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Generalized Bloch-Siegert shift in an artificial
trapped ion
M. Sc. Thesis
Iivari Pietikainen
University of Oulu
Department of Physics
Theoretical Physics
2014
Abstract
The purpose of this thesis is to study a two-level system that iscoupled non-linearly to a harmonic oscillator outside the Lamb-Dickeregime. These kind of non-linear couplings can be found betweenthe vibrational and electronic degrees of freedom in a laser-irradiatedtrapped ions. In this work we study a superconducting circuit withsimilar properties. The circuit consist of LC-circuit that is coupledto a single-Cooper-pair transistor. The single-Cooper-pair transistorbehaves as an artificial ion and the LC-circuit behaves as the electricpotential that is used to trap the ion. When the coupling is weak theinteraction can be approximated to be linear. The differences betweenthe absorption spectrum of the linear approximation and the wholesystem is known as generalized Bloch-Siegert shift. In this thesis wewill study the Bloch-Siegert shift as the coupling strength is increased.The non-linear coupling terms cause resonance shifts and additionalresonance in the absorption spectrum.
Contents
1 Introduction 3
2 Ion traps 4
2.1 Paul traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Superconducting electric circuits 9
3.1 Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . 103.1.1 Cooper pairs . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Josephson effect . . . . . . . . . . . . . . . . . . . . . 12
3.2 Quantum network theory . . . . . . . . . . . . . . . . . . . . 133.2.1 LC-circuit . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Single-Cooper-pair transistor . . . . . . . . . . . . . . 163.2.3 SCPT coupled to a LC-circuit . . . . . . . . . . . . . . 20
3.3 Superconducting circuit compared to ion trap . . . . . . . . . 23
4 Trapped ion Hamiltonian 23
4.1 Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . 244.2 Conventional Bloch-Siegert shift . . . . . . . . . . . . . . . . 264.3 Absorption spectrum . . . . . . . . . . . . . . . . . . . . . . . 27
4.3.1 Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . 274.4 Lamb-Dicke regime . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Calculations and Results 31
5.1 Correction of the d term . . . . . . . . . . . . . . . . . . . . . 355.1.1 Jaynes-Cummings model . . . . . . . . . . . . . . . . . 395.1.2 Bloch-Siegert correction . . . . . . . . . . . . . . . . . 40
5.2 Higher avoided crossings . . . . . . . . . . . . . . . . . . . . . 415.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6 Conclusion 47
2
1 Introduction
The first single-particle ion trap experiment was done in 1973 by D.Wineland,P. Ekstrom and H. Dehmelt [1]. Since then the ion traps have developedgreatly thanks to the improvement of laser technology allowing spectrallynarrower light sources to be used as the trapping fields. The ion trap exper-iments have provided important contributions in many fields of physics.
Trapping ions with electromagnetic fields causes the ionic motion to en-tangle with the internal electronic structure of the ion. The ion’s motionalstates behave as a harmonic oscillator and the internal electronic struc-ture can be thought as a two-level system. The interaction between thevibrational and electronic degrees of freedom is non-linear and generally notanalytically solvable.
If we assume that the ratio between the zero point motion of the ion andthe wave length of the electric field is small the interaction can be approxi-mated to be linear. This ratio is known as the Lamb-Dicke parameter. Thelinear models that we use are analytically solvable.
The linear models are accurate when the Lamb-Dicke parameter is small.There has been experiments done with strong coupling showing that in thecase of larger values of the Lamb-Dicke parameter the non-linear terms startto have a noticeable effect on the system and the linear approximations donot hold anymore [2]. In this work we look how accurate the approximationsare when the Lamb-Dicke parameter is increased.
Similar coupling can be achieved in superconducting circuits. By cou-pling an LC-circuit with a single-Cooper-pair transistor we can have a systemwith similar behavior as in the case of the trapped ion. This superconductingcircuit is what we will study in this thesis.
In chapter two we study our natural system which is the ion trap. Wederive the Hamiltonian for a specific ion trap, the Paul trap.
In chapter three we move to the artificial systems which in this caseare superconducting circuits. We introduce some important properties ofsuperconductivity. After that we introduce the quantum network theorythat is needed for the calculation of the quantum effects of the circuits.Then we look at some of the components used in our circuit and finally wederive the Hamiltonian for the circuit that is studied on this thesis.
In chapter four we concentrate on the Hamiltonian of our circuit. Westart by deriving the linear approximations of the interaction. Next weintroduce a weak probe to our system. This allows us to measure the ab-sorption spectrum of the system. We derive the equation for the absorptionspectrum.
Next we will do some calculations in chapter five. First we do calcula-tions on how well the approximations hold on different oscillator frequenciesand on different Lamb-Dicke parameter values. The differences betweenthe numerical calculations and the Jaynes-Cummings model are known as
3
the generalized Bloch-Siegert shift. Then we derive corrections to the ap-proximations that take account the asymmetry term that we have in theHamiltonian. Lastly we study the other avoided crossings and when theybecome visible in the absorption spectrum.
2 Ion traps
An ion trap is a device that can be used to trap a charged particle into aconfined region of space with electromagnetic fields. There are various typesof ion traps depending on the kind of electromagnetic fields that are used fortrapping. Most popular types are Penning trap that uses static electric andmagnetic fields and Paul trap that uses oscillating time-dependent electricfield.
Ion traps have several uses. They are used as mass spectrometers todetermine the mass to charge ratio of ions. This works by making the ionspass through the trap. The trap catches only ions with a particular mass tocharge ratio.
Ion traps can also be used for cooling particles. In the ion trap themotion of the ion is coupled with its internal electronic structure. With theright frequency of the electromagnetic field, this coupling can be used todecrease the kinetic energy of the ion.
One more possible use for ion traps is a quantum computer. If thefrequency of the electromagnetic field inducing the coupling is low enoughthe system does not have enough energy to excite the internal states of theion beyond the two lowest states. This means that the internal structure canbe thought as a two-level system. A quantum two-level system is called aquantum bit or a qubit. Quantum computers use qubits for the calculationinstead of classical bits as in classical computers. Systems consisting of afew trapped ions entangled together has been made [3]. If this could be donewith more ions it could be possible to build a quantum computer.
Even if we could not build quantum computers we could use the ion trapsto build simpler systems to be used for quantum simulations. Simulatingquantum systems using classical computers is complex. If we had a quantumsystem where we could control the interactions between the qubits we coulduse this system to do simulations of quantum systems that could not besimulated by classical computers. [4, 5, 6]
2.1 Paul traps
We shall now further study one of the ion traps, the Paul trap [4]. Wewill give the derivation of the Hamiltonian generated by this trap. Paultrap was developed by Wolfgang Paul in 1990 [7]. The trap uses a time-dependent field to confine the particle. The field is typically in the range ofradio-frequency (3 kHz to 300 GHz).
4
The electric potential of a Paul trap has a quadrupolar shape. Letus assume that the potential can be separated into a sinusoidally oscillat-ing time-dependent part and a time-independent part. If the potential isquadratic in all Cartesian coordinates the problem can be reduced to threeone-dimensional problems. The potential given by the electric field is
V (t) =m
8ω2rf [a+ 2q cos(ωrf t)]x
2 (1)
where m is the mass of the trapped particle, ωrf is the drive frequencyof the time-dependent field and a and q are constants that depend on theform of the trapping field and the charge of the trapped particle. The totalHamiltonian is
H(m) =p2
2m+m
8ω2rf [a+ 2q cos(ωrf t)]x
2 . (2)
In Heisenberg picture
˙x =1
i~[x, H(m)] =
p
m
˙p =1
i~[p, H(m)] = −m
4ω2rf [a+ 2q cos(ωrf t)]x .
(3)
Combining these gives
¨x+ω2rf
4[a+ 2q cos(ωrf t)]x = 0 . (4)
Further more by defining ξ = ωrf t/2 the equation is of the standard formof the Mathieu equation
∂2x
∂ξ2+ [a+ 2q cos(2ξ)]x = 0 . (5)
Replacing operator x with function u(t) and choosing boundary conditions
u(0) = 1 , u(0) = iν ,
we have the solution for the Mathieu equation as
u(t) = eiβωrf t/2∞∑
n=−∞
C2neinωrf t (6)
where β is a constant that depends on the coefficients a and q.Now we can construct an operator
C(t) =
√
m
2~νi(u(t) ˙x(t)− u(t)x(t)) (7)
5
that is in fact time-independent and of the same form as the annihilationoperator for harmonic oscillator of mass m and frequency ν
C(t) = C(0) =1√
2m~ν(mνx(0) + ip(0)) = a . (8)
The x(t) and p(t) operators can now be expressed as
x(t) =
√
~
2mν(au∗(t) + a†u(t))
p(t) =
√
~m
2ν(au∗(t) + a†u(t))
(9)
and now the time-dependence comes completely from u(t).By putting the solution (6) into the Mathieu equation (5) we get a re-
cursion relation for the coefficients C2n:
C2n−2 +D2nC2n + C2n+2 = 0 (10)
where D2n = [a− (2n+ β)2]/q. From this we obtain expressions
C2n+2 =C2n
D2n + 1D2n+2−
1...
C2n =C2n−2
D2n + 1D2n−2−
1...
.
(11)
Typically |a|, |q| ≪ 1 so we can truncate these expressions. By choosingC±4 = 0 and by using the recursion relations and the boundary conditionswe obtain
β ≈√
a+ q2/2
ν ≈ βωrf/2
u(t) ≈ eiνt1 + q
2 cos(ωrf t)
1 + q2
.
(12)
Since ν ≪ ωrf and q/2 ≪ 1 the q2 cos(ωrf t) term causes only fast, small
oscillation to the solution. This is called the micromotion. If we ignorethe micromotion and present the Hamiltonian H(m) with the creation andannihilation operators it becomes
H(m) =~ν
(
1 + q2
)2
(
1
2+ a†a
)
≈ ~ν
(
1
2+ a†a
)
. (13)
This is of the same form as the Hamiltonian of harmonic oscillator.
