Generalized Bloch-Siegert shift in an artificial trapped ion

Transcription

1 Generalized Bloch-Siegert shift in an artificial trapped ion M. Sc. Thesis Iivari Pietikäinen University of Oulu Department of Physics Theoretical Physics 4

2 Abstract The purpose of this thesis is to study a two-level system that is coupled non-linearly to a harmonic oscillator outside the Lamb-Dicke regime. These kind of non-linear couplings can be found between the vibrational and electronic degrees of freedom in a laser-irradiated trapped ions. In this work we study a superconducting circuit with similar properties. The circuit consist of LC-circuit that is coupled to a single-cooper-pair transistor. The single-cooper-pair transistor behaves as an artificial ion and the LC-circuit behaves as the electric potential that is used to trap the ion. When the coupling is weak the interaction can be approximated to be linear. The differences between the absorption spectrum of the linear approximation and the whole system is known as generalized Bloch-Siegert shift. In this thesis we will study the Bloch-Siegert shift as the coupling strength is increased. The non-linear coupling terms cause resonance shifts and additional resonance in the absorption spectrum. Contents Introduction 3 Ion traps 4. Paul traps Superconducting electric circuits 9 3. Superconductivity Cooper pairs Josephson effect Quantum network theory LC-circuit Single-Cooper-pair transistor SCPT coupled to a LC-circuit Superconducting circuit compared to ion trap Trapped ion Hamiltonian 3 4. Jaynes-Cummings model Conventional Bloch-Siegert shift Absorption spectrum Fermi s golden rule Lamb-Dicke regime Calculations and Results 3 5. Correction of the d term Jaynes-Cummings model Bloch-Siegert correction

3 5. Higher avoided crossings Discussion Conclusion 47

4 Introduction The first single-particle ion trap experiment was done in 973 by D. Wineland, P. Ekstrom and H. Dehmelt []. Since then the ion traps have developed greatly thanks to the improvement of laser technology allowing spectrally narrower light sources to be used as the trapping fields. The ion trap experiments have provided important contributions in many fields of physics. Trapping ions with electromagnetic fields causes the ionic motion to entangle with the internal electronic structure of the ion. The ion s motional states behave as a harmonic oscillator and the internal electronic structure can be thought as a two-level system. The interaction between the vibrational and electronic degrees of freedom is non-linear and generally not analytically solvable. If we assume that the ratio between the zero point motion of the ion and the wave length of the electric field is small the interaction can be approximated to be linear. This ratio is known as the Lamb-Dicke parameter. The linear 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 the case of larger values of the Lamb-Dicke parameter the non-linear terms start to have a noticeable effect on the system and the linear approximations do not hold anymore []. In this work we look how accurate the approximations are when the Lamb-Dicke parameter is increased. Similar coupling can be achieved in superconducting circuits. By coupling an LC-circuit with a single-cooper-pair transistor we can have a system with similar behavior as in the case of the trapped ion. This superconducting circuit is what we will study in this thesis. In chapter two we study our natural system which is the ion trap. We derive the Hamiltonian for a specific ion trap, the Paul trap. In chapter three we move to the artificial systems which in this case are superconducting circuits. We introduce some important properties of superconductivity. After that we introduce the quantum network theory that 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 we derive the Hamiltonian for the circuit that is studied on this thesis. In chapter four we concentrate on the Hamiltonian of our circuit. We start by deriving the linear approximations of the interaction. Next we introduce a weak probe to our system. This allows us to measure the absorption spectrum of the system. We derive the equation for the absorption spectrum. Next we will do some calculations in chapter five. First we do calculations on how well the approximations hold on different oscillator frequencies and on different Lamb-Dicke parameter values. The differences between the numerical calculations and the Jaynes-Cummings model are known as 3

5 the generalized Bloch-Siegert shift. Then we derive corrections to the approximations that take account the asymmetry term that we have in the Hamiltonian. Lastly we study the other avoided crossings and when they become visible in the absorption spectrum. Ion traps An ion trap is a device that can be used to trap a charged particle into a confined region of space with electromagnetic fields. There are various types of ion traps depending on the kind of electromagnetic fields that are used for trapping. Most popular types are Penning trap that uses static electric and magnetic fields and Paul trap that uses oscillating time-dependent electric field. Ion traps have several uses. They are used as mass spectrometers to determine the mass to charge ratio of ions. This works by making the ions pass through the trap. The trap catches only ions with a particular mass to charge ratio. Ion traps can also be used for cooling particles. In the ion trap the motion of the ion is coupled with its internal electronic structure. With the right frequency of the electromagnetic field, this coupling can be used to decrease the kinetic energy of the ion. One more possible use for ion traps is a quantum computer. If the frequency of the electromagnetic field inducing the coupling is low enough the system does not have enough energy to excite the internal states of the ion beyond the two lowest states. This means that the internal structure can be thought as a two-level system. A quantum two-level system is called a quantum bit or a qubit. Quantum computers use qubits for the calculation instead of classical bits as in classical computers. Systems consisting of a few trapped ions entangled together has been made [3]. If this could be done with more ions it could be possible to build a quantum computer. Evenifwecouldnotbuildquantumcomputerswecouldusetheiontraps to build simpler systems to be used for quantum simulations. Simulating quantum systems using classical computers is complex. If we had a quantum system where we could control the interactions between the qubits we could use this system to do simulations of quantum systems that could not be simulated by classical computers. [4, 5, 6]. Paul traps We shall now further study one of the ion traps, the Paul trap [4]. We will give the derivation of the Hamiltonian generated by this trap. Paul trap was developed by Wolfgang Paul in 99 [7]. The trap uses a timedependent field to confine the particle. The field is typically in the range of radio-frequency (3 khz to 3 GHz). 4

