## Abstract

According to Heisenberg’s uncertainty principle, measurement of a quantum observable introduces noise to this observable and thus limits the available precision of measurement. Quantum non-demolition measurements are designed to circumvent this limitation and have been demonstrated in detecting the photon flux of classical light beam. Quantum non-demolition measurement of a single photon is the ultimate goal because it is of great interest in fundamental physics and also a powerful tool for applications in quantum information processing. This chapter presents a brief introduction of the history and a review of the progress in quantum non-demolition measurement of light. In particular, a detailed description is presented for two works toward cavity-free schemes of quantum non-demolition measurement of single photons. Afterward, an outlook of the future in this direction is given.

### Keywords

- QND measurement
- single photon
- four-wave mixing
- Rabi oscillation

## 1. What is quantum non-demolition measurement?

Measurement of observables is at the very heart of quantum measurement. In the classical macroscopic world, measurement of a classical object can be conducted without introducing perturbation to the detected object. Repeating measurement of a classical object can improve the precision to arbitrarily accurate. Counterintuitively, the measurement of an observable of a quantum object cannot be arbitrarily precise in the microscopic world according to the well-known Heisenberg’s uncertainty principle [1], which roots in the wave nature of quantum mechanics. For non-commuting operators, A and B, described as physical quantities in the quantum formalism, a very precise measurement of A, resulting in a very small uncertainty

In response to this question, Braginsky and Vorontsov introduced in the 1970s the concept of “quantum non-demolition measurement” (QND) to evade the unwanted quantum back action in measurement [2]. Through studying the detectable minimum force on a quantum oscillator, they concluded that “Nondestructive recording of the n-quantum state of an oscillator is possible in principle.” Their measurement strategy opened a door for circumventing the issue of back action in quantum measurement. Thorne, Drever, Caves, Zimmermann, Sandberg, Unruh, and others developed the concept of QND measurement further [3, 4, 5]. The key point in the QND measurement is to keep the back-action noise confined to the unwanted observable quadrature, without being coupled back onto the quantity to be measured.

Although a great number of efforts have been made in various systems, quantum optics is particularly well suited for implementing QND measurement. The reason is threefold: (1) there are optical sources with very good quality; (2) photon detectors can be extremely sensitive, even being able to detect a single photon; and (3) a quantum system can be initialized with very high accuracy. The photon number and phase are two complementary observables of quantum light. They are associated with non-commuting operators. It means that QND measurement of photon number of a quantum field will inevitably add quantum noise to the phase quadrature. If only, in principle, the photon number of field remains unchanged during measurement, the measurement is QND. Of course, the real implementation of experiment may be imperfect, and this imperfection can cause noise to the variable of interest.

Throughout this chapter, we focus on the measurement of light according to the principle of quantum optics. In particular, we introduce the measurement of photon number of a light beam. In the conventional “direct” measurement, the light is absorbed. Therefore, the measurement completely changes the observable of photon number and causes a very large back action onto the light beam. In a QND measurement of photon number, it is required that the amount of photon number is *measured without changing*. Of course, the measurement still adds perturbation to the light. However, the perturbation is only confined to the phase of the photon but is not added to the photon flux of interest in measurement. In a restricted mathematical language, the condition for QND measurement is that

## 2. Classical measurement by absorbing photons

In the classical world, measurement of light always absorbs photons and then gets energy from them. In this way, the photon carried by a light beam disappears and is destroyed completely. This type of photon detector includes eyes, photoelectric converter, semiconductor photon detector, superconducting photon detector, and so on.

Eyes are photon detectors we use most often (Figure 1). It converts the energy of light into electric current and stimulates the nerve. Photons of light enter the eye through the cornea, that is the clear front “window” of the eye. Then light is bent by the cornea, passes freely through the pupil, the opening in the center of the iris, the eye’s natural crystalline lens, and then is focused into a sharp point on the retina. The retina is responsible for capturing all of the light rays, processing them into light impulses through millions of tiny eye nerve endings, and then converting these light impulses to signals which can be recognized by the optic nerve. In doing so, eyes convert light into bioelectric signals.

