Lecture 10: Superconducting Devices Introduction

Phys 769: Selected Topics in Condensed Matter Physics Summer 2010 Lecture 10: Superconducting devices

Lecturer: Anthony J. Leggett TA: Bill Coish

Introduction

So far, we described only the superconducting ground state. Before proceeding to the main topic of this lecture, we need to say a little about the simplest excited states. For this purpose it is easier to use the original representation of BCS, in which the condition of particle number conservation is relaxed and the ground state has the form of a product of states referring to the occupation of the pair of states (k ↑, −k ↓): in obvious notation,

2 2 ΨBCS = Φk, Φk = uk |00i + vk |11i , |uk| + |vk| = 1 (1) Yk where |00i is the state in which neither k ↑ nor −k ↓ is occupied, |11i is that in which both are, and uk,vk are complex coeﬃcients (though the usual convention is to take uk real). From the identiﬁcation (lecture 9, Eq. (30)) of the Fourier-transformed wave function

Fk with the “anomalous average” hN − 2| a−k↓ak↑ |Ni (which in the BCS representation becomes simply ha−k↓ak↑i we ﬁnd ∗ Fk = ukvk. (2)

In this representation we can associate a contribution to the kinetic and potential energy with each pair state (k ↑, −k ↓) separately (but must remember not to double-count). The contribution hV ik to the potential energy (excluding as usual the Hartree and Fock terms) is simply ∗ hV ik = −2∆kFk. (3) In the case of the kinetic energy, it is most convenient to calculate it relative to the value in the normal ground state (hnki = θ(kF − k)), which is 0 for ǫk > 0 and −2|ǫk| for ǫk < 0. This gives (cf. Eq. (31) of lecture 9)

2 δ hTki = |ǫk|−|ǫk| /Ek (4) independently of the sign of ǫk. Using Fk = ∆k/2Ek, we ﬁnd that the total change in the contribution to the ground-state energy of the pair state (k ↑, −vk, ↓) due to pairing is

1 |ǫk|− Ek. So far, we essentially have just an alternative notation for describing the ground state.

However, it is clear that the occupation space of the pair (k ↑, −k ↓) is spanned by the states |00i , |01i , |10i, and |11i, and is thus 4-dimensional. Thus, there must be three other energy eigenstates besides the state Φk of Eq. (1), and it is clear that within the general framework of the BCS ansatz1 these may be taken to be the “broken-pair” (BP) states |01i and |10i and the (unique) combination of |00i and |11i which is orthogonal to Φk, namely

∗ ∗ ′′ vk |00i− uk |11i (“excited pair states, (EP)) (5) we wish to evaluate the energy of these three states relative to the ground state (“ground pair”, GP) states.

In the case of the EP state the evaluation is straightforward, since it is easily veriﬁed that the transition from Eq. (5) to Eq. (3) involves the replacements of Fk with −Fk and of the − sign in Eq. (31) of lecture 9 by a + sign. Thus, the diﬀerence in energy between the GP and EP states is (again using Fk = ∆k/2Ek)

2 ∗ EEP − EGP = 2|ǫk| /Ek + 4∆kFk = 2Ek (6)

In calculating the energy necessary to excite the “broken pair” states from the GP one, we need to use the fact that this state does not contribute at all to the pairing energy, while its contribution to the kinetic energy (relative to the

normal ground state) is just |ǫk|. In view of Eq. (4), this means that the change in kinetic energy on going from the 2 GP to the broken-pair state is +|ǫk| /Ek, so that the total excitation energy is according to Eq. (3)

2 ∗ EBP − EGP = |ǫk| /Ek + 2∆kFk = Ek (7)

A standard representation in the literature is to regard the BP states |01i and |10i as each containing a single “Bogoliubov quasiparticle” while the EP state contains two such quasiparticles. If one takes this point of view, then it is easy to see that (for example) the operator which, acting on the ground-pair states (Eq. (1)) creates the (correctly normalized

1i.e. an ansatz in which the density matrix is factorized into terms associated with the diﬀerent pairs (k ↑, −k ↓).