6 The electric potential of a Paul trap has a quadrupolar shape. Let us assume that the potential can be separated into a sinusoidally oscillating time-dependent part and a time-independent part. If the potential is quadratic in all Cartesian coordinates the problem can be reduced to three one-dimensional problems. The potential given by the electric field is ˆV(t) = m 8 ω rf [a+qcos(ω rft)]ˆx () where m is the mass of the trapped particle, ω rf is the drive frequency of the time-dependent field and a and q are constants that depend on the form of the trapping field and the charge of the trapped particle. The total Hamiltonian is In Heisenberg picture Ĥ (m) = ˆp m + m 8 ω rf [a+qcos(ω rft)]ˆx. () Combining these gives ˆx = i [ˆx,Ĥ(m) ] = ˆp m ˆp = i [ˆp,Ĥ(m) ] = m 4 ω rf [a+qcos(ω rft)]ˆx. (3) ˆx+ ω rf 4 [a+qcos(ω rft)]ˆx =. (4) Further more by defining ξ = ω rf t/ the equation is of the standard form of the Mathieu equation ˆx +[a+qcos(ξ)]ˆx =. (5) ξ Replacing operator ˆx with function u(t) and choosing boundary conditions u() =, u() = iν, we have the solution for the Mathieu equation as u(t) = e iβω rft/ C n e inω rft n= (6) where β is a constant that depends on the coefficients a and q. Now we can construct an operator m Ĉ(t) = ν i(u(t) ˆx(t) u(t)ˆx(t)) (7) 5

7 that is in fact time-independent and of the same form as the annihilation operator for harmonic oscillator of mass m and frequency ν Ĉ(t) = Ĉ() = m ν (mνˆx()+iˆp()) = â. (8) The ˆx(t) and ˆp(t) operators can now be expressed as ˆx(t) = mν (âu (t)+â u(t)) m ˆp(t) = ν (â u (t)+â 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 recursion relation for the coefficients C n : C n +D n C n +C n+ = () where D n = [a (n+β) ]/q. From this we obtain expressions C n+ = C n = C n D n + D n+... C n D n + D n.... () Typically a, q so we can truncate these expressions. By choosing C ±4 = and by using the recursion relations and the boundary conditions we obtain β a+q / ν βω rf / u(t) e iνt+ q cos(ω rft) + q. () Since ν ω rf and q/ the q cos(ω rft) term causes only fast, small oscillation to the solution. This is called the micromotion. If we ignore the micromotion and present the Hamiltonian Ĥ(m) with the creation and annihilation operators it becomes ( ) Ĥ (m) ν = ( + q) +â â ν ( ) +â â. (3) This is of the same form as the Hamiltonian of harmonic oscillator. 6

8 Analogous to the harmonic oscillator we now define a basis { n } where n =,,,... The operator â operates on these states just like the ladder operator operates to the Fock states in harmonic oscillator â n = n n â n = n+ n+ ˆN n = n n (4) where ˆN = â â is the number operator. These states and the operator Ĉ are in the Heisenberg picture. We can change them to the Schrödinger picture by Ĉ S (t) = Û(t)ĈÛ (t) n,t = Û(t) n (5) where Û(t) = exp[ (i/ )Ĥ(m) t]. The Schrödinger picture operator ĈS(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 oscillator. It is important to note that these states are not energy eigenstates of the system since the harmonic oscillation of the electric field constantly changes the kinetic energy of the ion. These states are called quasistationary states. The coupling between the motional states and the internal structure states of the ion is caused by an additional electric field. If the frequency of this field is near resonance with two of the energy levels of the internal structure of the particle and the coupling strength is much smaller than the energy needed for the internal states to excite to a non-resonant state, we can approximate the internal structure of the ion as a two-level system. Let the resonance states be + and with energy difference ω = (ω + ω ). The two-level Hamiltonian is Ĥ (e) = (ω ω ) = ω + +ω ( )+ ω ( + + ). (6) By ignoring the constant energy term and using the Pauli matrices ( ) ( ) ( ) i ˆσ x =, ˆσ y =, ˆσ i z =, we have the two-level Hamiltonian as The total Hamiltonian is Ĥ (e) = ω ˆσ z. (7) Ĥ = Ĥ(m) +Ĥ(e) +Ĥ(i) (8) 7

9 whereĥ(m) andĥ(e) aretheparticle smotionalhamiltonianandthehamiltonian for the internal structure as we have discussed above. The Hamiltonian Ĥ(i) describes the interaction between the motional and internal structure states. The interaction is caused by the additional electric field. The coupling between electromagnetic field and charges is a complex subject and therefore we will not go into the details here. We will just state the form of the interaction term as Ĥ (i) = Ω( )(ei(kˆx ωt+φ) +e i(kˆx ωt+φ) ). (9) We can write + and + with the raising and lowering operators as + = ˆσ + = ( ), + = ˆσ = ( ). We will use the interaction picture with the free Hamiltonian being Ĥ = Ĥ (m) + Ĥ(e) and the interaction Ĥ(i). The transformation operator being Û = exp[ (i/ )Ĥt] we get the interaction into the form of Ĥ int = Û Ĥ(i) Û = Ωe(i/ )Ĥ(e)t (ˆσ + + ˆσ )e ( i/ )Ĥ(e)t e (i/ )Ĥ(m) t (e i(kˆx ωt+φ) +e i(kˆx ωt+φ) )e ( i/ )Ĥ(m) t = Ω(ˆσ +e iω t + ˆσ e iω t ) e (i/ )Ĥ(m)t (e i(kˆx ωt+φ) +e i(kˆx ωt+φ) )e ( i/ )Ĥ(m)t. () The transformation of the motional part into the interaction picture is the same as the transformation to Heisenberg picture. The Schrödinger picture operator ˆx will be replaced with the Heisenberg picture operator ˆx(t) from equation (9) kˆx(t) = η(âu (t)+â u(t)) () where we have now defined the Lamb-Dicke parameter η = k /(mν). Using the definition of u(t) we can expand the exponent terms in Ĥint. For the first one we get exp[i(η(âu (t)+â u(t))+(ω ω)t+φ)] ( = e i((ω ω)t+φ (iη) m ) âe iβω rft m! m= n= +â e iβω rft C ne inω rft n= C n e inω rft) m. () 8

