Physics Reports 492 (2010) 1–30

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

Landau–Zener–Stückelberg interferometry
S.N. Shevchenko a,b,∗ , S. Ashhab b,c , Franco Nori b,caB.Verkin Institute for Low Temperature Physics and Engineering, Kharkov, Ukraineb

RIKEN Advanced Science Institute, Wako-shi, Saitama, Japanc

Department of Physics, The University of Michigan, Ann Arbor, MI, USA

articleinfo

Article history:
Accepted 26 February 2010
Available online 24 March 2010
editor: S.N. Coppersmith
Keywords:
Landau–Zener transition
Stückelberg oscillations
Superconducting qubits
Multiphoton excitations
Spectroscopy
Interferometry

abstract
A transition between energy levels at an avoided crossing is known as a Landau–Zenertransition. When a two-level system (TLS) is subject to periodic driving with sufficiently
large amplitude, a sequence of transitions occurs. The phase accumulated between
transitions (commonly known as the Stückelberg phase) may result in constructive ordestructive interference. Accordingly, the physical observables of the system exhibit
periodic dependence on the various system parameters. This phenomenon is often
referred to as Landau–Zener–Stückelberg (LZS) interferometry. Phenomena related to LZS
interferometry occur in a variety of physical systems. In particular, recent experiments
on LZS interferometry in superconducting TLSs (qubits) have demonstrated the potential
for using this kind of interferometry as an effective tool for obtaining the parameterscharacterizing the TLS as well as its interaction with the control fields and with theenvironment. Furthermore, strong driving could allow for fast and reliable control of thequantum system. Here we review recent experimental results on LZS interferometry, andwe present related theory.
© 2010 Elsevier B.V. All rights reserved.

Contents
1.

2.

Introduction............................................................................................................................................................................................. 2
1.1.
Strongly driven TLSs: a brief historical review ......................................................................................................................... 2
1.1.1.
Atomic physics ............................................................................................................................................................. 2
1.1.2.
Other systems .............................................................................................................................................................. 3
1.1.3.
Theory of LZS interferometry...................................................................................................................................... 3
1.2.
Superconducting qubits ............................................................................................................................................................. 4
1.3.
Fano and Fabry–Perot interferometry using superconducting qubits .................................................................................... 4
1.4.
Hamiltonian and bases ............................................................................................................................................................... 4
Theory: adiabatic-impulse model .......................................................................................................................................................... 5
2.1.
Adiabatic evolution..................................................................................................................................................................... 6
2.2.
Single passage: Landau–Zener transition.................................................................................................................................. 6
2.3.
Double passage: Stückelberg phase........................................................................................................................................... 8
2.4.
Multiple passage ......................................................................................................................................................................... 8
2.4.1.
Slow-passage limit....................................................................................................................................................... 9
2.4.2.
Fast-passage limit ........................................................................................................................................................ 11
2.5.
Decoherence................................................................................................................................................................................ 14
2.6.
‘‘Breakdown’’ of the adiabatic theorem..................................................................................................................................... 15∗ Corresponding author at: B.Verkin Institute for Low Temperature Physics and Engineering, Kharkov, Ukraine.
E-mail address: sshevchenko@ilt.kharkov.ua (S.N. Shevchenko).
0370-1573/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.physrep.2010.03.002

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3.

4.

2.7.
Reversal of stimulated emission ................................................................................................................................................ 16
Recent experiments with strongly driven superconducting quantum circuits .................................................................................. 16
3.1.
Multiphoton transitions in superconducting qubits ................................................................................................................ 16
3.2.
LZS interferometry in superconducting qubits ......................................................................................................................... 17
3.3.
Multiphoton resonances in multi-level systems ...................................................................................................................... 19
Conclusions.............................................................................................................................................................................................. 19
Acknowledgements................................................................................................................................................................................. 20
Appendix A.
Solution of the Landau–Zener problem....................................................................................................................... 20
A.1.
Adiabatic wave function................................................................................................................................................. 20
A.2.
Non-adiabatic transition (evolution matrix for the Landau–Zener transition) .......................................................... 20
Appendix B.
Evolution of a periodically driven two-level system .................................................................................................. 22
Appendix C.
Rotating-wave approximation ..................................................................................................................................... 24
C.1.
Hamiltonian in the RWA-form....................................................................................................................................... 24
C.2.
Solving the Schrödinger equation in the absence of relaxation .................................................................................. 25
C.3.
Solving the Bloch equations with relaxation ................................................................................................................ 25
Appendix D.
Floquet theory............................................................................................................................................................... 26
Appendix E.
Dressed-state picture: quantized driving field ........................................................................................................... 27
References................................................................................................................................................................................................ 28

1. Introduction
1.1. Strongly driven TLSs: a brief historical review
The model of a quantum two-level system (TLS), with only two relevant quantum states, is used to describe a variety ofphysical systems. At first, it was used in relation to spins and atomic collisions, which describe natural microscopic systems.
It has been used to describe various other naturally occurring settings ever since. Furthermore, some artificial mesoscopic
systems that were realized recently in semiconductor quantum dots and superconducting circuits can operate as effective
TLSs. These solid-state TLSs have received increasing attention recently for two main reasons: they exhibit fundamental
quantum phenomena on macroscopic scales, and they are considered possible candidates to operate as quantum bits (qubits)
in future quantum information processing devices.
It is quite common that the two energy levels of a quantum TLS exhibit an avoided level crossing or anticrossing as someexternal parameter is varied, as shown in Fig. 1. The physical properties of the two energy eigenstates are typically exchanged
when going from one side of the avoided crossing to the other side. If the external control parameter is varied in time suchthat the system traverses the avoided crossing region, a non-adiabatic transition between the two energy levels can occur.
The transition probability is usually named after Landau and Zener (LZ) in recognition of their pioneering work on thissubject, Landau (1932b) and Zener (1932).
If a LZ transition is one step in a sequence of coherent processes occurring during the dynamics, not only the change inthe occupation probabilities but also the change in the relative phase between the quantum states is relevant. In particular,
in this review we shall focus on the situation where the control parameter is varied periodically and strongly, such that thesystem keeps going back and forth across the avoided crossing region. In this case, the physical observables exhibit periodic
dependence on the phase acquired between the transitions (Stückelberg, 1932; Landau, 1932a). This periodicity, which isthe basis of Landau–Zener–Stückelberg (LZS) interferometry, provides a useful tool that allows the characterization of theparameters defining the quantum TLS and its interaction with the control fields and the environment. We note here that theformula for the single-passage transition probability can also be found in the paper Majorana (1932), and it can alternatively
be named the Landau–Zener–Stückelberg–Majorana formula, as proposed in Di Giacomo and Nikitin (2005).
1.1.1. Atomic physics
The problem of strongly driven TLSs appears in a number of contexts in atomic physics. In atomic-collision problems,
the electronic state of the atom pair can experience a LZ transition from the ground states of the incoming atoms into a
hybridized intermediate state and then back to the electronic ground states of the outgoing atoms. In this case the avoided
crossing region is passed twice (Child, 1974; Nikitin and Umanskii, 1984), and the Stückelberg phase typically has large
variations and hence its interferometric consequences are washed out (Stückelberg, 1932; Landau, 1932a).
An atom in an intense laser field can also experience LZ transitions, but in contrast to atomic-collision problems, theatom is driven periodically in time and multiple LZ transitions occur (Delone and Krainov, 1985). In the case of an atom ina laser field, (regardless of the field intensity) if the atom is driven at a frequency that matches its energy-level splitting, it
can be resonantly excited to an upper level. Then, at low driving amplitudes, the level occupation probabilities experience
Rabi oscillations (Rabi, 1937). When the driving amplitude is increased, multiphoton processes become relevant. These
processes occur when the energy-level splitting matches the energy of an integer number of photons (Shirley, 1965). These
multiphoton processes have traditionally been described by making use of Floquet theory [ (Autler and Townes, 1955; Ritus,
1967; Sambe, 1973; Zel’dovich, 1973; Barone et al., 1977; Aravind and Hirschfelder, 1984); for a review see Chu and Tel’nov(2004)]. In addition to controlling atomic states, resonant driving of atoms has also been studied as a means for finding
atomic parameters, and the parameters of quantum systems in general (Coffey et al., 1969; Henry and Lang, 1977).

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

3Fig. 1. (Color online) (a) Energy levels E versus the bias ε . The two solid curves (red and blue) represent the adiabatic energy levels, E± , which display
avoided crossing with energy splitting ∆. The dashed lines show the crossing diabatic energy levels E↑,↓ , corresponding to the diabatic states ϕ↑ and ϕ↓ . (b)The bias ε represents the driving signal, and it oscillates between εmin = ε0 − A and εmax = ε0 + A with a sinusoidal time dependence: ε(t ) = ε0 + A sin ωt.

1.1.2. Other systems
In addition to atomic systems, LZS interference has been studied in a number of other systems. (Related discussions of
the intracycle and intercycle interference were given in Ashhab et al. (2007) and Arbo et al. (2010). For example, periodic
dependence and time-domain oscillations related to LZS interference have been predicted in the electronic transport insemiconductor superlattices (Rotvig et al., 1995, 1996). Stückelberg oscillations were also studied in superconducting
quantum point contacts biased with a dc voltage and exposed to periodic driving (Gorelik et al., 1998). This effect was
proposed as a sensitive tool for microwave-photon detection. Another example of interference in microscopic systems arisesin the study of interacting magnetic molecules (Wernsdorfer et al., 2000; Garanin, 2004), molecular nanomagnets (Calero
et al., 2005; Földi et al., 2007, 2008), and single nitrogen vacancy centers in diamond (Fuchs et al., 2009). Interference
effects have also been predicted to take place in the interlayer tunneling in quasi-one-dimensional layered materials, suchas organic conductors (Cooper and Yakovenko, 2006; Banerjee and Yakovenko, 2008), and in the tunneling of a periodicallydriven two-mode Bose–Einstein condensate (Zhang et al., 2008a,b). Recently, LZ tunneling and Stückelberg oscillations
were measured in a gas of ultracold molecules (Mark et al., 2007a,b) and in the dipole–dipole interaction between Rydberg
atoms (van Ditzhuijzen et al., 2009).
Avoided level crossings can also be used in the study of second-order phase transition problems (Damski and Zurek,
2006; Dziarmaga, 2009). Periodic driving around a phase transition can then result in interference fringes (Mukherjee andDutta, 2009). The LZS model has also been used to describe the singlet-triplet transition for the state of two electrons in asemiconductor double quantum dot (Ribeiro and Burkard, 2009; Petta et al., 2010; Burkard, 2010; Ribeiro et al., 2010) as wellas the transport of photons between two halves of an optical cavity divided by a dielectric membrane (Heinrich et al., 2010).
Recent technological advances have also allowed the realization of the strong-driving regime in superconducting qubits,
where such interference has been observed in several experiments (Oliver et al., 2005; Sillanpää et al., 2006; Wilson et al.,2007; Izmalkov et al., 2008; Sun et al., 2009; LaHaye et al., 2009). Such superconducting qubit systems will be discussed in
more detail below.
1.1.3. Theory of LZS interferometry
The theory of driven quantum TLSs has been studied in numerous papers [for reviews see Leggett et al. (1987), Grifoni
and Hänggi (1998), Thorwart et al. (2001), Vitanov et al. (2001), Zhu et al. (2001) and Frasca (2003)]. Some of the recent work
has focused on the impact of the environment, e.g. Ao and Rammer (1991), Thorwart et al. (2000), Wubs et al. (2006), Zueco
et al. (2008) and Nalbach and Thorwart (2009). In particular, some papers pointed out that if the relaxation in the system
is small enough, interference between multiple LZ transitions becomes relevant for the system’s dynamics, e.g. Shimshoni
and Gefen (1991), Saito and Kayanuma (2002) and Saito et al. (2007); when the avoided crossing is passed several times,
interference can take place and the dynamics is sensitive to the phase acquired between successive crossings, e.g. Kayanuma(1994).
LZS interference has also been recently proposed as a possible approach to achieve perfect population transfers and to
implement quantum gates for quantum-control and quantum-computing purposes (Teranishi and Nakamura, 1998; Zhu
et al., 2001; Gaitan, 2003; Zwanziger et al., 2003; Nagaya et al., 2007; Li et al., 2009) [related work on combining adiabatic

