NOON States Research Articles

Collapse in ultracold Bose Josephson junctions

We investigate how ultracold atoms in double-well potentials can be used to study and put bounds on models describing wave-function collapse. We refer in particular to the continuous spontaneous localization (CSL) model, which is the most well studied among dynamical reduction models. It modifies the Schr\"odinger equation in order to include the collapse of the wave function in its dynamics. We consider Bose Josephson junctions, where ultracold bosons are trapped in a double-well potential, since they can be experimentally controlled with high accuracy and are suited and used to study macroscopic quantum phenomena on a scale of microns, with a number of particles typically ranging from $\ensuremath{\sim}{10}^{2}--{10}^{3}$ to $\ensuremath{\sim}{10}^{5}--{10}^{6}$. We study the CSL dynamics of three atomic states showing macroscopic quantum coherence: the atomic coherent state, the superposition of two atomic coherent states, and the NOON state. We show that for the last two states, the suppression of quantum coherence induced by the CSL model increases exponentially with the number of atoms. We observe that in the case of optically trapped atoms, the spontaneous photon emission of the atoms induces a dynamics similar to the CSL one, and we conclude that magnetically trapped atoms may be more convenient to experimentally test the CSL model. Finally, we discuss decoherence effects in order to provide reasonable estimates on the bounds that it is (or will be) possible to obtain for the parameters of the CSL model in such class of experiments. As an example, we show that a NOON state with $N\ensuremath{\sim}{10}^{3}$ with a coherence time of $\ensuremath{\sim}1$ s can constrain the CSL parameters in a region where the other systems presently cannot.

Deterministic generations of photonic NOON states in cavities via shortcuts to adiabaticity

NOON states play the important roles in quantum information processings and quantum metrology, but the fidelities of these states generated previously are limited typically by the practically-unavoidable decoherence and operational imperfections. Here, we propose an efficient scheme to generate photonic NOON states alternatively by rapid population passage technique via shortcut to adiabaticity (STA), rather than the usual Rabi oscillations. Since the deterministic population passages based on the STAs are insensitive to details of the operations and can be implemented as fast as the Rabi oscillations, the fidelity of the generated NOON state could be satisfactorily high. The feasibility of the proposal is demonstrated specifically with the experimental circuit QED systems by rapidly driving two artifical qutrits.

Quantum multiparameter estimation with generalized balanced multimode NOON-like states

The simultaneous multi-parameter estimation problem using a class of multi-mode entangled states is investigated in this paper. Specifically, the problem of optical phase imaging is considered and the quantum probe is taken to be a balanced coherent superposition of components with an arbitrary quantum state in one mode and vacuum states in all the other modes, which is a generalization of the multi-mode NOON state. The analytical form for the quantum Cramer-Rao bound (QCRB) is presented, which shows the performance by providing a lower bound of the estimation uncertainty. It is shown that the NOON state has the worst performance among those in the class of the entangled states considered. We also analyze in detail four different scenarios, which are the NOON state, the entangled coherent state, the entangled squeezed coherent state, and the entangled squeezed vacuum state. From the comparison among these four states, we find that when the mean photon number is fixed, the squeezed vacuum state has the smallest QCRB, followed by the squeezed coherent state, entangled coherent state, and NOON state. We also illustrate that the balanced entangled state can perform better than a more generalized unbalanced form studied in previous works for certain scenarios. Finally, we give an experimental setup for producing a two-mode entangled state that can beat the NOON state in quantum metrology.

Generating double NOON states of photons in circuit QED

To generate a NOON state with a large photon number $N$, the number of operational steps could be large and the fidelity will decrease rapidly with $N$. Here we propose a method to generate a new type of quantum entangled states, $(|NN00\rangle+|00NN\rangle)/\sqrt{2}$ called "double NOON" states, with a setup of two superconducting flux qutrits and five circuit cavities. This scheme operates essentially by employing a two-photon process, i.e., two photons are simultaneously and separately emitted into two cavities when each coupler qutrit is initially in a higher-energy excited state. As a consequence, the "double" NOON state creation needs only $N$+2 operational steps. One application of double NOON states is to get a phase error of $1/(2N)$ in phase measurement. In comparison, to achieve the same error, a normal NOON state of the form $(|2N,0\rangle+|0,2N\rangle)/\sqrt{2}$ is needed, which requires at least $2N$ operational steps to prepare by using the existing schemes. Our numerical simulation demonstrates that high-fidelity generation of the double NOON states with $N\leq 10$ even for the imperfect devices is feasible with the present circuit QED technique.