10 We see that when ω ω = (l lβ)ω rf the â term will be varying slowly in time and similarly for the â when ω ω = (l lβ)ω rf. Here we use the rotating wave approximation and assume that the other terms are rotating so fast that they do not affect the time evolution of the system that much and can be ignored. The resonance frequency can be tuned to specific combination of l and l. Usually η meaning that the coupling strength vanishes quickly with higher l and l so the usual case is to choose l = which is what we are going to do now. We have the exponent term as exp[i(η(âu (t)+â u(t))+(ω ω)t+φ)] The total interaction Hamiltonian is = exp[i(η(âe iνt +â e iνt )+(ω ω)t+φ)]. (3) Ĥ int = Ω [ˆσ + exp[i(η(âe iνt +â e iνt )+(ω ω)t+φ)] + ˆσ + exp[i( η(âe iνt +â e iνt )+(ω +ω)t φ)] + ˆσ exp[i(η(âe iνt +â e iνt )+( ω ω)t+φ)] ] + ˆσ exp[i( η(âe iνt +â e iνt )+( ω +ω)t φ)]. (4) The interaction is time-dependent but in most cases the time-dependency causes very little change so we can neglect the time-dependency. By asserting t = into the interaction Hamiltonian we get Ĥ int = [ˆσ Ω + exp[i(η(â+â )+φ)]+ ˆσ + exp[ i(η(â+â )+φ)] ] + ˆσ exp[i(η(â+â )+φ)]+ ˆσ exp[ i( η(â+â )+φ)] ( ) = Ωcos η(â +â)+φ ˆσ x. (5) Now we have obtained the interaction part of the Hamiltonian. Now the Hamiltonian is ( ) Ĥ = ν +â â + ω ( ) ˆσ z + Ωcos η(â +â)+φ ˆσ x. (6) 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 mean systems that do not occur naturally but are intentionally built. In this case the artificial systems that we are going to study are superconducting circuits. 9

11 With current technology we can build superconducting circuits that are small enough for quantum effects to become visible. The artificially built quantum systems are an excellent way to study quantum phenomena because these system often offer a better control over the relevant parameters of the system. 3. Superconductivity Superconductivity was first discovered by Heike Kamerlingh Onnes in 9 [8]. He observed that under certain critical temperature mercury completely lost its electrical resistance. Kamerlingh Onnes observed the same phenomenon also in some other metals and after that various other materials exhibiting the phenomenon has been discovered. Superconductivity requires a very low temperature. For metals the transition temperature is usually less than K. There exist compounds and alloys with higher transition temperatures. In some cases it is even over K. Another characteristic of superconductivity was discovered by Walther Meissner and Robert Ochsenfeld in 933 [9] when they found out that there is no magnetic field in a superconductor. More importantly they discovered that a normal material will expel the magnetic field when it is cooled below its critical temperature. This is called Meissner effect. Meissner effect also implies that superconductivity can be broken by a critical magnetic field. In 957 John Bardeen, Leon Cooper and John Schrieffer proposed a theory explaining superconductivity []. This is known as the BCS theory. The theory is based on the idea of Cooper pairs. [] 3.. Cooper pairs A Cooper pair consist of two electrons bound together, thus forming a particle that behaves as a boson. The idea that there exists a weak attractive force that binds two electrons into a pair was first presented by Leon Cooper in 956 []. Let us consider two electrons added to a Fermi sea at temperature T =, and that the electrons interact only with each other not with any other electrons. We expect that the lowest energy state of the system has zero total momentum and that the two electrons must have equal and opposite momenta. This leads us to a wave function of the form ψ(r,r ) = k e ik (r r ). (7) Next we take account the spins of the electrons. Since we know that the total wave function for fermions has to be antisymmetric with respect to an exchange of two particles, the total ψ must be either the product of cos(k (r r )) and the antisymmetric spin state or

12 sin(k (r r )) and one of the symmetric spin states, + or. Since we are expecting an attractive interaction we assume that the antisymmetric spin state has lower energy, because the cosine term gives larger probability for the particles to be near each other. This means the wave function is [ ] ψ(r r ) = cos(k (r r )) ( ). (8) k Putting this into the Schrödinger equation of the system we get (E ǫ k )g k = k V kk g k (9) where ǫ k = k m and V kk = Ω V(r)e i(k k) r dr (3) where r is the distance between the electrons and Ω is the normalization volume. We make an approximation that V kk = V for energies less than ω c away from E F and elsewhere V kk =. E F is the energy that both of the electrons would have if there were no interaction between them. Now we have k g k = V g k. (3) E ǫ k Taking summation k over both sides and dividing by g k we get V = k We can replace the summation by integration E F = N k ǫ k E. (3) EF + ω c E F de (33) where N denotes the density of states for the electrons. Now EF + ω c V = N de ǫ k E = N ( ) ln EF E + ω c. (34) E F E Assuming a weak-coupling NV we have E E F ω c e NV. (35) From this we can see that the energy E for the bound electrons is smaller than the energy for the non-interacting electrons E F.

13 We have seen that an attractive interaction between two electrons could bound them forming Cooper pairs. The remaining question is where could this attractive force arise. We know that there is Coulomb interaction between two electrons but this interaction is repulsive. The attractive interaction comes when we take into account the motion of the ion cores. The first electron attracts the positive ions and these ions attract the other electron. This gives an effective attractive interaction between the electrons. It has been calculated that the attractive force created this way is about the same size as the repulsive Coulomb force. This means that it is possible for the total interaction to be attractive. [] 3.. Josephson effect A tunnel junction in electronics is a small barrier of insulating material between two conducting materials. Particles can move past the insulating layer by means of quantum tunneling. In 96 Brian Josephson proposed that a superconducting tunnel junction with no voltage difference across it would produce a supercurrent caused by the tunneling of Cooper pairs [3]. Further on, Josephson predicted that if a non-zero voltage V would go over the junction it would cause an alternating current with frequency ν = ev/h. In the case of a weakly coupled tunnel junction, the current going through the junction is I = I sin(φ) (36) where φ is the phase difference over the insulating barrier and I is the maximum supercurrent. The time evolution of the phase is given by dφ dt = ev. (37) From these we see that when V = we get a constant current over the junction. This is called the dc Josephson effect. For V the current oscillates with frequency ν = ev/h. This is called the ac Josephson effect. We can present the phase as the flux across the junction φ = π Φ Φ (38) where Φ = h/e is the flux quantum. We can get the potential energy of the Josephson junction by ( ) ( ) π π E = I(t)V(t)dt = I sin Φ Φdt = E J cos Φ (39) Φ Φ where we have defined E J = I /(e). The potential difference over the junction draws equal but opposite charges on the surfaces of the insulator so the junction has also capacitive

