QUANTUM & OPTICAL TECHNOLOGY

Research group home page

Historically, our laboratory has dealt with experimental quantum optics. We have made sig­nificant contributions to this field and re­cently expanded our research horizons. Now we are solving problems at the intersection of experimental physics, machine learning and ro­botics.

Research

Quantum Optical Experiment

There are multiple quantum systems that have a potential as the basis for future quantum information technology, and it is not known at present, which one is the best. Research groups all over the world are investigating advantages and disadvantages of various candidates. Our group’s effort is concentrated on one such candidate – quantum light, and its fundamental particle – the photon.

The most important, unique advantage of quantum light is its ability to be an information carrier. No matter what future quantum computers will be built of, they will almost certainly communicate by means of photons. This means that developing quantum optical information technology is essential for our quantum future. This is the goal of our group.

We define the following five basic construction blocks of quantum optical technology:

  1. Preparation of quantum states of light
  2. Manipulating them in a controlled manner
  3. Measuring them (quantum tomography)
  4. Interfacing quantum information between light and stationary media
  5. Bringing photons into controlled interaction with each other

There are multiple quantum systems that have a potential as the basis for future quantum information technology, and it is not known at present, which one is the best. Research groups all over the world are investigating advantages and disadvantages of various candidates. Our group’s effort is concentrated on one such candidate – quantum light, and its fundamental particle – the photon.

The most important, unique advantage of quantum light is its ability to be an information carrier. No matter what future quantum computers will be built of, they will almost certainly communicate by means of photons. This means that developing quantum optical information technology is essential for our quantum future. This is the goal of our group.

We define the following five basic construction blocks of quantum optical technology:

  1. Preparation of quantum states of light
  2. Manipulating them in a controlled manner
  3. Measuring them (quantum tomography)
  4. Interfacing quantum information between light and stationary media
  5. Bringing photons into controlled interaction with each other

Machine learning has made enormous progress during recent years, entering almost all spheres of technology, economy and our everyday life. Machines perform comparably to, or even surpass humans in playing board and computer games, driving cars, recognizing images, reading and comprehension. These developments however impose growing demand on our computing capabilities, including both the size of neural networks and the processing rate. With data centers already consuming 2-3% of the electric power produced in the world, and this number growing exponentially, we are in dire need of a new paradigm to continue progressing this technology.

This new paradigm is offered by optical neural networks (ONNs): implementing artificial neural networks using optics rather than electronics. The processing of information in a neural network consists of linear operations (matrix multiplication) combined with nonlinear activation functions applied to individual units. Both these operations can be implemented optically using lenses, spatial light modulators and nonlinear optical elements. Because all these computations in an ONN layer are performed in parallel, the fundamental processing time is independent of the size of the layer. This enables processing speeds and power efficiencies orders of magnitude beyond electronic computing.

Machine learning has made enormous progress during recent years, entering almost all spheres of technology, economy and our everyday life. Machines perform comparably to, or even surpass humans in playing board and computer games, driving cars, recognizing images, reading and comprehension. These developments however impose growing demand on our computing capabilities, including both the size of neural networks and the processing rate. With data centers already consuming 2-3% of the electric power produced in the world, and this number growing exponentially, we are in dire need of a new paradigm to continue progressing this technology.

This new paradigm is offered by optical neural networks (ONNs): implementing artificial neural networks using optics rather than electronics. The processing of information in a neural network consists of linear operations (matrix multiplication) combined with nonlinear activation functions applied to individual units. Both these operations can be implemented optically using lenses, spatial light modulators and nonlinear optical elements. Because all these computations in an ONN layer are performed in parallel, the fundamental processing time is independent of the size of the layer. This enables processing speeds and power efficiencies orders of magnitude beyond electronic computing.

Optical Neural Networks

Intelligent Robotics

Today’s neural networks outperform humans in environments about which they have complete information. The next frontier is our everyday world. Allowing machines to enter the natural environment, touch, experience, learn and change it in a way that humans do will give rise to a new phase of machine learning technology: smart robotics. This technology will revolutionize society by fulfilling the dream of many generations of philosophers, engineers and visionaries: eliminating physical labour from the range of necessary human activities.

We are engaged in a variety of research activities towards smart robotics. This includes developing reinforcement learning algorithms that allow robots to adapt themselves to solving a wide class of problems, applying these algorithms to “conventional” mechanical robots as well as robotic assistants in quantum optical experiments. Finally, we use optics to develop a new generation of tactile sensors that would enable a robotic sense of touch that is comparable in its sensitivity and versatility to that of human fingers.