6
Analogous to the harmonic oscillator we now define a basis |n〉 wheren = 0, 1, 2, .... The operator a operates on these states just like the ladderoperator operates to the Fock states in harmonic oscillator
a|n〉 =√n|n− 1〉
a†|n〉 =√n+ 1|n+ 1〉
N |n〉 = n|n〉(14)
where N = a†a is the number operator.These states and the operator C are in the Heisenberg picture. We can
change them to the Schrodinger picture by
CS(t) = U(t)CU †(t)
|n, t〉 = U(t)|n〉(15)
where U(t) = exp[−(i/~)H(m)t]. The Schrodinger picture operator CS(t)acts on the states |n, t〉 exactly as in the Heisenberg picture.
The states |n, t〉 behave similarly as the Fock states of harmonic oscil-lator. It is important to note that these states are not energy eigenstatesof the system since the harmonic oscillation of the electric field constantlychanges the kinetic energy of the ion. These states are called quasistationarystates.
The coupling between the motional states and the internal structurestates of the ion is caused by an additional electric field. If the frequencyof this field is near resonance with two of the energy levels of the internalstructure of the particle and the coupling strength is much smaller than theenergy needed for the internal states to excite to a non-resonant state, wecan approximate the internal structure of the ion as a two-level system.
Let the resonance states be |+〉 and |−〉 with energy difference ~ω0 =~(ω+ − ω−). The two-level Hamiltonian is
H(e) = ~(ω+|+〉〈+| + ω−|−〉〈−|)
= ~ω+ + ω−
2(|+〉〈+ |+ |−〉〈−|) + ~
ω0
2(|+〉〈+| − |−〉〈−|).
(16)
By ignoring the constant energy term and using the Pauli matrices
σx =
(
0 11 0
)
, σy =
(
0 −ii 0
)
, σz =
(
1 00 −1
)
,
we have the two-level Hamiltonian as
H(e) = ~ω0
2σz. (17)
The total Hamiltonian is
H = H(m) + H(e) + H(i) (18)
7
where H(m) and H(e) are the particle’s motional Hamiltonian and the Hamil-tonian for the internal structure as we have discussed above. The Hamilto-nian H(i) describes the interaction between the motional and internal struc-ture states. The interaction is caused by the additional electric field. Thecoupling between electromagnetic field and charges is a complex subject andtherefore we will not go into the details here. We will just state the form ofthe interaction term as
H(i) =~
2Ω(|−〉〈+|+ |+〉〈−|)(ei(kx−ωt+φ) + e−i(kx−ωt+φ)). (19)
We can write |+〉〈−| and |−〉〈+| with the raising and lowering operators as
|+〉〈−| = σ+ =
(
0 10 0
)
, |−〉〈+| = σ− =
(
0 01 0
)
.
We will use the interaction picture with the free Hamiltonian being H0 =H(m) + H(e) and the interaction H(i). The transformation operator beingU0 = exp[−(i/~)H0t] we get the interaction into the form of
Hint = U †0H
(i)U0 =~
2Ωe(i/~)H
(e)t(σ+ + σ−)e(−i/~)H(e)te(i/~)H
(m)t
× (ei(kx−ωt+φ) + e−i(kx−ωt+φ))e(−i/~)H(m)t
=~
2Ω(σ+e
iω0t + σ−e−iω0t)
× e(i/~)H(m)t(ei(kx−ωt+φ) + e−i(kx−ωt+φ))e(−i/~)H(m)t .
(20)
The transformation of the motional part into the interaction picture is thesame as the transformation to Heisenberg picture. The Schrodinger pictureoperator x will be replaced with the Heisenberg picture operator x(t) fromequation (9)
kx(t) = η(au∗(t) + a†u(t)) (21)
where we have now defined the Lamb-Dicke parameter η = k√
~/(2mν).
Using the definition of u(t) we can expand the exponent terms in Hint. Forthe first one we get
exp[i(η(au∗(t) + a†u(t)) + (ω0 − ω)t+ φ)]
= ei((ω0−ω)t+φ)
∞∑
m=0
(iη)m
m!
(
ae−iβωrf t∞∑
n=−∞
C∗2ne
−inωrf t
+ a†eiβωrf t∞∑
n=−∞
C2neinωrf t
)m
.
(22)
8
We see that when ω0 − ω = (l′ − lβ)ωrf the a term will be varying slowlyin time and similarly for the a† when ω − ω0 = (l′ − lβ)ωrf . Here weuse the rotating wave approximation and assume that the other terms arerotating so fast that they do not affect the time evolution of the system thatmuch and can be ignored. The resonance frequency can be tuned to specificcombination of l and l′. Usually η ≪ 1 meaning that the coupling strengthvanishes quickly with higher l and l′ so the usual case is to choose l′ = 0which is what we are going to do now. We have the exponent term as
exp[i(η(au∗(t) + a†u(t)) + (ω0 − ω)t+ φ)]
= exp[i(η(ae−i2νt + a†ei2νt) + (ω0 − ω)t+ φ)] .
(23)
The total interaction Hamiltonian is
Hint =~
2Ω[
σ+ exp[i(η(ae−i2νt + a†ei2νt) + (ω0 − ω)t+ φ)]
+ σ+ exp[i(−η(ae−i2νt + a†ei2νt) + (ω0 + ω)t− φ)]
+ σ− exp[i(η(ae−i2νt + a†ei2νt) + (−ω0 − ω)t+ φ)]
+ σ− exp[i(−η(ae−i2νt + a†ei2νt) + (−ω0 + ω)t− φ)]]
.
(24)
The interaction is time-dependent but in most cases the time-dependencycauses very little change so we can neglect the time-dependency. By assert-ing t = 0 into the interaction Hamiltonian we get
Hint =~
2Ω[
σ+ exp[i(η(a+ a†) + φ)] + σ+ exp[−i(η(a+ a†) + φ)]
+ σ− exp[i(η(a+ a†) + φ)] + σ− exp[−i(−η(a+ a†) + φ)]]
= ~Ωcos(
η(a† + a) + φ)
σx .
(25)
Now we have obtained the interaction part of the Hamiltonian. Now theHamiltonian is
H = ~ν
(
1
2+ a†a
)
+ ~ω0
2σz + ~Ωcos
(
η(a† + a) + φ)
σx . (26)
This is the total Hamiltonian of the Paul trap. [4, 7]
3 Superconducting electric circuits
Now we will look at the artificial system. By artificial systems we meansystems that do not occur naturally but are intentionally built. In this casethe artificial systems that we are going to study are superconducting circuits.
9
With current technology we can build superconducting circuits that aresmall enough for quantum effects to become visible. The artificially builtquantum systems are an excellent way to study quantum phenomena becausethese system often offer a better control over the relevant parameters of thesystem.
3.1 Superconductivity
Superconductivity was first discovered by Heike Kamerlingh Onnes in 1911[8]. He observed that under certain critical temperature mercury completelylost its electrical resistance. Kamerlingh Onnes observed the same phe-nomenon also in some other metals and after that various other materialsexhibiting the phenomenon has been discovered. Superconductivity requiresa very low temperature. For metals the transition temperature is usuallyless than 10 K. There exist compounds and alloys with higher transitiontemperatures. In some cases it is even over 100 K.
Another characteristic of superconductivity was discovered by WaltherMeissner and Robert Ochsenfeld in 1933 [9] when they found out that thereis no magnetic field in a superconductor. More importantly they discoveredthat a normal material will expel the magnetic field when it is cooled belowits critical temperature. This is called Meissner effect. Meissner effect alsoimplies that superconductivity can be broken by a critical magnetic field.
In 1957 John Bardeen, Leon Cooper and John Schrieffer proposed atheory explaining superconductivity [10]. This is known as the BCS theory.The theory is based on the idea of Cooper pairs. [11]
3.1.1 Cooper pairs
A Cooper pair consist of two electrons bound together, thus forming a par-ticle that behaves as a boson. The idea that there exists a weak attractiveforce that binds two electrons into a pair was first presented by Leon Cooperin 1956 [12].
Let us consider two electrons added to a Fermi sea at temperature T = 0,and that the electrons interact only with each other not with any otherelectrons. We expect that the lowest energy state of the system has zerototal momentum and that the two electrons must have equal and oppositemomenta. This leads us to a wave function of the form
ψ(r1, r2) =∑
k
eik·(r1−r2) . (27)
Next we take account the spins of the electrons. Since we know thatthe total wave function for fermions has to be antisymmetric with respectto an exchange of two particles, the total ψ must be either the prod-uct of cos (k · (r1 − r2)) and the antisymmetric spin state | ↑↓〉 − | ↓↑〉 or
10
sin (k · (r1 − r2)) and one of the symmetric spin states | ↑↑〉, | ↑↓〉 + | ↓↑〉or | ↓↓〉. Since we are expecting an attractive interaction we assume thatthe antisymmetric spin state has lower energy, because the cosine term giveslarger probability for the particles to be near each other. This means thewave function is
ψ(r1 − r2) =
[
∑
k
cos (k · (r1 − r2))
]
(| ↑↓〉 − | ↓↑〉) . (28)
Putting this into the Schrodinger equation of the system we get
(E − 2ǫk)gk =∑
k′
Vkk′gk′ (29)
where ǫk = ~2k2
2m and
Vkk′ = Ω−1
∫
V (r)ei(k′−k)·rdr (30)
where r is the distance between the electrons and Ω is the normalizationvolume.
We make an approximation that Vkk′ = −V for energies less than ~ωc
away from EF and elsewhere Vkk′ = 0. EF is the energy that both of theelectrons would have if there were no interaction between them. Now wehave
gk = −V∑
k′ gk′
E − 2ǫk. (31)
Taking summation∑
kover both sides and dividing by
∑
gk we get
1
V=∑
k
1
2ǫk − E. (32)
We can replace the summation by integration
∑
k
= N
∫ EF+~ωc
EF
dE (33)
where N denotes the density of states for the electrons. Now
1
V= N
∫ EF+~ωc
EF
dE1
2ǫk − E=N
2ln
(
2EF − E + 2~ωc
2EF − E
)
. (34)
Assuming a weak-coupling NV ≪ 1 we have
E ≈ 2EF − 2~ωce−2NV . (35)
From this we can see that the energy E for the bound electrons is smallerthan the energy for the non-interacting electrons 2EF .
