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Authors: P.M. Harrington, M. Naghiloo, D. Tan, K.W. Murch
First Author’s Primary Affiliation: Department of Physics, Washington University, Saint Louis, Missouri 63130, USA
Manuscript: Published in Physical Review A
Quantum systems are generally very sensitive, and upon interacting with the environment, their quantum properties can decohere. This essentially makes a given quantum system dissipate into purely classical behavior. However, in certain contexts it is possible to use dissipation in a controlled fashion to increase the control of quantum systems. A few examples of controlled dissipation in this way include laser cooling of atoms, cooling of low frequency mechanical oscillators, and for the control of quantum circuits. In this recent publication , the authors are able to demonstrate stabilization of superposition states in a superconducting qubit using a custom made photonic crystal loss channel. By considering how the photonic crystal induces loss on the system, the authors provide a master equation treatment which explains how the combination of a specialized drive applied to the qubit in addition to the dissipation provided via the photonic crystal allows for precise control of the qubit state for times much longer than standard qubit coherence times.
This experiment consists of a superconducting qubit whose dipole moment is coupled to the electric field inside of a three dimensional waveguide cavity. In this experiment, the role of the waveguide cavity is to provide microwave control of the qubit as well as reading out the state of the qubit. The superconducting qubit consists of two Josephson junctions in parallel, forming a superconducting quantum interference device (often referred to as a “SQUID”). This allows the authors to change the resonant frequency of the qubit by threading an external magnetic field through the SQUID loop. On the output of the waveguide cavity, the authors connect a photonic crystal to the circuit. This photonic crystal is made out of a regular coaxial cable which is mechanically deformed in a specific way in order to change the impedance of the cable. The result of the spatially varying impedance in the cable leads to the opening of a bandgap – leading to photon energies (or frequencies) where the photonic density of states is zero (see Fig. 1 for a schematic of the experimental setup). By changing the photonic density of states as a fucntion of energy, the decay of the qubit will also change as a function of frequency.
Qubit Decay Rates
In order to measure the decay rate of the qubit, the authors first excite the qubit into its excited state by applying a pulse of energy which is resonant with the qubit to the system. They then measure the probability of the qubit remaining in its excited state as a function of time after the pulse is applied. By fitting the measured probability to an exponential decay and extracting the decay constant, one is able to determine the qubit decay rate. The resonant frequency of the qubit is then adjusted by changing the external magnetic flux threading the SQUID loop, and measuring the qubit decay rate as a function of qubit frequency in order to investigate the impact of the photonic crystal on the qubit lifetime. The total decay rate of the qubit can be written as
In Eq. 1, is the measured decay rate of the qubit, is the linewidth of the microwave cavity, is the coupling strength between the qubit and the cavity, is the difference in resonant frequency between the qubit and the cavity, is the density of states of the photonic crystal at the qubit frequency, and represents decay of the qubit into dissipation channels other than the photonic crystal. By measuring the total qubit decay rate for various values of , it should be possible to extract information about the density of states of the photonic crystal! See Fig. 2 below for the resulting measurement
Dynamics and Emission of a Driven Qubit
After verifying that the density of states in the photonic crystal can shape the decay rate of the qubit, the authors now consider more carefully how the qubit actually emits energy. Specifically, a strong drive applied with amplitude which is detuned from the qubit energy by , where is the frequency of the drive and is the qubit energy is considered. If the amplitude of the drive is much larger than the loss rate of the qubit, the qubit will emit energy at three different frequencies , and , where is called the generalized Rabi frequency. This emission spectrum is called the Mollow triplet . See Fig. 3 for a schematic of the Mollow triplet emission.
Because the authors have observed that the photonic crystal shapes the loss rate of the qubit on a energy scale comparable to values of experimentally accessible , it is possible for one of the sidebands of the Mollow triplet to experience a high loss rate while the other sideband of the Mollow triplet experiences a low loss rate.
The next thing to consider is what the presence of an applied drive does to the energy spectrum of the qubit. In a frame rotating with the drive frequency, the qubit Hamiltonian is given by
where and are Pauli matrices. Since this Hamiltonian is not diagonal, it is convenient to rotate basis such that the Hamiltonian can be written in a new form
where the rotated Pauli Z matrix can be written as , and the rotation angle is defined as with . Because we have written the Hamiltonian in a rotated basis, we must also consider how the new eigenstates of the system rotate relative to the original eigenstates, which we will call and for ground and excited state, respectively.
At this point it’s probably useful to consider a useful example! In the case of a resonant drive, , which immediately informs us that , so we can rewrite the rotated eigenstates of the system as , and , which have the special property that . Because the state has a lower energy, it will emit energy corresponding to the lower energy sideband of the Mollow triplet and vice versa for the state . If the loss of the qubit is vastly different for either of these states, that will promote decay into either the state or ! Specifically, if the qubit is at a resonant frequency near 6.4766 GHz (see Fig. 2), the state at higher energy (corresponding to in this example) has a lower loss rate, so we should expect that while the drive is turned on, the qubit preferentially would decay into this state! This means that the expectation value would tend towards +1 in this scenario! In the case of a uniform loss spectrum, there would be no preferred decay of the qubit and one would expect that all of the qubit expectation values would decay to zero.
Lindblad Master Equation
In the presence of the combined drive and dissipation experienced by the qubit, the dynamics of the reduced density matrix which describes the qubit can be written according to the Lindblad Master equation :
Here, is the reduced density matrix for the qubit, the dissipation superoperator is also introduced as . The rate represents dephasing of the qubit in the basis rotated by , which couples to the operator and transitions between eigenstates in the rotated basis are driven by the “jump” operators which are related to the rates . Similar to the previous example, if the photonic crystal shapes the qubit loss such that , a corresponding rotating frame eigenstate will be stabilized.
In order to verify that the authors can use the combination of drive and dissipation to prepare and stabilize qubit states, they implement the following bath engineering protocol. First, the qubit is flux biased to a resonant frequency of 6.4766 GHz (as in our example). Then, a coherent drive is applied to the system for nearly (which is much longer than the qubit coherence times in the absence of drive!). During this time, the qubit should preferentially decay to an eigenstate of the rotated system if the Mollow triplet sidebands have different weights. Once the drive is shut off, the expectation value is measured for various combinations of drive parameters. Results are shown below as well in Fig. 4 as comparisons to numerical solutions to the master equation, the authors not only see that the qubit expectation values don’t decay to 0, but also fantastic agreement between the theory and the experiment! Additionally, we can recall our earlier example, and we see that along the linecut of , the expectation value approaches the value of +1 as we expected!
In conclusion, the authors are able to demonstrate the fabrication of a spatially changing impedance coaxial cable which acts as a photonic crystal, and in turn controlling the loss spectrum of a superconducting qubit. The authors are then able to leverage this shaped emission spectrum in the context of the master equation to prepare and stabilize non-trivial states of the qubit for times much longer than the coherence times of the qubit.
 P. M. Harrington, M. Naghiloo, D. Tan, and K. W. Murch, Bath engineering of a fluorescing artificial atom with a photonic crystal, Phys. Rev. A 99, 052126 (2019)
 B. R. Mollow, Power spectrum of light scattered by two-level systems, Phys. Rev. 188, 1969 (1969)
 G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976).