Matthew J. Filipovich, Aleksei Malyshev, A. I. Lvovsky




Aleksei Malyshev, Juan Miguel Arrazola, A. I. Lvovsky




James Spall, Xianxin Guo and A. I. Lvovsky




Jernej Frank, Alexander Duplinskiy, Kaden Bearne, and A. I. Lvovsky

Optica 10, 1147-1152 (2023)


Ekaterina Fedotova, Nikolai Kuznetsov, Egor Tiunov, A. I. Lvovsky




James Spall, Xianxin Guo and A. I. Lvovsky
M. K. Kurmapu, V. V. Tiunova, E. S. Tiunov, M. Ringbauer, C. Maier, R. Blatt, T. Monz, A. K. Fedorov and A. I. Lvovsky

Reconstructing complex states of a 20-qubit quantum simulator




James Spall, Xianxin Guo and A. I. Lvovsky

Hybrid training of optical neural networks

Optica 9, 803 (2022)



Dmitry A. Chermoshentsev, Artem E. Shitikov, Evgeny A. Lonshakov, Georgy V. Grechko, Ekaterina A. Sazhina, Nikita M. Kondratiev, Anatoly V. Masalov, Igor A. Bilenko, Alexander I. Lvovsky and Alexander E. Ulanov

Dual-laser self-injection locking to an integrated microresonator

Optics Express 10, 17094-17105 (2022)



A.K. Fedorov, N. Gisin, S.M. Beloussov and A.I. Lvovsky

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

Nature Machine Intelligence DOI 10.1038/s42256-022-00461-z



A. A. Pushkina, G. Maltese, J. I. Costa-Filho, P. Patel and A. I. Lvovsky

Superresolution Linear Optical Imaging in the Far Field

Physical Review Letters 127, 253602 (2021)


Stepan Makarenko, Dmitry Sorokin, Alexander Ulanov and A. I. Lvovsky

Aligning an optical interferometer with beam divergence control and continuous action space



Arsen Kuzhamuratov, Dmitry Sorokin, Alexander Ulanov and A. I. Lvovsky

Adaptation of Quadruped Robot Locomotion with Meta-Learning



D.A. Chermoshentsev, A.O. Malyshev, E.S. Tiunov, D. Mendoza, A. Aspuru-Guzik, A.K. Fedorov and A.I. Lvovsky

Polynomial unconstrained binary optimisation inspired by optical simulation



D. Beloborodov, A. E. Ulanov, J. Foerster, S. Whiteson and A. I. Lvovsky

Reinforcement learning enhanced quantum-inspired algorithm for combinatorial optimization

Machine Learning: Science and Technology, Volume 2, 025009 (2021)


Xianxin Guo, Thomas D. Barrett, Zhiming M. Wang and A.I. Lvovsky

Backpropagation through nonlinear units for the all-optical training of neural networks

Photonics Research 9, B71-B80 (2021)


D. Drahi, D. V. Sychev, K. K. Pirov, E. A. Sazhina, V. A. Novikov, I. A. Walmsley and A. I. Lvovsky

Entangled resource for interfacing single- and dual-rail optical qubits

Quantum 5, 416 (2021)


James Spall, Xianxin Guo, Thomas D. Barrett and A. I. Lvovsky

Fully reconfigurable coherent optical vector-matrix multiplication

Optics Letters 45, 5752-5755 (2020)



A. I. Lvovsky, P. Grangier, A. Ourjoumtsev, V. Parigi, M. Sasaki and R. Tualle-Brouri

Production and applications of non-Gaussian quantum states of light




G. S. Thekkadath, M. E. Mycroft, B. A. Bell, C. G. Wade, A. Eckstein, D. S. Phillips, R. B. Patel, A. Buraczewski, A. E. Lita, T. Gerrits, S. W. Nam, M. Stobińska, A. I. Lvovsky and I. A. Walmsley

Quantum-enhanced interferometry with large heralded photon-number states

npj Quantum Information 6, 89 (2020)



Dmitry Sorokin, Alexander Ulanov, Ekaterina Sazhina and Alexander Lvovsky

Interferobot: aligning an optical interferometer by a reinforcement learning agent

