## Periodic Reporting for period 3 - AQSuS (Analog Quantum Simulation using Superconducting Qubits)

Reporting period: 2020-04-01 to 2021-09-30

The remarkable progress in experimental physics over the last decade has enabled us to manipulate, control and detect the state of various quantum systems to a very high degree.The realization of a universal quantum computer though has proven to be a very demanding goal. As a consequence the focus has shifted to simulating specific quantum phenomena. The unifying idea is to build a well-controlled device that mimics a certain quantum system that is either very hard to or cannot be simulated with a classical computer.

In this ERC project we are implement an experimental platform for an analog quantum simulator of interacting spin systems using superconducting qubits. The scheme builds on the remarkable recent developments in circuit quantum electro dynamics systems to build interacting spins systems in one and two dimensions.

In the past 30 month we have installed and tested the experimental setup i.e. the cryostat and the microwave equipment. Furthermore we have implemented all fundamental building blocks such as superconducting resonators, qubits and waveguides and have built small systems of interacting qubits. Further details can be found in the section "work performed from the beginning of the project".

In this ERC project we are implement an experimental platform for an analog quantum simulator of interacting spin systems using superconducting qubits. The scheme builds on the remarkable recent developments in circuit quantum electro dynamics systems to build interacting spins systems in one and two dimensions.

In the past 30 month we have installed and tested the experimental setup i.e. the cryostat and the microwave equipment. Furthermore we have implemented all fundamental building blocks such as superconducting resonators, qubits and waveguides and have built small systems of interacting qubits. Further details can be found in the section "work performed from the beginning of the project".

The work that has been performed from the start of the project to the end of the 30 month period followed the proposed time plan of the project for most of the task. Due to some clean-room fabrication related problems, some tasks have been delayed by 3-4 month as outlined below. A summary of the achieved tasks can been seen by the checked boxes in Table 1 and the summary after:

MAIN RESULTS

Fast flux tuning of 3D transmon qubits

An important feature in investigating different spin models is the possibility to change the frequency of SQUID-based transmon qubits by applying magnetic flux. Recently, we demonstrated a new approach [2] for fast flux control of a transmon qubit in a 3D cavity architecture. We use a magnetic hose similar to the one jointly proposed by collaborators (A. Sanchez) in Barcelona and the Quantum Nanophysics, Optics and Information group at the IQOQI. This hose guides a fast flux pulse from the outside to the inside of a microwave cavity. The transition frequency can be tuned, using a simple square pulse, with a typical rise time of about 1 µs, which is much faster than the typical qubit decoherence. By measuring the transfer function of the system and applying a deconvolution kernel, the effective qubit response time can be further reduced, reaching about 50 ns (see Fig.1).

WAVEGUDIE QUANTUM ELECTRO DYNAMICS

While the use of a cavity for 3D transmon qubits constitutes a promising platform to study/simulate complex spin models, its size and lack of flexibility limits the number of qubits to about ten. More importantly, by using only a single cavity, the qubit state as well as the correlations functions can be challenging to measure. To address this problem, the cavity can be replaced by 3D waveguide. In [3], we demonstrate the experimental implementation of a 3D waveguide with a U-shaped microstrip resonator. This approach combines a planar design with the advantages of three dimensions. In the low temperature/low energy limit we measured an internal quality factor of up to one million for resonators made out of niobium, a value comparable with state-of-the-art systems.

QUANTUM SIMULATION OF SPIN CHAIN PHYSICS

Fig. 2. shows a conceptual schematic to implement a system for analog quantum simulation of a spin lattice, where the microstrip resonators serve as a means to readout the qubit state. As we can increase the length of the waveguide almost arbitrarily, up to the size of our dilution refrigerator, this will allow the systems size to be increased considerably. In addition to the qubits we will place stripline resonators in the waveguide (black U-shaped structures) and use them for a dispersive readout of the qubits they couple to. These resonators will provide means for locally probing the excitations and correlations in a spin chain or ladder geometry.

