing the impact of the external noise (Kenmoe et al., 2013,2015; Kenmoe and Fai, 2016; Li and Cen, 2018; Band and Avishai, 2019). Periodically driven four-level systems are usefulfor describing the charge states of two particles in DQDs, as realized in Shi et al. (2014), Chatterjee et al. (2018) and Mi
et al. (2018) and described in Pasek et al. (2018), Shevchenko et al. (2018), Zhao and Hu (2022) and Zhou et al. (2021).
A five-level model describes singlet–triplet states in spin-based DQDs (Stehlik et al., 2016b; Chen et al., 2017; Qi et al.,2017), which was also addressed by Zhao and Hu (2018a), Karami et al. (2019) and Ginzel et al. (2020).
As a further development, triple quantum dots were also studied (Aers et al., 2012; Poulin-Lamarre et al., 2015; GallegoMarcos et al., 2016; Łuczak and Bułka, 2016). These systems present more flexibility and controllability than DQDs while
representing a minimal model for a chain of quantum dots, which can be considered a highly controllable quantum
metamaterial (Rakhmanov et al., 2008; Smirnov et al., 2007). The interference effects, here appearing in periodically driven
coupled qubits, were extensively studied by Denisenko et al. (2010), Satanin et al. (2012), (Gramajo et al., 2017, 2018,2021), and Munyaev and Bastrakova (2021).
5. Quantum control
To summarize what we discussed above, LZSM transitions act as a tool to control a TLS’s occupation; that is, to control
single-qubit states and the states of multi-level systems. As our system is analogous to a two-slit layout, a single-passage
process presents a beam splitter for the energy-level occupation probabilities where the ratio of splitting can be controlled.
Besides creating the coherent superposition, repeated passage of the avoided-level crossing gives the option to control
the quantum system energy-level occupation probabilities by varying the accumulated dynamical phase.
42

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Having obtained information about the quantum system, we can go further by studying opportunities for its control.
In addition to other trends such as quantum simulations (Georgescu et al., 2014; Buluta and Nori, 2009) and quantum
sensing (Degen et al., 2017), coherent quantum control can be viewed as another direction in quantum engineering. Since
it is important for applications to have individual control of single quantum systems, we will present here studies of LZSM
non-adiabatic transitions and interference for the quantum control of few-level natural and artificial quantum systems.
5.1. Coherent control of microscopic and mesoscopic structures
The 2012 Nobel Prize in Physics was awarded for “experimental methods that enable measuring and manipulationof individual quantum systems”. To clarify, these systems were microscopic and consisted of atoms and photons. Morerecently, the variety of related quantum systems increased and also includes mesoscopic systems behaving as very large
atoms (Buluta et al., 2011). To demonstrate this, we present the following variety of systems and approaches where theunderlying effective Hamiltonian can be readily modified or engineered and their coherent quantum control was studied
in the frame of LZSM physics.
* In superconducting structures, the properties of non-adiabatic LZSM transitions can be used to control superposition
states in single qubits (Salmilehto and Möttönen, 2011) and the entanglement in coupled-qubit systems (Quintana et al.,2013). Several techniques were developed:• amplitude spectroscopy (Berns et al., 2008), which is applicable to multi-level systems, where increasing the driving
amplitude results in reaching more avoided-level crossings for a single driving frequency that may be orders ofmagnitude smaller than the energy scales being probed;• quantum phase tomography (Rudner et al., 2008), which is based on the Fourier transforms of the interferograms thatprovide additional visibility and information;• pulse imaging (Bylander et al., 2009), where the measured interferograms are used to image the actual wave formof the driving signal at the device, which in turn can be used to engineer the desired time evolution of a quantum
system. These and other similar methods provide means to characterize and manipulate states of superconducting
and other quantum systems.
* Quantum dots, which can be charge or spin qubits, can be controlled. The hyperfine interactions can be harnessed forquantum control in GaAs semiconductor quantum dots (Ribeiro et al., 2010). The visibility of quantum oscillations can beincreased by enhancing the adiabatic passage probability. For this purpose the researchers designed tailored pulses thatcombined both fast and slow rise-time ramps to minimize dissipation and enhance adiabaticity (Ribeiro et al., 2013a).
* A qubit can be coupled to a bath of TLSs describing noise in quantum circuits. Properly driving such TLSs can help
to dynamically decouple such systems, thus reducing dielectric losses. This was demonstrated with TLSs in depositedaluminum oxide by using it as the dielectric in a lumped-element LC -resonator (Matityahu et al., 2019). Using LZSM
interference in quantum dynamics can help to isolate a single particle (in a quantum dot) from a Fermi sea by closing
a tunnel barrier (Kashcheyevs and Timoshenko, 2012). The sensitivity of LZSM transitions to the parameters of thedissipative environment makes them useful for gauging a quantum heat bath (Wubs et al., 2006). Using repetitivetransitions with underlying interference allows accurately obtaining all relevant information about complex influences.
This was observed in Ref. Forster et al. (2014) on the example of a two-electron charge qubit defined in a lateral double
quantum dot.
* In composite or hybrid systems, non-adiabatic transitions can bring a system from one state to a desired state thatmight be entangled (Li et al., 2011). For a system of qubits, the transfer state between qubits was studied with a photonic
qubit (localized exciton in an optically active quantum dot) coupled to a spin qubit hosted in gate-defined quantum
dots (Joecker et al., 2019), with a composite system of Majorana-hosted semiconductor nanowire and superconducting
flux qubits (Zhang and Yu, 2013), and with three superconducting qubits (Li et al., 2019). For a hybrid system consistingof a qubit coupled to a photonic cavity, the transition results in the entanglement of the system and, the generation of
Schrödinger cat states in the photonic cavity (Lidal and Danon, 2020).
* Adiabatic passages and nonadiabatic transitions in a macroscopic, cylindrically shaped water sample were studied
using standard nuclear magnetic resonance (NMR) techniques with controlled linear ramping of the external magnetic
field (Hürlimann et al., 2020) . The pulse sequence used in this experiment was the standard in NMR applications: theCarr–Purcell–Meiboom–Gill sequence.
* The similarity between driven TLS dynamics and other processes can provide a basis for simulating these phenomena with a highly controllable TLS. The emulation of mesoscopic phenomena, such as weak localization and universal
conductance fluctuations, was demonstrated on a system of superconducting transmon qubits by Gramajo et al. (2020).
* Using the acquired phase of the wave function during single and repeated non-adiabatic transitions provides the basis
for optimal control theory (Zhdanov and Seideman, 2015). Depending on the purpose (e.g., to obtain complete populationtransfer, which we will also address later), the driving signal can be tailored (Larocca et al., 2018). The efficiency can bequantified with the methods of quantum metrology, and calculations of quantum Fisher information (Yang et al., 2017).
* A driven TLS provides a testbed quantum thermodynamics (Ono et al., 2020), where the non-adiabatic transitions
serve to change states between different regimes. This was studied in Otto heat engines (Halpern et al., 2019; Son et al.,2021), and in calorimetric methods and measurements of work (Solinas et al., 2013; Pekola et al., 2013).
The following three subsections present specific chapters in the quantum control with LZSM transitions.
43

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 16. LZSM gates. Dynamics of σ -matrix coefficients σx,y,z , which define the position on the Bloch sphere, under ε (t) = A cos ωt driving. (a) NOT
gate, which is the transfer |E− ⟩ → |E+ ⟩, here as performed by two consecutive LZSM transitions. For this, we assume the constructive-resonancecondition ζ + φS = 2π n (this defines the amplitude A) and take the transition probability P = 0.5. Specifically, we found A = 3.8872∆ and thefrequency was defined from ω = −π ∆2 [2h̄A ln P ]−1 . (b) Hadamard gate performed with the following parameters: P = 0.5, A = 4.309∆, which isdefined from the condition for destructive interference ζ + φS = 32π + 2π n.

⟨⟩5.2. Universal single- and two-qubit control
Most often, quantum control (logic gates including) is associated with resonant driving that then results in Rabi
oscillations (Kwon et al., 2021; Krantz et al., 2019). However, under resonant driving, certain limitations may arise,
including achievable gate speed and nonidealities, such as counter-rotating terms (Vitanov et al., 2003; Saito et al., 2006;
Longhi et al., 2019; Campbell et al., 2020). Coherent LZSM interference allows for a complementary approach to quantum
control based on nonresonant driving with the alternation of adiabatic evolution and nonadiabatic transitions. This enables
ultrafast qubit gates that are controlled solely using baseband pulses, hence alleviating the need for pulsed-microwave
control signals (Cao et al., 2013; Ota et al., 2018; Campbell et al., 2020; Zhang et al., 2021; Li et al., 2022).
Qubit evolution can be conveniently described by trajectories on the Bloch sphere. For this, the evolution matrices inthe transfer matrix (TM) approach should be rewritten using the Euler-angle decomposition (Sillanpää et al., 2006, 2007).
First, to describe the nonadiabatic transition, we rewrite the N matrix from Eq. (30) in the form(N =cos(θ /2)e−iφS
sin(θ /2)

− sin(θ/2)
cos(θ/2)eiφS)= Uz (φS ) Ux (θ) Uz (φS ) ,(95)

where sin2 (θ /2) = P and the matrices Ux,y,z (α ) = exp −iασx,y,z /2 describe their respective rotations. The adiabatic
evolution is characterized with the z-rotation U(t , ti ) = exp (−iζ σz ) in Eq. (27). Then, the single-passage evolution is
described by UN, Eq. (34). The repetition of these can correspond to a one-period evolution (with doubly passing the
avoided crossing), Ξ = N ⊤ U2 NU1 , Eq. (46), or to an n-cycle evolution with Ξ n in Eq. (51a). After writing this, we canequate the evolution matrix with the matrix for the Euler-angle decomposition U(α, β, γ ) = Uz (γ )Uy (β )Ux (α ) (Landau
and Lifshitz, 1965); as a result, we obtain the formulas for the designated rotation angles α , β , γ . With this, we can obtaina desired protocol and the parameters for single-qubit operations (Campbell et al., 2020). Here, the important claim is thatwith the pure LZSM interference technique, we can obtain any rotation and, correspondingly, the efficient single-qubit
and two-qubit control (Ribeiro et al., 2010). This approach allows for defining the parameters for a desired evolution ora unitary logic gate and for analyzing the resonance conditions (Ashhab et al., 2007).
As an illustrative example, consider first driving a TLS with pulses that result in a double passage of the avoided
crossing (Cao et al., 2013). This is described by the matrix Ξ = N ⊤ U2 NU1 , here with the alternation of adiabatic evolution(z-rotation) and nonadiabatic transition with the N rotation matrix. We note that the rotation angles, both θ and φS , aredefined by the adiabaticity parameter δ and, consequently, by the passage velocity v . Hence, adjusting the pulse shape(which defines this velocity v ), one can obtain any rotation and respective logic gate (Cao et al., 2013). For a fixed velocity,
the evolution is limited between the two trajectories defined by destructive and constructive interference (Ota et al., 2018).
In Fig. 16(a) we demonstrate the NOT operation, which is the rotation around the x axis by the angle π . For this,we consider driving with a cosine pulse and choose the parameters to perform the NOT operation. Namely, we use

(

)

44

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

P = 0.5 and the total phase changing during one LZSM transition φS + ζ = 2π n to have the constructive interference, see
Eqs. (28), (9). As a result, one can perform a fast and robust NOT operation, which in Fig. 16(a) illustrates the transition
from the ground state (logical 0) to the excited state (logical 1): |E− ⟩ → |E+ ⟩.
Next, consider the Hadamard gate. As above, this can be implemented with the double-passage process. However,with the chosen value P = 0.5, we can do this faster, here with a single-passage transition, which occurs during half theperiod. We illustrate this in Fig. 16(b). For performing the Hadamard√ gate, we use a single LZSM transition, with the total
phase changing φS + ζ = 3π/2 + 2π n, resulting in {|E− ⟩ + |E+ ⟩} / 2 → |E− ⟩. Hence, we have demonstrated two gates,
and this can be supplemented by, for example, the phase gate to obtain the universal set of single-qubit operations. The
phase gate can be realized by Uz (ζ ) = exp (−iζ σz ), which is idling (in the absence of driving).
In addition to single-qubit operations, two-qubit operations can also be done with LZSM transitions and interference.
Two-qubit operations are conveniently demonstrated by the conditional phase gate or controlled-Z (CZ) gate between two
qubits; four-level dynamics was studied by Rol et al. (2019) and Campbell et al. (2020) for superconducting qubits andby Huang et al. (2018) for Rydberg atoms. Similarly, another two-qubit operation—the controlled-NOT (CNOT) operation—
can be realized (Ribeiro et al., 2010; Wu et al., 2021). Importantly, this approach allows for fast execution of the two-qubit
operations with high fidelity. Provided one has single- and two-qubit operations, the next step could be the realizationof algorithms.
Destructive interference can be harnessed to perform unitary transformations based on cyclic evolutions, which refer tothe process where the state of the system returns to its original state after gate operations (Wang et al., 2016). As a resultof this evolution with destructive interference (i.e. coherent destruction of tunneling [CDT]), the state vector accumulates
a phase only. This phase acquired in the evolution contains both dynamic and geometric components; see the detailed
and pedagogical explanations in the supplementary material of Wang et al. (2016), and for relations to the Berry phase,see Lim et al. (2015a,b) and Bleu et al. (2018). The geometric phase can be related to the solid angle subtended by theevolution curve on the Bloch sphere. This analysis opens up new ways of interpretation and control; this is called geometric
LZSM interferometry (Gasparinetti et al., 2011). Because we can perform any logic operations using this, it provides the
instruments for geometric quantum computation. This is a promising approach to achieve robust control of a quantum
system (Tan et al., 2014; Zhuang et al., 2022; Barnes et al., 2022).

5.3. Shortcuts to adiabaticity
Two major aims of quantum control can be defined as either reaching a given target state or tracking the instantaneous
ground state of a system during its evolution (Bason et al., 2012). For each task, an optimum strategy can be designed
based on requirements such as the highest possible fidelity, fastest possible operation, and so forth.
For illustration, the authors of Ref. Bason et al. (2012) considered both a shortcut protocol that reaches the maximum
speed compatible with the Heisenberg uncertainty principle and the opposite limit of nearly perfectly following the
instantaneous adiabatic ground state. For the LZSM-form Hamiltonian H(t) = [−∆(t)σx − ε (t)σz ] /2, the authors designed
a protocol that drives the system through avoided crossing in such a way that at the end of evolution, the final state|ψfin ⟩ is as close as possible to the adiabatic ground state |E− (tfin )⟩, which here aims to reach a unit’s final fidelity|⟨ ψfin | E− (tfin )⟩|2 = 1. In general, there are infinitely numerous paths in Hilbert space connecting the initial state, forexample, |ψini ⟩ = |E− (tini )⟩ with the final one |E− (tfin )⟩. First, fixing ∆(t) = const, minimizing the evolution time results
in a problem known as the shortcut to adiabaticity (STA) (Guéry-Odelin et al., 2019; Chen et al., 2021b) or also by analogy
with the equivalent classical case, ‘‘quantum brachistochrone’’ (Oh et al., 2016). Here, the minimum time for reaching thetarget state is known as the quantum speed limit time:
Tqsl = 2 arccos |⟨ ψfin | ψini ⟩| /∆.

(96)

In the opposite formulation, aiming at perfect following the ground state |E− (t)⟩ is known as the counterdiabatic
protocol (Demirplak and Rice, 2003) or superadiabatic protocol (Berry, 2009). Details of this transitionless adiabatic
protocol are considered below.
As mentioned, there are two major routes for manipulating the state of a quantum system with interacting fields:
either using resonant pulses or adiabatic methods (Chen et al., 2010). The most popular methods for efficient populationtransfer are the adiabatic rapid passage (ARP; sometimes RAP), chirped-pulse excitation or Stark-chirped RAP (SCRAP),
and stimulated Raman adiabatic passage (STIRAP) (Vitanov et al., 2001; Goswami, 2003; Wei et al., 2008). The latter
technique permits the precise control of population transfer using partially overlapping pulses, from pump and Stokes
lasers (Bergmann et al., 1998). In its simplest form, for a three-level system, the pump pulse links the initial state |1⟩ withan intermediate state |2⟩, which, in turn, interacts via a Stokes (or dump) pulse with a target state |3⟩; the aim of thispump–dump technique is to achieve complete transfer of a population between states |1⟩ and |3⟩.
The STIRAP technique can be interpreted as pairs of LZSM transitions in the areas of avoided crossings of quasienergies (Yatsenko et al., 1998; Malinovsky and Krause, 2001). If the avoided crossing is passed repeatedly, the interference
has to be taken into account (Vitanov, 2018). This was also analyzed by Zhang and Liu (2019), in which the nonadiabatic
transitions were quantified by the upper-bound function, hence providing the optimal criteria for adiabatic control. In
relation to our previous discussion on the interrelation between LZSM interference and multi-photon processes, coherent
population transfer can also be described as multi-photon ARP (Maeda et al., 2006). The ARP strategy can be used for a45

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

high-quality source of single photons that are emitted because of spontaneous emission from the excited state, which isrobust against control errors and environmental fluctuations (Miao and Zheng, 2016b).
To understand the idea of counter-diabatic driving, consider this, following Demirplak and Rice (2003) and Theisen
et al. (2017): let our arbitrary n-level system be described by the Hamiltonian H(t), and let U(t) be a time-dependent
unitary transformation that makes the matrix U(t)H(t)U † (t) diagonal; it consists of the eigenvectors of H(t). Consider
now the full Hamiltonian H̃(t) = H(t) + H CD (t). Then, retaining the definition of U(t), consider the new Hamiltonian after
such transformation
H̃ ′ = U H̃U † − ih̄U U̇ † = UHU † + UHCD U † − ih̄U U̇ † .

)

(

(97)

†

Because a priori, UHU is diagonal, then the condition
HCD = ih̄U̇ † U(98)

guarantees that at all times, the population, which was placed initially in one of the adiabatic states, remains exactly inthat particular state.
Alternatively, the counter-diabatic Hamiltonian HCD (t) can be written following Berry (2009) in terms of the instantaneous eigenvalues En (t) and eigenstates |ψn (t)⟩ of the system Hamiltonian H(t)HCD = ih̄∑ ∑ |ψm ⟩ ⟨ψm | Ḣ |ψn ⟩ ⟨ψn |m̸ =n

En − Em

n

.

(99)

Consider this for a TLS with a generic Hamiltonian H(t) = − [∆(t)σx + ε (t)σz ] /2. This is written in the diabatic (timeindependent) basis {|0⟩ , |1⟩}, and the transformation to the instantaneous basis {|E− (t)⟩ , |E+ (t)⟩} is defined by the angle(t)θ (t) = arctan ∆ε(t). Then, Eq. (99) givesh̄θ̇ (t)σy .
(100)
2
We emphasize that this formula is valid for any ∆(t) and ε (t), leaving the system in an eigenstate of H(t). In particular,
˙ = 0 and ε̇ = v , we obtainfor the linear LZSM problem, with ∆HCD (t) =H̃(t) = −

∆
2

σx −

vt
2

σz +

∆2

v∆
σy .
+ v2 t 2(101)

Note that the time integration of θ̇ (t), from −∞ to ∞, gives π so that θ̇ (t) represents a π -pulse (Theisen et al., 2017).
More generally, for pulse areas equal to odd multiples of π , a complete population transfer to the excited state takes
place, whereas for pulse areas equal to even multiples of π , the system returns to the initial state (Vitanov et al., 2001;
Teranishi and Nakamura, 1999).
Therefore, for a perfect adiabatic behavior, the nonadiabatic transitions should be suppressed, for which the LZSM
model is central for testing control protocol. This was discussed for TLSs (Chen et al., 2010; Ahmadinouri et al., 2019;
Funo et al., 2021), for three-level systems (Theisen et al., 2017), and for n-level ones (Poggi et al., 2015; Xu et al., 2018). In
particular, it is possible to approximately decompose the transitionless quantum driving into the sum of separate singlecrossing corrections (Theisen et al., 2017). Analysis of the adiabatic condition can help further improve the coherent
population transfer, resulting in tangent-shaped pulses (Xu et al., 2019a).
In some cases, it is possible to reach diabaticity by excluding the σy component by certain system-specific transformation (Bason et al., 2012) or by applying additional fast and strong oscillating terms to a linear one, as in ε (t) (Chasseur
et al., 2015). Indeed, as it was scrutinized by Petiziol et al. (2018), adding a fast oscillation in the control parameters canapproximately (to arbitrary precision) cancel nonadiabatic effects. Hence, with this, transitionless dynamics can be realized
for finite-dimensional quantum systems without requiring additional Hamiltonian components that are not included in
the initial control setup; this was further developed for single and coupled qubits (creating entanglement between them),
as well as for three-level systems (Petiziol et al., 2018; Petiziol and Wimberger, 2019; Petiziol et al., 2019).
The techniques of population transfer that we considered here were tested and developed for diverse physicalrealizations: atoms and molecules controlled by lasers (Vitanov et al., 2001), Bose–Einstein condensate (Bason et al.,2012), Rydberg states (Shi et al., 2016; Bengs et al., 2022), photonic structures (Longhi, 2009) (see therein for classicalanalogies), NMR (Herrera et al., 2014), electron spin of a single NV center in diamond (Zhang et al., 2013), electron spin in
a QD (Shafiei et al., 2013) (see for chirping back and forth resulting in a double-passage dynamics; for theory see Teranishiand Nakamura (1999) and Nagaya et al. (2002)), ensemble of QDs to demonstrate an ideal molecular switch with the LZSM
formalism (Kaldewey et al., 2018), superconducting qubits (Wei et al., 2008), and topologically protected edge states in a
dimeric spin chain (Longhi et al., 2019).
5.4. Adiabatic quantum computation
Adiabatic quantum computation (AQC) is an approach to quantum computing that is an important alternative to thestandard circuit (gate) model. In principle AQC is expected to be as powerful as the circuit model of quantum computation46

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

(Albash and Lidar, 2018; Ashhab et al., 2006). The former is not very sensitive to dephasing, yet the huge cost of thisapproach is the lack of universality and problems with accurate calculations (Zagoskin, 2011). AQC is best suited for
optimization problems, where the requirement is to find the global minimum of a cost function. In AQC, the computation
proceeds from a simple initial Hamiltonian Hi , whose ground state is easy to prepare, to a final Hamiltonian Hf , whose
ground state encodes the solution to the computational problem. Hence, the task is to find the ground state of Hf , whose
unknown eigenvalues determine the cost function. The evolution is described by the Hamiltonian
H(s) = (1 − s)Hi + sHf ,(102)

so that at t = 0 with s = 0, we start from Hi , and at the end of computation at t = tf with s = 1, we have the desired
final Hamiltonian Hf (Cullimore et al., 2012). The eigenvalues and eigenstates of the Hamiltonian H(s) describing a system
of M qubits are given by H(s) |ϕm (s)⟩ = Em (s) |ϕm (s)⟩, with m ranging from 0 to 2M − 1. The suitable figure of merit is
the closeness of the state vector |ψ (s)⟩ at the end of the evolution to the desired state |ϕ0 (s)⟩. This is quantified by the
success probability |⟨ ϕ0 (1)| ψ (1)⟩|2 .
For adiabatic evolution, the cornerstone of this is the adiabatic theorem, which assumes no LZSM transitions (Johansson
et al., 2009). The avoided crossings are defined by the minima of ∆E(s) = E1 (s) − E0 (s). Hence, in the vicinity of theseminima, the low-energy Hamiltonian is equivalent to the one describing two wells that are connected by a tunneling rateand detuned from each other by a bias (Wild et al., 2016). In this case, LZSM transitions are usually considered a nuisancefor AQC (Grajcar et al., 2005). Also, LZSM theory tells us that, akin to a double-slit experiment, two consecutive transitions(at the minima of ∆E(s)) can generate interference effects (Lubin et al., 1990; Munoz-Bauza et al., 2019).
Therefore, studying LZSM physics in an open system with the Hamiltonian H(s) can help minimize undesired
nonadiabatic transitions (Wild et al., 2016). Because, on the one hand, the system has to evolve adiabatically slow to
avoid nonadiabatic transitions and, on the other hand, we want to make the calculations faster, there are several ways tosatisfy these two conflicting requirements. For this, we must make the control of the feedback in calculations, monitoring
the curvature of the ground state E0 (s). Then, the feedback-control scheme speeds up the computation process by allowing
for faster evolution of s(t) when the ground state is well separated from the excited one (Wilson et al., 2012). Alternatively,
the success probability can be increased by tailoring the driving pulses (Karanikolas and Kawabata, 2020). These quantum
annealing protocols, called pulsed quantum annealing, are reached by optimizing the pulse parameters. Also, in the context
of AQC, an additional periodic drive can be applied to suppress control errors by using a destructive LZSM interference,
which is known as coherent destruction of tunneling (CDT) (Atia et al., 2019).
Because the minimal gap between the ground and first excited state can decrease with an increasing number of qubits
M, in general, the probabilities of nonadiabatic transitions are nonzero and should be taken into account (Santoro, 2002).
Then, the question arises regarding how far the evolution can deviate from the ground state and still obtain useful results.
Taking these into account results in the approximate AQC. Particularly, estimating the efficiency in this case, here for a large
number of possible interlevel transitions, can be done by using the analogy of the evolution with the standard random
walk (Zagoskin, 2011). Indeed, let us denote by N ≫ 1 the average number of avoided-level crossings per one energy
level, κ = sN an estimate number of avoided-level crossings, q = q(n, κ ) the probability of finding the system in the
eigenstate |ϕn (s)⟩, and p the average LZSM transition probability. Then, assuming n ≫ 1, after passing an avoided-levelcrossing, the system can be found in the nth state if either it stays in this state or if it underwent a nonadiabatic transition
from a neighboring energy level:p(103)q(n, κ + 1) = (1 − p)q(n, κ ) + [q(n − 1, κ ) + q(n + 1, κ )] .
2
This is almost the equation for the standard random walk (Zagoskin, 2011), which allows us to use standard methods todescribe the statistics of multiple LZSM transitions in a multilevel system. For instance, the final dispersion reads⟨⟩(n − n0 )2 s=1 = pN ,(104)

which is proportional to the average transition probability p and the average number of quasicrossings N. Furthermore,
we note that for a linear bias, s = t /tf , the driving velocity v is inversely proportional to the evolution duration tf . Then,the LZSM probability gives an estimation for the accuracy of the approximate AQC:√⟨⟩
∆n2 ∝ exp (−α tf ) ,(105)

which is defined by the exponential dependence on tf with some nonuniversal rate α .
6. Related classical coherent phenomena
On many occasions, the similarities between classical wave optics and quantum wave mechanics have been highlighted (Dragoman, 2004; Longhi, 2009; Bondar et al., 2013; Bliokh et al., 2013; Dressel et al., 2014). In particular, in ourcontext, we can find numerous examples in the literature where nonadiabatic transitions between classical states havebeen interrelated with LZSM physics. Some examples that can be mapped to the precession of a spin-1/2 particle exhibiting
LZSM-type nonadiabatic transitions between two states, modes, and so forth, include the following: transformation ofelectromagnetic waves (Zheleznyakov et al., 1983; Bliokh and Grinyok, 2001), topological boundary states in acoustic
47

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 17. Different classical systems that behave analogously to quantum TLSs. In (a) the two eigenfrequencies, Ω± are shown to depend on the bias εand display avoided-level crossing at ε = 0 with a minimal distance ∆. (b–g) Several possible classical systems that have eigenfrequencies as in (a).
Namely, (b) two weakly coupled spring oscillators (Frimmer and Novotny, 2014), (c) two-mode nano-beam (Faust et al., 2012; Seitner et al., 2016),(d) optomechanical system with two cantilevers, one of which is coupled with an optical cavity (Fu et al., 2016, 2018), (e) two coupled electricalresonators (Alzar et al., 2002; Muirhead et al., 2016), (f) two polarization modes of light propagating in the counter-clockwise (ccw) direction tunedby the electro-optic modulators, EOM1 and EOM2, with the tuning parameter being the electric field inside EOM1 (Spreeuw et al., 1990; Beijersbergen
et al., 1992), and (g) two coupled curved waveguides in which an electromagnetic wave is spread between them (Longhi, 2009; Liu et al., 2019).

waveguides (Chen et al., 2021a,c), two coupled photonic crystal nanocavities (forming a photonic molecule) that are tuned
by RF surface acoustic waves (Kapfinger et al., 2015; Bliokh and Nori, 2019), magnetoresistance in quasi-one-dimensionalorganic conductors (Cooper and Yakovenko, 2006), the mixing of two tones in a ferrimagnetic sphere resonator (Mathai
et al., 2020), optical modes coupled by a moveable membrane in optomechanical systems (Wu et al., 2013; Jiang et al.,2020), two modes of a mechanical resonator or two directly-coupled resonators (Zhang et al., 2020), and a rolling sphereon a Cornu spiral (Rojo and Bloch, 2010). In these systems, different realizations of classical analogues of Rabi oscillationsand LZSM transitions and interference were demonstrated.
To be specific, in this section, we demonstrate that a system of two weakly coupled resonators can be describedanalogously to a quantum TLS. Namely, we demonstrate that under appropriate conditions, the Newton equations for thisclassical system are reduced to an equation formally analogous to the Schrödinger equation for a TLS. As a consequence,the classical system can display a classical analogue of LZSM transitions and interference. In addition, any physical systemthat can be mapped on a two-resonator system with weak coupling could behave similarly. In Fig. 17, we demonstrate avariety of such systems (Ivakhnenko et al., 2018).
6.1. From Newton to Schrödinger
We now consider how the classical Newton equations for two coupled resonators can be mapped to the Schrödinger
equation for a quantum TLS. Here, we follow Refs. Novotny (2010) and Frimmer and Novotny (2014). Consider the system
48

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

of classical Newton equations for two weakly coupled oscillators A and B, Fig. 17(b),{mA ẍA + γA ẋA + kA xA + kc (xA − xB ) = 0,(106)mB ẍB + γB ẋB + kB xB + kc (xA − xB ) = 0,

where xA,B are the coordinates, mA = mB = m are the masses, γA = γB = γ are the damping coefficients, kA,B are thespring coefficients, and kc is a coupling coefficient that is assumed to be weak, kc ≪ kA,B . To obtain an analogy with aquantum TLS, we choose the excitation in the form kA,B = k0 ± ∆k(t) with ∆k ≪ k0 . We denote the interaction-shifted
eigenfrequency Ω02 ≡ (k0 + kc )/m, and rewrite this system of classical Newton equations in matrix form, here using Paulimatrices:

(

d2

γ d

dt

m dt

+
2

+ Ω02

)[ ]

xA
−
xB

(

kc
m

σx +

∆k(t)
m

σz

)[ ]
xA
xB

= 0.

(107)

We use the ansatz
x̃A,B (t)x0

= ψA,B exp (iΩ0 t ),xA,B = x0 Re x̃A,B .

