Quantize electrical circuits

Transcription

1 Quantize electrical circuits a lecture in Quantum Informatics the 4th and 7th of September 017 Thilo Bauch and Göran Johansson In this lecture we will discuss how to derive the quantum mechanical hamiltonian governing the dynamics of an electrical circuit, starting from the circuit diagram. A classical electrical circuit is fully described by its nodes and the voltages across and currents through all branches of the circuit. In a analog to classical mechanics, these variables play the roles of velocities and forces, respectively. All branches should be given an orientation, to keep track of the sign of the voltage and current (see Fig. 1). The branch voltages and currents are not independent variables, but are constrained by the circuit topology and Kirchoff s rules, i.e. the sum of voltages across any loop is zero and sum of currents flowing into a node is zero. Since an Hamiltonian description of an Figure 1: From Devoret 1995 Les Houches lectures. electrical circuit requires the introduction of branch fluxes and branch charges (see e.g. the harmonic LC oscillator) we can define the branch flux Φ b (t) = t v b (t )dt, (1) and the branch charge Q b (t) = t i b (t )dt, () 1

2 1 Some circuit elements The different circuit elements are defined by different relations between the currents and voltages. We will mainly deal with three different elements. 1.1 Capacitance A capacitance is an element which stores charge and is characterized by its capacitance C. The voltage v(t) across the capacitance is proportional to the charge q(t) stored, v(t) = q(t) C. (3) The energy of the capacitance is stored in the electric field E C = Cv = q C. (4) We may rewrite the energy of the capacitance using branch flux variables E C = C Φ. (5) Figure : Circuit symbol for capacitance. 1. Inductance An inductance is an element which develops a voltage when you change the current through it according to, v(t) = L t i(t)., (6) where the proportionality constant L is the inductance. Equivalently, one can also talk about the magnetic flux Φ(t) = t v(t )dt, (7) which is the branch flux of the inductor. The inductance stores a flux Φ(t) proportional to the current i(t) through the inductance, i(t) = Φ(t) L. (8)

3 The energy of the inductance is stored in the magnetic field E L = Li = Φ L. (9) Figure 3: Circuit symbol for inductance. 1.3 Josephson junction The Josephson junction (JJ) is a circuit element made up of a tunnel junction between two superconductors. It is characterized by its capacitance C J and its so-called Josephson energy E J. According to the DC Josephson relation, the current through the JJ is determined by the phase difference φ between the superconducting condensates on the two sides of the junction, φ(t) = e h t 0 V (t )dt + φ(0), (10) which is given by the time-integral of the voltage across the junction. derivative of the above equation is the so-called AC Josephson relation: The φ(t) = e V (t). (11) h The time-integral of the voltage is also a magnetic flux Φ, and the normalization is sometimes written in therms of the magnetic flux quantum = h/e, φ = e h Φ = π Φ. (1) Now, the current through the JJ junction is given by the DC Josephson relation i J (t) = I c sin φ(t), (13) where the critical current I c is proportional to the Josephson energy We see that for small phase differences φ(t) 1, I c = e h E J = π E J. (14) ( ) π i J (t) I c φ(t) = E J Φ, (15) 3

4 Figure 4: Circuit symbol for a) a Josephson junction and b) an ideal Josephson junction in parallel with its capacitance. the JJ acts like an linear inductance with L J = 1 E J ( Φ0 π ), (16) while in general for larger phase-differences it is a non-linear inductance, giving rise to anharmonic spectra of the system where it is included. The energy stored in a Josephson junction is given by E JJ with E J = I c /π. = t dt I(t )V (t ) = t dt I c sin = ( ) Φ 0 dφ I c sin π Φ dφ = E J [1 cos (π ΦΦ0 ) dφ (π ΦΦ0 )], dt = The Lagrangian of an electrical circuit In the following we describe a recipe for deriving the Lagrangian and eventually the Hamiltonian of an electrical circuit (for details see Devoret 1995). An example of an electrical circuit is shown in Fig.5, where each branch is represented by either an inductor, capacitor, or Josephson junction. DC voltage sources can be represented by capacitors, taking the limit C and Q, while keeping the voltage V = Q/C constant. Similarly, a dc current source can be represented by an inductor. The branch fluxes and branch charges are not independent variables but are constraint by the circuit topology and the Kirchhoff s relations, which in flux and charge variables read Φ b = Φ l, (18) all branches b around loop (17) and Q b = Q n, (19) all branches b arriving at node n 4