2 state |10i has to be of the form2

† † αk↑ = ukak↑ − vka−k↓ (8) while that for |01i is † † α−k↓ = uka−k↓ + vkak↑ (9)

Apart from the form of the Bogoliubov operators, which we will need later in the context of the Josephson eﬀect, the most important conclusion of the above argument is simply that the minimum energy needed to break any Cooper pair is given by the quantity

2 2 1/2 Ek ≡ ǫk + |∆k| ≥|∆k|∼ ∆ (10) where at the last step we have assumed that the energy gap ∆ is approximately constant over the Fermi surface (true for the classic superconductors). Let us estimate the number of broken pairs (or, what is equivalent in this limit, the number of excited particles) in the limit kBT ≪ ∆ (fairly easily obtained in most of the classic (BCS) superconductors such as Al or Nb). In this limit we need not worry about the possibility of excitation of the EP state, and the probability of a broken pair(either |01i or |10i) is approximately exp −Ek/kBT . Hence the total number hnexci of excitations is

hnexci≃ 2 exp −Ek/kBT Xk ∞ ∞ dn 2 2 1/2 −1 2 2 1/2 = Ω dǫ exp − ǫ + ∆ /kBT ∼ Nǫ dǫ exp − ǫ + ∆ /kBT (11) dǫ Z F Z −∞ −∞ 2 2 1/2 1 2 or on expanding (ǫ + ∆ ) ≃ ∆+ 2 ǫ /∆,

1/2 ∆kBT hnexci∼ N 2 exp −∆/kBT (12) ǫF where N is the total number of electrons in the superconductor.

It is interesting to ask what is the condition to have nexc ≪ 1. Neglecting the rather weak prefactor, we see that a suﬃcient condition for this is kBT ≪ ∆/ ln N. So if we consider 3 e.g. a 1cm block of Nb, where ∆ ∼ 1.75kBTc ∼ 16 K, we see that the condition is roughly (ln 1023 ∼ 60) T ≪ 250 mK – almost routinely attainable with current cryogenic technology!

One might worry whether this argument could be vitiated by the existence of excitations, for example associated with the edge of the sample, which have energies ≪ ∆. Such excitations

2It is important to choose (and keep!) a deﬁnite convention for the state |11i relative to |00i. I choose it † † here so that |11i ≡ ak↑a−k↓ |00i.

3 could of course not propagate into the bullk, but it is possible that they might be a nuisance in the operation of a superconducting device, which as we shall see relies in some sense on having a “pure” condensate. Probably the best answer to this objection is a combination of experiment and the arguments given in lecture 2: any such excitations do not seem to affect the classical motion of the relevant macroscopic variable (the phase or flux, see below) very much, so it is not surprising that neither do they affect the quantum-coherent behavior.

One other point is worth mentioning: In the arguments of lecture 9, and above, it was implicitly assumed that the total number of electrons is even. What if it is odd? Actually, there are experiments which show that a well-insulated superconducting grain does “know the diﬀerence” between (e.g.) 109 and 109 + 1, and this is even exploited in some “Cooper- pair box” devices. However, in most realistic experimental setups the system is in electrical contact with some “normal” external reservoir – e.g. the metal of the normal leads – so that it is not the total particle number N but the chemical potential µ which is held constant. Under these conditions the energy necessary to add an extra Cooper pair is just 2µ, which is exactly the energy necessary to take two electrons out of the normal reservoir, so it is realistic to work in the grand canonical ensemble and regard the density matrix of the system as an incoherent mixture3 of states involving diﬀerent even numbers of electrons.

On the other hand, to add or subtract a single electron takes an energy µ + Ek, so single electrons approaching from the normal metal are Andreev-reﬂected and do not enter the superconductor. Thus the assumption that N =even is justiﬁed. Thus, in the remainder of this lecture it will be assumed unless otherwise stated that the bulk superconducting samples are in their pure quantum ground state, i.e. no electronic excitations whatsoever are present.

The Josephson eﬀect

The Josephson eﬀect occurs, rather generically, whenever two bulk superconductors are connected by a “weak link”, that is a region which prevents a much increased impedance to the transport of single electrons and hence a fortiori of Cooper pairs. There are many types of weak link (tunnel oxide junction, point contact, microbridge, SNS junction, . . . ) but the one which is mostly employed in superconducting devices is the original tunnel oxide junction discussed in Josephson’s paper, which compared to the other types is (a) better controlled and (b) subject to less dissipation. I will therefore conﬁne the discussion to that.

3Not, as many textbooks would have us believe, a coherent superposition of such states. The latter is a ﬁgment of the BCS approach.

