Effective Generation of Cat and Kitten States

The optical Schrödinger’s cat is highly anticipated for its potential to realize fault-tolerant quantum computing and quantum metrology. To attain a good performance in these processes, the amplitude of cat states is desired to be as high as possible. Here, we demonstrate the enlargement of a cat state through the superposition of two kitten states in coupled waveguides. Under reasonable superposition coefficients, the interference between two nearby coherent state components in the initial kittens will produce a coherent-like state with a larger amplitude, enabling the creation of larger cat-like states. Based on the above mechanism, we theoretically demonstrate the superposition of two kittens in coupled waveguides through conditional measurements. In an appropriate propagation length, a cat state, whose amplitude is enlarged beyond α02+β02, is achieved with high fidelity and a high success probability when the amplitudes of input kittens are α0 and iβ0, respectively. The mechanism we propose has the potential to be realized in other systems, and it could be extended to amplify other macroscopic quantum states. The physical realization based on coupled waveguides demonstrates the ability to efficiently amplify cat states and is expected to become a competitive approach for on-chip cat state preparation.

Recently, using conditioning approaches on the high-harmonic generation process induced by intense laser-atom interactions, we have developed a new method for the generation of optical Schrödinger cat states (Lewenstein et al. in Nat Phys, 17 1104–1108, 2021. https://doi/10.1038/s41567-021-01317-w). These quantum optical states have been proven to be very manageable as, by modifying the conditions under which harmonics are generated, one can interplay between kitten and genuine cat states. Here, we demonstrate that this method can also be used for the development of new schemes towards the creation of optical Schrödinger cat states, consisting of the superposition of three distinct coherent states. Apart from the interest these kind of states have on their own, we additionally propose a scheme for using them towards the generation of large cat states involving the sum of two different coherent states. The quantum properties of the obtained superpositions aim to significantly increase the applicability of optical Schrödinger cat states for quantum technology and quantum information processing.

Quantum superposition is a fundamental principle of quantum mechanics, which provides a crucial basis to observe phenomena beyond the predictions of classical physics. For example, a quantum entangled state can exhibit stronger correlation than classically possible one. In quantum state engineering, many new quantum states can be obtained from the superposition of many known states. In recent decades, the superposition of coherent states (CSs) with the same amplitude but two different phases has been a subject of great interest. This superposition state was often called Schrodinger cat state (here, we also name it 2-headed cat state (2HCS)), which becomes an important tool to study a lot of fundamental issues. Surprisingly, some studies have extended the quantum superposition to involving more than two component coherent states. In order to produce the superposition of three photons, people have considered the superposition of coherent states with three different phases (here, we also name it 3-headed cat state (3HCS)). Furthermore, in microwave cavity quantum electrodynamics of bang-bang quantum Zeno dynamics control, people have proposed the superposition of coherent states with four different phases (here, we also name it 4-headed cat state (4HCS)). In this paper, we make a detailed investigation on the quantum statistical properties of a phase-type 3HCS. These properties include photon number distribution, average photon number, sub-Poissionian distribution, squeezing effect, and Wigner function, etc. We derive their analytical expressions and make numerical simulations for these properties. The results are compared with the counterparts of the CS, the 2HCS and the 4HCS. The conclusions are obtained as follows. 1) The CS, the 2HCS, the 3HCS and the 4HCS have k, 2k, 3k and 4k photon number components, respectively (k is an integer); 2) small difference in average photon number among these quantum states in small-amplitude range can be observed, while their average photon numbers become almost equal in large-amplitude range; 3) the CS exhibits Poisson distribution, and the 2HCS, the 3HCS and the 4HCS exhibit super-Poisson distributions in most amplitude ranges, however, sub-Poisson distribution can be seen for the 3HCS and the 4HCS in some specific amplitude ranges; 4) except for the 2HCS that may have the squeezing property, no squeezing properties can be found in the CS, the 3HCS and the 4HCS; 5) negative values can exist in the Wigner functions for the 2HCS, the 3HCS and the 4HCS, while it is not found in the CS. Similar to the 2HCS and 4HCS, the Wigner function of the 3HCS has negative component, which implies its nonclassicality. Different from the 2HCS, the 3HCS exhibits sub-Poisson photon number distribution in a certain amplitude range, it is weaker than that of the 4HCS. At the same time, no squeezing is found in the 3 or 4HCS, which is another difference from the 2HCS.

Disentanglement and loss of quantum correlations due to one global collective noise effect are described for two-qubit Schrödinger cat and Werner states of a four level trapped ion quantum system. Once the Jaynes–Cummings ionic interactions are mapped onto a Dirac spinor structure, the elementary tools for computing quantum correlations of two-qubit ionic states are provided. With two-qubit quantum numbers related to the total angular momentum and to its projection onto the direction of an external magnetic field (which lifts the degeneracy of the ion’s internal levels), a complete analytical profile of entanglement for the Schrödinger cat and Werner states is obtained. Under vacuum noise (during spontaneous emission), the two-qubit entanglement in the Schrödinger cat states is shown to vanish asymptotically. Otherwise, the robustness of Werner states is concomitantly identified, with the entanglement content recovered by their noiseless-like evolution. Most importantly, our results point to a firstly reported sudden transition between classical and quantum decay regimes driven by a classical collective noise on the Schrödinger cat states, which has been quantified by the geometric discord.

