## Majorana representation of symmetric multiqudit states

2021/03/04 A while ago we introduced the Majorana representation for mixed states (or in fact any hermitian operator). Now, in a collaboration with colleagues in Mexico and Krakow, we figured out a Majorana representation of symmetric multi-qudit states. These are pure states of several spins-j (or pseudo-spins-j, i.e. just quantum systems with finite dimensional Hilbert space), with j arbitrary, generalizing Majorana's original work for j=1/2. Central to the construction is again a decomposition into SU(2) irreps, where now, however, typically several values of the total angular momentum J are needed (whereas Majorana's only needs the case of J=N/2, for N spins-1/2). As for the case of mixed quantum states, the relative weights and phases for these different SU(2) irreps are coded in a "spectator state". We use the new tool to explore a large number of interesting quantum states, including states that lead to optimal sensitivity for sensing rotations, either about a fixed axis or averaged over all axes.

## Completely synchronizing quantum channels

2021/03/02 Some of the most spectacular effects of quantum mechanics appear when a large number of subsystems are found in the same quantum state. We wrote previously about this effect in the context of quantum-phase synchronization. Together with colleagues from Krakow, we now solved the full problem of synchronizing the (reduced) quantum states of an arbirtary number of quantum systems with finite-dimensional Hilbert space, in the sense of finding necessary and sufficient conditions for such "quantum-state synchronization", proving tight lower bounds on the environment's Hilbert space dimension, and providing a construction for the joint unitary that propagates the subystem and the environment. And we show experimentally on an IBM quantum computer that this quantum-state synchronizer really works for the simplest case of two qubits with a single qubit as environment!

## Optimal estimation of time-dependent gravitational fields with quantum optomechanical systems

2020/09/01 We have a new paper in Phys. Rev. Research on how to best measure small gravitational fields using optomechanical systems. It builds on our previous work that allows us to solve the non-linear dynamics of a cavity with a movable mirror whose motion is light-pressure coupled to the electro-magnetic field inside the cavity, and which is driven by an external force. Based on our results we estimate that it should be possible in the near future to detect the gravitational field of an oscillating nanogram mass. We also identify the experimental parameter regime necessary for gravitational wave detection with a quantum optomechanical sensor.

## Entangling - or not?

2020/06/26 With the availability of first small-scale and yet imperfect quantum computers the question whether these devices are "really quantum" has become center stage. Now, if there is one thing that most people would agree upon is "really quantum", it is quantum entanglement. Indeed, it has been known for a long time that a pure-state quantum algorithm that does create only limited amounts of entanglement can be efficiently simulated classically. So the bare minimum one would request from a quantum computer is that it can entangle two qubits - and keep that entanglement indefinitely. While the existing quantum computers have no problem with the first part, they typically loose the entanglement again after a relatively short time (in fact, even keeping a single qubit "alive", i.e. in the chosen coherent superposition of its basis states over arbitrarily large time is still not possible at the time or writing!). Hence, finding out whether a quantum channel can create entanglement (the so-called quantum-channel separability problem) over a certain time is still an important task - and one that had no general solution so far. Now we provide a full solution of this problem by mapping it to a problem of "truncated moment sequences" - following the succssful recipe for quantum states that we developed earlier. The approach leads to a semi-definite algorithm that provides a definite "yes" or "no" answer in finite time. And while in practice the algorithm is very heavy on computational resources we provide examples where it succeeds where other simpler criteria have failed... Find out more in our new preprint!

## New research project approved

2020/06/05 Our new research project "Experimental demonstration of quantum chaotic sensors" together with Prof. Fortagh and Dr. Günther was approved for funding by the Federal Ministery of Economy and Energy.

## Lukas Fiderer successfully defends his PhD thesis

2020/05/13 Corona obliging, we had an online viva, but Lukas defended his PhD thesis with flying colors! Congratulations!!

## New research project approved

2020/03/27 Our new research project "Towards local measurements of space-time geometries with optical clocks" was approved for funding by the DFG. A post-doc position is available, see here!