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LZ crossings with coherent dynamics can be found in Goswami (2003), Vitanov et al. (2003), Kral et al. (2007), Oh et al.
(2008) and Wei et al. (2008)].
The purpose of the present review is to present the theory of strongly driven TLSs and LZS interferometry and recent
experimental results on the subject, particularly in relation to superconducting qubits.
1.2. Superconducting qubits
Superconducting qubits are Josephson-junction-based circuits that can behave as effective two-level systems. There
are several types of superconducting qubits, differing mostly by their topology and by their physical parameters. They are
generally called phase, charge, or flux qubits. For reviews see Makhlin et al. (2001), Devoret and Martinis (2004), You andNori (2005), Wendin and Shumeiko (2007), Zagoskin and Blais (2007) and Clarke and Wilhelm (2008).
Superconducting qubits have several properties that make them unique for studying quantum effects. The qubits havemesoscopic size, which means that they allow the observation of quantum effects on a macroscopic scale. They can be
integrated in electrical circuits, which provides different ways of controlling and probing their states. Their parameters areadjustable: they can be tuned by changing the applied bias current, gate voltage, or magnetic flux. The qubit’s state canbe probed by measuring the charge or current induced in it. These unique properties of superconducting qubits allow therealization of the strong-driving regime, and they have allowed the realization of several experiments on LZS interferometry
in the past few years.
1.3. Fano and Fabry–Perot interferometry using superconducting qubits
Superconducting qubits can be used for LZS interferometry, but also to study other types of interferometry, including
Fano and Fabry–Perot interferometry (Zhou et al., 2008a,b; Liao et al., 2010). These latter devices can be implemented using
quasi-one-dimensional open systems where photons are injected from the left and move towards the right side of the device.
Along the way, the photons interact with either one or two qubits acting as tunable mirrors, controlled by changing theapplied electric and/or magnetic fields on the qubits. These qubits, working as tunable mirrors, can change the reflection
and transmission coefficients of the photons confined in waveguides.
Let us first consider the case of a single superconducting qubit interacting with an incoming photon (Zhou et al., 2008a).
When the energy of the incoming photon matches the energy spacing of the qubit, the photon is reflected, otherwise it
is transmitted. This type of single-photon switch exhibits Breit–Wigner scattering: now in one dimension instead of thestandard three dimensional case for natural atoms. This Breit–Wigner scattering produces a symmetric Lorentzian peakin the reflection coefficient, versus frequency, of the photon. This situation occurs when the dispersion relation of theincoming photon is linear, as in a transmission line resonator, acting as a ‘‘rail’’ guiding the motion of the photons. Whenthe photon dispersion relation is nonlinear, for long-wavelength photons propagating in a quasi-one-dimensional array ofcoupled cavities, the reflection coefficient exhibits an asymmetric Fano line shape, due to the interference between thecontinuous mode of the incoming photon and the discrete energy levels of the qubit. Thus, for single-photon transport in a
one-dimensional waveguide, the photons can be partially or totally reflected by a controllable two-level system which can
act as a tunable mirror.
It is known that the Fabry–Perot interferometer, which consists of two highly reflecting planar mirrors, provides thesimplest cavity. It is then natural to ask the question: ‘‘is it possible to construct a quantum version of a Fabry–Perot
interferometer?’’ Namely, to build a resonator, in a one-dimensional waveguide, with two tunable-mirrors made of quantum
scatterers. Reference (Zhou et al., 2008b) focused on this question and studied quantum analogs of the Fabry–Perot
interferometer. They found that two separate two-level systems interacting with photons in a waveguide can also create,
between them, single-photon quasi-bound states. A recent work (Liao et al., 2010) studied a more complex form ofinterferometry, with the goal of controlling the transport of single photons by tuning the frequency of either one or two
cavities in an array of coupled cavities guiding the 1D motion of photons. In some regimes, the photons can be localizedaround the scatterers, which act as impurities, producing isolated states in the gap of the energy spectrum (Zhou et al.,2008b; Liao et al., 2010).
1.4. Hamiltonian and bases
The main subject of this review is a strongly and periodically driven TLS. A quantum TLS with energy bias ε and tunneling
amplitude ∆ is described by the Hamiltonian

ε(t )
σx −
σz
(1)
2
2
in terms of the Pauli matrices σx,z (h̄ = 1 is assumed throughout). One generally thinks of ∆ as being fixed by the systemproperties, and ε as being a tunable control parameter; hence ε is a function of time t and ∆ is not. We assume themonochromatic time-dependent bias
H (t ) = −∆ε(t ) = ε0 + A sin ωt ,with amplitude A, frequency ω and offset ε0 .

(2)

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

5Note that this problem with the time-dependent component along the z-axis can be related to the one with the timedependent component along the x-axis by a π /2-rotation around the y-axis:ππ

H 0 (t ) = e−i 4 σy H (t )ei 4 σy = −

∆
2

σz +

ε(t )
2

σx .

(3)

The latter Hamiltonian is typical for problems such as an atom in a laser field. It should also be noted that having different
signs in the Hamiltonian than the ones given above does not cause any nontrivial change in the results.
The instantaneous eigenvalues of H (t ) depend on the bias ε(t ) as follows:1
E± (t ) = ± Ω (t ),
2

(4)

p
Ω (t ) = ∆2 + ε(t )2 .

(5)The ε -dependence of the energy levels is shown in Fig. 1. In particular, it shows an avoided level crossing at ε = 0.
Although the TLS is a rather simple-looking model, concrete calculations on it can present significant difficulties.
Specifically, the Schrödinger equation can be written as a second-order differential equation with periodic coefficients, theHill equation, which is not solvable in analytic closed form (Grifoni and Hänggi, 1998). Nevertheless, different theoretical
approaches can be used to obtain approximate analytical results in various parameter regions.
As we shall explain in detail below, one key relation for purposes of deciding the suitability of a given theoretical
approach is that between the driving frequency ω and the minimal energy splitting ∆. In superconducting qubits, both
limits, ω  ∆ (Sillanpää et al., 2006) and ω  ∆ (Oliver et al., 2005), have been realized in recent experiments. Anotherkey relation concerns the velocity of passing the avoided level region, which is of the order of Aω (we also introduce theadiabaticity parameter δ later in Eqs. (19) and (20)). We will distinguish between the slow driving regime, Aω . ∆2 , and thefast driving regime, Aω  ∆2 . The natural representations for the two regimes are formulated in the adiabatic and diabatic
bases, respectively. The diabatic basis {ϕ↑ , ϕ↓ } is formed with the eigenstates of σz : σz ϕ↑ = ϕ↑ , σz ϕ↓ = −ϕ↓ (note thatthese states would be the eigenstates of the Hamiltonian if ∆ vanished). The adiabatic basis consists of the instantaneous
eigenstates of the time-dependent Hamiltonian: H (t )ϕ± (t ) = E± (t )ϕ± (t ). The two bases are related to each other by thetime-dependent coefficients β± (t ) (see Appendix A for details):

ϕ± (t ) = β∓ (t )ϕ↑ ∓ β± (t )ϕ↓ ,sΩ (t ) ± ε(t )
β± (t ) =.
2Ω (t )

(6)
(7)

Any given wave function of the quantum TLS can be decomposed in either one of these two bases; the coefficients in thequantum superposition are the probability amplitudes for the respective states.
2. Theory: adiabatic-impulse model
There are a number of theoretical approaches that can be used to study the problem of a strongly driven TLS. Among theseapproaches, the adiabatic-impulse model, which we shall explain in detail below (and in Appendices A and B), is perhaps themost intuitive whenever the TLS is repeatedly driven through the avoided crossing. We shall therefore put special emphasison this theoretical approach in this review. Three alternative theoretical approaches, which in certain parameter regimes
can be well suited for the study of this problem, will be discussed in Appendices C (rotating-wave approximation), D (Floquet
theory) and E (dressed-state picture).
The adiabatic energy levels E± introduced in Section 1.3 have a minimum distance of ∆, and this minimum distance isrealized at times t1,2 + 2π n/ω, where ωt1 = arcsin(−ε0 /A) and ωt2 = π − ωt1 , see Fig. 2; here n is an integer. Because
the energy eigenstates, i.e. the states of the adiabatic basis, change rather rapidly around the avoided-crossing region andare approximately constant far to the right or far to the left of that region, one would intuitively expect that the system will
evolve almost adiabatically far from the points of avoided crossing and the evolution becomes nonadiabatic in the vicinityof these points. One could therefore make the approximation that beyond a certain boundary the evolution is completely
adiabatic. However, it is convenient not to introduce an arbitrary boundary between the adiabatic and nonadiabatic regions
and instead consider the evolution to be adiabatic everywhere except at the points of minimum energy separation, where
the system experiences a sudden mixing in the populations of the two energy levels. Naturally, these sudden transitions do
not occur in reality. However, provided that the system follows a linear ramp traversing the avoided crossing region, one canstill obtain an accurate description of the net result of the adiabatic and nonadiabatic parts of the evolution. This discretizedpicture only simplifies the algebra that arises in the relevant calculations. The calculation presented in Appendix A can beused to obtain the appropriate assignment for the adiabatic phases and the mixing matrices at the crossing points in order toobtain the correct expressions describing the system’s dynamics. Following Damski and Zurek (2006) (see also Zurek (1996)
and Damski (2005)) we call this picture the adiabatic-impulse approximation to emphasize the two-stage character of themodel. In particular, this name emphasizes that the non-adiabatic transitions are described as instantaneous, which is just

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Fig. 2. (Color online) Time evolution of the energy levels during one period. The time-dependent adiabatic energy levels define a two-stage evolution:transitions at the non-adiabatic regions, described by the evolution matrix N, and the adiabatic evolution, described by the matrices U1,2 = exp(−iζ1,2 σz ).
The acquired phases ζ1,2 have a geometrical interpretation: they are equal to the area under the curves, shown by the yellow and blue regions. The diabatic
energy levels, ±ε(t )/2, are shown by the dashed lines.

a convenient description of the continuous dynamics; see also original articles, where the adiabatic-impulse theory wasdeveloped (Delone and Krainov, 1985; Averbukh and Perel’man, 1985; Vitanov and Garraway, 1996; Garraway and Vitanov,
1997; Damski and Zurek, 2006)].
2.1. Adiabatic evolution
The wave function ψ(t ) that describes the quantum state of the TLS as a function of time can be decomposed in the
adiabatic basis (see Appendix A):

ψ(t ) =

Xb± (t )ϕ± (t ) =

X

±

ζ =

1

Z

n



c± (t )ϕ± (t ) exp ∓i ζ +

±

π o
4

,

Ω (t )dt .

2

(8)

(9)

Since in the adiabatic approximation c± are time-independent coefficients between crossings, the adiabatic evolution fromt = ti to t = tf (assuming that no LZ transitions are encountered) can be described by the evolution matrix U, defined asfollowsb(tf ) = U (tf , ti )b(ti ),b(t ) ≡

b+b−

U (tf , ti ) =ζ (tf , ti ) =



(10)

,

(11)

 −iζ (t ,t )
f i

e

0
1

Z tf

2

ti

0
eiζ (tf ,ti )= e−iζ (tf ,ti )σz ,Ω (t )dt .

(12)

(13)

In particular, the phases acquired during the adiabatic stages (see Fig. 2) are given by:

ζ1 =

1

Z t2

2

t1

Ω (t )dt ,

ζ2 =

1
2

Z t1 +2π/ωΩ (t )dt .

(14)

t2

2.2. Single passage: Landau–Zener transition
Consider the nonadiabatic region in the vicinity of t1,2 : t = t1,2 + t 0 , ω t 0  1. Then the bias can be linearized:ε(t1,2 + t 0 ) ≈ ±v t 0 ,

(15)

where

r
v = Aω cos ωt1,2 = Aω 1 − ε 2
0

A

.