4 In the following it is convenient to introduce the notion of the “order parameter” Ψ of a bulk superconductor. We will consider a spatially uniform superconductor with no current ﬂow. Then up to an overall constant factor which is a matter of convention, we can deﬁne the order parameter by the equation

Ψ= Fk (13) Xk

It is then easy to see that it is (proportional to) the Cooper-pair wave function F (r) evalu- ated at r = 0, i.e. for the two electrons of the pair coincident. The original Ginzburg-Landau theory was formulated in terms of Ψ (though they did not know the post-BCS microscopic interpretation), and allows it to vary both in amplitude and in phase. However, it is easy to see that in equilibrium the amplitude within a single superconductor must have a ﬁxed value (corresponding to the solution of the BCS equation, suitably generalized to nonzero T ) and moreover the phase should be constant. However, these considerations do not ﬁx the relative phase of Ψ in two superconductors which are connected by a Josephson junction (that is, a weak link), and indeed if there were no contact at all across the junction the relative phase would not even be physically meaningful. If there is contact, then schematically we can write the (particle-conserving) many-body wave function in the form

N/2 ΨN = (aψL + bψR) (14) and the relative phase is then just arg(a∗b). Let’s denote this relative phase by ∆φ.

The interesting question is: How does the energy of the coupled system depend on ∆φ? Let’s follow the original reasoning of Ginzburg and Landau for the bulk energy and make an expansion in the two order parameters Ψ1 and Ψ2 of the bulk superconductors. Then symmetry and analyticity considerations imply that the lowest-order allowed term is of the form ∗ EJ = −const. (Ψ1Ψ2 +c.c.)= −const.|Ψ1|·|Ψ2| cos ∆φ ≡−EJ0 cos ∆φ (15) where EJ0 is in general a function of the system parameters and of T , and the − sign is included for convenience. Note that there is no a priori reason why EJ0 has to be positive, though we shall see that in most real-life experimental situations it is.

There is no a priori reason why the ∆φ-dependent coupling need be restricted to the simple form Eq. (15); for example, the next term allowed by symmetry is of the form

(2) ∗ 2 EJ = const. (Ψ1Ψ2) +c.c. ∼ const. cos 2∆φ (16) n o

5 More generally, given that EJ (∆φ) is real and must (for a nonhysteretic junction) be periodic in ∆φ, the most general expression is of the form

EJ (∆φ)= an cos n∆φ (17) Xn

4 However, a beautiful paper by Bloch shows that the coeﬃcient an is proportional to the (2n)th power of the single-particle tunnelling matrix element, so that in the limit of weak tunnelling (usually very well satisﬁed in tunnel oxide junctions) only the term corresponding to n = 1, i.e. Eq. (15), need be kept (In other kinds of junction, such as Nb point contacts, it may be necessary to keep higher-order terms). From now on I will assume, unless otherwise stated, that EJ (∆φ) is well approximated by Eq. (15).

Although in the context of superconducting devices it is usually adequate to regard the constant EJ0 as simply determined from experiment (from the critical current, see below), it may be useful to sketch brieﬂy how it is calculated within BCS theory5. One starts by describing the tunnelling of single electrons across the junction by the so-called Bardeen- Josephson Hamiltonian: ˆ † HT = Tkqakbq + H.c. (18) Xkq † † where ak (bq) creates an electron in superconductor 1 (2). Next, one inverts Eq. (8) † † and Eq. (9) to express the real-electron operators ak, bk, etc. in terms of Bogoliubov quasiparticle operators. Finally, one writes down the standard second-order perturbation theory expression for the energy shift due to HˆT :

2 hn| HˆT |0i ∆E = (19) E − E Xn 0 n In the present case the intermediate states n involve excitation of one Bogoliubov quasiparticle (or equivalently creation of a single broken-pair state) in each of the bulk supercon- 6 (1) (2) ductors , one thus has energies Ek + Eq ≥ 2∆. The crucial point, now, is that while any particular intermediate state n is excited by a † † † particular αk and a particular βq (related to the bq’s as αk is to the ak’s), the coefficient † † † of αkβq has two contributions: one from the term Tk−qakb−q (with coefficient ∼ ukvq) † 7 and one from the term T−kqbqa−k (with coefficient ∼ uqvk) If the tunnelling conserves

4F. Bloch, Phys. Rev. B 2, 109 (1970) 5V. Ambegaokar and A. Baratoﬀ, Phys. Rev. Lett. 10, 486 (1963) 6for simplicity assumed identical. 7It is of course necessary in a proper calculation to keep the spin indices and relative signs straight.

