Reconstructing complex states of a 20-qubit quantum simulator
Hybrid training of optical neural networks
Dual-laser self-injection locking to an integrated microresonator
Quantum computing at the quantum advantage threshold: a down-to-business review
Thomas D. Barrett, Aleksei Malyshev and A. I. Lvovsky
Autoregressive neural-network wavefunctions for ab initio quantum chemistry
Superresolution Linear Optical Imaging in the Far Field
Aligning an optical interferometer with beam divergence control and continuous action space
Adaptation of Quadruped Robot Locomotion with Meta-Learning
Polynomial unconstrained binary optimisation inspired by optical simulation
Reinforcement learning enhanced quantum-inspired algorithm for combinatorial optimization
Backpropagation through nonlinear units for the all-optical training of neural networks
Entangled resource for interfacing single- and dual-rail optical qubits
Fully reconfigurable coherent optical vector-matrix multiplication
Production and applications of non-Gaussian quantum states of light
Quantum-enhanced interferometry with large heralded photon-number states
Interferobot: aligning an optical interferometer by a reinforcement learning agent
Comprehensive model and performance optimization of phase-only spatial light modulators
Experimental quantum homodyne tomography via machine learning
Darkness of two-mode squeezed light in Lambda-type atomic system
Exploratory Combinatorial Optimization with Reinforcement Learning
An optical Eratosthenes’ sieve for large prime numbers
Engineering Schrödinger cat states with a photonic even-parity detector
Quantum-inspired annealers as Boltzmann generators for machine learning and statistical physics
Entanglement of macroscopically distinct states of light
Quantum technologies in Russia
Annealing by simulating the coherent Ising machine
Measuring fluorescence by observing field quadrature noise
Quantum computers put blockchain security at risk (Comment)
Entanglement and teleportation between polarization and wave-like encodings of an optical qubit
Two-level masers as heat-to-work converters
Optical nanofiber temperature monitoring via double heterodyne detection
Noise spectra in balanced optical detectors based on transimpedance amplifiers
Fisher information for far-field linear optical superresolution via homodyne or heterodyne detection in a higher-order local oscillator mode
Enlargement of optical Schrödinger’s cat states
Synthesis of the Einstein-Podolsky-Rosen entanglement in a sequence of two single-mode squeezers
Quantum Teleportation Between Discrete and Continuous Encodings of an Optical Qubit
Fisher information for far-field linear optical superresolution via homodyne or heterodyne detection in a higher-order local oscillator mode
Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode
Loss-tolerant quantum enhanced metrology and state engineering via the reverse Hong-Ou-Mandel effect
Undoing the effect of loss on quantum entanglement
Complete characterization of a multimode quantum black box
Complete temporal characterization of a single photon
Quantum vampire: collapse-free action at a distance by the photon annihilation operator
Efficiencies of Quantum Optical Detectors
Generation and tomography of arbitrary qubit states in a transient collective atomic excitation
Opto-mechanical micro-macro entanglement
Distillation of the two-mode squeezed state
Observation of micro-macro entanglement of light
The 20th anniversary of quantum state engineering
Creating and Detecting Micro-Macro Photon-Number Entanglement by Amplifying and Deamplifying a Single-Photon Entangled State
Observation of electromagnetically induced transparency in evanescent fields
Experimental characterization of bosonic creation and annihilation operators
A quantum delivery note (News and Views article)
Maximum-likelihood coherent-state quantum process tomography
Generation of arbitrary quantum states from atomic ensembles
Versatile Wideband Balanced Detector for Quantum Optical Homodyne Tomography
Tomography of a High-Purity Narrowband Photon From a Transient Atomic Collective Excitation
A monolithic filter cavity for experiments in quantum optics
Preservation of loss in linear-optical processing
Transverse multimode effects on the performance of photon-photon gates
Quantum process tomography with coherent states
A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution
A bridge between the single-photon and squeezed-vacuum state
Linear-optical processing cannot increase photon efficiency
Quantum-optical state engineering up to the two-photon level
Protecting quantum superpositions from the outside world
Optical quantum memory
Memory for Light as a Quantum Process