11
We have seen that an attractive interaction between two electrons couldbound them forming Cooper pairs. The remaining question is where couldthis attractive force arise. We know that there is Coulomb interaction be-tween two electrons but this interaction is repulsive. The attractive interac-tion comes when we take into account the motion of the ion cores. The firstelectron attracts the positive ions and these ions attract the other electron.This gives an effective attractive interaction between the electrons. It hasbeen calculated that the attractive force created this way is about the samesize as the repulsive Coulomb force. This means that it is possible for thetotal interaction to be attractive. [11]
3.1.2 Josephson effect
A tunnel junction in electronics is a small barrier of insulating materialbetween two conducting materials. Particles can move past the insulatinglayer by means of quantum tunneling. In 1962 Brian Josephson proposedthat a superconducting tunnel junction with no voltage difference acrossit would produce a supercurrent caused by the tunneling of Cooper pairs[13]. Further on, Josephson predicted that if a non-zero voltage V wouldgo over the junction it would cause an alternating current with frequencyν = 2eV/h.
In the case of a weakly coupled tunnel junction, the current going throughthe junction is
I = I0 sin(φ) (36)
where φ is the phase difference over the insulating barrier and I0 is themaximum supercurrent. The time evolution of the phase is given by
dφ
dt=
2eV
~. (37)
From these we see that when V = 0 we get a constant current over thejunction. This is called the dc Josephson effect. For V 6= 0 the currentoscillates with frequency ν = 2eV/h. This is called the ac Josephson effect.
We can present the phase as the flux across the junction
φ =2π
Φ0Φ (38)
where Φ0 = h/2e is the flux quantum. We can get the potential energy ofthe Josephson junction by
E =
∫
I(t)V (t)dt =
∫
I0 sin
(
2π
Φ0Φ
)
Φdt = −EJ cos
(
2π
Φ0Φ
)
(39)
where we have defined EJ = I0~/(2e).The potential difference over the junction draws equal but opposite
charges on the surfaces of the insulator so the junction has also capacitive
12
Figure 1: The circuit diagram of Josephson junction.
properties. In order to take into account the capacitance of the Josephsonjunction in circuit diagrams a capacitor is added parallel to the ideal Joseph-son element. The circuit diagram of Josephson junction is shown in figure1. [11, 14]
3.2 Quantum network theory
In order to study the quantum effects of the circuit we need to introduce thequantum network theory. The theory was developed by Bernard Yurke andJohn S. Denker [15] and Michel Devoret [16]. In Quantum network theory weuse flux Φ and charge Q as the canonical coordinates. They behave like thecanonical coordinates, position q and momentum p, in classical mechanics.
An electric circuit can be described as a network whose branches areelectrical components. A branch is a two-terminal electrical element such ascapacitor or inductor. Each branch b is associated with two variables: thevoltage Vb and the current Ib. They can be defined with electromagneticfields as
Vb =
∫ end of b
beginning of bE · ds (40)
Ib =1
µ0
∮
around bB · ds. (41)
For the Hamiltonian description of circuits we introduce branch flux andbranch charge which are defined as
Φb(t) =
∫ t
−∞
Vb(t′)dt′ (42)
Qb(t) =
∫ t
−∞
Ib(t′)dt′. (43)
13
We assume that at time t = −∞ there are no voltages or currents in the cir-cuit. With these definitions we get the currents for inductors and capacitorsas
Ib = Φb/L (44)
Ib = CΦb. (45)
The energy in a branch b is
Eb(t) =
∫ t
−∞
Vb(t′)Ib(t
′)dt′. (46)
This gives the energies for inductors
Eb =Φ2b
2L(47)
and for capacitors
Eb =C ′Φ2
b
2. (48)
The electrical circuits follow the Kirchhoff’s laws which state that thesum of voltages around a loop l should be zero as long as the flux Φl throughthe loop remains constant and that the sum of currents arriving at point nshould be zero as long as the charge of the point Qn remains constant,
∑
all b around l
Φb = Φl (49)
∑
all b arriving at n
Qb = Qn. (50)
With the Kirchhoff’s laws we could formulate the equations of motion forthe system with either using the fluxes or the charges. From these we couldget the Lagrangian of the circuit using the Euler-Lagrange equation.
A more convenient way to obtain the Lagrangian is by using the analogyof electric circuits and mechanical systems. For a mechanical system theLagrangian is L = T − V . This works also for the electric circuits. Inelectric circuits the kinetic energy terms T are the terms that depend on thechargeQ. This means the capacitive components. The potential terms V arethe terms that depend on the flux Φ. These are the inductive components.[16, 17]
3.2.1 LC-circuit
Let us now consider an LC-circuit pictured in figure 2. From the Kirchhoff’slaws we get Φ1 + Φ2 = Φ. The branch fluxes Φ1 and Φ2 can be expressed
14
Figure 2: LC-circuit. C and L are the capacitance and inductance. Φ is themagnetic flux that goes through the circuit loop. Φ1 and Φ2 are the branchfluxes going through the inductor and the capacitor. Φa is the node fluxand the other node is chosen as the ground.
in terms the node fluxes. One of the nodes can be chosen as the ground bydefining its flux as zero. This leaves only one non-zero node flux Φa. Nowwe have
Φ1 = Φa
Φ2 = Φ− Φa.(51)
In this circuit the capacitor C is the only capacitive term so the kineticenergy term is
T =1
2CΦ2
2 =1
2CΦ2
a . (52)
The only potential term is the inductor L therefore the potential energy is
V =Φ21
2L=
Φ2a
2L. (53)
This means that in this case the Lagrangian of the system is
L =1
2CΦ2
a −Φ2a
2L. (54)
The node charge can be defined as
Qa =∂L∂Φa
= CΦa (55)
and the Hamiltonian of the system is
H = Φa∂L∂Φa
− L =1
2CΦ2
a +Φ2a
2L=Q2
a
2C+
Φ2a
2L. (56)
15
Now we can move to the quantum mechanical description by replacing theclassical variables with operators
Φa → Φa
Qa → Qa
H → H .
(57)
Since the canonical variables of electrical circuits Φa and Qa are analogicalwith the canonical variables of mechanical systems they satisfy
[Φa, Qa] = i~ (58)
Qa = −i~ d
dΦa
. (59)
We can also define the annihilation and creation operators as
a =
√
1
2~Z0(Φa + iZ0Qa)
a† =
√
1
2~Z0(Φa − iZ0Qa)
(60)
where Z0 =√
L/C. From these we get
Φa =
√
~Z0
2(a† + a) (61)
Qa = i
√
~
2Z0(a† − a) . (62)
Inserting these into the Hamiltonian and using the commutation relation[a, a†] = 1 we get
H = ~ω0
(
a†a+1
2
)
(63)
where ω0 =√
1/LC. This is the Hamiltonian of a harmonic oscillator withω0 being the angular frequency of the oscillator. [16, 17]
3.2.2 Single-Cooper-pair transistor
The capacitors and inductors depend linearly on charge or flux and thecircuits made out of these are linear. Since the LC-circuit behaves as aharmonic oscillator it displays rather trivial quantum effects. In order tosee non-trivial quantum effects we need a non-linear circuit element. Thesimplest non-linear circuit element is the Josephson tunnel junction.
16
Figure 3: Circuit diagram of a single-Cooper-pair transistor. Vg and Cg arethe gate voltage and capacitance respectively. Φ is the magnetic flux thatgoes through the circuit loop and Φa is the node flux. C1 and C2 are thecapacitances associated with the Josephson junctions and EJ1 and EJ2 arethe maximum energies of the Josephson junctions.
Interrupting a superconducting loop with two Josephson junctions weinsulate a small island. The charge on the island can be controlled throughgate voltage. As long as the thermal fluctuation energy is smaller than theenergy required for a Cooper pair to tunnel through the insulating layerand there is no external disturbance in resonance with other than the lowesttransition, the single-Cooper-pair transistor can be regarded as a qubit.
The potential energy for Josephson tunnel junctions is
Eb = −EJ cos
(
2πΦb
Φ0
)
(64)
where Φ0 = h/2e is the flux quantum. Electrical circuit of a single-Cooper-pair transistor is presented in figure 3. Similarly as with the LC-circuit wecan obtain the Lagrangian as the difference between kinetic and potentialenergies:
L =1
2(C1+C2+Cg)Φ
2a+EJ1 cos
(
2πΦa
Φ0
)
+EJ2 cos
(
2π(Φ− Φa)
Φ0
)
. (65)
The node charge is
Qa =∂L∂Φa
= (C1 + C2 + Cg)Φa (66)
17
and the Hamiltonian
H =Q2
a
2CΣ− EJ1 cos
(
2πΦa
Φ0
)
− EJ2 cos
(
2π(Φ− Φa)
Φ0
)
(67)
where CΣ = C1 + C2 + Cg. Changing these into operators and defining
Φ =1
2(Φ1 − Φ2) = Φa −
1
2Φ (68)
we get the Hamiltonian to a form
H =Q2
a
2CΣ− EJ0
[
cos
(
2πΦ
Φ0
)
cos
(
πΦ
Φ0
)
+ d sin
(
2πΦ
Φ0
)
sin
(
πΦ
Φ0
)]
(69)
where EJ0 = EJ1 + EJ2 and d = (EJ1 − EJ2)/EJ0.The eigenstates of the charge operator Qa are
ψQa(Φ) = eiQaΦ/~ = ei(Q+Qg)Φ/~ (70)
where Qg = CgVg is the gate charge and Q = (C1 + C2)Φ is the charge ofthe island. Only way for its charge to change is if a Cooper pair tunnelsthrough one of the Josephson junctions. This means that the charge canonly change by multiples of the charge of a Cooper pair (±2e). If we nowlook at the energy of a Josephson tunnel junction it can be written in theoperator form as
−EJ cos2πΦ
Φ0= −EJ
2(ei2eΦ/~ + e−i2eΦ/~) . (71)
We see that if we operate with Josephson operator to a charge eigenstateit gives two different charge eigenstate that differ from the original chargestate by ±2e. This means that the Josephson tunnel junction transfers oneCooper pair from one side to another.
