Nonadiabatic Charge Pumping in Superconducting Circuits

Helsinki University of Technology Faculty of Information and Natural Sciences Engineering Physics and Mathematics Nonadiabatic Charge Pumping in Superconducting Circuits Bachelor s thesis May 23, 29 Janne Kokkala

Helsinki University of Technology Faculty of Information and Natural Sciences Abstract of bachelor s thesis Author: Degree Programme: Major Subject: Title: Title in Finnish: Chair: Supervisor: Instructor: Abstract: Janne Kokkala Engineering Physics and Mathematics Engineering Physics Nonadiabatic Charge Pumping in Superconducting Circuits Ei-adiabaattinen varauspumppaus suprajohtavissa piireissä Tfy-3 Physics Prof. Risto Nieminen Dr. Tech. Mikko Möttönen Superconducting circuits can potentially be used for both constructing a quantum bit and redefining the unit of electric current. This thesis considers possible control cycles to pump Cooper pairs nonadiabatically in two superconducting devices: the Cooper pair sluice and the Cooper pair transistor. Both devices consist of a superconducting island separated by two Josephson junctions or superconducting quantum interference devices (SQUIDs) and a gate capacitor. The pumping is carried out by first moving a Cooper pair out of the island, i.e. transferring the island from Cooper pair number eigenstate to eigenstate 1, through one of the junctions or SQUIDs and then moving a Cooper pair back to the island through the other one. Some idealizations are made, e.g. neglecting the influence of the environment. Twostate approximation is used to derive parameters that are used to pump Cooper pairs through the devices. Multi-state approximation is then used to numerically examine the evolution of the systems during several pumping cycles. In the Cooper pair sluice, the direction of the pumped current is controlled by adjusting the Josephson energies of the two SQUIDs. Even with some nonidealities, the pumped charge per cycle is constant, if the pumping frequency is adjusted properly. However, the required accuracy for metrological purposes is very high. Because the pumped charge per cycle depends strongly on the parameters, we did not find the nonadiabatic operation of the sluice optimal for metrology. In the Cooper pair transistor, pumping is carried out by maintaining a bias voltage over the device and adjusting the voltage in the gate. We found a set of parameters for which the pumping works nearly periodically for at least several hundred cycles. However, due to the limited time of this research, no nonidealities were considered. Nevertheless, the results are promising and further research is recommended. Number of pages: 31+3 Faculty fills Approved: Keywords: superconductivity, Cooper pair, sluice, CPT Library code:

Teknillinen korkeakoulu Informaatio- ja luonnontieteiden tiedekunta Kandidaatintyön tiivistelmä Tekijä: Tutkinto-ohjelma: Pääaine: Työn nimi: Työn nimi suomeksi: Professuurin koodi ja nimi: Työn valvoja: Työn ohjaaja: Tiivistelmä: Janne Kokkala Teknillinen fysiikka ja matematiikka Teknillinen fysiikka Nonadiabatic Charge Pumping in Superconducting Circuits Ei-adiabaattinen varauspumppaus suprajohtavissa piireissä Tfy-3 Fysiikka Prof. Risto Nieminen TkT Mikko Möttönen Suprajohtavia virtapiirejä voidaan mahdollisesti käyttää sekä kubitteina että sähkövirran yksikön uudelleenmäärittelemiseen. Tässä työssä käsitellään mahdollisia tapoja pumpata Cooperin pareja kahdessa suprajohtavassa laitteessa: Cooperin pari -sluicessa ja Cooperin pari -transistorissa (CPT). Molemmat laitteet koostuvat suprajohtavasta saarekkeesta, jotka on erotettu kahdella Josephson-liitoksella tai SQUID:lla (engl. superconducting quantum interference device), ja porttikondensaattorista. Pumppaus suoritetaan siirtämällä yksi Cooperin pari ulos saarekkeelta eli siirtämällä saari Cooperin parien lukumäärän ominaistilalta ominaistilalle -1 yhden liitoksen tai SQUID:n läpi ja sen jälkeen siirtämällä Cooperin pari takaisin toisen liitoksen tai SQUID:n läpi. Työssä tehdään joitakin idealisaatiota. Esimerkiksi ympäristön vaikutusta ei huomioida. Kontrollisyklin parametrit johdetaan käyttämällä kahden varaustilan approksimaatiota, minkä jälkeen monen tilan approksimaatiolla tutkitaan numeerisesti systeemin käyttäytymistä usean pumppaussyklin aikana. Sluicessa virran suuntaa hallitaan säätämällä SQUID:ien Josephson-energioita. Jopa hieman epäideaalisilla kontrollisykleillä pumpattu varaus yhtä sykliä kohti on vakio, jos pumppauksen taajuus säädetään oikein. Metrologinen virtalähde vaatii kuitenkin suuren tarkkuuden, ja koska sykliä kohti pumpattu varaus riippuu voimakkaasti parametreista, ei-adiabaattinen pumppaus sluicella ei liene optimaalinen metrologiaan. CPT:ssä Cooperin pareja pumpataan pitämällä vakiojännitettä laitteen yli ja säätämällä portin jännitettä. Parametrit voidaan valita siten että varauspumppaus toimii lähes jaksollisesti satoja pumppaussyklejä. Tutkimuksen rajoitetun ajan vuoksi mahdollisia epäideaalisuuksia ei tutkittu. Tulokset ovat kuitenkin lupaavia ja jatkotutkimusta suositellaan. Sivumäärä: 31+3 Täytetään tiedekunnassa Hyväksytty: Avainsanat: suprajohtavuus, Cooperin pari, sluice, CPT Kirjasto:

Contents 1 Introduction 1 2 Theoretical Background 3 2.1 Superconductivity......................... 3 2.2 The devices............................ 3 3 Numerical Methods 7 3.1 Finite-state approximation for the sluice............ 7 3.2 Finite-state approximation for the CPT............. 8 4 Cooper Pair Pumping with the Sluice 1 4.1 Pumping procedure........................ 1 4.2 Two-state approximation..................... 1 4.3 Multi-state approximation.................... 12 4.4 Effects of small residual energy................. 15 4.5 Finite ramp time of the Josephson energies........... 16 4.6 Error caused by inaccuracy of ϕ................. 18 5 Cooper Pair Pumping with the Cooper Pair Transistor 19 5.1 Pumping procedure........................ 19 5.2 Two-state approximation..................... 19 5.3 Multi-state approximation.................... 2 5.3.1 One pumping cycle.................... 21 5.3.2 Many pumping cycles................... 23 6 Discussion 27 6.1 Cooper pair sluice......................... 27 6.2 Cooper pair transistor...................... 27 A Suomenkielinen yhteenveto 32

List of Symbols and Abbreviations CPT Cooper pair transistor SQUID superconducting quantum interference device T, T π pumping cycle time and half of the pumping cycle time V g gate voltage C g gate capacitance k = l, r X k, k = l, r means that both X l and X r are considered, l meaning left and r meaning right E k Josephson energy of the left or right Josephson junction or SQUID, k = l, r C k capacitance of the left or right Josephson junction or SQUID, k = l, r ϕ k phase difference over the Josephson junction or SQUID, k = l, r φ phase operator of the island ϕ phase difference over the device n, ˆn number of Cooper pairs on the island and the corresponding operator n k, ˆn k number of Cooper pairs passed through the Josephson junction or SQUID k and the corresponding operator n, ˆn n l + n r and ˆn l + ˆn r, respectively n g gate charge in units of 2e C Σ total capacitance of the island E C charging energy of one excess Cooper pair on the island Φ magnetic flux through the Cooper pair sluice Φ k magnetic flux through the left or right SQUID, k = l, r EJ max maximum energy of the SQUIDs EJ res minimum energy of the SQUIDs k eigenstate of the operator ˆn corresponding to the eigenvalue k. k, k eigenstate of both the operator ˆn corresponding to the eigenvalue k and the operator ˆn corresponding to the eigenvalue k. ˆn ±, ˆn ± raising and lowering operators of n and n, respectively Ĥ, H Hamiltonian of the system and the corresponding matrix Î k, I k current operator through SQUID k and the corresponding matrix Q(t) average total charge pumped through the device at time t Q k (t) average total charge pumped through the Josephson junction or SQUID k at time t V b bias voltage over the ( Cooper ) pair transistor 1 σ x Pauli spin matrix 1