Instant single-photon Fock state tomography
Multimode electromagnetically-induced transparency on a single atomic line
Versatile digital GHz phase lock for external cavity diode lasers
Spatial and temporal characterization of a Bessel beam produced using a conical mirror
Continuous-variable optical quantum state tomography
Propagation of squeezed vacuum under electromagnetically induced transparency
Measurement of superluminal phase and group velocities of Bessel beams in free space
Complete Characterization of Quantum-Optical Processes
Matched Slow Pulses Using Double Electromagnetically Induced Transparency
Quantum memory for squeezed light
Photons as quasicharged particles
Adiabatic frequency conversion of quantum optical information in atomic vapor
Decomposing a pulsed optical parametric amplifier into independent squeezers
Electronic noise in optical homodyne tomography
Efficiency limits for linear optical processing of single photons and single-rail qubits
Diluted maximum-likelihood algorithm for quantum tomography
Time-resolved probing of the ground state coherence in rubidium
Raman adiabatic transfer of optical states
Interconvertibility of single-rail optical qubits
Decoherence of electromagnetically-induced transparency in atomic vapor
Pulsed squeezed light: simultaneous squeezing of multiple modes
Single qubit optical quantum fingerprinting
Classical and quantum fingerprinting with shared randomness and one-sided error
Remote preparation of a single-mode photonic qubit by measuring field quadrature noise
Iterative maximum-likelihood reconstruction in quantum homodyne tomography
Homodyne tomography characterization and nonlocality of a dual-mode optical qubit
Experimental Vacuum Squeezing in Rubidium Vapor via Self-Rotation
Quantum scissors: teleportation of single-mode optical states by means of a nonlocal single photon
Optical mode characterization of single photons prepared via conditional measurements on a biphoton state
Nonclassical character of statistical mixtures of the single-photon and vacuum optical states
Cabello’s nonlocality and linear optics
Synthesis and tomographic characterization of the displaced Fock state of light
Quantum-optical catalysis: generating nonclassical states of light by means of linear optics
Bulk contribution from isotropic media in surface sum-frequency generation
Superfluorescence-stimulated photon echo
High-resolution Doppler-free molecular spectroscopy using a continuous-wave optical parametric oscillator
Quantum State Reconstruction of the Single-Photon Fock State
Nonlinear optical studies of surface structures of rubbed polyimides and adsorbed liquid crystal monolayers
An ultra-sensitive pulsed balanced homodyne detector: Application to time-domain quantum measurements
Evaluation of surface vs. bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries
Sum-frequency generation (SFG) vibrational spectroscopy of side alkyl chain structures of polyimide surfaces
Coherent Fan Emissions
Time-delayed second-harmonic generation in cesium vapor
Photon echo modulation effects in cesium vapor
A homodyne tomography tutorial
How can one determine a quantum state’s Wigner function? Although it cannot be measured directly as a probability density, all its marginal distributions can. Once we know all the marginal distributions associated with different quadratures – i.e. the Wigner function projections upon various vertical planes – we can reconstruct the Wigner function. This reconstruction procedure is very similar to that used by medical doctors in computer tomography – where one reconstructs a three-dimensional picture of a tissue anomaly from a set of two-dimensional X-ray images acquired from different angles.
So our goal is to determine the marginal distributions for all different “points of view”. Suppose we can measure the coordinate of our oscillator (for example, by taking its photograph at a given time). Repeating this procedure many times for a set of identical quantum states we obtain the marginal distribution pr(X) for quadrature X. This is, however, insufficient for tomographic reconstruction of the Wigner function: we need distributions
for all values of q. In order to understand how to obtain these distributions we need to recall that our oscillator, well, oscillates. That is, our Wigner function rotates around the phase space origin with the oscillation frequency w. The marginal distribution pr(X) measured at the moment t = q / w is the same as the marginal distribution pr(Xq) measured at the moment t = 0. Therefore, instead of measuring the marginal distributions at a particular moment for all “view angles” we can measure marginal distribution associated with one view angle, but at different moments [Fig. 1].
