Structural instability of driven transmon circuit 

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Hamiltonian engineering with parametric pumping

In this manuscript, I investigate the behavior of circuits mediating parametric interac-tions between multiple modes in the large pumping regime. Indeed, by considering a non-linearity coupling multiple microwave modes and adding a periodic modulation on them (later called the “pump”), one can engineer and tailor specific interactions between the considered modes [18]. Such parametric interactions can be used for amplification using single-mode or two-mode squeezing, for frequency conversion or for more elaborate multi-photon interactions between the considered modes, as I will briefly review in this section.

Principles of parametric pumping

Parametric interactions are a tool of paramount importance in order to design Hamil-tonian interactions with a full control on the strength of the terms as well as being able to turn on and off the interaction easily. As pictured on Fig. 1.6, the core idea is not to directly couple the two modes and physically engineer the parameters of the system to control the interaction term, but rather to use an auxiliary mode (the pump mode) which will mediate the interaction. Provided that the pump frequency verifies a specific frequency matching condition, the resulting interaction Hamiltonian can be obtained from rotating-wave approximation [19]. Moreover, provided that the pump is non-resonant and that its bandwidth is larger than the one from other modes, the pump can be considered in the “stiff pump regime”, which is equivalent to consider that the pump is not affected by the interaction process and can therefore be treated classically. Such framework has the major advantage compared to in-situ interactions engineering that the pump drive fully controls both the strength as well as the phase of the interaction terms, allowing for near real-time control.
Figure 1.6: Interaction engineering through parametric interactions. Instead of considering a direct interaction between two modes a and b, we are using a third mode, c, to mediate interactions between a and b. This third mode is coupled to one or more AC drives (oscillating pumps) whose frequencies and amplitudes control the parametric interaction terms in the Hamiltonian.
Formally, let us consider the simplest Hamiltonian with two electromagnetic modes and a direct coupling between them, H = ℏωaa†a + ℏωbb†b + g ab† + a†b . (1.23)
In an interaction picture with respect to the harmonic oscillator part of this Hamiltonian, it becomes H = g ei(ωb−ωa)tab† + e−i(ωb−ωa)ta†b . (1.24) and the coupling term will remain in a rotating-wave approximation if and only if the two frequencies of the modes are detuned by a frequency mismatch much smaller than the coupling strength g. On the contrary, let us now consider a parametric interaction scheme with an extra mode c. Assuming there exists a physical implementation of it, the Hamiltonian could then read H = ℏωaa†a + ℏωbb†b + ℏωcc†c + g ab†c + a†bc† + ℏAp(t) c + c† (1.25) where Ap(t) = Ap cos(ωpt) is the amplitude of the parametric pump on the c mode. Considering a coherent displacement of the c mode to take into account the effect of the parametric pump (following section 1.2.3), the Hamiltonian reads H = ℏωaa†a + ℏωbb†b + ℏωcc˜†c˜ + g ab† (c˜ + ξ(t)) + a†b c˜† + ξ(t)∗ . (1.26)
In the interaction picture with respect to the harmonic part, the Hamiltonian reads e Apg ei(ωb−ωa−ωp)tab† + e−i(ωb−ωa−ωp)ta†b (1.27) H = 2 (ωc − ωp) where a partial trace has been used on the mode state. With such a parametric scheme, we recover of the form of Eq. (1.24) where ce which is very close to the vacuum an interaction term between a and b
• The resonance condition for this interaction to remain under rotating wave ap-proximations is now
ωp = ωb − ωa. (1.28)
It is important to note here that this frequency matching condition is a condition on the parametric pump frequency ωp (which can be tuned easily) and no longer impose a frequency match for the two electromagnetic modes.
• The coupling amplitude is no longer g but is scaled by a factor Ap/2(ωc − ωp). Therefore, increasing the pump strength increases the interaction strength.