(108)

Here, x0 is the initial deviation of the springs, which is used to normalize x̃A,B . We obtain the equation

(

d2

d

dt

dt

+ (γ + 2iΩ0 )2+ i Ω0 γ

)(

) (
)( )
kc
∆k(t)
ψA
ψA
= 0.
−
σx +
σz
ψB
ψB
m
m

(109)

We can simplify this equation by using two assumptionsγ ≪ Ω0 and kc , ∆k ≪ k0 .

(110)

The former assumption of small dissipation allows for neglecting γ in comparison with Ω0 . The latter assumption, together
with the ansatz (108), allows to use the slowly varying envelope approximation, which consists of neglecting the second
derivative. This means that ψA,B have small changes during the time span 2π/Ω0 ; in other words, the characteristic
evolution rate for ψA,B is muchthan Ω0 . Also, the assumption ∆k ≪ k0 means that we cannot reach a fast⏐
⏐ smallertransition, where v → ∞ and ⏐ψA2 (t)⏐ → θ (t)P . After using these assumptions, we obtain a Schrödinger-like equationiddt|ψ⟩ = H(t) |ψ⟩ − i

γ
2

|ψ⟩ ,

(111)

where |ψ⟩ = (ψA , ψB )⊤ and H(t) is the Hamiltonian of Eq. (1) with

∆=

kc
m Ω0

kc

≈ √

mk0

,

ε(t) =

∆k(t)
∆k(t)
.
≈ √
m Ω0
mk0

(112)

Much like qubits, the mechanical resonators can be driven via ∆k(t), to have both an offset and a periodic excitation,
we have ε (t) = ε0 + A cos ωt. Note that the problem of Eq. (111) can be described in terms of a non-Hermitian
Hamiltonian (Shen et al., 2019).
In the absence of dissipation, γ = 0, Eq. (111) formally coincides with the Schrödinger equation for a TLS, Eq. (15),in⏐ ⟩the natural measuring system, that is, assuming h̄ = 1. Dissipation can be⏐ eliminated⏐ by⟩ the substitution |ψ⟩ =⟩⏐ψ exp (−γ t /2); then, the classical Schrödinger-like equation (111) becomes i d ⏐ψ = H(t) ⏐ψ . In addition, the “density
dt()
matrix” can be introduced as ρ = |ψ⟩ ⟨ψ|, where ⟨ψ| := ψA∗ , ψB∗ . Then, for the derivative, we obtainρ̇ = −i [H , ρ ] − γ ρ.

(113)This coincides with the Bloch equation for a TLS, Eq. (85), for the Hamiltonian H(t), and assuming that h̄ = 1 and withequal relaxation rates, T1 = T2 = 1/γ .
This means that we can use the same methods for solving the classical and quantum systems. Interestingly, historically,
the system of nonadiabatic transitions between two states was first studied and developed for a quantum TLS, and only
later it was recognized that it has a classical doppelgänger. Particularly, the theory by Zener helped to describe the coupledpendulums in Ref. Maris and Xiong (1988).
In a system of two coupled classical oscillators, two eigenfrequencies are analogous to the two energy levels for thequantum TLS; these are the eigenvalues of the Hamiltonian above√Ω± = ± ∆2 + ε02 .

(114)

The probability ⏐of the⏐2 occupation of the energy levels in the quantum TLS is analogous to the squared amplitude of
the oscillations ⏐ψA,B ⏐ ; in other words, it is proportional to the amount of energy in that oscillation mode. To further
demonstrate the similarity of the classical problem to its quantum analogue, in Fig. 18, we consider a single-passage
LZSM-like transition with ε (t) = v t, and choose the following parameters: ∆c /k0 = 5 × 10−4 for the classical case and49

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 18. Single-passage transition in classical and quantum cases. We here compare the solutions of the classical Newton equations, Eq. (107), theSchrödinger equation, and Eq. (15) for a LZSM transition. For illustration purposes, we take the parameters so that⏐the final⏐ excitation probability isclose to unity, P ≈ 1. The classical transfer of energy from one harmonics to another is given by the black line ⏐ψA2 (t)⏐, which virtually coincides
with the quantum upper-level occupation probability P+ (t), which is the green line. Furthermore, the red curve shows the AIM, which is considered
above and is given here by θ (t)P ; the blue curve shows the classical oscillations x2A (t)/x20 .

δ = δcl ≡ k2c /4v = 3.57 × 10−4 for both cases. This demonstrates that under the conditions (110), the classical envelope|ψA (t)|2 corresponds to the quantum occupation probability P+ (t). This principle can be applied for any classical systemof two coupled resonators with weak coupling.
Coherent phenomena in classical mechanical systems were reliably demonstrated experimentally; see the caption ofFig. 17. In particular, LZSM interferometry was realized with mechanical resonators in Refs. Seitner et al. (2017), Zhou
et al. (2019) and Lorenz et al. (2022). For further theoretical study of such processes, see Refs. Chotorlishvili et al. (2011),
Parafilo et al. (2012), Parafilo and Kiselev (2018b) and Villazon et al. (2019).
6.2. Engineered photonic waveguides
Curved waveguides have recently provided a rich laboratory tool to visualize quantum coherent phenomena withclassical optical waves in the spatial domain (Longhi, 2009; Menchon-Enrich et al., 2016). To understand how thisbecomes possible, consider the propagation of monochromatic light waves of wavelength λ along a curved waveguidestructure. This could be a single waveguide, coupled ones, or multiple waveguides (Longhi, 2009; Morales-Molina andReyes, 2011; Reyes et al., 2012; Liu et al., 2019). Let the waveguide structure be planar, in the plane (x, z ); see Fig. 17(g).
There, z is the paraxial propagation distance, the axis bending profile ns , and the refractive index profile of the guiding
structure n(x), ω = kc0 . Using the scalar and paraxial approximations, we can write the electric field amplitude asψ (x, z) exp [i (kns z − ωt ) + c .c .], where the slow-varying field envelope ψ is described by the paraxial wave equation∂ψo2 ∂ 2 ψ=−+ U(x)ψ,(115)∂z
2ns ∂ x2[]
with o = λ/2π and U(x) = n2s − n2 (x) /2ns ≃ ns − n(x). This equation, Eq. (115), can be formally written in the form of
a 1D Schrödinger equation with the following substitutions: o → h̄, ns → m, and z → t. Furthermore, the eigenmodes
ioof two coupled waveguides can be described by a Hamiltonian formally analogous to the one of a driven TLS (Liu et al.,2019).
To conclude, electromagnetic waves propagating along waveguides have different advantages, such as long coherent
lengths and high control in preparation and measurement in a room-temperature environment. Given that the Maxwell
equation for these takes the form of the Schrödinger equation and the interwaveguide interactions are analogous to
the interaction between the quantum states, the light propagating along curved waveguides (namely, its intensity
distributions) in the spatial domain can simulate quantum driven dynamics in the time domain (Longhi, 2009; Liu et al.,2019; Shen et al., 2019).
7. Conclusion
An avoided-level crossing is described by the LZSM model, which includes adiabatic evolution, nonadiabatic transitions,and quantum interference. We considered this by starting from the original LZSM works. It was demonstrated that thisevergreen problem provides important tools for characterizing and controlling quantum systems.
50

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Our detailed consideration of the works by Landau, Zener, Stückelberg, and Majorana demonstrates that the nonadiabatic transitions should be attributed to all four names. When presenting these four approaches here, we wanted not
only to attract researchers’ attention to this, but to make these diverse approaches easily accessible and clear for modern
readers. We focused on Zener’s approach because this describes both the final wave function, including its phase, andthe dynamics. Both Zener’s and Majorana’s approaches can be developed to describe a few or many times passing of
avoided-level crossing, providing the basis for the AIM in terms of transfer matrices.
In describing single- and multiple-passage problems, we supplemented previous works and the review (Shevchenko
et al., 2010). In that review, besides theory, mainly experiments with superconducting quantum systems were described.
Here, we presented many more works, particularly the ones that have appeared since then, including on semiconducting
devices, graphene, ultracold gases, and many other systems.
Single and multiple LZSM transitions display rich physics of quantum systems, both on microscopic and mesoscopic
scales. Related physics can be harnessed to characterize their quantum dynamics (interferometry) and to control theirstates. Classical systems can also display analogous phenomena.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could haveappeared to influence the work reported in this paper.
Acknowledgments
We acknowledge useful discussions with S. Ashhab, T. Boolakee, B. Damski, S. Esposito, M.F. Gonzalez-Zalba, P.
Hommelhoff, P. Kofman, M. Liul, H. Nakamura, A. N. Omelyanchouk, K. Ono, A. Ryzhov, and A. Sotnikov. The research of
O.V.I. and S.N.S. was sponsored by the Army Research Office, United States and was accomplished under Grant Number
W911NF-20-1-0261. O.V.I. was supported by the RIKEN International Program Associates (IPA), Japan. S.N.S. was partially
supported by Grant No. 2/21-H from the NAS of Ukraine. F.N. is supported in part by: Nippon Telegraph and Telephone
Corporation (NTT) Research, Japan; the Japan Science and Technology Agency (JST) [via the Quantum Leap Flagship
Program (Q-LEAP), and the Moonshot R&D, Japan Grant Number JPMJMS2061]; the Japan Society for the Promotion of
Science (JSPS), Japan [via the Grants -in-Aid for Scientific Research (KAKENHI) Grant No. JP20H00134]; the Army Research
Office (ARO), United States (Grant No. W911NF-18-1-0358); the Asian Office of Aerospace Research and Development(AOARD), Japan (via Grant No. FA2386-20-1-4069); and the Foundational Questions Institute Fund (FQXi), United States
via Grant No. FQXi-IAF19-06.
Appendix A. LZSM transition probability
Here we present four ways to derive the LZSM formula, Eq. (7), following the four classic papers from 1932. In
our presentation we follow the original works and make those derivations clear for a broad audience. For this sake,some derivations are simplified and some notations changed. Readers interested in the original works can find them in
Refs. Landau (1932a), Landau (1932b), Zener (1932), Stückelberg (1932) and Majorana (1932) and translated into English
in Refs. Haar (1965), Cifarelli (2020a) and Stueckelberg (1970). Presenting these together, allows to see the interrelation
between these approaches. For example, C. Zener and E. Majorana solved the very same second-order differential equation,
both by referring to the same book “Modern Analysis” (Whittaker and Watson, 1920), but Zener made use of special
functions, while Majorana used the Laplace transform. Also, while following the LZSM papers, we will make several
remarks; for example, we will not only be interested in the upper-level excitation probability, but also in the changesof the wave function phase, which is important for quantum-mechanical interference.
A.1. Near-adiabatic limit (Landau)
First, we consider the LZSM transition following L.D. Landau’s two original papers (Landau, 1932a,b), which have beentranslated into English in Haar (1965), and the textbook (Landau and Lifshitz, 1965). This derivation of the LZSM formula
is arguably the simplest one.
Consider a slowly varying (with v ≪ ∆2 /h̄) time-dependent Hamiltonian (1). To describe the adiabatically slow
variations, one can consider time t as a parameter, and the system dynamics described by the Schrödinger equation
H(t) |ψ (t)⟩ = E |ψ (t)⟩ .

(A.1)

We substitute the wave function (3) in the Schrödinger equation (A.1) and obtain(H − E)(( )1 v t + 2E
α
=−
∆
β
2

∆
−v t + 2E

)( )
α
= 0.
β
51

(A.2)

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

For a nontrivial solution of this matrix system of equations, we need to equate the matrix determinant to zero. As a result,
we obtain the two energy levels of this TLS
E± (t) = ±1√2∆2 + ε (t)2 = ±

1√

∆E(t)

2

2

∆2 + (v t)2 ≡ ±

.

(A.3)

For slow excitations (v ≪ ∆2 /h̄), it is assumed that the evolution is adiabatic everywhere except for the avoidedcrossing region. Then, we determine the probability of the upper-energy-level occupation P+ (t → +∞), provided thatthe system was initially in its ground state |0⟩, which means P− (t → −∞) = 1.
∫
Slow perturbation implies that at large times of the “transition process” the change of the action S = E(t)dt is large.
This means that the dynamics has a quasi-classical character and it is fully analogous to the over-barrier reflection. Notethat we have the quasiclassical wave function as a function of time, not versus coordinates. Therefore, the avoided-crossing
region determines the transition probability between the energy levels. In this region, the energy levels do not have anintersection in real space, but they have an intersection in complex space, so the equation E+ (t0 ) = E− (t0 ) is fulfilled at
t0 = ±i∆/v . Therefore, we can obtain the probability of the transition by following section 53 in Ref. Landau and Lifshitz(1965):(

2
P+ = exp − Im
h̄

∫ t0

)
[E+ (t) − E− (t)]dt .

(A.4)

0

Using the substitution u = v t /∆, we obtain the integral∫ t0∆E(t) dt =

0

∆2
v

∫ i√

1 + u2 du = i

0

π ∆2
.
4v

(A.5)

Therefore, we obtain the excitation probability of the TLS for the linear perturbation, P+ = P = exp (−2πδ ), which is
Eq. (7).
In fact, in the original paper (Landau, 1932b), on the right-hand side of the formula (A.4), there should be a prefactorC ∼ 1. This happened to be exactly unity for the LZSM formula, as we can see from each of the three following subsections.
The formula (A.4) is very convenient for calculating the single-transition probability for any bias signal, and we consider
this in the main text for a nonlinear drive. Because the approach by Landau was generalized in Refs. Dykhne (1962) andDavis and Pechukas (1976) for calculating nonadiabatic transitions regarding the form of the Hamiltonian, this formula
is sometimes referred to as a Landau–Dykhne or Dykhne–Davis–Pechukas formula (Vitanov and Suominen, 1999; Lehto
and Suominen, 2015). Particularly, see the latter references for a generalization to the case when there are multiple zero
points.
A.2. Using parabolic cylinder functions (Zener)
In this section, we consider how to derive the probability of the LZSM transition by following the work by Clarence
Zener, Ref. Zener (1932). In fact we develop Zener’s approach by finding analytically the full wave function with thetime dependence (Vitanov and Garraway, 1996; Shevchenko et al., 2010). Here we pay more attention to find the wave
function’s phase change after the transition. In this approach, we obtain a straightforward solution of the Schrödinger
equation with time dependence in terms of a special function, known as the parabolic cylinder function. This method is
not so simple as the previous one, but it provides a more general and precise solution of this problem. In particular, there
will be no unknown prefactor C , as in the approach by Landau. For a theoretical description involving a generalization for
non-Hermitian Hamiltonians, see Shen et al. (2019) and Kam and Chen (2021).
Consider the time-dependent Schrödinger equation, Eq. (15). Using the wave function Eq. (3) and Hamiltonian in thematrix form Eq. (1), we obtain the system of equations{β,ih̄α̇ = − v2 t α − ∆
2

(A.6)

ih̄β̇ = − ∆α + v2 t β.
2This system of equations can be rewritten in the form of two second-order Weber equations, for either α or β :
d2 α
2

(
)
+ 2δ − i + τ 2 α = 0,

(A.7a)

2()+ 2δ + i + τ 2 β = 0,

(A.7b)

dτ
d2 β
dτ
where

τ =t

√

v
2h̄

,

(A.8)

and δ is the adiabaticity parameter, Eq. (8). The following replacementz=τ

√

2 exp (iπ /4)

(A.9)
52

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

allows us to obtain the so-called Weber equation in the canonical form
d2 α

(
)
1
z2
+
−
i
δ
−
−
α = 0,
dz 2
2
4
(
)
d2 β
1
z2
+
−
i
δ
+
−
β = 0.
2dz

2(A.10a)(A.10b)4

The solutions of the Weber equation are the combinations of the parabolic cylinder functions (Gradshteyn and Ryzhik,2007),

α = A+ D−1−iδ (z ) + A− D−1−iδ (−z ) ,β = B+ D−iδ (z ) + B− D−iδ (−z ) .

(A.11a)(A.11b)

We find relations between the coefficients A± and B± by inserting the solutions in the first-order equations (A.6):
B± = ∓

exp (−iπ/4)

√

δ

A± .

(A.12)

Now, using the relation (A.12) and the exact solution (A.11a), we can find A± from the initial conditions at z = zi withcertain α (zi ) and β (zi ) (Vitanov and Garraway, 1996):]Γ (1 + iδ ) [π √α (zi )D−iδ (−zi ) − β (zi )ei 4 δ D−1−iδ (−zi ) ,(A.13a)√2π]
Γ (1 + iδ ) [π √A− =(A.13b)α (zi )D−iδ (zi ) + β (zi )ei 4 δ D−1−iδ (zi ) .
√2πThis gives the exact analytical solution at any moment of time τ > τi for a single-passage LZSM transition for any initialcondition in terms of the evolution matrix Ξ e
(
) ( e
)(
)
(
)
e
α (z)
Ξ11 Ξ12α (zi )e α (zi )(A.14)
=
≡Ξ
,
e
e
β (z)
Ξ21
Ξ22
β (zi )β (zi )
A+ =where the time-dependent matrix elements of Ξ e are as followsΓ (1 + iδ )[D−iδ (−zi )D−1−iδ (z) + D−iδ (zi )D−1−iδ (−z)] ,√2π
Γ (1 + iδ ) iπ √ee 4 δ [−D−1−iδ (−zi )D−1−iδ (z) + D−1−iδ (zi )D−1−iδ (−z)] ,
Ξ12
= √
2π

e
Ξ11
=

(A.15a)(A.15b)iπΓ (1 + iδ ) e− 4
= √√ [−D−iδ (−zi )D−iδ (z) + D−iδ (zi )D−iδ (−z)] ,δ2πΓ (1 + iδ )e[D−1−iδ (−zi )D−iδ (z) + D−1−iδ (zi )D−iδ (−z)] .
Ξ22 = √
2πe
Ξ21(A.15c)(A.15d)From this transfer matrix (A.14), we can obtain the exact solution Eq. (A.11b) for the LZSM problem in terms of the wave
function (3) at any moment of time for linear drives. Of course, this exact solution coincides with the numerical solutionof Eq. (A.6). However if we want to know the wave function (3) at some moment of time, we do not need to calculate it for
all previous moments as we do for the numerical solution of Eq. (A.6). The exact solution transfer matrix (A.14) provides
the basis for the transfer matrix method describing the dynamics of the LZSM transition. Hence, now we explain how theexact transfer matrix Ξ e results in the transition matrix N, Eq. (30). For this, let us take the initial and final moments ofthe time symmetrically far from the quasicrossing region

τi = −τa , τf = τa ,√zi = τa 2e−3iπ /4 ,where τa ≫ 1;
zf = τa

√

iπ /4

2e

(A.16).

(A.17)

We now use approximations for the parabolic cylinder functions, with the argument tending to infinity (Gradshteyn andRyzhik, 2007) (see the chapter on parabolic cylinder functions, pp. 1092–1094),√

lim D−iδ−1 (z ) ≈ e−z 2 /4 −iδ−1z|z |→∞−

lim D−iδ−1 (−z ) ≈ e−z /4 z −iδ−1 ,2|z |→∞2π2eiπ (iδ+1) ez /4 z iδ ,Γ (iδ + 1)π
Arg(z) = ,4

Arg(z) = −

3π
4

,

(A.18a)(A.18b)and the following properties of the Gamma functionΓ (iδ ) =√π
exp (iArg[Γ (iδ )]),
δ sinh πδ

(A.19a)53

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Arg[Γ (iδ )] = −Arg[Γ (−iδ )] = −π2− Arg[Γ (1 − iδ )].

(A.19b)As a result, we find the asymptotic values of the single-passage evolution matrix in the diabatic basiseΞ11≈ exp [−π δ ] = T ,
√
{π
(√ )}
2π √ iπ iτ 2 (√ )2iδ − π δeδe 4 e a2τa
e 2 = R exp i
2τa .
Ξ12 ≈
+ Arg [Γ (1 − iδ )] + τa2 + 2δ ln
Γ (1 + iδ )4

(A.20a)(A.20b)ee
Other elements, Ξ21
and Ξ22, can be found from these two. Here,

T = e−π δ ,R =√1 − e−2π δ(A.21)

stand for the transition and reflection coefficients.
Let us now consider how to relate this to the adiabatic-impulse model; namely, let us find the nonadiabatic transition
matrix N. According to the TM method, Section 2.5, the full single-passage evolution for any initial condition consists ofthree stages:(a) adiabatic evolution before the transition,(b) the transition itself, and(c) adiabatic evolution after the transition.
Hence, for this, the natural description appears in the adiabatic basis. The three-stage evolution in the adiabatic
basis is described by the product of the three respective matrices Ξ a = U2 NU1 . Here, the adiabatic-stage matrices are
U1,2 = exp [iζ (±τa )σz /2] with1

ζ (±τa ) =∫ ±τa √2h̄

∆2 + 2h̄vτ 2 dτ ≈ Φ (τ ) − Φδ ,(A.22a)0whereΦ (τ ) =τ2+ δ ln

2

(√

)

2τ ,

1

Φδ =

2

δ (ln δ − 1) .

(A.22b)Therefore, we obtain the single-passage evolution matrix in the adiabatic basis

Ξ a = U2 NU1 =(R exp i − π4 − Arg [Γ (1 − iδ )] − 2Φ (τa )
−T

{

}

)

T
} .
R exp i 4 + Arg [Γ (1 − iδ )] + 2Φ (τa ){π(A.23)The evolution matrices in the adiabatic and diabatic bases are linked by the transition matrix
S(τ ) =

(

γ+
γ−

)
γ−
,
−γ+

(A.24)

where the coefficients γ± were defined in Eq. (6). The asymptotes of the matrix S(τ ) have the form(S(−τa ) =

)

0
1

1
,
0

S(τa ) =

(

)

1
0

0
.
−1

(A.25)

Then, we match the two representations
S(τa )Ξ e S(−τa ) = Ξ a = U2 NU1 ,(A.26)

from which we obtain the desired matrix for the nonadiabatic transition
N = U2 S(τa )Ξ S(−τa )U1
−1

−1

e

(
=

Re−iφS
T

−T

ReiφS

)

,

(A.27)

where the Stokes phase φS (δ ) was defined in Eq. (9). For more on this phase shift, see also Chapter 4 in Child (1996).
Note that the transition matrix N is given here in the adiabatic representation. In the diabatic one, we obtain
Nd = S(τa )NS(−τa ) =(T−Re−iφS

ReiφS
T

)

.

(A.28)√√

Consider the inverse transition, from τ = +τa to τ = −τa . Then, we have zi = τa 2eiπ/4 and zf = τa 2e−3iπ/4 . We
obtain that the transition matrices for this inverse transition are transposed to direct transitions matrices in the adiabaticand diabatic bases,
Ninverse = N⊤(=

Nd,inverse = Nd⊤ =(Re−iφS
−T

T
ReiφS

T
ReiφS

−Re

)

,
)
−iφ

T

S

(A.29a),(A.29b)54

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

where ⊤ denotes transposition. We emphasize that the inverse transition is described by the transposed matrices, which
can be obtained both from inverting time (t → −t) in the Schrödinger equation and from direct solution, as we checkedin Eq. (A.29a). Importantly, the above consideration bears a general character and can be given for any initial condition.
Particularly, in the case in which initially our TLS was in the lower (ground) energy level, with{α (zi ) = 0,
β (zi ) = 1,

(A.30)

we have{eα (z) = Ξ12,eβ (z) = Ξ22.

(A.31)

This expression defines the solution for any initial moment of time τi . Furthermore, the expressions for Ξ e are simplified
for −τi = τa ≫ 1. Then, we obtain the expression for the occupation of the upper diabatic energy level:πδ

Pd (z) = |α (z)|2 = δ e− 2 |D−1−iδ (−z)|2 .

(A.32)Taking into account the relationship between the diabatic and adiabatic bases, for the respective upper-level occupation
probability in the adiabatic basis, we have
Pa (z) = |β (z)γ+ − α (z)γ− |2 ,(A.33)and this results in the following time dependenceπδ⏐⏐

Pa (z) = e− 2 ⏐D−iδ (−z)γ+ −⏐2√ − iπ⏐δ e 4 D−1−iδ (−z)γ− ⏐ .

(A.34)

These expressions define the time dependence of the upper-level occupation probabilities. They are further simplified if weare interested in the asymptotic solution with τ ≫ 1.
Extending Zener’s 1932 approach: Consider now another, more conventional, way to solve this problem of developing
Zener’s approach. Below, we will extend Zener’s 1932 approach. We will study the asymptotics of α and β and findthe coefficients A′± after obtaining asymptotics from the initial conditions. Consider now the asymptotic solution with
Eq. (A.17), here with the asymptotics of the parabolic cylinder functions Eq. (A.18b).
We find the asymptotic values of α and β :α (−τa ) ≈ A′+ Θ1 exp (iΦ (τa )),()β (−τa ) ≈ −e−δπ /2 A′+ + eδπ /2 A′− Θ2 exp (−iΦ (τa )),(A.35a)(A.35b)α (τa ) ≈ A′− Θ1 exp (iΦ (τa )),()β (τa ) ≈ −eδπ /2 A′+ + e−δπ /2 A′− Θ2 exp (−iΦ (τa )),
where

√

(A.35d)( π )δ ,Γ (1 + iδ)
4
(
1
π )
π
Θ2 = √ exp −i − δ ,
4
4
δ
Θ1 =

2π

(A.35c)exp −(A.35e)(A.35f)and A′± are the asymptotic coefficients, which play the same role as A± for the exact solution. Let us match this asymptotic
solution with the transition from a diabatic to adiabatic basis, Eq. (5), at τa ≫ 1. Then, we obtain()α (−τa ),β (−τa )() ()
c+ (τa )β (τa )=,c− (τa )−α (τa ))c+ (−τa )c− (−τa )((A.36a)=(A.36b)where c± are the coefficients of the decomposition for a wave function |ψ⟩ with the adiabatic wave functions |ψ± ⟩,|ψ⟩ = c+ |ψ+ ⟩ + c− |ψ− ⟩ .

(A.37)

Then, we use the initial conditions (A.30), and we would like to describe the probability of finding the TLS in the upper
energy level, which is |c+ (τf )|2 ≈ |β (τf )|2 . For this, we need to find an absolute value of the coefficients A′± . We describe thecoefficients from the first initial condition (A.30) and the first equation in the system (A.6); hence, we obtain ih̄α̇ = − ∆.
2Then, before using the ansatz (A.11a), we need to rewrite the initial condition in terms of the variable z introduced in
Eq. (A.9)iπe− 4 dα

√

δ dz

= 1.

(A.38)
55

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

For the next step, we need to define the derivative for the parabolic cylinder function at infinity, as shown in Eq. (A.18b),
which reads (Gradshteyn and Ryzhik, 2007):limd|z |→∞ dz

12
D−iδ−1 (−z ) ≈ − e−z /4 z −iδ .
2(A.39)

Now, we determine the coefficients A′± , for which we use the ansatz (A.11a) and asymptotics of the parabolic cylinderfunctions (A.39):
iπ

e− 4

(√

τ2

√ A′± e−i 2
2 δ

2τ

)−iδ

πδ

e 4 = 1.

(A.40)

It follows that
A′+ = A′− = 2

[ 2]
[
√ ]
√ −δπ /4 i πτ4δee exp i
exp iδ ln( 2τ ) .
2(A.41)

Finally, we obtain the probability of the lower-energy-level occupation after the transition:|α (zf )|2 ≈2πδ

e−π δ =2

|Γ (1 + iδ) |2πδ e−π δδ 2 |Γ (iδ) |2= 1 − e−2πδ .

(A.42)

Here, we used Eq. (A.19a) with the normalization condition |α|2 + |β|2 = 1. Then, we obtain the probability of theupper-level occupation:|β (zf )|2 = exp [−2πδ ] = P .

(A.43)In addition to finding the final upper-level occupation probability, after defining the coefficients A′± , this approachgives us the asymptotics for the coefficients α and β of the wave function |ψ (t)⟩ = α (t) |0⟩ + β (t) |1⟩ after passing the
avoided-level region. If we consider the phase of α (zf ), using Eqs. (A.19a), (A.41) and (A.20b), we can find the transition
phase change:
Arg[α (zf )] =π
4

+ Arg [Γ (1 − iδ )] + 2Φ (τa ).

(A.44)(Note that Eq. (A.44) is not in Zener’s paper.) Hence, we obtained Eq. (7) twice using Zener’s method with the full singletransition evolution phase Eq. (A.20b). Then we can do the same transformation as in Eqs. (A.23), (A.27), and we obtainebecause of the initial conditions (A.30).
the same transition matrix with the Stokes phase Eq. (9). We have α (τa ) = Ξ12
Consequently, we obtain the same final result for the probability of excitation and the phase, as in Eq. (A.20b).
To summarize, in this section we have presented two approaches, generalizing the works by Zener (1932) and Vitanov
and Garraway (1996). We obtained the exact solution in terms of the parabolic cylinder special functions for a singlepassage problem in the general case in terms of the exact transfer matrix Ξ e , Eq. (A.14). Then, we obtained a simpler
limit variant of the transfer matrix, which describes the evolution from zi to zf , Eq. (30). As a particular case, we havethe single-passage problem, starting from the ground state and ending far from the avoided-level crossing; then, theprobability of the transition is given by the LZSM formula, while the phase difference is defined by the Stokes phase.
Zener obtained only the probability of the LZSM transition; apparently, he was not interested in the phase change of the
wave function. However, it happened to be straightforward to define it using Zener’s approach.
A.3. With contour integrals (Majorana)
Ettore Majorana studied the spin orientation in a dynamic magnetic field (Majorana, 1932) with components Hx = −∆,Hy = 0 and Hz = −v t. The English version of Majorana’s paper is available in the book (Bassani, 2006) and in the second
edition in Cifarelli (2020a), which is commented on by M. Inguscio (Cifarelli, 2020b); see also in Wilczek (2014). We noteagain that we use convenient and uniform notations for all the four LZSM approaches, which here differ from the ones
used by Majorana by only changing notation.
Note that Majorana studied first Eq. (7) and later Zener studied the same equation. In his approach, Majorana used theLaplace transform instead of using the special parabolic functions in Zener’s method. Given the importance of Majorana’s
approach, details can be found elsewhere (Kofman et al., 2022), while here we present its key aspects.
Following Majorana, the system of equations (A.6) can be transformed by the substitutions

α = f1 exp

(

i
2

τ

2

)

(
)
i 2
, β = f2 exp − τ ,(A.45)

2from which we obtain the system of two equations{√√f˙1 = i 2δ f2 exp (−iτ 2 ),(A.46)

f˙2 = i 2δ f1 exp (iτ 2 ),56

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

which can be split into two independent equations
d2 f1,2
dτ 2

± 2iτ

df1,2
dτ

+ 2δ f2,1 = 0.