Quantum-enhanced microscopy with binary-outcome photon counting

Polarized light microscopy using path-entangled $N$-photon states (i.e., the N00N states) has been demonstrated to surpass the shot-noise limit at very low light illumination. However, the microscopy images suffer from divergence of phase sensitivity, which inevitably reduces the image quality. Here, we show that due to experimental imperfections, such a singularity also takes place in the microscopy that uses twin-Fock states of light for illumination. We propose two schemes to completely eliminate this singularity: (i) locking the phase shift sensed by the beams at the optimal working point, by using a spatially dependent offset phase; (ii) a combination of two binary-outcome photon counting measurements, one with a fixed offset phase and the other without any offset phase. Our observations remain valid for any kind of binary-outcome measurement and may open the way for quantum-enhanced microscopy with high $N$ photon states.

NOON states via a quantum walk of bound particles

Tight-binding lattice models allow the creation of bound composite objects which, in the strong-interacting regime, are protected against dissociation. We show that a local impurity in the lattice potential can generate a coherent split of an incoming bound particle wave-packet which consequently produces a NOON state between the endpoints. This is non trivial because when finite lattices are involved, edge-localisation effects make their use for non-classical state generation and information transfer challenging. We derive an effective model to describe the propagation of bound particles in a Bose-Hubbard chain. We introduce local impurities in the lattice potential to inhibit localisation effects and to split the propagating bound particle, thus enabling the generation of distant NOON states. We analyse how minimal engineering transfer schemes improve the transfer fidelity and we quantify the robustness to typical decoherence effects in optical lattice implementations. Our scheme potentially have an impact on quantum-enhanced atomic interferometry in a lattice.

Quantifying the mesoscopic quantum coherence of approximate NOON states and spin-squeezed two-mode Bose-Einstein condensates

We examine how to signify and quantify the mesoscopic quantum coherence of approximate two-mode NOON states and spin-squeezed two-mode Bose-Einstein condensates (BEC). We identify two criteria that verify a nonzero quantum coherence between states with quantum number different by $n$. These criteria negate certain mixtures of quantum states, thereby signifying a generalised $n$-scopic Schrodinger cat-type paradox. The first criterion is the correlation $\langle\hat{a}^{\dagger n}\hat{b}^{n}\rangle\neq0$ (here $\hat{a}$ and $\hat{b}$ are the boson operators for each mode). The correlation manifests as interference fringes in $n$-particle detection probabilities and is also measurable via quadrature phase amplitude and spin squeezing measurements. Measurement of $\langle\hat{a}^{\dagger n}\hat{b}^{n}\rangle$ enables a quantification of the overall $n$-th order quantum coherence, thus providing an avenue for high efficiency verification of a high-fidelity photonic NOON states. The second criterion is based on a quantification of the measurable spin-squeezing parameter $\xi_{N}$. We apply the criteria to theoretical models of NOON states in lossy interferometers and double-well trapped BECs. By analysing existing BEC experiments, we demonstrate generalised atomic "kitten" states and atomic quantum coherence with $n\gtrapprox10$ atoms.

Spectral investigation of the noise influencing multiqubit states

Characterizing and understanding noise affecting quantum states has immense benefits in spectroscopy as well as in realizing quantum devices. Transverse relaxation times under a set of dynamical decoupling (DD) sequences with varying interpulse delays were earlier used for obtaining the noise spectral densities of single-qubit coherences. In this work, using a pair of homonuclear spins and NMR techniques, we experimentally characterize noise in certain decoherence-free subspaces. We also explore the noise of similar states in a heteronuclear spin pair. Further, using a 10-qubit system, we investigate noise profiles of various multiqubit coherences and study the scaling of noise with respect to the coherence order. Finally, using the experimentally obtained noise spectrum of the 10-qubit NOON state, we predict the performance of a Uhrig DD sequence and verify it experimentally.

Super-sensitive detection of quantum interferometer in atmospheric environment

With squeezed states or entangled states being the source, quantum metrology, imaging and sensing can break the standard quantum limit (SQL), even reach the Heisenberg limit (HL), which is difficult to achieve by traditional methods. However, the photon loss or phase fluctuation caused by the atmospheric attenuation and turbulence cannot be ignored in the actual application. Atmospheric transmittance and phase fluctuation are related to the detection distance, and the phase sensitivity becomes worse as the distance increases. As the functions of distance, the photon loss and phase fluctuation are uniformly expressed according to the introduction of atmospheric attenuation coefficient, turbulence structure constant and receive aperture size in this paper. The density matrixes and phase sensitivities of N00N states and M&M′ states in the atmospheric environment are proposed in terms of distance variables. Then the quantitative computation of super-sensitive distance is carried out. SQL-contour is proposed to describe the super-sensitive ability of the quantum interferometer for the affection from both photon loss and phase fluctuation. The simulation results show that, in atmospheric environment the super-sensitive distance can reach hundreds of meters. M&M′ states with less total photon number are more likely to reflect the advantage of super-sensitivity. SQL-contour can provide references for interferometric source choosing.

