Optimal Pure-State Qubit Tomography via Sequential Weak Measurements


A sequence of weak isotropic measurements can yield the optimal measurement of a collection of identical qubits — a POVM over the continuous set of spin-coherent states. We show here a numerical simulation for an independent sequence of collective isotropic weak measurements on an ensemble of 50 qubits. The vertical axis is a measure how close a POVM element is to a spin-coherent state projector with the value the “coherency” C=1 corresponding to a spin-coherent state. The olive region consists of the coherency for 50 samples of the measurement record as a function of time. The spheres are Husimi distributions of the POVM element (left) and the post-measurement state (right) for various times of the black sample trajectory. The POVM converges asymptotically, extracting the estimated direction of the unknown spin, whilst the state does not change drastically, because the measurements are weak.

 

It’s been known since the early days of quantum information science that the optimal way to gain information about a quantum state given many copies is to do a joint measurement of the whole ensemble. An example is seen in quantum state tomography. In 1995, Massar and Popescu proved that the optimal measurement for estimating an unknown direction of N copies of a pure qubit is a collective measurement. The realization for every N is the so-called spin-coherent-state POVM. After various attempts, by 2002 many researchers had come to the conclusion that this measurement was not practically realizable for large N. However, in recent work published in Physical Review Letters (give link) researchers at CQuIC showed that an implementation of the spin-coherent-state POVM could indeed be implemented via a sequence of collective and isotropic weak measurements without feedback or adaptivity.

Spin-coherent states are quantum states that transform in the same way as a vector under 3-dimensional rotations.This vector can be a literal direction in 3D real space but can also be an abstract direction such as in the Bloch sphere of a qubit. For N copies of a qubit, these directions are represented by states which sit in an irreducible subspace that spans N+1 of the 2^N dimensions of the total Hilbert space. This subspace is also known as the spin-J=N/2 representation of SU(2). The spin-coherent-state measurement of a spin-J system is a measurement where the outcomes are the continuum of directions which is represented by a POVM with elements that are projectors onto the spin-coherent states. The weak measurement record eventually converges on the estimated direction.

This protocol could be implemented on currently existing platforms, such can be done using the Faraday interaction to measure the collective spin projection of an atomic ensemble with a continuous laser probe.

With this new breakthrough, CQuIC graduate student Ezad Shojaee and postdoc Chris Jackson, together with Prof. Ivan Deutsch are working on applications for more generalized coherent-state measurements such as those with SU(1,1) symmetry and other semi-simple Lie group symmetries.

This work was published in the September 26, 2018 issue of Physical Review Letters:

Optimal Pure-State Qubit Tomography via Sequential Weak Measurements
Ezad Shojaee, Christopher S. Jackson, Carlos A. Riofrío, Amir Kalev, and Ivan H. Deutsch
Phys. Rev. Lett. 121, 130404

The full article can be found online.

 

Enhance atom-light coupling with a “weaker” local field

Strong atom-light coupling is the key to quantum information processing with atoms and photons. It is a common belief that a strong optical field is required to generate a strong atom-light coupling. Following this school of thought, great challenges including heating and decoherent photon scattering problems have been encountered in experiments to enhance atom-light coupling by placing atoms in the strongest trapping field, which hindered the implementing of quantum communication and quantum computing applications using atoms. Surprisingly, a recent theoretical study conducted by researchers in CQuIC and the Sandia National Labs demonstrated that, by placing the atoms at an azimuthal position where the guided probe mode of a waveguide has the lowest intensity, the atom-light coupling is the strongest for quantum measurement applications.

Fig 1. Schematic diagram of the QND measurement and spin squeezing protocol with nanophotonic waveguides based on Faraday effect.