Advances in Neural Information Processing Systems 33, 13238-13248 (2020)



A. A. Pushkina, J. I. Costa-Filho, G. Maltese and A. I. Lvovsky

Comprehensive model and performance optimization of phase-only spatial light modulators

Measurement Science and Technology 31, 125202 (2020)



E. S. Tiunov, V. V. Tiunova, A. E. Ulanov, A. I. Lvovsky and A. K. Fedorov

Experimental quantum homodyne tomography via machine learning

Optica 7, 448 (2020)



E. S. Moiseev, A. Tashchilina, S. A. Moiseev and A. I. Lvovsky

Darkness of two-mode squeezed light in Lambda-type atomic system

New Journal of Physics 22, 013014 (2020)



T. D. Barrett, W. R. Clements, J. N. Foerster and A. I. Lvovsky

Exploratory Combinatorial Optimization with Reinforcement Learning

Proceedings of Thirty-fourth AAAI conference on artificial intelligence, 3243-3250 (2020)



Bohan Li, G. Maltese, J. I. Costa-Filho, Anastasia A. Pushkina and A. I. Lvovsky

An optical Eratosthenes’ sieve for large prime numbers

Optics Express 28, 11965-11973 (2020)



G. S. Thekkadath, B. A. Bell, I. A. Walmsley and A. I. Lvovsky

Engineering Schrödinger cat states with a photonic even-parity detector

Quantum 4, 239 (2020)



Alexander E. Ulanov, Egor S. Tiunov and A. I. Lvovsky

Quantum-inspired annealers as Boltzmann generators for machine learning and statistical physics




Demid V. Sychev, Valeriy A. Novikov, Khurram K. Pirov, Christoph Simon and A. I. Lvovsky

Entanglement of macroscopically distinct states of light

Optica 6, 1425 (2019)



A.K. Fedorov, A.V. Akimov, J.D. Biamonte, A.V. Kavokin, F.Ya. Khalili, E.O. Kiktenko, N.N. Kolachevsky, Y.V. Kurochkin, A.I. Lvovsky, A.N. Rubtsov, G.V. Shlyapnikov, S.S. Straupe, A.V. Ustinov and A.M. Zheltikov

Quantum technologies in Russia

Quantum Science and Technologies, 4 (2019) 040501.



E. Tiunov, A. E. Ulanov and A. I. Lvovsky

Annealing by simulating the coherent Ising machine

Optics Express 7, 10288-10295 (2019)



S. Jalnapurkar, P. Anderson, E. S. Moiseev, P. Palittapongarnpim, A. Narayanan, P. E. Barclay and A. I. Lvovsky

Measuring fluorescence by observing field quadrature noise

Optics Letters 44, 1678-1681 (2019)



A. K. Fedorov, E. O. Kiktenko and A. I. Lvovsky

Quantum computers put blockchain security at risk (Comment)

Nature 56, 465-467 (2018)



D. V. Sychev, A. E. Ulanov, A. A. Pushkina, E. Tiunov, V. Novikov and A. I. Lvovsky

Entanglement and teleportation between polarization and wave-like encodings of an optical qubit

Nature Communications 9, 3672 (2018)



A. Ghosh, D. Gelbwaser-Klimovsky, W. Niedenzu, A. Lvovsky, I. Mazets, M. O. Scully and G. Kurizki

Two-level masers as heat-to-work converters

Proceedings of the National Academy of Sciences 115, 9941-9944 (2018)



E.O. Kiktenko, N.O. Pozhar, M.N. Anufriev, A.S. Trushechkin, R.R. Yunusov, Y.V. Kurochkin, A.I. Lvovsky and A.K. Fedorov

Quantum-secured blockchain

Quantum Science and Technology 3, 035004 (2018)



P. Anderson, S. Jalnapurkar, E. Moiseev, D. Chang, P. Barclay, A. Lezama and A.I. Lvovsky

Optical nanofiber temperature monitoring via double heterodyne detection

AIP Advances 8, 055005 (2018)



