Physical realization of cat-pumping interaction with a single Josephson junction

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Change of frame for numerical simulations

Non-perturbative numerical simulations of such a strongly driven nonlinear system are particularly challenging. They require the simulation of a master equation over a Hilbert space of large dimension and with times scales separated by many orders of magnitude [59]. The dimensionality issue can be overcame by changing the frame of reference, displacing the high excitation manifold into a tractable one. First, let us write quantum Langevin equations for the time evolution under Hamiltonian Eq. (2.9). To start and in order to simplify notations, I will ignore the influence of the charge offset Ng, setting Ng = 0. For the annihilation operator a, this gives da dt = −iωaa + gN+ Ap(t).

Simulations in the Floquet-Markov framework for weak dissipation

Non-perturbative numerical simulations of such a strongly driven nonlinear system are particularly challenging. They require the simulation of a master equation over a Hilbert space of large dimension and with times scales separated by many orders of magnitude [59]. The change of frame described in the previous section solves the dimension problem by displacing the high excitation manifold into a tractable one correctly. However, the full numerical simulation of the dynamics of this system, in absence of any rotating-wave approximation, remains difficult.
Indeed, usually, to simplify the dynamics, one starts by removing the fastest times scales through rotating-wave approximations. However, reliable simulations in the presence of strong drives require taking into account the counter-rotating terms in the Hamiltonian, as it had previously been noted in [40, 57]. In the study I conducted, I purposely avoided any time-averaging of the driven Hamiltonian. The Floquet-Markov framework [62] provides a useful and efficient framework to conduct these simulations for a system evolving under a periodic Hamiltonian and with weak dissipation.

Floquet-Markov framework for weak dissipation

Let us consider here a system evolving under a time-periodic Hamiltonian of period T = 2π/ωp. Such a system can be efficiently simulated using the tools from the Floquet theory [62, Section 2]. In this subsection, I recall some of the basic elements of the Floquet theory that are required to understand the numerical simulations method. The Schrödinger equation for such a quantum system is iℏ ∂ |Ψ⟩ ∂t = H(t) |Ψ(t)⟩.

Numerical simulations of the ac Stark shifts in absence of offset charge

In order to investigate the dynamics of the system Eqs. (2.16)-(2.10) for large pump amplitude, when the circulating photons number can reach a few hundreds, I performed Floquet-Markov simulations, assuming a white-noise spectrum for the bath (therefore taking J(ω) = 1 in Eq. (2.26)). In Fig. 2.4, I plot the populations of the true transmon eigenstates {|ηk⟩}∞ k=0, in ρss(0) and as a function of the pump power. The transmon eigenstates are those of the transmon Hamiltonian 4ECN2 − EJ cos(θ).
First, the populations of the mode ea remain close to the ground state over the whole range of pump values. This confirms that the actual state is well approximated by a coherent state and therefore, that the change of frame performed in section 2.2 is a correct approximation.

Comparison with experimental data

In parallel to the theoretical work described in previous section, an experimental investigation of the behavior of a single transmon embedded in a 3D copper cavity was conducted at Laboratoire Pierre Aigrain [32]. Such a circuit is well modeled by the Hamiltonian depicted in Eq. (2.4). The operating regime has comparable parameters of the system, that is EC/ℏ = 2π × 166 MHz (versus 150 MHz in simulations), EJ/ℏ = 2π × 23.3 GHz (versus 20 GHz in simulations), g/2π = 179 MHz (versus 140 MHz in simulations). The main difference lies in the cavity frequency under consideration, ωa/2π = 7.739 GHz (versus 5.5 GHz in simulations).

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Simulations in the Floquet-Markov framework for weak dissipation

I carried out numerical simulations of the driven dissipative system Eqs. (3.13)-(3.17) in the Floquet-Markov framework, similarly to the analysis done on the (unshunted) transmon circuit in section 2.3. Although the simulations are run here in the Fock states basis of the two modes ea and eb, I show and plot the results in the shunted transmon basis {|νk⟩} k (eigenstates of the Hamiltonian 4ECN2 + ELφ2/2 − EJ cosφ) to ease the comparison with results from the previous chapter. Here, the parameters are chosen such that the bare frequencies, impedances and coupling of the harmonic oscillator and the transmon mode coincide with those of chapter 2. The important change concerns the dilution of the nonlinearity by the addition of the harmonic shunt with an energy EL, about a factor 2 larger than EJ .
The populations of the ea mode remain very close to its ground state over the whole range of pump values, even more than in the regular transmon case. In a similar way, the populations of the eb mode, as shown on Fig. 3.2, remain very close to the ground state, with a slight increase with the pump power. Both modes remain much closer to their ground state than their counterpart in chapter 2. The state ρss(0) follows a very smooth behavior and, as shown on Fig. 3.3, the impurity of ρss (black crosses, right axis) remains close to zero (below 3% for the whole pump range).

Table of contents :

1 Introduction 
1.1 Quantum information processing with superconducting circuits
1.2 Superconducting circuits quantization
1.2.1 Circuit elements and notations
1.2.2 Circuit quantization
1.2.3 Useful Hamiltonian transformations
1.3 Hamiltonian engineering with parametric pumping
1.3.1 Principles of parametric pumping
1.3.2 Example of the Josephson Parametric Amplifier
1.4 Dissipation engineering
1.5 Using a cavity as a logical qubit: cat qubits
1.5.1 Two-photon cat qubits encoding
1.5.2 Stabilization of the cat-states manifold with dissipation engineering
1.5.3 Four-photon pumping extension
1.6 Physical realization of cat-pumping interaction with a single Josephson junction
1.7 Plan of the manuscript
2 Structural instability of driven transmon circuit 
2.1 Strongly driven transmon coupled to a cavity
2.2 Change of frame for numerical simulations
2.3 Simulations in the Floquet-Markov framework for weak dissipation
2.3.1 Floquet-Markov framework for weak dissipation
2.3.2 Numerical simulations of the ac Stark shifts in absence of offset charge
2.3.3 Influence of the offset charge
2.4 Comparison with experimental data
3 Inductively shunted transmon: a solution to dynamical instability 
3.1 Model of the driven shunted transmon circuit
3.2 Simulations in the Floquet-Markov framework for weak dissipation
3.3 Rotating-wave approximation results and comparison with numerical simulations
3.4 Choice of parameters for the shunted transmon circuit
4 Floquet-Markov simulations 
4.1 Steady-state computation framework
4.1.1 Encoding the circuit Hamiltonian
4.1.2 Floquet code
4.1.3 Running simulations
4.2 Analysis code
4.2.1 Resonant frequencies of the system
4.2.2 Induced Kerr strength
5 Asymmetric Josephson Ring Modulator 
5.1 Josephson Ring Modulator
5.1.1 Unshunted Josephson Ring Modulator
5.1.2 Josephson Ring Modulator with shunt inductances
5.2 Asymmetric Josephson Ring Modulator
5.2.1 Circuit Hamiltonian
5.2.2 Quantization of the AJRM circuit embedded in a microwave cavity
5.3 AJRM for the two-photon and four-photon pumping schemes
6 Conclusions and perspectives 
Bibliography

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