(16)

The linearized Hamiltonian
H (t 0 ) = −

∆
2

σx ∓

vt 0
2

σz

(17)

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

7Fig. 3. (Color online) One-passage LZ transition. The time dependence of the upper level occupation probability P+ (t ) is plotted for the following
parameters: A/∆ = 10, ω/∆ = 0.05, ε0 = 0. The blue solid curve presents the exact numerical solution, while red dashed curve corresponds to theanalytical solution. The time interval between the vertical green dotted lines corresponds to the crossover region, which is characterized by tLZ .

is exactly that of the LZ problem. The solution of the LZ problem in terms of the parabolic cylinder function is presented
in detail in Appendix A. For discussions of different theoretical approaches to the LZ problem, see e.g. Delos and Thorson(1972), Benderskii et al. (2003) and Wittig (2005). Here, in the main text, we also present the shortest solution, based on
the fact that the transition
under the adiabatic perturbation has quasiclassical character, where the change of the action
R(given by the integral E (t )dt) is large (Landau and Lifshitz, 1977). Then the problem of the transition under the adiabatic
perturbation is formally fully analogous to the problem of the quasiclassical over-barrier reflection.
Assuming that the system is initially in the lower energy level, the transition probability from the lower to the upper
level during the single-sweep process is described within the quasiclassical approximation as follows (Landau and Lifshitz,1977). The energy levels coincide at two points in the complex plane: at t 0 = ±t0 = ±i∆/v one finds that E+ (t0 ) = E− (t0 ).
Accordingly, the probability P+ that the system ends up in the upper level is determined by the contour integral in the planeof complex time:



Z t0

P+ = exp −2Im
[E+ (t 0 ) − E− (t 0 )]dt 0 .

(18)

0

Calculating this integral for 2E± = ± ∆2 + (v t 0 )2 and t0 = i∆/v , we obtain the LZ transition probability:pP+ = PLZ = exp (−2π δ) ,

δ=

∆2
.
4v

(19)
(20)

Eq. (19) describes the transition probability for an arbitrary value of the exponent (Nikitin, 1996). As the driving velocity v ischanged from 0 (adiabatic limit) to ∞ (sudden-change limit), the transition probability P+ varies from zero to unity. In Fig. 3
the red dashed curve shows the instantaneous transition of the adiabatic-impulse model, with the mixing probability given
by Eq. (19). The blue solid curve is calculated numerically by integrating the Schrödinger equation (Shevchenko et al., 2005).
Note that the LZ formula (19) accurately describes the final upper level occupation probability P+ , but it does not describe
the transient dynamics in the vicinity of the energy-level avoided crossing. This issue was studied in detail in Refs. Mullen
et al. (1989), Vitanov (1999) and Zenesini et al. (2009). In particular, it was shown that the duration of the transition (theregion between the green dotted lines in Fig. 3) can be estimated as follows:
tLZ ∼

2√

∆

δ max(1,

√
δ).

(21)

Here we would like to note that, as argued in Garraway and Vitanov (1997), Eq. (21) gives only the upper limit of thetransition time, and one can think of the transition time is being shorter than the one given in Eq. (21). The conditionsof the applicability of the adiabatic-impulse model, which describes the LZS interferometry, require that the relations ∆/ωand/or A/ω are large (Garraway and Vitanov, 1997):∆2 + A 2  ω 2 .

(22)

In particular, the multiphoton fringes were shown to be described by this model at A . ω . ∆ in Krainov and Yakovlev(1980).
The LZ formula (19) describes the upper level occupation probability P+ , which is the square of the absolute value of theprobability amplitude. However, when interference is relevant (as will be discussed in detail below), one has also to know
the change in the relative phase between the two components of the wave function as the crossing region is traversed. One

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S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

Fig. 4. (Color online) Double-passage transition. Adiabatic energy levels as in Fig. 2 are plotted. The lines with one and two arrows show the two
trajectories where the transition to the upper level happens during the first and the second passages of the avoided level crossing. Their respective transition
probabilities are given by PLZ (1 − PLZ ) and (1 − PLZ )PLZ , while the interference is described by Eq. (27).

can keep track of this phase by describing the LZ transition using the (non-adiabatic) unitary evolution matrix N, which isderived in Appendix A:b(t1,2 + 0) = Nb(t1,2 − 0),

p
N =

ϕS
1 −pPLZ e−ie

PLZ

eϕS = ϕS −
ϕS =

π

π
2

(23)

p



p − PLZ
ϕS ,1 − PLZ eie

,

(24)
(25)

+ δ(ln δ − 1) + arg Γ (1 − iδ),(26)
4where ϕS is the so-called Stokes phase (see e.g. Child (1974), Kayanuma (1997) and Wubs et al. (2005)) and Γ is the gammafunction.
2.3. Double passage: Stückelberg phase
Consider now the double-passage process, where the avoided crossing region is passed twice at the same speed. This
double-passage problem corresponds to one full driving cycle of the periodic driving that we are considering. The probability
for the upper level P+ after one period is given by Eq. (B.5) with n = 1:
P+ = 4PLZ (1 − PLZ ) sin2 ΦSt ,(27)

ΦSt = ζ2 + e
ϕS .

(28)

This result, that the probability of excitation P+ is an oscillating function of a certain phase ΦSt , was first obtained in(Landau, 1932a; Stückelberg, 1932). The Stückelberg phase ΦSt consists of two components: the first ζ2 is acquired during
the adiabatic evolution and the second e
ϕS during the non-adiabatic transitions.
Stückelberg oscillations were observed in atomic inelastic scattering cross-sections (Nikitin and Ovchinnikova, 1972)
and in the microwave excitation of Rydberg atoms (Baruch and Gallagher, 1992; Yoakum et al., 1992). However, in manycases these Stückelberg oscillations average out and can be neglected, and one only needs to consider the averaged probability (Nikitin, 1996)
P+ = 2PLZ (1 − PLZ ).

(29)

The above expression is the sum of two probabilities, which correspond to the system being excited at the first or at thesecond passage through the avoided-crossing region. These two components are schematically illustrated in Fig. 4 withtrajectories marked by single and double arrows, respectively.
The quantum-mechanical interference between the different LZ transitions has the result that the net excitation
probability after the double passage can range from 0 (destructive interference) to 4PLZ (1 − PLZ ) (constructive interference)
according to Eq. (27). The latter probability is twice as large as the one without interference, Eq. (29). This situation
is analogous to the one encountered in the Mach–Zehnder interferometer (Ji et al., 2003; Oliver et al., 2005; Pezze et al.,2007).
2.4. Multiple passage
Let us now study a multiple-passage process, where the system passes the avoided-crossing region periodically. Consider
first the relevant time scales. The characteristic time for an LZ transition tLZ can be estimated from Eq. (21). The time between

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

9Fig. 5. (Color online) Constructive (blue dashed line) versus destructive (red continuous line) interference. Time dependence of the upper level occupation
probability P+ with constructive (ζ1 = π/2 + kπ ) and destructive (ζ1 = kπ ) interference at ε0 = 0 and PLZ = 0.1. The TLS is taken to be initially in thelower level.

subsequent tunneling events is of the order of half driving period T /2 = π/ω. LZS interference takes place when (i)successive LZ transitions events do not overlap and (ii) the phase coherence (characterized by the dephasing time T2 ) ispreserved:
tLZ < T /2 < T2 .

(30)

In this subsection we assume T /2  T2 and ignore decoherence. For definiteness, as the initial condition we consider thesystem to be in the ground state at time t1 + 0. Then the state at time t, after n full periods, is described by the following
evolution matrices (see Fig. 2):


U


U

t , t1 +t , t2 +

2π n



ω
2π n

ω



(NU2 NU1 )n

for t −

NU1 (NU2 NU1 )n

2π n

ω

for t −∈ (t1 , t2 ) ,
2π n



2π
∈ t2 , t1 +,ωω

U1,2 = exp(−iζ1,2 σz ).

(31)

(32)
(33)

One can now derive expressions for the evolution matrices in terms of the parameters in the Hamiltonian (1), as explainedin Appendix B. Using these matrices one can analyze the dynamics for any given initial state. Fig. 5 illustrates two different
possibilities that correspond to constructive and destructive interference between consecutive crossings.1 In addition to
the time dependence, one can also derive expressions for the average populations of the two quantum states as functionsof the system parameters, since it is common to experimentally measure the steady-state populations of these states. In
general, one needs to make approximations in order to obtain clear analytical results. Below we consider two opposite limits,
depending on whether the individual crossings are slow (i.e., almost adiabatic) or fast (i.e., almost sudden). As explained inAppendix B (see also Ashhab et al. (2007)), in both cases a resonance condition can be derived that determines whether thesystem will exhibit oscillations between the two basis states. This condition is given by(1 − PLZ ) sin ζ+ − PLZ sin ζ− = 0,

(34)

where

ζ+ = ζ1 + ζ2 + 2eϕS ,
ζ− = ζ1 − ζ2 .

(35)
(36)

As we shall see shortly, this resonance condition results in drastically different patterns in the slow- and fast-passage limits.
2.4.1. Slow-passage limit
In the limit δ  1, PLZ  1 and the resonance condition reduces toζ+ = ζ1 + ζ2 + 2e
ϕS = kπ ,(37)

for any integer k, and eϕS ≈ −π /2. Going further and deriving an analytic expression in terms of ∆, ε0 and A is complicatedby the fact that the integrals that determine ζ1 and ζ2 cannot be evaluated in closed analytic form. Numerical calculations,

1 These cases are also clearly illustrated in the two computer animations available online at http://dml.riken.jp/animations.php.

10

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

Fig. 6. (Color online) Slow-driving LZS interferometry for Aω . ∆2 . The time averaged upper level occupation probability P+ as a function of the energy
bias ε0 and the driving amplitude A. The graph is calculated with Eq. (38) for ω/∆ = 0.32 < 1. The inclined red lines mark the region of the validity of thetheory: ε0 < A, which means that the system experiences avoided level crossings. Outside of this region the excitation probability is negligibly small. The
vertical dashed line shows the alteration of the excitation maxima and minima.
Table 1
Parameters used in different experiments studying LZS interferometry: tunneling amplitude ∆, maximal driving amplitude Amax , and driving frequency ω
maxin the units GHz×2π , minimal adiabaticity parameter δ min = ∆2 /(4ωAmax ), and maximal LZ probability PLZ= exp(−2πδ min ).

∆(Oliver et al., 2005)(Sillanpää et al., 2006)(Wilson et al., 2007)(Izmalkov et al., 2008)

Amax

0.004
12.5
2.6
3.5

24
95
62
40

ω
1.2
4
7
4

δ min

max
PLZ

−7

1
0.5
0.98
0.9

10
0.1
0.004
0.02

however, are straightforward. The resonance condition (37) describes arcs around the point A = ε0 = 0 (see Fig. 6), where
we treat ∆ as a fixed parameter. Taking the system to be initially in the lower energy level, the average occupation probabilityof the upper level is given by (see Eq. (B.13) in Appendix B):
P+ =PLZ (1 + cos ζ+ cos ζ− )sin2 ζ+ + 2PLZ (1 + cos ζ+ cos ζ− )

.

(38)

From this expression one can see that on resonance (i.e., when sin ζ+ = 0) the upper level occupation probability is maximal,
P+ = 1/2. One can also see that the widths of the resonance lines are modulated by the numerator in Eq. (38); it tends tozero at points wherecos ζ+ cos ζ− = −1.

(39)

Fig. 6 illustrates the interferometric pattern obtained from Eq. (38). In this figure we use the parameters of (Sillanpää
et al., 2006) (see also Table 1). Note that for any value of the energy bias offset ε0 , the maxima and minima alternate withincreasing driving amplitude A (see the color variation along the vertical dashed line in Fig. 6), as was studied by Shytov
et al. (2003) and Shevchenko et al. (2005).
In the case of zero offset ε0 = 0, one can go a little bit further with the analytic derivation. Taking ε0 = 0 we have
ζ1 + ζ2 ≈ 2A/ω and ζ1 − ζ2 = 0 and the resonance condition (Eq. (37)) is satisfied when
2A/ω = kπ .

