1. Introduction

The principles of quantum mechanics allow for a noise-free measurement of a quantum observable with arbitrarily high precision. A measurement scheme is usually referred to as back-action evading (BAE) if the noise induced by the back-action of the measurement process is decoupled from the measured observable and is entirely confined to the conjugate observable [1, 2, 3, 4]. If, in addition, the initial state is an eigenstate of the measurement operator, it is conserved during the evolution as it commutes with both the free and the interaction Hamiltonian, and the BAE measurement is called as quantum nondemolition (QND) measurement [1, 2]. This definition of QND implies repeated measurements on the measured observable without perturbing it, thus the measurement can attain high precision. A complete demonstration of an ideal QND measurement should consist a series of ideal BAE measurements. Only after this is satisfied, the main objective of QND measurements, that is to measure a signal observable of a certain quantum system repeatedly giving predictable results, could be achieved. This requirement makes QND a much stricter measurement than BAE. Once the above criteria are achieved, QND measurement becomes an ideal tool to perform projective measurement which is important for a variety of quantum information processing and quantum state preparation, to monitor the presence of a quantum signal, and to build controlled-not gates [5, 6].

Since the introduction in the 1970s [7, 8], QND and BAE have been widely investigated [9, 10, 11, 12, 13]. Quantum nondemolition measurement in the above strict definition is very difficult to achieve in practice because every macroscopic measurement setup, including the state preparation schemes, has unavoidable loss and noise, which leads to decoherence and irreversibility of the measurement, preventing one obtaining the result of the first measurement in the subsequent measurements. In order to evaluate the efficiency of QND and BAE in the presence of losses, a number of quantitative criteria have been developed [14, 15, 16]. Many experimental demonstrations have been performed in the fields of quantum optics [2, 17, 18, 19, 20, 21, 22, 23, 24, 25], atomic physics [26, 27, 28], and cavity-QED systems [29, 30]. Some of these experiments such as [20, 21] are good enough for QND but they lack quantum repeatability. The first repeated BAE experiment that demonstrates features of a complete QND measurement has been performed by Bencheikh et al. [22] followed by other limited number of experiments [23, 24, 29, 30].

In the optical domain, BAE and QND experiments have been performed using third-order nonlinear susceptibilities χ⁽³⁾ in optical fibers and second-order nonlinear susceptibilities χ⁽²⁾ in crystals. The simplicity and efficiency of the schemes utilizing χ⁽²⁾ allowed the first demonstration of repeated QND measurements. Due to significantly low-loss transmission of light and the presence of soliton regime, χ⁽³⁾ interaction in optical fibers was the first candidate for experimental realizations of QND. However, a complete QND measurement in optical fibers has not been performed due to the existence of other nonlinear effects, coupling losses and of parasitic noise sources [3]. Experiments with bright optical beams were realized in optical fibers [17, 18, 19] and in crystals [2, 20, 21, 22, 23, 24]. The strong nonlinearity required for measurements at low intensity beams has been demonstrated experimentally by either strong light-matter coupling that can be achieved in cavity Quantum Electrodynamics (QED) [29, 30] or measurement-induced nonlinearity [31]. While the former retains a distinct difficulty, the latter has low efficiency. Thus, efficient and practical measurement schemes which allow integration in quantum networks are in demand. Along these lines, photonic crystal waveguides have been proposed and shown to be feasible for QND measurement, although they are limited by material properties [32].

In an optical QND measurement using cross-phase modulation based on χ⁽³⁾, a light beam is split into two, one of which is referred to as the “probe” and the other as the “reference”. The probe is coupled to a “signal” beam in a medium with a high χ⁽³⁾ which induces a phase shift on the probe proportional to the intensity of the signal. This phase shift is measured by an optical interferometer, formed by recombining the probe and the reference at a symmetric balanced beamsplitter. Major impediments to QND measurement of light are the small value of nonlinearity in the available media and the photon absorption. Thus measurements with single-photon resolution, which require extremely strong coupling between the signal and the probe, during which the signal intensity remains unaltered becomes a challenge. In addition, one should consider effects such as self-phase modulation (SPM) and guided acoustic wave Brillouin scattering (GAWBS) [18, 32]. Therefore, materials to be used in these experiments should have very high nonlinearity and at the same time should minimize the absorption losses, technical noises and unwanted nonlinearities.

