Research ArticleHamiltonian Dynamics and Adiabatic Invariants forTime-Dependent Superconducting Qubit-Oscillators andResonators in Quantum Computing Systems

Jeong Ryeol Choi

Department of Radiologic Technology, Daegu Health College, Buk-gu, Daegu 702-722, Republic of Korea

Correspondence should be addressed to Jeong Ryeol Choi; choiardor@hanmail.net

Received 19 July 2015; Accepted 5 October 2015

Academic Editor: Boris G. Konopelchenko

Copyright © 2015 Jeong Ryeol Choi. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An adiabatic invariant, which is a conserved quantity, is useful for studying quantum and classical properties of dynamicalsystems. Adiabatic invariants for time-dependent superconducting qubit-oscillator systems and resonators are investigated usingthe Liouville-von Neumann equation. At first, we derive an invariant for a simple superconducting qubit-oscillator through theintroduction of its reduced Hamiltonian. Afterwards, an adiabatic invariant for a nanomechanical resonator linearly interfacedwith a superconducting circuit, via a coupling with a time-dependent strength, is evaluated using the technique of unitarytransformation. The accuracy of conservation for such invariant quantities is represented in detail. Based on the results of ourdevelopments in this paper, perturbation theory is applicable to the research of quantum characteristics of more complicated qubitsystems that are described by a time-dependent Hamiltonian involving nonlinear terms.

1. Introduction

Interactions of light wave with matter are ubiquitous andhave many applications in modern science and technology.An interesting research topic in this line is cavity quan-tum electrodynamics (cavity QED) which is relevant to theinteraction of an atom with light waves inside a cavity.Meanwhile, it has been shown that the role of the cavitycan be played by a superconducting device or others suchas a nanomechanical oscillator or a transmission line [1–3].Superconducting devices exhibit quantum interference andthe electrons in them have the ability to act like qubits that arenecessary in quantum computing systems. We mainly con-centrate on these devices in this research. It is demonstratedfrom corresponding experiments that entanglement and theRabi oscillations appear in a strongly coupled atom-cavitysystem as a result of quantum coherence between light andmatter [3]. These features of spatial correlations enable us todo specific tasks regarding quantum information techniquessuch as quantum computing [4].

Recently, the study of superconducting qubit-oscillatorsin quantum computing systems with time-dependent

parameters attracted considerable interest in the communityof physics and information science [5–9]. In general, ahigh ability of computation with a tremendous speed canbe achieved for particular mathematical problems fromquantum computation on the basis of new paradigmsknown as superposition and entanglement. This is the mostdistinguishing feature of quantum computers, which cannotbe mimicked by classical ones. The realization of producinguniversal quantum gates that fulfill quantum computationsmay require the use of geometric phase shifts of qubit systemsalong with the changes of suitable loops in the Hamiltonianunder efficient quantum control. This can be performedthrough the use of superconducting nanocircuits [10].

Meanwhile, obtaining conserved quantities, that is, adi-abatic invariants, is of importance in the research of thedynamics of quantum computing systems, because not onlyare they helpful for deepening the understanding of thesystem but also they can be used for studying the quantumand classical properties of the system itself. If the timederivative of a quantity results in zero, this quantity is akind of exact invariant. On the other hand, an adiabaticinvariant is a quantity in which its time derivative gives

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015, Article ID 120573, 5 pageshttp://dx.doi.org/10.1155/2015/120573

2 Advances in Mathematical Physics

nearly zero under a slow variation of the Hamiltonianwith time. The first proposal of adiabatic invariants wassprung by a pioneering work of Burgers [11], which is fora particular class of Hamiltonian systems. Soon after, manyother researches concerning adiabatic invariants were carriedout via the introduction of the Hamiltonian of a givensystem [12–18]. Adiabatic invariants for a linear harmonicoscillator have been studied by Dykhne in connection withquantum transitions [15]. Quantum algorithms for adiabaticquantum computation, which are robust to decoherenceand not affected by certain noises, were developed [19]. Ahybrid quantum simulationmodel composed of adiabaticallycontrolled simulation registers connected to a qubit wasproposed as well [20].