A. V. Masalov, A. Kuzhamuratov and A. I. Lvovsky

Noise spectra in balanced optical detectors based on transimpedance amplifiers

Review of Scientific Instruments 88, 113109 (2017)





D. Sychev, A. E. Ulanov, A. A. Pushkina, M. W. Richards, I. A. Fedorov and A. I. Lvovsky

Enlargement of optical Schrödinger’s cat states

Nature Photonics 11, 379-382 (2017)



I. A. Fedorov, A. E. Ulanov, Y. Kurochkin and A. I. Lvovsky

Synthesis of the Einstein-Podolsky-Rosen entanglement in a sequence of two single-mode squeezers

Optics Letters 42, 132 – 134 (2016)



A. E. Ulanov, D. Sychev, A. A. Pushkina, I. A. Fedorov and A. I. Lvovsky

Quantum Teleportation Between Discrete and Continuous Encodings of an Optical Qubit

Physical Review Letters 118, 160501 (2017)





A. E. Ulanov, I. A. Fedorov, A. A. Pushkina, Y. Kurochkin, T. C. Ralph and A. I. Lvovsky

Undoing the effect of loss on quantum entanglement

Nature Photonics 9, 764-769 (2015)



I. A. Fedorov, A. K. Fedorov, Y. V. Kurochkin and A. I. Lvovsky

Complete characterization of a multimode quantum black box

New Journal of Physics 17, 043063 (2015)



Z. Qin, A. S. Prasad, T. Brannan, A. MacRae, A. Lezama and A. I. Lvovsky

Complete temporal characterization of a single photon

Light: Science and Applications 4, e298 (2015)



I. A. Fedorov, A. E. Ulanov, Y. Kurochkin and A. I. Lvovsky

Quantum vampire: collapse-free action at a distance by the photon annihilation operator

Optica 2, 112 – 115 (2015)



A. I. Lvovsky

Squeezed light

Section in book: Photonics Volume 1: Fundamentals of Photonics & Physics, pp.121-164. Edited by D.Andrews. Wiley, West Sussex, 2015



D. Hogg, D. W. Berry and A. I. Lvovsky

Efficiencies of Quantum Optical Detectors

Physical Review A 90,053846 (2014)



T. Brannan, Z. Qin, A. MacRae and A. I. Lvovsky

Generation and tomography of arbitrary qubit states in a transient collective atomic excitation

Optics Letters 39, 5447 – 5450 (2014)



R. Ghobadi, S. Kumar, B. Pepper, D. Bouwmeester, A.I. Lvovsky and C. Simon

Opto-mechanical micro-macro entanglement

Physical Review Letters 112, 080503 (2014)



Y. Kurochkin, A. S. Prasad and A. I. Lvovsky

Distillation of the two-mode squeezed state

Physical Review Letters 112, 070402 (2014)
A. I. Lvovsky, R. Ghobadi, C. Simon, A. Chandra and A. S. Prasad

Observation of micro-macro entanglement of light

Nature Physics 9, 541-544 (2013)



R. Blatt, A. I. Lvovsky and G. J. Milburn

The 20th anniversary of quantum state engineering

Journal of Physics B: Atomic, Molecular and Optical Physics (special issue) 46, 100201 (2013)





R. Thomas, C. Kupchak, G. S. Agarwal and A. I. Lvovsky

Observation of electromagnetically induced transparency in evanescent fields

Optics Express 21, 6880 – 6888 (2013)



R. Kumar, E. Barrios, C. Kupchak and A. I. Lvovsky

Experimental characterization of bosonic creation and annihilation operators

Physical Review Letters 110, 130403 (2013)



A. I. Lvovsky

A quantum delivery note (News and Views article)

Nature Physics 9, 5 – 6 (2013)



A. Anis and A. I. Lvovsky

Maximum-likelihood coherent-state quantum process tomography

New Journal of Physics 14, 105021 (2012)



A. MacRae, T. Brannan, R. Achal and A. I. Lvovsky

Generation of arbitrary quantum states from atomic ensembles

Physics in Canada 68, 137 – 138 (2012)