1 Introduction In 1911, Heike Kamerlingh Onnes discovered a sudden disappearance of the electric resistance of mercury when it was cooled below 4.2 K and a return of the resistance above the limit. The phenomenon of lack of resistance was named superconductivity. Later, several other superconducting materials have been found, even at temperatures above 1 K. No measurable decrease have been observed in currents in superconducting loops after they have been flowing freely for a year. [1] Superconductivity of nanoscale circuits gives rise to quantum effects. The superconducting circuits can be used to create quantum systems with discrete energy states. Because of good integrability of these circuits with conventional electronics and ability to scale up the systems, the circuits provide a promising candidate for a quantum bit, a controllable two-level quantum system which is the basic building block of quantum computers. [2 4] Additionally, a possible use of the superconducting circuits is redefining the unit of electric current, the ampere. A quantum metrological definition would allow comparing the values of the natural constants and e and hence would play an important role in redefining the international system of units (SI) to be fully based on fundamental constants [5]. However, an accurate high-yield current source is required, and creating one requires typically extremely accurate control over the quantum device. Currently, the most accurate single electron pumping results have been achieved by electron counting in a 7- junction electron pump [6]. Adiabatic pumping using the Cooper pair sluice has also been studied [7 1]. Other suggestions include a hybrid normalmetal superconductor turnstile [11], use of surface acoustic waves where electrons travel as particles in a time-dependent potential instead of tunneling through barriers [12, 13], and a topological control cycle for a Cooper pair pump [14]. In this thesis, we consider two different devices, the Cooper pair sluice and the Cooper pair transistor (CPT). Some assumptions simplifying the theoretical treatment are used and thus further experimental research is needed. We introduce nonadiabatic control cycles for pumping charge in the devices and find theoretically with some approximations the parameters required for the pumping procedure. Finally, we use numerical simulations to examine the evolution of the systems. We begin Sec. 2 by briefly reviewing the theory of superconducting circuits. Furthermore, we present the devices used in this thesis. In Sec. 3, we present

the numerical methods used in this thesis. Sections 4 and 5 are devoted to the sluice and the CPT, respectively. We introduce the control cycle and use two-state approximations to find equations for the control parameters. Furthermore, we use numerical simulations to examine the systems with different sets of parameters. In Sec. 6, we present the conclusions and make suggestions for further research. 2

3 2 Theoretical Background 2.1 Superconductivity In 1957, John Bardeen, Leon Cooper, and John Schrieffer introduced a microscopic theory of superconductivity, called the BCS theory. An attractive interaction, for example a phonon interaction, between electrons causes electrons of opposite spin to form so-called Cooper pairs, the breaking of which requires a constant amount of energy which is not available at low temperatures. These Cooper pairs form a condensate which is analogous to the Bose-Einstein condensate for bosons at low temperatures. Seven years before the BCS theory, in 195, Vitaly Ginzburg and Lev Landau had introduced a pseudo-wavefunction describing the superconducting electrons. In 1959, Lev Gor kov showed that the phenomenological GL theory is actually a limiting form of the BCS theory, and nowadays the existence of a many-particle condensate wavefunction is regarded as the universal characteristic of the superconducting state. [1] In 1962, Brian Josephson predicted that Cooper pairs should tunnel between two superconducting electrodes even at zero voltage difference. The current depends on the gauge-invariant phase difference γ as I S = I C sin γ. Further, he predicted that if a voltage difference V occurred between the electrodes, the phase difference would change in time according to γ = 2eV. The free energy stored in the Josephson junction is F = E J cosγ, where E J = ( I C /2e) is called the Josephson energy of the junction. [1] Two parallel identical Josephson junctions form a superconducting quantum interference device, a SQUID. It is effectively a Josephson junction whose Josephson energy depends on the magnetic flux through the SQUID [1]. SQUIDs are commonly used for measuring small magnetic fluxes but they can as well be used as tunable Josephson junctions [2, 7]. 2.2 The devices In this thesis, we consider two different devices: the Cooper pair sluice [7] shown in Fig. 1 and the Cooper pair transistor (CPT) shown in Fig. 2. The Cooper pair sluice consists of an island that is separated from superconducting leads by two SQUIDs. In addition, the island is connected to a gate capacitor and a voltage is applied at the other end of the capacitor. In the CPT, the SQUIDs are replaced by single Josephson junctions, and a bias

4 V g C g Φ l ϕ l Φ Φ ϕ r r Figure 1: The Cooper pair sluice. The figure is copied from [8]. V g C g +V b /2 V b /2 ϕ l ϕ r Figure 2: The Cooper pair transistor. voltage is maintained across the device. In both devices, Cooper pairs are pumped through the device varying a few adjustable parameters periodically according to a simple pumping cycle, the period of which is denoted by T. The Cooper pairs are pumped from right to left, and thus the direction of the current is from left to right. In the figures, V g is the gate voltage, C g is the gate capacitance, E k and C k for k = l, r are the Josephson energy and the capacitance of the left and right SQUID or Josephson junction, respectively. The phase differences over the left and right SQUID or Josephson junction are denoted by ϕ l and ϕ r, respectively. The phase on the island is φ = (ϕ l ϕ r )/2 and the total phase difference over the device is ϕ = ϕ l + ϕ r. The phases φ, ϕ r and ϕ l are conjugate variables with the Cooper pair number operators of the island, ˆn, and of the right and the left Josephson junctions or SQUIDs, ˆn r and ˆn l, respectively [1, 7]. The operators ˆn r and ˆn l denote the number of Cooper pairs passed through the right and left SQUIDs, respectively, and ˆn = ˆn l ˆn r denotes the number of extra Cooper pairs on the island.

5 For simplicity, we define n g = V g C g /(2e) to be the gate charge in units of 2e. In addition, we define C Σ = C g +C l +C r +C, where C is the self-capacitance of the island, to be the total capacitance of the island. Denoting the Coulomb energy for one excess Cooper pair on the island as E C = 2e 2 /C Σ, the total charging energy of the island can be expressed as E C (n n g ) 2. In the sluice, the total phase difference ϕ over the device can be fixed arbitrarily by the magnetic flux Φ through the loop [8]. Thus, ϕ may be treated as a real number. The Josephson energies of the SQUIDs, E l and E r, can be adjusted by the magnetic fluxes through the left and right SQUID, Φ l, and Φ r, respectively, such that they can take values between some minimum residual value EJ res and a maximum value EJ max. By adjusting the control cycle, the minimum Josephson energy of a SQUID during the procedure can always be increased and the maximum Josephson energy can be decreased. Thus we may assume that the energies EJ res and EJ max are identical for each SQUID. We assume that the circuits do not interact with the surrounding environment. In the case of the CPT, we further assume that no quasiparticle tunneling occurs. Hence an upper bound for the absolute value of the bias voltage V b is 2 /e [1], where =.2 mev is the superconducting gap of aluminium [15 17]. Moreover, a small value of the Josephson energies will cause thermal fluctuations to be significant. Let k, where k is integer, denote the eigenfunctions of ˆn corresponding to the eigenvalue k, i.e. ˆn k = k k. Because ˆn and φ are conjugate variables, ˆn = i φ and e ±iφ k = k ± 1 [8]. We denote ˆn ± = e ±iφ. The Hamiltonian of the sluice can be written as [7] Ĥ = E C (ˆn n g ) 2 E l cosϕ l E r cosϕ r = E C (ˆn n g ) 2 E l cos(ϕ/2 + φ) E r cos(ϕ/2 φ) ( = E C (ˆn n g ) 2 El 2 e iϕ/2 + E ) r 2 eiϕ/2 ˆn ( El 2 eiϕ/2 + E r 2 e iϕ/2 ) ˆn +. (1) The current operators Îl and Îr through the left and right SQUIDs or Josephson junctions are defined as Îk = 2ie[ˆn k, Ĥ], for k = l, r [8]. This implies Î l = 2e Ĥ = 2eE l ϕ l sin(ϕ 2 + φ) = 2eE l ( e iϕ/2ˆn + e iϕ/2ˆn ), (2a) 2i

