The purpose of this Bachelor thesis is to study methods to engineer Schrödinger’s cat states in microwave resonators, as proposed in the paper Engineering quantum states of light by S. Puri et al. The thesis starts by introducing relevant notions and concepts. Thereafter, it provides several analytical as well as numerical derivations of the main equations and results in the paper by S. Puri et al. The Kerr non-linear resonator (KNR) is explored as a means to adiabatically generate cat states. The Hamiltonian of the KNR is modified by detuning and orthogonal drive modes to achieve faster initialisation. Some images are also replicated with Matlab and the Python toolbox QuTiP. Finally, we also construct a number of quantum gates using coherent states.
The purpose of this Bachelor thesis is to study methods to engineer Schrödinger’s cat states in microwave resonators, as proposed in the paper Engineering quantum states of light by S. Puri et al. The thesis starts by introducing relevant notions and concepts. Thereafter, it provides several analytical as well as numerical derivations of the main equations and results in the paper by S. Puri et al. The Kerr non-linear resonator (KNR) is explored as a means to adiabatically generate cat states. The Hamiltonian of the KNR is modified by detuning and orthogonal drive modes to achieve faster initialisation. Some images are also replicated with Matlab and the Python toolbox QuTiP. Finally, we also construct a number of quantum gates using coherent states.
Abstract Macroscopic cat states have been widely studied to illustrate fundamental principles of quantum physics as well as their applications in quantum information processing. In this paper, we propose a quantum speed-up method for the creation of cat states in a Kerr nonlinear resonator (KNR) via optimal adiabatic control. By simultaneously adiabatic tuning the cavity-field detuning and driving field strength, the width of the minimum energy gap between the target trajectory and non-adiabatic trajectory can be widened, which allows us to accelerate the evolution along the adiabatic path. Compared with the previous proposal, preparing cat states only by controlling two-photon pumping strength, our method can prepare the target state with a shorter time, a high-fidelity and a large non-classical volume. It is worth noting that the cat state prepared here is also robust against single-photon loss. Moreover, when we consider the KNR with a large initial detuning, our proposal will create a large-size cat state successfully. This proposal for preparing cat states can be implemented in superconducting quantum circuits, which provides a quantum state resource for quantum information encoding and fault-tolerant quantum computing.
Superconducting circuits provide an architecture upon which cavity quantum electrodynamics (QED) can be implemented at microwave frequencies in a highly tunable environment. Known as circuit QED, these systems can achieve larger nonlinearities, stronger coupling and greater controllability than can be achieved in cavity QED, all in a customisable, solid state device, making this technology an exciting test bed for both quantum optics and quantum information processing. These new parameter regimes open up new avenues for quantum technology, while also allowing older quantum optics results to finally be tested. In particular is is now possible to experimentally produce nonclassical states, such as squeezed and Schr\\"odinger cat states, relatively simply in these devices. Using open quantum systems methods, in this thesis we investigate four problems which involve the use of nonclassical states in circuit QED. First we investigate the effects of a Kerr nonlinearity on the ability to preserve transported squeezed states in a superconducting cavity, and whether this setup permits us to generate, and perform tomography, of a highly squeezed field using a qubit, with possible applications in the characterisation of sources of squeezed microwaves. Second, we present a novel scheme for the amplification of cat states using a coupled qubit and external microwave drives, inspired by the stimulated Raman adiabatic passage. This scheme differs from similar techniques in circuit QED in that it is deterministic and therefore compatible with a protocol for stabilising cat states without the need for complex dissipation engineering. Next we use solutions of Fokker-Planck equations to study the exact steady-state response of two nonlinear systems: a transmon qubit coupled to a readout resonator, where we find good agreement with experiments and see simultaneous bistability of the cavity and transmon; and a parametrically driven nonlinear resonator, where we compare the classical and quantum phases of the system and discuss applications in the generation of squeezed states and stabilisation of cat states. Finally, we investigate the use of two different types of superconducting qubits in a single experiment, seeing that this enables engineering of the self- and cross-Kerr effects in a line of cavities. This could provide a valuable means of entangling cavity states, in addition to a resource for quantum simulation.
