ac Stark Shift and Dephasing of a Superconducting Qubit Strongly Coupled to a Cavity Field

Abstract

We have spectroscopically measured the energy level separation of a superconducting charge qubit coupled non-resonantly to a single mode of the electromagnetic field of a superconducting on-chip resonator. The strong coupling leads to large shifts in the energy levels of both the qubit and the resonator in this circuit quantum electrodynamics system. The dispersive shift of the resonator frequency is used to non-destructively determine the qubit state and to map out the dependence of its energy levels on the bias parameters. The measurement induces an ac-Stark shift of 0.6 MHz per photon in the qubit level separation. Fluctuations in the photon number (shot noise) induce level fluctuations in the qubit leading to dephasing which is the characteristic back-action of the measurement. A cross-over from lorentzian to gaussian line shape with increasing measurement power is observed and theoretically explained. For weak measurement a long intrinsic dephasing time of T2 > 200 ns of the qubit is found.We have recently demonstrated that a superconducting quantum two-level system can be strongly coupled to a single microwave photon [1]. The strong coupling between a quantum solid state circuit and an individual photon, analogous to atomic cavity quantum electrodynamics (CQED) [2], has previously been envisaged by many authors, see Ref. 3 and references therein. Our circuit quantum electrodynamics architecture [3], in which a superconducting charge qubit, the Cooper pair box [4], is coupled strongly to a coplanar transmission line resonator, has great prospects both for performing quantum optics experiments [5] in solids and for realizing elements for quantum information processing [6] with superconducting circuits [7].In this letter we present spectroscopic measurements which demonstrate the non-resonant (dispersive) strong coupling between a Cooper pair box and a coherent microwave field in a high quality cavity. The quantum state of the Cooper pair box is controlled using resonant microwave radiation and is read out with a dispersive quantum non-demolition (QND) measurement [3,8,9]. The interaction between the Cooper pair box and the measurement field containing n photons on average gives rise to a large ac-Stark shift of the qubit energy levels, analogous to the one observed in CQED [10]. As a consequence of the strong coupling, quantum fluctuations in n induce a broadening of the transition line width, characterizing the back action of the measurement on the qubit.In our circuit QED architecture [3], see Fig. 1a, a split Cooper pair box [4], modelled by the two-level hamiltonian H a = −1/2 (E el σ x + E J σ z ) [11], is coupled capacitively to the electromagnetic field of a full wave (l = λ) transmission line resonator, described by a harmonic oscillator hamiltonian H r =hω r (a † a + 1/2). In the Cooper pair box, the energy difference E a =hω a = E 2 el + E 2 J between the ground state |↓ and the first excited state |↑ , see Fig. 1b, is determined by its electrostatic energy E el = 4E C (1 − n g ) and its Josephson ...

Strong-Field Effects in the Rabi Oscillations of the Superconducting Phase Qubit

“…A number of related experiments and calculations have appeared in the literature. Multiphoton Rabi oscillations of a superconducting qubit were first performed in a charge qubit device [8], while the ac Stark shift was recently measured in a qubitcavity system [9]. High-power multiphoton spectroscopy of a three-junction flux qubit has been accomplished (with up to 20 photons) [10].…”

Quantum systems under frequency modulation

“…In addition, the two Rabi doublets are asymmetric due to the interference between the doubly dressed states. [240] Quantum fluctuations in the cavity occupation n lead to photon shot noise [244][245][246]. As a consequence, the qubit linewidth is broadened and, thus, represents the cavity backaction on the qubit.…”

“…one obtains Lorentzian qubit resonance which is broadened as Γ 2 → Γ 2 + 2θ 2 0n κ, where [244,245]. In the above, we assume that the mean number of cavity quantan is small.…”

“…As a consequence, the qubit linewidth is broadened and, thus, represents the cavity backaction on the qubit. In references [244][245][246], the transmon setup with photon shot noise was considered in the dispersive strong coupling regime where κ < g |ω 0 − ω c |. Due to the shot noise δn(t), the qubit gains a relative phase…”

Characterization of hidden modes in networks of superconducting qubits

“…Quantum fluctuations in the number of photons in a resonator coupled to the qubit can create random phase differences between the |0 and |1 state of the qubit, which leads to dephasing. Measurement-induced-dephasing is a well known phenomenon [12][13][14][15][16] that has been characterized in experimental systems using tunable couplings between the qubit and its environment 17 and has served as a mechanism for measuring thermal noise [18][19][20] . Various efforts have been made to avoid populating resonators in qubit gates that depend on the qubit-resonator coupling 21,22 , to reduce qubit dephasing through measurement feedback [23][24][25] or tunable coupling to the readout resonator 26 .…”