Quantum Tomography - Methods of Quantum State Tomography - Linear Inversion - Example Homodyne Tomography.

Example Homodyne Tomography.

Field amplitudes or quadratures with high efficiencies can be measured with photodetectors together with temporal mode selectivity. Balanced homodyne tomography is a reliable technique of reconstructing quantum states in the optical domain. This technique combines the advantages of the high efficiencies of photodiodes in measuring the intensity or photon number of light, together with measuring the quantum features of light by a clever set-up called the homodyne tomography detector. This is explained by the following example. A laser is directed onto a 50-50% beamsplitter, splitting the laserbeam into two beams. One is used as local oscillator (LO) and the other is used to generate photons with a particular quantum state. The generation of quantum states can be realized, e.g. by directing the laser beam through a frequency doubling crystal and then onto a parametric down-conversion crystal. This crystal generates two photons in a certain quantum state. One of the photons is used as a trigger signal used to trigger (start) the readout event of the homodyne tomography detector. The other photon is directed into the homodyne tomography detector, in order to reconstruct its quantum state. Since the trigger and signal photons are entangled (this is explained by the Spontaneous parametric down-conversion article), it is important to realize, that the optical mode of the signal state is created nonlocal only when the trigger photon impinges the photodector (of the trigger event readout module) and is actually measured. More simply said, it is only when the trigger photon is measured, that the signal photon can be measured by the homodyne detector.

Now let us consider the homodyne tomography detector as depicted in figure 4. The signal photon (this is the quantum state we want to reconstruct) interferes with the local oscillator, when they are directed onto a 50-50% beamsplitter. Since the two beams originate from the same so called master laser, they have the same fixed phase relation. The local oscillator must be intense, compared to the signal so it provides a precise phase reference. The local oscillator is so intense, that we can treat it classically (a = α) and neglect the quantum fluctuations. The signal field is spatially and temporally controlled by the local oscillator, which has a controlled shape. Where the local oscillator is zero, the signal is rejected. Therefore, we have temporal-spatial mode selectivity of the signal. The beamsplitter redirects the two beams to two photodetectors. The photodetectors generate a electric current proportional to the photon number. The two detector currents are subtracted and the resulting current is proportional to the electric field operator in the signal mode, depended on relative optical phase of signal and local oscillator.

Since the electric field amplitude of the local oscillator is much higher than that of the signal the intensity or fluctuations in the signal field can be seen. The homodyne tomography system functions as an amplifier. The system can be seen as an interferometer with such a high intensity reference beam (the local oscillator) that unbalancing the interference by a single photon in the signal is measurable. This amplification is well above the photodetectors noise floor.

The measurement is reproduced a large number of times. Then the phase difference between the signal and local oscillator is changed in order to ‘scan’ a different angle in the phase space. This can be seen from figure 4. The measurement is repeated again a large number of times and a marginal distribution is retrieved from the current difference. The marginal distribution can be transformed into the density matrix and/or the Wigner function. Since the density matrix and the Wigner function give information about the quantum state of the photon, we have reconstructed the quantum state of the photon.

The advantage of this method is that this arrangement is insensitive to fluctuations in the frequency of the laser.

The quantum computations for retrieving the quadrature component from the current difference are performed as follows.

The photon number operator for the beams striking the photodetectors after the beamsplitter is given by:

,

where i is 1 and 2, for respectively beam one and two. The mode operators of the field emerging the beamsplitters are given by:

The denotes the annihilation operator of the signal and alpha the complex amplitude of the local oscillator. The number of photon difference is eventually proportional to the quadrature and given by:

,

Rewriting this with the relation:

Results in the following relation:

,

where we see clear relation between the photon number difference and the quadrature component . By keeping track of the sum current, one can recover information about the local oscillator’s intensity, since this is usually an unknown quantity, but an important quantity for calculating the quadrature component .