6 invariance under time reversal (the usual case) we have Tk,−q = T−k,q, and hence the matrix element hn| HˆT |0i is given by

hn| HˆT |0i = Tk,−q (ukvq + uqvk) . (20)

When we insert Eq. (20) into Eq. (19) we get two types of terms. First, there are terms 2 2 proportional to |uk| |vq| , etc. These are nonzero even when the bulk metals are in the normal phase: it turns out that the diﬀerence from the N-phase value is positive, let us call it K(∆). The second type of term (from the cross-term in the square of Eq. (20)) is

∗ ∗ (1) (2)∗ 2 ukvkuqvq 2 Fk Fq −Re |Tk,−q| ≡−Re |Tk,−q| (21) Ek + Eq Ek + Eq Xkq Xkq

Since the phase of all the Fk (Fq) is the same and equal to that of Ψ1 (Ψ2), it is clear that Eq. (21) is proportional to cos ∆φ. In fact, a proper calculation shows that the coeﬃcient

EJ0 is given (at T = 0) by π∆ EJ0 = (22) 2Rn where Rn is the resistance of the junction in the normal phase. Such a calculation also shows that for the “standard” model (3D bulk superconductors, constant DOS near the

Fermi surface, etc.) the constant K(∆) is exactly equal to EJ0. Thus the correct expression for the T = 0 coupling energy of two superconducting metals relative to its value in the normal phase is

EJ (∆φ)= EJ0(1 − cos ∆φ) (23) in other words, one cannot save energy by Josephson coupling! (This simple result seems to be overlooked in probably dozens of papers on the cuprate superconductors).

We now come to a crucial point: As well as the variable ∆φ (which is “macroscopic”, in the sense that it describes the common behavior of a large number of electrons) we have in this problem a second “macroscopic” variable, namely the imbalance ∆N of the number of 1 Cooper pairs across the junction (i.e. the value of 4 (N1 − N2), relative to (say) its value in the normal ground state, where Ni is the number of electrons in superconductor i). The crucial result is that ∆N and ∆φ are canonically conjugate variables

∆N,ˆ ∆φˆ = i (24) h i There are any number of ways of seeing this result, of which the most straightforward is probably to deﬁne operators corresponding to the Ψ’s:

ˆ † † ˆ † † Ψ1 ≡ ak↑a−k↓, Ψ2 ≡ bq↑b−q↓ (25) Xk Xq

7 Then evidently

Nˆi, Ψˆ j = 2δijΨˆ j (26) h i which is satisﬁed if we write Ψˆ j = |Ψj| exp iφˆj and require

Nˆi, φˆj = 2iδij (27) h i 8 Then setting φ1 = −φ2 = ∆φ/2 and using the deﬁnition of ∆N, the result (Eq. (24)) follows.

Armed with the fundamental commutation relation Eq. (24) and an expression for the total Hamiltonian of the junction as a function of ∆N and ∆φ, we can now easily obtain the equations of motion of these variables:

∂(∆Nˆ) ∂Hˆ ∂ ∂Hˆ ~ =+ , ~ ∆φˆ = − (28) ∂t ∂(∆φˆ) ∂t ∂ ∆Nˆ In the simple situation originally considered by Josephson, namely a single junction biased with an external current, the only term in the Hamiltonian apart from the coupling energy,

Eq. (23), is a term due to any external voltage bias across the junction, namely −eV (N1 −

N2) ≡ −2eV ∆N. Thus, taking (with Josephson) the semiclassical limit so that ∆N and

∆φ are regarded as c-numbers, and using the fact that the current IJ ﬂowing across the junction is just 2ed/dt(∆N), we get the two celebrated Josephson equations:

I = I sin ∆φ J c , I ≡ E /(~/2e) (≡ E /(Φ /2π)) (29) d ~ c J0 J0 0 dt (∆φ) = 2eV/

Thus we see that Ic is the maximum current which can be carried by Josephson tunneling of pairs across the junction.

These results are adequate to describe large junctions with negligible dissipation. However, in smaller junctions there may be a non-negligible effect due to the non-infinite capacitance. Moreover, under this condition we cannot in general automatically set the current through the junction, Is, equal to the externally imposed one. So we need to incorporate both these effects into our Hamiltonian description.

