Superconducting charge qubits from a microscopic many-body perspective
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Superconducting charge qubits from a microscopicmany-body perspective
D.A. Rodrigues † School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD,U.K.
T.P. Spiller
Hewlett Packard Laboratories, Filton Road, Bristol, BS34 8QZ, U.K.
J.F. Annett and B.L. Gy¨orffy
Department of Physics, Bristol University, Bristol, BS8 1TL, U.K.E-mail: † [email protected] Abstract.
The quantised Josephson junction equation that underpins the behaviourof charge qubits and other tunnel devices is usually derived through cannonicalquantisation of the classical macroscopic Josephson relations. However, this approachmay neglect effects due to the fact that the charge qubit consists of a superconductingisland of finite size connected to a large superconductor. We show that the wellknown quantised Josephson equation can be derived directly and simply from amicroscopic many-body Hamiltonian. By choosing the appropriate strong couplinglimit we produce a highly simplified Hamiltonian that nevertheless allows us to gobeyond the mean field limit and predict further finite-size terms in addition to thebasic equation.
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JPhysC harge qubits from a many-body perspective φ D across the junctions and apply cannonical quantisation rules to φ D asa ‘position’ variable. Of course, the classical Josephson equations for such a phasedifference are first derived from microscopic theory [1], and so this standard approachrepresents a ‘re-quantisation’ of ‘classical’ equations that were in turn derived fromquantum mechanical microscopic theory using a mean field theory that does not takeinto account the charging energy of the island. Consequently, in such approaches thedescription of quantum fluctuations is at best semi-phenomenological. In what followswe examine the limitations of the above procedure on the basis of a simple modelwhich permits an exact treatment of a superconducting island coupled weakly to a bulksuperconductor.To motivate our interest in the problem we note that much current experimentaland theoretical attention is focused on nanoscale superconducting grains coupled tolarge superconductors as such ‘Cooper Pair Boxes’ are becoming realistic candidates forbeing useful qubits in Quantum Information devices [3, 4, 5, 6]. Starting with the workof Nakamura et al. [7], over the last few years there have been a number of impressiveexperiments [8, 9, 10, 11, 12, 13] demonstrating appropriate charge qubit behaviour andmacroscopic tunnelling. As experiments continue to improve, it is now pertinent to re-examine the standard approach to describing quantum fluctuations in superconductingcharge qubits and related systems. Clearly, in the course of such investigation onewould expect to reproduce the basic quantum phenomenology from a fully microscopicapproach within a well controlled approximation, as there is already good experimentalsupport for this. Nevertheless, a generalised theory will also yield new additional termsdue to the finite size of the superconducting islands and for future experiments they couldhave significant consequences. To shed light on these, we examined a simple microscopicmodel of a Cooper Pair box, showing how the familiar phenomenology emerges, alongwith new finite size effects.