Today’s neural networks outperform humans in environments about which they have complete information. The next frontier is our everyday world. Allowing machines to enter the natural environment, touch, experience, learn and change it in a way that humans do will give rise to a new phase of machine learning technology: smart robotics. This technology will revolutionize society by fulfilling the dream of many generations of philosophers, engineers and visionaries: eliminating physical labour from the range of necessary human activities.

We are engaged in a variety of research activities towards smart robotics. This includes developing reinforcement learning algorithms that allow robots to adapt themselves to solving a wide class of problems, applying these algorithms to “conventional” mechanical robots as well as robotic assistants in quantum optical experiments. Finally, we use optics to develop a new generation of tactile sensors that would enable a robotic sense of touch that is comparable in its sensitivity and versatility to that of human fingers.

Quantum machine learning is an emerging interdisciplinary field that deals both with the application of quantum technology to accelerate the performance of neural networks, or, conversely, applying machine learning methods to solve problem in quantum physics.

We are interested in quantum variational optimization – the problem of finding the quantum state that best satisfies a certain criterion. Examples include determining the ground state of a certain Hamiltonian, quantum tomography (state estimation from measurements) and quantum chemistry. Hilbert space dimension, and hence the number of parameters describing the state of a quantum system, grows exponentially with its size and becomes unwieldy very quickly; hence the ability of machine learning algorithms to analyze and find regularities in large datasets is extremely useful.

The results of this research have a broad spectrum of applications, including drug and new material discovery, understanding biological processes, quantum computation and communications.

Quantum machine learning is an emerging interdisciplinary field that deals both with the application of quantum technology to accelerate the performance of neural networks, or, conversely, applying machine learning methods to solve problem in quantum physics.

We are interested in quantum variational optimization – the problem of finding the quantum state that best satisfies a certain criterion. Examples include determining the ground state of a certain Hamiltonian, quantum tomography (state estimation from measurements) and quantum chemistry. Hilbert space dimension, and hence the number of parameters describing the state of a quantum system, grows exponentially with its size and becomes unwieldy very quickly; hence the ability of machine learning algorithms to analyze and find regularities in large datasets is extremely useful.

The results of this research have a broad spectrum of applications, including drug and new material discovery, understanding biological processes, quantum computation and communications.

Quantum Machine Learning

Quantum-inspired superresolution imaging

Rayleigh’s criterion defines the minimum resolvable distance between two incoherent point sources as the diffraction-limited spot size. Enhancing the resolution beyond this limit has been a crucial outstanding problem for many years. A number of solutions that have been realized, such as those based on near-field imaging and nonlinear interactions, but they are expensive and not universally applicable. A recent theoretical breakthrough demonstrated that “Rayleigh’s curse” can be resolved by coherent detection of the image in certain transverse electromagnetic modes, rather than implementing the traditional imaging procedure.

To date, there exist proof-of-principle experimental results demonstrating the plausibility of this approach. Our goal is to test this approach in a variety of settings that are relevant for practical application, evaluate its advantages and limitations. If successful, it will result in a revolutionary imaging technology with a potential to change the faces of all fields of science and technology that involve optical imaging, including astronomy, biology, medicine and nanotechnology, as well as optomechanical industry.

Rayleigh’s criterion defines the minimum resolvable distance between two incoherent point sources as the diffraction-limited spot size. Enhancing the resolution beyond this limit has been a crucial outstanding problem for many years. A number of solutions that have been realized, such as those based on near-field imaging and nonlinear interactions, but they are expensive and not universally applicable. A recent theoretical breakthrough demonstrated that “Rayleigh’s curse” can be resolved by coherent detection of the image in certain transverse electromagnetic modes, rather than implementing the traditional imaging procedure.

To date, there exist proof-of-principle experimental results demonstrating the plausibility of this approach. Our goal is to test this approach in a variety of settings that are relevant for practical application, evaluate its advantages and limitations. If successful, it will result in a revolutionary imaging technology with a potential to change the faces of all fields of science and technology that involve optical imaging, including astronomy, biology, medicine and nanotechnology, as well as optomechanical industry.

Lab 5
Lab 4
Lab 3
Lab 2
Lab 1

Contacts

Group Leader

Alexander Lvovsky [web]
Alex.Lvovsky[at]physics.ox.ac.uk

Address

University of Oxford
Clarendon Laboratory
Parks Road
Oxford OX1 3PU, UK [map]

Want to work in our lab? Have ideas? Questions? Contact us.

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.