5 Figure 5: Example of an electrical circuit. Each branch is represented by either an inductor, capacitor, or Josephson junction. where Φ l and Q n are constants (see Fig.6). For circuits containing tunneling elements like Josephson junctions it is more convenient to use the node variable representation instead of the branch variable representation. The definition of node fluxes is shown in Fig.6. It is important to note the node flux variables depend on the particular description of the topology of the electrical circuit, i.e. the choice of reference node (ground) and spanning tree. After having chosen a reference node and spanning tree we first write down the Lagrangian in branch flux representation L(Φ b, Φ b ) = T V, (0) where T is the kinetic energy and V the potential energy. Since we are using flux variables, we can associate capacitive energies as kinetic energies (they depend on Φ) and inductive energies as potential energies (they depend on Φ). Note that the Josephson energy is also an inductive energy and hence a potential energy. Then we replace in the Lagrangian all branch fluxes and their time derivatives (Φ b, Φ b ) with node fluxes and their time derivatives (Φ n, Φ n ) L(Φ n, Φ n ) = T V, (1) taking also into account the constrains given by Kirchhoff s relation. From the above Lagrangian we can now define conjugate momenta of node fluxes by the usual relation q n = L(Φ n, Φ n ) Φ n, () with unit of charge. The physical meaning of a node charge q n is the sum of the charges on the capacitances connected to node n. The Hamiltonian of the circuit is obtained through the Legendre transformation of the Lagrangian H(Φ n, q n ) = n Φ n q n L, (3) 5

6 Figure 6: Definition of node variables of an electrical circuit. (a) The reference node (could be any node) is connected to ground. The spanning tree (set of branches shown in green) connects all other active nodes to ground (reference node) by only one path along the tree. The remaining branches are so-called closure branches. (b) The node fluxs Φ n at nodes n (shown in red) are defined as the sum of branch fluxes from ground to node n using a path through the spanning tree (keeping track of the branch flux orientation). The remaining closure branch fluxes are determined through the loop constrains, i.e. Kirchhoff s relation. where the sum goes over all active nodes n. Finally, we arrive at the quantum mechanical representation of the Hamiltonian be promoting the variables Φ n and q n to quantum mechanical operators, satisfying the canonical commutation relation [ˆΦn, ˆq n ] = ˆΦ nˆq n ˆq n ˆΦn = i h. (4) 3 An LC oscillator in a Lagrangian formulation Figure 7: An LC oscillator circuit. As in standard circuit theory, we can analyze the LC oscillator circuit sim- 6

7 ply by introducing the node voltages. However, in the case we have Josephson junctions in the circuit it is more convenient to use the node fluxes as coordinates. We consider a simple LC oscillator as in Fig. 7, where the lower node is grounded. In this case, we have only one active node, the upper one, with flux Φ. We get the equations of motion by equating the two currents: i) the sum of currents arriving from the inductive elements connected to that node and ii) the sum of currents going into the capacitive elements connected to that node: Φ = C v = C Φ. L Now, taking the mechanical analogue, where Φ is the coordinate of a particle, we see that this equation for current conservation is similar to Newton s equation of motion for a particle with a quadratic potential energy given by the inductance, and a kinetic energy E L = Φ L, (5) E C = Cv = C Φ, (6) given by the capacitance. The Lagrangian of that system is given by the kinetic energy minus the potential energy L = C Φ Φ L, written in terms of coordinate Φ and velocity Φ. To connect to quantum mechanics we then replace velocity by conjugate momentum q = L = C Φ, Φ which has the unit of charge. We get the hamiltonian through the Legendre transformation, H = Φ n q n L, n where the sum goes over all active nodes, which for our LC-oscillator gives ( H = Φq L = q C C Φ ) ( ) Φ = q q L C C Φ = q L C + Φ L, which we recognize as the harmonic oscillator hamiltonian. So far we haven t done anything more than rewritten the classical equations of motion for the circuit in terms of a hamiltonian. 7