Semiconductor photon detector is a sensitive man-made photodetector, which is made by using semiconductor materials. Two principal classes of semiconductor photodetectors are in common use: thermal detectors and photoelectric detectors. Thermal detectors convert photon energy into heat. Most thermal detectors are rather inefficient and relatively slow. Therefore, photoelectric detectors are widely used for optics. The operation of photoelectric detectors is based on the photoeffect. Similar to eyes, the detector absorbs photons from light, generating electronic current pulse which can be measured. The semiconductor photon detector is the most used photodetector in industry. The most common semiconductor-based devices are single-photon avalanche diode (SPAD) detectors and can reach sensitivity at the single photon level. The SPAD detector is reversely biased above the avalanche breakdown voltage in the Geiger mode. When a photon is captured by this SPAD detector, the absorbed photon generates an electron-hole pair which causes a self-sustaining avalanche, rapidly generating a measurable current pulse (Figure 2).

Superconducting nanowires have been used to detect single photons. It exploits a different principle in comparison with eyes and semiconductor photon detectors. It is designed in this way [6, 7]: a patterned superconducting nanowire is cooled below the transition temperature of the superconducting material. The superconducting nanowire is biased by an external current slightly smaller than the critical current at the operating temperature. When a single photon hits the nanowire, it creates a transient normal spot in the resistive state. As a result of loss of superconductivity, a nonzero voltage is induced between two terminals of the nanowire. Measuring this induced voltage can tell the arrival of the single photon. To date, superconducting single photon detectors have achieved a detection efficiency of more than 90% [8, 9].

The abovementioned are three representatives of photon detectors. All of them destroy photons in signals.

## 3. Measuring light intensity without absorption

QND measurement of light needs to keep the quantum average of the observable and its uncertainty unchanged after detection. In general quantum measurement, the observable of a signal system,

where

QND measurement requires (i)

It is quite straightforward to get the cross-Kerr effect in mind for QND measurement of photon flux,

where

The condition

The concept of QND measurement based on the cross-Kerr effect has been demonstrated in experiments for classical light including many photons [12]. However, QND measurement at the single photon level is still a challenging problem. The difficulty is twofold. Technically, the nonlinearity of normal materials is too weak to induce a large phase shift per photon. Although the cross-Kerr nonlinearity can be improved by orders by using atom system, typically, a single photon can only cause an mrad scale phase shift [13]. It is worth noting two recent experiments in cross-phase modulation [14, 15], which demonstrated the pi phase shift at the single photon level via the cross-Kerr nonlinearity of atoms. At first sight, the methods may be able to apply to QND measurement of single photons. Actually, they are yet to meet the criteria of QND measurement.

In the first work [14], by storing a single photon in a cloud of Rydberg atoms, Tiarks et al. achieved a

Alternatively, Liu et al. used a double-

At the fundamental level, the cross-Kerr-based QND measurement is found invalid when a continuous spatiotemporal multimode model [16] or a finite response time [17, 18, 19] is considered. In this sense, although many important progresses have been achieved, QND detection of a moving single photon still needs proposals.

## 4. Non-demolition measurement of photons with cavities

With the progress of cavity electrodynamics, in particular the ultrastrong coupling between a microwave cavity and an artificial atom, QND measurement of single mw photons have been realized via qubit-photon CNOT gate [20], ac Stark effect [21, 22, 23], and the intrinsic phase shift in Rabi oscillation [24]. Photon blockade has been demonstrated as a new effect to implement QND measurement of a single optical photon trapped in a high-quality optical cavity [25].

The first breakthrough of QND measurement of single photons was accomplished by Haroche et al. exploiting the intrinsic

## 5. Cavity-free schemes for non-demolition measurement of single photons

The concept of QND measurement and its realization in measuring classical light intensity have been introduced earlier. QND measurement of single photons is the ultimate goal. Single “static” photon in cavity has been detected nondestructively. Measuring “moving” single photons without destroying it is still far to be achieved. Two important progresses toward this direction are presented in the following.

