One example is one-way quantum computation with a quantum frequency comb 13 , 18 , which forms the most promising realization of scalable quantum information to date. This approach exploits the large bandwidth of frequency mode pairs from a single parametric oscillator two-mode squeezed vacuum as a set of quantum modes Q-modes , where coupling among near Q-modes demonstrated the largest entangled cluster states to date along with a complete set of quantum gate operations The number of parallel Q-modes is dictated by the squeezing bandwidth of the parametric oscillator, which can extend up to a full optical octave by rather simple means limited only by phase matching of the nonlinear interaction 19 , 20 , The limitation of this approach to quantum computation is the bandwidth of the measurement, where each Q-mode requires a separate homodyne detection using a precise pair of phase-correlated LOs.
A broad bandwidth of Q-modes requires a dense set of correlated LOs and multiple homodyne measurements, quickly multiplying the complexity to impracticality. In our experiment, we simultaneously measure the entire bandwidth of a broadband two-mode squeezed vacuum with only one LO—the pump field that generates the squeezed light to begin with. Another example is in quantum key distribution, where enhanced bandwidth was employed to increase the data rate by increasing the number of bits per photon. The concept here is to divide the photon readout time, which is limited by photo-detectors, into multiple short time-bins, which act as an additional time stamp for each photon or pair 22 , Here we present a different approach to optical homodyne, which resorts to a broadband optical nonlinearity—parametric amplification, as the field multiplier.
Using this method we measure the entire bandwidth simultaneously with a single homodyne device and a single LO.
Specifically, since parametric gain only amplifies one quadrature of the input signal but attenuates the other, analysis of the output spectrum enables evaluation of the input quadratures. Due to the parametric amplification of the quadrature of interest, our measurement is insensitive to detection inefficiency and to the added vacuum noise it introduces.
With sufficient parametric gain, any given x quadrature can be amplified to overwhelm the attenuated orthogonal y quadrature, even if it was originally squeezed, such that the resulting output signal is practically proportional only to the input x quadrature. Even if the parametric gain in the measurement is not high enough to completely diminish the y quadrature, measurement is simple, once the desired x quadrature is sufficiently enhanced above the vacuum level.
Specifically, two orthogonal measurements, one for each quadrature, enable extraction of both quadratures average over the entire optical bandwidth, as detailed hereon. The basic concept of our method for broadband homodyne detection is illustrated in Fig. To describe the effect of the parametric amplifier in Fig. We note that the parametric amplifier used in the measurement need not be ideal. Specifically, since the attenuated quadrature is not measured, it is not necessarily required to be squeezed below vacuum, only to be sufficiently suppressed compared to the amplified quadrature.
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Consequently, restrictions on the measurement amplifier are considerably relaxed compared to sources of squeezed light, allowing it to operate with much higher gain. The common source for squeezed light or squeezed vacuum, in our experiment, is also a parametric amplifier. If the amplification is spontaneous vacuum input , the amplifier attenuates one of the quadratures of the vacuum input state, squeezing its quantum uncertainty.
For measuring the squeezing, we exploit the same nonlinearity and the same pump that generates the squeezed state in the first place, thus guaranteeing a bandwidth match of the homodyne measurement to the squeezing process. The quadrature information over a broad frequency range is obtained simultaneously by measuring the spectrum of the light at the output of the detection parametric amplifier. The experimental demonstration of broadband parametric homodyne consists of two parts see Fig. We generate broadband squeezed vacuum by collinear FWM in a photonic crystal fiber PCF that is pumped by narrowband picosecond pulses near the zero dispersion wavelength of the PCF.
To measure the generated squeezing, we couple the light generated by the FWM process together with the pump into another PCF, which acts as the measurement parametric amplifier in the experiment this was the same PCF in the backward direction.
After this second measurement pass we record the parametric output spectrum to extract the quadrature information see Fig. Experimental schematic of the parametric homodyne. The experiment consists of two parts: 1 generation of broadband squeezed light and 2 homodyne measurement of the generated squeezing.
After generation, the pump is replaced by an appropriately delayed copy of the original pump light, via a narrowband filter, which allows independent intensity and phase control, to tune the parametric gain and to select the specific quadrature to be measured. After this second measurement pass through the amplifier, the pump is separated from the FWM light by a narrowband filter and the FWM light is measured by a spectrometer.
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The procedure of parametric homodyne. Measurement of the quadratures includes three stages: a raw output measurement, b calibration, and c quadrature extraction.
The specific quadrature to be amplified is defined by tuning the pump phase. The reduction of the raw output beneath the vacuum input level dashed green directly indicates squeezing. The inset shows the effect of loss at the input FWM light on the parametrically amplified output. As the loss is increased, the squeezing is reduced and the observed fringe minima rise towards the vacuum level vertical arrows even though the total input intensity is considerably decreased a non-classical signature. Since squeezed vacuum is a gaussian state, its quadrature distribution is completely defined by the second moment.
