Coherent dynamics of a flux qubit coupled to a harmonic oscillator

Abstract

In the emerging field of quantum computation 1 and quantum information, superconducting devices are promising candidates for the implementation of solidstate quantum bits or qubits. Single-qubit operations 2−6 , direct coupling between two qubits 7−10 , and the realization of a quantum gate 11 have been reported. However, complex manipulation of entangled states − such as the coupling of a two-level system to a quantum harmonic oscillator, as demonstrated in ion/atom-trap experiments 12,13 or cavity quantum electrodynamics 14 − has yet to be achieved for superconducting devices. Here we demonstrate entanglement between a superconducting flux qubit (a two-level system) and a superconducting quantum interference device (SQUID). The latter provides the measurement system for detecting the quantum states; it is also an effective inductance that, in parallel with an external shunt capacitance, acts as a harmonic oscillator. We achieve generation and control of the entangled state by performing microwave spectroscopy and detecting the resultant Rabi oscillations of the coupled system.The device was realized by electron-beam lithography and metal evaporation. The qubit-SQUID geometry is shown in Fig. 1a: a large loop interrupted by two Josephson junctions (the SQUID) is merged with the smaller loop on the right-hand side comprising three in-line Josephson junctions (the flux qubit) 15 . By applying a perpendicular external magnetic field, the qubit is biased around Φ 0 /2, where Φ 0 = h/2e is the flux quantum. Previous spectroscopy 16 and coherent timedomain experiments 6 have shown that the flux qubit is a controllable two-level system with 'spin-up/spin-down' states corresponding to persistent currents flowing in 'clockwise/anticlockwise' directions and coupled by tunneling. Here we show that a stronger qubit−SQUID coupling allows us to investigate the coupled dynamics of a 'qubit−harmonic oscillator' system.The qubit Hamiltonian is defined by the charging and Josephson energy of the qubit outer junctions (E C = e 2 /2C and E J = hI C /4e where C and I C are their capacitance and critical current) 16 . In a two-level truncation, the Hamiltonian becomes H q /h = −ǫσ z /2−∆σ x /2 where σ z,x are the Pauli matrices in the spin-up/spin-down basis, ∆ is the tunnel splitting and ǫ ∼ = I p Φ 0 (γ q − π)/hπ (I p is the qubit maximum persistent current and γ q is the superconductor phase across the three junctions). The resulting energy level spacing represents the qubit Larmor frequency F L = √ ∆ 2 + ǫ 2 . The SQUID dynamics is characterized by the Josephson inductance of the junctions L J ≈ 80 pH, shunt capacitance C sh ≈ 12 pF (see Fig. 1a) and self-inductances L sl ≈ 170 pH of the SQUID and shunt-lines. In our experiments, the SQUID circuit behaves like a harmonic oscillator described by H sq = hν p (a † a + 1/2), where 2πν p = 1/ (L J + L sl )C sh is called the plasma frequency and a (a † ) is the plasmon annihilation (creation) operator. Henceforth |βn represents the state with the qubit in the ground(β = 0) or excited ...

Circuit quantum electrodynamics in the ultrastrong-coupling regime

“…This unique possibility to experimentally explore the foundations of quantum physics has greatly evolved with the advent of circuit QED [4][5][6][7][8][9][10][11][12][13] , where on-chip superconducting qubits and oscillators play the roles of two-level atoms and cavities, respectively. In the strong coupling limit, atom and cavity can exchange a photon frequently before coherence is lost.…”

Josephson Junction Microscope for Low-Frequency Fluctuators

“…In particular, we show that spectroscopic transmission measurements of the junction resonator mode can reveal how the coupling magnitude between the junction and the TLSs varies with an external magnetic field applied in the plane of the tunnel barrier. The proposed experiments offer the possibility of clearly resolving the underlying coupling mechanism for these spurious TLSs, an important decoherence source limiting the quality of superconducting quantum devices.Superconducting quantum circuits have been intensively tested in various regimes in the past few years, from superconducting qubits demonstrating long coherence times, to superconducting transmission line cavities coherently coupled to a Single Cooper Pair box [1,2,3,4,5,6]. Such circuits are extremely sensitive to very small quanta and defect states, and hence have the ability to detect individual microwave photons, charged quasiparticles, as well as spurious TLSs within or near Josephson junction tunnel barriers [7,8,9,10,11].…”

Quantum downconversion and multipartite entanglement via a mesoscopic superconducting quantum interference device ring

“…Although this aspect of our investigations is an extension of the results of our previous papers, 12,14 it is important to show that this dependence remains valid for the kind of multicomponent circuits that would appear to be required in the quantum technologies currently being pursued. 1,2, [5][6][7][8][9][10][11]24 The Hamiltonian for the coupled system of Fig. 1 is made up of the uncoupled Hamiltonians for each mode together with a set of interaction terms.…”

“…[5][6][7][8][9][10] Furthermore, other recent experiments have been performed that show coherent dynamics of superconducting flux qubits coupled to individual harmonic oscillators. 11 Building on this background, and previous theoretical work by us on the coupling of SQUID rings to electromagnetic ͑em͒ fields, 12 we consider the process of photon downconversion via a mesoscopic SQUID ring. Specifically, by analogy with the field of quantum optics, we demonstrate that the ring may be viewed as a nonlinear medium which can downconvert photons from one frequency ͑input field oscillator mode͒ to another lower frequency and generate entangled photons ͑output field oscillator modes͒, as is often performed in quantum optics experiments.…”

“…This device was then coupled, quantum coherently, to a superconducting flux qubit. 11 However, although the physics of all these systems is similar, in this work we have chosen to consider a superconducting circuit coupled to quantum electrodynamic degrees of freedom, i.e., photons in a cavity. Recently it has been shown that a superconducting qubit can be quantum coherently coupled to an on-chip cavity.…”