## Quantum router of light

2020/03/09 We have a new preprint out on a quantum-router of light. It allows one to deflect weak light pulses with a single photon (or a coherent state) into an arbitrary direction in the plane that can be set fast and purely electronically, i.e. without using any mechanical elements such as moving mirrors. It works by generalizing existing quantum memories of light based on absorption in and re-emission from atomic clouds in the sense that after the storage process, when the light has been transformed into a spin wave in the atomic ensemble, a magnetic field gradient is applied that changes the phases of the components of the quantum state in Hilbert space. This leads effectively to a deflection of the spin-wave, and when it is transformed back into a pulse of light by switching back on a control-beam, the photon is re-emitted into a new direction in a controled way. This is a nice new example that shows how phases in Hilbert space control the dynamics of quantum objects in real space. After optimization the energy efficiency is comparable to the one in forward direction.

## Neural-Network Heuristics for Adaptive Bayesian Quantum Estimation

2020/03/04 We have a new preprint out which shows that machine learning can be used to efficiently design experiments that are optimized for maximal information gain. The setup is Bayesian quantum parameter estimation, where knowledge about parameters to be estimated is updated with every new result of a quantum measurement, and choosing the optimal next experiment depends on the knowledge already accumulated. One would like to have fast design rules with which one can calculate the optimal parameters for the next run, so-called "heuristics". We demonstrate that neural networks, once sufficiently trained (we use reinforcement learning, with the cross-entropy method and trusted-region-policy optimization) can provide such heuristics. We examine the approach in detail for the fundamental and technologically important problem of estimating precession frequency and damping of a single spin-1/2, for which we show that the trained neural network can outperform substantially the previously known heuristics. Commited to open science, we provide the source codes, which should allow experimentalists to easily integrate these heuristics into their measurement setup.

## Switching the condensate magnetization in a dipolar spinor gas

2020/02/20 Dipolar spinor gases of magnetic atoms carry a magnetization and can behave effectively as a ferro-magnet. Due to the relatively large distance between the atoms, the spins interact via the long-range dipole interaction rather than the usual Heisenberg interaction familiar in solid ferro-magnets. For sufficiently small sample sizes the sample behaves like a single domain, i.e. all magnetic dipoles point in the same direction. In a long cigar-shaped sample, they point preferably in the direction of the long axis, but can be switched by applying an external magnetic field. In a new preprint we show that the dynamics of the magnetization of a dipolar spinor Bose Einstein condensate is governed by a **modified Landau-Lifshitz Gilbert equation**, where the external field is modified by **an effective dipole field** arising from the magnetic dipoles themselves - if one makes the standard assumption of a scalar damping parameter. In a suitable parameter regime the dipole field **accelerates the switching substantially**. That´s great news for **measuring the damping constant in the gas**, of which so far only little is known, both theroetically and experimentally, as the typical damping time scale of density fluctuations is too long compared to the life-time of the gas in a typical experiment. We also show that in the limit of very slow switching one recovers the celebrated Stoner-Wolfarth model, one of the pilars of our understanding magnetization switching in small ferromagnets.

## Quantum metrology with opto-mechanical systems

2019/10/11 Opto-mechanical systems couple optical and mechanical degrees of freedom. One examples is a thin membrane that serves as mirror on one side of an optical cavity. The light pressure can, at sufficiently high intensity, push the elastic membrane a little bit to the side, modifying at the same time slightly the resonance frequency of the cavity, and so the light field and the mechanical object couple to each other. Even though a mode of the light and the elastically suspended membrane can be desrcibed by harmonic oscillators, the coupling between them is non-linear, which makes it difficult to solve the dynamics. Owing to our previous work leading to the exact analytical solution of the quantum dyamics of the system that makes use of a finite Lie-algebra created by the commutators of different parts of the hamiltonian, we now calculated the full expressions for the quantum Fisher information for all parameters of the system, including the possibility of linear or parametric driving with external forces. This allows us to predict what are the most precise measurements possible with such systems. For certain types of measurements, resonant driving can lead to a large improvement of sensitivity. Find out more in this preprint!