8 4 Quantization The act of quantizing the the circuit is now very simple. We simply promote the variables Φ and q to quantum mechanical operators, satisfying the canonical commutation relation [Φ, q] = Φq qφ = i h. We then have a quantum mechanical description of the LC circuit, which indeed is a quantum mechanical harmonic oscillator. For the simple LC-oscillator it was straightforward to guess what is the conjugate momentum to Φ, but in more complicated circuits the above scheme can be followed to systematically derive the hamiltonian of the circuit. In the exercise class you will see how to derive the hamiltonian of a current biased Josephson junction. 5 The superconducting charge qubit a single Cooper-pair box We now give an example of a real qubit, which is also studied at Chalmers both experimentally and theoretically, the single Cooper-pair box. The box is a very small metallic (Al) island connected via a tunnel junction to a metallic reservoir. The island is also capacitively coupled to a controllable gate voltage (see Fig. 8 and 9). Figure 8: A real single Cooper-pair box (SCB) qubit built at Chalmers. From T. Duty, D. Gunnarsson, K. Bladh, and P. Delsing, Physical Review B 69, (004). The box is described by the capacitances of the Josephson junction and the gate, C J and C g respectively, the gate voltage V g, and the Josephson energy E J. In Fig.9(c) a specific choice of the ground and spanning tree (including the orientation of the branch fluxes) of the SCB circuit is shown. Moreover, we have replaced the voltage source with a capacitor C 3. From the definition of the node fluxes (see Fig.9(c)) and the Kirchhoff s relation (constraint) we can rewrite the relevant branch flux variables in terms of node flux variables: 8

9 Figure 9: (a) A schematic single Cooper-pair box (SCB) qubit, from Yu. Makhlin, G. Schön, and A. Shnirman, Review of Modern Physics B 73 (001). (b) Circuit diagram of the SCB. (c) Circuit diagram of SCB with a specific choice of ground and spanning tree (green). The voltage source has been replaced by a capacitor C 3. The branch fluxes for the Josephson junction, junction capacitance, gate capacitance, and source capacitance are given by Φ J, Φ CJ, Φ Cg, and Φ 3, respectively. The node fluxes are Φ a = Φ J and Φ b = Φ J + Φ Cg. Φ J = Φ a (7) Φ CJ = Φ a + const. Φ CJ = Φ a (8) Φ J + Φ Cg = Φ b Φ Cg = Φ b Φ a (9) Φ 3 = Φ b + const. Φ 3 = Φ b (30) We first write the Lagrangian in branch flux representation L = C g Φ Cg + C J Φ CJ + C 3 Φ 3 ( ) π + E J cos Φ J. (31) Next we rewrite the Lagrangian in node flux representation using the above relations L = C g( Φ b Φ a ) + C Φ J a + C Φ 3 ( ) b π + E J cos Φ a. (3) We can now take the limit C 3, fixing Φ b = V g = const. (which makes sense, since we are dealing here with a voltage source). Dropping the constant 9

10 term C 3 Φ b / in the Lagrangian we obtain L(Φ a, Φ a ) = C g( Φ a V g ) + C J Φ a The conjugate momentum of the flux Φ a is given by ( ) π + E J cos Φ a. (33) q a = L Φ a = C g ( Φ a V g ) + C J Φa, (34) which is the charge on the island of the SCB. Through the Legendre transformation we arrive at the Hamiltonian H(Φ a, q a ) = q a Φa L = (q a + C g V g ) ( ) π E J cos Φ a, (35) C Σ with C Σ = C J + C g the island capacitance to ground. In the above derivation we have dropped constant terms which do not contribute to the dynamics of the system. Defining E C = e /C Σ the Coulomb energy of a single electron on the island, n = q a /e (with e C) the number of excess Cooper-pairs on the island, and n g = C g V g /e the dimensionless induced gate charge the Hamiltonian reads ( ) π H = 4E C (n n g ) E J cos Φ a. (36) We then proceed to quantize this system, promoting n and Φ a to operators fulfilling the commutation relation [ π ˆΦ ] a, ˆn = i, (37) with = h/e the superconductive flux quantum. Moreover, the eigenstates of ˆn define the charge basis ˆn n = n n. (38) Here n is again the number of excess Cooper-pairs on the island. Using the above commutation relation in combination with the Baker-Hausdorff lemma (see e.g. Sakurai text book) one can show that the exponential of the phase operator acts as a ladder operator in the charge basis e ±iπφa/φ0 n = n 1. (39) (It is a good exercise to check this.) Expressing the cosine as the sum of two exponentials, we then arrive at the quantized Hamiltonian of the Cooper-pair box, expressed in the charge basis H = n 4E C (n n g ) n n E J ( n n n + 1 n ). (40) 10