Detection-dependent six-photon Holland-Burnett state interference.

The NOON state, and its experimental approximation the Holland-Burnett state, have important applications in phase sensing measurement with enhanced sensitivity. However, most of the previous Holland-Burnett state interference (HBSI) experiments only investigated the area of the interference pattern in the region immediately around zero optical path length difference, while the full HBSI pattern over a wide range of optical path length differences has not yet been well explored. In this work, we experimentally and theoretically demonstrate up to six-photon HBSI and study the properties of the interference patterns over a wide range of optical path length differences. It was found that the shape, the coherence time and the visibility of the interference patterns were strongly dependent on the detection schemes. This work paves the way for applications which are based on the envelope of the HBSI pattern, such as quantum spectroscopy and quantum metrology.

Quantum optical feedback control for creating strong correlations in many-body systems

Light enables manipulating many-body states of matter, and atoms trapped in optical lattices is a prominent example. However, quantum properties of light are completely neglected in all quantum gas experiments. Extending methods of quantum optics to many-body physics will enable phenomena unobtainable in classical optical setups. We show how using the quantum optical feedback creates strong correlations in bosonic and fermionic systems. It balances two competing processes, originating from different fields: quantum backaction of weak optical measurement and many-body dynamics, resulting in stabilized density waves, antiferromagnetic and NOON states. Our approach is extendable to other systems promising for quantum technologies.

Hong-Ou-Mandel Interference between Two Deterministic Collective Excitations in an Atomic Ensemble.

We demonstrate deterministic generation of two distinct collective excitations in one atomic ensemble, and we realize the Hong-Ou-Mandel interference between them. Using Rydberg blockade we create single collective excitations in two different Zeeman levels, and we use stimulated Raman transitions to perform a beam-splitter operation between the excited atomic modes. By converting the atomic excitations into photons, the two-excitation interference is measured by photon coincidence detection with a visibility of 0.89(6). The Hong-Ou-Mandel interference witnesses an entangled NOON state of the collective atomic excitations, and we demonstrate its two times enhanced sensitivity to a magnetic field compared with a single excitation. Our work implements a minimal instance of boson sampling and paves the way for further multimode and multiexcitation studies with collective excitations of atomic ensembles.

Signifying the nonlocality of NOON states using Einstein-Podolsky-Rosen steering inequalities

We construct Einstein-Podolsky-Rosen (EPR) steering signatures for the nonlocality of the entangled superposition state described by $\frac{1}{\sqrt{2}}{|N\ensuremath{\rangle}|0\ensuremath{\rangle}+|0\ensuremath{\rangle}|N\ensuremath{\rangle}}$, called the two-mode NOON state. The signatures are a violation of an EPR steering inequality based on an uncertainty relation. The violation confirms an EPR steering between the two modes and involves certification of an intermode correlation for number, as well as quadrature phase amplitude measurements. We also explain how the signatures certify an $N\mathrm{th}$ order quantum coherence, so the system (for larger $N$) can be signified to be in a superposition of states distinct by a mesoscopic value of quantum number in each mode. Finally, we examine the limitations imposed for lossy scenarios, discussing how experimental realizations may be possible for $N=2,3$.

Optimal and near-optimal probe states for quantum metrology of number-conserving two-mode bosonic Hamiltonians

We derive families of optimal and near-optimal probe states for quantum estimation of the coupling constants of a general two-mode number-conserving bosonic Hamiltonian describing one-body and two-body dynamics. We find that the optimal states for estimating the dephasing of the modes, the self-interaction strength, and the contact interaction strength are related to the NOON states, whereas the optimal states for estimation of the intermode single particle tunneling amplitude are superpositions of antipodal SU(2) coherent states. For estimation of the amplitude of pair tunneling and the amplitude of density-dependent single particle tunneling processes, respectively, we introduce classes of variational superposition probe states that provide near perfect saturation of the corresponding quantum Cram\'{e}r-Rao bounds. We show that the ground state of the pair tunneling term in the Hamiltonian has a high fidelity with the optimal states for estimation of a single particle tunneling amplitude, suggesting that high-performance probes for tunneling amplitude estimation may be produced by tuning the two-mode system through a quantum phase transition.

