Multimode States Research Articles

Simulating realistic non-Gaussian state preparation

We consider conditional photonic non-Gaussian state preparation using multimode Gaussian states and photon-number-resolving detectors in the presence of photon loss. While simulation of such state preparation is often computationally challenging, we show that obtaining the required multimode Gaussian state Fock matrix elements can be reduced to the computation of matrix functions known as loop hafnians, and develop a tailored algorithm for their calculation that is faster than previously known methods. As an example of its utility, we use our algorithm to explore the loss parameter space for three specific non-Gaussian state preparation schemes: Fock state heralding, cat state heralding, and weak cubic-phase state heralding. We confirm that these schemes are fragile with respect to photon loss, yet find that there are regions in the loss parameter space that are potentially accessible in an experimental setting which correspond to heralded states with non-zero non-Gaussianity.

Mode-dependent-loss model for multimode photon-subtracted states

Multimode photon-subtraction provides an experimentally feasible option to construct large non-Gaussian quantum states in continuous-variable quantum optics. The non-Gaussian features of the state can lead towards the more exotic aspects of quantum theory, such as negativity of the Wigner function. However, the pay-off for states with such delicate quantum properties is their sensitivity to decoherence. In this paper, we present a general model that treats the most important source of decoherence in a purely optical setting: losses. We use the framework of open quantum systems and master equations to describe losses in n-photon-subtracted multimode states, where each photon can be subtracted in an arbitrary mode. As a main result, we find that mode-dependent losses and photon-subtraction generally do not commute. In particular, the losses do not only reduce the purity of the state, they also change the modal structure of its non-Gaussian features. We then conduct a detailed study of single-photon subtraction from a multimode Gaussian state, which is a setting that lies within the reach of present-day experiments.

Efficient methods for generating multi-mode NOON states

Two experimental setups to generate multi-mode NOON states are proposed. The first setup is a universal model, which can create a d-mode N-photon NOON state with an arbitrary mode number d and photon number N in principle in a lossless system with perfect optical devices, using states with non-deterministic distributions of photons as the input sources and post-selection of total N photons for detection. The second approach, on the other hand, can generate a 4-mode 4-photon NOON state using single photon sources. The core unit of both setups is the Fock state filter, which is used to filter out an arbitrary Fock state component from an incident state using a beam splitter with proper transmissivity. The intrinsic generation efficiencies of the NOON states produced by both methods, with the first setup particularly focused on using squeezed vacuum states as the inputs, are calculated. They are found to be at least two orders of magnitude larger than and scale more favorably to the previously proposed approach using coherent state inputs. We also show that the second setup has the best performance for the case with d=N=4, while the first method scales better for a larger N or d. These results manifest that multi-mode NOON states with 3 or 4 modes and 4 or 6 photons could feasibly be generated in experiments with current photonic technologies.

Optimal measurements for quantum fidelity between Gaussian states and its relevance to quantum metrology

Quantum fidelity is a measure to quantify the closeness of two quantum states. In an operational sense, it is defined as the minimal overlap between the probability distributions of measurement outcomes and the minimum is taken over all possible positive-operator valued measures (POVMs). Quantum fidelity has been investigated in various scientific fields, but the identification of associated optimal measurements has often been overlooked despite its great importance for practical purposes. We find here the optimal POVMs for quantum fidelity between multi-mode Gaussian states in a closed analytical form. Our general finding is specified for selected single-mode Gaussian states of particular interest and we identify three types of optimal measurements: a number-resolving detection, a projection on the eigenbasis of operator $\hat{x}\hat{p}+\hat{p}\hat{x}$, and a quadrature detection, each of which is applied to distinct types of single-mode Gaussian states. We also show the equivalence between optimal measurements for quantum fidelity and those for quantum parameter estimation, enabling one to easily find the optimal measurements for displacement, phase, squeezing, and loss parameter estimations using Gaussian states.