14 Figure : The circuit diagram of Josephson junction. properties. In order to take into account the capacitance of the Josephson junction in circuit diagrams a capacitor is added parallel to the ideal Josephson element. The circuit diagram of Josephson junction is shown in figure. [, 4] 3. Quantum network theory In order to study the quantum effects of the circuit we need to introduce the quantum network theory. The theory was developed by Bernard Yurke and John S. Denker[5] and Michel Devoret[6]. In Quantum network theory we use flux Φ and charge Q as the canonical coordinates. They behave like the canonical coordinates, position q and momentum p, in classical mechanics. An electric circuit can be described as a network whose branches are electrical components. A branch is a two-terminal electrical element such as capacitor or inductor. Each branch b is associated with two variables: the voltage V b and the current I b. They can be defined with electromagnetic fields as V b = endof b I b = µ beginning of b around b E ds (4) B ds. (4) For the Hamiltonian description of circuits we introduce branch flux and branch charge which are defined as Φ b (t) = Q b (t) = t t 3 V b (t )dt (4) I b (t )dt. (43)

15 We assume that at time t = there are no voltages or currents in the circuit. With these definitions we get the currents for inductors and capacitors as I b = Φ b /L (44) The energy in a branch b is E b (t) = I b = C Φ b. (45) t This gives the energies for inductors and for capacitors V b (t )I b (t )dt. (46) E b = Φ b L (47) E b = C Φ b. (48) The electrical circuits follow the Kirchhoff s laws which state that the sum of voltages around a loop l should be zero as long as the flux Φ l through the loop remains constant and that the sum of currents arriving at point n should be zero as long as the charge of the point Q n remains constant, Φ b = Φ l (49) allbaroundl allbarriving atn Q b = Q n. (5) With the Kirchhoff s laws we could formulate the equations of motion for the system with either using the fluxes or the charges. From these we could get the Lagrangian of the circuit using the Euler-Lagrange equation. A more convenient way to obtain the Lagrangian is by using the analogy of electric circuits and mechanical systems. For a mechanical system the Lagrangian is L = T V. This works also for the electric circuits. In electric circuits the kinetic energy terms T are the terms that depend on the charge Q. This means the capacitive components. The potential terms V are the terms that depend on the flux Φ. These are the inductive components. [6, 7] 3.. LC-circuit Let us now consider an LC-circuit pictured in figure. From the Kirchhoff s laws we get Φ + Φ = Φ. The branch fluxes Φ and Φ can be expressed 4

16 Figure : LC-circuit. C and L are the capacitance and inductance. Φ is the magnetic flux that goes through the circuit loop. Φ and Φ are the branch fluxes going through the inductor and the capacitor. Φ a is the node flux and the other node is chosen as the ground. in terms the node fluxes. One of the nodes can be chosen as the ground by defining its flux as zero. This leaves only one non-zero node flux Φ a. Now we have Φ = Φ a Φ = Φ Φ a. (5) In this circuit the capacitor C is the only capacitive term so the kinetic energy term is T = C Φ = C Φ a. (5) The only potential term is the inductor L therefore the potential energy is V = Φ L = Φ a L. (53) This means that in this case the Lagrangian of the system is L = C Φ a Φ a L. (54) The node charge can be defined as and the Hamiltonian of the system is Q a = L Φ a = C Φ a (55) H = Φ a L Φ a L = C Φ a + Φ a L = Q a C + Φ a L. (56) 5

17 Now we can move to the quantum mechanical description by replacing the classical variables with operators Φ a ˆΦ a Q a ˆQ a (57) H Ĥ. Since the canonical variables of electrical circuits Φ a and Q a are analogical with the canonical variables of mechanical systems they satisfy [ˆΦ a, ˆQ a ] = i (58) ˆQ a = i d dˆφ a. (59) We can also define the annihilation and creation operators as â = (ˆΦ a +iz ˆQa ) Z â = (ˆΦ a iz ˆQa ) Z (6) where Z = L/C. From these we get ˆΦ a = Z (â +â) (6) ˆQ a = i (â â). (6) Z Inserting these into the Hamiltonian and using the commutation relation [â,â ] = we get ( Ĥ = ω â â+ ) (63) where ω = /LC. This is the Hamiltonian of a harmonic oscillator with ω being the angular frequency of the oscillator. [6, 7] 3.. Single-Cooper-pair transistor The capacitors and inductors depend linearly on charge or flux and the circuits made out of these are linear. Since the LC-circuit behaves as a harmonic oscillator it displays rather trivial quantum effects. In order to see non-trivial quantum effects we need a non-linear circuit element. The simplest non-linear circuit element is the Josephson tunnel junction. 6

18 Figure 3: Circuit diagram of a single-cooper-pair transistor. V g and C g are the gate voltage and capacitance respectively. Φ is the magnetic flux that goes through the circuit loop and Φ a is the node flux. C and C are the capacitances associated with the Josephson junctions and E J and E J are the maximum energies of the Josephson junctions. Interrupting a superconducting loop with two Josephson junctions we insulate a small island. The charge on the island can be controlled through gate voltage. As long as the thermal fluctuation energy is smaller than the energy required for a Cooper pair to tunnel through the insulating layer and there is no external disturbance in resonance with other than the lowest transition, the single-cooper-pair transistor can be regarded as a qubit. The potential energy for Josephson tunnel junctions is ( ) πφb E b = E J cos (64) where Φ = h/e is the flux quantum. Electrical circuit of a single-cooperpair transistor is presented in figure 3. Similarly as with the LC-circuit we can obtain the Lagrangian as the difference between kinetic and potential energies: L = ( (C +C +C g ) Φ πφa a+e J cos Φ Φ ) ( +E J cos ) π( Φ Φ a ). (65) Φ The node charge is Q a = L Φ a = (C +C +C g ) Φ a (66) 7