1. Quantising the Josephson Relations
In the interest of clarity, we start our discussion by recalling, briefly, the usualphenomenological approach to the problem at hand. The standard way to obtain theHamiltonian describing a small superconductor connected through Josephson tunneljunctions to a bulk superconductor is by starting with the Josephson equations for φ D and V , the difference in phases of the two superconducting regions and the voltage harge qubits from a many-body perspective I = I C sin φ D (1) dφ D dt = 2 eV ~ (2)where I C is the critical current of the junction. Although derived from a quantummechanical microscopic treatment, as evidenced by the appearance of ~ , these equationscan be regarded as classical equations of motion. We follow the standard procedurefor canonical quantisation and first find the Lagrangian that leads to these equations.Namely, we take, L = 12 ~ C e (cid:18) dφ D dt (cid:19) + ~ I C e cos φ D (3)where we have introduced the total island capacitance, C . If we choose the phase φ D to be the canonical position variable, we can identify the canonical momentum π = ∂ L /∂ ˙ φ D , π = ~ C e ˙ φ D = ~ CV e = ~ ( N − n g ) , (4)and it is seen that the phase of the condensate and the excess number of Cooper pairson the island, ( N − n g ), are conjugate variables [3, 4]. The term n g represents an appliedgate voltage (in dimensionless units) and so the canonical momentum π = ( N − n g ) iseffectively the charge on the device, viewed as a capacitor, in units of 2 e . Note that if thetotal charge on the system is zero, this can be rewritten in terms of the charge difference.To quantise the system, we introduce the commutation relation between conjugatevariables [ φ D , π ] = i ~ and note that this can be satisfied by writing π = − i ~ ∂/∂φ D − n g (keeping the gate voltage explicit). Then the Hamiltonian, b H = π ˙ φ D − L , is given by, b H = E C (cid:18) i ∂∂φ D − n g (cid:19) − E J cos φ D . (5)where the charging energy is given by E C = 2 e /C and E J is defined as E J = ~ I C / e .The Schr¨odinger’s equation for the amplitude ψ ( φ D ) is then, b Hψ ( φ D ) = i ~ ddt ψ ( φ D ) . (6)Evidently, the probability that the phase difference takes on a certain value is given by | ψ ( φ D ) | .This is the desired standard quantum description of the Josephson Junction. In theremainder of this paper, we show how the above quantised Josephson junction equationcan be rederived directly from the microscopic theory in a way that includes finite sizeeffects.
2. Finite Superconductors As Spins
We wish to produce a description of a finite superconductor that is simple enough tosolve exactly but retains properties due to its finite size. Specifically, we wish to be able harge qubits from a many-body perspective H = X k, σ ǫ k c † k, σ c k, σ − X k, k ′ V k, k ′ c † k ↑ c †− k ↓ c − k ′ ↓ c k ′ ↑ , (7)where c † k, σ and c k, σ create and annihilate electrons, respectively, with spin σ in thestate k with energy ǫ k and the matrix element V k, k ′ describes an attractive two bodyinteraction. As we are discussing a finite superconducting island, the label k, ↑ does notrefer to a free electron wavevector but to a generic single-electron eigenstate, with − k, ↓ representing the corresponding time-reversed state.A common approximation to this equation is made by assuming the pairingpotential V k, k ′ is equal for all k, k ′ in a region around the Fermi energy determined bythe cutoff energy ~ ω c and zero outside this region. That is, V k, k ′ = V for | ǫ k − ǫ F | < ~ ω c and V k, k ′ = 0 otherwise. This greatly simplifies matters whilst retaining the essentialphysics. We now adopt a similar philosophy in making a further approximation, andtake all the single electron energy levels ǫ k within the cutoff region around the Fermienergy to be equal to the Fermi energy ǫ F .The interaction term, V k,k ′ acts only within the cutoff region around the Fermienergy. Outside this region the Hamiltonian is diagonal and trivially solved. Writingthe single electron energy as ǫ k = ǫ F + ( ǫ k − ǫ F ), we note that within the cutoff region | ǫ k − ǫ F | < ~ ω c , and thus if V ≫ ~ ω c , then | ǫ k − ǫ F | ≪ V and we can discard thevariation of ǫ k . Thus in this strong coupling approximation, our Hamiltonian becomes, b H = ǫ F ′ X k c † k, σ c k, σ − V ′ X k, k ′ c † k ↑ c †− k ↓ c − k ′ ↓ c k ′ ↑ (8)where the dashes on the sums indicate that they are only taken over states within thecutoff region.Although this caricature of a realistic Hamiltonian represents an uncontrolledapproximation, we will show that it allows us to derive a Josephson junction equationthat goes beyond mean field, and that the results it produces agree in the strong couplinglimit with known results in two important cases. Namely, the mean-field solution ofthis Hamiltonian agrees with the BCS solution, and the exact solution agrees with theRichardson solution[15].It should be noted that although many superconductors can be described ashaving strong coupling, the BCS Hamiltonian is not necessarily appropriate for theirdescription. Equation 8 is hence not intended as a description of this particular classof superconductors, but rather as a generic model that, although simplified, allows anexact solution and a description of the physics we are trying to capture.As a consequence of the above simplifications equation 8 can now be written interms of the three operators [16], b S Z = 12 ′ X k (cid:16) c † k ↑ c k ↑ + c †− k ↓ c − k ↓ − (cid:17) harge qubits from a many-body perspective b S + = ′ X k c † k ↑ c †− k ↓ b S − = ′ X k c − k ↓ c k ↑ , (9)Note that these operators obey the commutation relations, and therefore the algebra, ofquantum spin operators of size l/
2, where l is the number of levels in the cutoff region.Thus the main result of this section is the effective Hamiltonian, b H sp = 2( ǫ F − µ ) (cid:18) b S Z + l (cid:19) − V b S + b S − , (10)where we have introduced a chemical potential µ to describe coupling to a reservoir.