Entanglement condition for W type multimode states and schemes for experimental realization

We derive a class of inequality relations, using both the sum uncertainty relations of su(2) algebra operators and the Schrodinger–Robertson uncertainty relation of partially transposed su(1,1) algebra operators, to detect the three-mode entanglement of non-Gaussian states of electromagnetic field. These operators are quadratic in mode creation and annihilation operators. The inseparability condition obtained using su(2) algebra operators is shown to guarantee the violation of Schrodinger–Robertson uncertainty relation of partially transposed su(1,1) algebra operators. The obtained inseparability condition incorporates all the required bipartite entanglement conditions to detect the W type three-mode entangled states and hence it is shown to be a necessary condition for W type three-mode entangled states. A general form for a family of such inseparability conditions is provided. The results derived for three-mode systems are generalized to multimode systems. Experimental schemes using symmetric multiport beam splitters and multi-Λ EIT system are proposed to test the violation of multimode separability condition.

Efficient representation of Gaussian states for multimode non-Gaussian quantum state engineering via subtraction of arbitrary number of photons

We introduce a complete description of a multi-mode bosonic quantum state in the coherent-state basis (which in this work is denoted as "$K$" function ), which---up to a phase---is the square root of the well-known Husimi "$Q$" representation. We express the $K$ function of any $N$-mode Gaussian state as a function of its covariance matrix and displacement vector, and also that of a general continuous-variable cluster state in terms of the modal squeezing and graph topology of the cluster. This formalism lets us characterize the non Gaussian state left over when one measures a subset of modes of a Gaussian state using photon number resolving detection, the fidelity of the obtained non-Gaussian state with any target state, and the associated heralding probability, all analytically. We show that this probability can be expressed as a Hafnian, re-interpreting the output state of a circuit claimed to demonstrate quantum supremacy termed Gaussian boson sampling. As an example-application of our formalism, we propose a method to prepare a two-mode coherent-cat-basis Bell state with fidelity close to unity and success probability that is fundamentally higher than that of a well-known scheme that splits an approximate single-mode cat state---obtained by photon number subtraction on a squeezed vacuum mode---on a balanced beam splitter. This formalism could enable exploration of efficient generation of cat-basis entangled states, which are known to be useful for quantum error correction against photon loss.

Franck-Condon factors by counting perfect matchings of graphs with loops.

We show that the Franck-Condon Factor (FCF) associated with a transition between initial and final vibrational states in two different potential energy surfaces, having N and M vibrational quanta, respectively, is equivalent to calculating the number of perfect matchings of a weighted graph with loops that has P = N + M vertices. This last quantity is the loop hafnian of the (symmetric) adjacency matrix of the graph which can be calculated in O(P32P/2) steps. In the limit of small numbers of vibrational quanta per normal mode, our loop hafnian formula significantly improves the speed at which FCFs can be calculated. Our results more generally apply to the calculation of the matrix elements of a bosonic Gaussian unitary between two multimode Fock states having N and M photons in total and provide a useful link between certain calculations of quantum chemistry, quantum optics, and graph theory.

Quantum metrology with one-dimensional superradiant photonic states

Photonic states with large and fixed photon numbers, such as Fock states, enable quantum-enhanced metrology but remain an experimentally elusive resource. A potentially simple, deterministic, and scalable way to generate these states consists of fully exciting $N$ quantum emitters equally coupled to a common photonic reservoir, which leads to a collective decay known as Dicke superradiance. The emitted $N$-photon state turns out to be a highly entangled multimode state, and to characterize its metrological properties in this work we (i) develop theoretical tools to compute the quantum Fisher information of general multimode photonic states, (ii) use it to show that Dicke superradiant photons in one-dimensional waveguides achieve Heisenberg scaling, which can be saturated by a parity measurement, and (iii) study the robustness of these states to experimental limitations in state-of-the-art atom-waveguide QED setups.

Erratum: Characterization of Gram matrices of multimode coherent states [Phys. Rev. A 99 , 012346 (2019)

Received 22 February 2019DOI:https://doi.org/10.1103/PhysRevA.99.049903©2019 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum communicationQuantum protocolsQuantum Information

Means and covariances of photon numbers in multimode Gaussian states

In the field of continuous variables and particularly of Gaussian states an important role is played by the statistics of photon numbers. This issue has been considered in pioneering works for the one- and two-mode cases, but a systematic approach seems not to be available. In this paper, the most general multimode mixed Gaussian state, which is generated starting from a multimode thermal state and processed by the most general Gaussian unitary, is considered. In the $N$-mode the photon numbers are represented by $N$ random variables, one for each mode, and the target is the evaluation of the means and of the covariances. The means describe statistically the amount of photon numbers in each mode, while the covariances give the correlations between the photon numbers in the different modes. For both means and covariances a closed-form result is obtained, expressed in simple and compact formulas. The main tools are provided by the representation of Gaussian unitaries (given by a cascade of an $N$-mode squeeze operator, an $N$-mode rotation operator, and a parallel set of $N$ single-mode displacement operators) and by the corresponding Bogoliubov transformations.