We can now express the Hamiltonian (69) in the basis of the charge
states |n〉. The charge operator is now Q = −2en and ei2eΦ/~|n〉 = |n− 1〉.The Hamiltonian becomes
H =∞∑
n=−∞
4EC(Qg
2e− n)2|n〉〈n|
− EJ0
2
[
cos
(
πΦ
Φ0
)
+ id sin
(
πΦ
Φ0
)]
|n− 1〉〈n|
− EJ0
2
[
cos
(
πΦ
Φ0
)
− id sin
(
πΦ
Φ0
)]
|n+ 1〉〈n|.
(72)
18
−1 −0.5 0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
Qg/(2e)
Ene
rgy
[GH
z]
Figure 4: The energy states of a single-Cooper-pair transistor as a functionof Qg. The red lines are in the case where there is no coupling between thecharge states (EJ0 = 0) and the blue lines are the energy states when thereis coupling (EJ0 = 0.1 GHz). Parameters used in this figure are EC = 1GHz, φ = 0 and d = 0.
19
In figure 4 we can see the lowest energy states of the Hamiltonian (72)in the cases of no coupling between the charge states (EJ0 = 0) and weakcoupling between the charge states (EJ0/EC = 0.1). When there is nocoupling the energy states intersect at points Qg/(2e) = n + 0.5. With theweak interaction (EJ0 ≪ EC) there is a small gap between the intersectionpoints. When Qg/(2e) ≈ n + 0.5 the energy difference between the twolowest states is much smaller than the energy difference between any otherpair of states. We will note these states as |0〉 and |1〉. We can truncate theHamiltonian (72) to only contain the two lowest states. We get
H = 4EC(1−Qg
e)σz −
EJ0
2
[
cos
(
πΦ
Φ0
)
σx + d sin
(
πΦ
Φ0
)
σy
]
(73)
where the constant term has been left out.The qubit described here is a charge qubit. There are also flux [18] and
phase qubits [19]. [16, 17, 20]
3.2.3 SCPT coupled to a LC-circuit
Now we study a circuit where we have single-Cooper-pair transistor coupledto an LC-circuit. The circuit is shown in figure 5. We have the followingLagrangian
L =Cg
2Φ2a +
C1
2Φ2a +
C2
2(Φa − Φb)
2 +C
2Φ2b
− Φ2b
2L+ EJ1 cos
(
2πΦa
Φ0
)
+ EJ2 cos
(
2π(Φa − Φb − Φ)
Φ0
)
.(74)
Making a change of variables
Φ = Φb
Θ =Φ+ Φ
2− Φa
(75)
we get
L =Cg
2
(
Θ− Φ
2
)2
+C1
2
(
Θ− Φ
2
)2
+C2
2
(
Θ +Φ
2
)2
+C
2Φ2
− Φ2
2L+ EJ1 cos
(
2π(Θ− 12(Φ + Φ))
Φ0
)
+ EJ2 cos
(
2π(Θ + 12(Φ + Φ)
Φ0
)
.
(76)
20
Figure 5: The superconducting circuit of a two-level system coupled non-linearly with a harmonic oscillator. Vg and Cg are the gate voltage andcapacitance. Φ is the magnetic flux that goes through the circuit loop andΦa and Φb are the node fluxes. C1 and C2 are the capacitances associatedwith the Josephson junctions and EJ1 and EJ2 are the maximum energiesof the Josephson junctions. L and C are the inductance and capacitance ofthe LC-circuit.
21
Assuming C1 = C2 this can be written as
L =Cg
2
(
Θ− Φ
2
)2
+ C1
(
Θ2 +Φ2
4
)
+C
2Φ2 − Φ2
2L
+ EJ0
[
cos
(
2πΘ
Φ0
)
cos
(
π(Φ + Φ)
Φ0
)
+ d sin
(
2πΘ
Φ0
)
sin
(
π(Φ + Φ)
Φ0
)]
.
(77)
Now we calculate the charges corresponding to the canonical coordinates Θand Φ:
Q =∂L∂Θ
= (2C1 + Cg)Θ− Cg
2Φ
q =∂L∂Φ
=4C + 2C1 + Cg
4Φ− Cg
2Θ
(78)
and from these we get
Θ =1
4
(4C + 2C1 + Cg)Q+ 2Cgq
2CC1 + C21 + CCg + C1Cg
Φ =1
2
2(2C1 + Cg)q + CgQ
2CC1 + C21 + CCg + C1Cg
.
(79)
Now we can calculate the Hamiltonian as
H =∑
i
QiΦi − L =1
8
4C + 2C1 + Cg
2CC1 + C21 + CCg + C1Cg
Q2
+1
2
2C1 + Cg
2CC1 + C21 + CCg + C1Cg
q2 +1
2
Cg
2CC1 + C21 + CCg + C1Cg
Qq +Φ2
2L
− EJ0
[
cos
(
2πΘ
Φ0
)
cos
(
π(Φ + Φ)
Φ0
)
+ d sin
(
2πΘ
Φ0
)
sin
(
π(Φ + Φ)
Φ0
)]
.
(80)
By making the approximation C ≫ C1 ≫ Cg we get
4C + 2C1 + Cg
2CC1 + C21 + CCg + C1Cg
≈ 4C + 2C1
C1(2C + C1)≈ 2
C1
2C1 + Cg
2CC1 + C21 + CCg + C1Cg
≈ 2C1
C1(2C + C1)≈ 1
C
Cg
2CC1 + C21 + CCg + C1Cg
≈ 0
(81)
22
After the quantization, the Hamiltonian operator can be written as
H =q2
2C+
Φ2
2L+
Q2
4C1
− EJ0
[
cos
(
2πΘ
Φ0
)
cos
(
π(Φ + Φ)
Φ0
)
+ d sin
(
2πΘ
Φ0
)
sin
(
π(Φ + Φ)
Φ0
)]
.
(82)
We see that the first two terms are exactly the same as in the case of theLC-circuit. Just as in the case of the LC-circuit we can express q and Φwith annihilation and creation operators. The other terms are the same aswith the single-Cooper-pair transistor. Treating the operators Q and Θ aswith the SCPT-circuit we get the Hamiltonian into the form
H =~√LC
(
a†a+1
2
)
+ 4EC
(
1− Qg
e
)
σz
− EJ0
2
[
cos
√
πe2
h
√
L
C(a† + a) +
πΦ
Φ0
σx
+ d sin
√
πe2
h
√
L
C(a† + a) +
πΦ
Φ0
σy
]
.
(83)
This is similar to the Hamiltonian of the Paul trap. The main exception isthat this Hamiltonian contains the σy term. This term is due to the factthat the Josephson junctions have slightly different energies. The factor dthat is the relative difference between the energies is usually relatively smallbut not so small that it will not have an effect in our calculations. [16]
3.3 Superconducting circuit compared to ion trap
Because the circuit discussed above has almost the same Hamiltonian as thetrapped ion these systems are analogous. The advantage with the circuitcompared to the trapped ion is that with the circuit we can have bettercontrol of the parameters involved in the system. This way we can oftenstudy the system with a wider range of parameters than what the originalsystem would have allowed.
In the case of our circuit the gate voltage Vg and the flux Φ are pa-rameters that can be controlled after the building of the system. Otherparameters are fixed upon the building.
4 Trapped ion Hamiltonian
Now we will derive the approximations of the interaction and after that wewill introduce a probe to our system and derive the absorption spectrum
23
given by this. Let us start by writing the system’s Hamiltonian again
H = ~ωc(a†a+
1
2)+
~
2ω0σz−
~g
ηcos(η(a†+a)+φ)σx+
d~g
ηsin(η(a†+a)+φ)σy,
(84)where a† and a are the creation and annihilation operators of the harmonicoscillator, σi are the Pauli spin matrices, ωc is the characteristic frequency ofthe harmonic oscillator, ω0 is the level separation of the qubit, η is the Lamb-Dicke parameter, g is the strength of the coupling between the oscillator andqubit, d is asymmetry parameter, and φ is a control parameter.
In the circuit the qubit energy is ~ω0 = 4EC(1 − 2n0) where n0 =CgVg/2e. Because we can control the gate voltage Vg we can control thequbit energy. We can also control the static magnetic flux Φ that flowsthrough the induction loop. This affects parameter φ = πΦ/Φ0 where Φ0 =h/2e. The Lamb-Dicke parameter is η =
√
πZ0/RK where Z0 =√
L/C andRK = h/e2. The strength of the coupling depends on the Josephson energy~g = ηEJ/2.
There has been theoretical research done with the same trapped ionsystem in paper [21]. In that paper calculations are done for a specificmatrix elements of the Hamiltonian 〈n, a|H|n + k, a〉. In our case we willlook at the complete Hamiltonian.
4.1 Jaynes-Cummings model
The eigenvalues of the Hamiltonian (84) are not generally solvable analyt-ically. With certain approximations we can write the Hamiltonian in theJaynes-Cummings form. The Jaynes-Cummings model was introduced byE. Jaynes and F. Cummings in 1963 [22]. Originally it was used to studythe relationship between the quantum theory and the semi-classical theoryof radiation. Jaynes-Cummings model is commonly used to model a qubitinteracting with an optical cavity. The advance with the model comparedto the original Hamiltonian is that the eigenvalue problem of the Jaynes-Cummings Hamiltonian is analytically solvable.