(40)

One can alternatively analyze the dynamics in the time domain. When ε0 = 0, t1 = 0 and t2 = π /ω, and from Eq. (B.5) in
the first approximation in PLZ it follows that
P+ (t ) = PLZ

ζ1 =

1
2

sin2 nφsin2 φ

Z π/ω
0

≡

PLZ
cos2 ζ1 + PLZ sin2 ζ1

sin2 nφ,√∆2 + A2 eAΩ (t )dt =E √,ω
∆2 + A2

where eE (x) is the full elliptic integral of the second kind.

(41)

(42)

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

11

Let us now analyze the expression for P+ (t ), Eq. (41). The prefactor of sin2 nφ has a minimum at ζ1 = kπ (where thisprefactor is equal to PLZ ) and a maximum at ζ1 = π2 + kπ (where it is equal to 1). Thus, the former case (ζ1 = kπ ) corresponds
to destructive interference and the latter case (ζ1 = π2 + kπ ) corresponds to constructive interference. These two cases areillustrated in Fig. 5. Note that the latter case corresponds to the coherent destruction of tunneling, which has been studied
extensively in the literature (Grossmann et al., 1991a,b; Grossmann and Hänggi, 1992; Kayanuma and Saito, 2008; Eckel
et al., 2009; Ho et al., 2009; Wubs, 2010). From Eq. (41) for the cases of constructive and destructive interference we obtain:
constructive: P+ (t ) = sin2

pdestructive: P+ (t ) = PLZ sin2



PLZ n ,

π 
2

n ,

ζ1 =

π
2

+ kπ ,

ζ1 = kπ .

(43)
(44)

The dynamics (i.e., the time dependence of P+ ) in the cases of constructive and destructive interference is illustrated in Fig. 5.
The resonance condition ζ1 = π2 + kπ with Eq. (42) in the limit of strong driving (i.e., A  ∆) takes the form (Shevchenko
and Omelyanchouk, 2006):
2A/ω = π (2k + 1).

(45)

The positions of the resonances depend on the ratio A/ω but not on ∆, as was studied in detail by Shytov et al. (2003). Notethat Eq. (40) predicts more resonance peaks than Eq. (45). The explanation of this discrepancy is that at ε0 = 0 and for evenvalues of k the resonance peaks of Eq. (40) disappear according to Eq. (39), and we recover Eq. (45). Interestingly, if we takeEq. (42) in the limit of weak driving (i.e., A  ∆), we obtain:∆ = (2k + 1)ω.
This relation describes the odd multiphoton resonances, as studied by Shirley (1965) and Krainov and Yakovlev (1980),
where the energy level separation ∆ is an odd multiple of a photon energy ω. We have therefore obtained the correct
resonance condition, even though the weak-driving limit is outside the regime of validity of the above derivation, which isbased on the interference of multiple LZ crossings.
2.4.2. Fast-passage limit
In the limit δ  1, (1 − PLZ )  1 and the resonance condition reduces toζ− = ζ1 − ζ2 = k π .

(46)Unlike the slow-passage limit, one can now derive an approximate analytic expression for ζ1 − ζ2 in the large-amplitude
limit (the reason is that the parts of the yellow and blue areas in Fig. 2 that are difficult to calculate cancel when taking thedifference between the two areas):

ζ− = ζ1 − ζ2 ≈1

Z t1 +2π/ω2(ε0 + A sin ωt ) dt =

t1

π ε0
.
ω

(47)

The resonance condition is therefore given simply by

ε0 = kω.

(48)

If the system is driven on resonance (with a given value k), and taking the TLS to be initially in the lower diabatic state(i.e., ϕ↑ or ϕ↓ depending on the sign of ε0 ), the system exhibits full oscillations between ϕ↑ and ϕ↓ :(k)Pup(t ) =

1(k)1 − cos ΩR t .
2(49)

The oscillation frequency can be calculated from the effective rotation angle in the full-cycle evolution matrix (NU2 NU1 ), as
was done in (Ashhab et al., 2007). In the limit A  ε0 we find that(k)

ΩR ≈ ∆

r

2ω

πA


cos

A

ω

−k

π
2

−

π
4



.

(50)

As in the slow-passage limit, the Rabi frequency exhibits periodic behavior in the fast-passage limit as well. In particular, itvanishes whenever the cosine function in Eq. (50) vanishes. The interferometric pattern that results in this case is shown in
Fig. 7.
Above we have used the adiabatic-impulse approach to analyze the response of the TLS to the driving field. To demonstrate another approach, next we consider the results of the rotating-wave approximation (RWA); for more details see
Appendix C.

12

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

abFig. 7. (Color online) Fast-driving LZS interferometry for Aω  ∆2 : dependence of the time-averaged upper diabatic state occupation probability P up
on ε0 /ω and A/ω. The graphs are plotted using Eq. (57) for ω/∆ = 300  1, ωT1 /(2π) = 2.4 × 104 and ωT2 /(2π) = 24 (a) and ω/∆ = 1.14 > 1,
ωT1 /(2π) = ωT2 /(2π) = 6 (b). Several multiphoton resonances are shown by the vertical pink dotted lines at ε0 = kω (for k = 0, 1, 2, . . . , 5, only)
modulated by Bessel functions. The vertical red double-arrow in (b) shows the distance between two consecutive zeros of the Bessel function.

If decoherence processes are not taken into account, the TLS is described by the Schrödinger equation. Its solution gives
the probability of the upper diabatic state Pup , which exhibits k-photon Rabi oscillations:(k)Pup(t ) =∆k = ∆Jk
(k)

ΩR =

q

∆2k
A

ω



1 − cos ΩR t ,

2ΩR

 

(k)



(k)2

(51)

,

(52)

(kω − ε0 )2 + ∆2k ,(53)

where Jk is the Bessel function. Note that the k-photon Rabi frequency is described by a power law at small driving, A/ω  1;namely, at ε0 = kω:
(k)

ΩR

∆
= ∆k ≈
k!



k

A
2ω

.

(54)

Changing the system parameters (ε0 or ω), we pass through different k-photon resonances. Thus, the time-averaged
probability P up can be described by the sum of the Lorentzian-shape k-photon resonances:
P up =X(k)

P up =

k

∆2k

1X
2

k

(kω − ε0 )2 + ∆2k

.

(55)

The position of the resonances is (quasi-) periodic in the parameters ε0 , ω−1 , A; this corresponds to the multiphoton relation
ε0 = kω and to the quasi-periodic character of the Bessel functions Jk (A/ω) for large value of the argument:

r

 
Jk

A

≈

ω

2ω

πA


cos

A

ω

−

π
4


(2k + 1) .

(56)

Note that the oscillations described by Eq. (56) represent the counterpart to the Stückelberg oscillations of the doublepassage problem (Nikitin, 1996).
Note here that the results derived in the adiabatic-impulse approach and those of the RWA are essentially the same.
These results can also be derived using Floquet theory (Son et al., 2009) (see also Appendix D) or using the dressed-state
picture, which represent the hybridization of the qubit and photon degrees of freedom (Liu et al., 2006; Wilson et al., 2007)(see also Appendix E).
Relaxation can be described using the Bloch equations for the reduced density matrix; the influence of the environmentis taken into account using the phenomenological energy and phase relaxation times, T1 and T2 . In Appendix C we obtainthe stationary solution of the Bloch equations, which at zero temperature results in the excitation probability:
P up =

∆2k

1X
2

k

1
T1 T2

+

T2
T1

(kω − ε0 )2 + ∆2k

.

(57)

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

13

Fig. 8. (Color online) Crossover from the slow-passage limit (bottom part of the figure) to the fast-passage limit (top part of the figure) as the driving
amplitude A is increased. On the left the steady-state probability P + of the adiabatic excited state is plotted as a function of bias offset ε0 and driving
amplitude A. On the right the probability P up of the upper diabatic state is plotted. One can see that the resonance features are clearest in the adiabatic
basis for slow passage and in the diabatic basis for fast passage. The ratio ω/∆ is equal to 0.32, and there is no decoherence. Note that panel (a) differs from
Fig. 6 because that figure was generated using Eq. (38), whereas in this figure we numerically solve the Bloch equations.

In the limit where the dephasing and relaxation times are large and equal to each other, i.e., T1 = T2 → ∞, the above
expression (somewhat accidentally) coincides with the time averaged solution of the Schrödinger equation, Eq. (55). In
Fig. 7 we plot the probability P up with Eq. (57) for the parameters of Refs. Oliver et al. (2005) and Izmalkov et al. (2008) (seeTable 1).
We emphasize here that theq
distinction between the interference patterns in Figs. 6 and 7 is determined not by the ratioω/∆, but rather by the ratio ω A2 − ε02 /∆2 . This point is illustrated in Fig. 8, where one can see the crossover from theslow-passage to the fast-passage limit as the driving amplitude A is increased. Note that in order to see this crossover for a
fixed frequency, one needs to take a small value of ω/∆.
The analysis of the positions of the multiphoton resonances allows to do spectroscopy and to obtain the parameters of
the TLS. To calibrate the power of the generator one can use the distance between the minima or maxima in the Stückelberg
oscillations (one of them is shown in Fig. 7 with the vertical red double-arrow), or alternatively, the slope of the interference
fringes (shown by the orange dashed line in Fig. 7). The analysis of the interference fringes also allows to obtain theparameters that characterize the relaxation of the system due to the interaction with the environment. This was theoretically
discussed in Shytov et al. (2003) and experimentally realized in Oliver et al. (2005), Sillanpää et al. (2006), Wilson et al. (2007)
and Izmalkov et al. (2008). The longitudinal T1 and transverse T2 relaxation times can be extracted from the shapes of theresonances in Eq. (57). Moreover the effective temperature can be estimated using Eqs. (C.15) and (C.16) from the shape ofthe first resonance, where ∆E ≈ |ε0 | ≈ ω which is typically of the order of 1 GHz and smaller for superconducting TLSs.
In practice it can be useful to perform more detailed calculations beyond the stationary solution in Eq. (57). We note thatthe relaxation times, T1 and T2 , can be introduced phenomenologically as two fitting parameters (as in Oliver et al. (2005))
or calculated within certain model for dissipating environment (as in Sillanpää et al. (2006)). For superconducting qubits aconvenient model is the spin-boson model, which models the environment as a bath of harmonic oscillators (Makhlin et al.,
2001; Wendin and Shumeiko, 2007; McDermott, 2009; Wilson et al., 2010); in this model both T1 and T2 are parameterdependent and determined by the strength of dissipation. However in many cases the relaxation can be described byeffective parameter-independent times T1 and T2 ; in this paper for plotting Fig. 7 we have taken the relaxation times, T1
and T2 , for simplicity being constant. Also, the Bloch equations (C.13) can be solved exactly as, e.g., in Shevchenko et al.
(2008), which allows
qone to relax the assumption ∆  ω; the positions of the multiphoton resonances are then defined by
the relation: ∆E ≈∆2 + ε02 ≈ kω. Note that the inhomogeneity of the field due to low-frequency noise cannot be directly

described within the Bloch equations (Abragam, 1961); it results in the broadening of the Lorentzian-shaped resonances andcan be described as the additional term in the expression for the width of the resonances, as in Ref. Oliver et al. (2005).

14

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

a

b

c

d

Fig. 9. (Color online) Same as in Fig. 6 (i.e. LZS interferometry with low-frequency driving), but including the effects of decoherence. The time averaged
upper level occupation probability P+ was obtained numerically from the Bloch equations with the Hamiltonian (1). The dephasing time T2 is given by
ωT2 /(2π) = 0.1 in (a), 1 in (b), 5 in (c) and T2 = 2T1 in (d). The relaxation time is given by ωT1 /(2π) = 10.