Characterization of Gram matrices of multimode coherent states

Quantum communication protocols are typically formulated in terms of abstract qudit states and operations, leaving the question of an experimental realization open. Direct translation of these protocols, say into single photons with some d-dimensional degree of freedom, are typically challenging to realize. Multi-mode coherent states, on the other hand, can be easily generated experimentally. Reformulation of protocols in terms of these states has been a successful strategy for implementation of quantum protocols. Quantum key distribution and the quantum fingerprinting protocol have both followed this route. In this paper, we characterize the Gram matrices of multi-mode coherent states in an attempt to understand the class of communication protocols, which can be implemented using these states. As a side product, we are able to use this characterization to show that the Hadamard exponential of a Euclidean distance matrix is positive semidefinite. We also derive the closure of the Gram matrices, which can be implemented in this way, so that we also characterize those matrices, which can be approximated arbitrarily well using multi-mode coherent states. Using this we show that Gram matrices of mutually unbiased bases cannot be approximated arbitrarily well using multi-mode coherent states.

Characterizing the multipartite continuous-variable entanglement structure from squeezing coefficients and the Fisher information

Understanding the distribution of quantum entanglement over many parties is a fundamental challenge of quantum physics and is of practical relevance for several applications in the field of quantum information. The Fisher information is widely used in quantum metrology since it is related to the quantum gain in metrology measurements. Here, we use methods from quantum metrology to microscopically characterize the entanglement structure of multimode continuous-variable states in all possible multi-partitions and in all reduced distributions. From experimentally measured covariance matrices of Gaussian states with 2, 3, and 4 photonic modes with controllable losses, we extract the metrological sensitivity as well as an upper separability bound for each partition. An entanglement witness is constructed by comparing the two quantities. Our analysis demonstrates the usefulness of these methods for continuous-variable systems and provides a detailed geometric understanding of the robustness of cluster-state entanglement under photon losses.

Autocorrelation functions: a useful tool for both state and detector characterisation

Abstract The calculation of autocorrelation functions represents a routinely used tool to characterise quantum states of light. In this paper, we evaluate the g(2) function for detected photons in the case of mesoscopic multi-mode twin-beam states in order to fully investigate their statistical properties starting from measurable quantities. Moreover, we show that the second-order autocorrelation function is also useful to estimate the spurious effects affecting the employed Silicon-photomultiplier detectors.

Estimation of Gaussian quantum states

We derive several expressions for the quantum Fisher information matrix (QFIM) for the multi-parameter estimation of multi-mode Gaussian quantum states, the corresponding symmetric logarithmic derivatives, and conditions for saturability of the quantum Cramér–Rao bound. This bound determines the ultimate precision with which parameters encoded into quantum states can be estimated. We include expressions for mixed states, for the case when the Williamson decomposition of the covariance matrix is known, expressions in terms of infinite series, and expressions for pure states. We also discuss problematic behavior when some modes are pure, and present a method that allows the use of expressions that are defined only for mixed states, to compute QFIM for states with any number of pure modes.

Multiphoton Tomography with Linear Optics and Photon Counting.

Determining an unknown quantum state from an ensemble of identical systems is a fundamental, yet experimentally demanding, task in quantum science. Here we study the number of measurement bases needed to fully characterize an arbitrary multimode state containing a definite number of photons, or an arbitrary mixture of such states. We show this task can be achieved using only linear optics and photon counting, which yield a practical though nonuniversal set of projective measurements. We derive the minimum number of measurement settings required and numerically show that this lower bound is saturated with random linear optics configurations, such as when the corresponding unitary transformation is Haar random. Furthermore, we show that for N photons, any unitary 2N design can be used to derive an analytical, though nonoptimal, state reconstruction protocol.

