WAVEGUIDE-MEDIATED COOPERATIVE INTERACTIONS OF SUPERCONDUCTING QUBITS
4.2 Strong-coupling regime of waveguide QED: cavity QED with atom-like mirrorsmirrors
A central technical hurdle common to these research avenues—reaching the so- calledstrong coupling regime, in which atom-atom interactions dominate decay—is experimentally difficult, especially in waveguide QED, because while the waveguide facilitates infinite-range interactions between the atoms [196, 197], it also provides a dissipative channel [198]. Decoherence through this and other sources destroys the fragile many-body states of the system, which has limited the experimental state- of-the-art to spectroscopic probes of waveguide-mediated interactions [142, 147, 199]. However, by utilizing collective dark states, where the precise positioning of atoms protects them from substantial waveguide-emission decoherence, the strong coupling limit is predicted to be within reach [120]. Additionally, if the timescale of single-atom emission into the waveguide is long enough to permit measurement and manipulation of the system, the coherent dynamics can be driven and probed at the single-atom level. Here, we overcome these hurdles with a waveguide QED system consisting of transmon qubits coupled to a common microwave waveguide, strengthening opportunities for a range of waveguide QED physics.
As a demonstration of this tool, we construct such an emergent cavity QED system and probe its linear and non-linear dynamics. The system features an ancillary probe qubit and a cavity-like mode formed by the dark state of two single-qubit mirrors. Using waveguide transmission and individual addressing of the probe qubit,
a
· · ·
λ/4 Γ1D
λ/4
· · ·
Γ′ Γ′
Γ1D
Γ′p
Γ1D,p
m=1 m=-1
J
|e〉p|G〉
|g〉p|G〉
|g〉p|D〉 |g〉p|B〉 Γ1D,p 2Γ1D
J
Γ′p Γ′D Γ′B
g = J γ = Γ1D,p+ Γ′p
κ = Γ′D
b
Z3 Z5
Z6Z7
Z4 XY4
Input Output
1 mm
Q3 Q5 Q6 Q4
Q2
Q1 Q7
Z1
R4
Z2
200 μm Q1
Q3 Q2
Type I Type II Mirror qubits 100 μm
Q4 R4
Probe qubit
Figure 4.1: Waveguide-QED setup. a, The top schematic shows the cavity configuration of waveguide QED system consisting of an array of𝑁 mirror qubits (𝑁 =2 shown; green) coupled to the waveguide with an inter-qubit separation of𝜆0/2, with a probe qubit (red) at the center of the mirror array. The middle schematic shows the analogous cavity QED system with correspondence to waveguide parameters. The bottom panel shows the energy- level diagram of the system of three qubits (two mirror, one probe). The mirror dark state
|𝐷iis coupled to the excited state of the probe qubit|𝑒ip at a cooperatively enhanced rate of 2𝐽 = p
2Γ1DΓ1D,p. The bright state |𝐵iis decoupled from the probe qubit. b, Optical image of the fabricated waveguide QED circuit. Tunable transmon qubits interact via microwave photons in a superconducting CPW (false-color orange trace). The CPW is used for externally exciting the system and is terminated in a 50-Ωload. The insets show scanning electron microscope images of the different qubit designs used in our experiment. The probe qubit, designed to haveΓ1D,p/2𝜋=1 MHz, is accessible via a separate CPW (XY4; false- color blue trace) for state preparation, and is also coupled to a compact microwave resonator (R4; false-color cyan) for dispersive readout. The mirror qubits come in two types: type I, withΓ1D/2𝜋=20 MHz and type II, withΓ1D/2𝜋=100 MHz. The figure is adapted from Ref. [108].
we observe spectroscopic and time-domain signatures of the collective dynamics of the qubit array, including vacuum Rabi oscillations between the probe qubit and the cavity-like mode. These oscillations provide direct evidence of strong coupling between these modes as well as a natural method of efficiently creating and measuring dark states that are inaccessible through the waveguide. Unlike traditional cavity QED, our cavity-like mode is itself quantum nonlinear, as we show by characterizing the two-excitation dynamics of the array.
