2006

QUANTUM & OPTICAL TECHNOLOGY

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)

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 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.

QUANTUM & OPTICAL TECHNOLOGY

Raman adiabatic transfer of optical states

Interconvertibility of single-rail optical qubits

Decoherence of electromagnetically-induced transparency in atomic vapor

Pulsed squeezed light: simultaneous squeezing of multiple modes

Leave a Reply Cancel reply

A homodyne tomography tutorial

A galery of Wigner functions

Vacuum state

Coherent state

Thermal state

Squeezed vacuum state

Single-photon Fock state

Four-photon Fock state

Schrödinger cat state

Star state

Squeezed single-photon Fock state

Storage of light via electromagnetically-induced transparency

A Wigner function tutorial