6 and Î r = 2e Ĥ = 2eE r ϕ r sin(ϕ 2 φ) = 2eE r 2i ( e iϕ/2ˆn e iϕ/2ˆn + ). (2b) Starting from an initial state ψ() = ψ, the wave function evolves according to the Schrödinger equation t ψ(t) = i ψ(t), (3) Ĥ and the average pumped charge through SQUID k during the pumping procedure is Q k (t) = t ψ(τ) Îk(τ) ψ(τ) dτ. (4) If the system returns to its initial state after one pumping cycle is completed, the total pumped charge through each SQUID is equal. In the CPT, the constant bias voltage V b maintained over the device causes the phase difference ϕ over the device to change in time according to ϕ = 2eV b / [1]. In addition, the Hamiltonian contains the work done by the voltage source. Assuming C r = C l, the Hamiltonian becomes [18] Ĥ = E C (ˆn l ˆn r n g ) 2 1 2 V b( 2e)(ˆn l + ˆn r ) E l cosϕ l E r cosϕ r. Defining n = n l + n r, we can use a two-dimensional basis of n and n, the base vectors being n, n for all integers n and n. The Hamiltonian can be written as [18, 19] Ĥ = E C (ˆn n g ) 2 + 1 2 2eV bˆn E J 2 (ˆn + ˆn + )(ˆn + ˆn + ). (5) Here, the wave function evolves according to the Schrödinger equation. However, the total pumped charge through the left Josephson junction is obtained from Q l (t) = 2e ψ(t) ˆn l ψ(t) = 2e ψ(t) ˆn + ˆn 2 ψ(t), and the total pumped charge through the right Josephson junction from (6a) Q r (t) = 2e ψ(t) ˆn r ψ(t) = 2e ψ(t) ˆn ˆn 2 ψ(t). (6b)

7 3 Numerical Methods 3.1 Finite-state approximation for the sluice We use the eigenbasis of the charge operator ˆn. The pumped current is small and thus we can assume that the change in the expectation value of ˆn is small. This justifies the approximation of the system with only N charge eigenstates, m + 1,, m + N. Any wave function ψ(t) can now be represented as a vector c(t), for which ψ(t) = N c k (t) m + k. k=1 Here, the Hamiltonian in Eq. (1) can be represented as a time-dependent N N matrix H, for which H kk = E C (m + k n g ) 2, H (k 1)k = E l 2 e iϕ/2 E r 2 eiϕ/2, H (k+1)k = E l 2 eiϕ/2 E r 2 e iϕ/2, and the current operators in Eq. (2) are matrices I l and I r such that I l,(k 1)k = i 2eE l 2 e iϕ/2, I l,k(k 1) = i 2eE l 2 eiϕ/2, I r,(k 1)k = i 2eE r 2 eiϕ/2, I r,k(k 1) = i 2eE r 2 e iϕ/2. The Hamiltonian and the current matrices are tridiagonal, so this approximation fails if and only if c m+1 or c m+n are significant. Hence, we require that they are small. Starting from a known initial state ψ, the evolution of the wave function ψ(t) can be computed by solving the Schrödinger equation (3) for example using MATLAB s differential equation solvers. The cumulative pumped

8 charge through the left and right SQUIDs at time t can be computed simultaneously according to Eq. (4) using the current matrices. ċ(t) = i H(t)c Q k (t) = c(t) I k c(t) 3.2 Finite-state approximation for the CPT We use the eigenbasis of the operators ˆn and ˆn. The changes in the number of Cooper pairs on the island is small, and during one cycle, the change in the total number of pumped Cooper pairs is small. Thus we may approximate the system by N eigenstates of ˆn (n = m+1,, m+n) and N eigenstates of ˆn (n = m + 1,, m + N). The wavefunction can now be represented as a NN-vector c(t), for which where NN ψ(t) = c k (t) n k, n k, k=1 n k = m + 1 + (k 1 mod N), k 1 n k = m + 1 +. N The operators ˆn and ˆn are diagonal matrices whose kth elements are n k and n k, respectively. The Hamiltonian becomes a matrix H, for which H kk = E C (n k n g ) 2 + 1 2 2eV bn k, H kj = E J 2, if n k n j = n k n j = 1, H kj =, otherwise. After solving the Schrödinger equation, the pumped charge at time t is obtained according to Eq. (6). From the Hamiltonian (5), we observe that the states with odd n+n are not connected with the states with even n+n. Hence, if the pumping is initiated at eigenstate,, the states for which n+n is odd are negligible. Removing the corresponding states from the matrices and vectors, the dimension of the

9 vector space decreases to a half of its original value, and the computation is significantly faster. When one Cooper pair is pumped during one cycle, the expectation value of ˆn changes by 2. If N is small, this requires changing the eigenstates of n being used. If N is kept constant and m is changed by 2, a part of the wave function is lost and the remaining part is normalized to 1. We define γ k to be the norm of the remaining parts of the wave function before normalizing it. The effect of this error grows exponentially in time, and thus we use γ k as a measure of the accuracy of the N N-state approximation. If 1 γ k is high, simulating several cycles requires increasing N.

1 4 Cooper Pair Pumping with the Sluice 4.1 Pumping procedure In this section, we consider the Cooper pair sluice, shown in Fig. 1. During the pumping cycle, ϕ = and n g are kept constant and only the Josephson energies of the SQUIDs, E l and E r, are altered. The following pumping cycle, total time of which is T = 2T π, is used. Initially, the system is at charge eigenstate. 1. At t =, the right SQUID is rapidly opened (E r = EJ max ) and the left SQUID is rapidly closed (E l = EJ res). 2. At t = T π, the right SQUID is rapidly closed (E r = EJ res ) and the left SQUID is rapidly opened (E r = EJ max ). 3. At t = T = 2T π, the pumping cycle is repeated starting from 1. We find a naive estimate for the ideal pumped charge by assuming that with suitable parameters and EJ res =, exactly one Cooper pair is pumped through the SQUIDs per cycle. However, due to the fact that the SQUIDs cannot be perfectly closed, some charge is pumped in the opposite direction. The ratio of the current pumped in the right direction and the current pumped in the wrong direction is EJ max : EJ res. Thus, we conclude that the ideal pumped charge in one cycle is Q id (t) = Emax J EJ res. (7) 2e EJ max + EJ res 4.2 Two-state approximation Here, we consider a two state approximation in the basis {, 1 }. We wish to find a set of parameters for which in the first half of the pumping cycle, one Cooper pair is pumped into the island and in the second half, one Cooper pair is pumped out of the island. This is the case when ψ(t π ) = e iθ 1 for some θ and ψ(2t π ) = e iϑ for some ϑ. Since we set ϕ =, and E := E l + E r = EJ max Hamiltonian is H = E C n 2 g 1 2 E + EJ res 1 2 E E C(n g 1) 2, is constant, the system