19 and the Hamiltonian H = Q a C Σ E J cos ( πφa Φ ) ( ) π( Φ Φ a ) E J cos Φ (67) where C Σ = C +C +C g. Changing these into operators and defining ˆΦ = (ˆΦ ˆΦ ) = ˆΦ a Φ (68) we get the Hamiltonian to a form Ĥ = ˆQ [ ( ) a πˆφ E J cos cos C Σ Φ ( π Φ Φ ) ( ) ( )] πˆφ π Φ +dsin sin Φ Φ (69) where E J = E J +E J and d = (E J E J )/E J. The eigenstates of the charge operator ˆQ a are ψ Qa (Φ) = e iqaφ/ = e i(q+qg)φ/ (7) where Q g = C g V g is the gate charge and Q = (C + C ) Φ is the charge of the island. Only way for its charge to change is if a Cooper pair tunnels through one of the Josephson junctions. This means that the charge can only change by multiples of the charge of a Cooper pair (±e). If we now look at the energy of a Josephson tunnel junction it can be written in the operator form as E J cos πˆφ Φ = E J (eieˆφ/ +e ieˆφ/ ). (7) We see that if we operate with Josephson operator to a charge eigenstate it gives two different charge eigenstate that differ from the original charge state by ±e. This means that the Josephson tunnel junction transfers one Cooper 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 ˆQ = eˆn and e ieˆφ/ n = n. The Hamiltonian becomes Ĥ = 4E C ( Q g e n) n n n= [ ( ) ( )] E J π Φ π Φ cos +idsin n n Φ Φ [ ( ) ( )] E J π Φ π Φ cos idsin n+ n. Φ Φ (7) 8

20 .5 Energy [GHz] Q g /(e) Figure 4: The energy states of a single-cooper-pair transistor as a function of Q g. The red lines are in the case where there is no coupling between the charge states (E J = ) and the blue lines are the energy states when there is coupling (E J =. GHz). Parameters used in this figure are E C = GHz, φ = and d =. 9

21 In figure 4 we can see the lowest energy states of the Hamiltonian (7) in the cases of no coupling between the charge states (E J = ) and weak coupling between the charge states (E J /E C =.). When there is no coupling the energy states intersect at points Q g /(e) = n+.5. With the weak interaction (E J E C ) there is a small gap between the intersection points. When Q g /(e) n +.5 the energy difference between the two lowest states is much smaller than the energy difference between any other pair of states. We will note these states as and. We can truncate the Hamiltonian (7) to only contain the two lowest states. We get Ĥ = 4E C ( Q g e )ˆσ z E J [ cos ( π Φ Φ ) ˆσ x +dsin ( π Φ Φ )ˆσ y ] (73) where the constant term has been left out. The qubit described here is a charge qubit. There are also flux [8] and phase qubits [9]. [6, 7, ] 3..3 SCPT coupled to a LC-circuit Now we study a circuit where we have single-cooper-pair transistor coupled to an LC-circuit. The circuit is shown in figure 5. We have the following Lagrangian L = C g Φ a + C Φ a + C Φ b L +E Jcos ( Φ a Φ b ) + C Φ b ( ) ( πφa +E J cos Φ π(φ a Φ b Φ) Φ ). (74) Making a change of variables we get L = C g ( Θ Φ ) + C Φ L +E Jcos ( ( π(θ Φ = Φ b Θ = Φ+ Φ Θ Φ ) + C ) Φ (Φ+ Φ)) Φ a (75) ( Θ+ Φ ) + C Φ ( π(θ+ +E J cos Φ ) (Φ+ Φ). (76)

22 Figure 5: The superconducting circuit of a two-level system coupled nonlinearly with a harmonic oscillator. V g and C g are the gate voltage and capacitance. Φ is the magnetic flux that goes through the circuit loop and Φ a and Φ b are the node fluxes. C and C are the capacitances associated with the Josephson junctions and E J and E J are the maximum energies of the Josephson junctions. L and C are the inductance and capacitance of the LC-circuit.

23 Assuming C = C this can be written as ( L = C g Θ Φ ) ( +C Θ + Φ ) + C 4 Φ Φ L ( ) ( πθ π(φ+ +E J [cos cos Φ) ) ( πθ +dsin Φ Φ Φ ) ( sin π(φ+ Φ) )]. Φ (77) Now we calculate the charges corresponding to the canonical coordinates Θ and Φ: and from these we get Q = L Θ = (C +C g ) Θ C g Φ q = L Φ = 4C +C +C g Φ C g 4 Θ (78) Θ = (4C +C +C g )Q+C g q 4CC +C +CC g +C C g Φ = (C +C g )q +C g Q CC +C +CC. g +C C g (79) Now we can calculate the Hamiltonian as H = i Q i Φi L = 4C +C +C g 8CC +C +CC Q g +C C g + C +C g CC +C +CC q + C g g +C C g CC +C +CC Qq + Φ g +C C g L ( ) ( πθ π(φ+ E J [cos cos Φ) ) ( ) ( πθ π(φ+ +dsin sin Φ) )]. Φ Φ Φ Φ (8) By making the approximation C C C g we get 4C +C +C g CC +C +CC 4C +C g +C C g C (C +C ) C C +C g CC +C +CC g +C C g C g CC +C +CC g +C C g C C (C +C ) C (8)

24 After the quantization, the Hamiltonian operator can be written as Ĥ = ˆq C + ˆΦ L + ˆQ 4C E J [cos ( πˆθ Φ ) cos ( π(ˆφ+ Φ) ) ( +dsin Φ ) ( πˆθ sin Φ π(ˆφ+ Φ) )]. Φ (8) We see that the first two terms are exactly the same as in the case of the LC-circuit. Just as in the case of the LC-circuit we can express ˆq and ˆΦ with annihilation and creation operators. The other terms are the same as with the single-cooper-pair transistor. Treating the operators ˆQ and ˆΘ as with the SCPT-circuit we get the Hamiltonian into the form Ĥ = ( LC E J â â+ [ cos +dsin )+4E C ( πe L h πe L h Q g e ) ˆσ z C (â +â)+ π Φ ˆσ x C (â +â)+ π Φ Φ Φ ˆσ y ]. (83) This is similar to the Hamiltonian of the Paul trap. The main exception is that this Hamiltonian contains the ˆσ y term. This term is due to the fact that the Josephson junctions have slightly different energies. The factor d that is the relative difference between the energies is usually relatively small but not so small that it will not have an effect in our calculations. [6] 3.3 Superconducting circuit compared to ion trap Because the circuit discussed above has almost the same Hamiltonian as the trapped ion these systems are analogous. The advantage with the circuit compared to the trapped ion is that with the circuit we can have better control of the parameters involved in the system. This way we can often study the system with a wider range of parameters than what the original system would have allowed. In the case of our circuit the gate voltage V g and the flux Φ are parameters that can be controlled after the building of the system. Other parameters are fixed upon the building. 4 Trapped ion Hamiltonian Now we will derive the approximations of the interaction and after that we will introduce a probe to our system and derive the absorption spectrum 3