The non-inﬁnite capacitance is easy to incorporate: we have to add a term to the Hamilto- nian of the form of a standard capacitance energy

2 2 2 Ecap =+Q /2C = (2e) (∆N) /2C (30)

8This choice avoids irrelevant complications connected with the “overall” phase, which is not a physically meaningful quantity.

8 and this will then contribute to the rate of change of ∆φ.

The incorporation of the effect of the external current leads is a bit more tricky. Going back to the case of infinite capacitance, we require that under this condition, in a steady state, the condition Is = Iext is satisfied.

We can achieve this condition by adding to the Hamiltonian a term

Ebias = −Iextφ0∆φ (31)

so that the total ∆φ-dependent energy, that is Eq. (31) plus the Josephson term Eq. (23), has the form of the so-called “washboard potential”. This is quite useful for considering problems such as “escape” from the zero-voltage state, where we can afford to neglect what goes on for values of ∆φ differing by ∼ 2π. However, it is obvious that Eq. (31) does not respect the fact that states differing in ∆φ by 2π are (at least prima facie!) physically identical, and this poses some quite subtle problems when we eventually go back to a full quantum-mechanical interpretation of the results.

Using the results obtained so far, we can write down the complete Hamiltonian, as a function of the canonically conjugate variables ∆N, ∆φ, of a Josephson junction associated with a 2 nonzero capacitance and possibly biased by an external current Iext: (C˜ ≡ C/(2e) )

2 (∆N) Ic H(∆N, ∆φ)= − φ0 cos ∆φ − Iextφ0∆φ (32) 2C˜ 2π [∆N, ∆φ]= i (33)

Note that with the appropriate replacements this is just the Hamiltonian of a simple pendulum, with ∆φ the deviation from vertical and the term corresponding to the external current bias representing the eﬀect of an applied torque.

Substituting Eq. (32) into the equations of motion Eq. (28) and taking the quasiclassical limit produces an equation which can be written

Iext = IJ sin ∆φ + Icap (34) d I = CV˙ , V ≡ (~/2e) ∆φ (35) cap dt This has a simple intuitive interpretation: any part of the external current which cannot be accommodated by the junction has to ﬂow into the capacitance which shunts it.

9 However, we are still missing one thing: The Hamiltonian Eq. (32) is conservative, while in real life there is almost always some dissipation in any Josephson junction perturbed away from the steady-state situation Iext = IJ sin ∆φ (and in addition, there is prima facie nothing preventing us from imposing an (average) external current > Ic). In the semiclassical description this is usually modelled by adding to the RHS of the current continuity equation (Eq. (34)) a term Ires ≡ V/Rn, where Rn is considered as an eﬀective resistance “shunting” the junction, so that the circuit diagram looks like this:

The physical origin of Rn (or better of the shunting −1 conductance Rn ) is not always obvious. At T ∼ ∆ the dominant eﬀect probably comes from the tunnelling through the junction of excited Bogoliubov quasiparticles, but in the low-temperature limit we −1 are considering this eﬀect is negligible, so Rn must be due to something like a physical normal-metal shunt.

The incorporation of a dissipation mechanism in the Hamiltonian is not trivial, in fact it is a special case of the problem discussed in lecture 2. As there, we can argue that under rather general conditions we can model the mechanism of dissipation by adding to the Hamiltonian Eq. (32) terms corresponding to a bath of harmonic oscillators with a coupling linear in both the oscillator coordinates and also linear in the relevant “system” coordinate (in this case, ∆φ). Once we are satisﬁed that this description is adequate, we are in a position to carry out concrete quantum-mechanical calculations of quantities such as energy levels, decay rates etc.

To conclude this lecture, let’s briefly review the principal types of superconducting device which are currently envisaged as possible qubits. Of course, to use them in this way it is necessary to treat the macroscopic variables ∆N and ∆φ as quantum-mechanical operators, and this will be assumed in what follows. A rough feel- ing for how (un)important quantum-mechanical effects are likely to be for a given junction under “general” conditions is given by the “quasiclassicality parameter” λ defined by

3 ~2 1/2 λ ≡ (CIcφ0/ ) (36) which within a factor of order unity is the inverse of the ratio of the zero-point energy to the barrier height in the unbiased (Iext = 0) washboard potential: λ ≫ 1 corresponds to a

10 “very classical” case.

