Phys. Rev. Lett. 101, 080502 (2008)
http://www.eng.yale.edu/rslab/index.htmlControlling the Spontaneous Emission of a Superconducting Transmon Qubit
We present a detailed characterization of coherence in seven transmon qubits in a circuit QED architecture. We find that spontaneous emission rates are strongly influenced by far off-resonant modes of the cavity and can be understood within a semiclassical circuit model. A careful analysis of the spontaneous qubit decay into a microwave transmission-line cavity can accurately predict the qubit lifetimes over 2 orders of magnitude in time and more than an octave in frequency. Coherence times T1 and T*2 of more than a µs are reproducibly demonstrated.
Coherence poses the most important challenge for the development of a superconducting quantum computer. As the dephasing time T *2 can never exceed twice the relaxation time T1, it is the relaxation time which ultimately sets the limit on qubit coherence. Although T *2 turned out to be small compared to T1 in the earliest superconducting qubits [1], steady progress over the last decade has significantly reduced this gap [2, 3, 4, 5, 6]. Recently, the transmon, a new type of qubit immune to
1/f charge noise, has been shown to be nearly homogeneously broadened (T *2 \=2T1) [6]. Therefore, under-standing relaxation mechanisms is becoming critical to further improvements in both T1 and T *2 . Progress in this direction will be based on the accurate modeling ofcontributions to T1 and the reliable fabrication of many qubits reaching consistent coherence limits.
One of the main advantages of superconducting qubits is their strong interaction with the wires of an electrical circuit, making their integration with fast control and readout possible and allowing for large, controllable couplings between widely separated qubits [7]. The large coupling also implies a strong interaction between the qubits and their electromagnetic environment, which can lead to a short T1. However, careful control of the coupling to the environment has been shown to allow prevention of circuit dissipation [8, 9]. Relaxation times have been studied in a wide variety of superconducting qubits, created with different fabrication techniques, and measured with a multitude of readout schemes. Typically, values of T1 vary strongly from sample to sample as they can depend on many factors including materials, fabrication, and the design of both readout and control circuitry. In some instances a separation of these compo-nents has been achieved [10, 11, 12, 13], but typically it is difficult to understand the limiting factors, and T1 often varies strongly even among nominally identical qubit samples.
Here, we demonstrate that in a circuit quantum elecodynamics (QED) architecture, where qubits are emedded in a microwave transmission line cavity[3, 14], transmon qubits have reproducible and understandable relaxation times. Due to the simple and well-controlled fabrication of the qubit and the surrounding circuitry, involving only two lithography layers and a single cavity forboth control and readout, we are able to reliably understand and predict qubit lifetimes. This understanding extends to a wide variety of different qubit and cavity parameters. We find excellent agreement between theory and experiment for seven qubits over two orders of magnitude in relaxation time and more than an
octave infrequency. The relaxation times are set by either spontaneous emission through the cavity, called the
Purcell effect [15], or a
shared intrinsic limit consistent with a lossy dielectric. Surprisingly, relaxation times are often limited by electromagnetic modes of the circuit which are far detuned from the qubit frequency. In the circuitQED implementation studied here, the infinite set of cavity harmonics reduces the Purcell protection of the qubitat frequencies above the cavity frequency.
Generally, any discrete-level system coupled to the continuum of modes of the electromagnetic field is
subject to radiative decay. By placing an atom in a cavity, the rate of emission can be strongly enhanced [15, 16] or suppressed [17, 18, 19], depending on whether the cavity modes are resonant or off-resonant with the emitter’s transition frequency. This effect is named after E. M. Purcell [15], who considered the effect of a resonant electrical circuit on the lifetime of nuclear spins. Suppression of spontaneous emission provides effective protection from radiative qubit decay in the dispersive regime, where qubit and cavity are detuned [14]. Specifically, the Purcell rate for dispersive decay is given by γ_κ = (g/∆)^2*κ, where g denotes the coupling between qubit and cavity mode, ∆ their mutual detuning, and κ the average photon loss rate.
The suppression and enhancement of decay rates canalternatively be calculated within a circuit model. For concreteness, we consider the case of a qubit
capacitively coupled to an arbitrary environment with impedance Z_0(ω), see Fig. 1(a). This circuit may be reduced to a qubit coupled to an effective dissipative element, see Fig. 1(b). Specifically, replacing the coupling capacitor C_g and the environment impedance Z_0 by an effective resistor R = 1/Re[Y (ω)], one finds[8, 9] that the T1 is given by RC, where C is the qubit capacitance. Choosing a purely resistive environment, Z_0 = 50Ω, yields adecay rate γ \= ω^2*Z_0*C^2_g /C. If instead we couple to a parallel LRC resonator, the calculated radiation rate can be reduced to that of the atomic case, γ_κ = (g/∆)^2*κ, thusreproducing the Purcell effect.