When φ = π/2 and d = 0 the interaction part of the Hamiltonian (84)becomes
Hint =~g
ηsin(η(a† + a))σx. (85)
By assuming that η is small we can expand the sine into series and only takeinto account the lowest order term. Now we get
Hint ≈ ~g(a† + a)σx. (86)
Next we use the rotating wave approximation. Let us switch into an inter-
24
action picture. The interaction part of the Hamiltonian becomes
ei~H0tHinte
− i~H0t
= ei(ωc(a†a+12)+
ω02σz)t(~g(a† + a)σx)e
−i(ωc(a†a+12)+
ω02σz)t
= ~g(e−i(ωc+ω0)taσ− + ei(ωc+ω0)ta†σ+ + e−i(ωc−ω0)taσ+ + ei(ωc−ω0)ta†σ−).
(87)
Near resonance (ωc ∼ ω0) the terms e±i(ωc+ω0)t oscillate rapidly compared toterms e±i(ωc−ω0)t so we can assume that the rapidly rotating terms averageout in the relevant timescale.
After the rotating wave approximation we obtain the Hamiltonian intothe form
HJC ≈ ~ωc(a†a+
1
2) +
~
2ω0σz + ~g(a†σ− + aσ+). (88)
This is the Jaynes-Cummings model. We can see from the coupling termthat this couples the states |n ↑〉 and |n+ 1 ↓〉.
The eigenvalue problem of the Jaynes-Cummings model (88) can besolved exactly. The ground state |0〉 = |0 ↓〉 is not coupled with any otherstates and we get its eigenvalue as E0 = −~∆/2 where ∆ = ω0 − ωc. Pre-
senting the rest of the Hamiltonian as a matrix in a basis
(
|n ↑〉|n+ 1 ↓〉
)
where
n = 0, 1, 2, ... we get the Hamiltonian in the form of
HJC =
(
(n+ 12)~ωc +
~ω02 ~g
√n+ 1
~g√n+ 1 (n+ 3
2)~ωc − ~ω02
)
= (n+ 1)~ωc +~
2
(
ω0 − ωc 2g√n+ 1
2g√n+ 1 ωc − ω0
)
.
(89)
Defining sin θn = 2g√n+ 1/
√
∆2 + 4g2(n+ 1) and cos θn = ∆/√
∆2 + 4g2(n+ 1)we get the Hamiltonian (88) into the form
HJC = (n+ 1)~ωc +~
2
√
∆2 + 4g2(n+ 1)
(
cos θn sin θnsin θn − cos θn
)
. (90)
This can be diagonalized by making a unitary rotation U †HJCU with
U =
(
cos θn2 − sin θn
2
sin θn2 cos θn
2
)
. (91)
This gives the eigenenergies
E±,n = (n+ 1)~ωc ±~
2
√
∆2 + 4g2(n+ 1) (92)
25
and the eigenvectors
|+, n〉 = cos
(
θn2
)
|n ↑〉+ sin
(
θn2
)
|n+ 1 ↓〉,
|−, n〉 = − sin
(
θn2
)
|n ↑〉+ cos
(
θn2
)
|n+ 1 ↓〉.(93)
4.2 Conventional Bloch-Siegert shift
The Jaynes-Cummings model holds well when we are near resonance (ω0 ∼ωc) but once we go further from resonance we can no longer ignore thecounter-rotating terms e±i(ωc−ω0)t that we assumed to average out in therotating wave approximation. The difference in the eigenfrequencies causedby the counter-rotating terms is known as the conventional Bloch-Siegertshift. This was first shown to exist by F. Bloch and A. Siegert in 1940 [23].
We can add correction terms to the Jaynes-Cummings model in order totake into account the Bloch-Siegert shift. The term that was left out by therotating wave approximation in the Jaynes-Cummings model (88) was
HBS = ~g(aσ− + a†σ+). (94)
Still assuming that we are near resonance the term can be interpreted as asmall perturbation to the Jaynes-Cummings Hamiltonian. Let us make a
unitary transformation of eS(HJC+HBS)e−S where S = α(a†σ+− aσ−) and
α = g/(ωc + ω0). Expanding this with Baker-Campbell-Hausdorff formula
eSOe−S = O + [S, O] +1
2![S, [S, O]] +
1
3![S, [S, [S, O]]] + ... (95)
and using the commutation relation of the ladder operators [a, a†] = 1 wecan calculate
eS a†ae−S = a†a− α(aσ− + a†σ+)− α2[a†σ+, aσ−]
eS σze−S = σz − 2α(aσ− + a†σ+)− 2α2[a†σ+, aσ−]
eS(a†σ− + aσ+)e−S = a†σ− + aσ+
eS(aσ− + a†σ+)e−S = aσ− + a†σ+ + 2α[a†σ+, aσ−]
(96)
where we have neglected the terms that are higher than second order in g andthe two-photon processes. By two-photon process we mean processes wherethe Fock state of the harmonic oscillator changes by two. In other wordsthe terms containing a2 or (a†)2. The Hamiltonian after the transformationis
H = HJC +~g2
ωc + ω0[a†σ+, aσ−] = HJC +
~g2
ωc + ω0(a†aσz +
1
2σz −
1
2). (97)
26
The eigenvalue of the ground state |0〉 = |0 ↓〉 is E′0 = −~∆
2 − ~ωBS
where ωBS = g2/(ωc + ω0). By defining cos θ′n = ∆n/√
∆2n + 4g2(n+ 1),
sin θ′n = 2g√n+ 1/
√
∆2n + 4g2(n+ 1) and ∆n = ω0 − ωc + 2ωBS(n+ 1) we
can write the Hamiltonian (97) in a form
H = (n+ 1)~ωc − ~ωBS +~
2
√
∆2n + 4g2(n+ 1)
(
cos θ′n sin θ′nsin θ′n − cos θ′n
)
. (98)
This can be solved similarly as the Hamiltonian (90). We get the eigenen-ergies
E′±,n = (n+ 1)~ωc − ~ωBS ± ~
2
√
∆2n + 4g2(n+ 1) (99)
and eigenvectors
|+, n〉 = cos
(
θ′n2
)
|n ↑〉+ sin
(
θ′n2
)
|n+ 1 ↓〉,
|−, n〉 = − sin
(
θ′n2
)
|n ↑〉+ cos
(
θ′n2
)
|n+ 1 ↓〉.(100)
This is the Bloch-Siegert corrected Jaynes-Cumming model. [24]
4.3 Absorption spectrum
The way to get information from our system is by studying the absorptionspectrum from a weak harmonic perturbation. This is done by introducinga probe with Hamiltonian
HP (t) = ~gP (a† + a) cos(ωpt) (101)
to the system. We will use the Fermi’s golden rule to calculate the absorptionspectrum.
4.3.1 Fermi’s golden rule
Let us assume that we have a time-independent system. If we now intro-duce a weak time-dependent perturbation, Fermi’s golden rule gives us theprobability per time unit for a transition from one of the energy states ofthe system to another. For the Fermi’s golden rule we need to use thetime-dependent perturbation theory to the weak perturbation.
We now have a Hamiltonian of the form
H(t) = H0 + HP (t) (102)
where H0 is the time-independent Hamiltonian and HP (t) is the time-dependent weak perturbation. We have the solution for the time-independentproblem
H0|uq〉 = ǫq|uq〉 . (103)
27
Now we would like to solve the Schrodinger equation
i~∂
∂t|ψ(t)〉 = H(t)|ψ(t)〉 . (104)
First we switch into the interaction picture
|ψ(t)〉I = U(t)|ψ(t)〉H
(int)P (t) = U(t)HP (t)U
†(t)(105)
where U(t) = exp[(i/~)H0t]. Now we have the Schrodinger equation in thefrom
i~∂
∂t|ψ(t)〉I = H
(int)P (t)|ψ(t)〉I . (106)
Writing the states |ψ(t)〉I in the basis of the eigenstates of H0 as |ψ(t)〉I =∑
q aq(t)|uq〉 and plugging this into the Schrodinger equation we obtain
i~∑
q
∂aq(t)
∂t|uq〉 =
∑
i
ai(t)H(int)P (t)|ui〉 . (107)
Multiplying this from the left by 〈uf | we get
i~∂af (t)
∂t=∑
i
ai(t)〈uf |H(int)P (t)|ui〉
=∑
i
ai(t)〈uf |U(t)HP (t)U†(t)|ui〉
=∑
i
ai(t)ei~(ǫf−ǫi)t〈uf |HP (t)|ui〉 .
(108)
We can write the probe Hamiltonian as
HP = ~gP (a† + a)
1
2(eiωP t + e−iωP t) . (109)
With this we get the differential equation for the probability amplitude af (t)for the system to be in state |uf 〉 as
∂af (t)
∂t=gP2i
∑
i
ai(t)ei(ωfi−ωP )tFfi +
gP2i
∑
i
ai(t)ei(ωfi+ωP )tFfi (110)
where ~ωfi = ǫf − ǫi and Ffi = 〈uf |(a† + a)|ui〉.Assuming gP to be small we can do a perturbative expansion for equation
(110) and only take into account the terms up to first order in gP . Writing
ai(t) = a(0)i (t) + λa
(1)i (t) (111)
28
and replacing Ffi → λFfi. Now we get
∂a(0)f (t)
∂t= 0
∂a(1)f (t)
∂t=gP2i
∑
i
a(0)i (t)
(
ei(ωfi−ωP )tFfi + ei(ωfi+ωP )tFfi
)
.
(112)
If we assume that the probe is activated at t = 0 then a(1)f (0) = 0 and we
have
a(1)f (t) =
gP2i
∑
i
a(0)i Ffi
∫ t
0
(
ei(ωfi−ωP )t′ + ei(ωfi+ωP )t′)
dt′ . (113)
The transition amplitude for the system to move from state |ui〉 to state|uf 〉 is
γfi(t) =gP2ia(0)i Ffi
∫ t
0
(
ei(ωfi−ωP )t′ + ei(ωfi+ωP )t′)
dt′
=gP2a(0)i Ffi
(
ei(ωfi−ωP )t − 1
ωif − ωP+ei(ωfi+ωP )t − 1
ωif + ωP
)
.
(114)
We have two cases where the denominator goes to zero ωfi = ωP and ωfi =−ωP . The case ωfi = ωP can be written as ǫi = ǫf − ~ωP . This means thatthe system moves to a higher energy state i.e. the system absorbs energy.Similarly ωfi = −ωP is the case for emission.