(A.47)√

Note that there is some difference from Majorana (1932) in denoting the time variable τ = 2τ ′ . Following Majorana,
this equation can be solved by the two-sided Laplace transform; then, for the Laplace transforms of the functions f1,2 (τ ),one obtains
F1,2 (s) = Cδ exp(∓is2 /4)s∓1−iδ .

(A.48)

Here, the constant of integration Cδ should be defined from an initial condition, by assuming the system being initially in
the ground state. Then, we can find f1 (τ ) and f2 (τ ):f1,2 (τ ) =

∮
C1±,2

esτ F1,2 (s) ds,(A.49)

where C1±,2 are the steepest descent contours. The integrals have two contributions: the first one is from the saddle-point
and the second one is from the vicinity of zero. One can derive the contribution from the quasi-intersection points byusing the saddle-point method in complex space. We then obtain the approximate solutions of Eq. (A.47) for two cases,τ < 0 and τ > 0, here in their general form. Then using the substitutions (A.45), these can be written for the asymptotes
α (τ ) and β (τ ) at large τ()⎧√⎨ α (τ ) = Cδ 4π (−2iτ )−iδ−1 exp −i τ 2 − i π ,
4
√
(
)2
τ <0:
⎩ β (τ ) = Cδ 2π i (−2iτ )−iδ exp −i τ 2 ,
δ
2
(
)
( )
⎧
√
⎨ α (τ ) = Cδ 4π (−2iτ )−iδ−1 exp −i τ 2 − i π + Cδ 2π i τ iδ exp i τ 2 ,
24
Γ (iδ+1)
√
√
(
)
( 2)
τ >0:
⎩ β (τ ) = Cδ 2π i (−2iτ )−iδ exp −i τ 2 + Cδ δ 2π i τ iδ−1 exp i τ 2 ,δ
22 Γ (iδ+1)
2(A.50a)(A.50b)

where

√
Cδ =

(
)
δ
πδ
exp −
.
2π
2

(A.51)

From Eq. (A.50a), we can obtain the absolute values for the upper- and ground-diabatic-level occupations after the LZSM
transition:|α (τ → ∞)|2 = 1 − e−2π δ ,|β (τ → ∞)| = e
2

−2π δ

(A.52a).

(A.52b)In addition, we can find the phase shift, which is accumulated during the transition process:Arg[α (τ → ∞)] − Arg[α (−τ → ∞)] =π+ Arg [Γ (1 − iδ )] + 2Φ (τa ),(A.53)
4
We realize that this equation fully coincides with Eq. (A.44), so we can do the same transformations to obtain the Stokes
phase from it. As a result of this approach, we obtain the asymptotic formulas for the amplitude of the TLS wave function
after a single passage with the linear excitation in an elegant mathematical way, by using Laplace transformations andcontour integration.
A.4. Using the WKB approximation and the phase integral method (Stückelberg)
In his 54-page-long paper, Ref. Stückelberg (1932) (with the English translation in Stueckelberg (1970)), E.C.G. Stückelberg studied the transitions between two energy states. In brief, he considered the problem of inelastic collisions
and reduced the Schrödinger equation to a system of two coupled second-order differential equations, which he solved
developing the WKB approximation and phase integral methods. Here, the difficulty consists of analytic continuation of
the WKB solution through the so-called Stokes lines (Giacomo and Nikitin, 2005). In what follows, we give the mostimportant aspects of this, while the details can be found in Ref. Child (1974) and references therein.
Following Child (1974), we start from the two coupled wave equations]⎫+ k21 (r) u1 (r) = α (r)u2 (r),⎬
[ 2
]
d
2
+
k
(r)
u2 (r) = α (r)u1 (r),⎭22
dr(A.54a)k2i (r) = 2µ (E − Vi (r)) /h̄2 ,(A.54b)

[ 2

d
dr 2

where

57

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

α (r) = 2µV12 (r)/h̄2 .

(A.54c)We now reduce the Schrödinger equation, which describes the collision of atoms and molecules, by expanding the fullwave function with the ones of the electrons in the atoms and spherical functions (Kotova, 1969; Eu, 1970). Assumingthat the motion of the electrons has been solved and that the motion of the nuclei is slow with respect to that of theelectrons, this problem is reduced to two Eqs. (A.54a), for the nuclear wave functions ui (r), with r being the distance
between the nuclei, µ being their reduced mass, and the potential curves for the two states being V1 (r) and V2 (r), which
quasicross at r = R and that have a coupling V12 (r) between the states.
The approach by Stückelberg is done to eliminate u2 (r), resulting in a fourth-order equation for u1 (r). Developing thestandard WKB technique (Landau and Lifshitz, 1965), this equation can be solved by means of an expansion in powers ofh̄,[
u1 (r) = expih̄](S0 + h̄S1 + · · · ) .

(A.55)The zero-order terms give S0 /h̄ = ± k± (r)dr, where k± are determined by means of Eq. (A.54b) in terms of the adiabatic
potential functions V± (r), which are defined as

∫

V± =

1
2

(V1 − V2 ) ±

1
2

√

2
.
(V1 − V2 )2 + 4V12

(A.56)

Analysis of the first-iteration terms shows that this description becomes invalid near the classical turning points r = a± ,where V± (a± ) = E and near the transition points r = r± , where k+ (r± ) = k− (r± ). The latter takes place when the adiabatic
terms intersect, V+ (r± ) = V− (r± ); here, r± are the complex values with both having the same real part R = 12 (r+ + r− ).
The vicinities of these points define non-semi-classical regions that should be bridged by changes in the asymptotic
solutions along contours suitably defined in the complex r plane. Far from the turning-point regions, the solution becomes(∫ r
u1 (r) ≈ C sink± (r)dr +
a±

π
4

)

.

(A.57)

Around the transition zone, we obtain, to the left, at r ≪ R,

[

⎡ (±)
( ∫
)⎤
A
√1 exp ±i Rr k− dr
u1 (r)⎢ k⎥≈ ⎣ (±)−( ∫r)⎦
A2
u2 (r)√ exp ±i R k+ dr](A.58a)k+

and to the right, at r ≫ R,

⎡ (±)
( ∫
)⎤
B
√1 exp ±i Rr k+ dr
u1 (r)⎢ k⎥≈ ⎣ (±)+( ∫)⎦ .
B
u2 (r)√2 exp ±i Rr k− dr[](A.58b)k−Here, the upper and lower signs refer to positive- and negative-momentum solutions, respectively. Then, the left-handand right-hand coefficients become related[ (±) ]
B1

(±)

B2

√
√ P ±iφ=− 1 − Pe S

[

√

∓iφS
1−
√P e
P

] [ (±) ]A1(A.59)(±)

A2

with P = e−2π δ ,

δ=

1

π

{∫ r+
Im

}

(k− − k+ ) dr ,(A.60)

Rand certain phase factor φS . We write this as “certain” because the phase factor cannot be determined by the phase integral
method; this phase factor can be determined only by one of the differential equation methods (Child, 1974) (for moredetails, see the textbook (Child, 1996), particularly Appendices C and D therein).
In his paper, Stückelberg did not calculate the phase factor; later, it was shown that within such an approach, thisphase shift can only be defined in the diabatic limit (δ ≪ 1), with the result φS = π/4 (Thorson et al., 1971; Delos andThorson, 1972). (Note that our δ differs from the one in Child (1974) by a factor π , which is only a matter of notation.)
To interrelate with the linear drive problem, we can linearize the potential curves in the vicinity of the transition point
V1 (r) − V2 (r) ≈ − (F1 − F2 ) (r − R),

Fi = −⏐∂ Vi ⏐⏐,∂ r ⏐r =R
58(A.61)

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

and with V12 = ∆/2 = constant, we obtain the LZSM formula in the form of Eq. (7) with

δ=

∆2
,
4h̄v

v = ṙ |F1 − F2 | .

(A.62)For the double-passage evolution, we combine the nonadiabatic transitions, which we just considered above, and theadiabatic evolution described by the quasiclassical function in Eq. (A.57), hence resulting in Eq. (11) with

ΦSt =

∫ R

∫ R
k− dr −

k+ dr + φS .

(A.63)

a+a−

Therefore, we can see that this consists of two parts: the one accumulated during the adiabatic motion and the other one
φS , called the dynamical or Stokes phase, acquired during the single passage of the avoided-crossing region.
A.5. Duration of the LZSM transition
Transition dynamics matters not only for describing a single transition, but also for repeated processes. Specifically,
when describing periodic driving (which is the subject of Appendix B), if the time of a transition becomes larger than thetime span between subsequent transitions, tLZSM > Td /2, we cannot use the adiabatic-impulse model.
We can split the transition time tLZSM = tjump + trelax by two terms, where the jump time tjump describes the jumpof the probability from an initial value, to the vicinity of a final value, and where the relaxation time trelax states whenoscillations of the occupation probability decay around the final value P(t → ∞). Importantly, the transition times differ
in the adiabatic and diabatic bases; therefore, we define the relaxation times in both bases, following Vitanov (1999). In
the diabatic basis, Pd (∞) = 1 − P , and in the adiabatic basis, Pa (∞) = P . In this subsection, by P, we mean the upper-leveloccupation probabilities.
It is straightforward to define the jump time as follows:
tjump =P(∞)
P ′ (0).

(A.64)

Here, we should use the analytic solution for the probabilities in the LZSM problem, Eqs. (A.32), (A.34), expanding it in aseries around τ = 0. We find the value of this analytic solution in the zero-bias point:
Pd′ (0) =√2δ (1 − P ) cos χ,e−π δPa′ (0) = √8δ

(A.65a),(A.65b)whereχ (δ ) =

π
4

[ (
)]
[ (
)]
1
iδ
iδ
+ Arg Γ−− Arg Γ 1 −.
2

2(A.65c)2This very convenient definition, Eq. (A.64), is suitable for most cases, including diabatic dynamics and, in some cases,
adiabatic dynamics. However, as we see below, such definition becomes invalid for the adiabatic dynamics in the adiabatic
limit δ ≫ 1; see also Fig. 4. This is why we need to introduce a more elaborated definition.
For obtaining the transition times, we can use the expanded expression for P(τ > 0), Eqs. (A.32), (A.34), after passing
the avoided-crossing region in the strong-coupling regime with δ ≫ 1; then, we have
1

(

1

)

τ

√

2δ

√

cos ξ (τ ),τ + 2δ
τ + 2δ√√δδPa (τ > 0) ≈ P + (1 − 2P )+ P (1 − P )()3 sin ξ (τ ),2
38(τ + 2δ )2 τ 2 + 2δPd (τ > 0) ≈

2

+

2

−P

√

2

−

P (1 − P )2(A.66a)(A.66b)

where

[
)]
√
√
π
1 (
ξ (τ ) = −δ + 2δ ln √ τ + τ 2 + 2δ + τ τ 2 + 2δ + + Arg [Γ (1 − iδ)] .
4

2(A.66c)These expressions demonstrate that the approximate solution for the probability has two contributions. The first one
tends to P(∞) when τ tends to infinity; this term defines the jump time tjump . The second terms, with cos ξ (τ ) orsin ξ (τ ), describe decaying oscillations after passing the avoided-crossing region; these define the relaxation time trelax .
In particular, we can see that Pa′ (0) = 0 at δ ≫ 1. Hence, the above definition of the jump time becomes invalid.
(1)
Therefore, in general, the jump time tjump can be defined as the distance between the starting time tjump and ending time
(2)

(1)

tjump . The former can be defined as a moment when the probability leaves the vicinity of the initial probability, P(tjump ) =(2)), and the latter is defined as a moment when entering the vicinity of the final probability, P(tjump )

ηP(∞

59

− P(∞) = ηP(∞).

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Fig. 19. Transition times tjump and trelax versus the adiabaticity parameter δ . Here, the dimensionless time is τ = t [v/2h̄]1/2 . Note that in the adiabaticaabasis tjump≈ trelax, and also that these transition times grow exponentially in the adiabatic limit (δ ≫ 1). In the diabatic basis and the diabatic limitddd(δ ≪ 1) trelax> tjump, and there is no trelax
in the adiabatic limit. Note that τLZSM is plotted using Eqs. (21a) and (21b).

Here, η ≪ 1 is the small parameter that describes the magnitude of the vicinity near the initial and final probabilities.
Then, from Eqs. (A.32), (A.64), we obtain the jump time for the diabatic basis by expanding the parabolic cylinder functions
into a series (Gradshteyn and Ryzhik, 2007; Abadir, 1993):

√
1−P
d
tjump
= √
.
2δ cos χ (δ )(A.67a)From this, we can obtain the limit expressions:
d
tjump
≈

{ √
2 h̄π /v , δ ≪ 1,√
4 h̄δ/v , δ ≫ 1.

(A.67b)For the relaxation time in the diabatic basis, from Eq. (A.66a), we have

√ (
)
δ
1
d
trelax ≈ 2−1 .
v η2 (e2π δ − 1)

(A.67c)For the jump and relaxation times in the adiabatic basis, from Eqs. (A.34) and (A.64), it follows⎧ √⎨4 h̄δ , δ ≪ 1,
v
a
)1/6
tjump
≈ √ (
⎩ 2 1 8δ exp [4πδ ], δ ≫ 1,v η

(A.68a) [] 4δ ( e2π δ − 1 )1/3
atrelax ≈ √−1 .
v
16η2 δ 4(A.68b)andIn the limiting cases for the relaxation time, we then haveatrelax
≈

⎧√
2
1/3
⎪
⎪
⎨ v (2δ/η) , δ ≪ 1,

(A.68c)√ ()⎪⎪⎩ 2 exp[π δ] 1/3 (δ/2)1/6 , δ ≫ 1.
v
ηd
We show the transition times in Fig. 19. To plot these, we have used Eq. (A.67c) for t drelax and Eq. (A.67a) for tjump
inaaFig. 19(a); then, we used Eq. (A.68b) for trelax
and Eq. (A.68c) for tjump
in Fig. 19(b).
da≫ trelax. This difference in the relaxation times happens because the oscillations of Pd (t)Here, for δ ≪ 1, we have trelax
vanish proportionally to t −1 , but Pa (t) decays proportionally to t −3 .

60

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89adaa
For the case δ ≫ 1, we have the opposite situation: trelax≫ trelax≈ 0. Note that trelax≈ tjump; therefore, we can defineatLZSM ≈ 2trelax . As a result, the equations above can be summarized as Eqs. (21a), (21b) in the main text.

Appendix B. Description of a periodically driven two-level system

B.1. Adiabatic-impulse model (AIM)In this section, we describe the adiabatic-impulse model in more detail, expanding on Section 3.1.
B.1.1. Adiabatic evolution
Consider now the adiabatic evolution when the system passes in one of the adiabatic eigenstates |E± (t)⟩. This meansthat the adiabatic basis consists of the instantaneous eigenstates of the time-dependent Hamiltonian. We can obtain the
instantaneous eigenfunctions from the Schrödinger equation (A.1), where time t is a parameter:
H(t) |E± (t)⟩ = E± (t) |E± (t)⟩ .

(B.1)

To describe the system dynamics, we solve the dynamical form of the Schrödinger equation, Eq. (15), with the initialcondition|E± (ti )⟩ = |Ei± ⟩ .

(B.2)

We substitute Eq. (B.1) into the Schrödinger equation (15), solve this differential equation, and then obtain the evolutionof the upper- and lower-energy-level wave functions

(
)
∫ t
i
′
′
|E± (t)⟩ = exp −E± (t )dt |Ei± ⟩ = exp (∓iζ ) |Ei± ⟩ ,
h̄ ti
∫ t
1
ζ (t) =
∆E(t) dt .
2h̄

(B.3a)(B.3b)ti

As a result, we obtain the adiabatic evolution matrix, Eq. (27).
B.1.2. Multiple-passage evolution
Here we describe some aspects of the multiple-passage evolution in relation to Section 3.1. To raise the matrix Ξ to
the nth power [see Eq. (46)], we first need to diagonalize it. For this, we use the unitary matrix

(
A=

A11
A21

−A∗21

such that AΞ A† =

{

)

A∗11

AA† = 1,

,

( iφ
e
0

0

|A11 |2 + |A21 |2 = 1,

(B.4)

)

e−iφ

, where φ is the desired value. Consider the equation Ξ = A†

( iφ
e
0

0

e−iφ

)
A and obtainΞ11 = |A11 |2 eiφ + |A21 |2 e−iφ
Ξ21 = −2iA11 A21 sin φ.

(B.5)

From these, we define φ : cos φ =ReΞ11 . In the next step, we find the matrix Ξn of the evolution for n periods

Ξn = Ξ n = A†

( inφ
e

)

0

e−inφ

0

(
A=

Ξn11
Ξn21

)
∗
−Ξn21
,
∗
Ξn11

(B.6a)

where we simplify the obtained matrix elements, here taking into account Eq. (B.5):

Ξn11 = cos(nφ ) + iIm(Ξ11 )Ξn21 = Ξ21

sin (nφ )sin φsin (nφ )sin φ,(B.6b).

(B.6c)

Then, we obtain the probability of the upper-level occupation of the TLS during their respective time intervals:(1)

P+ (n) = |Ξn21 |2 = |Ξ21 |2(2)

P+ (n) = 2Q1

sin2 nφ
sin φ
2

+ Q2

sin2 nφsin2 φsin 2nφsin φ,

for (t − nTd ) ∈ (t2 , Td + t1 ),+ P cos 2nφ,for (t − nTd ) ∈ (t1 , t2 ),(B.7a)(B.7b)

where
Q1 = P P sin2 ζ− + (1 − P )(1 + cos ζ+ cos ζ− ) ,[](B.7c)
61

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Q2 = 2P (1 − P ) cos (ζ1 + φS ) cos (ζ2 + φS ).

(B.7d)For the time-averaged value, we need to average over many periods n ≫ 1, so we obtain
(1)

P+ =

1
|Ξ21 |2|Ξ21 |2=,2
2
2 |Ξ21 | + (ImΞ11 )2
2 sin φ
Q1

(2)

P+ =

sin2 φ(B.8a).

(B.8b)(1)(2)

We can neglect the difference between P+ and P+ in the case of the slow-passage limit P ≪ 1, that is, δ ≫ 1. Here, we
use P as a small parameter, and in the first approximation, we obtain Eq. (58).
B.2. Rotating-wave approximation (RWA)√

Consider now the case of strong excitation, ∆ ≪ Ah̄ω at kh̄ω ≈ ∆E, where the latter condition means that theenergy of k photons approximately equals the energy distance between the qubit levels ∆E. Here, we follow Refs. Silveri
et al. (2015) and Ono et al. (2019) presenting the formalism valid not only for the cosine driving but rather for a generic
excitation ε (t) = ε0 + ε̃ (t) for any function ε̃ (t) with the period Td = 2π/ω.
Let us first split the Hamiltonian into a stationary part H0 and a time-dependent part V (t):H = H0 + V (t) = −

∆

σx −

ε0

σz −

ε̃ (t)

σz .

(B.9)

2
2
2
To make it convenient to solve the Bloch equations, we perform the unitary transformation:[
U = exp −i

η(t)
2

]
η
η
σz = cos − iσz sin ,2η(t) =

2

1

∫ t

h̄

0

dt ′ ε̃ t ′ .

( )(B.10)We use the operator U(t) to link the wave function in the rotating and stationary coordinate systems, ψ = U(t)ψ ′ . Then,
we substitute this function into the Schrödinger equation and obtain ih̄U ψ̇ ′ + ih̄U̇ ψ ′ = HU ψ ′ . In the rotating coordinate
system, we need ih̄ψ̇ ′ = H ′ ψ ′ , thus obtaining the new HamiltonianH ′ = U † HU − ih̄U † U̇ = −

ε0
2

σz −

∆ ( iη
2e σ+ + h.c . ,)(B.11)

with σ+ = 2 σx + iσy . Then, the preparatory stage is finalized by the Fourier series expansion

(
1

)

∞
∑

∆eiη =∆m eimωt ,(B.12)m=−∞where the complex-valued amplitude is given by the inverse Fourier transformω∆m = ∆2π∫ 2π /ωdt e−imωt iη(t)

e

=∆

∫ 1

dτ ′ exp iη(τ ′ ) − i2π mτ ′ .

[](B.13)

00Then, the Hamiltonian becomes
′

H =−

ε0
2

σz +

∞
1 ∑ (

2

imωt

∆m e

σ+ + h.c . = −

m=−∞

)

∞
1 ∑

2

m=−∞

(

ε0

∆m e−imωt)∆m eimωt.
−ε0(B.14)

With this Hamiltonian, it is convenient to solve the Bloch equations for the density matrix,iρij − ρij(0)

h̄

τij

ρ̇ij = − [H , ρ ]ij −,(B.15)(0)(0)

where ρij is the density matrix, τij is the relaxation rate, ρ (0) is the equilibrium density operator with ρ01 = ρ10 = 0,

= tanh 2k∆ET , and kB is the Boltzmann constant. For this, we parameterize the density matrix
B
(
) (
)
) 1 1 + Z X − iY1(
ρ00 ρ01ρ=
1σ0 + X σx + Y σy + Z σz =≡.
1−Z
ρ10 ρ112
2 X + iY

Z (0) = ρ

(0)
00

− ρ

(0)
11

(B.16)For the moment, we assume that the system is driven close to a resonance, where ∆E ≈ |ε0 | ≈ kh̄ω. Then, we omit
the “fast-rotating” terms and leave only terms with m = k. We can write down the Bloch equations with the HamiltonianH ′ component-wise:Ż = −∆k2h̄Im(e−ikωt ) − (Z − Z (0) )Γ1 ,(B.17a)62

O.V. Ivakhnenko, S.N. Shevchenko and F. Noriρ̇10 = i∆k ikωt
e

2h̄

Z −i

ε0
h̄

Physics Reports 995 (2023) 1–89

ρ10 − ρ10 Γ2 ,(B.17b)

where
(0)

∆E

(0)

Z (0) = ρ00 − ρ11 = tanh

2kB T

.

(B.17c)Here ∆k was defined in Eq. (70). After the substitution ρ10 exp (−ikωt ) = X̃ + iỸ , we obtain the system of equations
X̃˙ =

(

ε0

kω +

h̄

(

Ỹ˙ = − kω +Ż = −∆k)ε0Ỹ − X̃ Γ2 ,
X̃ +

h̄

(B.18a)∆k)

Z − Ỹ Γ2 ,h̄

(B.18b)Ỹ − (Z − Z (0) )Γ1 .

h̄(B.18c)The stationary solution of these equations can be obtained after substituting X̃˙ = Ỹ˙ = Ż = 0. Consider now the lowtemperature case, T → 0, then Z (0) ≈ 1. Finally, we obtain the stationary value for the probability of the upper diabatic
state; summing all possible resonant terms, we obtain the qubit upper-level occupation probability
(k)

(k)

P up = ρ 11 =P up =

)
1(
(k)
,
1−Z
2

(k)

∑

P up =

k(B.19a)∞1 ∑2|∆k |2|∆k |2 + ΓΓ1 (kh̄ω − ε0 )2 + h̄2 Γ1 Γ2

k=−∞

.

(B.19b)2

Note that for a complex-valued ∆k , what matters is its absolute value.
Finally, to obtain the upper-level occupation probability for a given bias ε̃ (t), we must calculate the functions ∆k .
Although more examples can be seen in Silveri et al. (2015) and Ono et al. (2019), for the sinusoidal modulation, with
the Jacobi–Anger expansion following Eq. (70), it is straightforward to see that∆m (x) = ∆Jm (x),

x=

A
h̄ω

.

(B.20)

It is useful to recall here the asymptote at x ≫ 1:√
Jm (x) ≈2πx[

cos x −

πm
2

−

π]
4

.

(B.21)

This explicitly demonstrates that the occupation probability is quasiperiodic in driving the amplitude with the periodδ A = 2π h̄ω.
B.3. Floquet theory
From Floquet theory (Son et al., 2009), we can obtain a solution |ψ (t)⟩ for the Schrödinger equation (15) with any
time-periodic Hamiltonian H(t) = H(t + nTd ) for any integer n|ψ (t)⟩ = e−iϵ t |Φ (t)⟩ ,(B.22)

where |Φ (t)⟩ denote the time-periodic Floquet modes with the same period as the Hamiltonian H(t) and ϵ is the so-called
quasienergy. When we substitute Eq. (B.22) into Eq. (15), we can obtain an equation for the quasienergy(
H(t) − ih̄

d

)

dt

|Φ (t)⟩ = ϵ |Φ (t)⟩ .

(B.23)

Periodic functions can be decomposed into harmonic functions using the Fourier series expansion|Φ (t)⟩ =∞∑|Φn ⟩ einωt ,(B.24a)H [n] einωt ,(B.24b)n=−∞H(t) =∞∑n=−∞where the superscript [n] defines the number of the 2 × 2 part of the Hamiltonian matrix. For the next step, we rewrite
the Hamiltonian in its exponential form
H(t) = −

∆
2

σx −

ε0
2

σz −

A
4

(
)
σz eiωt + e−iωt ,(B.25)
63

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

and we obtain the system of stationary equations for the quasienergies using the Schrödinger equation (15) with thesubstitutions (B.22) and (B.24)∑∑
i⟨0, n|HF |i, m⟩⟨i, m|ϵj ⟩ = ϵj ⟨0, n|ϵj ⟩,(B.26)mwhere HF is the Floquet Hamiltonian, ϵj is the quasienergy eigenvalue, |ϵj ⟩ is an eigenvector, and |i, m⟩ is the mth Fouriercomponent of the ith energy level. We can also write it in terms of matrix elements

)
(
ε0
A
∆
ϵj Φj,n = − σx − σz + nω Φn − σz (Φn−1 + Φn+1 ) ,2

2(B.27)

4

where j defines the number of a state, which is either 0 or 1 for a TLS after decomposing the wave function, Eq. (3). Then,
we obtain the time-independent Floquet Hamiltonian, which is defined by⟨in | HF |jk ⟩ = H0[n,1−k] + nh̄ωδij δnk ,(B.28a)where i and j define the number of the state, or
HF,nk = H [n−k] + nh̄ωδnk I ,(B.28b)for 2 × 2 elements (I is 2 × 2 identity matrix) with three nonvanishing Fourier components of the Hamiltonian H [n−k] ,where n and k are the integer numbers corresponding to different states:
H [0 ] = −

ε0
2 ∆
1

(

)
(
1 A
∆
, H [+1] = H [−1] = −
−ε0
4 0

)

0
.
−A

(B.29)

This consists of the 2 × 2 submatrices, and Eq. (B.28b) defines its element in the nth column and the kth row. Hence,
the Floquet states matrix can be written as

⎛

..

⎞

⎜ .
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
HF = ⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

− ε20 + (n − 1)h̄ω

− ∆2

− A4

0

0

0

+ (n − 1)h̄ω

0

A
4

0

0

− A4

0

− ε20 + nh̄ω

− ∆2

− A4

0

0

A
4

− ∆2

+ nh̄ω

0

A
4

0

0

− A4

0

− ε20 + (n + 1)h̄ω

− ∆2

0

0

0

A
4

− ∆2

ε0

− ∆2

2

ε0
2

ε0
2

+ (n + 1)h̄ω
..

.

⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

(B.30)The eigenvalues of this matrix can be found numerically if we take a finite number of these 2 × 2 blocks. Then, we canobtain the time-averaged upper-level occupation probability
P up =∑∑ ⟨⟩⟨⟩ 2| 1n |ϵj ϵj |0(n=0) | .
n(B.31)j

The Schrödinger equation with the Hamiltonian HF in the general case cannot be solved analytically; therefore, we firstconsider the eigenvalue problem with ∆ = 0; next, we can use perturbation theory with the small parameter being ∆.
We denote H0 as the unperturbed Hamiltonian. In the situation when ∆ = 0, there is no coupling between the |0⟩ and|1⟩ states; therefore we can write the same Hamiltonian H0′ for both states:⎛⎜⎜⎜H̃0 (for state 0 or 1) = ⎜
⎜
⎝

..

.

⎞
b + (n − 1)h̄ωa0ab + nh̄ω
a

64

⎟
⎟
⎟
⎟
⎟
⎠

0
a

b + (n + 1)h̄ω..

.

(B.32)

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

ε

εwhere b = − 20 , a = − A4 for the |0⟩ state and b = 20 , a = A4 for the |1⟩ state. To find the eigenfunctions, we use the
Schrödinger equation (B.23) with Hamiltonian H̃0′ , in which ∆ = 0. As a result, using the Jacobi–Anger formula (B.20), we
obtain|Φ̃n (t)⟩ = einωt e−i(2a/h̄ω) sin ωt =

∞
∑

(
Jk

−

k=−∞

)

2a
h̄ω

ei(n+k)ωt =

∞
∑

(
Jk−n

)

A
2h̄ω

k=−∞

e−ikωt .

(B.33)The eigenstates of the unperturbed Hamiltonian H0 are the following:∞∑⏐0̃, n =

⏐

⟩

(
Jk−n

2h̄ω

k=−∞

∞
∑

⏐1̃, m =

⏐

⟩

)

A

(
Jk−m

k=−∞

−

|0, k⟩ ,

)

A
2h̄ω

(B.34a)|1, m⟩ .

(B.34b)For the Floquet Hamiltonian (B.30), using the addition theorem for Bessel functions
Jn (2z) =∑Jm (z) Jn−m (z),(B.35)

m

we can obtain the matrix elements for the Floquet Hamiltonian based on this (B.30):

( )
∞
∞
∑
∑
⟨ ′⏐ ⏐ ′ ⟩∆A
Jk−n (z) Jl−m (−z) ⟨0k | HF |1l ⟩ = −0n ⏐ HF ⏐1m =
Jm−n
,
2
h̄ω
k=−∞ l=−∞( )⟨ ′⏐ ⏐ ′ ⟩∆A
1n ⏐ HF ⏐0m = −
Jn−m
,
2
h̄ω
)
(
⟨ ′⏐ ⏐ ′ ⟩ε0+ nh̄ω δnm ,0n ⏐ HF ⏐0m = −2)⟨ ′ ⏐ ⏐ ′ ⟩ ( ε01n ⏐ HF ⏐1m =+ nh̄ω δnm .

(B.36a)(B.36b)(B.36c)(B.36d)2

From Eqs. (B.36), if we have the condition
resonances, ε0 ≈ nh̄ω, then we can neglect all other⏐ ⟩for the⏐ multi-photon⟩
coupling terms, except the one between ⏐0′0 and ⏐1′−n ; therefore, we obtain a 2 × 2 matrix Hamiltonian:

ε0
HRWA = −′2 ∆ J−n1(

A)∆ J−′ n−ε0 + nh̄2ω

)(B.37a)

where

(

′

Jn = Jnh̄ω.

(B.37b)Generalized Van Vleck perturbation theory: The RWA is the first approximation in Floquet theory; for the next
approximations, we can use generalized Van Vleck perturbation theory (GVVPT) (Son et al., 2009). The perturbation (small)parameter is

λ=−

∆
2

.