The simplest application of Eq. (32) is directly to a single Josephson junction biased by an external current (“current- biased junction”, CBJ). Typically, for such a case the quasiclassicality parameter λ is large, which means that to see 9 useful quantization effects one needs to bias close to Ic. Under these conditions the potential well is shallow and anharmonic, and may contain only two or three levels, in the latter case unequally spaced; the bottom of two can then be used as the qubit and the third, if present, for diagnostic purposes. In this configu- ration one does not normally have to worry about the periodicity with respect to ∆φ, since to sample a ∆φ ∼ 2π the system has to escape from the metastable well, and once it has done so the dissipation is usually sufficient to render its behavior semiclassical.

A system fairly closely related to the CBJ is an rf SQUID ring (flux qubit), which in fact is simply a CBJ inserted into an otherwise closed superconducting loop through which an external flux can be applied. Compared to the CBJ, this system has the conceptual advantage that one can work in terms of the total flux Φ through the loop, which is manifestly meaningful in an “absolute” sense (i.e. not, like ∆φ, only modulo 2π). The total flux Φ is related to the phase drop ∆φ across the junction by

Φ/Φ0 = n + ∆φ/2π (37) The biasing (last) term in Eq. (32) is now absent, but a new energy appears, namely that 2 associated with the self-inductance of the loop, Eind = (Φ − Φext) /2L. Thus the total Hamiltonian now reads Q2 I φ (Φ − Φ )2 H(Q, Φ) = + − c 0 cos (2πΦ/φ )+ ext (38) 2C 2π 0 2L with the commutation relation [Q, Φ] = i~ (39) where for convenience we substituted Q ≡ e∆N for ∆N. Again, most such systems have λ ≫ 1, so that to get any interesting quantum eﬀects one needs to choose L so that

βL ≡ (2πLIc/φ0) is close to 1 and moreover set Φext close to φ0/2: for βL slightly greater than 1 the graph of V (Φ) is as shown. The WKB exponent of the tunneling rate through the

9 For Iext ≪ Ic the spectrum of the low-lying states is essentially that of an SHO, so useless for quantum computing. 11 1/2 barrier between the wells is then of order λ(βL −1) , so by choosing βL suﬃciently large we can make it appreciable. The lowest even- and odd-parity states in the double well may then be suitable for use as a qubit. (In practice, one often pretties up the simple single-junction rf qubit shown in the ﬁgure by adding extra junctions or other bits of circuitry).

In most instantiations of the CBJ or the ﬂux qubit, the system even after the bias is still fairly “semiclassical”, in the sense that (∆φ)2 ≪ (∆N)2 (or the equivalent). The third major superconducting qubit candidate, the Cooper-pair box, operates in the op- posite regime: the two grains are small enough, and the junction weak enough, that the 2 ~2 1/2 quasiclassicality parameter λ ≡ (CIcφ0/ ) is already . 1 without the need for any special bias. The Hamiltonian Eq. (32) (with Iext = 0) is still a good description, but it is now more natural to use as a basis the eigenstates of ∆N and to treat the Josephson term as a perturbation (in the “pendulum” analogy, this corresponds to taking the angular momentum eigenstates as a basis and treating gravity as a perturbation). One has in this case the extra freedom to bias the system with an extra voltage Vext, thereby adding to the Hamiltonian a term

Ebias = 2e∆NVext. (40)

One often chooses Vext to nearly cancel the capacitance term, so that the states ∆N = 0 and ∆N = 1 are nearly degenerate: if the Josephson term is . the capacitance one, the phase diagram as a function of

Vext then looks as shown (note that we are now talk- ing about the total energy, not just the “potential” part), and the two states ∆N = 0 and ∆N = 1 (or the two energy eigenstates) then may be suitable as a qubit. Note by the way that a diagram similar to the above could be drawn for the flux qubit case, with the horizontal axis representing the external flux Φext; in fact, in many ways the two systems may be regarded as “dual” to one another. However, an important conceptual difference is that it is only in the flux-qubit case that the two basis states of the qubit are (even arguably) “macroscopically” distinct; in the Cooper-pair box case it is only a single pair which is transferred from one grain to the other, and (perhaps surprisingly) the other ∼ 109 electrons on the grains are almost insensitive to the transfer provided only that the box dimensions are ≫ the Debye screening length (a condition almost invariably fulfilled in practice).