In the previous calculation, I used an auxiliary c mode for clarity. However, the same parametric scheme could be achieved by pumping one of the two initial electromagnetic modes (a for example) and having an initial interaction with an extra power in this mode (an interaction of the form a2b† + h.c. for instance).
Various superconducting circuits implementing parametric interactions have been proposed to achieve amplification and conversion [20]. Starting from the Josephson Parametric Amplifier (JPA) with a single Josephson junction or two Josephson junctions in a SQUID configuration [21, 22, 23] which achieve amplification through single-mode squeezing with an interaction Hamiltonian of the form Hint = ℏg a2 + a† 2 (1.29) more refined circuits have been proposed. Among such circuits is the Josephson Ring Modulator (JRM) [24] (whose mode of operation is presented in details in Chapter 5) which consists in a loop of four Josephson junctions and offers an amplification through two-mode squeezing, using three-wave mixing terms of the form Hint = ℏg ab + a†b† (1.30)
As demonstrated in [25, 26], the stability of the Josephson Ring Modulator circuit can be further improved using inductive shunts and the resulting circuit can be used to engineer squeezing interaction Hamiltonian as well as conversion processes along with the quantum limited amplification. Similar methods have been used in [27] to implement a Q-SWITCH with a conversion Hamiltonian of the form Hint = ℏg ab† + a†b (1.31) which can be used to transfer the state of an electromagnetic mode of a cavity to a propagating electromagnetic mode or another remote cavity.
Parametric methods are also ubiquitous to the cat states encoding proposal from [16] which requires a two-photon or four-photon exchange Hamiltonian of the form Hint = ℏg adb† + a† d b (1.32) where d = 2 or d = 4. Superconducting circuits relying on parametric methods can engineer such Hamiltonians for the two-photon exchange process [28] or using higher order rotating-wave approximations to achieve four-photon exchange process [29].
Finally, these methods are not limited to standing electromagnetic modes but can also be used for amplification along a transmission line with travelling modes, such as in the Josephson Traveling-Wave Parametric Amplifier (JTWPA) [30].
At this point, it is worth noting that, usually, the strength of the engineered Hamilto-nian scales with the amplitude of the pump drive mediating the parametric interaction. Therefore, one usually wishes to increase the pump power, in order to reach better op-erating regimes. This, however, does not in general experimentally yield improvements indefinitely. Indeed, the interaction terms in the Hamiltonian which are selected thanks to the parametric pumping are only a subset of all the possible terms. For instance, in the strong pump regime, such terms which were at first either negligible or discarded through rotating-wave approximations [19] might actually no longer be negligible and should be taken into account. Additionally, in the strong pump regime, the system can suffer from phase slips or symmetry breaking (the total flux can no longer be symmet-rically divided across chains of Josephson junctions)[31], quasi-particle creation due to the strong electromagnetic fields breaking Cooper pairs[32, 33], or simply degrading the coherence times of the modes [34] (possibly through heating of the chip or by inducing other decay mechanisms which are not fully understood).
Figure 1.7: Josephson Parametric Amplifier circuit. The circuits consists of a RF Superconducting Quantum Interference Device (SQUID) with a parallel capacitor, capacitively coupled to a transmission line. The external flux threading the loop, Φext can be used to adjust the frequency of the resonator, ωa. Pumps (with time-dependent amplitude A1(t) and A2(t) and frequencies ω1 and ω2) are sent through the transmission line to achieve parametric pumping. A signal, at frequency ωa (matched with the RF-SQUID mode frequency) is sent through the same transmission line. Such a device can be used to achieve amplification through single-mode squeezing with the reflected signal, coming out of the transmission line.