## Majorana representation for hermitian operators

2019/10/17 In 1932, Ettore Majorana published a famous paper in which he showed how to visualize quantum states of any quantum system with finite-dimensional Hilbert space by a configuration of stars on the unit-sphere - the stellar representation was born. This was 6 years before he disappeared mysteriously on a boat trip from Palermo to Naples, where he had become a professor of theoretical physics just a year before. But his stellar representation lived on, owing to its beauty and invariance properties. E.g. when the physical system is a spin or angular momentum and you rotate the lab and then look at the state, its stellar configuration has rigidly rotated just in the same way as the lab, even though they exist in completely differen spaces, of course. For the most classical states, all the stars coincide. The more "quantum" the state, the more the stars are spread out. We have used the stellar representation earlier to visualize for example the "queens of quantum", the spin-states with the most "quantumness" possible.

Now we have generalized the stellar representation to mixed quantum states, i.e. states with a certain classical uncertainty, as for example in thermal states, and in fact even to any hermitian operator on a finite dimensional Hilbert space! We show how important operations such as taking the partial trace of the density matrix, or products of operators can be expressed in terms of the polynomials that determine the stars of the representation, compare representations of Schrödinger cat-states with their decohered mixtures, and show the connection to the tensor-representation of spin-states, as well as the Husimi- and P-functions, i.e. quasiprobability distributions familiar mostly from quantum optics that also describe the state of the system. Read more and discover some nice examples of stellar representations in this preprint...

## More sensitive quantum sensors with reinforcement learning

2019/08/22 At least since Alpha Go made headlines by beating a human world champion in the board game of Go, machine learning, and in particular reinforcement learning has been in the limelight of public and scientific interest. How we can profit from machine learning in all disciplines of science is a major long-term research topic, one that is highly relevant for society, and also part of the scientific program that won the University of Tübingen again its place in the nobel class of "Excellence Universities". In this new preprint we show that** reinforcement learning can strongly improve the sensitivity of quantum sensors**. These are devices with which one can measure e.g. magnetic fields already now extremely precisely - the current world record is below a femto Tesla per square root of Hertz (roughly hundred thousand million times smaller than the magnetic field of Earth at its surface) - and this by using a tiny little gas cell with about 10 billion atoms! Such sensors are already used for monitoring more precisely heart and brain activities than was possible before. Recently, we had found that these sensor can be improved substantially by "kicking" them periodically with laser pulses, rendering them chaotic. Now our analysis shows that even more sensitivity can be gained by adjusting the laser pulses via re-inforcement learning. In addition, the sensor becomes more robust against unavoidable decoherence: the machine learns by itself how to optimally shoot the laser pulses to maximize the sensitivity, and can even adapt to changing environments. We were stunned!

## Trade-off relations for operation entropy of complementary quantum channels

2019/08/09 Quantum channels map the quantum mechanical state of a system to another one. Non-unitary maps arise from interactions with an environment, which leads typically to decoherence and the destruction of quantum mechanical interference effects. Less studied is what happens to the environment in such a case. The map that propagates the system´s initial state to the environment´s final state is called the "complementary channel". In the case of a quantum communication channel you can think of it as propagating quantum information from the system to an evesdropper. The loss of quantum information measured by the entropy of the channel is zero if the system evolves unitarily, but can become large in the case of strong interaction. It turns out that there is a trade-off relation between between the entropies of the channel and the complementary channel: The sum of the entropies of the original channel and its complementary channel is bounded from below. Hence, if the original system evolves unitarily, its channel entropy is zero, whereas the entropy of the complementary channel is maximal, meaning that an eavesdropper gets no information, and vice versa. We prove the exact lower bound in the case of two qubits, find the channels that saturate it, and conjecture the bound for higher dimensional systems based on extensive numerical evidence. Apart from the mathematical insight into quantum information processing, this work also led to graphs of stunning beauty...

## Time-evolution of nonlinear optomechanical systems

2019/08/02 Non-linear systems are notoriously difficult to solve. However, there are exceptions. In this new preprint we describe how we can exactly solve the dynamics of an optomechanical system consisting of a cavity mode and a movable mirror, or any other mechanical oscillator such as a trapped micro-bead, to which the light couples through its light pressure. The solution is based on realizing that the operators in the Hamiltonian form a closed Lie algebra, which allows one decouple it into several parts. This works in principle even for arbitrary time-dependent parameters, with the caveat that the resulting differential equations that need to be solved may only be solvable numerically and might be unstable. Nevertheless, this method marks real progress compared to previous analyses of such a systems that approximated the dynamics by assuming that Gaussian initial states remain approximately Gaussian. This is something we can check now via our exact solutions, and we also investigate the interplay between non-Gaussianity and squeezing. This is a milestone towards rigorously investigating the usefulness of this system for quantum metrology, in particular the measurement of extremely weak forces.

