Photon-number measurement

Phase interference in optical or material systems can be utilized to achieve a type of quantum measmement, known as nondemolition measurements ([41], Chapter 19). The general objective is to make a measurement that does not change some property of the system at the expense of some other property(s) that is (are) changed. In optics, it is the phase that may act as a probe for determining the intensity (or photon number). The phase can change in the comse of the measurement, while the photon number does not [126]. 

The smallest unit (packet) of electromagnetic energy (a photon) is related to frequency by the formula, E = hv, in which E is the energy and h is Planck s constant. Alternatively, the relation can be written, E = hc/A,. Frequency (v) is a number with units of cycles per second (cps, the number of times a wavefront passes a given point in unit time, sec ) and is given the name Hertz (Hz), Planck s constant is a fundamental number, measured in J sec or erg-sec. 

Detect 100% of photons Photon detected as a delta function Large number of pixels Time tag for each photon Measure photon wavelength Measure photon polarization No detector noise fr Up to 99% detected fr One electron for each photon fr Over 377 million pixels 0 No - framing detectors 0 No - provided by optics 0 No - provided by optics 0 Readout noise and dark current... 

We can also count the total number of donor-emitted photons, or measure the corresponding analog intensity, in the presence and absence of energy transfer. From these intensities we can calculate the efficiency of energy transfer. The fluorescence intensity of the donor is proportional to the rate constant through the fluorescence pathway divided by the sum of the rates of leaving the excited state by all pathways. That is,... 

An important limitation that is sometimes encountered is due to the particulate nature of electricity (electrons, ions) and of radiation (photons). The measurement of radiation intensities is in certain cases (e.g., X rays) performed by counting particles or photons one at a time. The number A counted in a time interval of given magnitude is subject to statistical fluctuations a count of A is subject to an estimated standard error given by... 

In Floquet theory the exchange of photons can be analyzed from the temporal variation of the relative photon number. In experiments, one measures for instance the difference in intensity of the laser pulse before and after the interaction with the molecules. Denoting the initial condition (at t = to = 0) by 4>(x) % c(0), we describe the exchange of photons by... 

We can also get more precise information on the probability P(L, t) that L photons are exchanged If at time t 0 the photon field is in a photon number eigenstate eM and v /(i = 0) = 4o - 4 e k( then the probability that a measurement performed at time t yields that L photons have been exchanged is given by... 

One representative example of a covalently bound chromophore is the polymer 124. This material was subjected to nonlinear absorption measurements using the neat film. The TPA coefficient was 7 cm/GW [524], which can be considered large. Furthermore, this polymer shows a reduction of the photon-number noise of 0.1 dB (4.6%) during nonlinear transmission. This is important in order to minimize the optical loss at the laser wavelength. 

It is then both natural and even compelling to be able to explain the baryon asymmetry quantified by the baryon to the photon number density in the present universe. This ratio is the direct measure of the asymmetry prior to the cosmic disappearance of antimatter. This number is usually given in the form of the baryon to the entropy ratio ub/ns and is of order ft)-10, which seems too small at first one excess of baryon over 1010 B — B pairs led to the present B-dominated universe. But actually this number is often too large to be explained in theoretical models. With this large number the standard electroweak theory fails as the microscopic theory for the baryogenesis, as explained below. 

For our first measurement we act on the photon-number entangled states in mode 3. Since the NS operation is an interference effect, it only proceeds when the entangled photons and the ancilla photon arrive at BS2 within their coherence time rcoh- In this case, the operation performs the following ... 

Figure 2. A - Experimentally measured and theoretically calculated values of dns/dt, the number of Stokes photons per unit time emitted from the atomic vapor cell. For each plot, ns = f dt dns/dt represents the total number of photons emitted from the cell. The write laser power is varied from 25 mW to 100 mW. B - Experimentally measured and theoretically calculated values of dnAs/dt, the number of anti-Stokes photons per unit time emitted from the atomic vapor cell. The experimental pulse shapes correspond to a Stokes pulse with ns 3 photons, and the theoretical curves assume an initial spin wave with nspin = 3 excitations and an optical depth of 20. Each curve is labeled with the power of the retrieve laser. Inset theoretical calculation of the number of flipped spins per unit length dnspin/dt as a function of position in the atomic cell, for nspin = 3. C - Measured anti-Stokes pulse width (full-width at half-maximum) and total photon number as a function of the retrieve laser intensity. Lines are intended only to guide the eye.

Figure 4. Conditional nonclassical state generation. Conditional (f f (.4,9) as a function of the number of detected Stokes photons. Diamonds show experimentally measured values, which are calculated from the two arms of the anti-Stokes beam-splitter via g (A.S ) = (AS AS-2)/ AS ) AS-2) (see Fig. 1 C). The measured mean photons number on the Stokes and anti-Stokes channels were fis = 1.06 and has = 0.36 respectively. The solid line shows the result of a theoretical model including background and loss on both the Stokes and anti-Stokes channels. The overall detection efficiency (a) and number of background photons (hbg ) used in the model were as = 0.35, n G = 0.27 (qas = 0.1, rdfs = 0.12) on the Stokes (anti-Stokes) channel, and were estimated from experimental measurements. For these measurements an optically-pumped 87Rb cell was used to filter the Stokes photons from the write laser. The dotted line represents < ns (AS) corrected for loss and background on the anti-Stokes channel, obtained by setting the anti-Stokes channel loss and background to zero in this model. Inset measured mean anti-Stokes number n s conditioned on the Stokes photon number ns- The solid line represents n s as predicted by the model.

We have applied the above approach to a harmonic oscillator coupled to a spin by means of a photon number - nondemolition Hamiltonian. The spin is being measured periodically, whereas the measurement outcome is ignored. For a sufficiently high measurement frequency, the state of the harmonic oscillator evolves in a unitary manner which can be influenced by a choice of the meter basis. In practice however, the time interval At between two subsequent measurements always remains finite and, therefore, the system evolution is subject to decoherence. As an example of application, we have simulated the evolution of an initially coherent state of the harmonic oscillator into a Schrodinger cat-like superposition state. The state departs from the superposition as time increases. The simulations confirm that the decoherence rate increases dramatically with the amplitude of the initial coherent state, thus destroying very rapidly all macroscopic superposition states. 

The first term in the variances is the shot noise (SN) of fight. This can be measured in absence of the interaction where k 0. The quantum nature of the shot noise level is confirmed by checking the linear scaling with photon number of the pulse, see Eq. (4). The second term arises from the projection noise (PN) of atoms. Hence, we may calibrate i f to be the ratio i f = PN/ SN of atomic projection noise to shot noise of fight. Theoretically i f has the linear scaling i f a. /,. Sx T with the macroscopic spin size Jx which must... 

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