Wewill find adiabatic invariants for theHamiltonians thatdescribe the mechanics of qubit-oscillators and resonatorsin quantum computing systems that are characterized bytime-varying parameters. Concerning this, the identificationof adiabatic invariants for a time-dependent Hamiltoniansystem with complicated time-variable parameters may bea significant challenge. We will first address the invariantof a simple superconducting qubit-oscillator. Afterwards, anadiabatic invariant for a nanomechanical resonator coupledto a superconducting resonator, which is described by acomplicated Hamiltonian involving a cross term, will beinvestigated. The Liouville-von Neumann equation will beused in order to derive conservative quantities for suchdynamical systems.

2. A Simple Superconducting Qubit-Oscillator

Let us consider a simple superconducting qubit-oscillator ofwhich the effective reducedHamiltonian is given by Equation(2.2) of [21]. Typically, the two lowest energy levels ofquantized states in the anharmonic LC resonator are regardedas a set of a qubit. In this case, the system is described by theHamiltonian of the form [21]

̂

𝐻

𝐴= ℏ𝜔(𝑎

†𝑎 +

1

2

) +

√

ℏ𝛿 (𝑡) (𝑎 + 𝑎

†) , (1)

where 𝛿(𝑡) is the time-dependent coupling strength. Here, 𝑎is the annihilation operator of the simple harmonic oscillator:

𝑎 =√

𝐿𝜔

2ℏ

𝑞 +

𝑖̂𝑝

√2ℏ𝐿𝜔

,(2)

with ̂𝑝 = −𝑖ℏ𝜕/𝜕𝑞, where 𝐿 is the inductance of the LCresonator. According to the usual convention, 𝑎† is Hermitianadjoint of 𝑎, and this rule will also be applied to other ladderoperators that will be introduced later. In many cases ofprevious researches, the factor 1/2 given in (1) is neglected forsimplicity. Using (2) and its Hermitian adjoint, we can rewritethe Hamiltonian to be

̂

𝐻

𝐴=

̂𝑝

2

2𝐿

+

1

2

𝐿𝜔

2𝑞

2+√2𝐿𝜔𝛿 (𝑡) 𝑞.

(3)

The classical equation of motion for this system is given by

𝑑

2𝑞

𝑑𝑡

2+ 𝜔

2𝑞 = −

√

2𝜔

𝐿

𝛿 (𝑡) .(4)

Although an (adiabatic) invariant is obtained under theassumption that its total change during a physical processwith time-variable parameters is zero, it is represented interms of 𝑡. A quantum (adiabatic) invariant is obtained fromthe following Liouville-von Neumann equation:

𝑑

̂

𝐼

𝐴

𝑑𝑡

=

𝜕

̂

𝐼

𝐴

𝜕𝑡

+

[

̂

𝐼

𝐴,

̂

𝐻

𝐴]

𝑖ℏ

= 0.

(5)

From a straightforward evaluation with this equation using(3), we have

̂

𝐼

𝐴=

[̂𝑝 − 𝑃

𝑝 (𝑡)]

2

2𝐿

+

1

2

𝐿𝜔

2[𝑞 − 𝑄

𝑝 (𝑡)]

2

,

(6)

where 𝑄𝑝(𝑡) is a particular solution of (4) and 𝑃

𝑝(𝑡) =

𝐿𝑑𝑄

𝑝(𝑡)/𝑑𝑡. Most previous research findings for adiabatic

invariants, such as the research based on the method ofNoether’s theory [22], were performed under the suppositionof the slow change of time-variables; however, the derivationof the invariant given in (6) does not require such a conditionas you can see. This invariant in fact does not vary with time.It is an exact constant ofmotion for the given system, even forthe case that the parameter 𝛿(𝑡) undergoes fast change withtime.

3. A Nanoresonator Coupled toa Superconducting Resonator

The theorem of adiabatic invariants originally developed forslowly time-varying systems in one-dimension may also beextended to more complicated cases. Now let us consider ananomechanical resonator interfacedwith a superconductingcircuit in a linear way. In this case, the Hamiltonian isrepresented as (see Equation (2) of [23] and Equation (31)of [24])

̂

𝐻

𝐵= ℏ𝜔(𝑎

†

1𝑎

1+

1

2

) + ℏΩ(𝑎

†

2𝑎

2+

1

2

)