Meanwhile we have realized this setup experimentally as can be seen in Fig. 3. We are currently assessing the system capabilities and parameters. Furthermore, we are working on protocols to efficiently assess the Hamiltonian of the system.

OPEN SYSTEM QUANTUM DYNAMICS WITH INTERACTING SPIN SYSTEMS

To asses the feasibility of this setup to investigate open system dynamics of interacting spin system we have performed an experiment where we placed two pairs of qubits inside a copper waveguide. For low microwave powers, the qubits act as perfect mirros at their resonance frequency and can thus be easily seen in transmission if their frequencies are above the cutoff of the waveguide.A typical measurement curve can be seen in Fig.4. One can clearly see the avoided crossing between the two resonance curves which is a result of the direct dipole-dipole interaction between the qubits. At resonance both qubits hybridize with each other which results in subradiant and superradiant states. This experiment shows that the fundamental interactions between the qubits and the waveguide work out as intended.

With the second pair of qubits we are now working on encoding a qubit in a non-local Dark state, created by the waveguide mediated interaction. We have just started to characterize the coherence time of this logical qubit and we have already seen that we can suppress the coupling to the waveguide by a factor of 1000 or more. This results in coherence times of the Dark state qubit of several µs while each qubit individually is coupled to the waveguide with about 10MHz (see Fig. 4b).

DIRECT PHOTON WAVE FUNCTION MEASUREMENTS WITH TIME AND FREQUENCY DOMAIN RESOLVED SPECTROSCOPY

In collaboration with J. Garcia-Ripoll from the theory department of CSIC (Madrid, Spain) we proposed and realized experimentally a protocol for the direct measurement of a single photon wave function in the time and frequency domain. The results obtained experimentally are well understood from the point of view of a Wigner-Weisskopf theory as well as input-output theory and by an solving exact master equation (see Fig. 5). Since the qubit emission at short times depends on the spectral property of its environment, this approach can be used to probe the local density of optical states in the waveguide.

[1]Phys. Rev. B 92, 174507 (2015)

[2]arXiv:1811.10875

[3]AIP Advances 7, 085118 (2017)

MAIN RESULTS

Fast flux tuning of 3D transmon qubits

An important feature in investigating different spin models is the possibility to change the frequency of SQUID-based transmon qubits by applying magnetic flux. Recently, we demonstrated a new approach [2] for fast flux control of a transmon qubit in a 3D cavity architecture. We use a magnetic hose similar to the one jointly proposed by collaborators (A. Sanchez) in Barcelona and the Quantum Nanophysics, Optics and Information group at the IQOQI. This hose guides a fast flux pulse from the outside to the inside of a microwave cavity. The transition frequency can be tuned, using a simple square pulse, with a typical rise time of about 1 µs, which is much faster than the typical qubit decoherence. By measuring the transfer function of the system and applying a deconvolution kernel, the effective qubit response time can be further reduced, reaching about 50 ns (see Fig.1).

WAVEGUDIE QUANTUM ELECTRO DYNAMICS

While the use of a cavity for 3D transmon qubits constitutes a promising platform to study/simulate complex spin models, its size and lack of flexibility limits the number of qubits to about ten. More importantly, by using only a single cavity, the qubit state as well as the correlations functions can be challenging to measure. To address this problem, the cavity can be replaced by 3D waveguide. In [3], we demonstrate the experimental implementation of a 3D waveguide with a U-shaped microstrip resonator. This approach combines a planar design with the advantages of three dimensions. In the low temperature/low energy limit we measured an internal quality factor of up to one million for resonators made out of niobium, a value comparable with state-of-the-art systems.