3. Exact solution
The eigenstates of equation 10 are the eigenstates of the spin operator b S Z , | l , m N i ,where the component of the spin along the Z axis is given by m N = N − l/ N denotes the number of Cooper pairs on the island. The eigenenergies corresponding tothese eigenstates are, E N = 2( ǫ F − µ ) N − V N ( l − N + 1) . (11)A chemical potential allows us to specify the average number of Cooper Pairs on theisland in equilibrium. In the case of the exact solution, where N is a good quantumnumber, this means choosing a state with a particular value of N to be the ground state.We choose µ so that the ground state is the state with a chosen value of N , which welabel ¯ N . The eigenenergies E N therefore become, E N = − V N (2 ¯ N − N ) , (12)and the ground state is | l , m ¯ N i , with energy, E gs = − V ¯ N . (13)We can also easily see that although the pairing parameter h b S + i = 0 in all eigenstates,i.e. there is no symmetry breaking, we still have fluctuations as expected for a finitesuperconductor which are given by h b S + b S − i = N ( l − N + 1) for a general eigenstate N and h l , m ¯ N | b S + b S − | l , m ¯ N i = ¯ N ( l − ¯ N + 1) (14)for the ground state. We also see that the operator b S + which couples eigenstates,corresponds (when appropriately normalised) to the quasiparticle creation operator forthe system.
4. Comparison to Standard Results
To generate confidence in this simple model, we compare its solutions to the solutions ofthe full Hamiltonian (equation 7) in two ways. First, we find the mean field solution and harge qubits from a many-body perspective
The mean field approximation arises from the assumption that the operators b S ± remainclose to their expectation values, h b S ± i . We write b S ± = h b S ± i + ( b S ± − h b S ± i ), and discardterms to second order or higher in ( b S ± − h b S ± i ). Writing V h b S − i = ∆, we find the meanfield Hamiltonian for our model, b H MF = 2( ǫ F − µ ) (cid:18) b S Z + l (cid:19) − ∆ b S + − ∆ ∗ b S − + | ∆ | V (15)Apart from the constant term, this is a linear combination of the spin operators b S Z , b S Y and b S X and is therefore proportional to the projection of a spin operator on an unknowndirection specified by the unit vector ˆ n . Thus denoting b S. ˆ n by b S Z ˆ n , we may write, b H MF = γ b S Z ˆ n + | ∆ | V + 2( ǫ F − µ ) l , (16)and therefore the problem of diagonalising equation 15 is equivalent to the problem ofrotating the axis of quantisation for our effective spin operators. Requiring that thecommutation relations [ b S Z ˆ n , b S +ˆ n ] = b S +ˆ n hold for spin operators in the frame of referencewhere the axis of quantisation is along b n determines both an expression for b S +ˆ n and thevalue of γ , b S +ˆ n = 2∆ γ (cid:18) b S Z − ∆2 ξ F − γ b S + − ∆ ∗ ξ F + γ b S − (cid:19) γ = 2 p ( ξ F ) + | ∆ | (17)where ξ F = ǫ F − µ and we note that γ = 2 E F , the energy of a Cooper Pair evaluated atthe Fermi energy. The ground state of our Hamiltonian can now be trivially found, as itcorresponds to the m = − l/ b S Z ˆ n . Recalling that a maximal m state of a spin operator pointing in one direction is a spin coherent state [17] in anyother we find, | α i = 1 p (1 + | α | ) l l X N =0 ( α ∗ b S + ) N | l , m i N != 1 p (1 + | α | ) l Y k (1 + α ∗ c † k ↑ c †− k ↓ ) | l , m i (18)As one might expect, the second line is a way of writing the BCS ground statewavefunction in the limit where all the levels have equal probability of