Example of the Josephson Parametric Amplifier

As a real world example, let us focus on the simple case of the Josephson Paramet-ric Amplifier (JPA) whose circuit is pictured on Fig. 1.7. First, let us note that the Josephson junctions loop from the circuit is actually equivalent to a tunable Josephson junction. Assuming we operate at a fixed value of Φext, the Hamiltonian of this system reads, considering only up to quartic term in the Josephson cosine potential expansion, H = ℏωaa†a + K a + a† 4 + ℏA1(t) a + a† + ℏA2(t) a + a† (1.33) where K is a constant which can be expressed in terms of the capacitive and Josephson energy of the circuit elements, A1(t) = A1 cos(ω1t) and A2(t) = A2 cos(ω2t) are the time-dependent pump amplitudes.
Following the approach from equations (1.23)-(1.27), the Hamiltonian expressed in terms of displaced modes and in the interaction picture with the harmonic oscillator term reads H = K′ ae−iωat + ξ1e−iω1t + ξ2e−iω2t + h.c. 4 ′ a (1.34) with e K renormalized coupling strength, proportional to the pump amplitude, and ξ1,2 = A1,2/2(ωa − ω1,2).
Assuming a frequency matching condition ω1 + ω2 = 2ωa, (1.35) the remaining term of interest after a rotating-wave approximation is 12K′ a2ξ1∗ξ2∗ + h.c. (1.36) which can be used to implement amplification through single-mode squeezing. At this point, let us note that if we had only considered a single pump, the frequency matching condition would have been ω1 = ωa therefore requiring to have the pump in resonance with the oscillator mode and no longer being able to rely on the stiff pump approxima-tion. On the contrary, using two pumps here allows us to choose two pumps far detuned from the oscillator frequency (one red-detuned and the other one blue-detuned) while satisfying the frequency matching condition.
However, the term from Eq. (1.36) is not the only one remaining under rotating-wave approximation. Indeed, this term is coming from a fourth-order term in the expansion of the cosine potential, along with terms of the form ξ1∗,2ξ1,2 a† a2 (Kerr-type terms).
They would be resonant under the frequency matching condition Eq. (1.35) and can be detrimental depending on the envisioned application for the circuit. Additionally, one should note that these terms, just as the main term of interest for amplification through single-mode squeezing, have their amplitude proportional to the pump amplitudes.
These limitations when pushing the pump power lead to the theoretical investigation of the behavior of such systems in the strong pump regime, with a particular focus on the loss of the non-linearity when the pump strength becomes too high [35, 36, 37, 38, 39] or the reduction of the coherence time of the non-linear resonator as a function of the mean photon occupancy (T1 versus n¯) [40]. These previous analysis were relying on simplified models of the superconducting circuit in use, relying on Jaynes-Cumming model (only considering the two lowest state of the non-linear resonator), generalized Jaynes-Cumming (considering a few states of the non-linear resonator) or Duffing approximation (considering the non-linearity only to some order of its Taylor expansion). Throughout this manuscript, I will investigate the steady state dynamics of such a non-linear resonator (transmon circuit) coupled to a linear resonator (microwave cavity) and a parametric pump in the strong pumping regime. I am proposing a Floquet-Markov framework to carry out numerical investigation of this system in the strong pumping regime. This scheme and the associated change of basis let me consider a large transmon Hilbert space truncature, able to capture highly excited transmon states, and avoid rotating-wave approximations, which are not valid in this setting. The results presented in this thesis, although presented in the context of cat-states encoding, can be generalized to a wide variety of the previously mentioned parametric systems with a pump mediating a non-linear interaction.