+ ℏ𝜆 (𝑡) (𝑎1+ 𝑎

†

1) (𝑎

2+ 𝑎

†

2) ,

(7)

where 𝑎1and 𝑎

2are annihilation operators for the nanores-

onator and stripline, respectively, and 𝜆(𝑡) is the time-dependent coupling strength between the two oscillators. Wecan represent the two annihilation operators to be

𝑎

1=√

𝐿

𝑛𝜔

2ℏ

𝑞

1+

𝑖̂𝑝

1

√2ℏ𝐿

𝑛𝜔

,

𝑎

2=√

𝐿

𝑠Ω

2ℏ

𝑞

2+

𝑖̂𝑝

2

√2ℏ𝐿

𝑠Ω

,

(8)

with ̂𝑝𝑗= −𝑖ℏ𝜕/𝜕𝑞

𝑗, where 𝐿

𝑛and 𝐿

𝑠are inductances for the

nanomechanical resonator and the superconducting circuit,respectively. In terms of canonical variables, we can rewritethe Hamiltonian as

̂

𝐻

𝐵=

̂𝑝

2

1

2𝐿

𝑛

+

̂𝑝

2

2

2𝐿

𝑠

+

1

2

𝐿

𝑛𝜔

2𝑞

2

1+

1

2

𝐿

𝑠Ω

2𝑞

2

2

+ 2𝜆 (𝑡)√𝐿𝑛𝐿

𝑠𝜔Ω𝑞

1𝑞

2.

(9)

Advances in Mathematical Physics 3

If we introduce two linearly independent real classicalsolutions, 𝜌

𝑗,I(𝑡) and 𝜌𝑗,II(𝑡), of the following equations [14]

𝑑

2𝜌

𝑗

𝑑𝑡

2+ �̃�

2

𝑗(𝑡) 𝜌𝑗

= 0,

(10)

for each 𝑗, where 𝑗 = 1, 2, and

�̃�

2

1(𝑡) = 𝜔

2cos2𝜑 (𝑡) + Ω2sin2𝜑 (𝑡)

− 4𝜆 (𝑡)

√𝜔Ω cos𝜑 (𝑡) sin𝜑 (𝑡) ,

�̃�

2

2(𝑡) = 𝜔

2sin2𝜑 (𝑡) + Ω2cos2𝜑 (𝑡)

+ 4𝜆 (𝑡)

√𝜔Ω cos𝜑 (𝑡) sin𝜑 (𝑡) ,

𝜑 (𝑡) =

1

2

tan−1 (4𝜆 (𝑡)√𝜔Ω

Ω

2− 𝜔

2) ,

(11)

we can define the Wronskians in the form

𝑊

𝑗= 𝜌

𝑗,I (𝑡) ̇𝜌𝑗,II (𝑡) − ̇𝜌𝑗,I (𝑡) 𝜌𝑗,II (𝑡) , (12)

which are time-constants. The quantities 𝑊𝑗defined in this

way are necessary for evaluating the adiabatic invariant of thesystem. Hence, they will be used in the development of ourtheory for the adiabatic invariant later.

Let us start from the Hamiltonian given in (9) in order toderive an adiabatic invariant of the system.This Hamiltonianis somewhat complicated because it is not only dependent ontime, but also described in terms of a cross term 𝑞

1𝑞

2. Hence,

to simplify the problem, we will transform the Hamiltonianto a simple form using the unitary transformation method.To do so, let us introduce the following unitary operator:

̂

𝑈

𝐵= exp[ 𝑖

ℏ

(̂𝑝

1𝑞

1+ 𝑞

1̂𝑝

1) ln(

𝐿

𝑛

𝐿

𝑠

)

1/8

]

⋅ exp[ 𝑖ℏ

(̂𝑝

2𝑞

2+ 𝑞

2̂𝑝

2) ln(

𝐿

𝑠

𝐿

𝑛

)

1/8

]

⋅ exp [−𝑖𝜑 (𝑡)

ℏ

(̂𝑝

1𝑞

2−

̂𝑝

2𝑞

1)] .