occupation,i.e. u k /v k = α ∗ for all k . Making use of equation 16 and the fact that b S − ˆ n | α i = 0 forthe ground state, we find that α = ( ξ F − E F ) / ∆. To complete the calculation, we needto self-consistently determine the values ∆ and µ , which is relatively simple in the spinmodel and gives, | ∆ | = V ¯ N ( l − ¯ N ) (19) harge qubits from a many-body perspective ǫ F − µ = V ( l/ − ¯ N ) (20) H SMF | α i = − V ¯ N | α i (21)where ¯ N is the average occupation of the island. We find equations 18- 21 are exactlyequal to the expressions found if we were to solve the full BCS equation and then takethe weak coupling limit (this is shown in the appendix of [18]). Comparing the meanfield solution to the exact solution, we see that, surprisingly, the exact (equation 13)and mean field (equation 21) ground state energies are identical. However, we have anon-zero pairing parameter h S − i = ∆ /V and the expectation value of the mean fieldcoupling term, (cid:28) α (cid:12)(cid:12)(cid:12)(cid:12) − ∆ b S + − ∆ ∗ b S − + | ∆ | V (cid:12)(cid:12)(cid:12)(cid:12) α (cid:29) = − V ¯ N ( l − ¯ N ) , (22)neglects the quantum fluctuations present in the exact solution, (cid:28) l , m ¯ N (cid:12)(cid:12)(cid:12)(cid:12) − V b S + b S − (cid:12)(cid:12)(cid:12)(cid:12) l , m ¯ N (cid:29) = − V ¯ N ( l − ¯ N + 1) (23)in much the same way a classical spin neglects the fluctuations present in a quantumspin, i.e. the eigenvalues of b S are S ( S + 1) and not the classical values S . Unbeknownst to the condensed matter community for many years, there exists an exactsolution to the BCS Hamiltonian (equation 7) for finite superconductors, first discoveredin 1963 by Richardson[15, 19, 20] in the context of nuclear physics. It has been shownthat this solution reproduces the BCS result in the bulk limit, but it is difficult to workwith for any island occupied by more than a few Cooper pairs. The Richardson solutionrequires the introduction of operators that diagonalise the full (i.e. not mean field) BCSHamiltonian, b H = N X ν =1 E Jν b B + Jν b B − Jν (24) b B + Jν = X k c † k ↑ c †− k ↓ ǫ k − E Jν , (25)where the sum in equation 24 runs over ν up to the total number of Cooper pairs onthe island. The parameters E Jν are found by solving the equations,1 + 2 VE Jη − E Jν = V X k ǫ k − E Jν , (26)for all ν .Whilst the usual BCS theory has an essential singularity at V = 0, the theory iswell behaved near 1 /V ∼
0. Thus, following Altshuler et al. ,[21] we expand equation 26in powers of ~ ω c /V . Using ǫ k ∼ ǫ F leads to,1 V + N X ν =1 E Jη − E Jν = l ( E Jη − ǫ F ) + l X k =1 ǫ k − ǫ F )( E Jη − ǫ F ) . (27) harge qubits from a many-body perspective E Jη − ǫ F , and sumover the N parameters E Jη to obtain, N X η =1 E Jη − ǫ F V + N ( N −
1) + 0 =
N l (28)where the double sums over η, ν have either vanished, or gone to N ( N −
1) due tosymmetry. Finally, we recall that the energy of the Richardson solution is given bya sum over E Jη , and rewrite equation 28 to get the energy of an island containing NCooper pairs: E N = 2 ǫ F N − V N ( l − N + 1) . (29)As heralded in the introduction this result matches the exact energy of the spinHamiltonian as given in equation 11 for µ = 0.Thus we have shown that although we have made a significant approximation to theHamiltonian, the results thereby derived are consistent with results obtained by solvingthe full system in either the mean field approximation or exactly, and then taking theappropriate limit.