11 Operating the Cooper-pair box as a qubit implies limiting the range of the gate voltage so that only two charge states are reachable, e.g. limiting 0 < n g < 1 (see Fig. 10). We use the two charge states n = 0 and n = 1 for our charge qubit. (There are billions of Cooper-pairs on the island, so we choose n = 0 arbitrarily and n = 1 just implies one extra Cooper-pair in the box). In this two (charge) state basis the Hamiltonian reads H = 4E C (0 n g ) E C (1 n g ) E J ( ). (41) Using the Pauli matrices ( 0 1 σ x = 1 0 ), σ y = ( 0 i i 0 ) ( 1 0, and σ z = 0 1 ), (4) we can rewrite the Hamiltonian as H = 4E C [ n g n g + 1 ] 1 1 n g σ z E J σ x (43) where 1 is the unit (X) matrix. This term just determines a common reference of energy and is uninteresting in the present context, thus we neglect that term from here on. Note that any two-state Hamiltonian (X matrix) can be decomposed into a weighted sum of Pauli matrices and the unit matrix. Figure 10: The energy of a single Cooper-pair box (SCB), as a function of the number of Cooper pairs on the island n, and the induced gate charge n g. The dashed lines shows the case without Josephson coupling (E J = 0). 6 The transmission line Using the same method, we can derive the quantum mechanical hamiltonian of an one-dimensional transmission line with characteristic inductance/capacitance per unit length being L 0 /C 0. We consider a TL of length L and discretize it with M = N + 1 nodes numbered with integers from N to N with equal 11

12 Figure 11: The circuit of a discretized transmission line with characteristic inductance/capacitance per unit length being L 0 /C 0. (Adapted from Robert Johansson s thesis.) spacing x = L/N. From inspection of the discretized circuit in Fig. 11 we can write down the expression for the capacitive (kinetic) energy E C = xc 0 N Φ n, (44) as well as the potential (inductive) energy giving the Lagrangian L = E L = (Φ n Φ n 1 ) xl 0, (45) xc 0 Φ n (Φ n Φ n 1 ) xl 0, (46) where we used periodic boundary conditions Φ N 1 Φ N. The conjugate momentum to the node flux Φ n is q n = L Φ n = xc 0 Φn. (47) We get the hamiltonian from the Legendre transformation H = Φ n q n L = qn + (Φ n Φ n 1 ), (48) xc 0 xl 0 which is the hamiltonian of an infinite sum of coupled harmonic oscillators. 1

13 6.1 Diagonalizing the discrete hamiltonian We can diagonalize the discrete hamiltonian Eq. (48) by changing the coordinates from Φ n to the Fourier transformed ones Φ κ = 1 M N e iπ(κ/m)n Φ n Φ n = 1 M N κ= N e iπ(κ/m)n Φ κ, where for real Φ n we have Φ κ = Φ κ. This change of basis is unitary and we will use the orthogonality in the form of e iπ(κ/m)n = Mδ κ0, where δ mn is Kronecker s delta, which e.g. leads to Φ n = 1 M κ= N κ = N e iπ((κ+κ )n/m) Φκ Φκ = The conjugate momentum to the node flux Φ κ is For the potential energy we get = 1 M = (Φ n Φ n 1 ) = κ= N κ = N κ= N 1 e iπ(κ/m) Φκ κ= N Φ κ Φ κ = κ= N q κ = L Φ κ = xc 0 Φ κ. (49) Φ κ. ( 1 e iπ(κ/m)) ( 1 e iπ(κ /M) ) e iπ((κ+κ )n/m) Φ κ Φ κ = κ M ( ) πκ Φ κ, (50) M where the last approximation is OK if we include only modes with wavelengths (λ L/ κ ) much larger than the discretization length x, i.e. κ M. We now have a diagonalized Hamiltonian H = κ= N q κ + 1 e iπ(κ/m) Φ κ. (51) xc 0 xl 0 Now, in taking the continuum limit N we need to use not the charge q κ which goes to zero, but instead the charge density q κ = q κ / x, H x q κ ( ) πκ Φ κ +. C 0 M x L 0 κ M 13