Let us now remember that we work with light, not with mechanical oscillators. As we know from classical physics, these two physical systems are described by the same equations of motion. The role of the mechanical coordinate in the electromagnetic wave is played by the electric field. If we had an “electroscope” able to perform phase-sensitive measurements of the electric field in an electromagnetic mode, we would be able to reconstruct the “quantum portrait” of the mode’s quantum state – its Wigner function.
Unfortunately, such an “electroscope” does not exist. Typical oscillation frequencies of the light fields are hundreds of terahertz, and no one can build a detector that can follow such fast changes. Yet one can use a trick that allows to do phase-sensitive measurements of the electric field using regular, “slow” detectors. This trick is called balanced homodyne detection – here is its main idea.
Quantum objects are usually small, and electromagnetic fields that exhibit quantum properties are usually weak. Let us overlap the mode whose quantum properties we want to measure (the “signal” mode) with a relatively strong laser beam (the “local oscillator”) on a beam splitter [fig. 2]. If the electric fields in the two modes are Es and ELO, the fields emerging in the two beam splitter output ports are given by
Let us now have both beam coming out of the beam splitter hit high-efficiency photodiodes. A photodiode is a device which generates electric current proportional to the intensity (not the amplitude) of the electromagnetic field incident on it. Our two photodiodes produce photocurrents that we subtract from each other. The photocurrent difference is given by
since ELO >> Es, we neglect the quantum noise of the local oscillator field and find out that the photocurrent difference is proportional to the amplitude of the signal field – exactly the quantity we are looking for! By changing the relative optical phase of the local oscillator and signal waves, we measure the electric field at different phases. At each phase, we perform a multitude of electric field measurements (each time preparing an identical quantum state in the signal channel) thus obtaining the marginal distribution. A set of marginal distributions for various q’s will provide us with full information about the quantum state – and allows us to reconstruct its Wigner function and the density matrix.
An excellent introduction into nonclassical light and methods of its characterization is given in the textbook by U. Leonhardt “Measuring the quantum state of light”, CambridgeUniversity Press, 1997.
Our review paper on homodyne tomography
Gerd Breitenbach’s homodyne tomography page
A galery of Wigner functions
This is the simplest quantum state altogether: no light! But even when we switch off the light, the uncertainty principle holds valid, so both position and momentum exhibit some fluctuations. These are the vacuum fluctuations and they are the same for X and P – so the Wigner function is a round hill centered at the origin of phase space.
This state is emitted by a laser. Its Wigner function is the same as for the vacuum state, but displaced in the phase space so there is a nonzero average for the electric field.
This is the state of radiation a heated black body emits. It has no phase and its photon statistics is that of Bose-Einstein. By room temperature the thermal state is to a good approximation vacuum.
Squeezed vacuum state
The first nonclassical state on this page. The uncertainty in one quadrature is reduced at the other quadrature’s expense. The product of the two is however the same as for the vacuum: this is a minimum uncertainty state.
See our experimental work on generating squeezed light.
Single-photon Fock state
The name says it all – a state containing exactly one photon. We did an experimental project on this one, too!
Four-photon Fock state
Schrödinger cat state
This is a coherent superposition of two coherent states: |a> and |-a>. It is hard to generate experimentally, but the Wigner function is instructive to look at. We see two round hills at the top and at the bottom, associated with each coherent state involved. If our ensemble were an incoherent, statistical mixture of these two states, its Wigner function would just feature these two hills. But because the superposition is coherent, the Wigner function exhibits an additional, highly nonclassical feature in the middle: a fine structured interference pattern with negative regions.
This state is an analog of the squeezed state, but is generated (in theoreticians’ notebooks) by means of 3-photon, rather than 2-photon, down-conversion.
