Generation of squeezed vacuum in atomic ensembles

Jump to applications:
Generation of squeezed vacuum with optical vortex crossection
Superluminal propagation of squeezed vacuum
Spectral noise filtering with EIT

Any electromagnetic signal is subject to the laws of quantum mechanics.  This means that due to the Heisenberg uncertainty principle, any optical measurement will have on top of it, noise due to quantum fluctuations, even if you are measuring the vacuum itself.  This limits many measurements to having a minimum non-zero noise, called shot noise, or the standard quantum limit. 
This limit can be beaten however by using a special quantum state of light  called a “squeezed state”.  Squeezed light exhibits non-classical statistics where we can measure noise levels below the shot noise limit.  If we consider light in the quadrature picture, where the electric field is expressed as E(x, t) =X1(x,t)cos(ωt) +X2(x,t)sin(ωt) we can measure the quadratures X1 and X2 independently.  For classical fields like coherent lasers or the vacuum, the noise of these quadratures is equal and down to the shot noise limit.  But for squeezed states, we can “squeeze” the noise of the quadrature we are measuring while “stretching” the other quadrature in compensation.

Squeezed light finds applications in precision measurements as well as optical communications where the signal to noise needs to be as low as possible.  Due to it's special quantum nature, squeezing may also be applied in other related areas of quantum optics and quantum information.
Squeezed states of light may result from one of several nonlinear interactions of the light with matter such as those caused by parametric amplifiers, frequency doublers, and Kerr media.  The main squeezing source we study is a nonlinear light-atom interaction called polarization self-rotation.
The polarization self-rotation (PSR) effect occurs when elliptically polarized light propagates through an atomic medium (in our case 87Rb vapor) which causes the axis of ellipticity to rotate.  Since the intensity of the left and right circular polarization components are different in elliptically polarized light, this leads to unequal AC-Stark shifts and optical pumping of the different atomic Zeeman sublevels resulting in circular birefringence. Phase differences in the propagation of the two circular components of the light result in the polarization ellipse rotation.

In our experiment, we send linearly polarized laser light, made slightly elliptical by vacuum fluctuations of the horizontal polarization, through a glass cell containing hot 87Rb atoms.  This leads to polarization self-rotation as the light propagates, resulting in a squeezed vacuum state of the horizontal polarization at the output of the cell.  By mixing this squeezed vacuum field with a strong local oscillator field in a homodyne detection arrangement and sweeping the phase between these fields, we can measure the quadrature fluctuations of the squeezed vacuum.  Depending on experimental parameters, we have observed squeezing at several laser frequencies near the Rubidium atomic transitions with our highest level of squeezing to date being about 2 dB of noise suppression.
Our PSR and squeezing studies have concentrated on optimizing experimental conditions for the highest degree of noise suppression while gaining a better understanding of the squeezing process and how differing conditions will influence it.  We also have carried out studies of PSR and quadrature noise in cold Rubidium atoms held in a trap at temperatures of only hundreds of microkelvin.  The current installment of the squeezing experiment sends the squeezed vacuum through an atomic medium under the conditions of electromagnetically induced transparency (EIT) to study the effects of EIT noise filtering and leading up to the slowing and storage of squeezed vacuum.

Spatial Correlation of Squeezed Quantum Noise

While many experimentally measured characteristics of vacuum squeezing, based on polarization self-rotation in resonant atomic vapor are well described by the existing theory and numerical calculations, there is still a large unpredicted excess noise at higher atomic densities. One possible explanation is a distortion of the squeezing and/or pump field transverse profiles, since it deteriorate the mode matching between the squeezed vacuum field and the local oscillator (pump field).
To check this hypothesis, we have conducted simultaneous measurements of maximum measured degree of squeezing and the output waist of the laser beam, changing laser power and atomic density. In addition to a regular Gaussian laser beam, we used a first order optical vortex mask, placed before the cell, to produce a Laguerre-Gaussian beam (shown in Fig. 2.1). In this case, the laser beam becomes donut-shaped, with zero intensity in the center. It’s radial intensity distribution can be described by

, where w indicates the beam size. Because of this more complex geometry, we expect the vortex beam to be more sensitive to spatial variations. 
The results of the measurements are shown in Fig. 2.2, and provide some interesting insights. First of all, when the beam was spatially modified, the optimal conditions for generation of maximally squeezed optical field are different: for a Gaussian beam the maximum squeezing was observed at lower pump power and somewhat higher temperature. At the same time, both laser beam profiles showed similar trend in increasing size with atomic density. Such behavior seemed to be largely independent of laser intensity. Such behavior is a strong indication of the linear optics phenomenon, and thus it is unlikely that this self-defocusing affected two orthogonal optical fields differently, ruling out such mode mismatch as the main origin of the increasing quantum noise at high optical densities.