11 which is constant in time. Setting n g = 1, we obtain 2 H = E C 4 I E ( 1 2 σ x, where σ x = 1 ), which results in ψ(t) = e i(t/ )H = e it/ EC4 I E 2 σx. Setting T π = π /E, denoting θ 1 = πe C /(4E) and θ = π/2 θ 1, and using the relations e iπσx/2 = iσ x and σ x = 1, we obtain ψ(t π ) = e i(θ 1I π 2 σx) = e iθ 1I e i π 2 σx = (e iθ 1 I)(iσ x ) = e iθ 1. Also ψ(2t π ) = e i2θ, so the system returns to its initial state in the end of the cycle. Thus, n g = 1 and T 2 π = π /(EJ max + EJ res ) are the desired parameters. As examples, we consider the cases E max J = E C, EJ res =, and EJ max =.5E C,. In the first case, the pumped charge and wave function EJ res =.3EJ max evolution are shown in Fig. 3. Exactly 2 Cooper pairs are pumped during the first two pumping cycles. In the latter case, the pumped charge and wave function evolution are shown in Fig. 4. In that case, 1.769 2 Emax J EJ res EJ max +EJ res Cooper pairs are pumped. Q/(2e) 2 1.5 1 Right Left Ideal ψ k 2 1.8.6.4 State State 1.5.2 2 4 6 8 1 12 t/( (a) EC ) 2 4 6 8 1 12 t/( (b) EC ) Figure 3: (a) Pumped charge through the left and right SQUIDs and the ideal pumped charge according to Eq. (7) in the two-state approximation for EJ max = E C and EJ res =. (b) Wave function evolution. We conclude that Eq. (7) for the ideal pumped charge holds in the two-state approximation. If the pumping cycle is repeated, no error occurs.

12 1.5 1 Right Left Ideal 1.8 State State 1 Q/(2e).5 ψ k 2.6.4.2 -.5 5 1 15 t/( EC (a) 5 1 15 t/( EC (b) Figure 4: (a) Pumped charge through the left and right SQUIDs and the ideal pumped charge according to Eq. (7) in the two-state approximation for EJ max =.5E C and EJ res =.3EJ max. (b) Wave function evolution. 4.3 Multi-state approximation If more than two charge states are considered, the arguments in Sec. 4.2 are no longer valid. Nevertheless, we try to pump with the same parameters as in the previous case using a six-state approximation for the states ( 2,, 3 ). For simplicity, we first consider the ideal case EJ res =. First, we set EJ max = E C. The pumped charge and wave function evolution for the first few pumping cycles are shown in Fig. 5. For the boundary states (k = 2, 3), max t ( ψ(t) k 2 ) <.4, so they are not shown in Fig. 5(b). As we can see from the graphs, the direction of the current changes rapidly. This is due to the fact that the wave function evolution is slower than the pumping procedure, and after a few cycles, the system is at state 1 rather than at when the right SQUID is opened, and thus the effect of the pumping cycle is opposite. Sacrificing the amplitude of the pumped current, we may try treating this problem by decreasing EJ max. Say EJ max =.1E C. The pumped charge and wave function evolution graphs for the first two cycles are shown in Fig. 6. After two pumping cycles, the number of pumped Cooper pairs is 2.12 whereas the ideal number according to Eq. (7) is 2. However, there is still a slight difference between the period of the wave function evolution and T. This eventually results in change of the direction of the current, as we can see in Fig. 7 for the first 3 cycles.

13 4 1 3.5 3.8 Q/(2e) 2.5 2 1.5 1.5 Right Left Ideal ψ k 2.6.4.2 State 1 State State 1 State 2 1 2 3 4 5 t/( (a) EC ) 1 2 3 4 5 t/( (b) EC ) Figure 5: (a) Pumped charge through the left and right SQUIDs and the ideal pumped charge according to Eq. (7) in the six-state approximation for = E C with no residual energy. (b) Wave function evolution. E max J Even in 3 cycles, for the boundary states (k = 2, 3), max t ( ψ(t) k 2 ) < 1 6. Thus the six-state approximation is sufficiently accurate for this case. 2.5 2 Right Left Ideal 1.8 Q/(2e) 1.5 1 ψ k 2.6.4 State 1 State State 1 State 2.5.2 2 4 6 8 1 12 t/( (a) EC ) 2 4 6 8 1 12 t/( EC (b) Figure 6: (a) Pumped charge through the left and right SQUIDs and the ideal pumped charge according to Eq. (7) in the six-state approximation for =.1E C with no residual energy. (b) Wave function evolution. E max J Nevertheless, the problem can be treated in the following way. Because the Hamiltonian is constant in time, the pumping cycle time T does not affect the evolution of the wave function. Furhtermore, as we can observe from the wave function evolution shown in Fig. 7(b), the wave function is approximately periodic in the case E max J E C. We find the average period of ). Finally, the wave function, T w, and define a parameter α = T w /(2π /EJ max we set T π = απ /EJ max = T w.

14 Q/(2e) 3 25 2 15 1 5 Right Left Ideal ψ k 2 1.8.6.4.2 State 1 State State 1 State 2 5 1 15 t/( EC (a) 1.83 1.84 1.85 1.86 1.87 1.88 t/( ) 1 4 EC (b) Figure 7: (a) Pumped charge through the left and right SQUIDs and the ideal pumped charge according to Eq. (7) in the six-state approximation for EJ max =.1E C with no residual energy. (b) Wave function evolution during the last 1 pumping cycles. In the previous case, we get α 1.7. Now, after 3 cycles, the number of pumped Cooper pairs is approximately 299.4 whereas the ideal number is 3. This is significantly better than for α = 1. The graphs for the wave function evolution for the last 1 pumping cycles and the pumped charge are shown in Fig. 8. Also in this case, max t ( ψ(t) k 2 ) < 1 6 for the boundary states (k = 2, 3). Q/(2e) 3 25 2 15 1 5 Right Left Ideal ψ k 2 1.8.6.4.2 State 1 State State 1 State 2 5 1 15 t/( EC (a) 1.83 1.84 1.85 1.86 1.87 1.88 t/( ) 1 4 EC (b) Figure 8: (a) Pumped charge through the left and right SQUIDs and the ideal pumped charge according to Eq. (7) in the six-state approximation for EJ max =.1E C with no residual energy and cycle time fixed with α = 1.7. (b) Wave function evolution during the last 1 pumping cycles. Even better values of α can presumably be found. However, the required relative accuracy of α is high and thus even with accurate numerical methods,

15 the idealizations used in our simulations potentially cause an error in α. Thus, in practice, the value of α will have to be found experimentally. Recall that in the case EJ max = E C, the wave function was not as obviously periodic as in this case, and thus this method cannot be used in that case. 4.4 Effects of small residual energy In this subsection, we consider a more realistic case where the SQUIDs cannot be perfectly closed, i.e. < Eres J 1. For the solutions to be stable, EJ max we require that EJ max E C as in the previous case. We consider the case EJ max =.1E C and EJ res =.5EJ max and use T π = απ /(EJ max +EJ res ) as derived in the two-state approximation but now adjusted with α. First, we try α = 1 for 3 cycles. The wave function evolution for the last 1 pumping cycles and the pumped charge are shown in Fig. 9. The number of pumped Cooper pairs is approximately 2 whereas the ideal number according to Eq. (7) is 271.4. Q/(2e) 3 25 2 15 1 5 Right Left Ideal ψ k 2 1.8.6.4.2 State 1 State State 1 State 2 5 1 15 t/( EC (a) 1.74 1.75 1.76 1.77 1.78 1.79 t/( ) 1 4 EC (b) Figure 9: (a) Pumped charge through the left and right SQUIDs and the ideal pumped charge according to Eq. (7) in the six-state approximation for EJ max =.1E C and EJ res =.5EJ max. (b) Wave function evolution during the last 1 pumping cycles. Finding the wave function period yields α = 1.7, just as in the case with no residual energy. For this α, the wave function evolution and the pumped charge are shown in Fig. 1. The number of pumped Cooper pairs is 271.6 which is rather close to the ideal number 271.4. The relative error in the pumped current is thus 271.6 271.4 1 = 7 1 4. Again, max t ( ψ(t) k 2 ) <

