Hamiltonian and dissipation engineering with superconducting circuits

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Hamiltonian engineering

In the expansion of the non-linear terms in (2.9), every combination of annihilation and creation operators of the form (2.11) is present. In the regime of weakly anharmonic circuits where kjk 1, the low order ones, P i ki + `i small, are dominant.
These coupling terms fall into two categories. When 8i ki = `i, the term is called nonrotating, resonant or energy preserving, indeed it has as many creation as annihilation operators. Examples of such terms are Kerr terms a†2 i a2i that shift the mode frequency for each added excitation or cross-Kerr terms a† iaia† jaj that shift the frequency of mode i when j gets excited and conversely. However, the desired coupling Hamiltonian Hc is not resonant in general and rotates at frequency !c P i(ki − `i)!i that is of the order of the mode frequency. This means that when performing the rotating wave approximation (RWA) on the full Hamiltonian, this term has no effect on average on the dynamics. Indeed, it rotates much faster (GHz) than the typical evolution timescale of the state of the system (MHz). Still, this term can be made resonant by leveraging the AC biases. From now on, let us assume there is a single pump at frequency !p acting on the system such that ‘AC,j(t) = ‘j,0 cos(!pt). Depending on the parity of P i ki + `i, a term of the form ˜H c / ‘j,0ei!pt Y i a†ki i a`i i + h.c.

Full description

The potential energy of the transmon is a cosine function. In the former derivation, we expanded the cosine potential up to order 4 to get the low-energy behaviour of the transmon. One could expand it further to improved the description but a better approach is to modify the quantization procedure to take into account the periodic nature of the potential.
From the Cooper pair box to the Transmon The transmon is actually a direct descendant of the Cooper pair box [70]. A Cooper pair box is simply a superconducting island (delimited by the dashed lines in dashed lines in fig. 3.1) connected to ground via a Josephson junction which allows Cooper pairs to tunnel. It is conveniently described in the charge basis. The Hamiltonian of the superconducting island without the junction writes H = 4EC(N − Ng)2 .

Quantum dynamics of a driven transmon 1

As we explained previously, the transmon has an infinite number of energy levels that fall in two categories. First, those with energies smaller than the Josephson energy and that are confined by the Josephson potential. Second, those which lie above the cosine potential which we refer to as unconfined states. Due to their high energies, unconfined states have always been considered irrelevant for circuit dynamics and disregarded [71– 76].
In this section, we show that unconfined states play a central role in the dynamics of strongly driven Josephson circuits. This time-periodic system has periodic orbits known as Floquet states. In the dissipative steady state regime, the system converges to a statistical mixture of Floquet states. For weak pump powers, the steady state remains pure and occupies low energy confined states. Above a critical pump power, the steady state suddenly jumps to a statistical mixture of many Floquet states, with a significant population on unconfined states. The dramatic change in the steady state when the pump power is slightly increased above the threshold is a signature of structural instability [34].

Description of the experiment

Our device consists of a single transmon in a 3D copper cavity [77] coupled to a transmission line. The system is well modeled by the circuit depicted in fig. 3.3a, whose Hamiltonian is given by [69] H(t) = 4ECN2 − EJ cos () + ~!aa†a + ~gN a + a† + ~Ap(t) a + a† .
where N is the Cooper pair number operator and is the phase operator [20]. The operator cos () is the transfer operator for Cooper pairs across the junction while a† and a are the creation and annihilation operators for the resonator mode. The Josephson and charging energies are denoted EJ and EC, respectively. The angular frequency of the resonator in the absence of the JJ (EJ taken to be 0) is denoted !a and is known as the bare resonator frequency and g is the coupling rate between the transmon and the resonator. Note that this definition of g differs from the one usually used in the Jaynes-Cummings Hamiltonian [78]. The pump couples capacitively to the circuit, and is accounted for by the second line in (3.10), where Ap(t) = Ap cos(!pt), Ap and !p being the pump amplitude and angular frequency.
Our physical picture of the dynamics of a pumped transmon in a cavity is the following. When the pump power is sufficiently large, the transmon is excited above its bounded potential well (fig. 3.3c,d). This is reminiscent to an electron escaping from the bound states of the atomic Coulomb ptential and leaving the atom ionized. The transmon can occupy highly excited states which resemble charge states, or equivalently, plane waves in the phase representation. A classical analogy would be a strongly driven rotor which rotates indefinitely, making turns around its anchoring point (fig. 3.3b). When occupying such a state, the transmon behaves like a free particle which is almost not affected by the cosine potential. Thus the Josephson energy can be set to zero in Eq. (3.10). The Hamiltonian is now that of a superconducting island capacitively coupled to a resonator, with no Josephson junction. The island-resonator coupling term commutes with the Hamiltonian of the island, which consists of the charging energy only. The coupling is now longitudinal and therefore the cavity frequency is fixed to the bare frequency !a, independently of the island state.