## Outcoupling from a Bose-Einstein condensate in the strong-field limit

2019/06/12 Have you ever seen a laser that emmits not light, but cold, coherently propagating atoms? Have a look at our new preprint that investigates such matter waves both theoretically and experimentally! The bose laser works essentially by producing a Bose-Einstein condensate of atoms trapped in a harmonic potential, and then cutting a hole into the trap, such that atoms simply fall out. While none of this is new, lead author Caroline Arnold developed full 3D simulations and was able to probe the regime of strong outcoupling. She found that the intensity saturates in that regime - a purely theoretically finding first, that was then confirmed by experiments here in Tübingen.

## Quantum parameter-estimation of frequency and damping of a harmonic-oscillator

2019/05/20 The harmonic oscillator is one of the most important systems in all of physics, be it classical or quantum. Its importance arises from the fact that a.) for small amplitudes most stable systems can be approximated by a set of harmonic oscillators, and b.) that it can be solved exactly. Moreover, it has only two free parameters: its frequency and its damping. How precisely can one determine these in principle? While this question was answered many years ago for an undamped oscillator in this paper, in reality there is always some damping, and taking it into account can change the result a lot. In a new preprint, we solve the problem for Gaussian states, which are a broad class of experimentally relevant states. We predict that with existing carbon nanotube resonators it should be possible to achieve a mass sensitivity of the order of an electron mass per square root of Hz, orders of magnitude lower than the current record, due to the shift of frequency from an adsorbed mass.

## Maximal sensitivity for mixed states

2019/05/15 More often than not, quantum states come with a certain amount of uncertainty about themselves, which means in quantum mechanics jargon that they are "mixed states", rather than pure ones. This uncertainty reduces on average the nice quantum mechanical effects such as quantum interference that we want to use for enhancing e.g. sensitivity of sensors. The question is then, how to optimally prepare an initial mixed state through unitary transformations to obtain maximal sensitivity in a subsequent evolution that encodes the parameter to be measured in the state. This is an old open problem that we resolved completely in recent work. Here is how.

## Fast entanglement detection

13/2/2019 With the increasing size of existing quantum processors, the question of how to test their functionality as efficiently as possible has become a challenge. In a new preprint, "Optimal measurement strategies for fast entanglement detection" we show that one of the core goodies of quantum information processing, namely quantum entanglement, can be verified much more rapidly than through full state tomography. The approach is based on our work on truncated moment sequences, which deals very naturally with missing data. One set of measurements turns out to be particularly efficient.

## Farday effect and optical activity in the gravitational field of a laser beam

11/12/2018 Another chapter in our exploration of the gravitational interaction of light with light: Our preprint "Rotation of polarization in the gravitational field of a laser beam - Faraday effect and optical activity" is now finally out. It turns out that the gravitational field of a circularly polarized laser beam leads to the rotation of the polarization of a probe beam in its vicinity - a gravitational effect that is clearly beyound the scope of Newtonian gravity, but that can be precisely calculated using General Relativity. The rotation of polarization mixes effects reminiscent of the Faraday effect and optical activity in media which have been known for a long time in optics. In the latter, the effect is undone when the probe-beam propagates back, in the former, it is doubled. Unfortunately, the magnitude of the effect is very very small: of the order of the power of the laser beam divided by the Planck power, where the latter is the Planck energy divided by the Planck time. Optimistic estimates for the current most powerful lasers let us expect rotation angles of order 10^{-32 }- and that includes already an amplification using an optical cavity. Clearly not an effect that will be measured in the lab any time soon. Yet, the effect is of substantial fundamental interest, as it demonstrates gravitational spin-spin coupling in the well-defined and tested frame-work of General Relativity, while the effect is also predicted by certain quantum gravity theory candidates.