5. Phase Representation of the Spin Operators b S + and b S − The preceding sections have established our model of a finite superconducting systemas a large spin, as given in equation 10. We shall now go on to show how this model canbe used to derive a phase-representation description of the Josephson effect in a systemcomprising a small island coupled to a larger piece of bulk superconductor.We wish to convert to a representation in terms of the continuous phase variable φ ,i.e. convert from ket notation to wavefunction ψ ( φ ) and differential operator (such as ddφ ) notation. Thus, a ket | ψ a i will become a wavefunction h φ | ψ a i , and the differentialoperator must be consistent with this. Defining the state | φ i = (2 π ) − / P e iφN | l , m N i ,we find that the wavefunction corresponding to | l , m N i is (2 π ) − / e − iφN . We can thenexamine how the operators act on this wavefunction. b S Z h φ | l , m N i = h φ | ( N − l ) | l , m N i = ( N − l ) h φ | l , m N i = ( N − l ) e − iφN √ π b S Z ψ ( φ ) = (cid:18) i ∂∂φ − l (cid:19) ψ ( φ ) (30)Similarly, we find for the raising operator, b S + h φ | l , m N i = h φ | p ( N + 1)( l − N ) | l , m N i b S + ψ ( φ ) = e − iφ s(cid:18) i ∂∂φ + 1 (cid:19) (cid:18) l − i ∂∂φ (cid:19) ψ ( φ ) (31) harge qubits from a many-body perspective b S − h φ | l , m N i = h φ | p N )( l − N + 1) | l , m N i b S − ψ ( φ ) = e iφ s i ∂∂φ (cid:18) l − i ∂∂φ + 1 (cid:19) ψ ( φ ) (32)Collecting the differential forms for the operators and rewriting S + and S − into a moreconvenient form leaves us with, b S Z = (cid:18) i ∂∂φ − l (cid:19)b S ± = s(cid:18) l ± (cid:18) i ∂∂φ − l (cid:19)(cid:19) e ∓ iφ s(cid:18) l ∓ (cid:18) i ∂∂φ − l (cid:19)(cid:19) (33)The form of the raising and lowering operators can also be derived by requiring that thecommutation relations for quantum spins are enforced. We see that it is the (cid:16) i ∂∂φ − l (cid:17) terms in the S + , S − operators that take into account the finite size effects and ensurethat [ S + , S − ] = 0.Writing the operators in this form allows us to take the large size ( l → ∞ ) limit.In taking this limit we assume that S Z ≪ l . In the superconducting language, thiscorresponds to only states close to half filling being occupied. Specifically, we assume, |h l , m N | ψ i| ∼ f or | N − l | & (cid:18) l (cid:19) , (34)a condition which is fulfilled for coherent states with ¯ N set close to l . When this istrue, we can expand the square root in ( i ∂∂φ − l ) /l . We see that the leading order termsgive S ± = l e ∓ iφ , and we regain the semiclassical large-size limit for which [ S + , S − ] = 0as discussed by Lee and Scully[16].