### 5.1. QND measurement via Rabi-type photon-photon interaction

As mentioned earlier, although the optical cross-Kerr effect has been proposed for implementing intensity QND measurement of light, detection of light at the single photon level in a QND way is still a challenging task. In the cross-Kerr-based proposals [10], the signal photon changes the refractive index

where

To induce a Rabi-type interaction, the auxiliary mode is initially in a vacuum state. The signal field has at most one photon. The probe field is assumed to be weak that, to a good approximation, it can be considered as the superposition state of

To determine the phase shift of the probe field, a strong local bias is overlapped on the transmitted probe field via a highly reflective beam splitter. By properly choosing the bias field, the transmitted probe field presented to the detector is displaced by

To evaluate the performance of the QND measurement, only one investigates the response of system to the initial case of a single signal photon input,

In the presence of a single signal photon, the field presented to the detector is

The measured photon and the probe photon are “moving” pulse-shaped wavefunctions. The quantum Langevin equation describes the motion of system in the single mode regime, in which both the signal and the probe photons are treated as a single mode. In the real experiment, they are moving pulse including continuous spatiotemporal modes and can be confined in a one-dimensional (1D) waveguide. Therefore, a model accounting for the interaction of continuous spatiotemporal modes is required. The method developed by Fan et al. can model the interaction of the signal and probe photons in 1D real space [27]. In the Fan’s method, the photons are the wavefunctions of quantum fields propagating in 1D real space. The probability density of photon appearing at certain time (position) is the squared absolute value of wavefunctions. For the purpose of single-photon QND measurement, only one needs the fidelity and phase shift of a photon-pair input state

where

Solving Eqs. (2) and (3)) can simulate the evolution of the fields in medium. Without loss of generality, a Gaussian input is applied. For a single-photon pulse which is a quantum field, the photon can appear everywhere within the pulse with a probability density determined by the wave packet. This is the nonlocal nature of a single photon pulse. When the probe and signal fields propagate at the same group velocity in the medium as previous schemes, they have no necessity to interact with each other. Actually, with a large probability, they propagate independently as they never meet each other. The signal photon couples the probe photon only if they appear at the same position. As a result, only the central part of

By comparing two models, it can be seen that when the probe field has at most one photon, a unit fidelity for the transmitted signal mode is achieved. If the probe contains higher Fock states, then interaction with these high Fock states of probe mode prevents to achieve perfect non-demolition of the signal mode.

Rubidium vapor embedded in a hollow-core photonic crystal fiber [12] or a hollow antiresonant reflecting optical waveguide [29] can be a good experimental implementation for this QND measurement scheme. This setup, to a good approximation, can be modeled as a 1D nonlinear medium. The four-wave mixing can be effectively conducted using a diamond-level configuration as shown in Figure 9. The signal field can be slowed via EIT with the fifth level,

### 5.2. QND measurement with single emitters

Alternatively, Witthaut et al. proposed another scheme for QND measurement of single photons by using a single V-type emitter coupling to a 1D waveguide [30]. The configuration is depicted in Figure 10.

A V-type three-level emitter strongly couples to one end of semi-infinite waveguide. The signal photon drives the transition between

with

A passing resonant photon then introduces a phase shift if and only if the emitter is in state

Measuring the phase shift imprinted on an incident classical laser pulse can measure the state of emitter. The emitter in

For simplicity, set

## 6. A possible bright future

QND measurement opens a door for precise measurement and versatile applications in photon-based quantum information processing. In principle, QND measurement enables repeated measurement of photon number, n, of a light beam. Because QND measurement does not disturb the photon number of light, it allows one to measure the photon number many times. This can surpass the standard quantum limit bounded by the “shot-noise” and allows to measure light with ultrahigh sensitivity. QND measurement down to the single photon level further enables potential application in quantum information processing. Remarkably, when a single signal photon can induce a