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Dissipation engineering

The parametric methods presented in section 1.3 are of paramount interest to engineer specific dissipation terms, in particular non-linear dissipations or turning on and off dissipations on the fly in experiments. Indeed, parametric methods can be used to engineer a specific and controllable non-linear interaction between two electromagnetic modes. Then, in the limit where one of the two modes is strongly dissipative, adiabatic elimination procedures [41] can be used in order to get a simplified model without this extra dissipative mode. The engineered non-linear interaction between the two modes then turn into a non-linear dissipative terms for the remaining electromagnetic mode.
In the QUANTIC team at INRIA Paris, such methods are used and implemented in particular in the context of cat-states encoding, which I will more extensively describe in section 1.5 and which will be used as a recurring example for parametric methods and dissipation engineering considerations throughout this manuscript. I will now briefly describe the adiabatic elimination procedure for the specific case of the two-photon interaction a† 2 b + a2b† Hint = iℏg2−ph (1.37) which results in a non-linear dissipation of photons in pairs. Such an engineered dis-sipation is at the core of the two-photon cat-states encoding and the interest of such dissipation will be clear in section 1.5.
Starting from the two modes Lindblad master equation dρ = u hb† − b, ρi + κbD [b] (ρ) − ig2−ph a† 2 + La (ρ) b + a2b†, ρ (1.38) dt where D [A] (ρ) = AρA† − A†Aρ + ρA†A /2, u is the amplitude of the drive on the readout mode, κb is the (large) dissipation rate of the readout mode, g2−ph is the two-photon interaction rate and a and b are the storage (with a high quality factor) and readout (very dissipative) modes annihilation operators. Here, La (ρ) models parasitic Lindblad terms acting on the storage mode (for instance single photon dissipation, self-Kerr terms or dephasing noise). La (ρ) is supposed to be small compared to the other terms. Here, I assume that g2−ph/κb is small enough to be in the regime where the results from [41] are valid.
Eq. (1.38) can be recasted into dρ = −ig2−ph a2 − α2−ph2 † b − a2 − α2−ph 2 b†, ρ + κbD [b] (ρ) +εLa (ρ) (1.39) dt εLint(ρ) | {zb(ρ) } at the expense of introducing α2−ph = q . u/ig2−ph
The adiabatic elimination theorem from [41] can be applied on Eq. (1.39), as the dynamics can be split into two time scales:
• A rapid convergence of the fastest subsystem towards a given subspace (given by the dynamics in absence of coupling, with the time scale κb),
• A slow evolution of the other subsystem while maintaining the fastest subsystem in the vicinity of its steady state in absence of coupling.
Indeed, with ε = g2−ph/κb,
• Lb (ρ) = κbD [b] (ρ) is the fast dynamics
• εLa is the slow dynamics (which is supposed to be at most of order ε)
• εLint = −ig2−ph h a2 − α2−ph2 † b − a2 − α2−ph2 b†, ρi is an interaction Lind-bladian which verify the previous condition for the validity of the results from [41]. Let us note that in absence of coupling between the two modes (g2−ph = 0), the fast dynamics (of the mode b) rapidly converges to a coherent state |β given by β = 2u/κb. In presence of the coupling, however, the steady state lies in Span {|±α2−ph } ⊗ |0 .
Following the analysis from [41] and [42] and their notations, the steady state of the fast dynamics ρ¯b is given by ρ¯b = |0 0|. Then, the master equation at second order is given by1 dρs 4g2−ph2 D h 2 2 i 3 (1.40) = a − α2−ph (ρs) + La (ρs) + O(ε ) dt κb where ρs is a reduced density matrix. In absence of parasitic La Lindbladian, the reduced density matrix ρs obeys a master equation of the form dρ = κ2−phD ha2 − α2−ph2i (ρ) (1.41) dt with κ2−ph = 4g2−ph2/κb. This master equation and its implication for cat states pump-ing will be detailed in section 1.5.2.
As shown by Eq. (1.41), after an adiabatic elimination of the most dissipative mode, the non-linear parametric interaction from Eq. (1.38) translates in a non-linear two-photon dissipation on the remaining storage mode.
Moreover, in absence of La term, the full state of the system can be recovered from the fast dynamics steady state ρ¯b and the reduced density matrix ρs using the following Kraus map K (ρs) = ρs⊗ρ¯b+2ig2−ph h ρs a†2 − α2−ph∗2 ⊗ |0 1| − a2 − α2−ph2 ρs ⊗ |1 0|i .
At this point, it is worth noting that this Kraus map reduces to ρs ⊗ ρ¯b in the steady state of the system. Therefore, the Kraus map should be taken into account during the transient evolution but not in the steady state, where the ρs reduced density matrix coincides with the density matrix of the storage mode.

Using a cavity as a logical qubit: cat qubits

In [11], M. H. Devoret and R. J. Schoelkopf distinguish seven steps towards quantum information processing, from the realization of simple operations on individual physical qubits to the realization of a full fault-tolerant quantum computation on a large-scale qubits system. Systems based on superconducting qubits have successfully reached the third step, that is to be able to perform quantum nondemolition measurements (called QND measurements) for control and error correction purposes, and they are on the edge of reaching the fourth step, that is to be able to control logical qubits with lifetimes larger than individual physical qubits.
Classical information processing relies on storing the information on bits (two-states systems) and error correction is based on redundantly encoding the information and performing majority votes. In quantum information processing, information is stored on a qubit, a two-level system, quantum analogous of the classical information bit. The qubit being a physical system coupled to a noisy environment, it also experiences errors which should be corrected. These errors can be modelled as bit-flip errors (analogous to the classical bit inversion error, swapping the |0 and |1 states) and phase-flips errors (turning the |0 + |1 state into |0 −|1 ). In the quantum world, the no-cloning theorem and measurement back action prevents any scheme based on copying the information and exploiting redundancy (through majority votes) as simply as in the classical world. Therefore, more sophisticated error-correction schemes have to be implemented.
In order to implement quantum error correction, one can encode the (logical) qubit state on multiple (physical) qubits, which is the approach used by the Steane code[43] for instance. On the contrary, in this manuscript, I will focus on another approach relying on storing the information in a single high dimensional Hilbert space, such as an harmonic oscillator Hilbert space. The cat-code encoding presented in the next subsection uses this approach and relies on encoding the state of the qubit on superposition of coherent states. I will present the principle of the cat-state encoding and a cat-pumping scheme to further confine the state of the logical qubit to a manifold spanned by the useful coherent states, therefore extending the upper bound on the lifetime of the qubit. Then, I will present the current state of the art experiment for an experimental realization of this encoding scheme and an extension to a larger manifold allowing for better error tracking through carefully chosen projective measurements.

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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