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Solid-State Quantum Communication With Josephson Arrays NEST-INFM & Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy Department of Physics and Astronomy, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland Josephson junction arrays can be used as quantum channels to transfer quantum information between distant sites. In this work we discuss simple protocols to realize state transfer with highfidelity. The channels do not require complicate gating but use the natural dynamics of a properlydesigned array. We investigate the influence of static disorder both in the Josephson energies andin the coupling to the background gate charges, as well as the effect of dynamical noise. We alsoanalyze the readout process, and its backaction on the state transfer.
The transmission of a quantum state through a channel between distant parties is an important issue in quantumcommunication. In optical systems photons can be trans- ferred coherently over large distances [1]. However, it is also highly desirable to have similar protocols for quan- tum information transfer in solid-state environments. A possible solution would be to interface solid-state quan- tum hardware to optical systems [2]. Another possibility is to use flying qubits, i.e. to transfer the physical qubitsalong leads [3]. Inspired by the paper of Bose [4] the idea FIG. 1: Dashed box: one-dimensional Josephson array pro- of our work is to construct a genuine quantum transmis- posed for the transmission of quantum states. The crossed sion line using a Josephson junction array.
rectangles denote the Josephson junctions between the is-lands. The state prepared on the left-most island is transfered Recently, a spin chain with ferromagnetic Heisenberg to the right-most island by the time evolution generated by interactions has been proposed for quantum communica- the Hamiltonian. Left part: Cooper-pair box (charge qubit)used to prepare the state. Right part: SET transistor used as tion [4]. It was shown that Heisenberg chains can be used to transfer unknown quantum states over appreciable dis-tances (∼ 102 lattice sites) with high fidelity [4–7]. Bypreparing the state to be transferred at one end of the chain and waiting for a well-defined time interval, onecan reconstruct the state at the other end of the chain.
Even perfect transfer could be achieved over arbitrary distances in spin chains [8]. Quantum state transport through harmonic chains was considered in Ref. 9.
is the Hamiltonian of a one-dimensional Josephson junc- Josephson qubits are among the most promising candi- tion array [12] of length L, and φi,i+1 = φi − φi+1. The dates as building blocks of quantum information proces- other terms of the Hamiltonian describe the measurement sors [10, 11]. In this Letter, we extend their application apparatus and will be discussed later. The charge Qi range to quantum communication and show that a one- and phase φi are canonically conjugated. The first term dimensional Josephson array is a natural transmission in Eq. (1) is the charging energy, Cij is the capacitance line for systems with superconducting charge qubits. We matrix; the second is due to Josephson tunneling. An ex- calculate the transmission fidelity and investigate the ef- ternal gate voltage Vxi gives a contribution to the energy fect of static inhomogeneities and dynamical noise. We via the induced charges Qxi = 2eqxi = VxiCii. This ex- also analyze the readout process by a single-electron tran- ternal voltage can be either applied to the ground plane sistor (SET) at the end of the array. To our knowledge, or unintentionally caused by trapped charges in the sub- this is the first realizable and concrete implementation of strate (in this case Qxi will be a random variable). We a solid-state quantum communication protocol following assume that each island is coupled to its nearest neigh- bors by junction capacitances C and to the ground by acapacitance C0. In this case, the charging interaction has The model that we want to study is schematically il- lustrated in Fig. 1 and described by the Hamiltonian of the array [12]. In the following we put larger arrays a higher fidelity can be achieved (although approximately described by only two charge states for at larger times), see Fig. 3. The results of Figs. 2, 3 are each island. The chain Hamiltonian HJJ is equivalent to encouraging since they indicate that faithful state trans- an anisotropic XXZ spin-1/2 Heisenberg model [13, 14], mission using Josephson chains is already possible with the Josephson chain is thus different from the XY and anisotropy between the z-direction and the xy-plane.