Measurement of the instantaneous intensity distribution is possible with a shorter integration time, but not necessary for squeezed vacuum. Fringes appear across the output spectrum of the measurement parametric amplifier due to chromatic dispersion in the optical components filters, windows, etc. Thus, for some frequencies the stretched quadrature is amplified bright fringes while for others the squeezed quadrature is amplified relatively dark fringes , as seen in Fig. The specific quadrature to be amplified can be controlled by the pump phase see Methods for more details on the experiment.
The broadband squeezing is evident already from the raw output spectrum, shown in Fig. To verify this, we varied the squeezing by varying the loss of the input FWM field before the measurement second pass through the PCF.
As the loss is increased, the squeezing slowly vanishes, and even though the total power entering the fiber is diminished, the minimum fringes at the output of the measurement amplifier rise towards the vacuum input level, as shown in the inset of Fig. The extraction of the quadrature information from the measured parametric output assumes knowledge of the parametric gain.
The calibration of the parametric amplifier is simple, performed by recording the output spectrum for a set of known inputs Fig. For example, the vacuum level of the parametric amplifier is observed when both the signal and the idler-input fields are blocked I zsi —zero signal idler. This calibration process is fully described in the Methods. After calibration, we obtain the parametric homodyne results of Fig. The observed squeezing in our experiment is far from ideal, primarily due to the fact that the pump is pulsed, which induces an undesirable time dependence of both the magnitude and phase of the parametric gain in the squeezing process, as well as in the parametric homodyne detection via self-phase and cross-phase modulation—SPM and XPM.
Electromagnetic Noise and Quantum Optical Measurements (Advanced Texts in Physics)
Since our pump pulses are relatively long, their time dependence can be regarded as adiabatic, indicating that the instantaneous squeezing source and parametric amplification measurement are ideal, but the quadrature axis, squeezing level, and gain of the two amplifiers vary with time, not necessarily at the same rate. Thus, the measured spectrum, which represents a temporal average of the light intensity over the entire pulse, diminishes somewhat the expected squeezing see illustration in the Methods.
Even with a pulsed pump, however, the various homodyne and calibration measurements are consistent and unequivocal for weak enough pump intensity see Methods for further details on the pulse-averaging effects.
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With a pure CW pump, as is generally used in squeezing applications, this pulse-averaging limitation would not exist. We verified the properties of the parametric homodyne detection in several ways. Further increase of the pump does not improve the measured squeezing due to pulse effects, and the minimum uncertainty property deteriorates.
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Additional verification measurements of the broadband squeezing are presented and illustrated in the Methods. The generalization of the standard quadratures to two-mode quadratures requires some attention. As opposed to the standard quadrature operators, which are hermitian and represent the time-independent real amplitude of the cosine sine oscillation, the two-mode quadrature operators of Eq. Yet, the two-mode quadrature x is an observable quantity in contrast to the field operator a. Since the phase of the two-mode quadrature relates to commuting observables as opposed to the carrier phase , it does not reflect a non-classical property of the quantum light field, but rather defines the classical temporal mode in which the field is measured.
This offset, along with other mode parameters, such as polarization, spatial mode, carrier frequency, and so on define the mode of the LO. Of course, quantum entanglement is possible between the two envelope modes cosine or sine in direct equivalence to entanglement of a single photon or photon pair, or cat state between polarization modes, which is widely used for quantum information. For a broadband spectrum, standard two-mode homodyne requires a dense set of correlated pairs of LOs for each frequency component of the measurement.
As we have shown, however, in our experiment above, a single LO is sufficient to simultaneously extract a specific quadrature across the entire optical bandwidth, just as a single pump laser can simultaneously generate the entire bandwidth of quadrature squeezed mode pairs. Mathematically, our method relies on the similarity between the quadrature operators of interest Eq. Accordingly, the rotated complex quadrature operators Eq.
Parametric amplification directly amplifies one quadrature of the input and attenuates the other, as evident by expressing the field operators a s g at the output using the quadrature operators x , y of the input:. Finally, the parametric photon-number operator at the output is. When access is available simultaneously to the intensities of both the signal and the idler, their sum of intensities provides the cleanest measurement of the quadrature intensities.
Although the concept of parametric homodyne is conveniently understood in the limit of large gain, where the quadrature of interest dominates the output light field, parametric homodyne is equally effective with almost any finite gain. Indeed, the output intensity in this case will not directly reflect the quadrature intensity, but it still provides equivalent information about the quadrature at any finite gain, since two light intensity measurements along orthogonal axes uniquely infer the two quadrature intensities at any finite gain, indicating that the information content of a measurement of the output intensity is the same as that of the quadrature intensity.
An analytic derivation of this equivalence is provided in the Methods. Quantum state tomography is a major application of homodyne measurement. It allows reconstruction of an arbitrary quantum state or its density matrix or Wigner function from a set of quadrature measurements along varying quadrature axes 7. Unique reconstruction requires a complete measurement of the quadrature distribution function, which necessitates single-shot measurements of the instantaneous quadrature value, not just its average. Although both standard two-mode homodyne and parametric homodyne provide incomplete quadrature information in a single shot in somewhat different ways , they still allow reconstruction of the quantum state under some assumptions.