Superluminal squeezing propagation

Classically, an electromagnetic wave can be described with an amplitude and a phase, but on the quantum level it is a flux of photons. This makes it subject to quantum fluctuations – uncertainties in the amplitude and the phase. A typical laboratory laser will produce a beam, which can be closely described as a coherent state of light, a state with minimum uncertainty. Any measurement done on a coherent state will exhibit this minimum uncertainty by having so called shot noise. There are a lot of measurements, where it is the limiting factor. But luckily there is a way to reduce the noise below the shot noise limit.
A coherent state of light has equal uncertainties in the amplitude and the phase. Their product must be equal or greater than the minimum value of 1/4. But there are no limitations on the value of each particular quadrature. So one can, for example, reduce the amplitude noise at the expense of increasing the phase noise. If the uncertainty of one of the quadratures is below that of a coherent state, the state is called squeezed. We study a particular type of squeezed states called vacuum squeezed states, which have zero mean amplitude. For squeezed vacuum states uncertainties in the amplitude and the phase loose their meaning, and we talk about uncertainties in the quadratures X1 and X2.
It is well known, that one can use a resonant media that exhibits anomalous dispersion to produce superluminal classical signals (signals with the group velocity vg>c or vg<0). However, it is not clear from the theoretical point of view [7, 8], whether it is possible to observe the superluminal propagation of quantum fields. Here we experimentally demonstrate superluminal propagation of a squeezed vacuum state (a quantum field).


The experimental setup is shown on Fig. 1. The squeezed vacuum field is generated by shining a horizontally polarized laser beam with the wavelength of 795 nm into a glass cell, containing resonant Rubidium atoms. This laser beam is called pump. Because of the non-linear nature of the squeezing process, we focus the beam using lens L1 to make a more intense beam inside the squeezing cell. The beam is collimated back using lens L2. Under the right conditions the coherent vacuum in the vertical polarization becomes squeezed, and we are able to manipulate its group velocity in the second cell called interaction cell in Fig. 1. The strong horizontally polarized pump creates conditions for Faraday rotation in the second cell and for certain pump powers vertically polarized squeezed vacuum field experiences anomalous dispersion that leads to superluminal group velocities. We are able to bypass the interaction cell using two mirrors to determine the reference vg=c . We then detect the squeezed vacuum using a homodyne detector [9]. The signal is fed into a spectrum analyzer and recorded on a digital oscilloscope.

In order to determine the group velocity we modulate the level of squeezing by applying a small magnetic field to the squeezing cell [10]. This degrades the conditions for squeezing generation and leads to modulation show in Fig. 2. The modulation frequency was 3 kHz. We then fit the traces and determine the relative phase between the bypass channel and the interaction channel. This gives us the possibility to calculate the time difference Δ T between the signals arrival time. Values that are bigger than zero indicate a delay, while values that are less than zero indicate advancement. Our result for different pump powers is presented in Fig. 3.

In conclusion, we have experimentally demonstrated that it is possible to observe superluminal squeezed vacuum propagation. The largest measured advancement was 3 μs.

Spectral quantum noise filtering with EIT

The frequency-dependent transmission properties of an EIT medium can be used to manipulate to noise characteristics of a squeezed vacuum or squeezed light state. In Figure 4, we show an example of an EIT transmission window acting as a low-pass filter for the amplitude of squeezed and antisqueezed noise. Outside of the EIT resonance window, the squeezed vacuum photons are absorbed and so the noise amplitudes are filtered, becoming frequency-dependent. We have studied several such interactions of squeezed states of light with resonant atomic media to see how the quantum noise can be influenced. These studies are important to the implementation of squeezed states in quantum memory and quantum information protocols using resonant atoms.


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