(B.38)⏐ ⟩⏐ ⟩Here, we can rewrite the Floquet Hamiltonian in the basis of the states ⏐0′ and ⏐1′ :nmH̃F = H̃0 + λṼ ,′(B.39a)

where

⎛

..

⎜ .
⎜
⎜
⎜
⎜
⎜
′
H̃0 = ⎜
⎜
⎜
⎜
⎜
⎜
⎝

⎞
− ε20 − h̄ω
0
0
0
0
0

ε0
2

0

− h̄ω
0
0
0
0

0
0

− ε20
0
0
0

0
0

0
0

0
0

0

0
0

0
0
0

ε0
2

0
0

ε0

− 2 + h̄ω
0

65

ε0
2

+ h̄ω

..

.

⎟
⎟
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎟
⎟
⎠

(B.39b)

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori⎛..

Physics Reports 995 (2023) 1–89

⎞

.

⎜
⎜
⎜
⎜
⎜
⎜
Ṽ = ⎜
⎜
⎜
⎜
⎜
⎝

J0′
0

0
J0′
0
J1′
0
J2′

J1′
0
J0′
0

0

′
J−
1
0
′
J−
2
0

′
J−
1
0
J0′
0
J1′

0
′
J−
2
0
′
J−
1
0
J0′

′
J−
1

0

J2′
0
J1′
0
J0′
0

..

.

⎟
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎟
⎠

(B.39c)We can reduce the infinite-dimensional matrix of the Floquet Hamiltonian (B.30) to a 2 × 2 matrix Hamiltonian h by using
the nearly degenerate perturbation formalism in the GVVPT (Aravind and Hirschfelder, 1984). Following perturbation
theory, the 2 × 2 matrix of Hamiltonian h and its eigenstates solutions φ can be expanded in powers of λ
h =

∞
∑

λm h(m) ,(B.40a)λm φ (m) .

(B.40b)

m=0

φ =

∞
∑
m=0

For the n-photon resonance, the Floquet states are nearly degenerate, so⏐ ⟩⏐ ⟩φ−(0) = ⏐0̃0 and φ+(0) = ⏐1̃0

(B.41a)andh(0) =

(

1

)

0
,
ε0 − 2nh̄ω

−ε0
0

2

(B.41b)where the two states are nearly degenerate, that is, ε0 ≈ nh̄ω. Using GVVPT, a few high-order terms can be obtained:∑′

φ−(1) =kφ+(1) =

⏐ ⟩−Jk′
⏐1̃k ,
ε0 + kh̄ω

∑′

Jk′

k

ε0 + kh̄ω⏐1̃−n−k ,⏐⟩′h(1) = φ (0) |V ′ |φ (0) = J−⟨

(B.42a)⟩(0
1(B.42b))1,0h(2) = φ (0) |V ′ |φ (1) − h(1) φ (0) |φ (1) =

⟨

⟩

⟨

⟩

(B.42c)

∑′

Jk′2

(
−1

)

0
,
1

(B.42d)ε0 + kh̄ω 0
(
)(
)
′
∑′ ∑′
⟨ (1) ′ (1) ⟩ ⟨ (1) (1) ⟩ (1)Jk′ Jl′ Jk′ +l+nJk′2 J−0 1
n(3)
h = φ |V |φ− φ |φ h = −.
(B.42e)+1 0(ε0 + kh̄ω) (ε0 + lh̄ω) (ε0 + kh̄ω)2kl∑′Here, the summation
k is from −∞ to ∞, with the prime standing for k ̸ = −n. As a result of GVVPT, we obtain thekHamiltonian, which includes n-photon coupling channels
HGVVPT =−ε0 − δn
˜n
∆
2

1

)
˜n
∆
,
ε0 + δn − 2nh̄ω((B.43)where˜n = J−′ n 2λ −
∆
δn =

∑′

2

k

(
∑′ ∑′
k
Jk′2

ε0 + kh̄ωlJk′ Jl′ Jk′ +l+n

(ε0 + kh̄ω) (ε0 + lh̄ω)+′Jk′2 J−n(ε0 + kh̄ω)2λ2 + O(λ4 ).

)2λ3 + O(λ5 ),(B.44a)(B.44b)Then, we find the eigenvalues of the GVVPT Hamiltonian (B.43)

ϵ0,1 = −nh̄ω2± h̄Ω̃R ,(B.45a)

where
h̄Ω̃R

(

)2

˜2n .
= (nh̄ω − ε0 − δn )2 + ∆(B.45b)66

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

This gives the n-photon time-dependent upper-level occupation probability

˜2n
∆

1

(n)
(t) =
Pup

(

2 h̄Ω̃R 2()

1 − cos Ω̃R t ,)(B.46)

and averaged occupation probability
(n)
P up

1

∫ t

t →∞ t

0

= lim

(n) ′
(t )dt ′ =
Pup

˜2n
∆
.
˜2n + (nh̄ω − ε0 − δn )2
2∆
1(B.47)

When we take only the first term of this, we can reach the RWA limit(n)

P up =∆2n,
2 ∆2n + (nh̄ω − ε0 )21(B.48)

where ∆n is the same as in the RWA and as was defined in Eq. (B.20).
B.4. Rate-equation and white-noise approach
In this subsection, we describe a TLS dynamics modeling the decoherence by adding classical noise and then solving
the rate equation. We demonstrate that such approach gives exactly the same solution as the one by solving the Blochequation in the RWA.
B.4.1. Transition rate
Consider a transition rate in a driven TLS, following Berns et al. (2006) (see also Otxoa et al. (2019)). This is describedby the Hamiltonian

ε (t)
∆
2

)
∆
,
−ε (t)(B.49a)ε(t) = ε0 + A cos ωt + δε(t),(B.49b)H(t) = −1(

where δε (t) stands for classical noise, which models decoherence here.
After the unitary transformation(
U = exp i

φ (t)
2

)
σz ,

φ (t) =

1

∫ t

h̄

0

dt ′ ε (t ′ ),(B.50)∆(t)(B.51a)we obtain
H ′ (t) = U † HU − ih̄U † U̇ =(0∆∗ (t)

)

0

,

where

∆(t) = ∆e−iφ .

(B.51b)

Perturbation theory gives the transition rate:
W = lim⏐⏐⏐At ,t ′ ⏐2δtδ t →∞,

At ,t ′

=

1

∫ t′

2

t

dt ′ ∆(t ′ ),δt = t ′ − t .

(B.52)

Here, the limit implies δ t ≫ T2 . Perturbation theory is valid, provided that the change of the qubit population is slow on
the scale of T2 , which means that we have W ≪ T2 (Berns et al., 2006; Danon and Rudner, 2014).
We now add the averaging over white noise ⟨...⟩δε and rewrite this:
W =

1

∫ τ

4 δ t →∞ δ t

0

1

lim

dτ

∫ t′
t

dt ′ ∆(t + τ )∆∗ (t ′ ) δε .

⟨⟩(B.53)

Next, we use the Jacobi–Anger expansion, Eq. (B.20), to remove the sine from the exponent(

e−iφ (t) = exp −i

ε0
h̄

t −i

A
h̄ω

)

sin ωt − iδφ (t)ε0= e−i h̄ t e−iδφ (t)∞∑Jn (x)e−inωt .

(B.54)

n=−∞Here, we defineδφ (t) =

1

∫ t

h̄

0

dt ′ δε (t ′ ) and x =

A
h̄ω

.

(B.55)
67

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

If the noise is frequency independent (i.e., white) in a low-frequency region, the averaging gives (Du et al., 2013b)⟨

eiδφ (t) e−iδφ (t )
′

⟩
δε

⏐
⏐)
(
= exp −Γ2 ⏐t − t ′ ⏐ .

(B.56)

Then, calculating the integral

∫ ∞

dτ e

(
)
ε
−i h̄0 −nω τ −Γ2 |τ |

e

−∞

= (

2Γ2
ε0
h̄

− nω

)2

+ Γ22

,

(B.57)

we obtain the transition rate
∞
∑

W =

Wn ,

(B.58a)

∆2n
,
)2
ε0
− nω + Γ22h̄(B.58b)

n=−∞

1

Wn =

2

Γ2 (

∆n = ∆ Jn (x).

(B.58c)Note that the transition rate W consists of the partial contributions Wn , each of which is essential in the vicinity of thenth resonance, ε0 ∼ nh̄ω.
B.4.2. Rate equation
The rate equation for a TLS reads
dP+
dt

(
)
= W + Γ1′ P− − (W + Γ1 ) P+ .

(B.59)

Here, Γ1 refers to the relaxation rate from the excited to the ground state, and Γ1′ = Γ1 exp [−∆E /kB T ] stands for thethermally-excited inverse relaxation. From this, it is apparent how to take temperature into account. In the following, we
assume the low-temperature limit (T = 0) and adopt Γ1′ = 0.
As the second equation, we can write down the equation for dP− /dt, analogously to Eq. (B.59), or use the condition
P+ + P− = 1. Then, in the stationary regime, with the zero left-hand side in Eq. (B.59), we obtain
W

P+ =2W + Γ1.

(B.60)In the vicinity of the kth resonance, that is, at ε0 ∼ kh̄ω, the probability is given by the kth term Wk , Eq. (B.58b):
Wk

(k)

P+ =

2Wk + Γ1.

(B.61)The summation of all possible resonances gives the full upper-level occupation probability:
P+ =

∞
∑

(k)

P+ =

k=−∞

∞
1 ∑

∆2k

Γ1
2
2
k=−∞ ∆k + Γ2

( ε0
h̄

− kω

)2

+ Γ1 Γ2.

(B.62)

This exactly coincides with the solution of the Bloch equations in the RWA, Eq. (69).
B.4.3. From Bessel to Airy
It may be instructive to rewrite the Bessel function via the Airy function, as in Ref. Berns et al. (2006), (see also Ref. Malla
and Raikh (2019))( ) 132

Jk (x) ≈x(( ) 1
3
2

Ai

x

)
(k − x) ,(B.63)

which is valid for n ≫ 1 and n > x. Here, we can obtain Eq. (B.63) using Eqs. (8.455) and (8.433)( from )Ref. Gradshteyn

and Ryzhik (2007) and the definition of the Airy function of the first kind, Ai (u) = π1∫∞0dt cost3
3+ ut .

With this remark, we can proceed with the series in Eq. (B.58a). From the denominator in Eq. (B.58b), it follows that we
can make the replacement in the nominator: k −→ ε0 /h̄ω; then, the respective Airy function goes out of the summation.
We can execute the summation by expanding the fraction into two elementary fractions:
1

( ε0
h̄

− kω

)2

+ Γ22

=

1

Γ2 ω

Im

1
z−k

,

z=

ε0
Γ2
−i .
h̄ω
ω
68

(B.64)

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

To calculate the sum, consider the expansion of the cotangent (Gradshteyn and Ryzhik, 2007) and its approximation for|z | ≫ 1:
cot π z =

1

πz

∞
1 ∑

+

π

z
k (z − k)

k=−∞

≈

∞
1 ∑

π

1

k=−∞

z−k

.

(B.65)As a result, we obtain

π ∆2
W ≈
2h̄ω

(

2h̄ω

)2/3

(√
Ai

A

3

2

)

2h̄ω ε0 − A

[ (
)]
ε0
Γ2
× Im cot π
−i
.
h̄ω
ω

h̄ω

A

(B.66)

This formula, together with Eq. (B.60), is very convenient for further analysis of the upper-level occupation probability,of which the example follows.
B.4.4. Double-passage regime
Consider now the case when the decoherence rate is of the order of a driving period, Γ2 ≳ ω. Then, cot(...) ≈ i and,making use of Airy function asymptotics at a large negative argument (i.e., for the relevant A > ε0 ), we obtain:

π ∆2
W ≈
2h̄ω

(

4

) 13

2

(( ) 1
3
2

Ai

x2

x

ε0 − Ah̄ω)

1 ∆2 (≈2 A

1−ε0 )−1/2
A

[ √
2

cos

2 2 A (3h̄ω1−ε0 )3/2
A

−

π
4

]
.

(B.67)

Eq. (B.67) can be simplified to leading order in ε0 /A:
W ≈

1 ∆2
2 A

[ √
2

cos

√ ε0
π
− 2
−
3 h̄ω
h̄ω
4

2 2 A

]
.

(B.68)

Importantly, the above result can be obtained by directly considering the double-passage regime, as pointed out
by Berns et al. (2006). Indeed, from the adiabatic-impulse model, in the fast passage limit (∆2 /Ah̄ω < 1), for the upperlevel occupation probability, it follows [see Eq. (48) and also in Refs. Shevchenko et al. (2010) and Chatterjee et al. (2018)]:

doubleP+≃ 2π

ΦSt = −

∆2
Ah̄ω

(
1−

ε02

)−1/2

A2

ε0
A
ε0
+
arccosAh̄ωh̄ω

sin2 ΦSt ,

(
1−

ε02

(B.69a)

)1/2

A2

−

π
4

.

(B.69b)Eq. (B.69a) can be simplified to leading order in ε0 /A:
doubleP+≃

2π ∆ 2
h̄ω A

sin2

[

A
h̄ω

−

]
π ε0
π
−
.
2 h̄ω
4

(B.70)

Remarkably, the oscillationsin this latter formula are very√
√ close to the ones in Eq. (B.68). Compare the respective terms
in the two equations: 2 3 2 ≃ 0.94, which meets 1, and 2 ≃ 1.4, which meets π/2 ≃ 1.57. From Eq. (B.60), we haveP+ ≃ T1 W , and then, we have identical multiplier factors before sine in Eq. (B.70) and from Eq. (B.68), provided therelaxation time is equal to two driving periods, T1 = 2Td , where Td = 2π/ω.
References
Abadir, K.M., 1993. Expansions for some confluent hypergeometric functions. J. Phys. A: Math. Gen. 26 (16), 4059–4066.
Abovyan, G.A., Kryuchkyan, G.Y., 2016. Multiphoton resonant manipulation of qubits by train of pulses. J. Opt. Soc. Amer. B 33
(5), 971.
Aers, G.C., Studenikin, S.A., Granger, G., Kam, A., Zawadzki, P., Wasilewski, Z.R., Sachrajda, A.S., 2012. Coherent exchange anddouble beam splitter oscillations in a triple quantum dot. Phys. Rev. B 86 (4), 045316.
Agarwal, K., Ganeshan, S., Bhatt, R.N., 2017. Localization and transport in a strongly driven Anderson insulator. Phys. Rev. B 96(1), 014201.
Ahmadinouri, F., Hosseini, M., Sarreshtedari, F., 2019. Investigation of robust population transfer using quadratically chirped laserinteracting with a two-level system. Phys. Scr. 94 (10), 105404.
Akkermans, E., Dunne, G.V., 2012. Ramsey fringes and time-domain multiple-slit interference from vacuum. Phys. Rev. Lett. 108(3), 030401.
Albash, T., Lidar, D.A., 2018. Adiabatic quantum computation. Rev. Modern Phys. 90 (1), 015002.
Alzar, C.L.G., Martinez, M.A.G., Nussenzveig, P., 2002. Classical analog of electromagnetically induced transparency. Amer. J. Phys.
70 (1), 37–41.
Amin, M.H.S., 2009. Consistency of the adiabatic theorem. Phys. Rev. Lett. 102 (22), 220401.
69

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Anquez, M., Robbins, B.A., Bharath, H.M., Boguslawski, M., Hoang, T.M., Chapman, M.S., 2016. Quantum Kibble-Zurek mechanism
in a spin-1 Bose-Einstein condensate. Phys. Rev. Lett. 116 (15), 155301.
Ao, P., Rammer, J., 1989. Influence of dissipation on the Landau-Zener transition. Phys. Rev. Lett. 62 (25), 3004–3007.
Ao, P., Rammer, J., 1991. Quantum dynamics of a two-state system in a dissipative environment. Phys. Rev. B 43 (7), 5397–5418.
Aravind, P.K., Hirschfelder, J.O., 1984. Two-state systems in semiclassical and quantized fields. J. Phys. Chem. 88 (21), 4788–4801.
Arbó, D.G., Ishikawa, K.L., Schiessl, K., Persson, E., Burgdörfer, J., 2010. Intracycle and intercycle interferences in above-thresholdionization: The time grating. Phys. Rev. A 81 (2), 021403.
Ashhab, S., 2014. Landau-Zener transitions in a two-level system coupled to a finite-temperature harmonic oscillator. Phys. Rev.
A 90 (6), 062120.
Ashhab, S., 2016. Landau-Zener transitions in an open multilevel quantum system. Phys. Rev. A 94 (4), 042109.
Ashhab, S., 2017. Landau–Zener–Stueckelberg interferometry with driving fields in the quantum regime. J. Phys. A 50 (13), 134002.
Ashhab, S., Ilinskaya, O.A., Shevchenko, S.N., 2022. Nonlinear Landau-Zener-Stückelberg-Majorana problem. ArXiv, 2208.11416.
Ashhab, S., Johansson, J.R., Nori, F., 2006. Decoherence in a scalable adiabatic quantum computer. Phys. Rev. A 74 (5), 052330.
Ashhab, S., Johansson, J.R., Zagoskin, A.M., Nori, F., 2007. Two-level systems driven by large-amplitude fields. Phys. Rev. A 75 (6),063414.
Atia, Y., Oren, Y., Katz, N., 2019. Robust diabatic Grover search by Landau-Zener-Stückelberg oscillations. Entropy 21 (10), 937.
Autler, S.H., Townes, C.H., 1955. Stark effect in rapidly varying fields. Phys. Rev. 100 (2), 703–722.
Averbukh, I.S., Perel’man, N.F., 1985. Quasienergy and optical spectra of a two-level system in a low-frequency field of arbitrarystrength. JETP 61 (4), 665.
Band, Y.B., Avishai, Y., 2019. Three-level Landau-Zener dynamics. Phys. Rev. A 99 (3), 032112.
Band, Y.B., Japha, Y., 2022. Tuning the adiabaticity of spin dynamics in diamond nitrogen vacancy centers. J. Phys.: Condens.
Matter 34, 255503.
Barnes, E., 2013. Analytically solvable two-level quantum systems and Landau-Zener interferometry. Phys. Rev. A 88 (1), 013818.
Barnes, E., Calderon-Vargas, F.A., Dong, W., Li, B., Zeng, J., Zhuang, F., 2022. Dynamically corrected gates from geometric spacecurves. Quantum Sci. Technol. 7 (2), 023001.
Barone, S.R., Narcowich, M.A., Narcowich, F.J., 1977. Floquet theory and applications. Phys. Rev. A 15 (3), 1109–1125.
Barra, F., Esposito, M., 2016. Dissipation in small systems: Landau-Zener approach. Phys. Rev. E 93 (6), 062118.
Baruch, M.C., Gallagher, T.F., 1992. Ramsey interference fringes in single pulse microwave multiphoton transitions. Phys. Rev.
Lett. 68 (24), 3515–3518.
Bason, M.G., Viteau, M., Malossi, N., Huillery, P., Arimondo, E., Ciampini, D., Fazio, R., Giovannetti, V., Mannella, R., Morsch, O.,2012. High-fidelity quantum driving. Nat. Phys. 8 (2), 147–152.
Bassani, G.F. (Ed.), 2006. Ettore Majorana Scientific Papers. On Occasion of the Centenary of His Birth. Springer, Berlin, pp. 125–132,(Chapter).
Bäuerle, C., Bunkov, Y.M., Fisher, S.N., Godfrin, H., Pickett, G.R., 1996. Laboratory simulation of cosmic string formation in theearly universe using superfluid 3He. Nature 382 (6589), 332–334.
Béguin, A., Rodzinka, T., Vigué, J., Allard, B., Gauguet, A., 2022. Characterization of an atom interferometer in the quasi-Bragg
regime. Phys. Rev. A 105 (3), 033302.
Beijersbergen, M.W., Spreeuw, R.J.C., Allen, L., Woerdman, J.P., 1992. Multiphoton resonances and Bloch-Siegert shifts observedin a classical two-level system. Phys. Rev. A 45 (3), 1810–1815.
Benderskii, V.A., Vetoshkin, E.V., Kats, E.I., 2003. Instanton versus traditional WKB approach to the Landau-Zener problem. J. Exp.
Theor. Phys. 97 (2), 232–258.
Bendersky, D., Zangara, P.R., Pastawski, H.M., 2013. Fragility of superposition states evaluated by the Loschmidt echo. Phys. Rev.
A 88 (3), 032102.
Bengs, U., Patchkovskii, S., Ivanov, M., Zhavoronkov, N., 2022. All-optical Stückelberg spectroscopy of strongly driven Rydberg
states. Phys. Rev. Res. 4 (2), 023135.
Benisty, H., Piskunov, N., Kashkarov, P.K., Khayam, O., 2011. Crossing of manifolds leads to flat dispersion: Blazed Littrow
waveguides. Phys. Rev. A 84 (6), 063825.
Benito, M., Niklas, M., Kohler, S., 2016. Full-counting statistics of time-dependent conductors. Phys. Rev. B 94 (19), 195433.
Bera, T., Majumder, S., Sahu, S.K., Singh, V., 2021. Large flux-mediated coupling in hybrid electromechanical system with atransmon qubit. Commun. Phys. 4 (1), 12.
Bergmann, K., Theuer, H., Shore, B.W., 1998. Coherent population transfer among quantum states of atoms and molecules. Rev.
Modern Phys. 70 (3), 1003–1025.
Berns, D.M., Oliver, W.D., Valenzuela, S.O., Shytov, A.V., Berggren, K.K., Levitov, L.S., Orlando, T.P., 2006. Coherent quasiclassicaldynamics of a persistent current qubit. Phys. Rev. Lett. 97 (15), 150502.
Berns, D.M., Rudner, M.S., Valenzuela, S.O., Berggren, K.K., Oliver, W.D., Levitov, L.S., Orlando, T.P., 2008. Amplitude spectroscopy
of a solid-state artificial atom. Nature 455 (7209), 51–57.
Berry, M., 1995. Two-state quantum asymptotics. Ann. New York Acad. Sci. 755 (1), 303–317.
Berry, M.V., 2009. Transitionless quantum driving. J. Phys. A 42 (36), 365303.
Betthausen, C., Dollinger, T., Saarikoski, H., Kolkovsky, V., Karczewski, G., Wojtowicz, T., Richter, K., Weiss, D., 2012. Spin-transistor
action via tunable Landau-Zener transitions. Science 337 (6092), 324–327.
70

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Blattmann, R., Hänggi, P., Kohler, S., 2015. Qubit interference at avoided crossings: the role of driving shape and bath coupling.
Phys. Rev. A 91 (4), 042109.
Blattmann, R., Krenner, H.J., Kohler, S., Hänggi, P., 2014. Entanglement creation in a quantum-dot–nanocavity system by
Fourier-synthesized acoustic pulses. Phys. Rev. A 89 (1), 012327.
Bleu, O., Malpuech, G., Gao, Y., Solnyshkov, D.D., 2018. Effective theory of nonadiabatic quantum evolution based on the quantum
geometric tensor. Phys. Rev. Lett. 121 (2), 020401.
Bliokh, K.Y., Bekshaev, A.Y., Kofman, A.G., Nori, F., 2013. Photon trajectories, anomalous velocities and weak measurements: aclassical interpretation. New J. Phys. 15 (7), 073022.
Bliokh, K.Y., Grinyok, S.V., 2001. On the criterion for effectiveness of wave linear transformation in a smoothly inhomogeneous
medium and of nonadiabatic transitions during atomic collisions. J. Exp. Theor. Phys. 93 (1), 71–79.
Bliokh, K.Y., Nori, F., 2019. Klein-Gordon representation of acoustic waves and topological origin of surface acoustic modes. Phys.
Rev. Lett. 123 (5), 054301.
Bondar, D.I., Cabrera, R., Rabitz, H.A., 2013. Conceptual inconsistencies in finite-dimensional quantum and classical mechanics.
Phys. Rev. A 88 (1), 012116.
Bonifacio, M., Domínguez, D., Sánchez, M.J., 2020. Landau-Zener-Stückelberg interferometry in dissipative circuit quantum
electrodynamics. Phys. Rev. B 101 (24), 245415.
Boolakee, T., Heide, C., Wagner, F., Ott, C., Schlecht, M., Ristein, J., Weber, H.B., Hommelhoff, P., 2020. Length-dependence oflight-induced currents in graphene. J. Phys. B: At. Mol. Opt. Phys. 53 (15), 154001.
Boris, A.A., Krasnov, V.M., 2015. Quantization of the superconducting energy gap in an intense microwave field. Phys. Rev. B 92(17), 174506.
Braakman, F.R., Barthelemy, P., Reichl, C., Wegscheider, W., Vandersypen, L.M.K., 2013. Long-distance coherent coupling in aquantum dot array. Nat. Nanotechnol. 8 (6), 432–437.
Brataas, A., Rashba, E.I., 2011. Nuclear dynamics during Landau-Zener singlet-triplet transitions in double quantum dots. Phys.
Rev. B 84 (4), 045301.
Brundobler, S., Elser, V., 1993. S-matrix for generalized Landau-Zener problem. J. Phys. A: Math. Gen. 26 (5), 1211–1227.
Buluta, I., Ashhab, S., Nori, F., 2011. Natural and artificial atoms for quantum computation. Rep. Progr. Phys. 74 (10), 104401.
Buluta, I., Nori, F., 2009. Quantum simulators. Science 326 (5949), 108–111.
Burin, A.L., Maksymov, A.O., Osborn, K.D., 2014. Quantum coherent manipulation of two-level systems in superconducting circuits.
Supercond. Sci. Technol. 27 (8), 084001.
Burkard, G., 2010. Splitting spin states on a chip. Science 327 (5966), 650–651.
Burkard, G., Ladd, T.D., Nichol, J.M., Pan, A., Petta, J.R., 2021. Semiconductor spin qubits. ArXiv 2112.08863.
Bychkov, Y.A., Dykhne, A.M., 1970. Breakdown in semiconductors in an alternating electric field. Sov. Phys.—JETP 31 (5), 928–932.
Bylander, J., Rudner, M.S., Shytov, A.V., Valenzuela, S.O., Berns, D.M., Berggren, K.K., Levitov, L.S., Oliver, W.D., 2009. Pulse imaging
and nonadiabatic control of solid-state artificial atoms. Phys. Rev. B 80 (22), 220506.
Cahn, S.B., Ammon, J., Kirilov, E., Gurevich, Y.V., Murphree, D., Paolino, R., Rahmlow, D.A., Kozlov, M.G., DeMille, D., 2014.
Zeeman-tuned rotational level-crossing spectroscopy in a diatomic free radical. Phys. Rev. Lett. 112 (16), 163002.
Calero, C., Chudnovsky, E.M., Garanin, D.A., 2005. Quantum dynamics of a nanomagnet in a rotating field. Phys. Rev. B 72 (2),024409.
Campbell, D.L., Shim, Y.-P., Kannan, B., Winik, R., Kim, D.K., Melville, A., Niedzielski, B.M., Yoder, J.L., Tahan, C., Gustavsson, S.,
Oliver, W.D., 2020. Universal nonadiabatic control of small-gap superconducting qubits. Phys. Rev. X 10 (4), 041051.
Cao, G., Li, H.-O., Tu, T., Wang, L., Zhou, C., Xiao, M., Guo, G.-C., Jiang, H.-W., Guo, G.-P., 2013. Ultrafast universal quantum control
of a quantum-dot charge qubit using Landau–Zener–Stückelberg interference. Nature Commun. 4 (1), 1401.
Cao, Y.-F., Wei, L.-F., Hou, B.-P., 2010. Exact solutions to Landau–Zener problems by evolution operator method. Phys. Lett. A 374(22), 2281–2285.
Chang, Y.-H., Dubyna, D., Chien, W.-C., Chen, C.-H., Wu, C.-S., Kuo, W., 2020. Cavity quantum electrodynamics with dressed statesof a superconducting artificial atom. ArXiv 1906.06730.
Chasseur, T., Theis, L.S., Sanders, Y.R., Egger, D.J., Wilhelm, F.K., 2015. Engineering adiabaticity at an avoided crossing with optimal
control. Phys. Rev. A 91 (4), 043421.
Chatterjee, A., Shevchenko, S.N., Barraud, S., Otxoa, R.M., Nori, F., Morton, J.J.L., Gonzalez-Zalba, M.F., 2018. A silicon-based
single-electron interferometer coupled to a fermionic sea. Phys. Rev. B 97 (4), 045405.
Chen, H.-B., Lambert, N., Cheng, Y.-C., Chen, Y.-N., Nori, F., 2015. Using non-Markovian measures to evaluate quantum masterequations for photosynthesis. Sci. Rep. 5 (1), 12753.
Chen, X., Lizuain, I., Ruschhaupt, A., Guéry-Odelin, D., Muga, J.G., 2010. Shortcut to adiabatic passage in two- and three-levelatoms. Phys. Rev. Lett. 105 (12), 123003.
Chen, Y.-H., Qin, W., Wang, X., Miranowicz, A., Nori, F., 2021b. Shortcuts to adiabaticity for the quantum Rabi model: Efficient
generation of giant entangled cat states via parametric amplification. Phys. Rev. Lett. 126 (2), 023602.
Chen, Z.-G., Tang, W., Zhang, R.-Y., Chen, Z., Ma, G., 2021c. Landau-Zener transition in the dynamic transfer of acoustic topologicalstates. Phys. Rev. Lett. 126 (5), 054301.
Chen, B.-B., Wang, B.-C., Cao, G., Li, H.-O., Xiao, M., Guo, G.-C., Jiang, H.-W., Hu, X., Guo, G.-P., 2017. Spin blockade and coherent
dynamics of high-spin states in a three-electron double quantum dot. Phys. Rev. B 95 (3), 035408.
71

