Schroedinger Cat States Research Articles

Homodyne versus photon-counting quantum trajectories for dissipative Kerr resonators with two-photon driving

We investigate two different kinds of quantum trajectories for a nonlinear photon resonator subject to two-photon pumping, a configuration recently studied for the generation of photonic Schroedinger cat states. In the absence of feedback control and in the strong-driving limit, the steady-state density matrix is a statistical mixture of two states with equal weight. While along a single photon-counting trajectory the systems intermittently switches between an odd and an even cat state, we show that upon homodyne detection the situation is different. Indeed, homodyne quantum trajectories reveal switches between coherent states of opposite phase.

Enlargement of optical Schrödinger's cat states

Superpositions of macroscopically distinct quantum states, introduced in Schroedinger's famous Gedankenexperiment, are an epitome of quantum "strangeness" and a natural tool for determining the validity limits of quantum physics. The optical incarnation of Schroedinger's cat - the superposition of two opposite-amplitude coherent states - is also the backbone of quantum information processing in the continuous-variable domain. However, existing preparation methods limit the amplitudes of the component coherent states by about 2, which curtails the state's usefulness for fundamental and practical applications. Here we produce higher-amplitude optical Schroedinger's cats from two such states of lower amplitudes. The protocol consists in bringing the initial states into interference on a beamsplitter and a subsequent heralding quadrature measurement in one of the output channels. In the experiment, we convert a pair of negative squeezed Schroedinger's cat states of amplitude 1.25 to a single positive Schroedinger's cat of amplitude 2.15 with success probability of ~0.2. This amplitude is comparable to the highest values obtained for this state in any physical system. Our technique can be realized in an iterative manner, in principle allowing creation of Schroedinger's cat states of arbitrarily high amplitude.

Applications of the Fokker-Planck equation in circuit quantum electrodynamics

We study exact solutions of the steady state behaviour of several non-linear open quantum systems which can be applied to the field of circuit quantum electrodynamics. Using Fokker-Planck equations in the generalised P-representation we investigate the analytical solutions of two fundamental models. First, we solve for the steady-state response of a linear cavity that is coupled to an approximate transmon qubit and use this solution to study both the weak and strong driving regimes, using analytical expressions for the moments of both cavity and transmon fields, along with the Husimi Q-function for the transmon. Second, we revisit exact solutions of quantum Duffing oscillator which is driven both coherently and parametrically while also experiencing decoherence by the loss of single and pairs of photons. We use this solution to discuss both stabilisation of Schroedinger cat states and the generation of squeezed states in parametric amplifiers, in addition to studying the Q-functions of the different phases of the quantum system. The field of superconducting circuits, with its strong nonlinearities and couplings, has provided access to a new parameter regime in which returning to these exact quantum optics methods can provide valuable insights.

Observation of quantum state collapse and revival due to the single-photon Kerr effect

To create and manipulate non-classical states of light for quantum information protocols, a strong, nonlinear interaction at the single-photon level is required. One approach to the generation of suitable interactions is to couple photons to atoms, as in the strong coupling regime of cavity quantum electrodynamic systems. In these systems, however, the quantum state of the light is only indirectly controlled by manipulating the atoms. A direct photon-photon interaction occurs in so-called Kerr media, which typically induce only weak nonlinearity at the cost of significant loss. So far, it has not been possible to reach the single-photon Kerr regime, in which the interaction strength between individual photons exceeds the loss rate. Here, using a three-dimensional circuit quantum electrodynamic architecture, we engineer an artificial Kerr medium that enters this regime and allows the observation of new quantum effects. We realize a gedanken experiment in which the collapse and revival of a coherent state can be observed. This time evolution is a consequence of the quantization of the light field in the cavity and the nonlinear interaction between individual photons. During the evolution, non-classical superpositions of coherent states (that is, multi-component 'Schrödinger cat' states) are formed. We visualize this evolution by measuring the Husimi Q function and confirm the non-classical properties of these transient states by cavity state tomography. The ability to create and manipulate superpositions of coherent states in such a high-quality-factor photon mode opens perspectives for combining the physics of continuous variables with superconducting circuits. The single-photon Kerr effect could be used in quantum non-demolition measurement of photons, single-photon generation, autonomous quantum feedback schemes and quantum logic operations.

