Squeezed states of light

In quantum physics, light is in a squeezed state[1] if its electric field strength Ԑ for some phases

For a displaced coherent state, the expectation (mean) value of the electric field shows an oscillation, with an uncertainty independent of the phase (a).

The normalization chosen here has the nice property that the sum of the ground state variances directly provide the zero point excitation of the quantized harmonic oscillator

, span a phase space diagram, and the third axes provides the quasi probability of yielding a certain combination of

For squeezed states, the Wigner function has a Gaussian shape, with an elliptical contour line, see Fig.

In case of a single freely propagating monochromatic laser beam, the individual measurements are performed on consecutive time intervals of identical length.

A quantum statistical characterization through consecutive measurements on some sort of a carrier is thus always related to a specific frequency interval, for instance described by

If for instance the position of one interferometer mirror vibrates and thereby causes an oscillating path length difference, the output light has an amplitude modulation of the same frequency.

Independent of the existence of such a (classical) signal, a beam of light always carries at least the vacuum state uncertainty (see above).

This is the reason (in fact the only one) why Michelson interferometers for the detection of gravitational waves use very high optical power.

Mirror surfaces absorb parts of the light, become warmer, get thermally deformed and reduce the interferometer's interference contrast.

What has to be replaced is just the vacuum uncertainty in the difference of the phase quadrature amplitudes of the light fields in the arms, and only at modulation frequencies at which signals are expected.

For this the squeezed field has to be in the same mode as the bright light, i.e. has to have the same wavelength, same polarisation, same wavefront curvature, same beam radius, and, of course, the same directions of propagation in the interferometer arms.

For the squeezed-light enhancement of a Michelson interferometer operated at dark fringe, a polarising beam splitter in combination with a Faraday rotator is required.

In case of perfect quantum efficiency (100%), such a detector is supposed to convert every photon energy of incident light into exactly one photo electron.

Without any decoherence during generation, propagation and detection of squeezed light, the uncertainty product has its minimum value of 1/16 (see above).

If the inference shows an uncertainty smaller than that of the vacuum state, EPR correlations exist, see Fig.

A and B measure repeatedly and simultaneously (taking the different propagation times into account) one of two orthogonal quadrature amplitudes.

From the remaining data they make public a small but statistically significant amount to test whether B is able to precisely infer the measurement results at A.

In addition to conventional QKD, the test for EPR correlations not only characterizes the channel over which the light was sent (for instance a glas fibre) but also the measurement at the receiver site.

Typical materials are lithium niobate (LiNbO3) and (periodically poled) potassium titanyl phosphate (KTP).

of the steady-state light power inside the resonator gets transmitted towards the left and interferes with the beam that was retro-reflected directly.

For an empty loss-less resonator, 100% of the light power would eventually propagate towards the left, obeying energy conservation.

Squeezing resonators are usually operated slightly below threshold, for instance, to avoid damage to the photo diodes due to the bright down-converted field.

[32] Squeezed states of light can be fully characterized by a photo-electric detector that is able to (subsequently) measure the electric field strengths at any phase

Further components of the BHD are a balanced beam splitter and two photo diodes (of high quantum efficiency).

The two interference results in the beam splitter output ports are detected and the difference signal recorded (Fig.

Changing the differential propagation length before the beam splitter sets the quadrature angle to an arbitrary value.

The following should be stated at this point: Any information about the electro-magnetic wave can only be gathered in a quantized way, i.e. by absorbing light quanta (photons).

However, a BHD cannot resolve the discrete energy transfer from the light to the electric current, since in any small time interval a vast number of photons are detected.

Fig. 1(f): Left: Wigner function of a squeezed vacuum state. Right: Connection to Fig. 1 (e).
Fig. 2: Normalized variances of modulation states of the same carrier light beam versus modulation frequency . Here, the measurement band width is about 10 kHz. Each trace therefore describes about 200 mutually independent modulation modes.
Strain noise spectra of the LIGO (Hanford) detector in amplitude spectral density units for frequency-dependent squeezing (purple), frequency-independent squeezing (green) and no squeezing (black) [ 18 ] [ 19 ]
Fig. 3: Schematic of a laser interferometer for the detection of gravitational waves. Here, squeezed vacuum states are injected and overlapped with the bright field at the central beam splitter to improve the sensitivity.
Fig. 4: Photo voltages of a photo diode detecting light.
Fig. 5: Measurement results on two EPR entangled light fields. The measurement values taken on one subsystem (at A) and on the other subsystem (at B) vary a lot, i.e. show a large local uncertainty. Comparing the data as shown here reveals correlations (top, blue) or anti-correlations (bottom, blue). In this example, correlations as well as anti-correlations are stronger than the vacuum state uncertainty (black).
Fig. 6: Schematic of a squeezing resonator. The pumped nonlinear crystal inside the resonator attenuates the electric field at optical frequency . This leads to perfect destructive interference for one quadrature angle that is carried by the optical frequency and propagates towards the left (left side of resonator). The pump light enters from the right and is simply retro-reflected. If the pump light intensity is kept below the resonator's oscillation threshold, its input and output powers are basically identical.
Timeline of experimentally achieved light squeezing values in the laboratory. Since the first demonstration in 1985 values have steadily improved.
Fig. 7: Balanced homodyne detector. LO: local oscillator; PD: photo diode.