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Chen, M.-B., Wang, B.-C., Kohler, S., Kang, Y., Lin, T., Gu, S.-S., Li, H.-O., Guo, G.-C., Hu, X., Jiang, H.-W., Cao, G., Guo, G.-P., 2021a.
Floquet state depletion in ac-driven circuit QED. Phys. Rev. B 103 (20), 205428.
Chen, J.-D., Wen, X.-D., Sun, G.-Z., Yu, Y., 2011. Landau—Zener—Stückelberg interference in a multi-anticrossing system. Chin.
Phys. B 20 (8), 088501.
Chichinin, A.I., 2013. Comment on the Landau–Zener formula. J. Phys. Chem. B 117 (19), 6018.
Child, M.S., 1974. On the Stueckelberg formula for non-adiabatic transitions. Mol. Phys. 28 (2), 495–501.
Child, M.S., 1996. Molecular Collision Theory. Dover Publications.
Childress, L., McIntyre, J., 2010. Multifrequency spin resonance in diamond. Phys. Rev. A 82 (3), 033839.
Chin, C., Grimm, R., Julienne, P., Tiesinga, E., 2010. Feshbach resonances in ultracold gases. Rev. Modern Phys. 82 (2), 1225–1286.
Chotorlishvili, L., Ugulava, A., Mchedlishvili, G., Komnik, A., Wimberger, S., Berakdar, J., 2011. Nonlinear dynamics of two coupled
nano-electromechanical resonators. J. Phys. B: At., Mol. Opt. Phys. 44 (21), 215402.
Chruściński, D., Pascazio, S., 2017. A brief history of the GKLS equation. Open Systems & Information Dynamics 24 (03), 1740001.
Chu, S.-I., 1989. Generalized Floquet theoretical approaches to intense-field multiphoton and nonlinear optical processes. In: Adv.
in Chem. Phys.. John Wiley & Sons, Inc., pp. 739–799.
Chu, S.-I., Telnov, D.A., 2004. Beyond the Floquet theorem: generalized Floquet formalisms and quasienergy methods for atomic
and molecular multiphoton processes in intense laser fields. Phys. Rep. 390 (1–2), 1–131.
Cifarelli, L. (Ed.), 2020a. Scientific Papers of Ettore Majorana. Springer, Berlin, pp. 77–84, (Chapter). Oriented atoms in a variablemagnetic field.
Cifarelli, L. (Ed.), 2020b. Scientific Papers of Ettore Majorana. Springer, Berlin, pp. 85–88, (Chapter). Comment on: ‘‘Oriented atoms
in a variable magnetic field’’, by M. Inguscio.
Coffey, D., Lorents, D.C., Smith, F.T., 1969. Collision spectroscopy. II. Inelastic scattering of He+ by Ne. Phys. Rev. 187 (1), 201–220.
Cohen-Tannoudji, C., Dupont-Roc, J., Grynberg, G., 1998. Atom-Photon Interactions. Wiley-VCH Verlag GmbH.
Cong, S., Gao, M.-Y., Cao, G., Guo, G.-C., Guo, G.-P., 2015. Ultrafast manipulation of a double quantum-dot charge qubit using
Lyapunov-based control method. IEEE J. Quantum Electron. 51 (8), 1–8.
Cooper, B.K., Yakovenko, V.M., 2006. Interlayer Aharonov-Bohm interference in tilted magnetic fields in quasi-one-dimensionalorganic conductors. Phys. Rev. Lett. 96 (3), 037001.
Cucchietti, F.M., Damski, B., Dziarmaga, J., Zurek, W.H., 2007. Dynamics of the Bose-Hubbard model: Transition from a Mott
insulator to a superfluid. Phys. Rev. A 75 (2), 023603.
Cui, J.-M., Gómez-Ruiz, F.J., Huang, Y.-F., Li, C.-F., Guo, G.-C., del Campo, A., 2020. Experimentally testing quantum critical dynamics
beyond the Kibble–Zurek mechanism. Commun. Phys. 3 (1), 44.
Cui, J.-M., Huang, Y.-F., Wang, Z., Cao, D.-Y., Wang, J., Lv, W.-M., Luo, L., del Campo, A., Han, Y.-J., Li, C.-F., Guo, G.-C., 2016.
Experimental trapped-ion quantum simulation of the Kibble-Zurek dynamics in momentum space. Sci. Rep. 6 (1), 33381.
Cullimore, M., Everitt, M.J., Ormerod, M.A., Samson, J.H., Wilson, R.D., Zagoskin, A.M., 2012. Relationship between minimum gap
and success probability in adiabatic quantum computing. J. Phys. A 45 (50), 505305.
Dai, X., Trappen, R., Chen, H., Melanson, D., Yurtalan, M.A., Tennant, D.M., Martinez, A.J., Tang, Y., Mozgunov, E., Gibson, J.,
Grover, J.A., Disseler, S.M., Basham, J.I., Novikov, S., Das, R., Melville, A.J., Niedzielski, B.M., Hirjibehedin, C.F., Serniak, K.,
Weber, S.J., Yoder, J.L., Oliver, W.D., Zick, K.M., Lidar, D.A., Lupascu, A., 2022. Dissipative Landau-Zener tunneling: crossoverfrom weak to strong environment coupling. ArXiv 2207.02017.
Damski, B., 2005. The simplest quantum model supporting the Kibble-Zurek mechanism of topological defect production:
Landau-Zener transitions from a new perspective. Phys. Rev. Lett. 95 (3), 035701.
Damski, B., Zurek, W.H., 2006. Adiabatic-impulse approximation for avoided level crossings: from phase-transition dynamics toLandau-Zener evolutions and back again. Phys. Rev. A 73 (6), 063405.
Damski, B., Zurek, W.H., 2009. Quantum phase transition in space in a ferromagnetic spin-1 Bose–Einstein condensate. New J.
Phys. 11 (6), 063014.
Danga, J.E., Kenfack, S.C., Fai, L.C., 2016. Quantum wire and magnetic control of a spin qubit in the Landau–Zener–Stückelberg
interferometry transition. J. Phys. A 49 (19), 195306.
Danon, J., Rudner, M.S., 2014. Multilevel interference resonances in strongly driven three-level systems. Phys. Rev. Lett. 113 (24),247002.
Das, A., 2010. Exotic freezing of response in a quantum many-body system. Phys. Rev. B 82 (17), 172402.
Davis, J.P., Pechukas, P., 1976. Nonadiabatic transitions induced by a time-dependent Hamiltonian in the semiclassical/adiabatic
limit: The two-state case. J. Chem. Phys. 64 (8), 3129.
Degen, C.L., Reinhard, F., Cappellaro, P., 2017. Quantum sensing. Rev. Modern Phys. 89 (3), 035002.
Delone, N.B., Krainov, V.P., 2012. Atoms in Strong Light Fields. Springer-Verlag GmbH.
Delos, J.B., Thorson, W.R., 1972. Studies of the potential-curve-crossing problem. II. General theory and a model for close crossings.
Phys. Rev. A 6 (2), 728–745.
Demirplak, M., Rice, S.A., 2003. Adiabatic population transfer with control fields. J. Phys. Chem. A 107 (46), 9937–9945.
Demkov, Y.N., Osherov, V.I., 1968. Stationary and nonstationary problems in quantum mechanics that can be solved by means ofcontour integration. Sov. Phys.—JETP 26 (916), 1.
Demkov, Y.N., Ostrovsky, V.N., 1995. Crossing of two bands of potential curves. J. Phys. B: At. Mol. Opt. Phys. 28 (3), 403–414.
Deng, C., Orgiazzi, J.-L., Shen, F., Ashhab, S., Lupascu, A., 2015. Observation of Floquet states in a strongly driven artificial atom.
Phys. Rev. Lett. 115 (13), 133601.
72

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Deng, C., Shen, F., Ashhab, S., Lupascu, A., 2016. Dynamics of a two-level system under strong driving: quantum-gate optimization
based on Floquet theory. Phys. Rev. A 94 (3), 032323.
Denisenko, M.V., Satanin, A.M., Ashhab, S., Nori, F., 2010. Dynamics of interacting qubits in a strong alternating electromagnetic
field. Phys. Solid State 52 (11), 2281–2286.
van Ditzhuijzen, C.S.E., Tauschinsky, A., van Linden van den Heuvell, H.B., 2009. Observation of Stückelberg oscillations in
dipole-dipole interactions. Phys. Rev. A 80 (6), 063407.
Divakaran, U., Mukherjee, V., Dutta, A., Sen, D., 2009. Defect production due to quenching through a multicritical point. J. Stat.
Mech. Theory Exp. 2009 (02), P02007.
Dodin, A., Garmon, S., Simine, L., Segal, D., 2014. Landau-Zener transitions mediated by an environment: population transfer andenergy dissipation. J. Chem. Phys. 140 (12), 124709.
Douçot, B., Danneau, R., Yang, K., Caputo, J.-G., Mélin, R., 2020. Berry phase in superconducting multiterminal quantum dots. Phys.
Rev. B 101 (3), 035411.
Dragoman, M., 2004. Quantum-Classical Analogies. Springer Berlin Heidelberg.
Dressel, J., Bliokh, K.Y., Nori, F., 2014. Classical field approach to quantum weak measurements. Phys. Rev. Lett. 112 (11), 110407.
Du, L., Lan, D., Yu, Y., 2013a. Low-frequency Landau–Zener–Stückelberg interference in dissipative superconducting qubits. Low
Temp. Phys. 39 (8), 665–679.
Du, L., Wang, M., Yu, Y., 2010. Landau-Zener-Stückelberg interferometry in the presence of quantum noise. Phys. Rev. B 82 (4),045128.
Du, L., Yu, Y., 2010. Noise effects on Landau-Zener-Stückelberg interferometry in multilevel artificial atoms. Phys. Rev. B 82 (14),144524.
Du, L., Yu, Y., Lan, D., 2013b. Electromagnetically induced interference in a superconducting flux qubit. Low Temp. Phys. 39 (6),503–514.
Duan, H.-G., Miller, R.J.D., Thorwart, M., 2016. Impact of vibrational coherence on the quantum yield at a conical intersection. J.
Phys. Chem. Lett. 7 (17), 3491–3496.
Dupont-Ferrier, E., Roche, B., Voisin, B., Jehl, X., Wacquez, R., Vinet, M., Sanquer, M., Franceschi, S.D., 2013. Coherent coupling of
two dopants in a silicon nanowire probed by Landau-Zener-Stückelberg interferometry. Phys. Rev. Lett. 110 (13), 136802.
Dutta, A., Aeppli, G., Chakrabarti, B.K., Divakaran, U., Rosenbaum, T.F., Sen, D., 2015a. Quantum Phase Transitions in Transverse
Field Spin Models. Cambr. Univ. Press.
Dutta, A., Das, A., Sengupta, K., 2015b. Statistics of work distribution in periodically driven closed quantum systems. Phys. Rev. E
92 (1), 012104.
Dykhne, A.M., 1962. Adiabatic perturbation of discrete spectrum states. Sov. Phys.—JETP 14 (4), 941.
Dziarmaga, J., 2005. Dynamics of a quantum phase transition: Exact solution of the quantum Ising model. Phys. Rev. Lett. 95 (24),245701.
Dziarmaga, J., 2010. Dynamics of a quantum phase transition and relaxation to a steady state. Adv. Phys. 59 (6), 1063–1189.
Eckardt, A., 2017. Colloquium: Atomic quantum gases in periodically driven optical lattices. Rev. Modern Phys. 89 (1), 011004.
Eckel, J., Reina, J.H., Thorwart, M., 2009. Coherent control of an effective two-level system in a non-Markovian biomolecularenvironment. New J. Phys. 11 (8), 085001.
Emmanouilidou, A., Zhao, X.-G., Ao, P., Niu, Q., 2000. Steering an eigenstate to a destination. Phys. Rev. Lett. 85 (8), 1626–1629.
Esposito, S., 2014. The Physics of Ettore Majorana: Theoretical, Mathematical, and Phenomenological. Cambridge University Press.
Esposito, S., 2017. Ettore Majorana. Springer-Verlag GmbH.
Eu, B.C., 1970. Theory of inelastic collisions: The WKB-type general solutions. J. Chem. Phys. 52 (4), 1882–1893.
Fai, L.C., Tchoffo, M., Jipdi, M.N., 2015. Landau Zener scenario in a trapped atomic gas: multi-level multi-particle model. Eur. Phys.
J. B 88 (7), 181.
Faust, T., Rieger, J., Seitner, M.J., Krenn, P., Kotthaus, J.P., Weig, E.M., 2012. Nonadiabatic dynamics of two strongly coupled
nanomechanical resonator modes. Phys. Rev. Lett. 109 (3), 037205.
Feng, J.-J., Huang, Z., Wang, Z., Niu, Q., 2018. Hysteresis from nonlinear dynamics of Majorana modes in topological Josephson
junctions. Phys. Rev. B 98 (13), 134515.
Ferrón, A., Domínguez, D., Sánchez, M.J., 2010. Large-amplitude harmonic driving of highly coherent flux qubits. Phys. Rev. B 82
(13), 134522.
Ferrón, A., Domínguez, D., Sánchez, M.J., 2012. Tailoring population inversion in Landau-Zener-Stückelberg interferometry of flux
qubits. Phys. Rev. Lett. 109 (23), 237005.
Ferrón, A., Domínguez, D., Sánchez, M.J., 2016. Dynamic transition in Landau-Zener-Stückelberg interferometry of dissipative
systems: The case of the flux qubit. Phys. Rev. B 93 (6), 064521.
Fillion-Gourdeau, F., Gagnon, D., Lefebvre, C., MacLean, S., 2016. Time-domain quantum interference in graphene. Phys. Rev. B 94
(12), 125423.
Fink, J.M., Baur, M., Bianchetti, R., Filipp, S., Göppl, M., Leek, P.J., Steffen, L., Blais, A., Wallraff, A., 2009. Thermal excitation ofmulti-photon dressed states in circuit quantum electrodynamics. Phys. Scr. T137, 014013.
Fogarty, M.A., Chan, K.W., Hensen, B., Huang, W., Tanttu, T., Yang, C.H., Laucht, A., Veldhorst, M., Hudson, F.E., Itoh, K.M., Culcer, D.,
Ladd, T.D., Morello, A., Dzurak, A.S., 2018. Integrated silicon qubit platform with single-spin addressability, exchange control
and single-shot singlet-triplet readout. Nature Commun. 9 (1), 4370.
73

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Földi, P., Benedict, M.G., Peeters, F.M., 2008. Dynamics of periodic anticrossings: Decoherence, pointer states, and hysteresis curves.
Phys. Rev. A 77 (1), 013406.
Földi, P., Benedict, M.G., Pereira, J.M., Peeters, F.M., 2007. Dynamics of molecular nanomagnets in time-dependent external
magnetic fields: Beyond the Landau-Zener-Stückelberg model. Phys. Rev. B 75 (10), 104430.
Førre, M., 2004. Landau-Zener-Stueckelberg theory for multiphoton intrashell transitions in Rydberg atoms: Bloch-Siegert shifts
and widths. Phys. Rev. A 70 (1), 013406.
Forster, F., Mühlbacher, M., Blattmann, R., Schuh, D., Wegscheider, W., Ludwig, S., Kohler, S., 2015. Landau-Zener interference atbichromatic driving. Phys. Rev. B 92 (24), 245422.
Forster, F., Petersen, G., Manus, S., Hänggi, P., Schuh, D., Wegscheider, W., Kohler, S., Ludwig, S., 2014. Characterization of qubit
dephasing by Landau-Zener-Stückelberg-Majorana interferometry. Phys. Rev. Lett. 112 (11), 116803.
Fouokeng, G.C., Tchoffo, M., Ateuafack, M.E., Fai, L.C., 2014. Dynamics of a central electron spin coupled to an anti-ferromagnetic
spin bath driven by a variable magnetic field in the Landau-Zener scenario. Eur. Phys. J. Plus 129 (7), 151.
Frimmer, M., Novotny, L., 2014. The classical Bloch equations. Am. J. Phys. 82 (10), 947–954.
Frisch, R., Segre, E., 1933. Uber die einstellung der richtungsquantelung. II. Z. Phys. 80 (9–10), 610–616.
Fu, H., Gong, Z.-C., Mao, T.-H., Sun, C.-P., Yi, S., Li, Y., Cao, G.-Y., 2016. Classical analog of Stückelberg interferometry in a
two-coupled-cantilever–based optomechanical system. Phys. Rev. A 94 (4), 043855.
Fu, H., Zhi-cheng, G., Yang, L.-P., Mao, T.-H., Sun, C.-P., Yi, S., Li, Y., Cao, G.-Y., 2018. Coherent optomechanical switch for motion
transduction based on dynamically localized mechanical modes. Phys. Rev. App. 9 (5), 054024.
Fuchs, G.D., Burkard, G., Klimov, P.V., Awschalom, D.D., 2011. A quantum memory intrinsic to single nitrogen–vacancy centres indiamond. Nat. Phys. 7 (10), 789–793.
Fuchs, J.-N., Lim, L.-K., Montambaux, G., 2012. Interband tunneling near the merging transition of Dirac cones. Phys. Rev. A 86(6), 063613.
Funo, K., Lambert, N., Nori, F., 2021. General bound on the performance of counter-diabatic driving acting on dissipative spin
systems. Phys. Rev. Lett. 127 (15), 150401.
Gagge, A., Larson, J., 2018. Bloch-like energy oscillations. Phys. Rev. A 98 (5), 053820.
Gagnon, D., Fillion-Gourdeau, F., Dumont, J., Lefebvre, C., MacLean, S., 2016. Coherent destruction of tunneling in graphene
irradiated by elliptically polarized lasers. J. Phys.: Condens. Matter 29 (3), 035501.
Gagnon, D., Fillion-Gourdeau, F., Dumont, J., Lefebvre, C., MacLean, S., 2017. Suppression of multiphoton resonances in driven
quantum systems via pulse shape optimization. Phys. Rev. Lett. 119 (5), 053203.
Gallego-Marcos, F., Sánchez, R., Platero, G., 2015. Photon assisted long-range tunneling. J. Appl. Phys. 117 (11), 112808.
Gallego-Marcos, F., Sánchez, R., Platero, G., 2016. Coupled Landau-Zener-Stückelberg quantum dot interferometers. Phys. Rev. B
93 (7), 075424.
Ganeshan, S., Barnes, E., Sarma, S.D., 2013. Exact classification of Landau-Majorana-Stückelberg-Zener resonances by Floquet
determinants. Phys. Rev. Lett. 111 (13), 130405.
Gao, Z.-P., Zhang, D.-W., Yu, Y., Zhu, S.-L., 2017. Anti-Kibble-Zurek behavior of a noisy transverse-field XY chain and its quantum
simulation with two-level systems. Phys. Rev. B 95 (22), 224303.
Garanin, D.A., 2003. Landau-Zener-Stueckelberg effect in a model of interacting tunneling systems. Phys. Rev. B 68 (1), 014414.
Garanin, D.A., 2004. Interference effect in the Landau-Zener tunneling of the antiferromagnetically coupled dimer ofsingle-molecule magnets. Phys. Rev. B 70 (21), 212403.
Garanin, D.A., Neb, R., Schilling, R., 2008. Effect of environmental spins on Landau-Zener transitions. Phys. Rev. B 78 (9), 094405.
Garanin, D.A., Schilling, R., 2002a. Effects of nonlinear sweep in the Landau-Zener-Stueckelberg effect. Phys. Rev. B 66 (17), 174438.
Garanin, D.A., Schilling, R., 2002b. Inverse problem for the Landau-Zener effect. Europhys. Lett. 59 (1), 7–13.
Garraway, B.M., Perrin, H., 2016. Recent developments in trapping and manipulation of atoms with adiabatic potentials. J. Phys.
B: At. Mol. Opt. Phys. 49 (17), 172001.
Garraway, B.M., Stenholm, S., 1992a. Interferometer within a molecule. Phys. Rev. A 46 (3), 1413–1420.
Garraway, B.M., Stenholm, S., 1992b. Population transfer at periodically repeated level crossings. Phys. Rev. A 45 (1), 364–373.
Garraway, B.M., Suominen, K.A., 1995. Wave-packet dynamics: new physics and chemistry in femto-time. Rep. Progr. Phys. 58(4), 365–419.
Garraway, B.M., Vitanov, N.V., 1997. Population dynamics and phase effects in periodic level crossings. Phys. Rev. A 55 (6),4418–4432.
Gasparinetti, S., Solinas, P., Pekola, J.P., 2011. Geometric Landau-Zener interferometry. Phys. Rev. Lett. 107 (20), 207002.
Gasparinetti, S., Solinas, P., Yoon, Y., Pekola, J.P., 2012. Single Cooper-pair pumping in the adiabatic limit and beyond. Phys. Rev.
B 86 (6), 060502.
Gaudreau, L., Granger, G., Kam, A., Aers, G.C., Studenikin, S.A., Zawadzki, P., Pioro-Ladrière, M., Wasilewski, Z.R., Sachrajda, A.S.,
2011. Coherent control of three-spin states in a triple quantum dot. Nat. Phys. 8 (1), 54–58.
Gauthey, F.I., Garraway, B.M., Knight, P.L., 1997. High harmonic generation and periodic level crossings. Phys. Rev. A 56 (4),3093–3096.
Gefen, Y., Thouless, D.J., 1987. Zener transitions and energy dissipation in small driven systems. Phys. Rev. Lett. 59 (15),1752–1755.
Georgescu, I.M., Ashhab, S., Nori, F., 2014. Quantum simulation. Rev. Modern Phys. 86 (1), 153–185.
74

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Giacomo, F.D., Nikitin, E.E., 2005. The Majorana formula and the Landau–Zener–Stückelberg treatment of the avoided crossing
problem. Phys.-Usp. 48 (5), 515–517.
Giavaras, G., Tokura, Y., 2019a. Probing the singlet-triplet splitting in double quantum dots: Implications of the ac field amplitude.
Phys. Rev. B 100 (19), 195421.
Giavaras, G., Tokura, Y., 2019b. Spectroscopy of double quantum dot two-spin states by tuning the interdot barrier. Phys. Rev. B
99 (7), 075412.
Ginzel, F., Mills, A.R., Petta, J.R., Burkard, G., 2020. Spin shuttling in a silicon double quantum dot. Phys. Rev. B 102 (19), 195418.
Glutsch, S., 2004. Nonresonant and resonant Zener tunneling. Phys. Rev. B 69 (23), 235317.
Goldman, N., Dalibard, J., 2014. Periodically driven quantum systems: Effective Hamiltonians and engineered gauge fields. Phys.
Rev. X 4 (3), 031027.
Gong, M., Wen, X., Sun, G., Zhang, D.-W., Lan, D., Zhou, Y., Fan, Y., Liu, Y., Tan, X., Yu, H., Yu, Y., Zhu, S.-L., Han, S., Wu, P., 2016a.
Simulating the Kibble-Zurek mechanism of the Ising model with a superconducting qubit system. Sci. Rep. 6 (1), 22667.
Gong, M., Zhou, Y., Lan, D., Fan, Y., Pan, J., Yu, H., Chen, J., Sun, G., Yu, Y., Han, S., Wu, P., 2016b. Landau-Zener-Stückelberg-Majorana
interference in a 3D transmon driven by a chirped microwave. Appl. Phys. Lett. 108 (11), 112602.
Gonzalez-Zalba, M.F., Shevchenko, S.N., Barraud, S., Johansson, J.R., Ferguson, A.J., Nori, F., Betz, A.C., 2016. Gate-sensing coherent
charge oscillations in a silicon field-effect transistor. Nano Lett. 16 (3), 1614–1619.
Gopalakrishnan, S., Knap, M., Demler, E., 2016. Regimes of heating and dynamical response in driven many-body localized systems.
Phys. Rev. B 94 (9), 094201.
Góral, K., Köhler, T., Gardiner, S.A., Tiesinga, E., Julienne, P.S., 2004. Adiabatic association of ultracold molecules via magnetic-field
tunable interactions. J. Phys. B: At., Mol. Opt. Phys. 37 (17), 3457–3500.
Gorelik, L.Y., Lundin, N.I., Shumeiko, V.S., Shekhter, R.I., Jonson, M., 1998. Superconducting single-mode contact as a
microwave-activated quantum interferometer. Phys. Rev. Lett. 81 (12), 2538–2541.
Goswami, D., 2003. Optical pulse shaping approaches to coherent control. Phys. Rep. 374 (6), 385–481.
de Graaf, S.E., Leppäkangas, J., Adamyan, A., Danilov, A.V., Lindström, T., Fogelström, M., Bauch, T., Johansson, G., Kubatkin, S.E.,
2013. Charge qubit coupled to an intense microwave electromagnetic field in a superconducting Nb device: evidence forphoton-assisted quasiparticle tunneling. Phys. Rev. Lett. 111 (13), 137002.
Gradshteyn, I.S., Ryzhik, I.M., 2007. Table of Integrals, Series, and Products, seventh ed. Academic Press is an imprint of Elsevier,pp. 1028–1031, Chap. 9,
Grajcar, M., Izmalkov, A., Il’ichev, E., 2005. Possible implementation of adiabatic quantum algorithm with superconducting flux
qubits. Phys. Rev. B 71 (14), 144501.
Gramajo, A.L., Campbell, D., Kannan, B., Kim, D.K., Melville, A., Niedzielski, B.M., Yoder, J.L., Sánchez, M.J., Domínguez, D.,
Gustavsson, S., Oliver, W.D., 2020. Quantum emulation of coherent backscattering in a system of superconducting qubits.
Phys. Rev. Appl. 14 (1), 014047.
Gramajo, A.L., Domínguez, D., Sánchez, M.J., 2017. Entanglement generation through the interplay of harmonic driving andinteraction in coupled superconducting qubits. Eur. Phys. J. B 90 (12), 255.
Gramajo, A.L., Domínguez, D., Sánchez, M.J., 2018. Amplitude tuning of steady-state entanglement in strongly driven coupled
qubits. Phys. Rev. A 98 (4), 042337.
Gramajo, A.L., Domínguez, D., Sánchez, M.J., 2019. Revealing the system-bath coupling via Landau-Zener-Stückelberg
interferometry in superconducting qubits. Phys. Rev. B 100 (7), 075410.
Gramajo, A.L., Domínguez, D., Sánchez, M.J., 2021. Efficient steady-state-entanglement generation in strongly driven coupled
qubits. Phys. Rev. A 104 (3), 032410.
Granger, G., Aers, G.C., Studenikin, S.A., Kam, A., Zawadzki, P., Wasilewski, Z.R., Sachrajda, A.S., 2015. Visibility study of S −T+
Landau-Zener-Stückelberg oscillations without applied initialization. Phys. Rev. B 91 (11), 115309.
Greenberg, Y.S., 2007. Low-frequency Rabi spectroscopy of dissipative two-level systems: Dressed-state approach. Phys. Rev. B
76 (10), 104520.
Grifoni, M., Hänggi, P., 1998. Driven quantum tunneling. Phys. Rep. 304 (5–6), 229–354.
Grimaudo, R., Nakazato, H., Messina, A., Vitanov, N.V., 2020. Dzyaloshinskii-Moriya and dipole-dipole interactions affectcoupling-based Landau-Majorana-Stückelberg-Zener transitions. Phys. Rev. Res. 2 (3), 033092.
Grimaudo, R., Vitanov, N.V., de Castro, A.S.M., Valenti, D., Messina, A., 2022. Greenberger-Horne-Zeilinger-state generation in
qubit-chains via a single Landau-Majorana-Stückelberg-Zener π /2-pulse. Fortschr. Phys. 70 (5), 2200010.
Grimaudo, R., Vitanov, N.V., Messina, A., 2019. Coupling-assisted Landau-Majorana-Stückelberg-Zener transition in a system of
two interacting spin qubits. Phys. Rev. B 99 (17), 174416.
Grossmann, F., Dittrich, T., Jung, P., Hänggi, P., 1991. Coherent destruction of tunneling. Phys. Rev. Lett. 67 (4), 516–519.
Gu, X., Kockum, A.F., Miranowicz, A., Liu, Y.-X., Nori, F., 2017. Microwave photonics with superconducting quantum circuits. Phys.
Rep. 718–719, 1–102.
Guéry-Odelin, D., Ruschhaupt, A., Kiely, A., Torrontegui, E., Martínez-Garaot, S., Muga, J.G., 2019. Shortcuts to adiabaticity:
Concepts, methods, and applications. Rev. Modern Phys. 91 (4), 045001.
Haar, D.T., 1965. Collected Papers of L. D. Landau. Intl Pub Distributor Inc, Chap. 7. A theory of energy transfer, 9. A theory ofenergy transfer II.
Hagedorn, G.A., 1991. Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gaps. Comm. Math. Phys.
136 (3), 433–449.
75