25 given by this. Let us start by writing the system s Hamiltonian again Ĥ = ω c (â â+ )+ ω ˆσ z g η cos(η(â +â)+φ)ˆσ x + d g η sin(η(â +â)+φ)ˆσ y, (84) where â and â are the creation and annihilation operators of the harmonic oscillator, ˆσ i are the Pauli spin matrices, ω c is the characteristic frequency of theharmonicoscillator, ω isthelevelseparationofthequbit, η isthelamb- Dicke parameter, g is the strength of the coupling between the oscillator and qubit, d is asymmetry parameter, and φ is a control parameter. In the circuit the qubit energy is ω = 4E C ( n ) where n = C g V g /e. Because we can control the gate voltage V g we can control the qubit energy. We can also control the static magnetic flux Φ that flows through the induction loop. This affects parameter φ = π Φ/Φ where Φ = h/e. The Lamb-Dicke parameter is η = πz /R K where Z = L/C and R K = h/e. The strength of the coupling depends on the Josephson energy g = ηe J /. There has been theoretical research done with the same trapped ion system in paper []. In that paper calculations are done for a specific matrix elements of the Hamiltonian n,a Ĥ n + k,a. In our case we will look at the complete Hamiltonian. 4. Jaynes-Cummings model The eigenvalues of the Hamiltonian (84) are not generally solvable analytically. With certain approximations we can write the Hamiltonian in the Jaynes-Cummings form. The Jaynes-Cummings model was introduced by E. Jaynes and F. Cummings in 963 []. Originally it was used to study the relationship between the quantum theory and the semi-classical theory of radiation. Jaynes-Cummings model is commonly used to model a qubit interacting with an optical cavity. The advance with the model compared to the original Hamiltonian is that the eigenvalue problem of the Jaynes- Cummings Hamiltonian is analytically solvable. When φ = π/ and d = the interaction part of the Hamiltonian (84) becomes Ĥ int = g η sin(η(â +â))ˆσ x. (85) By assuming that η is small we can expand the sine into series and only take into account the lowest order term. Now we get Ĥ int g(â +â)ˆσ x. (86) Next we use the rotating wave approximation. Let us switch into an inter- 4

26 action picture. The interaction part of the Hamiltonian becomes e i Ĥt Ĥ int e i Ĥt = e i(ωc(â â+ )+ω ˆσ z)t ( g(â +â)ˆσ x )e i(ωc(â â+ )+ω ˆσ z)t = g(e i(ωc+ω )tâˆσ +e i(ωc+ω )tâ ˆσ + +e i(ωc ω )tâˆσ + +e i(ωc ω )tâ ˆσ ). (87) Near resonance (ω c ω ) the terms e ±i(ωc+ω )t oscillate rapidly compared to terms e ±i(ωc ω )t so we can assume that the rapidly rotating terms average out in the relevant timescale. After the rotating wave approximation we obtain the Hamiltonian into the form Ĥ JC ω c (â â+ )+ ω ˆσ z + g(â ˆσ +âˆσ + ). (88) This is the Jaynes-Cummings model. We can see from the coupling term that this couples the states n and n+. The eigenvalue problem of the Jaynes-Cummings model (88) can be solved exactly. The ground state = is not coupled with any other states and we get its eigenvalue as E = / where ( = ω ω ) c. Presenting the rest of the Hamiltonian as a matrix in a basis where n n+ n =,,,... we get the Hamiltonian in the form of ( (n+ Ĥ JC = ) ω c + ω g ) n+ g n+ (n+ 3 ) ω c ω = (n+) ω c + ( ω ω c g ) (89) n+ g. n+ ω c ω Definingsinθ n = g n+/ +4g (n+)andcosθ n = / +4g (n+) we get the Hamiltonian (88) into the form Ĥ JC = (n+) ω c + ( ) +4g cosθn sinθ (n+) n. (9) sinθ n cosθ n This can be diagonalized by making a unitary rotation Û Ĥ JC Û with This gives the eigenenergies Û = ( cos θ n sin θn sin θn cos θn ). (9) E ±,n = (n+) ω c ± +4g (n+) (9) 5

27 and the eigenvectors ( ) ( ) θn θn +,n = cos n +sin n+, ( ) ( ) θn θn,n = sin n +cos n+. (93) 4. Conventional Bloch-Siegert shift The Jaynes-Cummings model holds well when we are near resonance (ω ω c ) but once we go further from resonance we can no longer ignore the counter-rotating terms e ±i(ωc ω )t that we assumed to average out in the rotating wave approximation. The difference in the eigenfrequencies caused by the counter-rotating terms is known as the conventional Bloch-Siegert shift. This was first shown to exist by F. Bloch and A. Siegert in 94 [3]. We can add correction terms to the Jaynes-Cummings model in order to take into account the Bloch-Siegert shift. The term that was left out by the rotating wave approximation in the Jaynes-Cummings model (88) was Ĥ BS = g(âˆσ +â ˆσ + ). (94) Still assuming that we are near resonance the term can be interpreted as a small perturbation to the Jaynes-Cummings Hamiltonian. Let us make a unitary transformation of eŝ(ĥjc+ĥbs)e Ŝ where Ŝ = α(â ˆσ + âˆσ ) and α = g/(ω c +ω ). Expanding this with Baker-Campbell-Hausdorff formula eŝôe Ŝ = Ô +[Ŝ,Ô]+! [Ŝ,[Ŝ,Ô]]+ [Ŝ,[Ŝ,[Ŝ,Ô]]]+... (95) 3! and using the commutation relation of the ladder operators [â,â ] = we can calculate eŝâ âe Ŝ = â â α(âˆσ +â ˆσ + ) α [â ˆσ +,âˆσ ] eŝˆσ z e Ŝ = ˆσ z α(âˆσ +â ˆσ + ) α [â ˆσ +,âˆσ ] eŝ(â ˆσ +âˆσ + )e Ŝ = â ˆσ +âˆσ + eŝ(âˆσ +â ˆσ + )e Ŝ = âˆσ +â ˆσ + +α[â ˆσ +,âˆσ ] (96) wherewehaveneglectedthetermsthatarehigherthansecondordering and the two-photon processes. By two-photon process we mean processes where the Fock state of the harmonic oscillator changes by two. In other words the terms containing â or (â ). The Hamiltonian after the transformation is Ĥ = ĤJC + g [â ˆσ +,âˆσ ] = ω c +ω ĤJC + g (â âˆσ z + ω c +ω ˆσ z ). (97) 6

