Dissipative dynamics of a qubit-oscillator system in the ultrastrong coupling and driving regimes [Elektronische Ressource] / vorgelegt von Johannes Hausinger

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• Physik

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• Publié par: universitat_regensburg
• Publié le: 01 janvier 2010
• Nombre de lectures: 7
• Langue: English
• Poids de l'ouvrage: 6 Mo

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Dissipative dynamics of a qubit-oscillator system
in the ultrastrong coupling and driving regimes
DISSERTATION
zur Erlangung des
DOKTORGRADES DER NATURWISSENSCHAFTEN (DR. RER. NAT.)
der Naturwissenschaftlichen Fakult¨at II - Physik
der Universita¨t Regensburg
vorgelegt von
Johannes Hausinger
aus
Offenberg
im Jahr 2010Das Promotionsgesuch wurde am 30.06.2010 eingereicht.
Das Kolloquium fand am 27.10.2010 statt.
Die Arbeit wurde von Prof. Dr. Milena Grifoni angeleitet.
Pru¨fungsausschuß:
Vorsitzender: Prof. Dr. D. Bougeard
1. Gutachter: Prof. Dr. M. Grifoni
2. Gutachter: Prof. Dr. S. Kohler
weiterer Pru¨fer: Prof. Dr. A. Scha¨ferContents
1. Introduction 7
1.1. Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2. Qubit-oscillator systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3. Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2. Analytical methods beyond the rotating-wave approximation 21
2.1. Rotating-wave approximation and Jaynes-Cummings model . . . . . . . . . . 22
2.2. Perturbation theory in g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3. Perturbation theory in Δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3. Dissipation and the quantum master equation 53
3.1. The classical Langevin equation . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2. The Caldeira-Leggett model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3. Elimination of the heat bath . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4. Spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5. The master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4. The dissipative qubit-oscillator system 65
4.1. The qubit-oscillator-bath system . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2. Dissipative dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3. Discussion of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4. Comparison with the Jaynes-Cummings model . . . . . . . . . . . . . . . . . 81
4.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5. The dissipative, driven two-level system 85
5.1. Driven quantum systems and Floquet theory . . . . . . . . . . . . . . . . . . 86
5.2. The nondissipative, driven two-level system . . . . . . . . . . . . . . . . . . . 92
5.3. The dissipative system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6. The driven qubit coupled to an oscillator 115
6.1. Dressed Floquet states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2. Quasienergy spectrum for finite Δ . . . . . . . . . . . . . . . . . . . . . . . . 118
6.3. Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7. Summary and open questions 127A.Van Vleck perturbation theory 131
A.1. Perturbation in g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.2. Perturbation in Δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
B.Dynamics 137
B.1. Dynamics of the qubit-oscillator system . . . . . . . . . . . . . . . . . . . . . 137
B.2. Dynamics of the driven TLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
C.Dissipative qubit-oscillator system 143
C.1. Oscillator matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
C.2. Rate coefficients for the off-diagonal density matrix elements . . . . . . . . . 144
C.3. Diagonal reduced density matrix elements . . . . . . . . . . . . . . . . . . . . 145
References 147
Nomenclature 1531 Introduction
Most phenomena of our daily life seem to be governed completely by classical physics. The
trajectory of an apple falling from a tree is fairly well described by Newtonian physics. Even
events involving much smaller objects, like theBrownian motion of a small particle in a drop
of water, obey classical laws [1]. Similar to the theory of general and special relativity, which
on the first glimpse seems to play a role only in the realms of free space and at velocities
beyond human capabilities, quantum mechanics might yield the impression to be of impor-
tanceonlyintheregimeofmicroscopicscales. Surely,weallhaveaccepted thatboththeories
play a non-negligible role in our modern world. Let us just think, for instance, of the Global
Positioning System,whichneedsindeedto take intoaccount relativistic effects ontheatomic
clocks in its satellites in order to accurately determine a position on earth [2]. Also quantum
mechanics entered our daily life through various technical gadgets. For example, the laser
finds application in many different devices; transistors play an important role in modern
computers; ferromagnetism makes storage of information possible; the photoelectric effect
is used for energy production, etc. All these applications are based on quantum mechanical
effects. Nevertheless, they seem to take place only on a level which is not perceptible to our
human eye, not having a direct influence on our macroscopic world. One of these effects is
the superposition of physical states. As a student one encounters usually at the beginning
of a lecture on quantum mechanics the famous gedankenexperiment of Schro¨dinger’s cat [3].
In the beginning it is a very puzzling idea that the condition (or state if one wishes) of a cat,
which is hidden in a box from the observer’s eyes and subject to a lethal device obeying the
laws of radioactive decay and thus of quantum mechanics, is not well defined, but - said in a
sloppy way - rather a superposition of life and dead. The idea looses a big deal of its fasci-
nation and seems to be of more philosophical nature, when we take into account that under
observation, i.e., by opening the box and examining the cat, its state becomes determinate.
It seems that we have no means to find out how the cat’s state was before the measurement.
For microscopic objects the superposition principle has been confirmed in various cases, see,
e.g., the interference experiments on electrons by Jo¨nsson [4] and Tonomura et al. [5]. Quite
naturally the question arises if the superposition principle and other quantum mechanical
laws can be extended into macroscopic realms. This issue was put forward by Leggett in
the early 80s [6, 7]. At this point it is important to define what is meant by “macroscopic
quantum effects”. Let us consider for example the Josephson effect [8]: even with no voltage
applied aresistiveless currentismeasured through two superconductingelectrodes which are
separated by a thin oxide layer. This is a macroscopic effect in the sense that the current
consists of many electronic degrees of freedom. It is a quantum effect, as the current results
from the tunneling of pairs of electrons, so-called Cooper-pairs, whose wavefunction consists
of a superposition of pairs being localized on either side of the barrier. However, the effect
relies not on the superposition of two macroscopic degrees of freedom like in the example of
Schro¨dinger’s cat. Rather it can be seen as collective behavior of many microscopic degrees8| 1. Introduction
of freedom experiencing quantum effects on the microscopic scale.
To test quantum mechanics acting on macroscopic degrees of freedom, Leggett suggested
to investigate the effects of “macroscopic quantum tunneling” (MQT) and “macroscopic
quantum coherence” (MQC) in a superconducting ring interrupted by a Josephson junction,
a so-called “radio frequency superconducting quantum interference device” (rf-SQUID). In
this system, the phase difference across the Josephson junction or the corresponding mag-
1netic flux through the ring can be seen as macroscopic parameter. For a high enough
self-inductance of the loop, it behaves like a particle being trapped in one of the minima of
a double-well potential. Under certain conditions tunneling out of the well (MQT) and even
coherent quantum oscillations (MQC) between the two minima were predicted theoretically.
The main obstacles to observe these effects consist in thermal escape from the well (a too
high temperature also makes a distinction of the separate quantum levels impossible) and
coupling of the macroscopic degree of freedom to microscopic ones, which act on the system
of interest like a constant measurement and thus destroy quantum coherence. When Leggett
wrote his articles, at least the first problem seemed to be feasible, owed to big achievements
in cooling techniques. Concerning the second one, superconducting devices seemed to be
most promising candidates being less sensitive to dissipative effects. Caldeira and Leggett
[11] could theoretically show that dissipation leads merely to a reduction in the tunneling
rate, and indeed MQT could already be experimentally realized soon after Leggett’s pro-
posal (see, e.g., [1, 12] and references therein). However, as pointed out in [6], MQT is not a
sufficient proof for the superposition of macroscopic states, becau