Operational Resource Theory of Continuous-Variable Nonclassicality

Genuinely quantum states of a harmonic oscillator may be distinguished from their classical counterparts by the Glauber-Sudarshan P-representation -- a state lacking a positive P-function is said to be nonclassical. In this paper, we propose a general operational framework for studying nonclassicality as a resource in networks of passive linear elements and measurements with feed-forward. Within this setting, we define new measures of nonclassicality based on the quantum fluctuations of quadratures, as well as the quantum Fisher information of quadrature displacements. These lead to fundamental constraints on the manipulation of nonclassicality, especially its concentration into subsystems, that apply to generic multi-mode non-Gaussian states. Special cases of our framework include no-go results in the concentration of squeezing and a complete hierarchy of nonclassicality for single mode Gaussian states.

Quantum model for traveling-wave electro-optical phase modulator

We develop a quantum model of a traveling-wave electro-optical phase modulator by considering a parametric interaction of the optical and radiofrequency waves in the modulator medium by means of the electro-optic effect. The model includes a possible dispersion of the modulator medium in the optical domain. In the case of negligible dispersion, we derive a simple analytic expression for the evolution operator of the optical field, which has been proposed previously in the literature on the basis of an analogy to classical field transformation. We study by means of a perturbative approach the influence of small dispersion on the field evolution operator and formulate conditions of negligibility of modulator dispersion. We demonstrate the signature of dispersion in phase modulation: breaking the symmetry of sidebands with respect to the mean photon number. We obtain the law of transformation of a multimode coherent state by a phase modulator, which is important for understanding the structure of signal states in quantum cryptographic schemes utilizing phase modulation of coherent optical beams.

Generating Multimode Entangled Microwaves with a Superconducting Parametric Cavity

In this Letter, we demonstrate the generation of multimode entangled states of propagating microwaves. The entangled states are generated by parametrically pumping a multimode superconducting cavity. By combining different pump frequencies, applied simultaneously to the device, we can produce different entanglement structures in a programable fashion. The Gaussian output states are fully characterized by measuring the full covariance matrices of the modes. The covariance matrices are absolutely calibrated using an in situ microwave calibration source, a shot noise tunnel junction. Applying a variety of entanglement measures, we demonstrate both full inseparability and genuine tripartite entanglement of the states. Our method is easily extensible to more modes.

Can nonclassical correlations survive in the presence of asymmetric lossy channels?

Nonclassical correlations are a fundamental resource for applications involving multipartite systems. To assess nonclassicality, several different criteria can be applied to measured data. Here we compare three inequalities by applying them to a multi-mode twin-beam state, in which one of the two parties is transmitted through a varying lossy channel. We demonstrate, both theoretically and experimentally, that the asymmetry introduced by losses affects the three criteria in different ways, thus offering a tool for quantum optics applications.

Analytical study of static beyond-Fröhlich Bose polarons in one dimension

Grusdt et al. [New J. Phys. 19, 103035 (2017)] recently made a renormalization group study of a one-dimensional Bose polaron in cold atoms. Their study went beyond the usual Frohlich description, which includes only single-phonon processes, by including two-phonon processes in which two phonons are simultaneously absorbed or emitted during impurity scattering [Shchadilova et al. Phys. Rev. Lett. 117, 113002 (2016)]. We study this same beyond-Frohlich model, but in the static impurity limit where the ground state is described by a multimode squeezed state instead of the multimode coherent state in the static Frohlich model. We solve the system exactly by applying the generalized Bogoliubov transformation, an approach that can be straightforwardly adapted to higher dimensions. Using our exact solution, we obtain a polaron energy free of infrared divergences and construct analytically the polaron phase diagram. We find that the repulsive polaron is stable on the positive side of the impurity-boson interaction but is always thermodynamically unstable on the negative side of the impurity-boson interaction, featuring a bound state, whose binding energy we obtain analytically. We find that the attractive polaron is always dynamically unstable, featuring a pair of imaginary energies which we obtain analytically. We expect the multimode squeezed state to help with studies that go not only beyond the Frohlich paradigm but also beyond Bogoliubov theory, just as the multimode coherent state has helped with the study of Frohlich polarons.