We are now only concerned about absorption so we leave out the emissionterm. The transition probability for the system to move from state |ui〉 tostate |uf 〉 is
Pi→f = |γfi(t)|2 =g2P4|a(0)i |2|Ffi|2
(ei(ωfi−ωP )t − 1)(e−i(ωfi−ωP )t − 1)
(ωfi − ωP )2
=g2P4|a(0)i |2|Ffi|2
2− 2 cos((ωfi − ωP )t)
(ωfi − ωP )2
=g2P |a
(0)i |2|Ffi|2
(ωfi − ωP )2sin2
(
(ωfi − ωP )t
2
)
.
(115)
This function is highly peaked at ωfi = ωP . We know that
limt→∞
1
π
sin2 ωt
tω2= δ(x) . (116)
We assume that the time scale is such that approximating Pi→f as a deltafunction is appropriate. Since delta functions are difficult to detect in the
29
absorption spectrum we are going to broaden the spikes using Lorentzianline shape. We have
4
t
sin2(
12(ωfi − ωP )t
)
(ωfi − ωP )2≈ 2πδ(ωfi − ωP ) ≈
γif
(ωfi − ωP )2 +14γ
2if
(117)
where γif is just a constant used for determining the width of the spike.The absorption rate P is the absorption probability per time unit
Pi→f =g2P4pi
γif |Ffi|2(ωfi − ωP )2 +
14γ
2if
(118)
where pi = |a(0)i |2 is the occupation probability of state |ui〉. The totalabsorption spectrum is achieved when all individual absorption rates areadded together assuming γif to be so small that the resonances do notoverlap
Pa =g2P4
∑
i,f
piγ|〈uf |(a† + a)|ui〉|2(ωfi − ωP )2 +
14γ
2(119)
where we have for simplicity assumed that all the resonances have the samewidth γif = γ. The occupation probabilities can be assumed to follow thethermal occupation
pi =e−Ei/kBT
∑
n e−En/kBT
. (120)
Because the temperature in the system is often very low, the occupationprobability pi is much higher for the ground state |0 ↓〉 than for the otherstates. Because of that, we assume that the system is initially at the groundstate and the occupation probability becomes
pi =
1, i = 0
0, i 6= 0.(121)
With this the absorption spectrum is
Pa =g2P4
∑
f
γ|〈uf |(a† + a)|0 ↓〉|2(ωf0 − ωP )2 +
14γ
2. (122)
Now we can calculate the absorption spectrum for the Jaynes-Cummingsmodel (88). We have the eigenvectors (92) and eigenenergies (93). Now wejust plug them into equation (122) and get
Pa =g2P4
(
γ|〈+, 0|(a† + a)|0〉|2(ω+0 − ωP )2 +
14γ
2+γ|〈−, 0|(a† + a)|0〉|2(ω−0 − ωP )2 +
14γ
2
)
=γg2P4
(
sin2 θ02
(ω+0 − ωP )2 +14γ
2+
cos2 θ02
(ω−0 − ωP )2 +14γ
2
) (123)
30
where ~ω±0 = E±,0 − E0.Same thing for the Bloch-Siegert corrected Jaynes-Cummings model (97)
gives us the absorption spectrum
Pa =γg2P4
[
sin2(θ′02 )
(ω′+0 − ωP )2 +
14γ
2+
cos2(θ′02 )
(ω′−0 − ωP )2 +
14γ
2
]
, (124)
where ~ω′±0 = E′
±,0 − E′0.
It is good to notice that the absorption spectrum calculated here is anapproximation. In order to get the exact absorption spectrum we wouldneed to solve numerically the master equation of the driven system. As longas the width of the resonances is small enough for them to not overlap thisapproximation should be accurate. [25, 26]
4.4 Lamb-Dicke regime
Lamb-Dicke regime in ion trapping is the area where the coupling betweenthe internal and motional ion states is so small that the transitions where themotional state is changed by more than one quantum number are suppressed.The Lamb-Dicke regime is expressed by inequality
η√
〈(a+ a†)2〉 ≪ 1 . (125)
Trapped ion experiments are traditionally within the Lamb-Dicke regime.With the superconducting circuits it is possible to build a system with Lamb-Dicke parameter big enough to go outside the Lamb-Dicke regime.
Both the Jaynes-Cummings model and the Bloch-Siegert corrected Jaynes-Cummings model are based on the assumption that we are in the Lamb-Dicke regime. Now we would like to study how the system behaves outsidethe regime and at what point when increasing the Lamb-Dicke parameter theapproximations start to deviate from the numerically calculated solutions.[4]
5 Calculations and Results
Paper [27] shows experiments done with the same electric circuit as discussedin here. From this we get the parameter values used in the experiments:EJ/h = 27.0 GHz, d = 0.19, L = 410 pH, C = 10 pF and ωc = 3.5 GHz.These give η ≈ 0.03 and g ≈ 0.4. We know that these are values that can beachieved in experiments and that is why in our calculations we use parametervalues that are close to these. Recently in a slightly modified system [28],η ≈ 0.1 was observed, and we expect that with a little optimization in thefabrication process, one can obtain even higher values.
31
−3 −2 −1 0 1 2 3 4 5 6 7 8−2
−1
0
1
2
3
4
5
6
7
8
∆ [GHz]
Ene
rgy
[GH
z]
Figure 6: Eigenenergies of the lowest energy states of the Hamiltonian (84) asa function of ∆. The blue lines are the energies of the numerical solution, redlines are those of the Jaynes-Cummings model, green lines are the energies ofthe Bloch-Siegert corrected Jaynes-Cummings model and the black dashedlines are the energy states of Hamiltonian (84) in the case where there is nointeraction (η = 0). The parameters used in this figure are d = 0, η = 0.1,g = 1 GHz, φ = π/2 and ωc = 3.5 GHz.
32
In figure 6 we have plotted the eigenenergies of the system as a functionof ∆. The black dashed lines in the figure are the eigenenergies of theuncoupled Hamiltonian. The decreasing lines correspond to the states |n ↓〉and the increasing lines to |n ↑〉. We can see that in the uncoupled system theeigenenergies of the second and third lowest states intersect at ∆ = 0. Whenthe interaction is added the energies form a gap around the intersection pointand there is no intersection. This is called an avoided crossing.
At the intersections of the third and fourth states and fourth and fifthstates there is no avoided crossings shown in the approximations. Thisis due to the fact that there is no coupling between these states. The cou-pling between these states would come from the multi-photon processes thatwere left out when the interaction was approximated as linear. There is noavoided crossing in the numerical solution at the crossing of the third andfourth states either but there is a small gap at the crossing of the fourth andfifth states. This is because we have used the parameters values d = 0 andφ = π/2. With these the interaction term is
Hint =~g
ηsin(η(a† + a))σx . (126)
The series expansion of sine has only the odd order terms. The couplingbetween the third and fourth states comes from the two-photon interactions,the coupling between the fourth and fifth states comes from the three-photoninteractions and so on. Because of this every other avoided crossing is notseen in the numerical solution.
In figure 7 we have plotted the numerically calculated absorption spec-trum of Hamiltonian (84) and the resonance curves of both the Jaynes-Cummings and the Bloch-Siegert corrected Jaynes-Cummings models forη = 0.1. The red dashed lines in the figure are the resonance frequencyof the qubit (ωP = ω0) and the first resonance frequency of the oscillator(ωP = ωc). These correspond to the transitions from state |0 ↓〉 to states|0 ↑〉 and |1 ↓〉 respectively. The avoided crossing seen in this figure is theone between the first and second exited states seen in figure 6. The ab-sorption is strongest from the ground state to the state |1 ↓〉. Near theresonance the state |0 ↑〉 is coupled with the state |1 ↓〉 and there is alsostrong absorption along the resonance frequency of the qubit. Further fromresonance the coupling is weaker and this absorption diminishes.
The numerical absorption spectrum is calculated by presenting the Hamil-tonian (84) as a matrix in the basis |n, a〉 = |n〉⊗|a〉|n = 0, 1, 2, ... and a =↑, ↓ where |n〉 are the oscillator eigenstates and |a〉 the qubit eigenstates.The matrix is truncated to include only twenty lowest oscillator eigenstatesmaking it a 40×40 matrix. The eigenstates and eigenenergies of this matrixare numerically calculated and using these we get the absorption spectrumfrom equation (122).
33
ωP [GHz]
∆ [G
Hz]
2 4 6 8
−3
−2
−1
0
1
2
3
4
10−
4 [GH
z]
1
2262
−2 0 2 40
0.05
0.1
0.15
0.2
0.25
∆ [GHz]
Err
or [G
Hz]
Figure 7: On the left figure is the absorption spectrum plotted as a functionsof ∆ and ωP . The red lines are the resonance curves of the Jaynes-Cummingsmodel and the green lines denote the resonances of the Bloch-Siegert cor-rected Jaynes-Cummings model. The dashed lines are ωP = ωc (vertical)and ωP = ω0. On the right figure, we plot the absolute errors between theresonance curves of the approximations and the numerical solution. Thered lines are the errors of the Jaynes-Cummings approximation and thegreen lines are the errors of the Bloch-Siegert corrected approximation. Thedashed lines are the errors of the lower energy resonance and the continuouslines the errors of the higher energy resonance. The parameter values usedin this figure are d = 0, η = 0.1 GHz, g = 1 GHz, φ = π/2, ωc = 3.5 GHz,gP = 0.05 GHz and γ = 0.1 GHz.
34
The absorption maxima for the approximations are obtained from equa-tions (123) and (124). These have maximum values at the minima of thedenominator, that is, when ωP = ω+0 and ωP = ω−0 for Jaynes-Cummingsmodel and similarly for the Bloch-Siegert corrected approximation.