28 The eigenvalue of the ground state = is E = ω BS where ω BS = g /(ω c + ω ). By defining cosθ n = n / n +4g (n+), sinθ n = g n+/ n +4g (n+) and n = ω ω c +ω BS (n+) we can write the Hamiltonian (97) in a form Ĥ = (n+) ω c ω BS + cosθ n +4g (n+)( n sinθ n ) sinθ n cosθ n. (98) This can be solved similarly as the Hamiltonian (9). We get the eigenenergies E ±,n = (n+) ω c ω BS ± n +4g (n+) (99) and eigenvectors ( ) ( ) θ +,n = cos n θ n +sin n n+, ( ) ( ) θ,n = sin n θ n +cos n n+. () This is the Bloch-Siegert corrected Jaynes-Cumming model. [4] 4.3 Absorption spectrum The way to get information from our system is by studying the absorption spectrum from a weak harmonic perturbation. This is done by introducing a probe with Hamiltonian Ĥ P (t) = g P (â +â)cos(ω p t) () tothesystem. WewillusetheFermi sgoldenruletocalculatetheabsorption spectrum Fermi s golden rule Let us assume that we have a time-independent system. If we now introduce a weak time-dependent perturbation, Fermi s golden rule gives us the probability per time unit for a transition from one of the energy states of the system to another. For the Fermi s golden rule we need to use the time-dependent perturbation theory to the weak perturbation. We now have a Hamiltonian of the form Ĥ(t) = Ĥ +ĤP(t) () where Ĥ is the time-independent Hamiltonian and ĤP(t) is the timedependent weak perturbation. We have the solution for the time-independent problem Ĥ u q = ǫ q u q. (3) 7

29 Now we would like to solve the Schrödinger equation First we switch into the interaction picture i ψ(t) = Ĥ(t) ψ(t). (4) t ψ(t) I = Û(t) ψ(t) Ĥ (int) P (t) = Û(t)ĤP(t)Û (t) (5) where Û(t) = exp[(i/ )Ĥt]. Now we have the Schrödinger equation in the from i t ψ(t) I = Ĥ(int) P (t) ψ(t) I. (6) Writing the states ψ(t) I in the basis of the eigenstates of Ĥ as ψ(t) I = q a q(t) u q and plugging this into the Schrödinger equation we obtain i q a q (t) u q = t i a i (t)ĥ(int) P (t) u i. (7) Multiplying this from the left by u f we get i a f(t) t = i a i (t) u f Ĥ(int) P (t) u i = i a i (t) u f Û(t)ĤP(t)Û (t) u i (8) = i a i (t)e i (ǫ f ǫ i )t u f ĤP(t) u i. We can write the probe Hamiltonian as Ĥ P = g P (â +â) (eiω Pt +e iω Pt ). (9) With this we get the differential equation for the probability amplitude a f (t) for the system to be in state u f as a f (t) t = g P i i a i (t)e i(ω fi ω P )t F fi + g P i a i (t)e i(ω fi+ω P )t F fi () i where ω fi = ǫ f ǫ i and F fi = u f (â +â) u i. Assumingg P tobesmallwecandoaperturbativeexpansionforequation () and only take into account the terms up to first order in g P. Writing a i (t) = a () i (t)+λa () i (t) () 8

30 and replacing F fi λf fi. Now we get a () f (t) = t a () f (t) = g P t i i ) a () i (t) (e i(ω fi ω P )t F fi +e i(ω fi+ω P )t F fi. () If we assume that the probe is activated at t = then a () f () = and we have a () f (t) = g P i i a () i F fi t ( e i(ω fi ω P )t +e i(ω fi+ω P )t ) dt. (3) The transition amplitude for the system to move from state u i to state u f is γ fi (t) = g t P i a() i F fi = g P a() i F fi ( e i(ω fi ω P )t +e i(ω fi+ω P )t ) dt ( e i(ω fi ω P )t + ei(ωfi+ωp)t ω if ω P ω if +ω P ). (4) We have two cases where the denominator goes to zero ω fi = ω P and ω fi = ω P. The case ω fi = ω P can be written as ǫ i = ǫ f ω P. This means that the system moves to a higher energy state i.e. the system absorbs energy. Similarly ω fi = ω P is the case for emission. Wearenowonlyconcernedaboutabsorptionsoweleaveouttheemission term. The transition probability for the system to move from state u i to state u f is P i f = γ fi (t) = g P 4 a() i F fi (ei(ω fi ω P )t )(e i(ω fi ω P )t ) (ω fi ω P ) = g P 4 a() i F fi cos((ω fi ω P )t) (ω fi ω P ) = g P a() i F fi ( ) (ω fi ω P ) sin (ωfi ω P )t. (5) This function is highly peaked at ω fi = ω P. We know that sin ωt lim t π tω = δ(x). (6) We assume that the time scale is such that approximating P i f as a delta function is appropriate. Since delta functions are difficult to detect in the 9