Deterministic synthesis of mechanical NOON states in ultrastrong optomechanics

We propose a protocol for the deterministic preparation of entangled NOON mechanical states. The system is constituted by two identical, optically coupled optomechanical systems. The protocol consists of two steps. In the first, one of the two optical resonators is excited by a resonant external $\ensuremath{\pi}$-like Gaussian optical pulse. When the optical excitation coherently partly transfers to the second cavity, the second step starts. It consists of sending simultaneously two additional $\ensuremath{\pi}$-like Gaussian optical pulses, one at each optical resonator, with specific frequencies. In the optomechanical ultrastrong coupling regime, when the coupling strength becomes a significant fraction of the mechanical frequency, we show that NOON mechanical states with quite high Fock states can be deterministically obtained. The operating range of this protocol is carefully analyzed. Calculations have been carried out taking into account the presence of decoherence, thermal noise, and imperfect cooling.

Quantum error correction against photon loss using NOON states

The so-called NOON states are quantum optical resources known to be useful especially for quantum lithography and metrology. At the same time, they are known to be very sensitive to photon losses and rather hard to produce experimentally. Concerning the former, here we present a scheme where NOON states are the elementary resources for building quantum error correction codes against photon losses, thus demonstrating that such resources can also be useful to suppress the effect of loss. Our NOON-code is an exact code that can be systematically extended from one-photon to higher-number losses. Its loss scaling depending on the codeword photon number is the same as for some existing, exact loss codes such as bosonic and quantum parity codes, but its codeword mode number is intermediate between that of the other codes. Another generalisation of the NOON-code is given for arbitrary logical qudits instead of logical qubits. While, in general, the final codewords are always obtainable from multi-mode NOON states through application of beam splitters, both codewords for the one-photon-loss qubit NOON-code can be simply created from single-photon states with beam splitters. We give various examples and also discuss a potential application of our qudit code for quantum communication.

Coherent Absorption of N00N States.

Recent results in deeply subwavelength thickness films demonstrate coherent control and logical gate operations with both classical and single-photon light sources. However, quantum processing and devices typically involve more than one photon and nontrivial input quantum states. Here we experimentally investigate two-photon N00N state coherent absorption in a multilayer graphene film. Depending on the N00N state input phase, it is possible to selectively choose between single- or two-photon absorption of the input state in the graphene film. These results demonstrate that coherent absorption in the quantum regime exhibits unique features, opening up applications in multiphoton spectroscopy and imaging.

Generation of atomic NOON states via shortcuts to adiabatic passage

Based on Lewis---Riesenfeld invariants and quantum Zeno dynamics, we propose an effective scheme for generating atomic NOON states via shortcuts to adiabatic passage. The photon losses are efficiently suppressed by engineering shortcuts to adiabatic passage in the scheme. The numerical simulation shows that the atomic NOON states can be generated with high fidelity.

Loss-tolerant state engineering for quantum-enhanced metrology via the reverse Hong-Ou-Mandel effect.

Highly entangled quantum states, shared by remote parties, are vital for quantum communications and metrology. Particularly promising are the N00N states—entangled N-photon wavepackets delocalized between two different locations—which outperform coherent states in measurement sensitivity. However, these states are notoriously vulnerable to losses, making them difficult to both share them between remote locations and recombine in order to exploit interference effects. Here we address this challenge by utilizing the reverse Hong–Ou–Mandel effect to prepare a high-fidelity two-photon N00N state shared between two parties connected by a lossy optical medium. We measure the prepared state by two-mode homodyne tomography, thereby demonstrating that the enhanced phase sensitivity can be exploited without recombining the two parts of the N00N state. Finally, we demonstrate the application of our method to remotely prepare superpositions of coherent states, known as Schrödinger's cat states.

Quantum-enhanced spectroscopy with entangled multiphoton states

Traditionally, spectroscopy is performed by examining the position of absorption lines. However, at frequencies near the transition frequency, additional information can be obtained from the phase shift. In this work we consider the information about the transition frequency obtained from both the absorption and the phase shift, as quantified by the Fisher information in an interferometric measurement. We examine the use of multiple single-photon states, NOON states, and numerically optimized states that are entangled and have multiple photons. We find the optimized states that improve over the standard quantum limit set by independent single photons for some atom number densities.