2.5. Decoherence
When studying the effect of decoherence on the dynamics of a quantum system, it is perhaps most common to start fromthe case of zero decoherence and slowly increase the decoherence rates. In the problem of LZS interferometry, we find it
simpler to think about the effect of coherence rather than the effect of decoherence. We study the effect of coherence byconsidering the steady-state populations of the two quantum states of the TLS (as was done in Berns et al. (2006)). We notehere that in the slow-passage limit we analyze the populations of the adiabatic-basis states whereas in the fast-passage limit
we analyze the populations of the diabatic-basis states.
Starting from the limit of strong decoherence, no interference between the different LZ transitions occurs. The result isthen simple. If the driving amplitude is large enough to take the TLS through the avoided crossing, each crossing will act
as a small ‘‘kick’’ to the populations of the two states, and these kicks will eventually add up such that the TLS reaches asteady state with equal population of the two energy levels (the ground state can have a higher population if relaxation issubstantial). If the driving amplitude A is smaller than the bias offset |ε0 |, no LZ transitions will occur, and the system willremain in the lower level. This situation is depicted in Figs. 9(a) and 10(a).
We now increase the coherence time such that the two transitions in a single driving period are separated by a timethat is smaller than the coherence time. The transitions induced by different driving cycles are separated by more thanthe coherence time, and they therefore act as independent kicks. As in the case of strong decoherence, a sufficiently large
amplitude is required to drive transitions between the two quantum states. However, if the phase accumulated between thetwo interfering transitions corresponds to destructive interference, the net effect of a full driving period will be no mixing
at all. As a result, we now find lines in the A–ε0 plane where the TLS is driven back and forth across the avoided crossing
but the system remains in the lower level (in the fast-passage limit, the TLS remains in the diabatic state that corresponds
to the lower level for the given value of ε0 ). This situation is depicted in Figs. 9(b) and 10(b).

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

a

b

c

d

15

Fig. 10. (Color online) Same as in Fig. 7 (i.e. LZS interferometry with high-frequency driving), but including the effects of decoherence. The time-averaged
upper diabatic state occupation probability P up is obtained numerically by solving the Bloch equations with the Hamiltonian (1). The dephasing time T2 isgiven by ωT2 /(2π) = 0.1 in (a), 0.5 in (b), 1 in (c) and T2 = 2T1 in (d). The relaxation time is given by ωT1 /(2π) = 103 .

We now increase the coherence time further, such that interference between transitions from a few successive driving
cycles occurs. We now start to see the resonance lines forming, or rather the mixing between the two states being suppressed
whenever the resonance condition is not satisfied (Figs. 9(c) and 10(c)). Finally, in Figs. 9(d) and 10 (d) we take the case ofnegligible decoherence, and we find that the resonance lines are now well defined. Deviation from the resonance condition
leads to the absence of any substantial mixing between the two states.
2.6. ‘‘Breakdown’’ of the adiabatic theorem
The adiabatic theorem is used in various subfields of physics. The basic idea of the theorem is the following (Landau andLifshitz, 1977): if a physical system starts in one of the eigenstates of the system’s Hamiltonian, the Hamiltonian is varied
slowly enough and there are no degeneracies between the eigenstates during the variation of the Hamiltonian, then thesystem will end up in an eigenstate of the Hamiltonian whose order (starting from the ground state counting upwards) is
the same as that of the initial state.
As the term ‘‘theorem’’ suggests, the above statement is a rigorously provable statement. The tricky point is in deciding
how slow the Hamiltonian variation must be in order to ensure adiabaticity. The most common form of the adiabatic
condition is obtained by inspecting the LZ formula (Eq. (19)): δ  1, and slightly generalized forms of this condition. If thecondition δ  1 is satisfied in the LZ problem, the system will almost certainly stay in the ground state, assuming it started in
the ground state. The problem is that the above condition was derived for a specific problem, namely the simple LZ problem,and not from general considerations that apply to any quantum system with any form of Hamiltonian variation. This issue
has been the subject of a number of recent papers (Marzlin and Sanders, 2004; Sarandy et al., 2004; Tong et al., 2005, 2007;
Amin, 2009). For example, if a many-level system starts in an eigenstate of the Hamiltonian, and then during the variationof the Hamiltonian an enormously large number of avoided level crossings are encountered (in what can colloquially be16

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

called a ‘‘spaghetti’’ energy-level structure), as e.g. in Pokrovsky and Sinitsyn (2002, 2004), Shytov (2004) and Garanin et al.
(2008), knowing that each crossing depletes the population of the initial eigenstate by a small amount does not guaranteethat the total depletion of the state from all the crossings will also be small.
Another example where the above simple condition does not guarantee adiabaticity over the full time evolution occurs
in the simple LZS interferometry setup considered in this review. If the system is repeatedly driven back and forth acrossthe avoided-crossing point and the phase accumulated between successive crossings is designed to produce constructive
interference, the small transition amplitudes of the different crossings will add up to produce a large transition probability.
For example, if we take δ  1, ε0 = 0 and A/ω = π /2 + kπ (i.e., we choose the parameters to satisfy the resonancecondition, Eq. (43)), we know that it is only a matter of time until the ground-state population is transferred completely tothe excited state and vice versa. Finally, it should be emphasized that one does not need to satisfy any resonance condition of
the kind given in Eq. (37). An arbitrary-looking driving signal that produces suitable phase accumulation between successive
crossings will also eventually result in full population transfer between the ground and excited states. Even random phases
will result in diffusion-like dynamics that ultimately result in large mixing between eigenstates.
When discussing the adiabatic theorem, it is worth digressing for a moment to mention an idea that has received
considerable attention recently, namely adiabatic quantum computing (AQC) (Farhi et al., 2001). The basic idea of AQC is asfollows: one subjects a physical system to a simple initial Hamiltonian, such that it is guaranteed that the system will relax
to its ground state. One then adiabatically changes the Hamiltonian to reach a final Hamiltonian whose ground state encodes
the answer a given problem (the specifics of the problem are encoded in the parameters of the final Hamiltonian). Measuring
the final state then provides the answer to the problem. During the evolution from the initial to the final Hamiltonian, one
can expect that avoided crossing structures will be encountered, and the size and number of these avoided crossings will
determine the requirements on the sweep rate. The smaller the energy gaps and the larger the number of crossings, the more
slowly one has to sweep the parameters of the Hamiltonian. The traversal of the avoided crossings is governed by LZ physics,
and non-adiabatic LZ transitions play a destructive role and should be avoided; this was studied for a superconducting qubit
e.g. in Grajcar et al. (2005). It is currently unknown how the running time scales with system size for different problems. The
answer to this question will determine whether AQC provides a computational advantage over classical computers or not. In
a study that comes a step closer to the subject of the present article, it was argued in Ashhab et al. (2006) that the interaction
between an AQC system and its surrounding environment could lead to a situation where the environment effectively drives
the system back and forth across the avoided crossings, thus reducing the probability of remaining in the ground state.
2.7. Reversal of stimulated emission
It is well known that when an atom is placed in a resonant cavity and the atom is initialized in its excited state, it emits aphoton into the cavity with a rate that is proportional to the number of photons in the cavity (more accurately, the emission
rate is proportional to n + 1, where n is the number of photons in the cavity). Recently, this situation was realized in anexperiment that uses a superconducting charge qubit whose energy-level structure is essentially the one that we study
here (Astafiev et al., 2007). The qubit is biased such that it experiences an effective inverted relaxation taking it from theground to the excited state. The qubit therefore constantly emits photons into the resonant cavity, and a lasing state isrealized.
As discussed in Ashhab et al. (2009), for small photon numbers the emission rate is proportional to the number of photons
in the cavity, which can be seen as being a result of the fact that the Rabi frequency is proportional to the square-root of thephoton number. It is interesting to consider what happens if the photon number is so large that the cavity field effectively
drives the qubit back and forth across the quasi-crossing region. In this case the Rabi frequency is no longer proportional tothe driving amplitude (which is proportional to the square-root of the photon number), but it follows the Bessel-function
dependence that is characteristic of LZS interferometry (see Appendix C). One therefore has the situation where√ the emission
rate is no longer proportional to the photon number in the cavity n, but rather it is proportional to J12 (α n), where α is aconversion coefficient. Stimulated emission therefore exhibits oscillatory behavior, where the emission rate has regions
where it increases with increasing photon number and regions where it decreases with increasing photon number.
One consequence of the quasi-periodicity in the Bessel function is that, with a suitable choice of parameters, there canexist several stable steady-state photon numbers in the cavity. These solutions can be obtained by comparing the emission
rate into the cavity (which follows the Bessel-function dependence) and the loss rate out of the cavity (which is proportional
to the photon number). This situation concerning the existence of multiple steady-state solutions in this system was studiedin Rodrigues et al. (2007a,b).
3. Recent experiments with strongly driven superconducting quantum circuits
3.1. Multiphoton transitions in superconducting qubits
If a quantum system is strongly driven, it can be excited by a multiphoton process. One way to observe the multiphoton
resonances is to consider the frequency dependence, which is convenient for atoms exposed to a periodic electromagnetic
field (Gallagher, 1994; Fregenal et al., 2004; Førre, 2004; Maeda et al., 2006). Another way is to tune the energy difference at

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

17

fixed driving frequency. In superconducting qubits the energy difference is tuned by an external source. The observation of
the resonances allows for multiphoton spectroscopy. Furthermore, the strong driving provides a tool to control the energy
level population, with the driving frequency being a fraction of the energy level difference. This is important for operatingsuperconducting qubits, where the energy difference is of the order of 0.1 to 10 GHz. The multiphoton resonances have
Lorentzian shapes (Tornes and Stroud, 2008), as described by Eqs. (55) or (57), which provide a way to estimate the relaxation
rates in the system.
Multiphoton resonances were observed in several superconducting qubits with a small number of photons. They wereobserved in charge qubits (Nakamura et al., 2001), where the energy-level difference is controlled by a gate voltage, as
well as in the charge-phase qubits, where the energy difference is controlled by both a gate voltage and an external
magnetic flux (Shnyrkov et al., 2006, 2009). Up to five-photon transitions were reported for the phase qubit (Wallraff
et al., 2003), where the energy difference is tuned by the bias current. Multiphoton transitions were also observed in aflux qubit (Saito et al., 2004) controlled by an external magnetic flux. Further improvement of the coherence characteristicsof the superconducting qubits allowed to observe resonant excitations with more photons, at higher driving amplitudes.
These results are presented below.
3.2. LZS interferometry in superconducting qubits
The observation of LZ tunneling in superconducting qubits was first reported by Izmalkov et al. (2004) and thenby Ithier et al. (2005) and Johansson et al. (2009) (see also Ankerhold and Grabert (2003) for the theoretical background).
Recently LZS interferometry was demonstrated by several groups on different superconducting qubits with complementarymeasurement techniques (Oliver et al., 2005; Sillanpää et al., 2006; Wilson et al., 2007; Izmalkov et al., 2008; LaHaye et al.,2009). Below we briefly describe these works and present the major results in Fig. 11. There, the left column shows thecircuits used in the experimental setups and the right columns shows their corresponding experimental LZS interferometry.
The latter shows a measurable quantity versus driving power (amplitude) and energy bias (which defines the energy
level distance). Also recently the LZS interferometry was demonstrated with the rf SQUID qubit in Sun et al. (2009) [seealso Sun et al. (2006) for experimental details]. One important difference from the previous works is that the system isessentially a multi-level system with several different avoided crossings involved (see also next subsection). This allows for
additional interesting and useful phenomena such as controllable population inversion, as convincingly demonstrated bothexperimentally and theoretically in Sun et al. (2009).
The parameters used in different experiments studying LZS interferometry with driven superconducting qubits are given
in Table 1. Remarkable is the fact that in all of these experiments the√relation of the characteristic time of the LZ transition(Eq. (21)) estimated with maximal driving amplitude Amax , tLZ ∼ 1/ Amax ω, to the driving period T was almost identical:
tLZ /T ∼ 0.05. Note that the exact value of tLZ /T does not affect the analysis or results, provided that this ratio is muchsmaller than one (see Eq. (30)).
(1) In Ref. Oliver et al. (2005) a superconducting flux qubit was subjected to a strong microwave drive and LZS multiple
interference fringes were observed. The flux qubit was a niobium superconducting ring with three Josephson junctions.
The circuit and results are shown in Fig. 11(a) to the left and to the right respectively. The qubit was biased with both dc
and microwave magnetic fluxes. The level population was probed with a dc SQUID. The switching probability of the SQUID
to the resistive state measures the state of the qubit in the diabatic basis {ϕ↑ , ϕ↓ }. These measurements showed resonantpeaks and dips when the energy of k photons matched the qubit’s energy-level difference. The latter was controlled withthe dc flux (flux detuning) and multiphoton resonances with k up to 20 were observed. Later the observation of multiphoton
resonances of order up to 45 was reported (Berns et al., 2006). The interference fringes shown in Fig. 11 are described by
the theory presented in Section 2.4.2; cf. also Fig. 7(a). The fringes exhibit the Bessel-function dependence, according to Eqs.
(52) and (55), and the steplike pattern in Fig. 11(a) was called a ‘‘Bessel staircase’’ (Oliver et al., 2005); the inclined whitelines in Fig. 11(a) show the maxima for Jk2 (A/ω) which were numbered with the Roman numbers I to VI. For more details
see Rudner et al. (2008), Bylander et al. (2009) and Oliver and Valenzuela (2009).
(2) In Ref. Sillanpää et al. (2006) an interferometer-type charge qubit was studied. The qubit consisted of two nearby
Josephson junctions embedded in a loop (Fig. 11(b)). The charge on the island between the junctions (the so-called Cooperpair box) was controlled via the applied gate voltage Vg = eng /Cg , ng (t ) = ng0 + δ nrf sin(ωrf t ). The effective Josephsonenergy of the two junctions was tuned using the applied magnetic flux Φ through the loop. The energy levels of the qubit
were controlled by the flux Φ and by the dc component (ng0 ) of the voltage, while the energy-level populations were
controlled by the ac component of the voltage. The qubit was coupled in parallel to an LC -circuit. The effective capacitance
Ceff of the qubit modifies the resonant frequency of the LC -circuit (Sillanpää et al., 2005). This provides a method for thecontinuous-time monitoring of the qubit’s effective capacitance, which is related to the occupations of the qubit’s energy
eigenstates. The resulting LZS interference pattern is shown to the right in Fig. 11(b); this is described by the theory presented
in Section 2.4.1; cf. also Fig. 6. For more details see also Paila et al. (2009) and Sillanpää et al. (2007). Also very recently LZSinterferometry with an interferometer-type charge qubit was demonstrated in LaHaye et al. (2009). The parameters of thequbit were close to the ones in Ref. Sillanpää et al. (2006); consequently, LZS interference fringes were similar to the ones
shown in Fig. 11(b). The important difference is in the measuring device: the qubit was coupled to a nanoelectromechanicalresonator, as studied theoretically in Irish and Schwab (2003), and the nanoresonator frequency shift was measured.