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Table of contents :

1 Introduction 
1.1 Quantum information
1.2 Quantum error correction
1.2.1 Classical error correction
1.2.2 Bit-flip Correction
1.3 Classical computer memories
1.3.1 Dynamic RAM
1.3.2 Static RAM
1.3.3 Conclusions
1.4 Bistable systems in classical mechanics
1.4.1 Driven oscillator
1.4.2 Parametric oscillator
1.5 Superconducting circuits
1.6 Outline
2 Hamiltonian and dissipation engineering with superconducting circuits
2.1 Circuit quantization
2.1.1 Circuit description
2.1.2 Equations of motion
2.1.3 Quantization
2.1.4 AC Biases
2.2 Engineering dynamics
2.2.1 Hamiltonian engineering
2.2.2 Dissipation engineering
3 Parametric pumping with a transmon 
3.1 The Transmon qubit
3.1.1 Low energy behaviour
3.1.2 Full description
3.2 Quantum dynamics of a driven transmon
3.2.1 Description of the experiment
3.2.2 Observing the transmon escape into unconfined states .
3.2.3 Effect of pump-induced quasiparticles
3.3 Additional Material
3.3.1 Sample fabrication
3.3.2 Photon number calibration
3.3.3 Multi-photon qubit drive
3.3.4 Charge noise
3.3.5 Numerical simulation
3.3.6 Quasiparticle generation
4 Exponential bit-flip suppression in a cat-qubit 
4.1 Protecting quantum information in a resonator
4.1.1 General considerations
4.1.2 The bosonic codes
4.2 Exponential bit-flip suppression in a cat-qubit
4.2.1 The dissipative cat-qubits
4.2.2 Engineering two-photon coupling to a bath
4.2.3 Coherence time measurements
4.3 Additional Material
4.3.1 Full device and wiring
4.3.2 Hamiltonian derivation
4.3.3 Circuit parameters
4.3.4 Semi-classical analysis
4.3.5 Bit-flip time simulation
4.3.6 Tuning the cat-qubit
5 Itinerant microwave photon detection 
5.1 Dissipation engineering for photo-detection
5.1.1 Engineering the Qubit-Cavity dissipator
5.1.2 Efficiency
5.1.3 Dark Counts
5.1.4 Detector Reset
5.1.5 QNDness
5.2 Additional Material
5.2.1 Circuit parameters
5.2.2 Purcell Filter
5.2.3 System Hamiltonian derivation
5.2.4 Adiabatic elimination of the waste mode
5.2.5 Qubit dynamics and detection efficiency
5.2.6 Reset protocol
5.2.7 Spectroscopic characterization of the detector
5.2.8 Tomography of the itinerant transmitted photon
Conclusion and perspectives
A Circuit quantization 
A.1 LC-oscillator
A.2 General Circuit quantization
B Native couplings in superconducting circuits 
B.1 Capacitively coupled resonators
B.2 Mutual inductances in superconducting loops
B.3 Inductively coupled resonators
C Useful Formulae 
C.1 Miscellaneous
C.2 Standard changes of frame
D Classical mechanical analogy to the stabilisation of cat-qubits 
E Fabrication Recipes 
E.1 Wafer preparation
E.2 Circuit patterning
E.3 Junction fabrication


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