(13)

Then, the Hamiltonian can be transformed using the relation

̂

𝐻

𝐵=

̂

𝑈

−1

𝐵̂

𝐻

𝐵̂

𝑈

𝐵− 𝑖ℏ

̂

𝑈

−1

𝐵

𝜕

̂

𝑈

𝐵

𝜕𝑡

.(14)

Through a little evaluation using fundamental properties ofquantum operators, we derive the transformed Hamiltonianin the form

̂

𝐻

𝐵=

̂𝑝

2

1+

̂𝑝

2

2

2√𝐿

𝑛𝐿

𝑠

+

1

2

√𝐿

𝑛𝐿�̃�

2

1(𝑡) 𝑞

2

1+

1

2

√𝐿

𝑛𝐿�̃�

2

2(𝑡) 𝑞

2

2

+ 𝛽 (𝑡)

̇

𝜆 (𝑡) (̂𝑝

1𝑞

2−

̂𝑝

2𝑞

1) ,

(15)

where

𝛽 (𝑡) = −

2√𝜔Ω

Ω

2− 𝜔

2+ 16𝜔Ω𝜆

2(𝑡) / (Ω

2− 𝜔

2)

. (16)

If we consider the adiabatic condition ̇𝜆(𝑡) ≃ 0, thisHamiltonian reduces to

̂

𝐻

𝐵≃

̂𝑝

2

1+

̂𝑝

2

2

2√𝐿

𝑛𝐿

𝑠

+

1

2

√𝐿

𝑛𝐿

𝑠�̃�

2

1(𝑡) 𝑞

2

1

+

1

2

√𝐿

𝑛𝐿

𝑠�̃�

2

2(𝑡) 𝑞

2

2.

(17)

This decoupled Hamiltonian is much more simple than theoriginal one given in (9). We will use this Hamiltonian in thesubsequent development of our quantum theory, instead of(15), for convenience.

Similar to the previous case, the invariant operator in thetransformed system should satisfy the relation

𝑑

̂

𝐼

𝐵

𝑑𝑡

=

𝜕

̂

𝐼

𝐵

𝜕𝑡

+

[

̂

𝐼

𝐵,

̂

𝐻

𝐵]

𝑖ℏ

= 0.

(18)

The direct evaluation of the invariant operator using thisequation with (17) leads to

̂

𝐼

𝐵=

√𝐿

𝑛𝐿

𝑠

2

⋅ ∑

𝑗=1,2

[

𝑊

2

𝑗

𝜌

2

𝑗(𝑡)

𝑞

2

𝑗+ ( ̇𝜌

𝑗 (𝑡) 𝑞𝑗

−

𝜌

𝑗 (𝑡)

√𝐿

𝑛𝐿

𝑠

̂𝑝

𝑗)

2

] ,

(19)

where

𝜌

𝑗 (𝑡) = [𝜌

2

𝑗,I (𝑡) + 𝜌2

𝑗,II (𝑡)]1/2

.(20)

From the inverse transformation of (19) using ̂𝐼𝐵=

̂

𝑈

𝐵̂

𝐼

𝐵̂

𝑈

−1

𝐵,

we can obtain the adiabatic invariant in the original system as

̂

𝐼

𝐵=

√𝐿

𝑛𝐿

𝑠

2

⋅ ∑

𝑗=1,2

[

𝑊

2

𝑗

𝜌

2

𝑗(𝑡)

̂

𝑄

2

𝑗+ ( ̇𝜌

𝑗 (𝑡)

̂

𝑄

𝑗−

𝜌

𝑗 (𝑡)

√𝐿

𝑛𝐿

𝑠

̂

𝑃

𝑗)

2

] ,

(21)

where

̂

𝑄

1= (

𝐿

𝑛

𝐿

𝑠

)

1/4

cos𝜑 (𝑡) 𝑞1 − (𝐿

𝑠

𝐿

𝑛

)

1/4

sin𝜑 (𝑡) 𝑞2,

̂

𝑄

2= (

𝐿

𝑛

𝐿

𝑠

)

1/4

sin𝜑 (𝑡) 𝑞1 + (𝐿

𝑠

𝐿

𝑛

)

1/4

cos𝜑 (𝑡) 𝑞2,

̂

𝑃

1= (

𝐿

𝑠

𝐿

𝑛

)

1/4

cos𝜑 (𝑡) ̂𝑝1 − (𝐿

𝑛

𝐿

𝑠

)

1/4

sin𝜑 (𝑡) ̂𝑝2,

̂

𝑃

2= (

𝐿

𝑠

𝐿

𝑛

)

1/4

sin𝜑 (𝑡) ̂𝑝1 + (𝐿

𝑛

𝐿

𝑠

)

1/4

cos𝜑 (𝑡) ̂𝑝2.