In this study, the authors consider an atom-nanophotonic waveguide interface, where atoms are trapped in the evanescent field of a waveguide which has an effective diameter (a few hundred nano-meters) less than the wavelength of the light and supports two orthogonal guided modes. They use cooperativity to quantitatively characterize the atom-light coupling and study the enhancement of cooperativity in the atom-light interface near a nanophotonic waveguide for application to quantum nondemolition (QND) measurement of atomic spins. Here the
cooperativity per atom is determined by the ratio between the measurement strength and the decoherence rate. The two orthogonal guided modes are adiabatically connected to two orthogonal linearly polarized modes as the cross-section of the waveguide become large compared to optical wavelength. In the QND measurement protocol, as shown in the figure above, a horizontally polarized light is sent into the waveguide and becomes the H-mode in the interaction region where the atoms are trapped. By preparing the trapped alkali atoms at a certain state known as a spin coherent state, the light detected at the measurement equipment will no longer be horizontally polarized due to the Faraday interaction with the atoms. How much the polarization state of the light
is changed is a signature of the state of the collective spin of the atoms. Following the principles of quantum mechanics, when a measurement is done there is “backaction” on the state. By measuring the direction of the collective spin projected to the z-axis we dramatically reduced our uncertainty in this value from the superposition that existed in the initial spin coherent state. On the other hand, the uncertainty of the collective spin state projected to the orthogonal direction, the y-direction in this case, on the Bloch sphere has been elongated. This state is called a spin squeezed state, which has applications in precision measurements and other quantum information processing protocols. How much the spin is squeezing we can generation is determined by how strong the atoms are coupled to the probe light.

Fig 2. Spin squeezing illustrated on the Bloch sphere.

Fig 3. (a) Field components of Rel[Ex], Re[Ey] and Im[Ez] distributions for a nanofiber and a square waveguide. (b) Atoms (black stars) are better to put at the weak field positions.

What is surprising about this study is that the authors discovered that the optimal configuration to generate the strongest spin squeezing effect is defined by placing the atoms at the weakest probe mode position, which is along the vertical direction–not the strongest mode position as the top 1 common guess of people–which is the horizontal direction in the xy-plane. This arises because the QND measurement strength relies on the interference between the probe and scattered light guided into an orthogonal polarization–V-mode, while the decoherence rate depends on the local intensity of the probe defined by the H-mode.  At the position of the atom the vacuum V-Mode is strongest. Thus, by placing the atoms on the y-axis, the ratio of good to bad scattering can be strongly enhanced for highly anisotropic modes. The researchers of this work apply this idea to study spin squeezing resulting from QND measurement of spin projection noise via the Faraday effect in two nanophotonic geometries, a cylindrical nanofiber and a square waveguide. They find, with about 2500 atoms using realistic experimental parameters, about 6.3 dB and 13 dB of squeezing can be achieved on the nanofiber and square waveguide, respectively. In contrast, to generate the same amount of spin squeezing using an atomic cloud trapped by Gaussian laser beams in free space, it might require billions of atoms.

This study gives a new perspective on the design of the atom-nanophotonic waveguide interface.  Based on the approach taken here, one can prepare and readout collective spin squeezing while avoiding strongly disturbing the mechanical motion and internal states of atoms, which are often big challenges if atoms are sitting at strong field spots of the probe. By using some additional techniques, such as adding an optical cavity on the waveguide, one can generate highly nonclassical nonGaussian states, which have a variety of applications in quantum information processing. This work also marks a milestone towards atom-nanophotonic interface based quantum simulations and quantum computing applications.

Related paperPhys. Rev. A 97, 033829 – Published 16 March 2018

arXiv version: arXiv:1712.02916.

Related presentation: SQuInT 2018 workshop at Santa Fe, NM, USA.

Data and source files repository: on GitHub (comments are welcome).

Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms

THIS PAPER WAS CHOSEN AS AN “EDITORS’ SUGGESTION” IN PHYSICAL REVIEW A, Vol. 95, Iss. 4 — April 2017.

Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms

Jongmin Lee, Michael J. Martin, Yuan-Yu Jau, Tyler Keating, Ivan H. Deutsch, and Grant Biedermann

The Jaynes-Cummings model, a widely employed theoretical framework in cavity quantum electrodynamics, is experimentally tested on a platform involving Rydberg-blockaded atomic ensembles. The work opens the way to a richer exploration of protocols for quantum control or, more broadly, quantum computing.

The full article is available online at Phys. Rev. A 95, 041801(R) (2017)

 

FIG 1.  Experimental setup.  The Rydberg laser and the Raman lasers are  aligned along the x axis.  Two optical tweezers are formed by two lasers  with an angular separation θ..  In this setup, eight electrodes control the electric fields near the trapped atoms.  The bias magnetic field is applied along the x axis.