16 1 6 for the boundary states (k = 2, 3), so the six-state approximation is suitable. Q/(2e) 3 25 2 15 1 5 Right Left Ideal 5 1 15 t/( (a) EC ) ψ k 2 1.8.6.4.2 State 1 State State 1 State 2 1.74 1.75 1.76 1.77 1.78 1.79 t/( 1 4 (b) EC ) Figure 1: (a) Pumped charge through the left and right SQUIDs and the ideal pumped charge according to Eq. (7) in the six-state approximation for EJ max =.1E C, EJ res =.5EJ max, and the cycle time fixed with α = 1.7. (b) Wave function evolution during the last 1 pumping cycles. 4.5 Finite ramp time of the Josephson energies Above, we have assumed that the Josephson energies of the SQUIDs are changed instantaneously. However, in the real physcal system, the finite changing time causes a significant difference in the pumping procedure. Nevertheless, as long as ϕ =, the arguments in Sec. 4.2 are valid for a two-state approximation if Tπ E r (t) + E l (t) dt = π. (8) We model the finite changing time assuming the change is linear and is done in time T C. We consider the following three different pumping cycles, illustrated also in Fig. 11. 1. The Josephson energies of the SQUIDs are changed simultaneously, keeping E r + E l constant. 2. First, the other SQUID is closed and then the other SQUID is opened. 3. First, the other SQUID is opened and then the other SQUID is closed. For these cases, calculating the integral in (8) gives us

17.1 E r.1 E r.8 E l.8 E l Ek/EC.6.4 Ek/EC.6.4.2.2 1 2 3 4 t/t π (a) 1 2 3 4 t/t π (b).1.8 E r E l Ek/EC.6.4.2 1 2 3 4 t/t π (c) Figure 11: Three different ways to ramp the Josephson energies of the SQUIDs with finite ramp time T C. 1. T π = π E res J +Emax J π 2. T π = EJ res+emax J 3. T π = π EJ res+emax J, EJ res EJ max +EJ res EJ + T max C EJ res EJ max +EJ res EJ T max C, and. However, taking more than the two states into account requires multiplying T π by the factor α. For finite T C, no simple ideal formula such as Eq. (7) can be found for the pumped charge. Below, we compare the actual pumped charge to the ideal charge by Eq. (7) which is achieved for T C =. As an example, we set the parameters to EJ max =.1E C, EJ res =.5EJ max and T C = 5 E C. The ideal pumped charge according to Eq. (7) is 361.9 Cooper pairs for 4 cycles. For each of the three cases, we find the factor α and calculate the corresponding pumped charge.

18 In the first case, α = 1.7 and T π 29.94 E C. The pumped charge was 358. Cooper pairs. The pumped charge per one cycle was nearly constant throughout the pumping procedure. In the second case, α = 1.5 and T π 34.46 E C. The pumped charge was 36.3 Cooper pairs. The pumped charge per one cycle was nearly constant throughout the pumping procedure. In the third case, α = 1.12 and T π 25.43 E C. The pumped charge was 38.5 Cooper pairs. The pumped charge per one cycle was nearly constant throughout the pumping procedure. In all the cases, the cumulative pumped charge was nearly linear in time and the factor α was close to 1, so apart from the factor α, the parameters derived using the two-state approximation are suitable for the E k (t) s considered. The real evolution of the Josephson energies differs from the sample cases considered, but our results suggest that the pumping can be done similarly in that case, too, as long as the value of T π is set correctly. 4.6 Error caused by inaccuracy of ϕ Using the third pumping cycle in the last section, in an extreme case ϕ =.3, the pumped charge during the first 4 pumping cycles was 317.2 which is little higher than 38.5 for ϕ =. The pumped charge per one cycle remained nearly constant. We conclude that an error in ϕ, i.e. an error in the magnetic flux through the whole circuit, Φ, changes the pumped current slightly but the pumped charge remains linear in time.

5 Cooper Pair Pumping with the Cooper Pair Transistor 5.1 Pumping procedure In the CPT, shown in Fig. 2, E r = E l = E J are constant in time. Troughout the pumping procedure, a constant bias voltage V b is maintained over the device. In the real physical device, a small E J will make the thermal fluctuations significantly large, and a bias voltage V b larger than 4 µv will break the Cooper pairs. These limitations are discussed in Sec. 6.2. Starting from the charge eigenstate ψ =,, the following pumping procedure is used. 1. At t =, the gate charge n g is rapidly set to n max g. 2. At t = T π = T/2, the gate charge is rapidly set to n min g. 3. At t = T, the pumping cycle is repeated starting from 1. Now, V b, n max g, n min g, and T π are parameters that can be adjusted prior to the pumping procedure. During the simulations, we noticed that MATLAB s ordinary differential equation solver behaves strangely around the discontinuities of the Hamiltonian H. Thus we constructed an infinitely smooth n g (t) by taking its convolution with the Gaussian kernel f(t) = 1 4πs e 4s, t2 with s chosen so that f(t π /1) = 1 1, that is n g (t) = f(τ)n g,old (t τ)dτ. If the change from n max g to n min g was made shaper, the results varied less than.1 % unless a numerical error occurred. 19 5.2 Two-state approximation We use a two-state approximation to find an approximation for the correct choise of parameters. We note again that the parity of n+n does not change. Therefore it is sufficient to consider only states n, n where n + n is even.

2 For the first half of the pumping cycle, taking into account only states, and 1, 1, the Hamiltonian is a 2 2 matrix E C (n max g ) 2 E J H = 2. E J 2 E C ( 1 n max g ) 2 1 2 2eV b As in the two state approximation of the sluice, the system evolves from state, to e iθ 1, 1 for some θ in time π E J if the diagonal elements are equal, i.e. E C (n max g n max g ) 2 = E C ( 1 n max ) 2 1 2 2eV b = 2eV b 4E C 1 2. Assuming the two-state approximation is exact and setting T π = π E J n max g g and = 2eV b 4E C 1 2, the system is at state ψ(t) = eiθ 1, 1 at t = T π. Hence, during the first half of the cycle, one Cooper pair is pumped out of the island through the left jucntion. Consequently, considering the states 1, 1 and, 2, the Hamiltonian is expressed as E C ( 1 n min g ) 2 1 H = 2 2eV b E J 2. E J 2 E C (n min g ) 2 2eV b Requiring that the diagonal elements are equal, we get n min g = n max g 1 = 2eV b 4E C 1. Now, during another T 2 π, the system will evolve from state e iθ 1, 1 to state e iϑ, 2 for some ϑ, i.e. one Cooper pair is pumped into the island through the right junction. We found that the pumping cycle time is T = 2 π E J. If the pumped charge per cycle is exactly 2e, the pumped current is I = ee J π = I C 2π. 5.3 Multi-state approximation The arguments in the previous section are generally not valid if more states are considered. Nevertheless, the parameters derived using the two-state

21 approximation may be close to the optimal values in multi-state approximation. In this section, we find values of E J and V b (always n max g = 2eV b 4E C 1 and 2 n min g = 2eV b 4E C 1 ) for which the pumped charge per cycle remains constant 2 after several cycles, i.e. the cumulative pumped charge is linear in time. Increasing the bias voltage V b increases the difference between energies of states with different n s and thus it may result in more stable pumping cycle. Moreover, decreasing the value of E J will decrease the effect of the other charge eigenstates than those used in the two-state approximation. However, as mentioned above, the values of E J and V b are limited in the real physical device. 5.3.1 One pumping cycle A periodic pumping procedure is achieved when the wave function is close to, 2 after one pumping cycle. Here, we use E J =.1E C and simulate one pumping cycle with different values of the bias voltage V b. As an example, we simulate 1 pumping cycle for V b = E C /(2e) using a 5 5-state approximation, n = 2,, 2, n = 3,, 1. The wave function evolution and current graphs are shown in Fig. 12. We notice that ψ(t), 2 =.9928. This small difference can be crucial if the pumping cycle is repeated several times. 1.8 Right Left 1.8 Q/(2e).6.4.2 ψ n, n 2.6.4.2 State, 2 State 1, 1 State, State 1, 1 State 1, 1 1 2 3 4 5 6 t/( (a) EC ) 1 2 3 4 5 6 t/( (b) EC ) Figure 12: (a) Pumped charge through left and right Josephson junctions in the 5 5-state approximation for V b = E C /(2e) and E J =.1E C. (b) Wave function evolution. Next, using a 5 5-state approximation, we find the value of ψ(t), 2 as a function of the bias voltage V b. This is shown in Fig. 13. A large