Since experimental arrays are never completely ho- Moreover the z-coupling has a range which depends on mogeneous, we now consider the case in which a small the electrostatic energy and can extend over several lat- amount of static disorder is present. In general, imperfec- tions will reduce the fidelity. In Fig. 4 we show both the At t = 0, the chain is initialized in the state |ψ0 = effect of bond disorder (Josephson couplings distributed |ψ, 000.0 , where |0 (|2 ) denotes the state of an is- around an average value) and site disorder (mimicking land without (with) an excess Cooper pair, and |ψ = the effect of static background charges and/or different cos (θ/2)|0 + eiφ sin (θ/2)|2 is the state that has been capacitances) and compare to the case without disorder.
prepared in the left-most island. This initial state is not The effect of charge disorder appears to be more dis- an eigenstate of the Hamiltonian, it will evolve as a func- ruptive: this is because additional frequencies enter the tion of time. In fact, as the total charge Q = dynamical evolution making the reconstruction of the ad- a conserved quantity, the dynamics is restricted to the ditional wave-function more difficult. The dotted line in L + 1-dimensional space H = H0 ⊕ H2 of total charge Fig. 4 shows the variance of the fidelity for the case with- zero, H0, and charge two, H2 = span{|j }, where |j , out disorder. Around the maxima of the fidelity, the vari- 1 ≤ j ≤ L is the state with an excess Cooper pair on the ance is small, i.e., the transmission quality for any given j-th site. In this basis the Hamiltonian reads Dynamical fluctuations play a different role.
arise from gate-voltage fluctuations and are described by adding stochastic terms to the gate voltages, qxi → jL)|j + 1 + (1 − δj1)|j − 1 ) .
qxi + ξi(t) in the Hamiltonian in Eq. (1). Here we choosea very simple model and assume the ξi(t) to be indepen- We first calculate the fidelity of transmission and the dently gaussian distributed: ξi(t) = 0, ξi(t)ξj (t ) = time required for the transfer of information as a function γδijδ(t − t ). Nevertheless, due to capacitive coupling of the coupling constants of the Josephson chain. The between separated sites, such stochastic factors result in quality of the transmission is quantified by the fidelity of correlated stochastic terms in the effective Hamiltonian the (mixed) state ρL of the right-most island (site L) to Eq. (2), Hnoise|j = HJJ|j − 2Ξj(t)|j , where the zero- averaging gaussian functions Ξi(t), are uniquely fixed by Ξi(t)Ξj (t ) = γ[(C−1)2]ij δ(t − t ). Averaging out the stochastic terms leads to the master equation for the den- This definition gives the fidelity averaged over all possibleinitial states on the Bloch sphere, 1/2 ≤ FL ≤ 1.
The fidelity is a strongly oscillating function of time.
Only at well-defined times the state is transferred faith- fully through the chain. This does not necessarily corre- spond to the time in which a Cooper pair has been trans- ferred, since also the relative phases of the state have to be reconstructed. In Fig. 2 we show the value of the first fidelity maximum and the time at which it is reached as a function of the length L of the array and for different values of the ratio C/C0. For the parameters considered,the fidelity is never smaller than 75%. For longer chains, of the fidelity is considerably reduced. Another option is to fix a threshold for the fidelity of transmission and seekfor the first local maximum above the threshold. The FIG. 2: Maximum value of the fidelity as a function of the time at which these maxima occur increases exponen- length of the chain for two different values of C/C0 tially with the chain length. The value of the fidelity (2e)2/(EJ C0) = 10. Inset: time at which the maximum is does not necessarily decrease on increasing L, and for where the operators Qi projected on the space H are a completely incoherent mixture in which any chargedstate is equally probable, ρ The average fidelity as defined in Eq. (3) is reduced toF∞ = 1/2 + 1/(6L), corresponding to an almost unfaith- FIG. 4: Fidelity as a function of time for an array of length ful transmission. The time dependence of the fidelity in L = 7, (2e)2/(EJ C0) = 10 and C = 0, i.e. a junction capac- the noisy system is presented in Fig. 5 where it is com- itance much smaller than the ground capacitance. Disorder pared with the fidelity in the absence of noise. The peaks parameters: relative variance ∆EJ /EJ = 0.1 for bond dis-order, absolute variance ∆Q of the fidelity are not smeared out by noise. The dom- Dotted line (right axis): variance of the fidelity for the case inant effect of the coupling to the environment is the relaxation of the fidelity amplitude towards the station-ary value (independent on the initial state). Numerically,such relaxation takes place on a characteristic time scale where γ, γ† are annihilation and creation operators of ∼ 1/(Lγ). Thus, to observe high values of the fidelity it quasiparticles in the grain L and in the leads, u and d (see is important to have a maximum at a short time. As a quasiparticles tunneling into the grain with an associated EJ /L2 is required to have a high value for the first charge increasing e−i(φL−ϕb)/2. Qb is the total charge entering the chain from the up or down reservoirs. We Finally we discuss how the fidelity can be measured assume non-vanishing quasiparticle tunneling only across in a practical setup. To do this, we assume that the the upper junction (eVd = 0, eVu = −eV ≈ −2∆) and right-most island (site L) is part of a SET transistor.
