Phys. Rev. Lett. 101, 080502 (2008)
http://www.eng.yale.edu/rslab/index.html
Controlling 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.
Two Number Theory Items (and Woody Allen)
4 hours ago
11 comments:
The qualitative features of the Purcell effect are apparent in measurements of T1, shown for 3 qubits in Fig.2, measured with a dispersive readout by varying a delay time between qubit excitation and measurement[6, 20].
Near the cavity resonance at 5.2GHz, spontaneous emission is Purcell-enhanced and T1 is short. Away from resonance, the cavity protects the qubit from decay and the relaxation time is substantially longer than expected for decay into a continuum. However, at detunings above the
cavity frequency, the measured T1 deviates significantly from the single-mode Purcell prediction. This deviation can be directly attributed to the breakdown of the single-mode approximation.
The cavity does not just support a single electromagnetic mode, but also all higher harmonics of the fundamental mode. This has a striking impact on relaxation
times. At first glance, it would appear that the effects of higher modes could be ignored when the qubit is close to the fundamental frequency and detuned from all higher modes. However, the coupling g_n to the n^th mode of
the cavity increases with mode number, g_n = g_0*sqr(n + 1).
In addition, the input and output capacitors act as frequency-dependent mirrors, so that the decay rate of the n^th harmonic, κ_n = (n + 1)^2*κ, is larger than that of the fundamental. As a result, higher modes significantly
contribute to the qubit decay rate, and the simple single-mode quantum model turns out to be inadequate for understanding the T1 of the system. The naive attempt
to treat the fundamental and harmonics in terms of a multi-mode Jaynes-Cummings Hamiltonian faces problems with divergences. Work on developing a consistent quantum model is currently under way [21]
Here, we follow the alternative route of calculating T1 semiclassically, on the basis of the full underlying circuit,and show that this accurately reproduces the measured T1. The relationship between the classical admittance Y(ω) of a circuit and its dissipation has long been known
[8, 9], providing a practical means of understanding relaxation rates [12]. The full calculation includes a transmission line cavity rather than a simple LRC resonator, see Fig. 1(c). The results from this are shown in Fig. 2, and reveal two striking differences as compared to the single-mode model: First, there is a strong asymmetry between relaxation times for qubit frequencies above (positive detuning) and below (negative detuning) the fundamental
cavity frequency. While the single-mode model predicts identical relaxation times for corresponding positive and negative detunings, T1can be two orders of magnitude
shorter for positive detunings than for negative detunings in the circuit model. Second, the circuit model shows a surprising dependence of T1 on the qubit position in the cavity. While qubits located at opposite ends of the cavity have the same T1 within the single-mode model, the circuit model correctly captures the asymmetry induced by the differing input and output coupling capacitors and leads to vastly different T1. The circuit model accurately resolves the discrepancy between the experimental data and the single-mode model, see Fig. 2.
The predictive power of the circuit model extends to all of our transmon qubits. Here, we present T1 measurements on a representative selection of seven qubits. The qubits were fabricated on both oxidized high-resistivity
silicon and sapphire substrates, and coupled to microwave cavities with various decay rates and resonant frequencies. Table 1 provides parameters for each of
the seven qubits. Qubits are fabricated via electron beam lithography and a double-angle evaporation process(25nm and 80nm layers of aluminum), while cavities are fabricated by optical lithography with either lifted-off Al or dry-etched Nb on a Si or sapphire substrate [22].
Predictions from the circuit model are in excellent agreement with observed qubit lifetimes (see Fig. 3), up to a Q = 70, 000 for qubits on sapphire. The agreement is valid over more than two orders of magnitude in qubit lifetime and more than an octave of frequency variation. We emphasize that the circuit model does not correspond
to a fit to the data, but rather constitutes a prediction based on the independently measured cavity parameters ω_r and κ, and the coupling g.
In the qubits on silicon,coherence times of no more than 100 ns are observed above the cavity resonance,far below predictions from the single-mode model, but
consistent with the circuit model. Initially, this caused concern for the transmon qubit: it appeared as if the transmon solved the 1/f-noise dephasing problem for charge qubits, but introduced a new relaxation problem[23, 24, 25]. However, with the circuit model of relaxation, it is now clear that the 100 ns limit originated from
the surprisingly large spontaneous emission rate due to higher cavity modes. By working at negative detunings instead, it is possible to achieve long relaxation times,
here observed up to 4µs.
All qubits on sapphire substrates reach a shared intrinsic limit of Q = 70, 000 when not otherwise Purcell limited. The constant-Q frequency dependence of the intrinsic limit (T1 ∝ 1/ω) is suggestive of dielectric loss
as the likely culprit. While the observed loss tangent tan δ \\10−5 is worse than can be achieved for sapphire,it is not unreasonable depending on the type and density
of surface dopants that might be present[26, 27]. The overall reproducibility of the intrinsic limit gives hope that future experiments may isolate its cause and reveal a solution. It is instructive to reexpress the relaxation times in terms of a parasitic resistance, see Fig. 3. Note that here a T1 of a microsecond roughly corresponds to a resistance of 20MΩ. To build more complex circuits with still longer T1, all dissipation due to parasitic couplings must be at the GΩ level.
Transmon qubits benefit greatly from the increased relaxation times, as they are insensitive to 1/f-charge noise, the primary source of dephasing in other charge
qubits. As a result, coherence is limited primarily by energy relaxation and transmons are nearly homogeneously
broadened (T*_2 \= 2T1). Improvements in T1 thus translate directly into improvements in dephasing times T*_2. This is demonstrated in Fig. 4, showing a comparison of relaxation and dephasing times. Here, T*_2 is measured in a pulsed Ramsey experiment and without echo [6]. The gain in coherence time is most striking in samples with a higher-frequency cavity, ω_r/(2π)\\ 7GHz, where it is easier to operate at negative detunings and attain long
T1. In all these samples, we observe consistently long dephasing times of nearly a microsecond, with the largest
T*_2 exceeding two microseconds without echo.
There are two main effects determining the observed dependence of T*_2 on the qubit frequency. First, away from the maximum frequency for each qubit, i.e. the flux sweet spot[2], the sensitivity to flux noise increases. This can cause additional inhomogeneous broadening. Despite this, T*_2 remains close to two microseconds, even away from the flux sweet spot. Second, tuning the qubit frequency via EJ directly affects the ratio of Josephson to charging energy, EJ/EC, which dictates the sensitivity to charge noise. At low qubit frequencies, the qubits regain the charge sensitivity of the Cooper Pair Box, thus ex-
plaining the strong drop in dephasing times seen in Fig.4.
Future improvements in T*_2 require further improvements in T1. The accurate modeling of relaxation processes will be essential as quantum circuits become more complicated. In particular, the addition of multiple cavities and individual control lines may introduce accidental electromagnetic resonances. As we have shown here, even far off-resonant modes of a circuit can have a dramatic impact on qubit lifetimes. However, with careful circuit design, it should be possible not only to avoid additional accidental resonances, but to utilize the circuit model of relaxation to build filters to minimize dissipation. The concise understanding of spontaneous emission lifetimes in our system, and the reproducibility of intrin-
sic lifetimes open up vistas for a systematic exploration of limits on coherence.
This work was supported in part by Yale University via a Quantum Information and Mesoscopic Physics Fellowship (AAH, JK), by CNR-Istituto di Cibernetica (LF), by LPS/NSA under ARO Contract No. W911NF-05-1-0365, and the NSF under Grants Nos. DMR-0653377 and DMR-0603369.
Post a Comment