22 value of ψ(t), 2 corresponds to a stable pumping cycle. The 5 5- state approximation is very inaccurate when ψ(t), 2 is low, but in these cases we already know that the pumping cycle does not work, and a simulation with a higher number of states is worthless. 1 1 ψ(t), 2 1 1 1 2 1 3 1 1 1 1 1 1 2 V b / EC 2e Figure 13: Value of 1 ψ(t), 2 as a function of the bias voltage V b in the 5 5-state approximation. A smaller value corresponds to more stable pumping cycle. Some values of V b cause some charge eigenstates to have the same energy than the states, and 1, 1. For example if V b = 2E C /(2e) and n max g =, the energies of states 1, 1,,, and 1, 1 are equal. Therefore the two-state approximation is not valid and the wave function evolves to a superposition of these three states during the first half of the cycle, as seen in Fig. 14. Q/(2e).7.6.5.4.3.2.1 Right Left ψ n, n 2 1.8.6.4.2 State 1, 1 State 1, 1 State 2, State, State 2, 2 State, 2 State 1, 1 1 2 3 4 5 6 t/( (a) EC ) 1 2 3 4 5 6 t/( EC (b) Figure 14: (a) Pumped charge through left and right Josephson junctions in the 5 5-state approximation for V b = 2E C /(2e). (b) Wave function evolution. Moreover, a higher bias voltage V b yields a more stable pumping cycle. In the following, we consider the first half of the pumping cycle, n g = n max g. For

23 the second half, n g = n min g, the arguments are valid for n = 1 n and n = n 1. Using V b = 2E C (2n g + 1)/(2e), the distance of the energies of states n, n and n ± 1, n 1 is ( E C (n ± 1 n g ) 2 1 ) 2 V b( 2e)(n 1) ( E C (n n g ) 2 1 ) 2 V b( 2e)(n) =E C (±2n + 1 2n g ) ± 1 2 V b( 2e) =E C (±2n + 1 2n g 2n g 1) =E C (1 ± (2n 1 4n g )), which is large for large V b. Therefore the states n, n for which n n = 2 are further away from the degeneracy of states, and 1, 1 than with a small bias voltage. Hence, the wavefunction stays more close to a linear combination of states n, n where n = n, and thus the first half of the pumping cycle finishes more closely to state 1, 1. 5.3.2 Many pumping cycles Let us study how the pumping procedure works if the pumping cycle is repeated several times. We begin with E J =.1E C and consider three different cases; V b = E C /(2e), V b = 1E C /(2e), and V b = 1E C /(2e). As mentioned earlier, simulating several pumping cycles is prone to error. We define γ k as explained in Sec. 3.2. The value 1 k γ k is the maximum relative error caused by the finite state approximation. Nevertheless, the ignored parts of the wave function should also contribute to the total pumped charge, so the numerical error in pumped current is much lower than 1 k γ k. First, we consider the case V b = E C /(2e). For the first 1 cycles, a 7 7- approximation (n = 3,, 3, n = 4,, 2) yields min γ k =.9944. The error is significant for more cycles, but even with a 7 8-approximation, the value of γ k does not change significantly. Therefore simulating several cycles will require using a high number of states. We use a 6 11-state approximation with states n = 3,, 2 and n = 5,, 5 for 2 cycles. Figure 15 represents ψ(nt), 2N as a function of the number of cycles N. If the pumping is linear, the value should be close to 1. However, this clearly is not the case, and even the numerical error does not explain this difference. We conclude that a higher bias voltage is needed if E J is kept unchanged.

24 1 ψ(nt), 2N.8.6.4.2 5 1 15 2 N Figure 15: Value of ψ(nt), 2N as a function of the number of cycles N in the case E J =.1E C and V b = E C /(2e). In the 5 5-state approximation for V b = 1E C /(2e), min γ k >.9971 for the first 4 cycles, and at the end of the 4th cycle, ψ, 8 =.9987. However, 4 k=1 γ k =.393, so the approximation is rather inaccurate. In the approximation with 6 7 states, the value of ψ(nt), 2N decreases as in the case V b = E C /(2e). The value of ψ(nt), 2N as a function of the number of cycles N is represented in Fig. 16. ψ(nt), 2N 1.95.9.85.8.75.7.65 1 2 3 4 N Figure 16: Value of ψ(nt), 2N as a function of the number of cycles N in the case E J =.1E C and V b = 1E C /(2e). Finally, we consider the case V b = 1E C /(2e). Solving the Schrödinger equation is now significantly slower than in the previous cases, and thus we use less cycles than above. For the first 2 cycles in the 5 5 -state approximation, minγ k =.9975, and ψ, 4 =.9961. Using 5 6 states yields again min γ k =.9975 and ψ, 4 =.9946. We conclude that using V b = 1E C /(2e) is not more stable than using V b = 1E C /(2e).

25 The results suggest that a smaller value of E J is required. Next, we try E J =.1E C. Since the value of E J /E C decreases and E J is limited below by the finite temperature of the system, it is possible that a larger E C is needed and thus the upper bound of V b /E C is smaller. The effect of E J on the stability of the wave function is exponential whereas the effect of V b is only polynomial. Hence we may try smaller values of V b /E C, say V b =.2E C /(2e), V b = E C /(2e), and V b = 5E C /(2e). We use a 6 7-state approximation in all the cases below. First, we consider V b =.2E C /(2e). For the first 4 cycles, γ k >.9994 and 4 k=1 γ k =.9266, so the accuracy of the numerical approximation is much higher than above. The value of ψ(nt), 2N as a function of the number of cycles N is represented in Fig. 17. At the end of the 4th cycle, ψ, 8 =.7735, which is small, so a higher bias voltage V b is needed. ψ(nt), 2N 1.95.9.85.8.75.7.65 1 2 3 4 N Figure 17: Value of ψ(nt), 2N as a function of the number of cycles N in the case E J =.1E C and V b =.2E C /(2e). Next, we try V b = E C /(2e). For the first 4 cycles, γ k >.9999 and 4 k=1 γ k =.975. At the end of the 4th cycle, ψ, 8 =.9994. This suggests that the pumping works fine for several cycles. The pumped current through the left and right Josephson junctions during the 4 cycles and also a closer look at the 4th cycle is shown in Fig. 18. At the end of the 4th cycle, the total pumped charge through both Josephson junctions is 399.9987 Cooper pairs. This implies that the relative error in pumped current is only 1 399.9987 = 3.25 1 6. However, the numerical error caused 4 by the finite state approximation may be higher. Finally, we try V b = 5E C /(2e). For the first 4 cycles, γ k >.9998 and 4 k=1 γ k =.9323. Finally, after 4 cycles, ψ, 8 =.9987. This is lower than in the case V b = E C /(2e) and we conclude that the pumping is not more stable than in the case V b = E C /(2e).