We have also plotted the errors between the location of the absorp-tion maxima of the numerical solution and the resonance curves of the ap-proximations. Near the resonance (∆ = 0) the Jaynes-Cummings modelis the most accurate since in that area the rotating wave approximation(|∆| ≪ ωc + ω0) holds the best. Further from the resonance the counter-rotating terms become so big that the Bloch-Siegert correction, that takesinto account these terms, is needed.
In figure 8 we have plotted the absorption spectra and errors as a functionof η. The two absorption spikes seen also at the lower values of Lamb-Dickeparameter are the same as the ones in figure 7. The additional spike seenonly in higher values of Lamb-Dicke parameter comes from the state |3 ↓〉.This state is coupled to state |1 ↓〉 so weakly that the transition becomesvisible only at the higher values of η. This is the same avoided crossing thatis visible in the figure 6 between the fourth and fifth lowest states.
The approximations break down at the point where the lower resonancesgoes to zero. Beyond this point the coupling strength g is so big that E−,0 <E0 and |0 ↓〉 is no longer the lowest energy state in the approximation. Theaccuracy of the approximations is quite poor well before this limit. Ataround η = 0.2 we can see the other avoided crossing starting to shift thesecond absorption spike away from the approximations.
The difference between the Jaynes-Cummings model and the numericalsolution is known as the generalized Bloch-Siegert shift. The Bloch-Siegertcorrection takes into account the conventional Bloch-Siegert shift but outsidethe Lamb-Dicke regime this correction is not enough.
5.1 Correction of the d term
In figure 8 we have used d = 0. This condition was assumed in the derivationof the approximations. If we place d 6= 0 the accuracy of the approximationwill decrease greatly. Figure 9 is the same as figure 8 except that the pa-rameter d = 0.19. We see that the approximations do not match with thenumerical solution even at η = 0. This is due to the fact that when φ = π/2the d term of the Hamiltonian has a η-independent part
d~g
ηsin(
η(a† + a) +π
2
)
σy =d~g
ηcos(
η(a† + a))
σy
=d~g
η
[
1 +∞∑
n=1
(−1)n(
η(a† + a))2n
2n!
]
σy =d~g
ησy +O(η)σy
(127)
where in the last part the first term is independent of η since g = ηEJ/(2~)and O(η) is the η-dependent part.
35
ω0 = 3.5 [GHz]
ωP [GHz]
η
5 10 15
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.40
1
2
3
ηE
rror
[GH
z]ω
0 = 2.0 [GHz]
ωP [GHz]
η
5 10 15
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.40
1
2
3
η
Err
or [G
Hz]
10−
4 [GH
z]0
1
2694
10−
4 [GH
z]
0
1
2719
Figure 8: The absorption spectra and the errors of the approximations plot-ted as a function of η for ω0 = 3.5 and ω0 = 2.0. The left figures are theabsorption spectra. The red lines are the resonance curves of the Jaynes-Cummings model and the green lines denote the resonances of the Bloch-Siegert corrected Jaynes-Cummings model. On the right figures are theabsolute errors between the resonances of the approximations and the nu-merical solution. Red lines are the errors of the Jaynes-Cummings approx-imation and the green lines are the errors of the Bloch-Siegert correctedapproximation. The dashed lines are the errors of the lower energy reso-nance and the continuous lines the errors of the higher energy resonance.The parameter values used for this figure are d = 0, φ = π/2, ωc = 3.5 GHz,gP = 0.05 GHz, γ = 0.1 GHz and g = 10η GHz.
36
ω0 = 3.5 [GHz]
ωP [GHz]
η
5 10 15
0.1
0.2
0.3
0.4
0.5
10−
4 [GH
z]0
1
2626
0 0.1 0.2 0.3 0.40
1
2
3
ηE
rror
[GH
z]
ω0 = 2.0 [GHz]
ωP [GHz]
η
5 10 15
0.1
0.2
0.3
0.4
0.5
10−
4 [GH
z]
0
1
2663
0 0.1 0.2 0.3 0.40
1
2
3
4
η
Err
or [G
Hz]
Figure 9: The absorption spectra and the errors of the approximations plot-ted as a function of η for ω0 = 3.5 and ω0 = 2.0. The left figures are theabsorption spectra. The red lines are the resonance curves of the Jaynes-Cummings model and the green lines correspond to those of the Bloch-Siegert corrected Jaynes-Cummings model. On the right figures are theabsolute errors between the resonances of the approximations and the nu-merical solution. Red lines are the errors of the Jaynes-Cummings approx-imation and the green lines are the errors of the Bloch-Siegert correctedapproximation. The dashed lines are the errors of the resonance with lowerenergy and the continuous lines the errors of the higher energy resonance.The parameter values used in this figure are d = 0.19, φ = π/2, ωc = 3.5GHz, gP = 0.05 GHz, γ = 0.1 GHz and g = 10η GHz.
37
In the case of φ = π/2 and η = 0 we have the whole Hamiltonian as
H = ~ωc(a†a+
1
2) +
~
2ω0σz +
d~g
ησy . (128)
This system has the eigenenergies of
E±,n =
(
n+1
2
)
~ωc ±~
2
√
ω20 + 4
(
dg
η
)2
(129)
and the eigenstates
|n, e〉 = − idg
ηω′0
√
12 − ω0
2ω′0
|n ↑〉+√
1
2− ω0
2ω′0
|n ↓〉
|n, g〉 = idg
ηω′0
√
12 + ω0
2ω′0
|n ↑〉+√
1
2+
ω0
2ω′0
|n ↓〉(130)
where
ω′0 =
√
ω20 + 4
(
dg
η
)2
. (131)
We now take into account the terms of the first order of η. Our Hamil-tonian is
H = ~ωc(a†a+
1
2) +
~
2ω0σz +
d~g
ησy + ~g(a† + a)σx . (132)
Now we make a rotation to the eigenbasis of the states (130). For this weneed the transformation matrix
U =
− idg
ηω′0
√
12−
ω02ω′
0
idg
ηω′0
√
12+
ω02ω′
0√
12 − ω0
2ω′0
√
12 + ω0
2ω′0
. (133)
With thisU †σxU = −σy (134)
and the Hamiltonian in the new basis is
H = ~ωc(a†a+
1
2) +
~
2ω′0σz − ~g(a† + a)σy . (135)
38
5.1.1 Jaynes-Cummings model
Now we derive the Hamiltonian in the Jaynes-Cummings model. First wedo the rotating wave approximation for the interaction term
−~g(a† + a)σy ≈ i~g(aσ+ − a†σ−) . (136)
The ground state is |0〉 = |0, g〉 with E0 = −~∆/2 where ∆ = ω′0 − ωc. The
rest of the eigenenergies are
E±,n = (n+ 1)~ωc ±~
2
√
∆2 + 4g2(n+ 1) (137)
where n = 0, 1, 2, .... The eigenenergies are the same as in the Jaynes-Cummings model with d = 0 except with the difference that now ω0 isreplaced with ω′
0. Defining
cos θn =∆
√
∆2 + 4g2(n+ 1)
sin θn =2g
√n+ 1
√
∆2 + 4g2(n+ 1)
(138)
we have the Hamiltonian as
HJC = (n+ 1)~ωc +~
2
√
∆2 + 4g2(n+ 1)
(
cos θn i sin θn−i sin θn − cos θn
)
. (139)
This can be diagonalized using a unitary transformation
U =
(
sin θn2 sin θn
2
cos θn2 sin θn
2
−i sin θn2 i cos θn
2
)
. (140)
The eigenstates are
|+, n〉 = sin θn
2 sin θn2
|n, e〉 − i sinθn2|n+ 1, g〉,
|−, n〉 = cos θn
2 sin θn2
|n, e〉+ i cosθn2|n+ 1, g〉.
(141)
For the absorption spectrum we need the terms
|〈+, 0|(a† + a)|0〉|2 = sin2θn2
|〈−, 0|(a† + a)|0〉|2 = cos2θn2.
(142)
39
5.1.2 Bloch-Siegert correction
For the Bloch-Siegert correction we need to include the term
HBS = i~g(a†σ+ − aσ−) (143)
as a small perturbation. This will be done similarly as earlier with a unitary
transformation eS(HJC + HBS)e−S . This time the transformation matrix is
S = α(a†σ+ + aσ−) where α = ig/(ωc + ω′0). Now we have
eS a†ae−S = a†a+ α(aσ− − a†σ+) + α2[a†σ+, aσ−]
eS σze−S = σz + 2α(aσ− − a†σ+) + 2α2[a†σ+, aσ−]
eS(aσ+ − a†σ−)e−S = aσ+ − a†σ−
eS(a†σ+ − aσ−)e−S = a†σ+ − aσ− − 2α[a†σ+, aσ−]
(144)
where we have again ignored the third and higher order terms of g and thetwo-photon transitions. With this the Hamiltonian is
H = HJC +~g2
ωc + ω′0
(a†aσz +1
2σz −
1
2) . (145)
The Bloch-Siegert correction term is the same as in the case of d = 0 withthe only exception that the ω0 is being replaced with ω′
0.Now by defining
cos θ′n =∆n
√
∆2n + 4g2(n+ 1)
sin θ′n =2g
√n+ 1
√
∆2n + 4g2(n+ 1)
(146)
where ∆n = ω′0 − ωc + 2ωBS(n + 1) and ωBS = g2/(ωc + ω′
0) we get theeigenenergies as
E′0 = −~∆
2− ~ωBS
E′±,n = (n+ 1)~ωc − ~ωBS ± ~
2
√
∆2n + 4g2(n+ 1)
(147)
and the eigenstates
|0〉 = |0, g〉
|+, n〉 = sin θ′n
2 sin θ′n2
|n, e〉 − i sinθ′n2|n+ 1, g〉,
|−, n〉 = cos θ′n
2 sin θ′n2
|n, e〉+ i cosθ′n2|n+ 1, g〉.
(148)
40
The absorption spectrum given by this is
Pa =γg2P4
[
sin2(θ′02 )
(ω′+0 − ωP )2 +
14γ
2+
cos2(θ′02 )
(ω′−0 − ωP )2 +
14γ
2
]
, (149)
where ~ω′±0 = E′
±,0 − E′0. This is the same as in the d = 0 case with ω0
being replaced with ω′0.