18

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

a

b

c

d

Fig. 11. (Color online) Experimentally realized Landau–Zener–Stückelberg (LZS) interferometry. The panels from top to bottom present the results of thefollowing articles: (a) Oliver et al. (2005), (b) Sillanpää et al. (2006), (c) Wilson et al. (2007), (d) Izmalkov et al. (2008). Schematic diagrams of the circuits
used are shown to the left, while results for the LZS interferometry are presented to the right. A more detailed description of the experiments can be foundin the main text and, of course, in the respective original articles. Figure (a) is reprinted from Oliver et al. (2005) with permission from AAAS. Figure (b) isreprinted from Sillanpää et al. (2006) with permission; copyright (2006) by APS. Figure (c) is reprinted from Wilson et al. (2007) with permission; copyright(2007) by APS. Figure (d) is reprinted from Izmalkov et al. (2008) with permission; copyright (2008) by APS.

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

19(3) References (Wilson et al., 2007, 2010) studied a single Cooper-pair box (SCB), which was composed of an aluminium
island connected to a reservoir via a Josephson junction (Fig. 11(c)). The state of the qubit was controlled by the gate voltage
Vg (ng = Cg Vg /2e) and by the magnetic field in the dc SQUID loop connected to the island, which tunes the Josephsonenergy EJ . The qubit was driven through the gate with the microwave amplitude Aµ . The qubit state was probed with anrf oscillator (Duty et al., 2005; Persson et al., 2010). The color image in Fig. 11(c) presents the phase of the rf reflection
coefficient Γ . The ground and excited states contributed phase shifts with opposite signs. The results were interpreted interms of the dressed states of the qubit coupled to the driving microwave field (Liu et al., 2006; Greenberg, 2007).
(4) In Ref. Izmalkov et al. (2008) a superconducting flux qubit was driven by an external magnetic flux, analogously toRef. Oliver et al. (2005). The qubit was weakly coupled to a classical resonant tank circuit, Fig. 11(d). The tank circuit wasbiased with an rf current Ibias and the measurable quantity was the phase shift Θ between the voltage V and the bias current
in the tank circuit. The inductive coupling of the qubit to the tank circuit resulted in a phase shift determined by the effective
inductance of the qubit (Greenberg et al., 2002; Il’ichev et al., 2009). The resonant excitation of the flux qubit changes the
direction of the current in the qubit’s loop, which resonantly changes the effective inductance of the qubit. The respective
resonances were observed in the phase shift dependence as a sequence of the ridges and troughs (Shevchenko et al., 2008).
The LZS interference pattern is shown in Fig. 11(d) in the dependence of the tank phase shift Θ on the dc (Φb ) and ac (Φac )components of the magnetic flux; the interference fringes are described by the theory presented in Section 2: see Fig. 7(b).
Thus, strongly driven superconducting qubits were studied by different, complementary methods; their resonant
characteristics were shown to be periodic in the bias ε0 and driving amplitude A due to Stückelberg oscillations. The different
experimental measurements allowed to obtain the following parameters: (i) those which characterize the qubit and enter inthe pseudo-spin Hamiltonian, (ii) the driving microwave amplitude as felt by the qubit (which is the calibration of a generator
signal), (iii) the relaxation and dephasing rates, which describe the impact of the environment on the qubit dynamics (via
the Bloch equations).
3.3. Multiphoton resonances in multi-level systems
Most qubits are formed by the lowest two levels in a multi-level structure. Moreover, a qubit can be coupled to anotherquantum system, in particular, to another qubit, so that the overall structure is also multi-level. High-amplitude driving
allows not only to reach the upper level of a TLS (qubit) with the absorption of several photons, but also to involve more of
the qudit’s energy levels in the process.
Multiphoton resonant excitations were studied in superconducting multi-level systems. One-, two-, and three-photontransitions were driven in the flux qubit not only between the two lowest levels (which form the qubit), but rather between
five levels in Yu et al. (2005). In this case multiphoton spectroscopy allowed to obtain the system parameters and to visualize
the energy-level diagram. The low-frequency and extremely strong-driving regime was studied in Berns et al. (2008) [seealso Wilhelm (2008) and Wen and Yu (2009)]. Berns et al. demonstrated that their method, which they called amplitude
spectroscopy, allows probing higher levels that are three orders of magnitude higher in energy than the driving frequency.
Recently interference fringes associated with LZ transitions at two nearby avoided crossings were also observed in a multilevel structure of the rf-SQUID qubit driven by large-amplitude and high-frequency microwave field (Wang et al., 2010) [seealso Sun et al. (2009) for more details]. Also, multiphoton multi-level Rabi oscillations were studied in a phase qubit (Dutta
et al., 2008). Multiphoton transitions in the multilevel system formed by the dressed states of a qubit coupled to a quantum
resonator were demonstrated in Fink et al. (2009) and Bishop et al. (2009). Another situation where a multi-level structureappears, is a system of coupled qubits. Direct and ladder-type multiphoton excitations were recently demonstrated in a
two-flux-qubit system in Il’ichev et al. (2010).
4. Conclusions
Strong driving of a TLS results in a periodic dependence of its response on the bias, driving frequency and amplitude. Thisdependence was recently observed in superconducting qubit systems by several groups using complementary measurement
techniques, both in flux qubits (Oliver et al., 2005; Izmalkov et al., 2008) and in charge qubits (Sillanpää et al., 2006; Wilson
et al., 2007). Very recently LZS interferometry was also realized in a rf SQUID qubit (Sun et al., 2009). No related effect has
been reported in experiments using phase qubits. This is because it is difficult to produce avoided energy-level structuresin phase qubits without involving higher levels.
LZS interferometry allows, first, to control the qubit state, and, second, to obtain the system parameters. In particular,from the positions of the multiphoton resonances, the qubit parameters can be obtained (spectroscopy); the Stückelberg
oscillations can be used for the definition of the driving amplitude (calibration of power); and the width and shape of theinterference fringes can be used to estimate the relaxation rates. These properties make LZS interferometry a very powerfultool for studying TLSs (see also LaHaye et al. (2009), where LZS interferometry was realized recently for a superconducting
qubit coupled to a nanoelectromechanical resonator).
We have reviewed recent experimental results on LZS interferometry, and we have presented the theoretical descriptionof this problem. In particular, we have demonstrated the applicability of the adiabatic-impulse model for both slow andfast driving regimes. This model is a powerful tool in the study of non-adiabatic transitions at avoided crossings and theinterference between these transitions.

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S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

Acknowledgements
We would like to thank A.N. Omelyanchouk, A.M. Zagoskin, A.A. Soroka and D.A. Rodrigues for useful discussions.
We would also like to thank J.R. Johansson for useful discussions and for providing the numerical calculations used in
Figs. 8–10. This work was supported in part by the National Security Agency (NSA), Laboratory Physical Science (LPS),
Army Research Office (ARO), National Science Foundation (NSF) grant number 0726909, JSPS-RFBR grant 09-02-92114,
Fundamental Researches State Fund (grant number F28. 2/019), and NAS of Ukraine (project number 02/09-N).
Appendix A. Solution of the Landau–Zener problem
A.1. Adiabatic wave function
In this Appendix we consider the driven two-level system in the adiabatic representation, assuming the evolution to beadiabatic far from the avoided level crossings, with non-adiabatic transitions occurring around these points.
The wave function in the adiabatic approximation takes the form

ψ=

 
a1
a2



,

Z

a1,2 = A1,2 exp −i

E (t )dt



.

(A.1)

Then the Shrödinger equation H ψ = E ψ gives the instantaneous eigenvalues, Eq. (4), and eigenvectors:πψ± = ϕ± e∓i(ζ + 4 ) ,
ϕ± =



ζ =

1
2

r


β∓
,
∓β±

β± =

Z

Ω (t )dt ,

Ω ±ε
.
2Ω

(A.2a)(A.2b)

Here the term π /4 arises when the next term in the quasiclassical (adiabatic) approximation is taken into account (Delone
and Krainov, 1985).
Let us consider the original LZ problem described by the Hamiltonian (1) with ε = v t {see also Eq. (17); we consider tinstead of t 0 in this Appendix}:
H (t ) = −

∆
2

σx −

vt
2

σz .

(A.3)

Consider now the adiabatic wave function in the region far from the avoided level crossing, where |ε(t )| = v |t |  ∆.
Note that this latter condition is consistent with the condition ω |t |  1 used for the linearization of the harmonic-driving
problem, Eq. (15), at large enough driving amplitudes. Then we have:

ϕ+ ≈
ϕ+ ≈

 

1
,
0



ϕ− ≈



0
,
−1

ζ (±ta ) =1

Z ± ta

2

0

 

ϕ− ≈

0
1

 
1
0

if ta < 0,

(A.4a)

if ta > 0,

(A.4b)

 r vΩ (t ) dt ≈ ± Φta − Φδ .

(A.5)

2

The subscript ‘‘a’’ indicates that we now consider the time within the asymptotic region, where |ε(t )|  ∆, and the acquired
phase was split into time-dependent and independent parts:

Φ (za ) =
Φδ =

1
2

za2
2

√
+ δ ln( 2za ),δ(ln δ − 1).

(A.6)
(A.7)

A.2. Non-adiabatic transition (evolution matrix for the Landau–Zener transition)
Let us now consider the non-adiabatic transition between the energy levels in the vicinity of the point of avoided levelcrossings, following Zener (1932). Then the Schrödinger equation iψ̇ = H ψ takes the form:

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30v∆iȧ1 = − ta1 − a2 ,
2

2

2

2

21

(A.8)

v
∆

iȧ2 = − a1 + ta2 .

This can be written in the form of the second-order Weber equations:d 2 a1 , 2+ (2δ ∓ i + z 2 )a1,2 = 0,

dz 2

r
z=

v
2

(A.9)

t.

(A.10)The solutions are combinations of parabolic cylinder functions (Gradshteyn and Ryzhik, 1994):
a1 =

X

√

A± D−1−iδ (± 2eiπ/4 z ),(A.11a)

±

a2 =

X

√

B± D−iδ (± 2eiπ/4 z ).