26 Q/(2e) 4 35 3 25 2 15 1 5 Right Left.5 1 1.5 2 2.5 t/( 1 5 (a) EC ) Q/(2e) 4 Right Left 399 2.57 2.58 2.59 2.51 2.511 2.512 2.513 t/( 1 5 (b) EC ) 1.8 ψ n, n 2.6.4.2 State, 8 State 1, 799 State, 798 2.57 2.58 2.59 2.51 2.511 2.512 2.513 t/( 1 5 (c) Figure 18: (a) Pumped charge through left and right Josephson junctions during the first 4 cycles in the 6 7-state approximation for E J =.1E C and V b = E C /(2e). (b) Pumped charge during the 4th pumping cycle. (c) Wave function evolution during the 4th pumping cycle. For other states, max t ( ψ(t) n, n 2 ) < 1 3, and hence they are not shown. EC )

27 6 Discussion Neglecting the interference of the environment, nonadiabatic control cycles were found to be suitable for pumping Cooper pairs. Finding the accuracy in which these assumptions, and thus the results derived, hold in the real physical system is beyond the scope of this thesis and will require future research. 6.1 Cooper pair sluice Even with some nonidealities, the pumped charge per cycle is constant if the pumping frequency is adjusted properly. A realistic value for the charging energy is E C = k B 1 K. With EJ max =.1E C and EJ res =.5EJ max, this yields T.2 ns. The average pumped charge per cycle in that case was 271.6/3 Cooper pairs, and thus the average current is I 1 na. The error in pumped charge compared to the ideal charge taking into account the residual energy was 7 1 4. However, the average pumped charge per cycle depends highly on the value of EJ res/emax J and the finite ramp time of the Jospephson energies. Hence the pumping procedure here is hardly optimal for metrology. Two possible error sources were considered: the overall phase difference ϕ, and a finite ramp time of the Josephson energies. In both cases, an error caused a significant (1% 2%) change in the average current, but the pumped charge per cycle remained constant. The required accuracy of the control cycle time T is very high. If the properties of the device are not known in this accuracy, the value of T needs to be found experimentally. An alternative way to treat the problems may be using arbitrarily accurate composite pulse sequences [2 23]. However, this is left for future research. 6.2 Cooper pair transistor The simulations were done assuming identical Josephson junctions and neglecting quasiparticle tunneling. Further, the gate voltage V g was assumed to change nearly instantaneously. The pumping procedure was found to work well when E J =.1E C and V b = E C /(2e). With, say E C = k B 9 K, we have E J = k B 9 mk and V b = 387.8 µv whereas the upper bound of the bias

28 voltage is approximately 4 µv for aluminium. This yields T.5 ns and since exactly one Cooper pair was pumped per cycle, the average current is I.6 na. In our numerical approximation in this case, the relative error of the pumped current was 3 1 6. However, it is possible that the numerical error is higher. The current is linearly dependent on E J. However, if E J /E C is kept constant, a larger value of E J requires higher bias voltage V b. Nevertheless, decreasing E J /E C will allow smaller values of V b /E C, and a suitable set of parameters may be found. In this thesis, we considered an ideal control cycle and an ideal device. The Hamiltonian for asymmetric CPT (nonidentical Josephson junctions) is given for example in Ref. [18]. In addition, the effect of the finite ramp time of the gate voltage was not considered. Nevertheless, the results in this thesis are promising and further research is recommended.

29 References [1] M. Tinkham. Introduction to Superconductivity. Dover Publications Inc., Mineola, NY, 24. [2] Y. Makhlin, G. Schön, and A. Shnirman. Quantum-state engineering with josephson-junction devices. Rev. Mod. Phys., 73(2):357 4, 21. [3] I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, and J. E. Mooij. Coherent Quantum Dynamics of a Superconducting Flux Qubit. Science, 299(5614):1869 1871, 23. [4] J. Clarke and F. K. Wilhelm. Superconducting quantum bits. Nature, 453(7198):131 142, 28. [5] F. Piquemal, A. Bounouh, L. Devoille, N. Feltin, O. Thevenot, and G. Trapon. Fundamental electrical standards and the quantum metrological triangle. Comptes Rendus Physique, 5(8):857 879, 24. [6] M. W. Keller, J. M. Martinis, N. M. Zimmerman, and A.H. Steinbach. Accuracy of electron counting using a 7-junction electron pump. Appl. Phys. Lett., 69(12):184 186, 1996. [7] A. O. Niskanen, J. P. Pekola, and H. Seppä. Fast and accurate singleisland charge pump: Implementation of a cooper pair pump. Phys. Rev. Lett., 91(17):1773, 23. [8] M. Möttönen, J. P. Pekola, J. J. Vartiainen, V. Brosco, and F. W. J. Hekking. Measurement scheme of the Berry phase in superconducting circuits. Phys. Rev. B, 73(21):214523, 26. [9] J. J. Vartiainen, M. Möttönen, J. P. Pekola, and A. Kemppinen. Nanoampere pumping of cooper pairs. Appl. Phys. Lett., 9(8):8212, 27. [1] M. Möttönen, J. J. Vartiainen, and J. P. Pekola. Experimental determination of the berry phase in a superconducting charge pump. Phys. Rev. Lett., 1(17):17721, 28. [11] J. P. Pekola, J. J. Vartiainen, M. Möttönen, O-P. Saira, M. Meschke, and D. V. Averin. Hybrid single-electron transistor as a source of quantized electric current. Nature Phys., 4(2):12 124, 28.

3 [12] M. D. Blumenthal, B. Kaestner, L. Li, S. Giblin, T. J. B. M. Janssen, M. Pepper, D. Anderson, G. Jones, and D. A. Ritchie. Gigahertz quantized charge pumping. Nature Phys., 3(5):343 347, 27. [13] A. Fujiwara, K. Nishiguchi, and Y. Ono. Nanoampere charge pump by single-electron ratchet using silicon nanowire metal-oxide-semiconductor field-effect transistor. Appl. Phys. Lett., 92(4):4212, 28. [14] R. Leone, L. P. Lévy, and P. Lafarge. Cooper-pair pump as a quantized current source. Phys. Rev. Lett., 1(11):1171, 28. [15] T. A. Fulton, P. L. Gammel, D. J. Bishop, L. N. Dunkleberger, and G. J. Dolan. Observation of combined josephson and charging effects in small tunnel junction circuits. Phys. Rev. Lett., 63(12):137 131, 1989. [16] Y. Harada, D. B. Haviland, P. Delsing, C. D. Chen, and T. Claeson. Fabrication and measurement of a Nb based superconducting single electron transistor. Appl. Phys. Lett., 65(5):636 638, 1994. [17] M. T. Tuominen, J. M. Hergenrother, T. S. Tighe, and M. Tinkham. Experimental evidence for parity-based 2e periodicity in a superconducting single-electron tunneling transistor. Phys. Rev. Lett., 69(13):1997 2, 1992. [18] J. Leppäkangas and E. Thuneberg. The effect of decoherence on resonant Cooper-pair tunneling in a voltage-biased single-cooper-pair transistor. ArXiv e-prints, 86, 28. [19] A. M. van den Brink, A. A. Odintsov, P. A. Bobbert, and G. Schön. Coherent cooper pair tunneling in systems of josephson junctions: Effects of quasiparticle tunneling and of the electromagnetic environment. Z Phys. B, 85(3):459 467, 1991. [2] K. R. Brown, A. W. Harrow, and I. L. Chuang. Arbitrarily accurate composite pulse sequences. Phys. Rev. A, 7(5):52318, 24. [21] J. J. L. Morton, A. M. Tyryshkin, A. Ardavan, K. Porfyrakis, S. A. Lyon, and G. A. D. Briggs. Measuring errors in single-qubit rotations by pulsed electron paramagnetic resonance. Phys. Rev. A, 71(1):12332, 25. [22] J. J. L. Morton, A. M. Tyryshkin, A. Ardavan, K. Porfyrakis, S. A. Lyon, and G. A. D. Briggs. High fidelity single qubit operations using pulsed electron paramagnetic resonance. Phys. Rev. Lett., 95(2):251, 25.