31 absorption spectrum we are going to broaden the spikes using Lorentzian line shape. We have 4sin ( (ω fi ω P )t ) γ if t (ω fi ω P ) πδ(ω fi ω P ) (ω fi ω P ) + (7) 4 γ if where γ if is just a constant used for determining the width of the spike. The absorption rate P is the absorption probability per time unit P i f = g P 4 p γ if F fi i (ω fi ω P ) + 4 γ if (8) where p i = a () i is the occupation probability of state u i. The total absorption spectrum is achieved when all individual absorption rates are added together assuming γ if to be so small that the resonances do not overlap P a = g P 4 i,f p i γ u f (â +â) u i (ω fi ω P ) + 4 γ (9) where we have for simplicity assumed that all the resonances have the same width γ if = γ. The occupation probabilities can be assumed to follow the thermal occupation p i = e E i/k B T n e En/k BT. () Because the temperature in the system is often very low, the occupation probability p i is much higher for the ground state than for the other states. Because of that, we assume that the system is initially at the ground state and the occupation probability becomes {, i = p i = (), i. With this the absorption spectrum is P a = g P 4 γ u f (â +â) (ω f ω P ) +. () 4γ f Now we can calculate the absorption spectrum for the Jaynes-Cummings model (88). We have the eigenvectors (9) and eigenenergies (93). Now we just plug them into equation () and get P a = g P 4 = γg P 4 ( γ +, (â +â) (ω + ω P ) + + γ, (â +â) 4 γ (ω ω P ) + 4 γ ( sin θ (ω + ω P ) + 4 γ + cos θ (ω ω P ) + 4 γ 3 ) ) (3)

32 where ω ± = E ±, E. Same thing for the Bloch-Siegert corrected Jaynes-Cummings model(97) gives us the absorption spectrum [ ] P a = γg P 4 θ ) sin ( θ ) (ω + ω P) + + cos ( 4 γ (ω ω P) + 4 γ, (4) where ω ± = E ±, E. It is good to notice that the absorption spectrum calculated here is an approximation. In order to get the exact absorption spectrum we would need to solve numerically the master equation of the driven system. As long as the width of the resonances is small enough for them to not overlap this approximation should be accurate. [5, 6] 4.4 Lamb-Dicke regime Lamb-Dicke regime in ion trapping is the area where the coupling between the internal and motional ion states is so small that the transitions where the motional state is changed by more than one quantum number are suppressed. The Lamb-Dicke regime is expressed by inequality η (â+â ). (5) 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 outside the regime and at what point when increasing the Lamb-Dicke parameter the approximations start to deviate from the numerically calculated solutions. [4] 5 Calculations and Results Paper[7] shows experiments done with the same electric circuit as discussed in here. From this we get the parameter values used in the experiments: E J /h = 7. GHz, d =.9, L = 4 ph, C = pf and ω c = 3.5 GHz. These give η.3 and g.4. We know that these are values that can be achieved in experiments and that is why in our calculations we use parameter values that are close to these. Recently in a slightly modified system [8], η. was observed, and we expect that with a little optimization in the fabrication process, one can obtain even higher values. 3

33 Energy [GHz] [GHz] Figure 6: Eigenenergies of the lowest energy states of the Hamiltonian(84) as afunctionof. Thebluelinesaretheenergiesofthenumericalsolution, red lines are those of the Jaynes-Cummings model, green lines are the energies of the Bloch-Siegert corrected Jaynes-Cummings model and the black dashed lines are the energy states of Hamiltonian (84) in the case where there is no interaction (η = ). The parameters used in this figure are d =, η =., g = GHz, φ = π/ and ω c = 3.5 GHz. 3

34 In figure 6 we have plotted the eigenenergies of the system as a function of. The black dashed lines in the figure are the eigenenergies of the uncoupled Hamiltonian. The decreasing lines correspond to the states n andtheincreasinglinesto n. Wecanseethatintheuncoupledsystemthe eigenenergies of the second and third lowest states intersect at =. When the interaction is added the energies form a gap around the intersection point and there is no intersection. This is called an avoided crossing. At the intersections of the third and fourth states and fourth and fifth states there is no avoided crossings shown in the approximations. This is due to the fact that there is no coupling between these states. The coupling between these states would come from the multi-photon processes that were left out when the interaction was approximated as linear. There is no avoided crossing in the numerical solution at the crossing of the third and fourth states either but there is a small gap at the crossing of the fourth and fifth states. This is because we have used the parameters values d = and φ = π/. With these the interaction term is Ĥ int = g η sin(η(â +â))ˆσ x. (6) The series expansion of sine has only the odd order terms. The coupling between the third and fourth states comes from the two-photon interactions, the coupling between the fourth and fifth states comes from the three-photon interactions and so on. Because of this every other avoided crossing is not seen in the numerical solution. In figure 7 we have plotted the numerically calculated absorption spectrum of Hamiltonian (84) and the resonance curves of both the Jaynes- Cummings and the Bloch-Siegert corrected Jaynes-Cummings models for η =.. The red dashed lines in the figure are the resonance frequency of the qubit (ω P = ω ) and the first resonance frequency of the oscillator (ω P = ω c ). These correspond to the transitions from state to states and respectively. The avoided crossing seen in this figure is the one between the first and second exited states seen in figure 6. The absorption is strongest from the ground state to the state. Near the resonance the state is coupled with the state and there is also strong absorption along the resonance frequency of the qubit. Further from resonance the coupling is weaker and this absorption diminishes. The numerical absorption spectrum is calculated by presenting the Hamiltonian (84) as a matrix in the basis { n,a = n a n =,,,...anda =, } where n are the oscillator eigenstates and a the qubit eigenstates. The matrix is truncated to include only twenty lowest oscillator eigenstates making it a 4 4 matrix. The eigenstates and eigenenergies of this matrix are numerically calculated and using these we get the absorption spectrum from equation (). 33

35 [GHz] [GHz] Error [GHz] ω P [GHz] 4 [GHz] Figure 7: On the left figure is the absorption spectrum plotted as a functions of andω P. TheredlinesaretheresonancecurvesoftheJaynes-Cummings model and the green lines denote the resonances of the Bloch-Siegert corrected Jaynes-Cummings model. The dashed lines are ω P = ω c (vertical) and ω P = ω. On the right figure, we plot the absolute errors between the resonance curves of the approximations and the numerical solution. The red lines are the errors of the Jaynes-Cummings approximation and the green lines are the errors of the Bloch-Siegert corrected approximation. The dashed lines are the errors of the lower energy resonance and the continuous lines the errors of the higher energy resonance. The parameter values used in this figure are d =, η =. GHz, g = GHz, φ = π/, ω c = 3.5 GHz, g P =.5 GHz and γ =. GHz. 34