As we can see in the case of φ = π/2 when adding the η-independentterm (~dg/η)σy into the Hamiltonian this changes the approximations byonly replacing the parameter ω0 with ω′
0 =√
ω20 + 4(dg/η)2.
These calculations could have been done for an arbitrary φ. In this casethe η-independent part would be
g
η
(
− cos(φ)σx + d sin(φ)σy
)
. (150)
In the special case φ = π/2 the linear η term is transformed to the newbasis as ~g(a† + a)σx → −~g(a† + a)σy. In the general case the linear partwould have a much more complex form in the new basis. Because of thisthe calculations of the approximations would get more complicated.
Now we can plot figure 9 again except this time using the new approx-imations. This is done in figure 10. We can see that with the d-correctedapproximations the accuracy of the resonance curves is nearly the same asin the d = 0 case.
5.2 Higher avoided crossings
The avoided crossing between the first and second exited states can be seenin the absorption spectrum in figure 11. The points where the dashed redlines intersect are the points where the resonance frequency of the qubitis the same as a resonance frequency of the cavity. These are the pointswhere we could expect to see avoided crossings. There are no other avoidedcrossings in figure 11 because the Lamb-Dicke parameter is too small.
In figure 12 we have the absorption spectra plotted with different φ. Thesix curves are the transition energies from ground state to the six lowestexited states. This means the curves are the possible absorption spikes thatcould be seen. Far from resonance (ω0 ≫ ωP ) these curves are the same asthe lines ωP = nωc.
When φ = π/2 the curves (2n+1)ωc have highest transition probabilityand the absorptions to states 2nωc are being ignored. This is because whenφ = π/2 and d = 0 all the eigenstates are of the form
|ψ0〉 =∞∑
n=0
A2n+1↑|2n+ 1 ↑〉+A2n↓|2n ↓〉 (151)
41
ω0 = 3.5 [GHz]
ωP [GHz]
η
5 10 15
0.1
0.2
0.3
0.4
0.5
10−
4 [GH
z]0
1
2626
0 0.1 0.2 0.3 0.40
0.5
1
1.5
2
ηE
rror
[GH
z]ω
0 = 2.0 [GHz]
ωP [GHz]
η
5 10 15
0.1
0.2
0.3
0.4
0.5
10−
4 [GH
z]
0
1
2663
0 0.1 0.2 0.3 0.40
0.5
1
1.5
2
η
Err
or [G
Hz]
Figure 10: The absorption spectra and the errors of the approximationsplotted as a function of η for ω0 = 3.5 and ω0 = 2.0. The approximation arenow the ones including the corrections of the η-independent d term. The leftfigures are the absorption spectra. The red lines are the resonance curvesof the Jaynes-Cummings model and the green lines correspond to those ofthe Bloch-Siegert corrected Jaynes-Cummings model. On the right figuresare the absolute errors between the spikes of the numerical solution andthe resonance curves of the approximations. Red lines are the errors of theJaynes-Cummings approximation and the green lines are the errors of theBloch-Siegert corrected approximation. The dashed lines are the errors ofthe lower energy resonance and the continuous lines the errors of the higherenergy resonance. The parameter values used in this figure are d = 0.19,φ = π/2, ωc = 3.5 GHz, gP = 0.05 GHz, γ = 0.1 GHz and g = 10η GHz.
42
ωP [GHz]
∆ [G
Hz]
2 4 6 8 10 12 14
−2
0
2
4
6
8
10
12
10−
4 [GH
z]
0
1
2
263
Figure 11: The absorption spectrum of the Hamiltonian (84) plotted as afunctions of ∆ and ωP . The vertical dashed lines are the resonance frequen-cies of the cavity ωP = ωc, 2ωc and 3ωc. The diagonal dashed line is theresonance frequency of the qubit ωP = ω0. The parameters used in thisfigure are d = 0, ωc = 3.5 GHz, η = 0.1, g = 1 GHz, gP = 0.05 GHz andγ = 0.1 GHz.
43
φ = π/2
ωP [GHz]
∆ [G
Hz]
5 10 15 20
0
5
10
15
10−
4 [GH
z]
0
1
2480
φ = 3π/8
ωP [GHz]
∆ [G
Hz]
5 10 15 20
0
5
10
15
10−
4 [GH
z]
0
1
2255
φ = 1.45
ωP [GHz]
∆ [G
Hz]
5 10 15 20
0
5
10
15
10−
4 [GH
z]
0
1
2331
Figure 12: The absorption spectra of the numerical solution of the Hamilto-nian (84) plotted as a functions of ∆ and ωP with different values of φ. Thesix curves in all plots are the transition energies from the ground state to thesix lowest excited states. The red dashed lines are ωP = ωc (vertical) andωP = ω0. The parameter values used in this figure are d = 0.19, ωc = 3.5GHz, gP = 0.05 GHz, γ = 0.1 GHz, η = 0.3 and g = 3 GHz.
44
or
|ψ1〉 =∞∑
n=0
A2n↑|2n ↑〉+A2n+1↓|2n+ 1 ↓〉. (152)
This is due to the fact that the interaction is of the form (126). This meansthat all the interaction terms are either a2n+1σx or (a†)2n+1σx. Because theground state |0〉 is of the from |ψ0〉 we have 〈ψ0|(a†+ a)|0〉 = 0. This is whyonly half of the transitions are seen in the absorption spectrum. With d 6= 0and d ≪ 1 the probabilities for the other transitions becomes non-zero butstill so small that they are not visible in the figure.
In the figure for φ = 3π/8 in figure 12 we see that now the absorption isstronger to the states 2nωc. The absorption to states (2n+1)ωc is strongestat φ = π/2 but the other states have minima at this point. The states 2nωc
have absorption maxima at φ ≈ π/2 ± 0.2. The value φ = 1.45 in the lastplot in figure 12 is chosen such that all multi-photon resonances would bevisible in the absorption spectrum.
In figure 13 we have plotted the absorption spetrum with different η. Itcan be seen that when η is increased more avoided crossings become visible.
5.3 Discussion
The superconducting analog of the ion trap experiment has an extra termin the Hamiltonian that is caused by the asymmetry d of the Josephsonjunction. When this extra term is taken into account the previous analyticapproximations work poorly. We showed that this could be fixed quite easilyat least in the case of φ = π/2.
We can see that the errors in the approximations increase as η is in-creased. From figure 8 we can see that at around η = 0.2 the two-photoncoupling starts to have an effect on the transition energy between the groundand the second exited states. The relative error of the higher resonance is∆E/E+ = 0.1 for the Bloch-Siegert corrected Jaynes-Cummings model, atη ≈ 0.2. Here ∆E is the error between the resonances of the approximationand the numerical solution and E+ is the resonance energy of the numeri-cal solution. The approximations do not work above the limit η = 0.2. Inmost cases, we are dealing with Lamb-Dicke parameters well under this. Forthe ion trap it is possible to have a Lamb-Dicke parameter value up to 0.2[29, 30]. With the artificial ion traps we anticipate even higher values of theLamb-Dicke parameter.
We detected that some of the multi-photon avoided crossings are notvisible at φ = π/2. In order to detect all the possible resonances we need tochoose the value of the control parameter φ carefully.
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η = 0.1
ωP [GHz]
∆ [G
Hz]
5 10 15 20
0
5
10
15
10−
4 [GH
z]
0
1
2261
η = 0.25
ωP [GHz]
∆ [G
Hz]
5 10 15 20
0
5
10
15
10−
4 [GH
z]
0
1
2313
η = 0.3
ωP [GHz]
∆ [G
Hz]
5 10 15 20
0
5
10
15
10−
4 [GH
z]
0
1
2331
η = 0.5
ωP [GHz]
∆ [G
Hz]
5 10 15 20
0
5
10
15
10−
4 [GH
z]0
1
2375
Figure 13: The absorption spectra of the numerical solution of the Hamilto-nian (84) plotted as a functions of ∆ and ωP with different values of η. Thesix curves in all plots are the transition energies from the ground state to thesix lowest excited states. The red dashed lines are ωP = ωc (vertical) andωP = ω0. The parameter values used in this figure are d = 0.19, φ = 1.45,ωc = 3.5 GHz, gP = 0.05 GHz, γ = 0.1 GHz and g = 10η GHz.
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6 Conclusion
With current fabrication methods it is possible to build electric circuits insuch a small scale that the quantum effects become essential. The greatbenefit of using circuits instead their natural counterparts is that they oftenallows the use of much wider range of parameter values.
The circuit used in this thesis had a LC-circuit coupled with a single-Cooper-pair transistor. This produces a Hamiltonian that is almost identicalto that of a single particle in a Paul trap. The only difference being that thecircuit has the asymmetry term in the interaction. This motivates a newstudy of the ion trap physics, a field that was started already in 1970’s andhas various uses in several fields of physics.
In this thesis we studied the accuracy of the Jaynes-Cummings modeloutside the Lamb-Dicke regime. We numerically calculated the absorptionspectrum for the system and compared this to the absorption spectrum givenby the approximations. We noticed that the Bloch-Siegert correction shouldbe taken into account unless we are very near the resonance (|∆|/ωc ≪ 0.15).We also discovered that once the Lamb-Dicke parameter is close to η = 0.2the generalized Bloch-Siegert shift is so big that we start to see other avoidedcrossings and the approximations are not good anymore.
We saw that the asymmetry term had a noticeable effect on the absorp-tion spectrum and we re-derived the approximations, this time taking intoaccount the η-independent asymmetry.
In order to make the approximations hold for bigger values of the Lamb-Dicke parameter we would need to take account the second order interactionterms of the Hamiltonian. This would include two-photon transition termsand result in additional resonances.
The circuit studied here could offer advances also in the field of quantumsimulations if we were able to couple these systems together. Speciallywith circuits well outside the Lamb-Dicke regime we could do simulations ofsystems with non-linear couplings.
For further studying one could derive the absorption spectrum moreaccurately by starting from the master equation of the probed system.
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