Squeezed single-photon Fock state
A limiting case of the “odd” Schrödinger cat state |α> – |-α> for small α.
Gerd Breitenbach’s page featuring another gallery of quantum states.
Storage of light via electromagnetically-induced transparency
This tutorial gives a technical description of the light storage phenomenon. The physical description – i.e. not what happens, but why it happens, is much more involved and is hard to describe in the semi-popular language.
Consider a glass cell filled with a gas of so-called Lambda-type atoms – a three-level system with two ground and one excited states:
We assume the gas to be optically dense on the transition between levels 1 and 3, that is, a laser beam of the transition wavelength will be absorbed by the gas. If we now, at the same time, apply an additional, strong laser field to the transition 2 – 3 to exhibit a narrow transparency window in the absorption spectrum. Moreover, if the blue laser is pulsed, the pulse will propagate through the cell at a speed much less than the vacuum speed of light – possibly slower than a bicycle! This phenomenon is called electromagnetically-induced transparency (EIT).
Let us now turn the control (red) laser off slowly while the signal (blue) pulse is entirely within the sample making its way through. As we reduce the control field, the signal pulse will propagate slower and slower and will eventually stop. The quantum information contained in the pulse will be transferred to the atoms – more precisely, to the coherent superposition of the two ground levels 1 and 2. Because both are low energy states, their superposition is long-lived, that is – subject to little decay. It can remain unchanged for hundreds of microseconds – an epoch on the atomic time scale!
Theoreticians say, this storage procedure is completely reversible. When we wish to read our memory, we slowly turn the control field back on. The signal pulse will be reemitted with its original shape, phase and in its original quantum state – thus completing the storage/release procedure.
Here is a little movie showing the process of light storage. Notice how the EIT absorption and dispersion, as well as the group velocity of the light pulse change as a function of the control field.
A Wigner function tutorial
Consider a classical harmonic oscillator. Its motion can be completely described by a point in the phase space – the two-dimensional space with the particle’s coordinate X and momentum P as dimensions (quadratures) [Fig. 1(a)]. For a large number of identical classical oscillators, one can talk about the phase-space probability distribution – a function W(X, P) which indicates the probability of finding a particle at a certain point in the phase space [Fig. 1(b)]. This function must, of course, be non-negative and normalized: its integral over the entire phase space must be equal to one. This classical probability distribution has another important property. Consider a series of measurements in which we only measure the oscillator’s coordinate but not the momentum. After a large number of such measurements, one obtains the probability distribution associated with the coordinate – we call this a marginal distribution pr(X). This distribution is related to the phase-space probability density in the following way:
In other words, a marginal distribution is just a density projection of W(X, P) onto a plane associated with the given quadrature [Fig 1(d)].
In the quantum world [Fig. 1(c)], the notion of a “certain point in the phase space” does not make sense because the position and the momentum cannot be measured simultaneously (Heisenberg’s uncertainty principle). Neither does the phase-space probability density. However, even in the quantum domain one can perform quantum measurements of a single quadrature – be it X, P, or their linear combination. A multiple measurement of a quadrature on a set of identical quantum states will yield a probability density associated with this quadrature, i.e. a marginal distribution. Is there any connection between marginal distributions for different quadratures?
In the classical world this connection exists – through the phase-space probability density as discussed above. The amazing fact is that even in the quantum domain there exists so called phase space quasiprobability density – called the Wigner function – with exactly the same property. A marginal distribution associated with a particular quantum state and a particular quadrature is a projection of the state’s Wigner function upon the relevant vertical plane.
The Wigner function of a given state can be calculated from its density matrix. For each quantum ensemble there exists a Wigner function. Just as the classical phase-space probability density, it is real and normalized. However, the Wigned function has one very important difference from its classical analogue. Because by itself it does not have a meaning of a probability density, it does not have to be positive definite. An example is provided by so-called Fock states of harmonic oscillators – the states of definite energy. No matter what the energy is, the phase space has regions where the Wigner function takes on negative values.