6. Quantised Josephson Junction Equation
We can now use the forms of the operators given in equations 33 to write down thequantised Josephson equation. We begin with a Hamiltonian that describes a finitesuperconducting island coupled to a superconducting reservoir, b H = b H I + b H R + b H C + b H T (35)where b H I and b H R are the BCS Hamiltonians on the island and reservoir respectively,and we introduce Hamiltonians representing the charging energy of the island, b H C = 4 e C ( b N I − n g ) (36)and the tunnelling between island and reservoir, b H T = − T X k,q c † k c †− k c − q c q + c † q c †− q c − k c k . (37) harge qubits from a many-body perspective T is the standard tunnelling matrix element for Cooper Pairs [22], which weassume for simplicity to be real and independent of k, q . If we make our strong-couplingapproximation and assume that all the electronic energy levels can be considered equal,we can write these Hamiltonians in terms of spin operators, as follows: b H I = 2( ǫ F I − µ I ) b S ZI − V I b S + I b S − b H R = 2( ǫ F R − µ R ) b S ZR − V R b S + R b S − R b H C = 4 e C ( b S ZI + l I / − n g ) b H T = − T (cid:16) b S + I b S − R + b S − I b S + R (cid:17) . (38)Inserting these expressions into equation 35, we obtain b H = E ′ C ( b S ZI − n ′ g ) − T ( b S + I b S − R + b S − I b S + R ) , (39)where we have incorporated the terms from b H I linear and quadratic in b S ZI into therenormalised charging energy and gate voltage represented by E ′ C and n ′ g respectively.In the limit that both the reservoir and the island can be considered infinite, we regainthe standard form for the quantised Josephson junction Hamiltonian, b H = E ′ C (cid:18) i ∂∂φ I − n ′ g (cid:19) − T l R l I φ I − φ R ) , (40)suggesting that equation 6 can be considered as a large-size limit where the finite sizeof the island can be neglected. However, we are now able to obtain, using equation 33,the next terms in the series expansion, b H = E ′ C (cid:18) i ∂∂φ I − n ′ g (cid:19) − T l R n ( l I + 1) cos( φ I − φ R ) − l I (cid:18) i ∂∂φ I − l I (cid:19) cos( φ I − φ R ) − l I i (cid:18) i ∂∂φ I − l I (cid:19) sin( φ I − φ R ) + · · · o (41)We find that the new terms involve products of both the phase and the charge operators.Thus the Josephson tunnelling term effectively depends on the island charge. Makingan analogy with a particle in a potential, with a position corresponding to the phase,we see that the extra terms in equation 41 can be thought of as a velocity-dependentpotential. This effect also breaks the periodicity of the island energy with n ′ g , i.e. theenergy now depends on the absolute value of n ′ g , rather than merely its value modulo1. In this derivation we have assumed that the single electron energies are all equal,and thus the occupations u k /v k are all equal. However, these occupations maintain asimilar order of magnitude when different, and so we would expect the extra terms ineq. 41 to be of a similar size when the strong-coupling approximation is relaxed.We can make an estimate of the size of these effects for an island of a given size bycalculating l I , the number of electrons within the cutoff region, by comparing the level harge qubits from a many-body perspective l I ≈ × , and thus the additional terms in equation 41 (proportionalto 1 /l I are unlikely to be significant. If we consider instead a nanograin of the typedescribed in [19, 20, 23], we find that l I ≈
7. Conclusions
We have shown how the quantum Josephson Junction equation, usually derived by re-quantising the mean field equations of motion, can be directly derived from a microscopicdescription of a superconducting island. We used a simplified Hamiltonian in which theenergy of the individual microscopic electron levels is considered equal that allowedan exact solution to be found. We have shown how a mean field approximation leadsto a solution that corresponds to a spin coherent state, which is the BCS state inthe appropriate limit. As well as illustrating how the familiar phenomenology emergesthrough the mean field approximation, we showed we can describe effects beyond themean field, such as quantum fluctuations. We went on to rederive the JosephsonJunction equation and describe size dependent corrections to the familiar terms.
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