R. Kumar, E. Barrios, A. MacRae, E. Cairns, E. H. Huntington and A. I. Lvovsky

Versatile Wideband Balanced Detector for Quantum Optical Homodyne Tomography

Optics Communications 285, 5259 – 5267 (2012)



A. MacRae, T. Brannan, R. Achal and A. I. Lvovsky

Tomography of a High-Purity Narrowband Photon From a Transient Atomic Collective Excitation

Physical Review Letters 109, 033601 (2012)



P. Palittapongarnpim, A. MacRae and A. I. Lvovsky

A monolithic filter cavity for experiments in quantum optics

Review of Scientific Instruments 83, 066101 (2012)



D. W. Berry and A. I. Lvovsky

Preservation of loss in linear-optical processing

Physical Review A 84, 042304 (2011)



B. He, A. MacRae, Y. Han, A. I. Lvovsky and C. Simon

Transverse multimode effects on the performance of photon-photon gates

Physical Review A 83, 022312 (2011)



S. Rahimi-Keshari, A. Scherer, A. Mann, A. T. Rezakhani, A. I. Lvovsky and B. C. Sanders

Quantum process tomography with coherent states

New Journal of Physics 13, 013006 (2011) [Selected for New Journal of Physics Highlights of 2011]



Y. Chi, B. Qi, W. Zhu, L. Qian, H.-K. Lo, S.-H. Youn, A. I. Lvovsky and L. Tian

A balanced homodyne detector for high-rate Gaussian-modulated coherent-state quantum key distribution

New Journal of Physics 13, 013003 (2011)





A. I. Lvovsky, B. C. Sanders and W. Tittel

Optical quantum memory

Nature Photonics 3, 706 – 714 (2009)



M. Lobino, C. Kupchak, E. Figueroa and A. I. Lvovsky

Memory for Light as a Quantum Process

Physical Review Letters 102, 203601 (2009)



S. R. Huisman, Nitin Jain, S. A. Babichev, Frank Vewinger, A. N. Zhang, S. H. Youn and A. I. Lvovsky

Instant single-photon Fock state tomography

Optics Letters 34, 2739 – 2741 (2009)



G. Campbell, A. Ordog and A. I. Lvovsky

Multimode electromagnetically-induced transparency on a single atomic line

New Journal of Physics 11, 103021 (2009)



J. Appel, A. MacRae and A. I. Lvovsky

Versatile digital GHz phase lock for external cavity diode lasers

Measurement Science and Technology 20, 055302 (2009)



K. Kuntz, B. Braverman, S.-H. Youn, M. Lobino, E. M. Pessina and A. I. Lvovsky

Spatial and temporal characterization of a Bessel beam produced using a conical mirror

Physical Review A 79, 043802 (2009)



A.I.Lvovsky and M.G.Raymer

Continuous-variable optical quantum state tomography

Reviews of Modern Physics 81, 299 (2009)



E. Figueroa, M. Lobino, D. Korystov, J. Appel and A. I. Lvovsky

Propagation of squeezed vacuum under electromagnetically induced transparency

New Journal of Physics 11, 013044 (2009)



B. Braverman, K. Kuntz, M. Lobino, E. M. Pessina and A. I. Lvovsky

Measurement of superluminal phase and group velocities of Bessel beams in free space




M. Lobino, D. Korystov, C. Kupchak, E. Figueroa, B. C. Sanders and A. I. Lvovsky

Complete Characterization of Quantum-Optical Processes

Science 322, 563 (2008)



A. MacRae, G. Campbell and A. I. Lvovsky

Matched Slow Pulses Using Double Electromagnetically Induced Transparency

Optics Letters 33, 2659 (2008)



J. Appel, E. Figueroa, D. Korystov, M. Lobino and A. I. Lvovsky

Quantum memory for squeezed light

Physical Review Letters 100, 093602 (2008)



P. Marzlin, J. Appel and A. I. Lvovsky

Photons as quasicharged particles

Physical Review A 77, 043813 (2008)