Impurity in a Bose-Einstein condensate in a double well

We compare and contrast the mean-field and many-body properties of a Bose-Einstein condensate trapped in a double well potential with a single impurity atom. The mean-field solutions display a rich structure of bifurcations as parameters such as the boson-impurity interaction strength and the tilt between the two wells are varied. In particular, we study a pitchfork bifurcation in the lowest mean-field stationary solution which occurs when the boson-impurity interaction exceeds a critical magnitude. This bifurcation, which is present for both repulsive and attractive boson-impurity interactions, corresponds to the spontaneous formation of an imbalance in the number of particles between the two wells. If the boson-impurity interaction is large, the bifurcation is associated with the onset of a Schroedinger cat state in the many-body ground state. We calculate the coherence and number fluctuations between the two wells, and also the entanglement entropy between the bosons and the impurity. We find that the coherence can be greatly enhanced at the bifurcation.

Physical realization of quantum teleportation for a nonmaximal entangled state

Recently, Kossakowski and Ohya (K-O) proposed a new teleportation scheme which enables perfect teleportation even for a nonmaximal entangled state [A. Kossakowski and M. Ohya, Infinite Dimensional Analysis Quantum Probability and Related Topics 10, 411 (2007)]. To discuss a physical realization of the K-O scheme, we propose a model based on quantum optics. In our model, we take a superposition of Schroedinger's cat states as an input state being sent from Alice to Bob, and their entangled state is generated by a photon number state through a beam splitter. When the average photon number for our input states is equal to half the number of photons into the beam splitter, our model has high fidelity.

Reversal of the circulation of a vortex by quantum tunneling in trapped Bose systems

We study the quantum dynamics of a model for a vortex in a Bose gas with repulsive interactions in an anisotropic, harmonic trap. By solving the Schroedinger equation numerically, we show that the circulation of the vortex can undergo periodic reversals by quantum-mechanical tunneling. With increasing interaction strength or particle number, vortices become increasingly stable, and the period for reversals increases. Tunneling between vortex and antivortex states is shown to be described to a good approximation by a superposition of vortex and antivortex states (Schroedinger cat state), rather than the mean-field state, and we derive an analytical expression for the oscillation period. The problem is shown to be equivalent to that of the two-site Bose-Hubbard model with attractive interactions.

Fourier multiport devices

A formalism for multiport devices is developed in which the creation and annihilation operators at the output are related with those at the input through a finite Fourier transform. By considering two specific inputs of number states and coherent states, we show how these devices can be used to produce multimode Schroedinger cat states and can be combined to create multipath interferometers. We introduce uncertainty relations for the distributions of particles in the input modes and the correlations between the output modes. These allow us to determine how the phase resolution of the output state scales with the number of ports and particles. This scaling is of fundamental interest for schemes that seek to use multipath interferometers for enhanced measurement precision.

Effect of initial correlations on short-time decoherence

We study the effect of initial correlations on the short-time decoherence of a particle linearly coupled to a bath of harmonic oscillators. We analytically evaluate the attenuation coefficient of a Schroedinger cat state both for a free and a harmonically bound particle, with and without initial thermal correlations between the particle and the bath. While short-time decoherence appears to be independent of the system in the absence of initial correlations, we find on the contrary that, for initial thermal correlations, decoherence becomes system dependent even for times much shorter than the characteristic time of the system. The temperature behavior of this system dependence is discussed.

Schroedinger cat states and optimum universal quantum cloning by entangled parametric amplification

The new process of quantum-injection by a single-photon in a pure quantum superposition state into an optical parametric amplifier operating in entangled configuration is adopted to generate an all optical multiphoton Schroedinger-cat state which is largely detectable against the squeezed-vacuum noise. The invariance properties of the OPA interaction hamiltonian show that, under certain conditions, the device may act as a universal quantum cloning machine (UQCM) of the input qubits. Preliminary results here reported show the first experimental realization of such a device based on stimulated emission process in an optical parametric amplifier.

Recovering coherence from decoherence: A method of quantum-state reconstruction

We present a feasible scheme for reconstructing the quantum state of a field prepared inside a lossy cavity. Quantum coherences are normally destroyed by dissipation, but we show that at zero temperature we are able to retrieve enough information about the initial state, making possible to recover its Wigner function as well as other quasiprobabilities. We provide a numerical simulation of a Schroedinger cat state reconstruction.

Sampling quantum phase space with squeezed states

We study the application of squeezed states in a quantum optical scheme for direct sampling of the phase space by photon counting. We prove that the detection setup with a squeezed coherent probe field is equivalent to the probing of the squeezed signal field with a coherent state. An example of the Schr odinger cat state measurement shows that the use of squeezed states allows one to detect clearly the interference between distinct phase space components despite losses through the unused output port of the setup.

Planning of nonclassical vibrational states

Methods are analyzed to find the pulse shape and chirp of the exciting light pulse to generate a given vibrational state of a molecule during Franck-Condon transitions. An example is given for generation of vibrational Schroedinger-cat states. Their remarkable feature is that they have simultaneously two distinguishable distances of the atoms in a diatomic molecule due to quantum mechanical superposition.