14 Thus, the hamiltonian consists of a sum of independent harmonic oscillators characterized with their quantum number κ. In real space, the eigenstates are plane waves e ±ikx±iω kt with wave vector k = πκ M x = κπ L, and frequency ω k = v k, where v = 1/ L 0 C 0 is the wave velocity of the waveguide. Introducing creation and annihilation operators for these modes the hamiltonian can be written ( H = hω k a k a k + 1 ) dk. (5) 6. The massless Klein-Gordon field From the discrete hamiltonian Eq. (48) one may also derive equations of motion for the field, [ q n = q n, x (Φ n+1 Φ n ) (Φ n Φ n 1 ) ] L 0 ( x) q(x, t) = dx Φ(x, t) L 0 x (53) where q(x, t) = C 0 dx Φ(x, t). (54) The phase in the transmission lines thus obeys the massless scalar Klein-Gordon equation Φ(x, t) t 1 Φ(x, t) L 0 C 0 x = 0, (55) which has solutions in the form of right and leftgoing travelling waves which formally can be written as ( x ) ( Φ = Φ v + t + Φ x ) v + t, (56) where v = 1/ L 0 C 0 is the velocity of the waves in the transmission line. The Klein-Gordon equation of motion is easily solved by introducing the Fourier transformed operators in the transmission line Φ(x, t) = 1 π Φ(k, t)e ikx dk, (57) and we get an ordinary differential equation for each value of the wave-number k. Solving Eq. (55) is reduced to the problem of solving Φ(k, t) + v k Φ(k, t) = 0, (58) which is the classical equation of motion for a harmonic oscillator with the dispersion relation ω k = v k. In analogy with the ordinary harmonic oscillator it is convenient to work with the creation and annihilation operators a and 14

15 a instead of charges and phases. However, since we are dealing with a field equation each Fourier component is treated independently and we write h ( Φ(k, t) = a k (t) + a k ), C 0 ω (t) (59) k where a and a obey the canonical commutation relations [a k, a k ] = δ(k k ) and [a k, a k ] = 0, which give the statistics of the transmission line excitations (photons). The harmonic oscillator nature of the transmission line makes the phase of the creation and annihilation operators rotate with angular frequency ω k, a k (t) = a k e iωkt, a k (t) = a k eiωkt and we arrive at the final expression for the left and right going fields Φ and Φ in terms of a k and a k h dk Φ (x, t) = (a ) k e i(ωkt kx) + (a k ) e i(ω kt kx), (60) 4πC 0 ωk Φ (x, t) = h 4πC dk ωk (a k e i(ω kt+kx) + (a k ) e i(ω kt+kx) ), (61) where we used only positive k > 0 and indicate the direction with arrow indices. One may also use frequencies instead of wavenumbers Φ (x, t) = Φ (x, t) = hz0 4π hz0 4π 0 0 dω ) (a ω ω e i(ωt kωx) + H.C., (6) dω ) (a ω ω e i(ωt+kωx) + H.C., (63) where Z 0 = L 0 /C 0 is the characteristic impedance of the transmission line. 7 Transmission Lines Modeling Dissipation In a em closed quantum system the time-evolution is unitary. Different initial states thus maps to different final states. In a system with dissipation we may have that all initial states will evolve into the same final state, e.g. the ground state in the case of relaxation. To model dissipation we need an open quantum system, e.g. by including a semi-infinite transmission line. A qubit can then relax by emitting photons into the transmission line, which then propagate away from the qubit and never comes back. Different initial states will give different photon states in the transmission line, but they will a finally leave the qubit in the ground state. By tracing out the transmission line, ignoring the different photon states, one may then describe the dissipative dynamics of the qubit. One may note that in order for the photons never to come back, the transmission line needs to be semi-infinite, having an dense set of photon modes. 15

16 8 References and Reading suggestions Quantum Fluctuations in Electrical Circuits, M. H. Devoret, Les Houches Session LXIII, Quantum Fluctuations p (1995). ( fluctuations.pdf ) Superconducting Quantum Circuits, Qubits and Computing, G. Wendin, V.S. Shumeiko, Handbook of Theoretical and Computational Nanotechnology, ( arxiv:cond-mat/ ) The introduction to Robert Johansson s thesis. Quantum network theory, B. Yurke and J.S. Denker, Phys. Rev. A 9, (1984) An Introduction to Quantum Field Theory by Peskin and Schröder, Chapter (?) about the Klein-Gordon field 16