Quantum State Characterization

Quantum states of light show characteristics beyond the classical properties of the electromagnetic field. In particular, non-classical states such as single photon Fock states or entangled states can be seen as resources for many quantum information tasks. In our group, we use photon number resolving measurements. These enable us to generate, on the one hand, highly non-lassical states and, on the other hand, offer promising new methods for quantum state characterization.

Phase space representation of light

Every quantum state can be described as an expansion in the photon number basis |n〉. Alternatively, the phase space representation of light is an equivalent description that incorporates all quantum effects, but resembles the classical concept of amplitude and phase. Starting from the creation and annihilation operators ↠and â, one can define quadrature operators x̂ = (1 / 2) (â + â†) andŷ = (-i / 2) (â - â†). They obey the Heisenberg commutation relation and can be interpreted as position and momentum. The quantum state in phase space can then be characterized by their quasi-probability distribution, such as the Wigner function W(x, y).

The Wigner function becomes a classical distribution of amplitude and phase in the limit of bright fields. Weak fields have a blurred distribution in accordance with the Heisenberg uncertainty. It is possible to narrow the distribution in one quadrature, at the cost of widening the other. Light states which feature these distributions are called squeezed states. Moreover, the Wigner function can, for certain non-classical states, have negative values.

Figure 1: Coherent state with amplitude A and phase ϕ (left), vacuum state (center) and squeezed state (right) Bild vergrößern
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Characterization of highly non-classical states of light

The standard technique for characterizing quantum states of light, homodyne detection, measures the distribution of field quadratures but relies on the tomographic reconstruction of the Wigner function, since the Heisenberg uncertainty principle precludes simultaneous exact measurement of the field quadratures.

Nevertheless, it is possible to measure the value of the Wigner function at single points in phase space by utilizing direct probing [5,9]. The value at the origin(x = y = 0) can be reconstructed from the photon number distribution and a displacement of the state in phase space gives access to the value of the Wigner function at any point (x, y). We have implemented such a scheme and measured the Wigner function of free-propagating spectrally broadband, pulsed single photons at individual points in phase space [7], see graphic.

Figure 2: Left: Experimental data for the photon number distribution for different displacements, i.e. different distances from the origin in phase space. Right: Reconstructed Wigner function from photon number probabilities. The deviation from the theoretical curve originates from an imperfect purity of the state as well as non-ideal mode overlap with the displacement field. However, the negativity clearly shows the highly non-classical character of the state. Bild vergrößern

Photon-number resolving measurement

The implementation of the above scheme requires photo detectors with photon number resolution. Avalanche photo diodes (APDs) can detect single photons, but can only distinguish between 0 photons and more than 0 photons. We are using a time multiplexing detector (TMD), which has been introduced by Achilles et al. [2] and employed for several parametric down conversion (PDC) experiments [3,1]. It combines APDs with a fiber network, distributing incoming light into many (up to 16) time bins such that the average photon number per bin is less than 1. Post processing of the data allows for reconstruction of the true photon number probabilities independent of losses. We have also utilized a TMD to measure high order loss independent correlation functions [4], by evaluating single-event coincidences rather than averaged photon numbers. All in all, the TMD provides us with a powerful tool for state characterization.

Figure 3: Time multiplexing detector. The right part is an 8 bin time multiplexing detector. Photons are split at the first 50/50 beam splitters, delayed by different times and merged together at the next beam splitter. The number of time bins scales exponentially with the number of beam splitters. The left part is the realization of a displacement operator. Bild vergrößern
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Two mode squeezed states

Adding a second degree of freedom like polarization or space, the former single mode quantum state becomes a two mode state. Entanglement between these two modes can be generated by interfering two orthogonally polarized squeezed states at a single beam splitter as shown in the right figure [8]. The resulting Einstein Podolsky Rosen (EPR) state has a four dimensional Wigner function W( x1, y1, x2, y2) with correlations between the two modes. That means: After having measured one specific value of(x1, y1), the conditional distribution of (x2, y2) becomes narrower than the Heisenberg uncertainty (this is indicated in the right picture by the small circles whereas the unconditioned, wider distribution is indicated by the bigger circles). This is not a violation of the Heisenberg uncertainty principle, but the result of entanglement, producing stronger correlations than classically possible.

With specially engineered waveguides, we are able to produce nearly pure, two mode squeezed states [6]. We aim at characterizing these states by implementing our photon counting scheme. Measuring specific points of the two mode Wigner function would show directly the strong correlations between the two modes. A setup is sketched on the right to measure W(αreal, αimag, βreal, βimag).

Author: Georg Harder

References:

  1. Daryl Achilles, Christine Silberhorn, and Ian A. Walmsley. Direct, Loss- Tolerant characterization of nonclassical photon statistics. Physical Re- view Letters, 97(4):043602, July 2006.
  2. Daryl Achilles, Christine Silberhorn, Cezary ‘liwa, Konrad Banaszek, and Ian A. Walmsley. Fiber-assisted detection with photon number resolution. Optics Letters, 28(23):23872389, December 2003.
  3. M. Avenhaus, H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, I. A. Walmsley, and C. Silberhorn. Photon number statistics of multimode parametric Down-Conversion. Physical Review Letters, 101(5):053601, 2008.
  4. M. Avenhaus, K. Laiho, M. V. Chekhova, and C. Silberhorn. Accessing higher order correlations in quantum optical states by time multiplexing. Physical Review Letters, 104(6):063602, February 2010.
  5. Konrad Banaszek and Kryzysztof Wódkiewicz. Direct probing of quantum phase space by photon counting. Physical Review Letters, 76(23):4344, June 1996.
  6. Andreas Eckstein, Andreas Christ, Peter J. Mosley, and Christine Silberhorn. Highly ecient Single-Pass source of pulsed Single-Mode twin beams of light. Physical Review Letters, 106(1):013603, January 2011.
  7. Kaisa Laiho, Katiúscia N. Cassemiro, David Gross, and Christine Silberhorn. Probing the negative wigner function of a pulsed single photon point by point. Physical Review Letters, 105(25):253603, December 2010.
  8. Ch. Silberhorn, P. K. Lam, O. Weiÿ, F. König, N. Korolkova, and G. Leuchs. Generation of continuous variable Einstein-Podolsky-Rosen entanglement via the kerr nonlinearity in an optical ber. Physical Re- view Letters, 86(19):42674270, May 2001.
  9. S. Wallentowitz and W. Vogel. Unbalanced homodyning for quantum state measurements. Physical Review A, 53(6):45284533, June 1996.