QUANTUM SIMULATION OF SPIN CHAIN PHYSICS

Fig. 2. shows a conceptual schematic to implement a system for analog quantum simulation of a spin lattice, where the microstrip resonators serve as a means to readout the qubit state. As we can increase the length of the waveguide almost arbitrarily, up to the size of our dilution refrigerator, this will allow the systems size to be increased considerably. In addition to the qubits we will place stripline resonators in the waveguide (black U-shaped structures) and use them for a dispersive readout of the qubits they couple to. These resonators will provide means for locally probing the excitations and correlations in a spin chain or ladder geometry.

Meanwhile we have realized this setup experimentally as can be seen in Fig. 3. We are currently assessing the system capabilities and parameters. Furthermore, we are working on protocols to efficiently assess the Hamiltonian of the system.

OPEN SYSTEM QUANTUM DYNAMICS WITH INTERACTING SPIN SYSTEMS

To asses the feasibility of this setup to investigate open system dynamics of interacting spin system we have performed an experiment where we placed two pairs of qubits inside a copper waveguide. For low microwave powers, the qubits act as perfect mirros at their resonance frequency and can thus be easily seen in transmission if their frequencies are above the cutoff of the waveguide.A typical measurement curve can be seen in Fig.4. One can clearly see the avoided crossing between the two resonance curves which is a result of the direct dipole-dipole interaction between the qubits. At resonance both qubits hybridize with each other which results in subradiant and superradiant states. This experiment shows that the fundamental interactions between the qubits and the waveguide work out as intended.

With the second pair of qubits we are now working on encoding a qubit in a non-local Dark state, created by the waveguide mediated interaction. We have just started to characterize the coherence time of this logical qubit and we have already seen that we can suppress the coupling to the waveguide by a factor of 1000 or more. This results in coherence times of the Dark state qubit of several µs while each qubit individually is coupled to the waveguide with about 10MHz (see Fig. 4b).

DIRECT PHOTON WAVE FUNCTION MEASUREMENTS WITH TIME AND FREQUENCY DOMAIN RESOLVED SPECTROSCOPY

In collaboration with J. Garcia-Ripoll from the theory department of CSIC (Madrid, Spain) we proposed and realized experimentally a protocol for the direct measurement of a single photon wave function in the time and frequency domain. The results obtained experimentally are well understood from the point of view of a Wigner-Weisskopf theory as well as input-output theory and by an solving exact master equation (see Fig. 5). Since the qubit emission at short times depends on the spectral property of its environment, this approach can be used to probe the local density of optical states in the waveguide.

[1]Phys. Rev. B 92, 174507 (2015)

[2]arXiv:1811.10875

[3]AIP Advances 7, 085118 (2017)

Progress beyond state of the art

In summary the main innovations are:

• Natural implementation of strongly interacting 2D spin models with arbitrary lattice geometries.

• Investigate currently inaccessible quantum phenomena in 2D systems.

• Combine open system dynamics with strongly interacting spin systems which opens up a new approach to investigate interacting quantum many body problems.

• Provide valuable insight into strongly interacting superconducting qubit systems which will affect design considerations in cQED quantum information processing and quantum simulation applications.

• Develop a novel method to realize fast flux tuning inside a 3D superconducting cavity

The impact of the expected results of this project will range from condensed matter to quantum optics, quantum information and solid state physics and will allow the investigation of quantum many body problems which are inaccessible with current technology.

In summary the main innovations are:

• Natural implementation of strongly interacting 2D spin models with arbitrary lattice geometries.

• Investigate currently inaccessible quantum phenomena in 2D systems.

• Combine open system dynamics with strongly interacting spin systems which opens up a new approach to investigate interacting quantum many body problems.

• Provide valuable insight into strongly interacting superconducting qubit systems which will affect design considerations in cQED quantum information processing and quantum simulation applications.

• Develop a novel method to realize fast flux tuning inside a 3D superconducting cavity

The impact of the expected results of this project will range from condensed matter to quantum optics, quantum information and solid state physics and will allow the investigation of quantum many body problems which are inaccessible with current technology.