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Haikka, P., Mølmer, K., 2014. Dissipative Landau-Zener level crossing subject to continuous measurement: Excitation despite
decay. Phys. Rev. A 89 (5), 052114.
Halpern, N.Y., White, C.D., Gopalakrishnan, S., Refael, G., 2019. Quantum engine based on many-body localization. Phys. Rev. B
99 (2), 024203.
Han, Y., Luo, X.-Q., Li, T.-F., Zhang, W., 2020. Analytical double-unitary-transformation approach for strongly and periodicallydriven three-level systems. Phys. Rev. A 101 (2), 022108.
Han, Y., Luo, X.-Q., Li, T.-F., Zhang, W., Wang, S.-P., Tsai, J.S., Nori, F., You, J.Q., 2019. Time-domain grating with a periodicallydriven qutrit. Phys. Rev. Appl. 11 (1), 014053.
Harvey-Collard, P., Jacobson, N.T., Bureau-Oxton, C., Jock, R.M., Srinivasa, V., Mounce, A.M., Ward, D.R., Anderson, J.M.,
Manginell, R.P., Wendt, J.R., Pluym, T., Lilly, M.P., Luhman, D.R., Pioro-Ladrière, M., Carroll, M.S., 2019. Spin-orbit interactions
for singlet-triplet qubits in silicon. Phys. Rev. Lett. 122, 217702.
Hatomura, T., Kato, G., 2020. Bounds for nonadiabatic transitions. Phys. Rev. A 102 (1), 012216.
Hausinger, J., Grifoni, M., 2010. Dissipative two-level system under strong ac driving: A combination of Floquet and Van Vleck
perturbation theory. Phys. Rev. A 81 (2), 022117.
He, P., Li, Z., 2020. Nonlinear Bloch-Zener oscillations for Bose-Einstein condensates in a Lieb optical lattice. New J. Phys. 22 (6),063031.
Hedvall, P., Larson, J., 2017. Dynamics of non-equilibrium steady state quantum phase transitions. arXiv:1712.01560.
Heide, C., Boolakee, T., Eckstein, T., Hommelhoff, P., 2021a. Optical current generation in graphene: CEP control vs. ω + 2ω control.
Nanophotonics 10 (14), 3701–3707.
Heide, C., Boolakee, T., Higuchi, T., Hommelhoff, P., 2020. Sub-cycle temporal evolution of light-induced electron dynamics inhexagonal 2D materials. J. Phys. Phot. 2 (2), 024004.
Heide, C., Boolakee, T., Higuchi, T., Hommelhoff, P., 2021b. Adiabaticity parameters for the categorization of light-matterinteraction: From weak to strong driving. Phys. Rev. A 104 (2), 023103.
Heide, C., Boolakee, T., Higuchi, T., Weber, H.B., Hommelhoff, P., 2019. Interaction of carrier envelope phase-stable laser pulses
with graphene: the transition from the weak-field to the strong-field regime. New J. Phys. 21 (4), 045003.
Heide, C., Higuchi, T., Weber, H.B., Hommelhoff, P., 2018. Coherent electron trajectory control in graphene. Phys. Rev. Lett. 121(20), 207401.
Heinrich, G., Harris, J.G.E., Marquardt, F., 2010. Photon shuttle: Landau-Zener-Stückelberg dynamics in an optomechanical system.
Phys. Rev. A 81 (1), 011801.
Hendry, P.C., Lawson, N.S., Lee, R.A.M., McClintock, P.V.E., Williams, C.D.H., 1994. Generation of defects in superfluid 4He as ananalogue of the formation of cosmic strings. Nature 368 (6469), 315–317.
Henriet, L., Hur, K.L., 2016. Quantum sweeps, synchronization, and Kibble-Zurek physics in dissipative quantum spin systems.
Phys. Rev. B 93 (6), 064411.
Henriet, L., Ristivojevic, Z., Orth, P.P., Hur, K.L., 2014. Quantum dynamics of the driven and dissipative Rabi model. Phys. Rev. A
90 (2), 023820.
Henry, C.H., Lang, D.V., 1977. Nonradiative capture and recombination by multiphonon emission in GaAs and GaP. Phys. Rev. B
15 (2), 989–1016.
Herrera, M., Sarandy, M.S., Duzzioni, E.I., Serra, R.M., 2014. Nonadiabatic quantum state engineering driven by fast quench
dynamics. Phys. Rev. A 89 (2), 022323.
Higuchi, T., Heide, C., Ullmann, K., Weber, H.B., Hommelhoff, P., 2017. Light-field-driven currents in graphene. Nature 550 (7675),
224–228.
Higuera-Quintero, S., Rodríguez, F.J., Quiroga, L., Gómez-Ruiz, F.J., 2022. Testing the Kibble-Zurek mechanism through a digitalquantum computer. Front. Quantum. Sci. Technol. 1, 1026025.
Hijii, K., Miyashita, S., 2010. Symmetry for the nonadiabatic transition in Floquet states. Phys. Rev. A 81 (1), 013403.
Ho, L.T.A., Chibotaru, L.F., 2014. A simple derivation of the Landau–Zener formula. Phys. Chem. Chem. Phys. 16 (15), 6942.
Holthaus, M., 2000. Bloch oscillations and Zener breakdown in an optical lattice. J. Opt. B: Quantum Semiclass. Opt 2 (5), 589–604.
Hu, X., Sun, S., Zheng, Y., 2022. Witnessing localization of a quantum state via quantum speed limits in a driven avoided-levelcrossing system. J. Chem. Phys. 156 (13), 134113.
Huang, X.-R., Ding, Z.-X., Hu, C.-S., Shen, L.-T., Li, W., Wu, H., Zheng, S.-B., 2018. Robust Rydberg gate via Landau-Zener control of
Förster resonance. Phys. Rev. A 98 (5), 052324.
Huang, P., Hu, X., 2021. Electric-dipole-induced resonance and decoherence of a dressed spin in a quantum dot. arXiv:2103.05817.
Huang, W.-C., Liang, Q.-F., Yao, D.-X., Wang, Z., 2015. Manipulating the Majorana qubit with Landau-Zener-Stückelberg
interference. Phys. Rev. A 92 (1), 012308.
Huang, J.-F., Liao, J.-Q., Tian, L., Kuang, L.-M., 2017. Manipulating counter-rotating interactions in the quantum Rabi model via
modulation of the transition frequency of the two-level system. Phys. Rev. A 96 (4), 043849.
Huang, B., Novičenko, V., Eckardt, A., Juzeliūnas, G., 2021. Floquet chiral hinge modes and their interplay with Weyl physics in athree-dimensional lattice. Phys. Rev. B 104 (10), 104312.
Huang, P., Zhou, J., Fang, F., Kong, X., Xu, X., Ju, C., Du, J., 2011. Landau-Zener-Stückelberg interferometry of a single electronic
spin in a noisy environment. Phys. Rev. X 1 (1), 011003.
Hürlimann, M.D., Utsuzawa, S., Hou, C.-Y., 2020. Adiabatic and nonadiabatic behavior of the Carr-Purcell-Meiboom-Gill sequencein time-dependent magnetic fields. Phys. Rev. Res. 2 (3), 033352.
76

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Ikeda, T.N., Tanaka, S., Kayanuma, Y., 2022. Floquet-Landau-Zener interferometry: Usefulness of the Floquet theory inpulse-laser-driven systems. Phys. Rev. Res. 4 (3), 033075.
Il’ichev, E., Greenberg, Y.S., 2007. Flux qubit as a sensor of magnetic flux. Eur. Phys. Lett. 77 (5), 58005.
Il’ichev, E., Oukhanski, N., Wagner, T., Meyer, H.-G., Smirnov, A.Y., Grajcar, M., Izmalkov, A., Born, D., Krech, W., Zagoskin, A.,
2004. Radio-frequency method for investigation of quantum properties of superconducting structures. Low Temp. Phys. 30(7), 620–628.
Ishikawa, K.L., 2013. Electronic response of graphene to an ultrashort intense terahertz radiation pulse. New J. Phys. 15 (5),055021.
Ishkhanyan, A., Sokhoyan, R., Suominen, K.A., Leroy, C., Jauslin, H.R., 2010. Quadratic-nonlinear Landau-Zener transition for
association of an atomic Bose-Einstein condensate with inter-particle elastic interactions included. Eur. Phys. J. D 56 (3),421–429.
Ithier, G., Collin, E., Joyez, P., Vion, D., Esteve, D., Ankerhold, J., Grabert, H., 2005. Zener enhancement of quantum tunneling in a
two-level superconducting circuit. Phys. Rev. Lett. 94 (5), 057004.
Ivakhnenko, O.V., Shevchenko, S.N., Nori, F., 2018. Simulating quantum dynamical phenomena using classical oscillators:
Landau-Zener-Stückelberg-Majorana interferometry, latching modulation, and motional averaging. Sci. Rep. 8 (1), 12218.
Izmalkov, A., Grajcar, M., Il’ichev, E., Oukhanski, N., Wagner, T., Meyer, H.-G., Krech, W., Amin, M.H.S., van den Brink, A.M.,
Zagoskin, A.M., 2004. Observation of macroscopic Landau-Zener transitions in a superconducting device. Europhys. Lett. 65(6), 844–849.
Izmalkov, A., van der Ploeg, S.H.W., Shevchenko, S.N., Grajcar, M., Il’ichev, E., Hübner, U., Omelyanchouk, A.N., Meyer, H.-G., 2008.
Consistency of ground state and spectroscopic measurements on flux qubits. Phys. Rev. Lett. 101 (1), 017003.
Jehl, X., Voisin, B., Roche, B., Dupont-Ferrier, E., Franceschi, S.D., Sanquer, M., Cobian, M., Niquet, Y.-M., Sklénard, B., Cueto, O.,
Wacquez, R., Vinet, M., 2015. The coupled atom transistor. J. Phys.: Condens. Matter 27 (15), 154206.
Jiang, W., Sarabalis, C.J., Dahmani, Y.D., Patel, R.N., Mayor, F.M., McKenna, T.P., Laer, R.V., Safavi-Naeini, A.H., 2020. Efficientbidirectional piezo-optomechanical transduction between microwave and optical frequency. Nature Commun. 11 (1), 1166.
Jin, J.-Z., Liang, H., Xiao, X.-R., Wang, M.-X., Chen, S.-G., Wu, X.-Y., Gong, Q., Peng, L.-Y., 2018. Michelson interferometry ofhigh-order harmonic generation in solids. J. Phys. B: At. Mol. Opt. Phys. 51 (16), 16LT01.
Jirovec, D., Mutter, P.M., Hofmann, A., Crippa, A., Rychetsky, M., Craig, D.L., Kukucka, J., Martins, F., Ballabio, A., Ares, N.,
Chrastina, D., Isella, G., Burkard, G., Katsaros, G., 2022. Dynamics of hole singlet-triplet qubits with large g-factor differences.
Phys. Rev. Lett. 128 (12), 126803.
Joe, A., 1994. Proof of the Landau-Zener formula. Asympt. Anal. 9 (3), 209–258.
Joecker, B., Cerfontaine, P., Haupt, F., Schreiber, L.R., Kardynał, B.E., Bluhm, H., 2019. Transfer of a quantum state from a photonic
qubit to a gate-defined quantum dot. Phys. Rev. B 99, 205415.
Johansson, J., Amin, M.H.S., Berkley, A.J., Bunyk, P., Choi, V., Harris, R., Johnson, M.W., Lanting, T.M., Lloyd, S., Rose, G., 2009.
Landau-Zener transitions in a superconducting flux qubit. Phys. Rev. B 80 (1), 012507.
Johansson, J.R., Nation, P.D., Nori, F., 2012. QuTiP: An open-source Python framework for the dynamics of open quantum systems.
Comput. Phys. Comm. 183 (8), 1760–1772.
Johansson, J.R., Nation, P.D., Nori, F., 2013. QuTiP 2: A Python framework for the dynamics of open quantum systems. Comput.
Phys. Comm. 184 (4), 1234–1240.
Kaldewey, T., Kuhlmann, A.V., Valentin, S.R., Ludwig, A., Wieck, A.D., Warburton, R.J., 2018. Far-field nanoscopy on a semiconductor
quantum dot via a rapid-adiabatic-passage-based switch. Nat. Photon. 12 (2), 68–72.
Kam, C.-F., Chen, Y., 2020. Analytical results for the dynamics of parabolic level-crossing model. New J. Phys. 22 (2), 023021.
Kam, C.-F., Chen, Y., 2021. Non-adiabatic transitions in parabolic and super-parabolic PT-symmetric non-Hermitian systems in
1D optical waveguides. Ann. Physics 533 (2), 2000349.
Kane, E., 1960. Zener tunneling in semiconductors. J. Phys. Chem. Solids 12 (2), 181–188.
Kang, Y.-H., Chen, Y.-H., Wang, X., Song, J., Xia, Y., Miranowicz, A., Zheng, S.-B., Nori, F., 2022. Nonadiabatic geometric quantum
computation with cat-state qubits via invariant-based reverse engineering. Phys. Rev. Res. 4 (1), 013233.
Kapfinger, S., Reichert, T., Lichtmannecker, S., Müller, K., Finley, J.J., Wixforth, A., Kaniber, M., Krenner, H.J., 2015. Dynamic
acousto-optic control of a strongly coupled photonic molecule. Nature Commun. 6 (1), 8540.
Kaprálová-Žďánská, P.R., 2022. Complex time method for quantum dynamics when an exceptional point is encircled in theparameter space. Ann. Physics 443, 168939.
Kar, S., Mukherjee, B., Sengupta, K., 2016. Tuning towards dynamic freezing using a two-rate protocol. Phys. Rev. B 94 (7), 075130.
Karami, M., Javdani, A., Karami, K., Jamshidi, F., 2019. Occurrence of multiphoton resonances in five-level systems bylarge-amplitude driving fields. Eur. Phys. J. D 73 (8), 179.
Karanikolas, V., Kawabata, S., 2020. Pulsed quantum annealing. J. Phys. Soc. Japan 89 (9), 094003.
Kashcheyevs, V., Timoshenko, J., 2012. Quantum fluctuations and coherence in high-precision single-electron capture. Phys. Rev.
Lett. 109 (21), 216801.
Kayanuma, Y., 1993. Phase coherence and nonadiabatic transition at a level crossing in a periodically driven two-level system.
Phys. Rev. B 47 (15), 9940–9943.
Kayanuma, Y., 1994. Role of phase coherence in the transition dynamics of a periodically driven two-level system. Phys. Rev. A
50 (1), 843–845.
Kayanuma, Y., 1997. Stokes phase and geometrical phase in a driven two-level system. Phys. Rev. A 55 (4), R2495–R2498.
77

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Kayanuma, Y., Nakayama, H., 1998. Nonadiabatic transition at a level crossing with dissipation. Phys. Rev. B 57 (20), 13099–13112.
Kayanuma, Y., Saito, K., 2008. Coherent destruction of tunneling, dynamic localization, and the Landau-Zener formula. Phys. Rev.
A 77 (1), 010101.
Kazantsev, A.P., Ryabenko, G.A., Surdutovich, G.I., Yakovlev, V.P., 1985. Scattering of atoms by light. Phys. Rep. 129 (2), 75–144.
Ke, Y., Qin, X., Zhong, H., Huang, J., He, C., Lee, C., 2015. Bloch-Landau-Zener dynamics in single-particle Wannier-Zeeman systems.
Phys. Rev. A 91 (5), 053409.
Kenmoe, M.B., Fai, L.C., 2016. Periodically driven three-level systems. Phys. Rev. B 94 (12), 125101.
Kenmoe, M.B., Phien, H.N., Kiselev, M.N., Fai, L.C., 2013. Effects of colored noise on Landau-Zener transitions: two- and three-levelsystems. Phys. Rev. B 87 (22), 224301.
Kenmoe, M.B., Tchouobiap, S.E.M., Sadem, C.K., Tchapda, A.B., Fai, L.C., 2015. Non-adiabatic and adiabatic transitions at levelcrossing with decay: two- and three-level systems. J. Phys. A 48 (9), 095303.
Kervinen, M., Ramírez-Muñoz, J.E., Välimaa, A., Sillanpää, M.A., 2019. Landau-Zener-Stückelberg interference in a multimodeelectromechanical system in the quantum regime. Phys. Rev. Lett. 123, 240401.
Khivrich, I., Ilani, S., 2020. Atomic-like charge qubit in a carbon nanotube enabling electric and magnetic field nano-sensing. Nat.
Commun. 11 (1), 2299.
Khomeriki, R., Flach, S., 2016. Landau-Zener Bloch oscillations with perturbed flat bands. Phys. Rev. Lett. 116 (24), 245301.
Khomitsky, D.V., Studenikin, S.A., 2022. Single spin Landau-Zener-Stückelberg-Majorana interferometry of zeeman-split stateswith strong spin-orbit interaction in a double quantum dot. arXiv:2208.14965.
Kibble, T.W.B., 1976. Topology of cosmic domains and strings. J. Phys. A: Math. Gen. 9 (8), 1387–1398.
Kiselev, M.N., Kikoin, K., Kenmoe, M.B., 2013. SU(3) Landau-Zener interferometry. Europhys. Lett. 104 (5), 57004.
Kjaergaard, M., Schwartz, M.E., Braumüller, J., Krantz, P., Wang, J.I.-J., Gustavsson, S., Oliver, W.D., 2020. Superconducting qubits:
Current state of play. Annu. Rev. Condens. Matter Phys. 11 (1), 369–395.
Kling, S., Salger, T., Grossert, C., Weitz, M., 2010. Atomic Bloch-Zener oscillations and Stückelberg interferometry in optical lattices.
Phys. Rev. Lett. 105 (21), 215301.
Kmetic, M.A., Meath, W.J., 1985. Permanent dipole moments and multi-photon resonances. Phys. Lett. A 108 (7), 340–343.
Kofman, P.O., Ivakhnenko, O.V., Shevchenko, S.N., Nori, F., 2022. Majorana’s approach to nonadiabatic transitions validates theadiabatic-impulse approximation. arXiv:2208.00481.
Kohler, S., 2017. Dispersive readout of adiabatic phases. Phys. Rev. Lett. 119 (19), 196802.
Kohler, S., 2018. Dispersive readout: Universal theory beyond the rotating-wave approximation. Phys. Rev. A 98 (2), 023849.
Köhler, T., Góral, K., Julienne, P.S., 2006. Production of cold molecules via magnetically tunable Feshbach resonances. Rev. Modern
Phys. 78 (4), 1311–1361.
Koski, J.V., Landig, A.J., Pályi, A., Scarlino, P., Reichl, C., Wegscheider, W., Burkard, G., Wallraff, A., Ensslin, K., Ihn, T., 2018. Floquet
spectroscopy of a strongly driven quantum dot charge qubit with a microwave resonator. Phys. Rev. Lett. 121 (4), 043603.
Kotova, L.P., 1969. Angular distribution in inelastic atomic collisions. Sov. Phys.—JETP 28, 719–722.
Krainov, V.P., 1976. Theory of resonance multiphoton transitions in a three-level system under the influence of a strong
electromagnetic field. J. Exp. Theor. Phys. 43, 622.
Krainov, V.P., Yakovlev, V.P., 1980. Quasienergy states of a two-level atom in a strong low-frequency electromagnetic field. J. Exp.
Theor. Phys. 78, 2204.
Krantz, P., Kjaergaard, M., Yan, F., Orlando, T.P., Gustavsson, S., Oliver, W.D., 2019. A quantum engineer’s guide to superconducting
qubits. Appl. Phys. Rev. 6 (2), 021318.
Krueckl, V., Richter, K., 2012. Bloch-Zener oscillations in graphene and topological insulators. Phys. Rev. B 85 (11), 115433.
Krzywda, J.A., Cywiński, Ł., 2020. Adiabatic electron charge transfer between two quantum dots in presence of 1/f noise. Phys.
Rev. B 101 (3), 035303.
Krzywda, J.A., Cywiński, Ł., 2021. Interplay of charge noise and coupling to phonons in adiabatic electron transfer between
quantum dots. Phys. Rev. B 104 (7), 075439.
Kuno, Y., 2019. Non-adiabatic extension of the Zak phase and charge pumping in the Rice–Mele model. Eur. Phys. J. B 92 (9), 195.
Kwon, S., Tomonaga, A., Bhai, G.L., Devitt, S.J., Tsai, J.-S., 2021. Gate-based superconducting quantum computing. J. Appl. Phys.
129 (4), 041102.
LaHaye, M.D., Suh, J., Echternach, P.M., Schwab, K.C., Roukes, M.L., 2009. Nanomechanical measurements of a superconducting
qubit. Nature 459 (7249), 960–964.
Lambert, N., Ahmed, S., Cirio, M., Nori, F., 2019. Modelling the ultra-strongly coupled spin-boson model with unphysical modes.
Nat. Commun. 10 (1), 3721.
Lambert, N., Raheja, T., Cross, S., Menczel, P., Ahmed, S., Pitchford, A., Burgarth, D., Nori, F., 2020. QuTiP-BoFiN: A bosonicand fermionic numerical hierarchical-equations-of-motion library with applications in light-harvesting, quantum control, andsingle-molecule electronics. arXiv:2010.10806.
Landau, L., 1932a. Zur theorie der energieubertragung. Phyz. Z. Sowjetunion 1, 88.
Landau, L., 1932b. Zur theorie der energieubertragung II. Phyz. Z. Sowjetunion 2, 46–51.
Landau, L.D., Lifshitz, E.M., 1965. Quantum Mechanics, Non-Relativistic Theory, second ed. Pergamon Press, Oxford, pp. 185–187,
Chap. 53 Transition under the action of adiabatic perturbation.
Lang, F., Straten, P.v.d., Brandstätter, B., Thalhammer, G., Winkler, K., Julienne, P.S., Grimm, R., Denschlag, J.H., 2008. Cruisingthrough molecular bound-state manifolds with radiofrequency. Nat. Phys. 4 (3), 223–226.
78

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Larocca, M., Poggi, P.M., Wisniacki, D.A., 2018. Quantum control landscape for a two-level system near the quantum speed limit.
J. Phys. A 51 (38), 385305.
Larson, J., 2014. Interaction-induced Landau-Zener transitions. Europhys. Lett. 107 (3), 30007.
Lefebvre, C., Gagnon, D., Fillion-Gourdeau, F., MacLean, S., 2018. Carrier-envelope phase effects in graphene. J. Opt. Soc. Amer. B
35 (4), 958.
Leggett, A.J., Chakravarty, S., Dorsey, A.T., Fisher, M.P.A., Garg, A., Zwerger, W., 1987. Dynamics of the dissipative two-state system.
Rev. Modern Phys. 59 (1), 1–85.
Lehto, J., Suominen, K.-A., 2012. Superparabolic level-glancing models for two-state quantum systems. Phys. Rev. A 86 (3), 033415.
Lehto, J.M.S., Suominen, K.-A., 2015. Time-dependent two-level models and zero-area pulses. Phys. Scr. 91 (1), 013005.
Leppäkangas, J., de Graaf, S.E., Adamyan, A., Fogelström, M., Danilov, A.V., Lindström, T., Kubatkin, S.E., Johansson, G., 2013. Effectsof quasiparticle tunnelling in a circuit-QED realization of a strongly driven two-level system. J. Phys. B: At. Mol. Opt. Phys. 46(22), 224019.
Levi, A.C., 2013. Free will with afterthoughts: A quasichemical model. J. Theor. Chem. 2013, 902423.
Li, B., Ahmed, S., Saraogi, S., Lambert, N., Nori, F., Pitchford, A., Shammah, N., 2022. Pulse-level noisy quantum circuits with QuTiP.
Quantum 6, 630.
Li, W., Cen, L.-X., 2018. Dynamical transitions in a modulated Landau–Zener model with finite driving fields. Ann. Physics 389,
1–10.
Li, Q.Z., Elliott, P., Dewhurst, J.K., Sharma, S., Shallcross, S., 2021. Ab initio study of ultrafast charge dynamics in graphene. Phys.
Rev. B 103 (8), L081102.
Li, S.-C., Fu, L.-B., 2020. Rosen-Zener-Stückelberg interferometry of ultracold atoms. Phys. Rev. A 101 (2), 023618.
Li, S.-C., Fu, L.-B., Liu, J., 2018. Nonlinear Landau-Zener-Stückelberg-Majorana interferometry. Phys. Rev. A 98 (1), 013601.
Li, Z., Li, D., Li, M., Yang, X., Song, S., Han, Z., Yang, Z., Chu, J., Tan, X., Lan, D., Yu, H., Yu, Y., 2019. Quantum state transfer via
multi-passage Landau–Zener–Stückelberg interferometry in a three-qubit system. Phys. Status Solidi b 257 (1), 1900459.
Li, J., Silveri, M.P., Kumar, K.S., Pirkkalainen, J.-M., Vepsäläinen, A., Chien, W.C., Tuorila, J., Sillanpää, M.A., Hakonen, P.J.,
Thuneberg, E.V., Paraoanu, G.S., 2013. Motional averaging in a superconducting qubit. Nature Commun. 4 (1), 1420.
Li, J.-W., Wu, C.-W., Dai, H.-Y., 2011. Quantum information transfer in circuit QED with Landau–Zener tunneling. Chin. Phys. Lett.
28 (9), 090302.
Liang, S., Li, Z.-C., Zhang, W., Zhou, L., Lan, Z., 2020. Stückelberg interferometry using spin-orbit-coupled cold atoms in an opticallattice. Phys. Rev. A 102 (3), 033332.
Lidal, J., Danon, J., 2020. Generation of Schrödinger-cat states through photon-assisted Landau-Zener-Stückelberg interferometry.
Phys. Rev. A 102 (4), 043717.
Lim, L.-K., Fuchs, J.-N., Montambaux, G., 2012. Bloch-Zener oscillations across a merging transition of Dirac points. Phys. Rev. Lett.
108 (17), 175303.
Lim, L.-K., Fuchs, J.-N., Montambaux, G., 2014. Mass and chirality inversion of a Dirac cone pair in Stückelberg interferometry.
Phys. Rev. Lett. 112 (15), 155302.
Lim, L.-K., Fuchs, J.-N., Montambaux, G., 2015a. Geometric phase in Stückelberg interferometry. Phys. Rev. A 91 (4), 042119.
Lim, L.-K., Fuchs, J.-N., Montambaux, G., 2015b. Geometry of Bloch states probed by Stückelberg interferometry. Phys. Rev. A 92(6), 063627.
Liu, H., Dai, M., Wei, L.F., 2019. Simulating the Landau-Zener transitions and Landau-Zener-Stückelberg interferometers withcompacted optical waveguides: an invariant method. Phys. Rev. A 99 (1), 013820.
Liu, J., Fu, L., Ou, B.-Y., Chen, S.-G., Choi, D.-I., Wu, B., Niu, Q., 2002. Theory of nonlinear Landau-Zener tunneling. Phys. Rev. A 66(2), 023404.
Liu, X.-Z., Tian, D.-P., Chong, B., 2016. Transition time of nonlinear Landau–Zener model in adiabatic limit. Modern Phys. Lett. B
30 (15), 1650194.
Liu, W.-X., Wang, T., Zhang, X.-F., Li, W.-D., 2021. Time-domain Landau-Zener-Stückelberg-Majorana interference in an opticallattice clock. Phys. Rev. A 104 (5), 053318.
Liul, M.P., Shevchenko, S.N., 2022. Rate-equation approach for multi-level quantum systems. arXiv:2209.13505.
Llorente, J.M.G., Plata, J., 1992. Tunneling control in a two-level system. Phys. Rev. A 45 (10), R6958–R6961.
Long, Y., Ren, J., 2017. Topological Landau-Zener Bloch oscillations in photonic floquet Lieb lattices. arXiv:1706.01107.
Longhi, S., 2009. Quantum-optical analogies using photonic structures. Laser Phot. Rev. 3 (3), 243–261.
Longhi, S., Giorgi, G.L., Zambrini, R., 2019. Landau-Zener topological quantum state transfer. Adv. Q. Tech. 2, 1800090.
Lopez-Castillo, J.-M., Filali-Mouhim, A., Jay-Gerin, J.-P., 1992. Periodic motion around the crossing point in a two-level system. J.
Chem. Phys. 97 (3), 1905–1910.
Lorenz, H., Kohler, S., Parafilo, A., Kiselev, M., Ludwig, S., 2022. Visualized wave mechanics by coupled macroscopic pendula:Classical analogue to driven quantum bits. arXiv:2207.09296.
Lubin, D., Gefen, Y., Goldhirsch, I., 1990. Mesoscopic rings driven by time-dependent magnetic flux: Level correlations andlocalization in energy space. Phys. Rev. B 41 (7), 4441–4455.
Łuczak, J., Bułka, B.R., 2016. Landau–Zener transitions in spin qubit encoded in three quantum dots. Quantum Inf. Process. 16 (1),10.
Ma, J., Wang, X., Sun, C.P., Nori, F., 2011. Quantum spin squeezing. Phys. Rep. 509 (2–3), 89–165.
79

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Maeda, H., Gurian, J.H., Norum, D.V.L., Gallagher, T.F., 2006. Coherent population transfer in an atom by multiphoton adiabatic
rapid passage. Phys. Rev. Lett. 96 (7), 073002.
Magazzù, L., Forn-Díaz, P., Belyansky, R., Orgiazzi, J.-L., Yurtalan, M.A., Otto, M.R., Lupascu, A., Wilson, C.M., Grifoni, M., 2018.
Probing the strongly driven spin-boson model in a superconducting quantum circuit. Nature Commun. 9 (1), 1403.
Majorana, E., 1932. Atomi orientati in campo magnetico variabile. Il Nuovo Cimento 9 (2), 43–50.
Makhlin, Y., Schön, G., Shnirman, A., 2001. Quantum-state engineering with Josephson-junction devices. Rev. Modern Phys. 73(2), 357–400.
Malinovsky, V.S., Krause, J.L., 2001. General theory of population transfer by adiabatic rapid passage with intense, chirped laser
pulses. Eur. Phys. J. D 14 (2), 147–155.
Malla, R.K., Chernyak, V.Y., Sinitsyn, N.A., 2021. Nonadiabatic transitions in Landau-Zener grids: Integrability and semiclassicaltheory. Phys. Rev. B 103 (14), 144301.
Malla, R.K., Raikh, M.E., 2018. Landau-Zener transition in a two-level system coupled to a single highly excited oscillator. Phys.
Rev. B 97 (3), 035428.
Malla, R.K., Raikh, M.E., 2019. High Landau levels of two-dimensional electrons near the topological transition caused by interplayof spin-orbit and Zeeman energy shifts. Phys. Rev. B 99, 205426.
Mallavarapu, S.K., Niranjan, A., Li, W., Wüster, S., Nath, R., 2021. Population trapping in a pair of periodically driven Rydberg
atoms. Phys. Rev. A 103 (2), 023335.
Maris, H.J., Xiong, Q., 1988. Adiabatic and nonadiabatic processes in classical and quantum mechanics. Am. J. Phys. 56 (12),1114–1117.
Mark, M., Ferlaino, F., Knoop, S., Danzl, J.G., Kraemer, T., Chin, C., Nägerl, H.-C., Grimm, R., 2007a. Spectroscopy of ultracold trapped
cesium Feshbach molecules. Phys. Rev. A 76 (4), 042514.
Mark, M., Kraemer, T., Waldburger, P., Herbig, J., Chin, C., Nägerl, H.-C., Grimm, R., 2007b. ‘‘Stückelberg interferometry’’ withultracold molecules. Phys. Rev. Lett. 99 (11), 113201.
Marzlin, K.-P., Sanders, B.C., 2004. Inconsistency in the application of the adiabatic theorem. Phys. Rev. Lett. 93 (16), 160408.
Mathai, C., Masis, S., Shtempluck, O., Hacohen-Gourgy, S., Buks, E., 2020. Frequency mixing in a ferrimagnetic sphere resonator.
Eur. Phys. Lett. 131 (6), 67001.
Matityahu, S., Schmidt, H., Bilmes, A., Shnirman, A., Weiss, G., Ustinov, A.V., Schechter, M., Lisenfeld, J., 2019. Dynamical decouplingof quantum two-level systems by coherent multiple Landau–Zener transitions. NPJ Quantum Inf. 5 (1), 114.
Mavalankar, A., Pei, T., Gauger, E.M., Warner, J.H., Briggs, G.A.D., Laird, E.A., 2016. Photon-assisted tunneling and charge dephasing
in a carbon nanotube double quantum dot. Phys. Rev. B 93 (23), 235428.
Mehl, S., DiVincenzo, D.P., 2013. Noise-protected gate for six-electron double-dot qubit. Phys. Rev. B 88 (16), 161408.
Mélin, R., Danneau, R., Yang, K., Caputo, J.-G., Douçot, B., 2019. Engineering the Floquet spectrum of superconducting multiterminal
quantum dots. Phys. Rev. B 100, 035450.
Mélin, R., Douçot, B., 2020. Inversion in a four-terminal superconducting device on the quartet line. II. Quantum dot and Floquet
theory. Phys. Rev. B 102 (24), 245436.
Menchon-Enrich, R., Benseny, A., Ahufinger, V., Greentree, A.D., Busch, T., Mompart, J., 2016. Spatial adiabatic passage: a review
of recent progress. Rep. Progr. Phys. 79 (7), 074401.
Mi, X., Kohler, S., Petta, J.R., 2018. Landau-Zener interferometry of valley-orbit states in Si/SiGe double quantum dots. Phys. Rev.
B 98 (16), 161404.
Miao, K.C., Bourassa, A., Anderson, C.P., Whiteley, S.J., Crook, A.L., Bayliss, S.L., Wolfowicz, G., Thiering, G., Udvarhelyi, P., Ivády, V.,
Abe, H., Ohshima, T., Gali, Á., Awschalom, D.D., 2019. Electrically driven optical interferometry with spins in silicon carbide.
Sci. Adv. 5 (11), 0527.
Miao, Q., Zheng, Y., 2016a. Coherent destruction of tunneling in two-level system driven across avoided crossing via photon
statistics. Sci. Rep. 6 (1), 28959.
Miao, Q., Zheng, Y., 2016b. Single-photon emission of two-level system via rapid adiabatic passage. Sci. Rep. 6 (1), 32827.
Mies, F.H., Tiesinga, E., Julienne, P.S., 2000. Manipulation of Feshbach resonances in ultracold atomic collisions using
time-dependent magnetic fields. Phys. Rev. A 61 (2), 022721.
Monaco, R., Mygind, J., Rivers, R.J., Koshelets, V.P., 2009. Spontaneous fluxoid formation in superconducting loops. Phys. Rev. B
80 (18), 180501.
Montambaux, G., 2018. Artificial graphenes: Dirac matter beyond condensed matter. C. R. Phys. 19 (5), 285–305.
Montambaux, G., Jérome, D., 2016. Rapid magnetic oscillations and magnetic breakdown in quasi-1D conductors. C. R. Phys. 17(3–4), 376–388.
Morales-Molina, L., Reyes, S.A., 2011. Non-adiabatic transitions in a non-symmetric optical lattice. J. Phys. B: At. Mol. Opt. Phys.
44 (20), 205403.
Mori, T., 2022. Floquet states in open quantum systems. arXiv:2203.16358.
Muirhead, C.M., Gunupudi, B., Colclough, M.S., 2016. Photon transfer in a system of coupled superconducting microwave
resonators. J. Appl. Phys. 120 (8), 084904.
Mukherjee, V., Divakaran, U., Dutta, A., Sen, D., 2007. Quenching dynamics of a quantum XY spin-1/2 chain in a transverse field.
Phys. Rev. B 76 (17), 174303.
Mukherjee, V., Dutta, A., 2009. Effects of interference in the dynamics of a spin-1/2 transverse XY chain driven periodically throughquantum critical points. J. Stat. Mech. Theory Exp. 2009 (05), P05005.
80