(22)

Thus, the adiabatic invariant of the system is completelyidentified now. Because we have used the condition ̇𝜆(𝑡) ≃ 0in the derivation of the invariant quantity, (21) is valid onlywhen 𝜆(𝑡) varies sufficiently slowly. Hence, the accuracy ofconservation for ̂𝐼

𝐵is determined by the degree of time-

variance of 𝜆(𝑡). Such accuracy for a conserved quantity is oftheoretical interest in the general physics literature [16–18].

4 Advances in Mathematical Physics

4. Conclusion

Adiabatic invariants for superconducting qubit-oscillatorsystems and nanomechanical resonators have been inves-tigated. An exact invariant for a simple superconductingqubit-oscillator was evaluated. An adiabatic invariant for ananoresonator coupled to a superconducting resonator wasalso obtained as shown in (21). The invariant ̂𝐼

𝐵for the latter

case is more complicated than the former ̂𝐼𝐴. Notice that ̂𝐼

𝐵

was derived under the assumption that the time-dependentparameter 𝜆(𝑡) varies sufficiently slowly, while the derivationof ̂𝐼𝐴required no such assumption. The concept of adiabatic

invariants is important in both quantum and classical pointsof view [25, 26], because we can study the characteristicsof the system by making use of them. In particular, theseinvariants enable us to derive Schrödinger solutions of thedynamical systems [13].

For a more complicated system, the evaluation of adia-batic invariants may be not an easy task. We do not knowhow to derive adiabatic invariants for the case that theHamiltonian involves higher order terms. For such example,let us consider the Hamiltonian:

̂

𝐻

𝐶= ℏ𝜔(𝑎

†

1𝑎

1+

1

2

) + ℏΩ(𝑎

†

2𝑎

2+

1

2

)

+ ℏ

3/2𝑔 (𝑡) 𝑎

†

2𝑎

2(𝑎

1+ 𝑎

†

1) .

(23)

The last term in this equation represents a nonlinear inter-action which typically appears in a mechanical resonatorcoupled on superconducting stripline resonator [23]. In thiscase, it is difficult to know the adiabatic invariant due tothe nonlinear term, and hence, to our knowledge, the use ofperturbation theory is the best policy for studying quantummechanical properties of this system. For a more detailedillustration for this, let us rewrite (23) as

̂

𝐻

𝐶=

̂

𝐻

𝐴+ 𝑔 (𝑡)

√

𝐿

𝑛𝜔

2

(𝐿

𝑠Ω𝑞

2

2𝑞

1+

1

𝐿

𝑠Ω

̂𝑝

2

2𝑞

1) , (24)

where

̂

𝐻

𝐴=

̂𝑝

2

1

2𝐿

𝑛

+

̂𝑝

2

2

2𝐿

𝑠

+

1

2

𝐿

𝑛𝜔

2𝑞

2

1+

1

2

𝐿

𝑠Ω

2𝑞

2

2

− ℏ𝑔 (𝑡)

√

𝐿

𝑛𝜔

2

𝑞

1.

(25)

By applying the method of previous development given fora simple superconducting resonator cavity, we can easilyderive the adiabatic invariant of ̂𝐻

𝐴. If it is possible to derive

quantum solutions of the system described by ̂𝐻𝐴, we can use

perturbation theory in order to derive the quantum solutionsof ̂𝐻𝐶after regarding the last term in (24) as the perturbation

term [27].The accuracy of conservation for invariant quantities is

crucial for securing the validity of the associated analysesof classical and quantum mechanical characteristics of thesystem fulfilled on the basis of such invariants [16]. In thisresearch, ̂𝐼

𝐴given in (6) is an exact invariant while ̂𝐼

𝐵given

in (21) is an adiabatic invariant valid under the condition that𝜆(𝑡) is a slowly varying function. Both invariants are useful forinvestigating the classical and the quantum properties of eachrespective system. As a further task in the future, it may be agood subject to investigate quantum mechanical propertiesof superconducting qubit-oscillator systems employing thetheory of invariants developed in this work.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This research was supported by the Basic Science ResearchProgram of the year 2015 through the National ResearchFoundation of Korea (NRF) funded by the Ministry ofEducation (Grant no. NRF-2013R1A1A2062907).

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