Cotunneling in pairs of coupled flux qubits

Abstract

We report measurements of macroscopic resonant tunneling between the two lowest energy states of a pair of magnetically coupled rf-SQUID flux qubits. This technique provides a direct means of observing two-qubit dynamics and a probe of the environment coupled to the pair of qubits. Measurements of the tunneling rate as a function of qubit flux bias show a Gaussian line shape that is well matched to theoretical predictions. Moreover, the peak widths indicate that each qubit is coupled to a local environment whose fluctuations are uncorrelated with that of the other qubit.Superconducting circuits have played an essential role in realizing quantum mechanical phenomena in macroscopic systems. One such example is the observation of macroscopic resonant tunneling (MRT) of magnetic flux between the lowest energy states of single rf-SQUID flux qubits, as demonstrated by several groups [1][2][3][4]. These measurements provide both a clear signature of quantum mechanics in a macroscopic circuit at a finite temperature and in the presence of noise and a direct means of determining the tunneling energy between states. Theoretical descriptions of the MRT rate have been presented [5,6] and indicate a direct connection between the profile of the MRT rate peaks and properties of the environment. Analogous measurements of the tunneling of magnetization in nanomagnets [7,8] suggest that MRT is responsible for dynamics in these materials as well.In this work, we extend measurements of MRT to inductively coupled pairs of flux qubits. We present experimental observations of tunneling between the two lowest energy states of the coupled system for several coupling strengths. These data yield two-qubit energy gaps that match those predicted by the independently calibrated Hamiltonian of the coupled system. Moreover, measurements of the two-qubit energy gap are used to infer single qubit energy gaps at ∼ h × 10 9 Hz without the use of microwave lines. Finally, the profile of the MRT rate versus qubit flux bias has a Gaussian lineshape with a width that is a factor of √ 2 larger than that of a single qubit. We argue that this observation indicates that the environments coupled to each qubit are uncorrelated.For a single flux qubit, an MRT experiment consists of measuring the rate of tunneling of flux between two wells of the double-well potential of the rf SQUID when the lowest energy levels of each well are closely aligned. Restricting the dynamics of the single rf SQUID to its two lowest energy states allows one to map the physics of this device onto the canonical qubit Hamiltonian:where σ x,z are Pauli matrices, ǫ ≡ 2I p (Φ (1) is valid when |ǫ|, ∆ ≪hω p , wherehω p is the energy spacing to the next excited state of the rf SQUID. For a non-Markovian environment [9], the initial tunneling rate from |0 to |1 (eigenstates of σ z ) versus ǫ has a Gaussian profile, as given by Eq. (2) in Ref. [2].A natural extension to the single qubit MRT experiment is to add a second qubit that is inductively coupled to a first qubit via a mutual induct...

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“…This assumption is justified experimentally [27] and also theoretically [3]. The dissipative functions f H and g H are calculated in appendix C. The total reorganization energy, , e e º a is defined by (38), so that .…”

“…L H e e e = + Here ε L is defined in (7), and H e is the HF component of the reorganization energy (38). For the Ohmic spectrum (44) of the bath, we have .…”

“…Here the matrix elements m s a and n s a are taken at time t, whereas the elements m s a ¢ and n s a ¢ depend on the time t¢. The total reorganization energy ε α is defined by equation (38).…”

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