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Mukherjee, B., Mohan, P., Sen, D., Sengupta, K., 2018. Low-frequency phase diagram of irradiated graphene and a periodicallydriven spin- 12 XY chain. Phys. Rev. B 97 (20), 205415.
Mukherjee, B., Sen, A., Sen, D., Sengupta, K., 2016. Signatures and conditions for phase band crossings in periodically driven
integrable systems. Phys. Rev. B 94 (15), 155122.
Mullen, K., Ben-Jacob, E., Gefen, Y., Schuss, Z., 1989. Time of Zener tunneling. Phys. Rev. Lett. 62 (21), 2543–2546.
Munoz-Bauza, H., Chen, H., Lidar, D., 2019. A double-slit proposal for quantum annealing. NPJ Quantum Inf. 5 (1), 51.
Munsch, M., Wüst, G., Kuhlmann, A.V., Xue, F., Ludwig, A., Reuter, D., Wieck, A.D., Poggio, M., Warburton, R.J., 2014. Manipulationof the nuclear spin ensemble in a quantum dot with chirped magnetic resonance pulses. Nat. Nanotechnol. 9 (9), 671–675.
Munyaev, V.O., Bastrakova, M.V., 2021. Control of spectroscopic features of multiphoton transitions in two coupled qubits bydriving fields. Phys. Rev. A 104 (1), 012613.
Nagaya, K., Teranishi, Y., Nakamura, H., 2002. Control of molecular processes by a sequence of linearly chirped pulses. J. Chem.
Phys. 117 (21), 9588–9604.
Nagaya, K., Zhu, C., Lin, S.H., 2007. Nonlinear responses of degenerate two-level systems to intense few-cycle pulses. J. Chem.
Phys. 127 (9), 094304.
Nakajima, T., Delbecq, M.R., Otsuka, T., Amaha, S., Yoneda, J., Noiri, A., Takeda, K., Allison, G., Ludwig, A., Wieck, A.D., Hu, X.,
Nori, F., Tarucha, S., 2018. Coherent transfer of electron spin correlations assisted by dephasing noise. Nature Commun. 9 (1),2133.
Nakamura, H., 2012. Nonadiabatic Transition. World Scientific.
Nakamura, H., 2019. Introduction to Nonadiabatic Dynamics. World Scientific.
Nakamura, Y., Pashkin, Y.A., Tsai, J.S., 2001. Rabi oscillations in a Josephson-junction charge two-level system. Phys. Rev. Lett. 87(24), 246601.
Nalbach, P., Javanbakht, S., Stahl, C., Thorwart, M., 2016. Stueckelberg oscillations in a two-state two-path model of a conicalintersection. Ann. Physics 529 (9), 1600147.
Nalbach, P., Knörzer, J., Ludwig, S., 2013. Nonequilibrium Landau-Zener-Stueckelberg spectroscopy in a double quantum dot. Phys.
Rev. B 87 (16), 165425.
Nalbach, P., Thorwart, M., 2009. Landau-Zener transitions in a dissipative environment: Numerically exact results. Phys. Rev. Lett.
103 (22), 220401.
Neilinger, P., Shevchenko, S.N., Bogár, J., Rehák, M., Oelsner, G., Karpov, D.S., Hübner, U., Astafiev, O., Grajcar, M., Il’ichev, E., 2016.
Landau-Zener-Stückelberg-Majorana lasing in circuit quantum electrodynamics. Phys. Rev. B 94 (9), 094519.
Nikitin, E.E., 1999. Nonadiabatic transitions: What we learned from old masters and how much we owe them. Annu. Rev. Phys.
Chem. 50 (1), 1–21.
Nikitin, E., 2006. Adiabatic and diabatic collision processes at low energies. In: Springer Handbook of Atomic, Molecular, andOptical Physics. pp. 741–752.
Nikitin, E.E., Reznikov, A.I., 1972. Calculation of transition probabilities using the Landau-Zener model. Phys. Rev. A 6 (1), 522–523.
Niranjan, A., Li, W., Nath, R., 2020. Landau-Zener transitions and adiabatic impulse approximation in an array of two Rydberg
atoms with time-dependent detuning. Phys. Rev. A 101 (6), 063415.
Novotny, L., 2010. Strong coupling, energy splitting, and level crossings: A classical perspective. Am. J. Phys. 78 (11), 1199–1202.
Oh, S., Hu, X., Nori, F., Kais, S., 2016. Singularity of the time-energy uncertainty in adiabatic perturbation and cycloids on a Bloch
sphere. Sci. Rep. 6 (1), 20824.
O’Keeffe, M.F., Chudnovsky, E.M., Garanin, D.A., 2013. Landau-Zener dynamics of a nanoresonator containing a tunneling spin.
Phys. Rev. B 87 (17), 174418.
Oliver, W.D., Valenzuela, S.O., 2009. Large-amplitude driving of a superconducting artificial atom. Quantum Inf. Process. 8 (2–3),
261–281.
Oliver, W.D., Yu, Y., Lee, J.C., Berggren, K.K., Levitov, L.S., Orlando, T.P., 2005. Mach-Zehnder interferometry in a strongly driven
superconducting qubit. Science 310 (5754), 1653–1657.
Ono, K., Shevchenko, S.N., Mori, T., Moriyama, S., Nori, F., 2019. Quantum interferometry with a g -factor-tunable spin qubit. Phys.
Rev. Lett. 122 (20), 207703.
Ono, K., Shevchenko, S.N., Mori, T., Moriyama, S., Nori, F., 2020. Analog of a quantum heat engine using a single-spin qubit. Phys.
Rev. Lett. 125 (16), 166802.
Oriekhov, D.O., Cheipesh, Y., Beenakker, C.W.J., 2021. Voltage staircase in a current-biased quantum-dot Josephson junction. Phys.
Rev. B 103 (9), 094518.
Orth, P.P., Imambekov, A., Hur, K.L., 2013. Nonperturbative stochastic method for driven spin-boson model. Phys. Rev. B 87 (1),014305.
Osika, E.N., Szafran, B., 2017. Spin-valley resolved photon-assisted tunneling in carbon nanotube double quantum dots. Phys. Rev.
B 95 (20), 205305.
Ostrovsky, V.N., Nakamura, H., 1997. Exact analytical solution of the N-level Landau - Zener-type bow-tie model. J. Phys. A: Math.
Gen. 30 (19), 6939–6950.
Ostrovsky, V.N., Volkov, M.V., 2006. Landau-Zener transitions in a system of interacting spins: Exact results for demagnetization
probability. Phys. Rev. B 73 (6), 060405.
Ostrovsky, V.N., Volkov, M.V., Hansen, J.P., Selstø, S., 2007. Four-state nonstationary models in multistate Landau-Zener theory.
Phys. Rev. B 75 (1), 014441.
81

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Ota, T., Hitachi, K., Muraki, K., 2018. Landau-Zener-Stückelberg interference in coherent charge oscillations of a one-electron
double quantum dot. Sci. Rep. 8 (1), 5491.
Ota, T., Hitachi, K., Muraki, K., Fujisawa, T., 2017. Dissipative Landau–Zener transition in double quantum dot under sinusoidal
potential modulation. Appl. Phys. Express 10 (11), 115201.
Otxoa, R.M., Chatterjee, A., Shevchenko, S.N., Barraud, S., Nori, F., Gonzalez-Zalba, M.F., 2019. Quantum interference capacitor
based on double-passage Landau-Zener-Stückelberg-Majorana interferometry. Phys. Rev. B 100, 205425.
Paila, A., Tuorila, J., Sillanpää, M., Gunnarsson, D., Sarkar, J., Makhlin, Y., Thuneberg, E., Hakonen, P., 2009. Interband transitions
and interference effects in superconducting qubits. Quantum Inf. Process. 8 (2–3), 245–259.
Parafilo, A.V., Kiselev, M.N., 2018a. Landau–Zener transitions and Rabi oscillations in a Cooper-pair box: beyond two-level models.
Low Temp. Phys. 44 (12), 1325–1330.
Parafilo, A.V., Kiselev, M.N., 2018b. Tunable RKKY interaction in a double quantum dot nanoelectromechanical device. Phys. Rev.
B 97 (3), 035418.
Parafilo, A.V., Krive, I.V., Shekhter, R.I., Jonson, M., 2012. Nanoelectromechanics of superconducting weak links (Review Article).
Low Temp. Phys. 38 (4), 273–282.
Pasek, W.J., Maialle, M.Z., Degani, M.H., 2018. Application of the Landau-Zener-Stückelberg-Majorana dynamics to the electricallydriven flip of a hole spin. Phys. Rev. B 97 (11), 115417.
Paul, N., Crowley, P.J.D., Devakul, T., Fu, L., 2022. Moiré Landau fans and magic zeros. Phys. Rev. Lett. 129 (11), 116804.
Pavlyukh, Y., 2020. Toroidal spin states in molecular magnets. Phys. Rev. B 101 (14), 144408.
Pekola, J.P., Solinas, P., Shnirman, A., Averin, D.V., 2013. Calorimetric measurement of work in a quantum system. New J. Phys.
15 (11), 115006.
Penfold-Fitch, Z., Sfigakis, F., Buitelaar, M., 2017. Microwave spectroscopy of a carbon nanotube charge qubit. Phys. Rev. Appl. 7(5), 054017.
Petersson, K.D., Petta, J.R., Lu, H., Gossard, A.C., 2010. Quantum coherence in a one-electron semiconductor charge qubit. Phys.
Rev. Lett. 105 (24), 246804.
Petiziol, F., Dive, B., Carretta, S., Mannella, R., Mintert, F., Wimberger, S., 2019. Accelerating adiabatic protocols for entangling two
qubits in circuit QED. Phys. Rev. A 99 (4), 042315.
Petiziol, F., Dive, B., Mintert, F., Wimberger, S., 2018. Fast adiabatic evolution by oscillating initial Hamiltonians. Phys. Rev. A 98(4), 043436.
Petiziol, F., Wimberger, S., 2019. Effect of phase errors on a quantum control protocol using fast oscillations. Condens. Matter 4(1), 34.
Petta, J.R., Lu, H., Gossard, A.C., 2010. A coherent beam splitter for electronic spin states. Science 327 (5966), 669–672.
Platero, G., Aguado, R., 2004. Photon-assisted transport in semiconductor nanostructures. Phys. Rep. 395 (1–2), 1–157.
Plötz, P., Wimberger, S., 2011. Stückelberg-interferometry with ultra-cold atoms. Eur. Phys. J. D 65 (1–2), 199–205.
Poggi, P.M., Lombardo, F.C., Wisniacki, D.A., 2015. Time-optimal control fields for quantum systems with multiple avoided
crossings. Phys. Rev. A 92 (5), 053411.
Polkovnikov, A., 2005. Universal adiabatic dynamics in the vicinity of a quantum critical point. Phys. Rev. B 72 (16), 161201.
Poulin-Lamarre, G., Thorgrimson, J., Studenikin, S.A., Aers, G.C., Kam, A., Zawadzki, P., Wasilewski, Z.R., Sachrajda, A.S., 2015.
Three-spin coherent oscillations and interference. Phys. Rev. B 91 (12), 125417.
Pu, H., Raghavan, S., Bigelow, N.P., 2000. Manipulating spinor condensates with magnetic fields: Stochastization, metastability,and dynamical spin localization. Phys. Rev. A 61 (2), 023602.
Pyka, K., Keller, J., Partner, H.L., Nigmatullin, R., Burgermeister, T., Meier, D.M., Kuhlmann, K., Retzker, A., Plenio, M.B., Zurek, W.H.,
del Campo, A., Mehlstäubler, T.E., 2013. Topological defect formation and spontaneous symmetry breaking in ion Coulomb
crystals. Nature Commun. 4 (1), 2291.
Qi, Z., Wu, X., Ward, D.R., Prance, J.R., Kim, D., Gamble, J.K., Mohr, R.T., Shi, Z., Savage, D.E., Lagally, M.G., Eriksson, M.A., Friesen, M.,
Coppersmith, S.N., Vavilov, M.G., 2017. Effects of charge noise on a pulse-gated singlet-triplet S-T- qubit. Phys. Rev. B 96 (11),115305.
Qin, X.-K., 2016. How to control the coherent oscillations in Landau-Zener-Stueckelberg dynamics of three-level system. Modern
Phys. Lett. B 30 (09), 1650149.
Quan, H.T., Zurek, W.H., 2010. Testing quantum adiabaticity with quench echo. New J. Phys. 12 (9), 093025.
Quintana, C.M., Petersson, K.D., McFaul, L.W., Srinivasan, S.J., Houck, A.A., Petta, J.R., 2013. Cavity-mediated entanglement
generation via Landau-Zener interferometry. Phys. Rev. Lett. 110 (17), 173603.
Rabi, I.I., 1937. Space quantization in a gyrating magnetic field. Phys. Rev. 51 (8), 652–654.
Rakhmanov, A.L., Zagoskin, A.M., Savel’ev, S., Nori, F., 2008. Quantum metamaterials: Electromagnetic waves in a Josephson qubit
line. Phys. Rev. B 77 (14), 144507.
Reid, B., Moreno-Cardoner, M., Sherson, J., Chiara, G.D., 2016. Manipulating matter waves in an optical superlattice. Phys. Rev. A
94 (6), 063629.
Restrepo, S., Cerrillo, J., Bastidas, V., Angelakis, D., Brandes, T., 2016. Driven open quantum systems and Floquet stroboscopic
dynamics. Phys. Rev. Lett. 117 (25), 250401.
Reyes, S.A., Olivares, F.A., Morales-Molina, L., 2012. Landau–Zener–Stückelberg interferometry in PT-symmetric opticalwaveguides. J. Phys. A 45 (44), 444027.
82

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Reynoso, A.A., Flensberg, K., 2012. Dephasing and hyperfine interaction in carbon nanotubes double quantum dots: Disorderedcase. Phys. Rev. B 85 (19), 195441.
Ribeiro, H., Burkard, G., Petta, J.R., Lu, H., Gossard, A.C., 2013a. Coherent adiabatic spin control in the presence of charge noiseusing tailored pulses. Phys. Rev. Lett. 110 (8), 086804.
Ribeiro, H., Petta, J.R., Burkard, G., 2010. Harnessing the GaAs quantum dot nuclear spin bath for quantum control. Phys. Rev. B
82 (11), 115445.
Ribeiro, H., Petta, J.R., Burkard, G., 2013b. Interplay of charge and spin coherence in Landau-Zener-Stückelberg-Majorana
interferometry. Phys. Rev. B 87 (23), 235318.
Rodionov, Y.I., Kugel, K.I., Nori, F., 2016. Floquet spectrum and driven conductance in Dirac materials: Effects of
Landau-Zener-Stückelberg-Majorana interferometry. Phys. Rev. B 94 (19), 195108.
Rodriguez-Vega, M., Lentz, M., Seradjeh, B., 2018. Floquet perturbation theory: formalism and application to low-frequency limit.
New J. Phys. 20 (9), 093022.
Rodriguez-Vega, M., Vogl, M., Fiete, G.A., 2021. Low-frequency and Moiré–Floquet engineering: A review. Ann. Physics 435,
168434.
Rojo, A.G., Bloch, A.M., 2010. The rolling sphere, the quantum spin, and a simple view of the Landau–Zener problem. Am. J. Phys.
78 (10), 1014–1022.
Rol, M.A., Battistel, F., Malinowski, F.K., Bultink, C.C., Tarasinski, B.M., Vollmer, R., Haider, N., Muthusubramanian, N., Bruno, A.,
Terhal, B.M., DiCarlo, L., 2019. Fast, high-fidelity conditional-phase gate exploiting leakage interference in weakly anharmonicsuperconducting qubits. Phys. Rev. Lett. 123, 120502.
Romanova, J.Y., Demidov, E.V., Mourokh, L.G., Romanov, Y.A., 2011. Zener tunneling in semiconductor superlattices. J. Phys.:
Condens. Matter 23 (30), 305801.
Roósz, G., Divakaran, U., Rieger, H., Iglói, F., 2014. Nonequilibrium quantum relaxation across a localization-delocalizationtransition. Phys. Rev. B 90 (18), 184202.
Rotvig, J., Jauho, A.-P., Smith, H., 1995. Bloch oscillations, Zener tunneling, and Wannier-Stark ladders in the time domain. Phys.
Rev. Lett. 74 (10), 1831–1834.
Roy, A., 2013. Nonequilibrium dynamics of ultracold Fermi superfluids. Eur. Phys. J. Spec. Top. 222 (3–4), 975–993.
Rozhkov, A., Sboychakov, A., Rakhmanov, A., Nori, F., 2016. Electronic properties of graphene-based bilayer systems. Phys. Rep.
648, 1–104.
Rubbmark, J.R., Kash, M.M., Littman, M.G., Kleppner, D., 1981. Dynamical effects at avoided level crossings: a study of the
Landau-Zener effect using Rydberg atoms. Phys. Rev. A 23 (6), 3107–3117.
Rudner, M.S., Shytov, A.V., Levitov, L.S., Berns, D.M., Oliver, W.D., Valenzuela, S.O., Orlando, T.P., 2008. Quantum phase tomographyof a strongly driven qubit. Phys. Rev. Lett. 101 (19), 190502.
Russomanno, A., Pugnetti, S., Brosco, V., Fazio, R., 2011. Floquet theory of Cooper pair pumping. Phys. Rev. B 83 (21), 214508.
Ruutu, V.M.H., Eltsov, V.B., Gill, A.J., Kibble, T.W.B., Krusius, M., Makhlin, Y.G., Plaçais, B., Volovik, G.E., Xu, W., 1996. Vortex
formation in neutron-irradiated superfluid 3 He as an analogue of cosmological defect formation. Nature 382 (6589), 334–336.
Saiko, A.P., Fedaruk, R., Markevich, S.A., 2014. Relaxation, decoherence, and steady-state population inversion in qubits doubly
dressed by microwave and radiofrequency fields. J. Phys. B: At. Mol. Opt. Phys. 47 (15), 155502.
Saiko, A.P., Fedoruk, G.G., Markevich, S.A., 2007. Multiphoton transitions in a spin system driven by strong bichromatic field. J.
Exp. Theor. Phys. 105 (5), 893–899.
Saiko, A.P., Markevich, S.A., Fedaruk, R., 2016. Dissipative two-level systems under ultrastrong off-resonant driving. Phys. Rev. A
93 (6), 063834.
Saiko, A.P., Markevich, S.A., Fedaruk, R., 2018. Multiphoton Raman transitions and Rabi oscillations in driven spin systems. Phys.
Rev. A 98 (4), 043814.
Saiko, A.P., Markevich, S.A., Fedaruk, R., 2019. Possibility of direct observation of the Bloch-Siegert shift in coherent dynamics ofmultiphoton Raman transitions. JETP Lett. 110 (7), 441–446.
Saito, K., Kayanuma, Y., 2002. Nonadiabatic transition probabilities in the presence of strong dissipation at an avoided-levelcrossing point. Phys. Rev. A 65 (3), 033407.
Saito, S., Thorwart, M., Tanaka, H., Ueda, M., Nakano, H., Semba, K., Takayanagi, H., 2004. Multiphoton transitions in a macroscopic
quantum two-state system. Phys. Rev. Lett. 93 (3), 037001.
Saito, K., Wubs, M., Kohler, S., Hänggi, P., Kayanuma, Y., 2006. Quantum state preparation in circuit QED via Landau-Zenertunneling. Europhys. Lett. 76 (1), 22–28.
Saito, K., Wubs, M., Kohler, S., Kayanuma, Y., Hänggi, P., 2007. Dissipative Landau-Zener transitions of a qubit: Bath-specific anduniversal behavior. Phys. Rev. B 75 (21), 214308.
Salmilehto, J., Möttönen, M., 2011. Superadiabatic theory for Cooper pair pumping under decoherence. Phys. Rev. B 84 (17),174507.
Sambe, H., 1973. Steady states and quasienergies of a quantum-mechanical system in an oscillating field. Phys. Rev. A 7 (6),2203–2213.
Santoro, G.E., 2002. Theory of quantum annealing of an Ising spin glass. Science 295 (5564), 2427–2430.
Sarandy, M.S., Wu, L.-A., Lidar, D.A., 2004. Consistency of the adiabatic theorem. Quantum Inf. Process. 3 (6), 331–349.
Särkkä, J., Harju, A., 2011. Spin dynamics at the singlet–triplet crossings in a double quantum dot. New J. Phys. 13 (4), 043010.
83

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Satanin, A.M., Denisenko, M.V., Ashhab, S., Nori, F., 2012. Amplitude spectroscopy of two coupled qubits. Phys. Rev. B 85 (18),184524.
Satanin, A.M., Denisenko, M.V., Gelman, A.I., Nori, F., 2014. Amplitude and phase effects in Josephson qubits driven by a biharmonic
electromagnetic field. Phys. Rev. B 90 (10), 104516.
Saxon, R.P., Olson, R.E., 1975. Quantum calculations and classical interpretation of a model curve-crossing problem. Phys. Rev. A
12 (3), 830–834.
Scala, M., Militello, B., Messina, A., Vitanov, N.V., 2011. Microscopic description of dissipative dynamics of a level-crossing
transition. Phys. Rev. A 84 (2), 023416.
Scheuer, J., Schwartz, I., Müller, S., Chen, Q., Dhand, I., Plenio, M.B., Naydenov, B., Jelezko, F., 2017. Robust techniques forpolarization and detection of nuclear spin ensembles. Phys. Rev. B 96 (17), 174436.
Seitner, M.J., Ribeiro, H., Kölbl, J., Faust, T., Kotthaus, J.P., Weig, E.M., 2016. Classical Stückelberg interferometry of ananomechanical two-mode system. Phys. Rev. B 94 (24), 245406.
Seitner, M.J., Ribeiro, H., Kölbl, J., Faust, T., Weig, E.M., 2017. Finite-time Stückelberg interferometry with nanomechanical modes.
New J. Phys. 19 (3), 033011.
Sen, A., Sen, D., Sengupta, K., 2021. Analytic approaches to periodically driven closed quantum systems: methods and applications.
J. Phys.: Condens. Matter 33 (44), 443003.
Setiawan, F., Sengupta, K., Spielman, I., Sau, J.D., 2015. Dynamical detection of topological phase transitions in short-lived atomic
systems. Phys. Rev. Lett. 115 (19), 190401.
Shafiei, M., Nowack, K.C., Reichl, C., Wegscheider, W., Vandersypen, L.M.K., 2013. Resolving spin-orbit- and hyperfine-mediatedelectric dipole spin resonance in a quantum dot. Phys. Rev. Lett. 110 (10), 107601.
Shaikhaidarov, R.S., Kim, K.H., Dunstan, J.W., Antonov, I.V., Linzen, S., Ziegler, M., Golubev, D.S., Antonov, V.N., Il’ichev, E.V.,
Astafiev, O.V., 2022. Quantized current steps due to the a.c. coherent quantum phase-slip effect. Nature 608, 45–49.
Shammah, N., Ahmed, S., Lambert, N., Liberato, S.D., Nori, F., 2018. Open quantum systems with local and collective incoherent
processes: Efficient numerical simulations using permutational invariance. Phys. Rev. A 98 (6), 063815.
Shen, W., Devereaux, T.P., Freericks, J.K., 2014. Beyond Planck-Einstein quanta: Amplitude-driven quantum excitation. Phys. Rev.
B 90 (19), 195104.
Shen, X., Wang, F., Li, Z., Wu, Z., 2019. Landau-Zener-Stückelberg interferometry in PT -symmetric non-hermitian models. Phys.
Rev. A 100, 062514.
Shevchenko, S.N., 2008. Impedance measurement technique for quantum systems. Eur. Phys. J. B 61 (2), 187–191.
Shevchenko, S.N., 2019. Mesoscopic Physics Meets Quantum Engineering. World Scientific Pub Co Inc.
Shevchenko, S.N., Ashhab, S., Nori, F., 2010. Landau–Zener–Stückelberg interferometry. Phys. Rep. 492 (1), 1–30.
Shevchenko, S.N., Ashhab, S., Nori, F., 2012a. Inverse Landau-Zener-Stückelberg problem for qubit-resonator systems. Phys. Rev.
B 85 (9), 094502.
Shevchenko, S.N., Kiyko, A.S., Omelyanchouk, A.N., Krech, W., 2005. Dynamic behavior of Josephson-junction qubits: crossover
between Rabi oscillations and Landau–Zener transitions. Low Temp. Phys. 31 (7), 569–576.
Shevchenko, S.N., Omelyanchouk, A.N., Il’ichev, E., 2012b. Multiphoton transitions in Josephson-junction qubits (Review Article).
Low Temp. Phys. 38 (4), 283–300.
Shevchenko, S.N., van der Ploeg, S.H.W., Grajcar, M., Il’ichev, E., Omelyanchouk, A.N., Meyer, H.-G., 2008. Resonant excitations of
single and two-qubit systems coupled to a tank circuit. Phys. Rev. B 78 (17), 174527.
Shevchenko, S.N., Ryzhov, A.I., Nori, F., 2018. Low-frequency spectroscopy for quantum multilevel systems. Phys. Rev. B 98 (19),195434.
Shi, Z.-C., Chen, Y.-H., Qin, W., Xia, Y., Yi, X.X., Zheng, S.-B., Nori, F., 2021. Two-level systems with periodic N-step driving fields:
Exact dynamics and quantum state manipulations. Phys. Rev. A 104 (5), 053101.
Shi, Z., Simmons, C.B., Ward, D.R., Prance, J.R., Wu, X., Koh, T.S., Gamble, J.K., Savage, D.E., Lagally, M.G., Friesen, M., Coppersmith, S.N., Eriksson, M.A., 2014. Fast coherent manipulation of three-electron states in a double quantum dot. Nature
Commun. 5 (1), 3020.
Shi, Z.C., Wang, W., Yi, X.X., 2016. Population transfer driven by far-off-resonant fields. Opt. Express 24 (19), 21971.
Shimshoni, E., Gefen, Y., 1991. Onset of dissipation in Zener dynamics: relaxation versus dephasing. Ann. Physics 210 (1), 16–80.
Shirley, J.H., 1965. Solution of the Schrödinger equation with a Hamiltonian periodic in time. Phys. Rev. 138 (4B), B979–B987.
Shytov, A.V., 2004. Landau-Zener transitions in a multilevel system: An exact result. Phys. Rev. A 70 (5), 052708.
Shytov, A.V., Ivanov, D.A., Feigel’man, M.V., 2003. Landau-Zener interferometry for qubits. Eur. Phys. J. B 36 (2), 263–269.
Sillanpää, M., Lehtinen, T., Paila, A., Makhlin, Y., Hakonen, P., 2006. Continuous-time monitoring of Landau-Zener interference ina Cooper-pair box. Phys. Rev. Lett. 96 (18), 187002.
Sillanpää, M., Lehtinen, T., Paila, A., Makhlin, Y., Hakonen, P.J., 2007. Landau–Zener interferometry in a Cooper-pair box. J. Low
Temp. Phys. 146 (1–2), 253–262.
Sillanpää, M.A., Lehtinen, T., Paila, A., Makhlin, Y., Roschier, L., Hakonen, P.J., 2005. Direct observation of Josephson capacitance.
Phys. Rev. Lett. 95 (20), 206806.
Silveri, M.P., Kumar, K.S., Tuorila, J., Li, J., Vepsäläinen, A., Thuneberg, E.V., Paraoanu, G.S., 2015. Stückelberg interference in asuperconducting qubit under periodic latching modulation. New J. Phys. 17 (4), 043058.
Silveri, M., Tuorila, J., Kemppainen, M., Thuneberg, E., 2013. Probe spectroscopy of quasienergy states. Phys. Rev. B 87 (13), 134505.
84