Abstract Quantum superpositions of macroscopically distinct classical states—so-called Schrödinger cat states—are a resource for quantum metrology, quantum communication and quantum computation. In particular, the superpositions of two opposite-phase coherent states in an oscillator encode a qubit protected against phase-flip errors1, 2. However, several challenges have to be overcome for this concept to become a practical way to encode and manipulate error-protected quantum information. The protection must be maintained by stabilizing these highly excited states and, at the same time, the system has to be compatible with fast gates on the encoded qubit and a quantum non-demolition readout of the encoded information. Here we experimentally demonstrate a method for the generation and stabilization of Schrödinger cat states based on the interplay between Kerr nonlinearity and single-mode squeezing1, 3 in a superconducting microwave resonator4. We show an increase in the transverse relaxation time of the stabilized, error-protected qubit of more than one order of magnitude compared with the single-photon Fock-state encoding. We perform all single-qubit gate operations on timescales more than sixty times faster than the shortest coherence time and demonstrate single-shot readout of the protected qubit under stabilization. Our results showcase the combination of fast quantum control and robustness against errors, which is intrinsic to stabilized macroscopic states, as well as the potential of these states as resources in quantum information processing5–8.
Schrödinger cat states, quantum superpositions of macroscopically distinct classical states, are an important resource for quantum communication, quantum metrology and quantum computation. Especially, cat states in a phase space protected against phase-flip errors can be used as a logical qubit. However, cat states, normally generated in three-dimensional cavities and/or strong multi-photon drives, are facing the challenges of scalability and controllability. Here, we present a strategy to generate and preserve cat states in a coplanar superconducting circuit by the fast modulation of Kerr nonlinearity. At the Kerr-free work point, our cat states are passively preserved due to the vanishing Kerr effect. We are able to prepare a 2-component cat state in our chip-based device with a fidelity reaching 89.1% under a 96 ns gate time. Our scheme shows an excellent route to constructing a chip-based bosonic quantum processor.
In this thesis work, after a brief introduction of some basic concepts of quantum optics, some experimental techniques useful for the manipulation of quantum states of light are presented. In particular, I focused on two transformations operated at the single-photon level, like the addition and the subtraction of a single photon from a travelling light beam. Both these methods share a probabilistic approach, that is their implementation has a non unitary success probability. Anyway, the detection of an additional photon in an ancillary mode exactly announces when the operation has been actually implemented. That's why this type of implementation is generally called heralded. Starting from these basic operations, a way to implement some of their superpositions is then analyzed. In the second part of this thesis, I describe two state-engineering experiments based on these fundamental operations. The first one realizes a so-called universal orthogonalizer, an universal strategy that, given an arbitrary input state, can be used to produce at the output an orthogonal one. The adjective universal refers to the fact that the procedure works equally well for arbitrary input fields. The other experiment uses coherent superpositions of basic quantum operations to emulate a strong Kerr nonlinearity at the few-photon level, and could be used to implement new logic gates for quantum computation. The third part of this thesis is dedicated to a different experimental approach, in some sense complementary to state engineering: mode engineering. Indeed, while in state engineering the purpose is the detection of an unknown state after its manipulation, in mode engineering experiments the perspective is completely reversed: the final state is assumed to be known and what has to be inferred are its spatio-temporal features. By following this alternative experimental approach, the interaction of broadband single photons with resonant atomic vapours is analyzed by observing the traces left on the photon temporal shape.
Abstract Photonic cat states stored in high-Q resonators show great promise for hardware efficient universal quantum computing. We propose an approach to efficiently prepare such cat states in a Kerr-nonlinear resonator by the use of a two-photon drive. Significantly, we show that this preparation is robust against single-photon loss. An outcome of this observation is that a two-photon drive can eliminate undesirable phase evolution induced by a Kerr nonlinearity. By exploiting the concept of transitionless quantum driving, we moreover demonstrate how non-adiabatic initialization of cat states is possible. Finally, we present a universal set of quantum logical gates that can be performed on the engineered eigenspace of such a two-photon driven resonator and discuss a possible realization using superconducting circuits. The robustness of the engineered subspace to higher-order circuit nonlinearities makes this implementation favorable for scalable quantum computation.