We proposed and analyzed a scheme to generate large-size Schr\"odinger cat states based on linear operations of Fock states, squeezed vacuum states, and conditional measurements. By conducting conditional measurements via photon number detectors, two unbalanced small-amplitude Schr\"odinger kitten states combined by a beam splitter can be amplified to a large-size cat state with the same parity. According to simulation results, two Schr\"odinger odd kitten states with amplitudes of $|\ensuremath{\beta}|=1.06$ and $|\ensuremath{\beta}|=1.11$ generated from one-photon-subtracted 3 dB squeezed vacuum states, are amplified to an odd cat state of $|\ensuremath{\beta}|=1.73$ with a fidelity of $F=99%$. A large-size Schr\"odinger odd cat state with $|\ensuremath{\beta}|=2.51$ and $F=97.30%$ is predicted when 5.91 dB squeezed vacuum states are employed. According to the analysis on the impacts of experimental imperfections in practice, Schr\"odinger odd cat states of $|\ensuremath{\beta}|>2$ are available. A feasible configuration based on a quantum frequency comb is developed to realize the large-size cat state generation scheme we proposed.

Schrödinger’s cat originates from the famous thought experiment querying the counterintuitive quantum superposition of macroscopic objects. As a natural extension, several “cats” (quasi-classical objects) can be prepared into coherent quantum superposition states, which is known as multipartite cat states demonstrating quantum entanglement among macroscopically distinct objects. Here, we present a highly scalable approach to deterministically create flying multipartite Schrödinger’s cat states by reflecting coherent-state photons from a microwave cavity containing a superconducting qubit. We perform full quantum state tomography on the cat states with up to four photonic modes and confirm the existence of quantum entanglement among them. We also witness the hybrid entanglement between discrete-variable states (the qubit) and continuous-variable states (the flying multipartite cat) through a joint quantum state tomography. Our work provides an enabling step for implementing a series of quantum metrology and quantum information processing protocols based on cat states.

We show that in some cases the coherent state can have a larger violation of the Leggett-Garg inequality (LGI) than the cat state by numerical calculations. To achieve this result, we consider the LGI of the cavity mode weakly coupled to a zero-temperature environment as a practical instance of the physical system. We assume that the bosonic mode undergoes dissipation because of an interaction with the environment but is not affected by dephasing. Solving the master equation exactly, we derive an explicit form of the violation of the inequality for both systems prepared initially in the coherent state $|\alpha\rangle$ and the cat state $(|\alpha\rangle+|-\alpha\rangle)$. For the evaluation of the inequality, we choose the displaced parity operators characterized by a complex number $\beta$. We look for the optimum parameter $\beta$ that lets the upper bound of the inequality be maximum numerically. Contrary to our expectations, the coherent state occasionally exhibits quantum quality more strongly than the cat state for the upper bound of the violation of the LGI in a specific range of three equally spaced measurement times (spacing $\tau$). Moreover, as we let $\tau$ approach zero, the optimized parameter $\beta$ diverges and the LGI reveals intense singularity.

We investigate the effect of giant negative magnetoresistance in ultrahigh-mobility (μ≫107cm2V−1s−1) two-dimensional electron systems. These systems present a dramatic drop in the mangetoresistance at low magnetic fields (B∼0.1 T) and temperatures (T∼0.1 K). This effect is reversed by increasing the temperature or the presence of an in-plane magnetic field. The motivation for the present work is to develop a microscopical model to explain the experimental evidence, based on coherent states and Schródinger cat states of the quantum harmonic oscillator. Thus, we approach the giant negative magnetoresistance effect based on the description of ultrahigh-mobility two-dimensional electron systems in terms of Schrödinger cat states (superposition of coherent states of the quantum harmonic oscillator). We explain the experimental results in terms of the increasing disorder in the sample due to the rising temperature or the in-plane magnetic field, breaking up the Schrödinger cat states and giving rise to mere coherent states, which hold magnetoresistance in lower-mobility samples. The latter, jointly with the description of ultrahigh-mobility samples with Schrödinger cat states, accounts for the main contribution. The most interesting application of this novel description of such systems would be in the implementation of qubits for quantum computing based on bosonic models.

Schrödinger's cat states and their nonlinear solitonic analogues

A method for producing macroscopic quantum superposition states (generally known as Schr\"odinger cat) states for optical fields is presented. The proposed method involves two modes of the field interacting dispersively in a Kerr medium where one of the modes is an arm of a Mach-Zehnder interferometer and the other mode is external to it. If the external mode initially contains a macroscopic quantum state, such as a coherent state, and the vacuum and a single photon state are the inputs to the interferometer, the external field state becomes entangled with the number states associated with the two paths of the interferometer. Selective measurement at the output ports of the interferometer project the external mode into the desired cat states. It is pointed out that the method can also be used to generate cat states out of multimode states initially containing correlations.