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Silveri, M., Tuorila, J., Sillanpää, M., Thuneberg, E., Makhlin, Y., Hakonen, P., 2012. Basis dependence of approximative energy
levels in a strongly driven two-level system. J. Phys. Conf. Ser. 400 (4), 042054.
Silveri, M.P., Tuorila, J.A., Thuneberg, E.V., Paraoanu, G.S., 2017. Quantum systems under frequency modulation. Rep. Progr. Phys.
80 (5), 056002.
Singh, A.K., Chotorlishvili, L., Srivastava, S., Tralle, I., Toklikishvili, Z., Berakdar, J., Mishra, S.K., 2020. Generation of coherence inan exactly solvable nonlinear nanomechanical system. Phys. Rev. B 101 (10), 104311.
Sinitsyn, N.A., 2002. Multiparticle Landau-Zener problem: Application to quantum dots. Phys. Rev. B 66 (20), 205303.
Sinitsyn, N.A., 2015. Solvable four-state Landau-Zener model of two interacting qubits with path interference. Phys. Rev. B 92(20), 205431.
Sinitsyn, N.A., Chernyak, V.Y., 2017. The quest for solvable multistate Landau-Zener models. J. Phys. A 50 (25), 255203.
Skelt, A.H., Godby, R.W., D’Amico, I., 2018. Measuring adiabaticity in nonequilibrium quantum systems. Phys. Rev. A 98 (1), 012104.
Smirnov, A.Y., Savel’ev, S., Mourokh, L.G., Nori, F., 2007. Modelling chemical reactions using semiconductor quantum dots.
Europhys. Lett. 80 (6), 67008.
Snyder, R., Trimble, C., Rong, C., Folkes, P., Taylor, P., Williams, J., 2018. Weak-link Josephson junctions made from topologicalcrystalline insulators. Phys. Rev. Lett. 121 (9), 097701.
Solinas, P., Averin, D.V., Pekola, J.P., 2013. Work and its fluctuations in a driven quantum system. Phys. Rev. B 87 (6), 060508.
Son, S.-K., Han, S., Chu, S.-I., 2009. Floquet formulation for the investigation of multiphoton quantum interference in asuperconducting qubit driven by a strong ac field. Phys. Rev. A 79 (3), 032301.
Son, J., Talkner, P., Thingna, J., 2021. Monitoring quantum Otto engines. Phys. Rev. X Quantum 2 (4), 040328.
Spreeuw, R.J.C., van Druten, N.J., Beijersbergen, M.W., Eliel, E.R., Woerdman, J.P., 1990. Classical realization of a strongly driven
two-level system. Phys. Rev. Lett. 65 (21), 2642–2645.
Stehlik, J., Dovzhenko, Y., Petta, J.R., Johansson, J.R., Nori, F., Lu, H., Gossard, A.C., 2012. Landau-Zener-Stückelberg interferometryof a single electron charge qubit. Phys. Rev. B 86 (12), 121303.
Stehlik, J., Liu, Y.-Y., Eichler, C., Hartke, T., Mi, X., Gullans, M., Taylor, J., Petta, J., 2016a. Double quantum dot Floquet gain medium.
Phys. Rev. X 6 (4), 041027.
Stehlik, J., Maialle, M.Z., Degani, M.H., Petta, J.R., 2016b. Role of multilevel Landau-Zener interference in extreme harmonic
generation. Phys. Rev. B 94 (7), 075307.
Stehlik, J., Schroer, M., Maialle, M., Degani, M., Petta, J., 2014. Extreme harmonic generation in electrically driven spin resonance.
Phys. Rev. Lett. 112 (22), 227601.
Stoof, T.H., Nazarov, Y.V., 1996. Time-dependent resonant tunneling via two discrete states. Phys. Rev. B 53 (3), 1050–1053.
Stückelberg, E.C.G., 1932. Theorie der unelastischen Stösse zwischen atomen. Helv. Phys. Acta 5, 369–423.
Studenikin, S.A., Aers, G.C., Granger, G., Gaudreau, L., Kam, A., Zawadzki, P., Wasilewski, Z.R., Sachrajda, A.S., 2012. Quantum
interference between three two-spin states in a double quantum dot. Phys. Rev. Lett. 108 (22), 226802.
Stueckelberg, E.C.G., 1970. Theory of inelastic collisions between atoms. Nasa Tech. Dock. 19720003957.
Sun, Y., Leykam, D., Nenni, S., Song, D., Chen, H., Chong, Y., Chen, Z., 2018. Observation of valley Landau-Zener-Bloch oscillations
and pseudospin imbalance in photonic graphene. Phys. Rev. Lett. 121 (3), 033904.
Sun, Z., Ma, J., Wang, X., Nori, F., 2012. Photon-assisted Landau-Zener transition: role of coherent superposition states. Phys. Rev.
A 86 (1), 012107.
Sun, G., Wen, X., Gong, M., Zhang, D.-W., Yu, Y., Zhu, S.-L., Chen, J., Wu, P., Han, S., 2015. Observation of coherent oscillation insingle-passage Landau-Zener transitions. Sci. Rep. 5 (1), 8463.
Sun, G., Wen, X., Mao, B., Chen, J., Yu, Y., Wu, P., Han, S., 2010. Tunable quantum beam splitters for coherent manipulation of asolid-state tripartite qubit system. Nature Commun. 1 (5), 51.
Sun, G., Wen, X., Mao, B., Yu, Y., Chen, J., Xu, W., Kang, L., Wu, P., Han, S., 2011. Landau-Zener-Stückelberg interference of
microwave-dressed states of a superconducting phase qubit. Phys. Rev. B 83 (18), 180507.
Sun, G., Wen, X., Wang, Y., Cong, S., Chen, J., Kang, L., Xu, W., Yu, Y., Han, S., Wu, P., 2009. Population inversion induced byLandau–Zener transition in a strongly driven rf superconducting quantum interference device. Appl. Phys. Lett. 94 (10), 102502.
Sun, Z., Zhou, L., Xiao, G., Poletti, D., Gong, J., 2016. Finite-time Landau-Zener processes and counterdiabatic driving in open
systems: Beyond Born, Markov, and rotating-wave approximations. Phys. Rev. A 93 (1), 012121.
Suominen, K.-A., Garraway, B., Stenholm, S., 1991. The adiabatic limit of level crossing models. Opt. Commun. 82 (3–4), 260–266.
Suzuki, T., Nakazato, H., 2022. Generalized adiabatic impulse approximation. Phys. Rev. A 105 (2), 022211.
Suzuki, T.T., Sakai, O., 2016. Oscillatory spin asymmetric scattering of low-energy He+ ions on Sn surfaces. Nucl. Instrum. Methods
Phys. Res. B 382, 2–6.
Suzuki, T.T., Yamauchi, Y., 2010. Stückelberg oscillations in nonadiabatic spin transitions of an electron spin-polarized 4 He+ ion
beam. Phys. Rev. A 82 (4), 042709.
Syzranov, S.V., Rodionov, Y.I., Kugel, K.I., Nori, F., 2013. Strongly anisotropic Dirac quasiparticles in irradiated graphene. Phys. Rev.
B 88 (24), 241112.
Tagliaferri, M.L.V., Bavdaz, P.L., Huang, W., Dzurak, A.S., Culcer, D., Veldhorst, M., 2018. Impact of valley phase and splitting on
readout of silicon spin qubits. Phys. Rev. B 97 (24), 245412.
Takahashi, R., Sugimoto, N., 2018. Semiclassical lattice effects on interband tunneling of a two-state system. J. Phys. Soc. Japan87 (10), 104701.
85

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Tan, X., Yu, H., Yu, Y., Han, S., 2015. Rapid characterization of microscopic two-level systems using Landau-Zener transitions in asuperconducting qubit. Appl. Phys. Lett. 107 (10), 102601.
Tan, X., Zhang, D.-W., Zhang, Z., Yu, Y., Han, S., Zhu, S.-L., 2014. Demonstration of geometric Landau-Zener interferometry in asuperconducting qubit. Phys. Rev. Lett. 112 (2), 027001.
Taya, H., Hongo, M., Ikeda, T.N., 2021. Analytical WKB theory for high-harmonic generation and its application to massive Dirac
electrons. Phys. Rev. B 104 (14), l140305.
Tayebirad, G., Zenesini, A., Ciampini, D., Mannella, R., Morsch, O., Arimondo, E., Lörch, N., Wimberger, S., 2010. Time-resolved
measurement of Landau-Zener tunneling in different bases. Phys. Rev. A 82 (1), 013633.
Tchouobiap, S.E.M., Danga, J.E., Tsiaze, R.M.K., Fai, L.C., 2018. Coherent nonlinear low-frequency Landau–Zener tunneling inducedby magnetic control of a spin qubit in a quantum wire. Int. J. Quantum Inf. 16 (06), 1850049.
Teixeira, W., Semiao, F.L., Tuorila, J., Möttönen, M., 2021. Assessment of weak-coupling approximations on a driven two-levelsystem under dissipation. New J. Phys. 24, 013005.
Teranishi, Y., Nakamura, H., 1998. Control of time-dependent nonadiabatic processes by an external field. Phys. Rev. Lett. 81 (10),2032–2035.
Teranishi, Y., Nakamura, H., 1999. New way of controlling molecular processes by time-dependent external fields. J. Chem. Phys.
111 (4), 1415–1426.
Thalhammer, G., Winkler, K., Lang, F., Schmid, S., Grimm, R., Denschlag, J.H., 2006. Long-lived Feshbach molecules in athree-dimensional optical lattice. Phys. Rev. Lett. 96 (5), 050402.
Theisen, M., Petiziol, F., Carretta, S., Santini, P., Wimberger, S., 2017. Superadiabatic driving of a three-level quantum system.
Phys. Rev. A 96 (1), 013431.
Thingna, J., Barra, F., Esposito, M., 2017. Kinetics and thermodynamics of a driven open quantum system. Phys. Rev. E 96 (5),052132.
Thingna, J., Esposito, M., Barra, F., 2019. Landau-Zener Lindblad equation and work extraction from coherences. Phys. Rev. E 99(4), 042142.
Thorson, W.R., Delos, J.B., Boorstein, S.A., 1971. Studies of the potential-curve crossing problem. I. Analysis of Stueckelberg method.
Phys. Rev. A 4 (3), 1052–1066.
Tien, P.K., Gordon, J.P., 1963. Multiphoton process observed in the interaction of microwave fields with the tunneling between
superconductor films. Phys. Rev. 129 (2), 647–651.
Tomka, M., Venuti, L.C., Zanardi, P., 2018. Accuracy of the adiabatic-impulse approximation for closed and open quantum systems.
Phys. Rev. A 97 (3), 032121.
Tornes, I., Stroud, D., 2008. Model of multiphoton transitions in a current-biased Josephson junction. Phys. Rev. B 77 (22), 224513.
Tschischik, W., Haque, M., Moessner, R., 2012. Nonequilibrium dynamics in Bose-Hubbard ladders. Phys. Rev. A 86 (6), 063633.
Tuorila, J., Silveri, M., Sillanpää, M., Thuneberg, E., Makhlin, Y., Hakonen, P., 2010. Stark effect and generalized Bloch-Siegert shift
in a strongly driven two-level system. Phys. Rev. Lett. 105 (25), 257003.
Tuorila, J., Silveri, M., Sillanpää, M., Thuneberg, E., Makhlin, Y., Hakonen, P., 2013. Charge qubit driven via the Josephson
nonlinearity. Supercond. Sci. Technol. 26 (12), 124001.
Uehlinger, T., Greif, D., Jotzu, G., Tarruell, L., Esslinger, T., Wang, L., Troyer, M., 2013. Double transfer through Dirac points in atunable honeycomb optical lattice. Eur. Phys. J. Spec. Top. 217 (1), 121–133.
Vandersypen, L.M.K., Bluhm, H., Clarke, J.S., Dzurak, A.S., Ishihara, R., Morello, A., Reilly, D.J., Schreiber, L.R., Veldhorst, M., 2017.
Interfacing spin qubits in quantum dots and donors—hot, dense, and coherent. Npj Quantum Inf. 3 (1), 34.
Vasilev, G.S., Ivanov, S.S., Vitanov, N.V., 2007. Degenerate Landau-Zener model: Analytical solution. Phys. Rev. A 75 (1), 013417.
Villazon, T., Claeys, P.W., Polkovnikov, A., Chandran, A., 2021. Shortcuts to dynamic polarization. Phys. Rev. B 103 (7), 075118.
Villazon, T., Polkovnikov, A., Chandran, A., 2019. Swift heat transfer by fast-forward driving in open quantum systems. Phys. Rev.
A 100 (1), 012126.
Vitanov, N.V., 1999. Transition times in the Landau-Zener model. Phys. Rev. A 59 (2), 988–994.
Vitanov, N.V., 2018. Relations between the single-pass and double-pass transition probabilities in quantum systems with two andthree states. Phys. Rev. A 97 (5), 053409.
Vitanov, N.V., Garraway, B.M., 1996. Landau-Zener model: Effects of finite coupling duration. Phys. Rev. A 53 (6), 4288–4304.
Vitanov, N.V., Halfmann, T., Shore, B.W., Bergmann, K., 2001. Laser-induced population transfer by adiabatic passage techniques.
Annu. Rev. Phys. Chem. 52 (1), 763–809.
Vitanov, N.V., Suominen, K.A., 1999. Nonlinear level-crossing models. Phys. Rev. A 59 (6), 4580–4588.
Vitanov, N.V., Yatsenko, L.P., Bergmann, K., 2003. Population transfer by an amplitude-modulated pulse. Phys. Rev. A 68 (4),043401.
Vogelsberger, M., Garanin, D.A., 2006. Butterfly hysteresis curves generated by adiabatic Landau-Zener transitions. Phys. Rev. B
73 (9), 092412.
Wallraff, A., Duty, T., Lukashenko, A., Ustinov, A.V., 2003. Multiphoton transitions between energy levels in a current-biased
Josephson tunnel junction. Phys. Rev. Lett. 90 (3), 037003.
Wang, L.V., 2022. Multi-stage Stern-Gerlach experiment modeled. arXiv:2208.06471.
Wang, Y., Cong, S., Wen, X., Pan, C., Sun, G., Chen, J., Kang, L., Xu, W., Yu, Y., Wu, P., 2010. Quantum interference induced bymultiple Landau-Zener transitions in a strongly driven rf-SQUID qubit. Phys. Rev. B 81 (14), 144505.
86

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Wang, Z., Huang, W.-C., Liang, Q.-F., Hu, X., 2018. Landau-Zener-Stückelberg interferometry for Majorana qubit. Sci. Rep. 8 (1),7920.
Wang, G., Li, C., Cappellaro, P., 2021. Observation of symmetry-protected selection rules in periodically driven quantum systems.
Phys. Rev. Lett. 127 (14), 140604.
Wang, X., Miranowicz, A., Li, H.-R., Nori, F., 2017. Observing pure effects of counter-rotating terms without ultrastrong coupling:
A single photon can simultaneously excite two qubits. Phys. Rev. A 96 (6), 063820.
Wang, L., Tu, T., Gong, B., Zhou, C., Guo, G.-C., 2016. Experimental realization of non-adiabatic universal quantum gates using
geometric Landau-Zener-Stückelberg interferometry. Sci. Rep. 6 (1), 19048.
Wang, L., Zhou, C., Tu, T., Jiang, H.-W., Guo, G.-P., Guo, G.-C., 2014. Quantum simulation of the Kibble-Zurek mechanism using asemiconductor electron charge qubit. Phys. Rev. A 89 (2), 022337.
Ward, D.R., Kim, D., Savage, D.E., Lagally, M.G., Foote, R.H., Friesen, M., Coppersmith, S.N., Eriksson, M.A., 2016. State-conditional
coherent charge qubit oscillations in a Si/SiGe quadruple quantum dot. Npj Quantum Inf. 2 (1), 16032.
Wei, L.F., Johansson, J.R., Cen, L.X., Ashhab, S., Nori, F., 2008. Controllable coherent population transfers in superconducting qubits
for quantum computing. Phys. Rev. Lett. 100 (11), 113601.
Wen, P.Y., Ivakhnenko, O.V., Nakonechnyi, M.A., Suri, B., Lin, J.-J., Lin, W.-J., Chen, J.C., Shevchenko, S.N., Nori, F., Hoi, I.-C., 2020.
Landau-Zener-Stückelberg-Majorana interferometry of a superconducting qubit in front of a mirror. Phys. Rev. B 102 (7),075448.
Wen, X., Wang, Y., Cong, S., Sun, G., Chen, J., Kang, L., Xu, W., Yu, Y., Wu, P., Han, S., 2010. Landau-Zener-Stuckelberg interferometry
in multilevel superconducting flux qubit. arXiv:1003.2025.
Wen, X., Wang, Y., Yu, Y., 2009. Play building blocks on population distribution of multilevel superconducting flux qubit withquantum interference. arXiv:0912.0881.
Wertnik, M., Chin, A., Nori, F., Lambert, N., 2018. Optimizing co-operative multi-environment dynamics in a dark-state-enhanced
photosynthetic heat engine. J. Chem. Phys. 149 (8), 084112.
Whitney, R.S., Clusel, M., Ziman, T., 2011. Temperature can enhance coherent oscillations at a Landau-Zener transition. Phys. Rev.
Lett. 107 (21), 210402.
Whittaker, E.T., Watson, G.N., 1920. A Course of Modern Analysis, 196, third ed. Cambridge University Press, pp. 347–349, Chap.
The parabolic cylinder functions. Weber’s equation.
Wilczek, F., 2014. Majorana and condensed matter physics. In: Esposito, S. (Ed.), The Physics of Ettore Majorana. Cambridge Univ.
Press, pp. 279–302.
Wild, D.S., Gopalakrishnan, S., Knap, M., Yao, N.Y., Lukin, M.D., 2016. Adiabatic quantum search in open systems. Phys. Rev. Lett.
117 (15), 150501.
Wilson, C.M., Duty, T., Persson, F., Sandberg, M., Johansson, G., Delsing, P., 2007. Coherence times of dressed states of asuperconducting qubit under extreme driving. Phys. Rev. Lett. 98 (25), 257003.
Wilson, C.M., Johansson, G., Duty, T., Persson, F., Sandberg, M., Delsing, P., 2010. Dressed relaxation and dephasing in a strongly
driven two-level system. Phys. Rev. B 81 (2), 024520.
Wilson, R.D., Zagoskin, A.M., Savel’ev, S., Everitt, M.J., Nori, F., 2012. Feedback-controlled adiabatic quantum computation. Phys.
Rev. A 86 (5), 052306.
Witthaut, D., Trimborn, F., Kegel, V., Korsch, H.J., 2011. Quantum dynamics of Bose-Einstein condensates in tilted and driven
bichromatic optical lattices. Phys. Rev. A 83 (1), 013609.
Wittig, C., 2005. The Landau-Zener formula. J. Phys. Chem. B 109 (17), 8428–8430.
Wu, H., Heinrich, G., Marquardt, F., 2013. The effect of Landau–Zener dynamics on phonon lasing. New J. Phys. 15 (12), 123022.
Wu, B., Niu, Q., 2003. Superfluidity of Bose–Einstein condensate in an optical lattice: Landau–Zener tunnelling and dynamical
instability. New J. Phys. 5, 104.
Wu, J.-L., Wang, Y., Han, J.-X., Su, S.-L., Xia, Y., Jiang, Y., Song, J., 2021. Resilient quantum gates on periodically driven Rydberg
atoms. Phys. Rev. A 103 (1), 012601.
Wu, X., Ward, D.R., Prance, J.R., Kim, D., Gamble, J.K., Mohr, R.T., Shi, Z., Savage, D.E., Lagally, M.G., Friesen, M., Coppersmith, S.N.,
Eriksson, M.A., 2014. Two-axis control of a singlet-triplet qubit with an integrated micromagnet. Proc. Natl. Acad. Sci. USA
111 (33), 11938–11942.
Wu, E., Zhang, C., Wang, Z., Liu, C., 2020. Waveform control of currents in graphene by chirped few-cycle lasers. New J. Phys. 22(3), 033016.
Wu, T., Zhou, Y., Xu, Y., Liu, S., Li, J., 2019. Landau-Zener-Stückelberg interference in nonlinear regime. Chin. Phys. Lett. 36 (12),124204.
Wubs, M., 2010. Instantaneous coherent destruction of tunneling and fast quantum state preparation for strongly pulsed spin
qubits in diamond. Chem. Phys. 375 (2–3), 163–169.
Wubs, M., Saito, K., Kohler, S., Hänggi, P., Kayanuma, Y., 2006. Gauging a quantum heat bath with dissipative Landau-Zenertransitions. Phys. Rev. Lett. 97 (20), 200404.
Wubs, M., Saito, K., Kohler, S., Kayanuma, Y., Hänggi, P., 2005. Landau–Zener transitions in qubits controlled by electromagnetic
fields. New J. Phys. 7, 218.
Xia, S., Zhang, Y., Li, Z., Qin, L., Yang, C., Lu, H., Zhang, J., Zhao, X., Zhu, Z., 2021. Band evolution and Landau-Zener Bloch oscillations
in strained photonic rhombic lattices. Opt. Express 29 (23), 37503.
87

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Xiang, Z.-L., Ashhab, S., You, J.Q., Nori, F., 2013. Hybrid quantum circuits: superconducting circuits interacting with other quantum
systems. Rev. Modern Phys. 85 (2), 623–653.
Xie, Q., 2018. Exactly solvable model for Landau-Zener-Stückelberg-Majorana interferometry. Phys. Rev. A 97 (2), 022113.
Xu, J., 2014. Probing the energy spectrum in the vicinity of Dirac points with Landau—Zener transition. Commun. Theor. Phys. 62(3), 343–347.
Xu, J., Du, Y.-X., Huang, W., 2019a. Improving coherent population transfer via a stricter adiabatic condition. Phys. Rev. A 100,023848.
Xu, X.-Y., Han, Y.-J., Sun, K., Xu, J.-S., Tang, J.-S., Li, C.-F., Guo, G.-C., 2014b. Quantum simulation of Landau-Zener model dynamics
supporting the Kibble-Zurek mechanism. Phys. Rev. Lett. 112 (3), 035701.
Xu, C., Poudel, A., Vavilov, M.G., 2014a. Nonadiabatic dynamics of a slowly driven dissipative two-level system. Phys. Rev. A 89(5), 052102.
Xu, K., Xie, T., Shi, F., Wang, Z.-Y., Xu, X., Wang, P., Wang, Y., Plenio, M.B., Du, J., 2019b. Breaking the quantum adiabatic speed
limit by jumping along geodesics. Sci. Adv. 5 (6), 3800.
Xu, Z.-Y., You, W.-L., Dong, Y.-L., Zhang, C., Yang, W.L., 2018. Generalized speed and cost rate in transitionless quantum driving.
Phys. Rev. A 97, 032115.
Xu, S., Yu, Y., Sun, G., 2010. Landau-Zener-Stückelberg spectroscopy of a superconducting flux qubit. Phys. Rev. B 82 (14), 144526.
Yan, Y., Lü, Z., Luo, J., Zheng, H., 2017. Effects of counter-rotating couplings of the Rabi model with frequency modulation. Phys.
Rev. A 96 (3), 033802.
Yang, J., Pang, S., Jordan, A.N., 2017. Quantum parameter estimation with the Landau-Zener transition. Phys. Rev. A 96 (2), 020301.
Yatsenko, L.P., Shore, B.W., Bergmann, K., Romanenko, V.I., 1998. Stimulated Raman adiabatic passage with amplitude modulated
fields. Eur. Phys. J. D 4 (1), 47–56.
Yoakum, S., Sirko, L., Koch, P.M., 1992. Stueckelberg oscillations in the multiphoton excitation of helium Rydberg atoms:
Observation with a pulse of coherent field and suppression by additive noise. Phys. Rev. Lett. 69 (13), 1919–1922.
You, J.Q., Nori, F., 2005. Superconducting circuits and quantum information. Phys. Today 58 (11), 42–47.
You, J.Q., Nori, F., 2011. Atomic physics and quantum optics using superconducting circuits. Nature 474 (7353), 589–597.
You, J.Q., Wang, Z.D., Zhang, W., Nori, F., 2014. Encoding a qubit with Majorana modes in superconducting circuits. Sci. Rep. 4(1), 5535.
Zagoskin, A.M., 2011. Quantum Engineering. Cambridge University Press.
Zangara, P.R., Henshaw, J., Pagliero, D., Ajoy, A., Reimer, J.A., Pines, A., Meriles, C.A., 2019. Two-electron-spin ratchets as a platform
for microwave-free dynamic nuclear polarization of arbitrary material targets. Nano Lett. 19 (4), 2389–2396.
Zazunov, A., Egger, R., Gefen, Y., 2020. Multi-particle interferometry in the time-energy domain with localized topologicalquasiparticles. Phys. Rev. Res. 2 (2), 023054.
Zel’dovich, Y.B., 1967. The quasienergy of a quantum-mechanical system subjected to a periodic action. JETP 24, 1006.
Zel’dovich, Y.B., 1973. Scattering and emission of a quantum system in a strong electromagnetic wave. Sov. Phys. Usp. 16 (3),427–433.
Zener, C., 1932. Non-adiabatic crossing of energy levels. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 137 (833), 696–702.
Zener, C., 1934. A theory of the electrical breakdown of solid dielectrics. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 145 (855),
523–529.
Zenesini, A., Ciampini, D., Morsch, O., Arimondo, E., 2010. Observation of Stückelberg oscillations in accelerated optical lattices.
Phys. Rev. A 82 (6), 065601.
Zenesini, A., Lignier, H., Tayebirad, G., Radogostowicz, J., Ciampini, D., Mannella, R., Wimberger, S., Morsch, O., Arimondo, E., 2009.
Time-resolved measurement of Landau-Zener tunneling in periodic potentials. Phys. Rev. Lett. 103 (9), 090403.
Zhang, H., Chakram, S., Roy, T., Earnest, N., Lu, Y., Huang, Z., Weiss, D., Koch, J., Schuster, D.I., 2021. Universal fast-flux control ofa coherent, low-frequency qubit. Phys. Rev. X 11 (1), 011010.
Zhang, J., Cucchietti, F.M., Laflamme, R., Suter, D., 2017. Defect production in non-equilibrium phase transitions: experimental
investigation of the Kibble–Zurek mechanism in a two-qubit quantum simulator. New J. Phys. 19 (4), 043001.
Zhang, Q., Hänggi, P., Gong, J., 2008. Nonlinear Landau–Zener processes in a periodic driving field. New J. Phys. 10 (7), 073008.
Zhang, L., Liu, J., 2019. The upper bound function of nonadiabatic dynamics in parametric driving quantum systems. Chin. Phys.
B 28 (8), 080301.
Zhang, Z.-T., Liu, D.E., 2021. Readout of Majorana bound states via Landau-Zener transition. Phys. Rev. B 103 (19), 195401.
Zhang, J., Shim, J.H., Niemeyer, I., Taniguchi, T., Teraji, T., Abe, H., Onoda, S., Yamamoto, T., Ohshima, T., Isoya, J., Suter, D., 2013.
Experimental implementation of assisted quantum adiabatic passage in a single spin. Phys. Rev. Lett. 110 (24), 240501.
Zhang, Z.-Z., Song, X.-X., Luo, G., Su, Z.-J., Wang, K.-L., Cao, G., Li, H.-O., Xiao, M., Guo, G.-C., Tian, L., Deng, G.-W., Guo, G.-P., 2020.
Coherent phonon dynamics in spatially separated graphene mechanical resonators. Proc. Natl. Acad. Sci. 117 (11), 5582–5587.
Zhang, J.-N., Sun, C.-P., Yi, S., Nori, F., 2011. Spatial Landau-Zener-Stückelberg interference in spinor Bose-Einstein condensates.
Phys. Rev. A 83 (3), 033614.
Zhang, Z.-T., Yu, Y., 2013. Processing quantum information in a hybrid topological qubit and superconducting flux qubit system.
Phys. Rev. A 87 (3), 032327.
Zhang, J., Zhang, J., Zhang, X., Kim, K., 2014. Realization of geometric Landau-Zener-Stückelberg interferometry. Phys. Rev. A 89(1), 013608.
88

O.V. Ivakhnenko, S.N. Shevchenko and F. Nori

Physics Reports 995 (2023) 1–89

Zhao, L., Goryachev, M., Bourhill, J., Tobar, M.E., 2017. Indirect methods to control population distribution in a large spin system.
New J. Phys. 19 (3), 033016.
Zhao, X., Hu, X., 2018a. Coherent electron transport in silicon quantum dots. arXiv:1803.00749.
Zhao, X., Hu, X., 2018b. Toward high-fidelity coherent electron spin transport in a GaAs double quantum dot. Sci. Rep. 8 (1),13968.
Zhao, X., Hu, X., 2022. Measurement of tunnel coupling in a Si double quantum dot based on charge sensing. Phys. Rev. Appl. 17(6), 064043.
Zhao, P.-J., Li, W., Cao, H., Yao, S.-W., Cen, L.-X., 2018. Exotic dynamical evolution in a secant-pulse-driven quantum system. Phys.
Rev. A 98 (2), 022136.
Zhao, Y.-J., Liu, Y.-L., xi Liu, Y., Nori, F., 2015. Generating nonclassical photon states via longitudinal couplings between
superconducting qubits and microwave fields. Phys. Rev. A 91 (5), 053820.
Zhdanov, D.V., Seideman, T., 2015. Role of control constraints in quantum optimal control. Phys. Rev. A 92 (5), 052109.
Zheleznyakov, V.V., Kocharovskii, V.V., Kocharovskii, V.V., 1983. Linear coupling of electromagnetic waves in inhomogeneous
weakly-ionized media. Phys.-Uspekhi 141 (10), 257.
Zheng, F., Shen, Y., Sun, K., Zhao, Y., 2021. Photon-assisted Landau–Zener transitions in a periodically driven Rabi dimer coupled
to a dissipative mode. J. Chem. Phys. 154 (4), 044102.
Zhou, Y., Gu, S., Wang, K., Cao, G., Hu, X., Gong, M., Li, H.-O., Guo, G.-P., 2021. Frequency-selective effective model of driven
multilevel systems in gate-defined quantum dots. arXiv:2110.09852.
Zhou, J., Huang, P., Zhang, Q., Wang, Z., Tan, T., Xu, X., Shi, F., Rong, X., Ashhab, S., Du, J., 2014. Observation of time-domain Rabi
oscillations in the Landau-Zener regime with a single electronic spin. Phys. Rev. Lett. 112 (1), 010503.
Zhou, X., Zhao, C., Xiao, D., Sun, J., Sobreviela, G., Gerrard, D.D., Chen, Y., Flader, I., Kenny, T.W., Wu, X., Seshia, A.A., 2019. Dynamic
modulation of modal coupling in microelectromechanical gyroscopic ring resonators. Nature Com. 10 (1), 4980.
Zhu, C., Teranishi, Y., Nakamura, H., 2007. Nonadiabatic transitions due to curve crossings: complete solutions of the
Landau-Zener-Stueckelberg problems and their applications. In: Adv. Chem. Phys. Vol. 117. John Wiley & Sons, Inc., pp.
127–233.
Zhuang, F., Zeng, J., Economou, S.E., Barnes, E., 2022. Noise-resistant Landau-Zener sweeps from geometrical curves. Quantum 6,639.
Zudov, M.A., Du, R.R., Pfeiffer, L.N., West, K.W., 2006. Multiphoton processes in microwave photoresistance of two-dimensional
electron systems. Phys. Rev. B 73, 041303.
Zueco, D., Hänggi, P., Kohler, S., 2008. Landau–Zener tunnelling in dissipative circuit QED. New J. Phys. 10 (11), 115012.
Zurek, W.H., 1985. Cosmological experiments in superfluid helium? Nature 317 (6037), 505–508.
Zurek, W.H., Dorner, U., Zoller, P., 2005. Dynamics of a quantum phase transition. Phys. Rev. Lett. 95 (10), 105701.
Zvyagin, A.A., 2018. Quantum quench for the biaxial spin system. Low Temp. Phys. 44 (11), 1173–1179.
Zwanenburg, F.A., Dzurak, A.S., Morello, A., Simmons, M.Y., Hollenberg, L.C.L., Klimeck, G., Rogge, S., Coppersmith, S.N.,
Eriksson, M.A., 2013. Silicon quantum electronics. Rev. Modern Phys. 85 (3), 961–1019.

89