The collective evolution of an array of resonant qubits coupled to a 1D waveguide can be formally described by a master equation [116, 120] of the form
¤ˆ 𝜌=−𝑖
ℏ[𝐻ˆeff,𝜌ˆ] +Õ
𝑚,𝑛
Γ𝑚,𝑛𝜎ˆ𝑚
ge𝜌ˆ𝜎ˆ𝑛
eg, (4.1)
where ˆ𝜎𝑚
ge = |𝑔𝑚ih𝑒𝑚|, |𝑔i and h𝑒|are a qubit’s ground and excited states, respec- tively, and𝑚and𝑛represent the indices of the qubit array. Within the Born-Markov approximation, the effective Hamiltonian can be written in the interaction picture as
ˆ
𝐻eff= ℏÕ
𝑚,𝑛
𝐽𝑚,𝑛−𝑖 Γ𝑚,𝑛
2
ˆ 𝜎𝑚
eg𝜎ˆ𝑛
ge, (4.2)
whereℏ = ℎ/2𝜋 is the reduced Planck constant. Figure 4.1a depicts the waveguide QED system considered in this work. The system consists of an array of 𝑁 qubits separated by distance𝑑 =𝜆0/2 and a separate probe qubit centered in the middle of the array with one-dimensional waveguide decay rate Γ1D,p, and where 𝜆0 = 𝑐/𝑓0 is the wavelength of the field in the waveguide at the transition frequency of the qubits 𝑓0. In this configuration, the effective Hamiltonian can be simplified in the single-excitation manifold to
ˆ
𝐻eff=−𝑖 𝑁ℏΓ1D 2
𝑆ˆ†
B𝑆ˆB−
𝑖ℏΓ1D,p
2 𝜎ˆee(p)+ℏ𝐽
ˆ 𝜎ge(p)𝑆ˆ†
D+H.c.
, (4.3)
where ˆ𝑆B,𝑆ˆD = 1/√ 𝑁Í
𝑚 >0(𝜎ˆ𝑚
ge∓ 𝜎ˆ−𝑚
ge ) (−1)𝑚 are the lowering operators of the bright collective state |𝐵i and the fully-symmetric dark collective state |𝐷i of the qubit array, as shown in Fig. 4.1a, and where 𝑚 > 0 and 𝑚 < 0 denote qubits to the right and left of the probe qubit, respectively. As shown by the last term in the Hamiltonian, the probe qubit is coupled to this dark state at a cooperatively enhanced rate 2𝐽 = √
𝑁p
Γ1DΓ1D,p. (H.c. is the Hermitian conjugate.) The bright state super-radiantly emits into the waveguide at a rate of𝑁Γ1D. The collective dark state has no coupling to the waveguide, and a decoherence rateΓ0
D which is set by parasitic damping and dephasing not captured in the simple waveguide QED model (see Sec. C.2 and Sec. C.3). In addition to the bright and dark collective states described above, there exist an additional 𝑁 −2 collective states of the qubit array with no coupling to either the probe qubit or the waveguide [120].
The subsystem consisting of a coupled probe qubit and symmetric dark state of the mirror qubit array can be described as an analog to a cavity QED system [120].
In this depiction, the probe qubit plays the role of a two-level atom and the dark state mimics a high-finesse cavity, with the qubits in the𝜆0/2-spaced array acting as atomic mirrors (see Fig. 4.1a). In general, provided that the fraction of excited array qubits remains small as𝑁 increases, ˆ𝑆D stays nearly bosonic and the analogy to the Jaynes-Cummings model remains valid. By mapping the waveguide parameters to those of a cavity QED system, the cooperativity between probe qubit and atomic cavity can be written asC =(2𝐽)2/(Γ1D,p+Γ0p)Γ0
D ≈ 𝑁 𝑃1D. Here𝑃1D = Γ1D/Γ0
Dis
the single qubit Purcell factor, which quantifies the ratio of the waveguide emission rate to the parasitic damping and dephasing rates. AttainingC >1 is a prerequisite for observing coherent quantum effects. Referring to the energy level diagram of Fig. 4.1a, by sufficiently reducing the waveguide coupling rate of the probe qubit one can also realize a situation in which 𝐽 > (Γ1D,p+Γ0p),Γ0
D, corresponding to the strong coupling regime of cavity QED between excited state of the probe qubit (|𝑒ip|𝐺i) and a single photon in the atomic cavity (|𝑔ip|𝐷i) (see Sec. C.2). This mapping of a waveguide QED system onto a cavity QED analog therefore allows us to use cavity QED techniques to efficiently probe the dark states of the qubit array with single-photon precision.
The fabricated superconducting circuit used to realize the waveguide-QED system is shown in Fig. 4.1b. The circuit consists of seven transmon qubits (Q𝑗, where 𝑗 = 1-7), all of which are side-coupled to the same coplanar waveguide (CPW).
Each qubit’s transition frequency is tunable via an external flux bias port (Z1-Z7).
We use the top-center qubit in the circuit (Q4) as a probe qubit. This qubit can be independently excited via a weakly coupled CPW drive line (XY4), and is coupled to a lumped-element microwave cavity (R4) for dispersive readout of its state. The other six qubits are mirror qubits. The mirror qubits come in two different types (I and II), which are designed with different waveguide coupling rates (Γ1D,I/2𝜋 = 20 MHz andΓ1D,II/2𝜋 = 100 MHz) in order to provide access to a range of Purcell factors.
Type I mirror qubits also lie in pairs across the CPW waveguide and have rather large (∼50 MHz) direct coupling.