[23] M. Möttönen, R. de Sousa, J. Zhang, and K. B. Whaley. High-fidelity one-qubit operations under random telegraph noise. Phys. Rev. A, 73(2):22332, 26. 31

32 A Suomenkielinen yhteenveto Joissakin materiaaleissa sähköinen resistanssi katoaa kokonaan tietyn lämpötilarajan alapuolella. Tätä vuonna 1911 löydettyä ilmiötä kutsutaan suprajohtavuudeksi. Vuonna 1957 John Bardeen, Leon Cooper ja John Schrieffer kehittivät BCS-toerian suprajohtavuudelle. Sen mukaan elektronit muodostavat suprajohteissa niin sanottuja Cooperin pareja. Nämä parit pääsevät virtaamaan vapaasti ja niiden hajottamiseen tarvitaan ylimääräistä energiaa, jota ei matalissa lämpötiloissa ole. Nanometrin suuruusluokkaa olevissa virtapiireissä suprajohtavuus aiheuttaa havaittavissa olevia kvanttifysiikasta johtuvia ilmiöitä. Suprajohtavilla virtapiireillä voidaan muodostaa systeemi, jonka energiatilojen kvantittuminen voidaan havaita. Näitä virtapiirejä voidaan yhdistää toisiinsa ja perinteisiin virtapiireihin. Tämän takia suprajohtavat virtapiirit ovat lupaava vaihtoehto kvanttitietokoneen rakenteeksi. Koska suprajohtavat virtapiirit mahdollistavat yksittäisten Cooperin parien manipuloinnin, myös virran yksikön ampeerin uudelleenmäärittely on niille mahdollinen sovelluskohde. Yksikön määrittelemiseen vaadittavaa tarkkaa virtalähdettä ei ole kuitenkaan vielä pystytty rakentamaan. Tässä työssä tutkitaan teoreettisesti ja laskennallisesti Cooperin parien pumppausta kahdella laitteella, Cooperin pari -sululla (engl. sluice) ja Cooperin pari -transistorilla (myöhemmin CPT). Pumppaus suoritetaan toistamalla kontrollisykliä, jonka aikana siirretään vuorotellen yksi Cooperin pari ulos saarekkeelta ja sen jälkeen yksi Cooperin pari takaisin saarekkeelle toiselta puolelta. Työssä tehdään joitakin ideaalisuusoletuksia. Esimerkiksi ympäristön vaikutusta ei huomioida. Kontrollisyklin parametrit johdetaan käyttämällä kahden varaustilan approksimaatiota, minkä jälkeen monen tilan approksimaatiolla tutkitaan MATLABilla numeerisesti systeemin käyttäytymistä usean pumppaussyklin aikana. Tämän työn laitteissa Cooperin parien liikkuminen perustuu Josephson-liitoksiin. Josephson-liitoksessa kahden suprajohtavan osan välillä on eristekerros, jonka läpi Cooperin parit voivat tunneloitua. Kaksi rinnakkain kytkettyä Josephsonliitosta muodostavat SQUIDin (superconducting quantum interference device), joka voidaan avata ja sulkea magneettikenttää säätämällä. Cooperin pari -sulun rakenne on esitetty kuvassa 19. Laite muodostuu saarekkeesta, joka on kiinnitetty virtasilmukkaan kahdella SQUIDillä. Lisäksi saareke on kondensaattorin kautta kiinni säädettävässä jännitelähteessä. Laitetta kuvaavan aaltofunktion vaihe-ero saarekkeen yli voidaan säätää silmukan läpi

33 V g C g Φ l ϕ l Φ Φ ϕ r r Figure 19: Cooperin pari -sulku. Kuva on otettu [8]:sta. olevaa magneettikenttää muuttamalla. Cooperin parien pumppaus tapahtuu pitämällä porttijännite V g ja magneettikenttä Φ vakioina ja avaamalla ja sulkemalla SQUIDejä vuorotellen. Laitteen tilaa voidaan kvanttifysiikassa kuvata varauksen ominaistiloilla: tila k vastaa tilaa, jossa saarekkeella on k ylimääräistä Cooperin paria. Porttijännite valitaan siten että tilojen ja 1 energia on sama. Tällöin kun toinen SQUID on auki, saarekkeelle virtaa sen läpi yksi Cooperin pari tietyssä ajassa T π eli systeemi siirtyy tilalta tilalle 1. Tämän jälkeen toinen SQUID suljetaan ja toinen avataan. Nyt yksi Cooperin pari virtaa saarekkeelta pois. Kahden tilan approksimaatiolla saadaan T π :lle teoreettinen arvo. Käytännössä muut varaustilat ja se, että SQUIDien täydellinen sulkeminen ei onnistu, vaikuttavat aikaan, joka Cooperin parilla kestää siirtyä pois saarekkeelta. Tässä työssä löydettiin simulaatioilla T π :lle tarkempi arvo. Tämä arvo riippuu kuitenkin huomattavasti laitteen ominaisuuksista, ja se pitäisi siksi etsiä jokaiselle laitteelle kokeellisesti. Koska SQUIDien täydellinen sulkeminen ei onnistu, osa Cooperin pareista kulkee väärään suuntaan, ja siksi tämä kontrollisykli tuskin sopii sähkövirran yksikön määrittelemiseen. Pumpatulle kokonaisvirralle realistinen suuruus on noin 1 na. Cooperin pari -transistorin rakenne on esitetty kuvassa 2. Cooperin pari -sulun tapaan CPT koostuu saarekkeesta, joka on kytketty kondensaattorilla jännitelähteeseen. SQUIDien tilalla on Josephson-liitokset ja virtasilmukan tilalla on laitteen yli kytketty vakioesijännite V b. Pumppaus suoritetaan vaihtamalla porttijännitteen V g arvoa kahden arvon välillä. Esijännitteen V b takia Cooperin parin siirtyminen saarekkeen puolelta toiselle vaatii energiaa. Tämän takia systeemin tilan kuvaamiseen tarvitaan saarek-

34 V g C g +V b /2 V b /2 ϕ l ϕ r Figure 2: Cooperin pari -transistori. keen varaustilan k lisäksi summa kummankin Josephson-liitoksen yli kulkeneiden Cooperin parien lukumäärästä k. Systeemin tila voidaan esittää muodossa k, k. Pumppaussyklin ensimmäisessä vaiheessa porttijännite asetetaan arvoon Vg max, jolla tilojen, ja 1, 1 energia on sama. Nyt kahden tilan approksimaatiossa tietyn ajan T π kuluttua yksi Cooperin pari on kulkenut pois saarekkeelta vasemmanpuoleisen liitoksen läpi. Tämän jälkeen portijännite asetetaan arvoon Vg min, jolla tilojen 1, 1 ja, 2 energia on sama. Ajan T π kuluttua systeemi on tilalla, 2 eli yksi Cooperin pari on kulkenut laitteen läpi. Cooperin pari -transistorinkin tapauksessa muiden varaustilojen vaikutus on kuitenkin merkittävä ja ne vaikuttavat systeemin tilan aikakehitykseen. Muiden tilojen vaikutusta saadaan vähennettyä suurentamalla esijännitettä, mutta toisaalta liian suuri esijännite voi hajottaa Cooperin pareja. Työssä löydettiin realistiset parametrit, joilla laite toimii vähintään satoja syklejä. Pumpattavan virran suuruus on tällöin noin, 6 na ja suhteellinen tarkkuus on noin 3 1 6. Virran suuruus riittäisi sähkövirran yksikön määrittelemiseen, mutta simulaatioiden numeerisen epätarkkuuden ja työssä tehtyjen yksinkertaistusten takia virran tarkkuus voi todellisuudessa olla hyvinkin heikko. Tulokset ovat silti lupaavia ja jatkotutkimusta suositellaan.