conversely we allow coherent Cooper pairs tunneling only We therefore specify the effective coupling Hamiltonian between the right-most island and the leads [15], voltages are chosen so that the SET is off resonance andwe can therefore neglect the stationary current through the SET due to the Cooper-pair quasiparticle cycle [15].
The measurement device modifies the dynamics of Cooper pairs on the chain and requires taking into ac- count quasiparticle excitations on the L-th site of thechain. By neglecting quasiparticle tunneling we would have a coherent dynamics for the charges in the chain described by the Hamiltonian H0 = H(Tqk → 0). Trac-ing out the quasiparticle degrees of freedom results, in- FIG. 3: Maximum value of the fidelity as a function of thelength of the chain for three different values of C/C FIG. 5: Fidelity versus time in presence of gate voltage fluc- e2/(EJ C0) = 10, C/C0 = 0.1, γ = 0.01.
As at time t > t the SET is disconnected from the rest of the chain and is out of resonance, the measured current does not provide a tomography for the state of the right- most site: the measured current does not depend on thecoherences of ρL(t ), to which the fidelity is sensitive.
Nevertheless, the peaks in the current correspond exactly to the maxima of the fidelity as shown in Fig. 6. The cur-rent decay in time due to quasiparticle tunneling happens FIG. 6: Full line: time dependence of the current (in units ofe/T ) through the SET. Dashed line: fidelity of the isolated on a time scale ∼ 1/Γ irrespective of the length of the chain. Γ = 0.05, all other parameters as in Fig. 5.
chain. Therefore, our measurement scheme can be usedalso for long chains, the main constraint are disorder andgate voltage fluctuations. In this sense the current mea- stead, in an incoherent dynamics described by a mas- surement allows to check the theoretical prediction for ter equation for the reduced density matrix ˜ In conclusion, we have proposed to use a Josephson H0|α = Eα|α [17], the master equation reads [16] junction chain as a solid-state quantum communicationchannel. We have analyzed the state preparation, its ˙˜ραβ(t) = −i α| [H0, ˜ρ] |β − propagation (with a model appropriate for Josephson nanocircuits), the role of the measuring apparatus andthe effect of noise and imperfections. This, we believe, is The prime indicates that the sum has to be performed an important and necessary step towards the experimen- over states with energies such that |Eα−Eβ −Eµ+Eν| tal realization of quantum communication in solid-state 1/∆t, ∆t being the time over which the coarse-graining systems. Present-day technology should allow the real- ization of quantum channels of the type described here.
As we are interested in the time evolution over short We acknowledge fruitful discussions with G. De times, let us discuss in some detail our approxima- Chiara, C. Macchiavello, G. M. Palma, and S. Mon- evaluating the kernel R, we neglect the Josephson cou- SQUBIT2, RTN-Nanoscale Dynamics), by Fondazione pling to the leads [18]. In this case the spectrum of Silvio Tronchetti Provera, and by the Swiss NSF and the NCCR Nanoscience. During completion of this work we became aware of Ref. 19 which discusses state transmis- sion in a setup using SQUID loops coupled to resonators.
mixes population and coherences of the density ma-trix only in the subspace of ¯ the coarse-grained dynamics of Eq. (7) can resolve thetime scales of order 1/EJ that we are interested in.
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