F. Vewinger, J. Appel, E. Figueroa and A. I. Lvovsky

Adiabatic frequency conversion of quantum optical information in atomic vapor

Optics Letters 32, 2771 (2007)



A. I. Lvovsky, W. Wasilewski and K. Banaszek

Decomposing a pulsed optical parametric amplifier into independent squeezers

Journal of Modern Optics 54, 721 (2007)



J. Appel, D. Hoffman, E. Figueroa and A. I. Lvovsky

Electronic noise in optical homodyne tomography

Physical Review A 75, 035802 (2007)



D. W. Berry, A. I. Lvovsky and B. C. Sanders

Efficiency limits for linear optical processing of single photons and single-rail qubits

Journal of the Optical Society of America B 24, 189 (2007)



J. Řeháček, Z. Hradil, E. Knill and A. I. Lvovsky

Diluted maximum-likelihood algorithm for quantum tomography

Physical Review A 75, 042108 (2007)



M. Oberst, F. Vewinger and A. I. Lvovsky

Time-resolved probing of the ground state coherence in rubidium

Optics Letters 32, 1755 (2007)



JJ. Appel, K.-P. Marzlin and A. I. Lvovsky

Raman adiabatic transfer of optical states

Physical Review A 73, 013804 (2006)



D. W. Berry, A. I. Lvovsky and B. C. Sanders

Interconvertibility of single-rail optical qubits

Optics Letters 31, 107 (2006)



E. Figueroa, F. Vewinger, J. Appel and A. I. Lvovsky

Decoherence of electromagnetically-induced transparency in atomic vapor

Optics Letters 31, 2625 (2006)



W. Wasilewski, A. I. Lvovsky, K. Banaszek and C. Radzewicz

Pulsed squeezed light: simultaneous squeezing of multiple modes

Physical Review A 73, 063819 (2006)



T. Aichele, A. I. Lvovsky and S. Schiller

Optical mode characterization of single photons prepared via conditional measurements on a biphoton state

European Physical Journal D 18, 237 (2002)



A. I. Lvovsky and J. H. Shapiro

Nonclassical character of statistical mixtures of the single-photon and vacuum optical states

Physical Review A 65, 033830 (2002)



A. I. Lvovsky

Cabello’s nonlocality and linear optics

Physical Review Letters 88, 098901 (2002)



A. I. Lvovsky and S. A. Babichev

Synthesis and tomographic characterization of the displaced Fock state of light

Physical Review A (Rapid communications) 66, 011801 (2002)



A. I. Lvovsky and J. Mlynek

Quantum-optical catalysis: generating nonclassical states of light by means of linear optics

Physical Review Letters 88, 250401 (2002)



H. Held, A. I. Lvovsky, X. Wei and Y. R. Shen

Bulk contribution from isotropic media in surface sum-frequency generation

Physical Review B 66, 205110 (2002)



A. I. Lvovsky, S. R. Hartmann and F. Moshary

Superfluorescence-stimulated photon echo

Physical Review Letters 89, 263602 (2002)



E.V. Kovalchuk, D. Dekorsy, A. I. Lvovsky, C. Braxmaier, A. Peters, J. Mlynek and S. Schiller

High-resolution Doppler-free molecular spectroscopy using a continuous-wave optical parametric oscillator

Optics Letters 26, 1430 (2001)



A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek and S. Schiller

Quantum State Reconstruction of the Single-Photon Fock State

Physical Review Letters 87, 050402 (2001)



M. Oh-e, A. I. Lvovsky, X. Wei, D. Kim and Y. R. Shen

Nonlinear optical studies of surface structures of rubbed polyimides and adsorbed liquid crystal monolayers

Molecular Crystals and Liquid Crystals 364, 427 (2001)



H. Hansen, T. Aichele, C. Hettich, P. Lodahl, A. I. Lvovsky, J. Mlynek and S. Schiller

An ultra-sensitive pulsed balanced homodyne detector: Application to time-domain quantum measurements

Optics Letters 26, 1714 (2001)



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.

Fig. 1

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].

Fig. 2

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

Vacuum state

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.

Coherent state

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.

Thermal state

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.

Star state

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.