In this paper we study the entanglement in symmetric N-quDit systems. In particular we use generalizations to U(D) of spin U(2) coherent states (CSs) and their projections on definite parity c∈Z2D−1 (multicomponent Schrödinger cat) states and we analyse their reduced density matrices when tracing out M < N quDits. The eigenvalues (or Schmidt coefficients) of these reduced density matrices are completely characterized, allowing to prove a theorem for the decomposition of a N-quDit Schrödinger cat state with a given parity c into a sum over all possible parities of tensor products of Schrödinger cat states of N − M and M particles. Diverse asymptotic properties of the Schmidt eigenvalues are studied and, in particular, for the (rescaled) double thermodynamic limit ( N,M→∞,M/N fixed), we reproduce and generalize to quDits known results for photon loss of parity adapted CSs of the harmonic oscillator, thus providing an unified Schmidt decomposition for both multi-quDits and (multi-mode) photons. These results allow to determine the entanglement properties of these states and also their decoherence properties under quDit loss, where we demonstrate the robustness of these states.

A cat-state is a superposition of two coherent states with amplitudes\n$\\alpha_{0}$ and $-\\alpha_{0}$. Recent experiments create cat states in a\nmicrowave cavity field using superconducting circuits. As with degenerate\nparametric oscillation (DPO) in an adiabatic and highly nonlinear limit, the\nstates are formed in a signal cavity mode via a two-photon dissipative process\ninduced by the down conversion of a pump field to generate pairs of signal\nphotons. The damping of the signal and the presence of thermal fluctuations\nrapidly decoheres the state, and the effect on the dynamics is to either\ndestroy the possibility of a cat state, or else to sharply reduce the lifetime\nand size of the cat-states that can be formed. In this paper, we study the\neffect on both the DPO and microwave systems of a squeezed reservoir coupled to\nthe cavity. While the threshold nonlinearity is not altered, we show that the\nuse of squeezed states significantly lengthens the lifetime of the cat states.\nThis improves the feasibility of generating cat states of large amplitude and\nwith a greater degree of quantum macroscopic coherence, which is necessary for\nmany quantum technology applications. Using current experimental parameters for\nthe microwave set-up, which requires a modified Hamiltonian, we further\ndemonstrate how squeezed states enhance the quality of the cat states that\ncould be formed in this regime. Squeezing also combats the significant\ndecoherence due to thermal noise, which is relevant for microwave fields at\nfinite temperature. By modeling a thermal squeezed reservoir, we show that the\nthermal decoherence of the dynamical cat states can be inhibited by a careful\ncontrol of the squeezing of the reservoir. To signify the quality of the cat\nstate, we consider different signatures including fringes and negativity, and\nthe $C_{l_{1}}$ measure of quantum coherence.\n

A system of two-level noninteracting atoms driven by superposition of two Glauber coherent photonic states (a cat state) is studied. The field state is continuously restored by a source explicitly incorporated into the model. Due to its nature, the cat state changes phase by $\ensuremath{\pi}$ upon stimulated excitation of any atom, a peculiar kind of coherent quantum feedback. That results in correlations between photonic and atomic subsystems. In the limit of a strong field, the ansatz for the system's density matrix is proposed and an approximate analytical solution to the master equation is obtained in the case of a large number of atoms and slow spontaneous emission. Based on this solution, the steady-state second-order correlation function of atomic photoemissions is evaluated and investigated. The results demonstrate a remarkable difference from the case of the classical field (i.e., the field in a Glauber coherent state).

We explore the non-classicality and precision limits of coherent superposition mechanical states generated in a Mach-Zehnder interferometer (MZI). In this setup, a photonic NOON state interacts with a mechanical oscillator, producing such coherent superposition states that give rise to Schrödinger cat (SC) states at specific times and phase differences. The non-classicality is quantified via the negativity volume (NV) of the Wigner function, and the precision limit of the mechanical state is characterized by the quantum Fisher information (QFI). We observe that NV and QFI exhibit correlated trends at selected times and phase differences, indicating that higher non-classicality may enhance the precision limits. Notably, for certain instances, QFI can remain high even when fidelity between the coherent superposition state and the target SC state is low, indicating that conditional states can encode substantial phase information. Using parity detection, we demonstrate that the quantum Cramér-Rao bound (CRB) can be saturated, reaching the Heisenberg limit (HL) for certain evolution times and confirming it as the optimal detection strategy. These results provide insights into the quantum dynamics of optomechanical systems for high-precision measurements and enable the creation of high-fidelity SC states.

The evolution of the input coherent state in the Kerr medium is studied. Upon the derivation we know that the output state at time t which corresponds to a rational number is Schrodinger's cat state. Using Fourier integral of the evolution operator, we obtain the integral expression of the output state at time t which corresponds to an irrational number. It is a kind of one-dimensional continuous superposition of coherent states, not an ordinary Schrodinger's cat state.