(A.11b)±

We find the relation between the coefficients A± and B± from the first-order equation (by inserting the solutions for a1,2 inthe first equation of the system (A.8)) making use of the recurrence formula:e−iπ/4
B± = ∓ √ A± .

(A.12)

δ

Now with the asymptotes of the parabolic cylinder functions (Gradshteyn and Ryzhik, 1994), we find a1,2 far from thetransition point (where z = 0), at z = ±za with za  1:a1 (−za ) ≈ A+ Ξ1 eiΦ (za ) ,a2 (−za ) ≈ (−e− π2 δ

a1 (za ) ≈ A− Ξ1 e

(A.13a)πδ

A+ + e 2 A− )Ξ2 eiΦ (za )−iΦ (za ),(A.13b),πδ

a2 (za ) ≈ (−e 2 A+ + e(A.13c)− π2 δA− )Ξ2 e−iΦ (za ),(A.13d)

where

√

2π

Ξ1 ≡

 π 
δ ,

exp −

(A.14a)Γ (1 + iδ)
4

π 
1
π
Ξ2 ≡ √ exp −i − δ .
44δLet us match this asymptotic solution with the adiabatic one, ψ(t ) =




b+
b−



b−
−b +

 
=

a1
a2



 
=

(A.14b)P

b± ϕ± , from the previous subsection:

at ta < 0,a1
a2

at ta > 0.

(A.15a)(A.15b)

Here the time dependence in the l.h.s. and in the r.h.s. is the same, see respectively Eqs. (A.5) and (A.13). This allows to
describe the evolution from t = −ta to ta as a sequence of adiabatic evolutions from t = −ta to t = −0 and from t = +0
to t = ta , with the nonadiabatic transition in between (described by the time-independent matrix N), namely:b(ta ) = U (ta , +0)NU (−0, −ta ) b(−ta ) = e−iσz ζ (ta ) Ne−iσz ζ (ta ) b(−ta ).

(A.16)

Then collecting this with Eqs. (A.15) and (A.13) and equating multipliers with A+ and A− , we obtain the expression (24) forthe matrix N. This matrix describes the transition between the two adiabatic states with the probability given by Eq. (19)and also includes the phase jump e
ϕS . We note that the matrix N at t = t2 is the same as at t = t1 for the nonadiabatic
transitions.
The argument of the asymptotes of the gamma function Γ are (Gradshteyn and Ryzhik, 1994):(
arg Γ (1 − iδ) ≈

−

π
4

C δ,

δ  1,− δ(ln δ − 1), δ  1,

(A.17)

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S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

where C ≈ 0.58 is the Euler constant. Thus, the phase jump ϕS (δ) is a monotonous function, which changes from 0 in the
adiabatic limit (δ  1) to π /4 in the diabatic (fast driving) limit (δ  1).
It is worth making a comment on bases here. In this review we have used the energy eigenbasis for describing the
dynamics of the TLS. Furthermore, we have followed the reasonable convention where the ground state far to the rightof the avoided crossing region coincides with the excited state far to the left of the avoided crossing region, and vice versa.
In principle, we could have defined these pairs of states such that they differ by some phase factors. If we follow such a
convention, there would be some asymmetry between the different matrices (denoted by N) that describe the different
LZ transitions. However, since such a convention seems rather unnatural, we do not comment on it further. A different
convention from ours that appears more frequently in the literature is the use of the diabatic states for describing the
dynamics. In this case the matrices that describe the evolution from time −ta to time ta for a single passage depend on thesigns that appear in the Hamiltonian, Eq. (17). The part describing the dynamical phase accumulated during the adiabatic
parts of the evolution is straightforward. The LZ transition matrices are less obvious, and we give the rules for how todetermine them explicitly here. For the Hamiltonian in Eq. (17) with negative signs in front of both terms in the Hamiltonian,
the LZ transition matrix takes the form:

p



p

PLZ

− 1 − PLZ e−iϕS
p

iϕS
1−
p PLZ e

PLZ.

(A.18)

If the sign in front of the second term in the Hamiltonian is reversed, the off-diagonal matrix elements in Eq. (A.18) arereversed. If the sign in front of the first term in the Hamiltonian is reversed, only the minus sign at the beginning of thebottom-left matrix element is moved to the top-right matrix element. Combining the two rules above, if both signs in the
Hamiltonian are reversed, only the minus sign in front of the Stokes phase is moved from the bottom-left matrix element to
the top-right matrix element. Note that in the diabatic basis there will always be two relevant signs in front of the second
term in the Hamiltonian, depending on whether the avoided crossing is traversed from left to right or from right to left. As
a result, there will be two different matrices describing the different crossings.
Appendix B. Evolution of a periodically driven two-level system
The time evolution of a periodically driven TLS is described by Eqs. (31) and (32). Consider first the matrix for a one-period
evolution:

α
γ


−γ ∗
,
∗
α


NU2 NU1 =(B.1a)

α = (1 − PLZ )e−iζ+ − PLZ e−iζ− ,p
γ = PLZ (1 − PLZ )eieϕS (e−iζ+ + e−iζ− ),(B.2b)

ζ+ = ζ1 + ζ2 + 2e
ϕS ,(B.3a)

ζ− = ζ1 − ζ2 .

(B.3b)(B.2a)

After diagonalizing the matrix, it is straightforward to write down its nth power, e.g. Bychkov and Dykhne (1970). This
unitary evolution matrix (NU2 NU1 )n has the form

(NU2 NU1 ) =
n



u11
u21

−u∗21



u∗11

,

(B.3c)

with elements:
u11 = cos nφ + i(Imα)
u21 = γsin nφsin φ

sin nφsin φ,(B.4a),(B.4b)

cos φ = Reα.

(B.4c)

Then with Eq. (31) for (t − 2π n/ω) ∈ (t1 , t2 ), we obtain the upper-level occupation probability, P+ = |b+ | ,2(I)

P+ (n) = |γ |2

sin2 nφsin2 φ.

(B.5)

And for (t − 2π n/ω) ∈ (t2 , t1 + 2π /ω) with Eq. (32) we obtain:(II)

P+ (n) = 2Q1

sin2 nφ
sin φ
2

+ Q2

sin 2nφsin φ+ PLZ cos 2nφ,(B.6)

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

23

Q1 = PLZ PLZ sin2 ζ− + (1 − PLZ )(1 + cos ζ+ cos ζ− ) ,(B.7a)

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

(B.8)





Now the time-averaging corresponds to averaging over n  1 periods and, from Eqs. (B.5) and (B.6), we obtain:

|γ |2

(I)

P+ =

2 sin2 φ
Q1

(II)

P+ =

sin2 φ=|γ |2,2 |γ |2 + (Imα)2
1

(B.9)

.

(B.10)

From Eq. (B.9) it follows that the upper-level occupation probability is maximal at Imα = 0. This results in the general
resonance condition:(1 − PLZ ) sin ζ+ − PLZ sin ζ− = 0.

(B.11)

In particular, in the slow and fast limits the resonance condition takes the form:

ζ1 + ζ2 = kπ for 2π δ  1,ζ1 − ζ2 = kπ for 2π δ  1.

(B.12a)(B.12b)

The resonance condition (B.11) was derived in Ashhab et al. (2007) using an alternative, geometric visualization of the
dynamics. There, the evolution matrix for one full driving period, i.e. (NU2 NU1 ), is decomposed into two rotations, a rotation
around an axis in the xy plane followed by a rotation around the z-axis. The resonance condition is then identified as therequirement that the z-axis rotation does not affect the dynamics, i.e. the z-axis rotation is a rotation by a multiple of 2π .
Straightforward algebra then gives Eq. (B.11). A related alternative derivation of the resonance condition relies on visualizing
the evolution matrix for one full driving period as a single rotation: on resonance the rotation axis lies in the xy-plane.
First, let us consider the case of the slow-passage limit, where δ  1 and PLZ  1. We now neglect the difference between
(I)

(II)

P+ and P+ and, from Eq. (B.10) to the lowest order approximation in PLZ , we obtain:
P+ =PLZ (1 + cos ζ+ cos ζ− )sin2 ζ+ + 2PLZ (1 + cos ζ+ cos ζ− ).

(B.13)For small offset ε0 /A  1 the expressions for ζ± can be simplified: ζ1 + ζ2 ≈ 2A/ω, ζ1 − ζ2 ≈ π ε0 /ω. This makes Eq.
(B.13) analogous to the result obtained in Shytov et al. (2003) (their Eq. (20) at u = w → 0). But for arbitrary (not small)offset ε0 one has to make use of Eq. (B.13) with ζ± given by Eqs. (B.3) and (14); the result, such as Fig. 6 is significantly
dependent on this correction.
One can also obtain an idea about the form of the resonance condition in the zeroth in ∆/ε0 approximation:

ζ1 ≈

1

Z t2

2

t1

ε(t )dtq

A − ε0π ε0ε0ε0− cos−1
+
,
ω
ω
A
ω
q
A2 − ε02ε0ε0ζ2 ≈ − cos−1
+
.
ω
A
ω
2

2

=

(B.14a)(B.14b)Eq. (B.12a) then givesq

A2 − ε02
ε0
ε0
ε0
+2
−2
cos−1
= k.
ω
πω
πω
A

(B.15)

We emphasize, however, that the above equation should be seen as a limiting case for small ∆ that demonstrates thecomplicated form of the resonance condition.
Now consider the case of fast passage, where (1 − PLZ ) ≈ 2π δ  1. The fast limit (δ  1) assumes a small value of ∆.
In this limit there is a large probability (PLZ ∼ 1) for transitions between the adiabatic states in a single-passage; while thetransition probability between diabatic states is small, (1 − PLZ )  1. Hence, we will consider the time-averaged probabilityof the upper diabatic state Pup . Note that upper diabatic state is ϕ↑ for ε0 < 0 and ϕ↓ for ε0 > 0. Let us consider the latter
(I )

(II )

case, as shown in Fig. 1. Then we can relate Pup to P+ and P+ as follows:(I )Pup ≈ P+for (t − 2π n/ω) ∈ (t1 , t2 ),(II )Pup ≈ 1 − P+

for (t − 2π n/ω) ∈ (t2 , t1 + 2π /ω).

(B.16)(B.17)

24

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

(II )

(I )

Then we note that 1 − P+ ≈ P+ and obtain:
P up =

4 cos2 (ζ2 − π /4)

1

2 sin ζ− + 4 cos2 (ζ2 − π /4)
2,(B.18)

where ζ2 is given by Eq. (B.14b). On resonance ζ− = kπ and P up = 1/2, then with ζ− ≈ π ε0 /ω for small offset (ε0 /A  1)we obtain the resonance frequency ω(k) = ε0 /k. Consider now the vicinity of the kth resonance, introducing a small
frequency detuning as follows

δω = kω − ε0 .

(B.19)

Then to first approximation in ε0 /A, we obtain:
(k)

∆2k

1

,
+ δω2
r


2ω
A
π
∆k = ∆cos− (2k + 1) .
πAω
4
P up =(B.20a)2 ∆2k(B.20b)(k)

The total probability P up is obtained as the sum of the partial contributions P up . This result, obtained in the adiabaticimpulse model, coincides with the one obtained within the RWA, Eqs. (55) and (56).
Appendix C. Rotating-wave approximation
C.1. Hamiltonian in the RWA-form
Following Pegg and Series (1973), Lopez-Castillo et al. (1992), Oliver et al. (2005) and Ashhab et al. (2007) we consider
the rotating-wave approximation (RWA).
First, we split the Hamiltonian into time-independent and time-dependent parts:
H =−∆H0 = −

ε0 + A cos ωt

σx −

2

∆
2

V (t ) = −

2

σx −

ε0

2
A cos ωtσz = H0 + V (t ),

(C.1)

σz ,

(C.2a)σz .
(C.2b)
2
Here we consider the time-dependent term with a cosine function (in contrast to the sine function used in Eq. (2)) because
this choice agrees with the standard convention in the literature. The difference between the two choices is simply a shiftof ωt by π /2, and it does not change the time-averaged probabilities which are of most interest to us.
Then we make a transformation to a rotating frame with the operator W :

W (t ) = exp −i

η(t ) =AωZ

V (t )dt= exp i

η(t )
2



σz ,

sin ωt .

(C.3)
(C.4)

This operator matches the wave function in the reference frame ψ with the wave function in the rotating frame ψ 0 :ψ = W (t )ψ 0 . Then the Shrödinger equation in the rotating frame reads: iψ̇ 0 = H 0 ψ 0 with
H 0 = W Ď H0 W = −∆(e−iη(t ) σ+ + h.c .) −2σ+ = σx + iσy . Making use of the formula
eix sin τ =

∞
X

ε0
2

Jn (x) einτ ,

σz ,

(C.5)

(C.6)

n=−∞

the new Hamiltonian takes the form
H0 = −

∞
X
∆n
n=−∞

2

∆n = ∆Jn (A/ω).

e−inωt σ+ + h.c . −



ε0
2

σz ,

(C.7)
(C.8)

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

25

Note that the unitary transformation W does not change the level occupation probabilities (the absolute values of the spinor
components):



ψ=

ψ1
ψ2



= Wψ = e0ηi 2 σz



ψ10
ψ20

η


=

ei 2 ψ10!
.

η

e−i 2 ψ20

(C.9)

C.2. Solving the Schrödinger equation in the absence of relaxation
Let us now study first the system without relaxation. We now look for a solution of the Schrödinger equation iψ̇ 0 = H 0 ψ 0in the form:

ψ =
0

!
ψ100 exp −i kω2 t .
ψ200 exp i kω2 t(C.10)e (t )ψ 0 with We (t ) = exp(i kωt σz ).
Actually, this corresponds to the transformation: ψ 00 = W
2Consider now parameters close to the k-photon resonance, where kω ≈ ∆E = |ε0 |. In other words, we now consider thefrequency detuning δω = kω − ε0 to be small. The RWA consists in the assumption that, in the vicinity of the kth resonance,
fast oscillating terms with n 6= k can be neglected. Then the Schrödinger equation can be readily solved. The probability of
the upper diabatic state (if the system was initially in the lower diabatic state) is given by:(k)

P up (t ) = |ψ2 (t )| =
2

∆2k

1


1 − cos2 δω2 + ∆2k

q

δω +
2

∆2k t



.

(C.11)

C.3. Solving the Bloch equations with relaxation
Now let us take into account relaxation by using the Bloch equations, which include two phenomenological relaxation
times, T1 and T2 (Abragam, 1961). A convenient parametrization for the density matrix is the following:

ρ=

1
2



1+Z
X + iYX − iY1−Z.

(C.12)

Here we note that the Bloch equations have to be written in the adiabatic (energy) representation, so that the constants
T1 and T2 describe relaxation to the equilibrium values: Z → Z0 and ρ21 = X + iY → 0. This representation differs fromthe diabatic representation (with the states {ϕ↑ , ϕ↓ }) by a small rotation with the angle ∆/ε0  1. Here we ignore thisdifference in the bases and consider the diabatic representation. Then for Z and ρ21 = X + iY , the Bloch equations with the
Hamiltonian H 0 read:Ż = −∆k Im e−ikωt ρ21 −

Z − Z0ρ̇21 = i∆k eikωt Z − iε0 ρ21 −

T1

ρ21
T2

,

(C.13a).

(C.13b)E(Again, we have neglected fast oscillating terms with n 6= k.) Here Z0 = tanh ∆describes the equilibrium energy level
2T

distribution at temperature T . After the substitution eρ21 = ρ21 exp(ikωt ) = e
X + ieY , we obtain the system of equations:.
eXe
X = δωeY−,(C.14a)

T2

.
e
Y
e
Y = −δωe
X + ∆k Z −,(C.14b)T2

Y−
Ż = −∆ke

Z − Z0T1.

(C.14c).

.

The stationary solution of these equations is obtained after substituting e
X =e
Y = Ż = 0. Then the stationary value for the
probability of the upper diabatic state is given by(k)

P up = ρ 22 =
Z =

1

1
2

(1 − Z ),1 + T22 δω2

2 1 + T22 δω2 + T1 T2 ∆2k(C.15)
Z0 .

(C.16)

26

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

At low temperatures we can substitute Z0 ≈ 1 and obtain from Eqs. (C.15) and (C.16) the following formula
P up =X(k)

P up =

k

∆2k

1X
2

k

1
T1 T2

+ TT21 δω2 + ∆2k,(C.17)

which is useful for the description of multiphoton resonances at ∆/ε0  1.
Appendix D. Floquet theory
Floquet theory can be applied to any quantum system governed by a time-periodic Hamiltonian. The main idea ofapplying this theoretical approach is as follows: the time-dependent problem is turned into a time-independent one at
the expense of increasing the effective number of states in the Hilbert space to an infinite number of states. Approximations
are then made to reduce the problem into a finite-dimensional one, and analytic or numerical results are obtained. One
advantage of Floquet theory is that it allows for the inclusion of dissipative processes relatively easily into the analysis. In thisAppendix we briefly outline the application of Floquet theory to the problem of LZS interferometry, closely following (Sonet al., 2009). More detailed discussions can be found in Grifoni and Hänggi (1998), Chu and Tel’nov (2004) and Son et al.
(2009).
From Floquet’s theorem we know that, since the Hamiltonian given by Eqs. (1) and (2) is periodic, there exist two solutions
to the Schrödinger equation with the property thatej (t )
ψj (t ) = e−iej t ψ(D.1)

ej (t ) is a periodic function of t with the same period as the Hamiltonian, i.e., 2π /ω. Furthermore, these two quasiand ψ
periodic solutions are orthogonal and form a complete basis. As a result, knowledge of these states and of the quasi-energiese allows one to describe the dynamics for any given initial state.
Finding the Floquet states and quasi-energies is typically achieved using Fourier analysis. Writing+∞Xej (t ) =ψξj,n einωt ,

(D.2)

n=−∞

where each ξj,n = (ξj,n,+ , ξj,n,− )T is a two-component vector (with components ξj,n,+ and ξj,n,− ), and writing the
Hamiltonian (C.1) as
H (t ) = −

∆

σx −

ε0


A
σz − σz eiωt + e−iωt ,

(D.3)

2
2
4
one obtains the following set of equations governing ξjn :eεξj,n = −

∆
2

σx −

ε0
2



A

A

4

2

σz + nω ξjn − σz ξj,n−1 − σz ξj,n+1 .

(D.4)

The task is now to find ξj,n for all values of n (noting that ξj,n is composed of the components ξj,n,− and ξj,n,+ ) and along withthem the quasi-energy e
εj (note that the above equations apply to both values of j). If one constructs the infinite-dimensional
vector eξj = {..., ξj,n−1,− , ξj,n−1,+ , ξj,n,− , ξj,n,+ , ξj,n+1,− , ξj,n+1,+ , . . .}, then the above set of equations governing ξj,n turns intoa single eigenvalue problemeεjeξj = HFe
ξj

(D.5)

where

.












HF = 












..


−

ε0

+ (n − 1)ω
2
∆
−
−

2
A
4

0

−
ε0
2

∆

−

2

+ (n − 1)ω
0
A
4

A

0

4

A

0

−

ε0
2

−

+ nω
∆
2
A

0

0

−

0

0

0

4

4

∆

−
2
ε0
+ nω

A
4

0

0

0

−

A

0

4

A

0

2

0

0

−

4

ε0

+ (n + 1)ω
2
∆
−
2

−
ε0
2

∆
2

+ (n + 1)ω













 . (D.6)









.. 
.

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

27

This eigenvalue problem has an infinite number of solutions, even though we started by looking for only two solutions. Thisissue is resolved by noting that this infinite number of solutions can be divided into two groups, each of which contains an
infinite number of equivalent solutions; clearly, if there is a solution to the original Floquet problem with quasi-energy eε,then essentially the same solution can be re-written such that the quasi-energy is given by e
ε + nω, with any integer n.
The eigenvalue problem derived above cannot be solved in analytic closed form, and approximations have to be made. It is
common to treat ∆ as a small parameter in a perturbation-theory-like calculation. Taking ∆ = 0, one finds the (unperturbed)
eigenvectors (Son et al., 2009)










ξk,−,n+k−1,−





ξk,−,n+k−1,+




ξ
(
0
)
e
ξk,− : k,−,n+k,−ξk,−,n+k,+





ξk,−,n+k+1,−






ξk,−,n+k+1,+





..
.
=
=
=


J1
0
J0

=

0

=

J1

=
..
.

0

A



2ω
A


0)
e
ξk(,+
:

2ω
A



2ω








ξk,+,n+k−1,−






ξk,+,n+k−1,+





ξk,+,n+k,−ξk,+,n+k,+






ξk,+,n+k+1,−






ξk,+,n+k+1,+







..
.
=

0

=

J1

=

0

=

J0

=

0

=
..
.

J1

−

−

−

A



2ω
A



2ω
A

,



2ω

with quasienergies eεk,± = ±ε0 /2 − kω (note that the above eigenvectors are normalized to unity). When two quasienergies
are almost equal (specifically e
ε0,− and eεk,+ ), and assuming that ∆/ω is small, all others are far off resonance from these two,
one can make a rotating-wave approximation and construct an effective two-level problem that involves the two respective
Floquet states. Using the relation∞X

Jn (x)Jk−n (x) = Jk (2x),

(D.7)

n=−∞

one can calculate the effective coupling strength between the two relevant states. The resulting 2 × 2 effective Hamiltonianis given by
HF ,RWA =

1



−ε0
−
∆
J
−
k (A/ω)2−∆J−k (A/ω).
ε0 − 2kω

(D.8)

Using this effective Hamiltonian, it is now straightforward to find the Floquet states and quasi-energies. The separation
between the quasienergies on resonance (i.e., when ε0 = kω) is given by |∆Jk (A/ω)| and determines the oscillation
frequency between the two states of the TLS, in clear analogy to the RWA calculation of Appendix C. One can proceed withthe calculation in order to obtain more accurate approximations to the Floquet states and quasienergies (Son et al., 2009),
but we shall not do this here.
In this Appendix we have re-derived results that we had already derived using the adiabatic-impulse model and theRWA. The use of Floquet theory in this context can therefore be seen as just an alternative approach that can be used toobtain results that can be obtained using other methods. One advantage that the Floquet approach has over the adiabaticimpulse model and the RWA is that decoherence can be introduced into the problem in the Floquet approach following astandard formalism (Grifoni and Hänggi, 1998), whereas the Bloch equations that we have used in Sec. II can be seen as aphenomenological approach to including decoherence in the theoretical analysis. The question of the effects of decoherence
on LZS interferometry in the Floquet approach is currently under study (Hausinger and Grifoni, 2010).
Appendix E. Dressed-state picture: quantized driving field
In all our discussion above, we have treated the driving field as an externally applied, classical driving signal. An
alternative theoretical approach that can be used to study the problem of a driven quantum system treats the driving field
quantum mechanically. The quantum system under study is then enlarged from simply a two-level system to a two-levelsystem and a harmonic oscillator that can have any number of excitations (i.e., photons). As in the Floquet-theory approach,a major advantage that justifies the expansion of the Hilbert space is the simplification associated with turning the timedependent Hamiltonian into a time-independent one. In fact, the algebra encountered in the Floquet-theory calculation is
identical to the one encountered in the dressed-state analysis. The difference between these two approaches is therefore
simply the physical interpretation of the mechanisms at play during the system dynamics.
The equivalence between the Floquet approach and the dressed-state approach can be seen by considering the
Hamiltonian of the dressed-state picture:
H =−

∆
2

σx −

ε0
2


A
σz + ωaĎ a − σz a + aĎ ,
2

(E.1)

28

S.N. Shevchenko et al. / Physics Reports 492 (2010) 1–30

where a and aĎ are the annihilation and creation operators for photons in the electromagnetic field. When written in matrix
form, the above Hamiltonian is given by Eq. (D.6) derived in the Floquet approach. More detailed studies of the dressed-state
picture can be found in Liu et al. (2006), Greenberg (2007), Wilson et al. (2007) and Wilson et al. (2010).
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