Motivated by the observation of optical Kerr nonlinearity in ultra-high-Q silica toroidal microcavities on a silicon chip [33] and their possible use for circuit cavity QED systems [34], we propose and discuss the feasibility of QND measurement of photon number using cross-phase modulation (XPM) due to optical Kerr effect in toroidal silica microcavity. Unlike the proposed and demonstrated schemes mentioned above, the signal and the probe beams are not directly interacting in the nonlinear medium, but rather, they are coupled to each other in the nonlinear medium through their whispering gallery mode (WGM) fields. Although χ⁽³⁾ of isotropic silica glass is very weak [35], the toroid-shaped cavities in micro-scale dramatically enhance the nonlinear effects for QND measurement. First, it allows light confinement in the micro-scale mode volume which produces enormous enhancement in the circulating optical power within a 3λ² cross section. For instance, the coupling of a light field with a power of 1 mW can produce a circulating power of GW/cm² in a microtoroid with a diameter of 50 micron and a high quality factor of Q ∼ 𝒪(10⁸). Thus, the threshold power for the nonlinear effects is reduced significantly. Second, the long photon lifetime in the ultra-high-Q microcavities effectively increases the nonlinear interaction length. Third, GAWBS is unlikely to occur in microtoroid cavities as the strong azimuthal confinement reduces the mode spectrum of WGM (free spectral range on the order of terahertz) eliminating the possibility of its overlap with the gain bandwidth (< 100MHz) of GAWBS [33, 40]. Therefore, such micro-scale cavities would provide great potential for the study of nonlinear optical phenomena and in particular for QND measurements using low-intensity excitations [33, 36, 37, 38, 39, 40, 41] of this work.

This paper is organized as follows. In Sec. 2, we introduce our model and construct the equation of motion for the evolution of the cavity field. Moreover, we find the conditions to maintain a stable operation with maximal intracavity intensity. In Sec. 3, we discuss the detection scheme and the accuracy of QND measurement. Section 4 contains a discussion of possible sources of errors and their effect on a practical implementation. Finally, in Sect. 5, we summarize and discuss our results and future prospects.

2. Theoretical model of taper-coupled microtoroid system

We propose to measure the photon number of a signal wave by measuring the phase of a probe wave via the optical Kerr effect in a toroidal microcavity [42]. Figure 1 shows a schematic of our model which is based on the nonlinear coupling between two bosonic modes similar to the Imoto-Haus-Yamamoto configuration [9]. The significant difference is that, in our model, the signal and the probe are not directly interacting in the nonlinear medium but rather they are coupled into the toroid microcavity using tapered optical fibers leading to the excitation of the signal and probe WGMs. Subsequently the excited signal and probe WGMs interact via the χ⁽³⁾ nonlinear effect. Thus, it is expected that even the signal fields from small photon numbers would induce detectable phase shift on the probe.

 

Fig. 1 Schematic of QND measurement with an ultra-high-Q toroidal microcavity. BS1 and BS2 are beamspitters with reflection coefficients r and 1/2, respectively, while M1 and M2 are mirrors with unit reflectivity for the probe field and unit transmissivity for the signal wave. Signal and probe waves are introduced into the same fiber waveguide and then coupled to the microcavity. The change in the phase of the probe field is finally detected by balanced homodyne detection using two photon detectors D₁ and D₂.

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The Heisenberg equations of motion describing the dynamics of the internal cavity mode annihilation (creation) operators a_j ( $a_j^{†}$) can be written as [45, 46]

Substituting Eq. (3) into Eq. (1), we find the equations of motion as

In order to proceed further, the nonlinear equations need to be linearized at least to first order in K. We find the steady-state solutions by employing the transformations a_j = A_je^−iφ_j, $a_j^{\text{in}}=A_j^{\text{in}}{e}^{-{\varphi }_j}$, and $a_j^{\text{out}}=A_j^{\text{out}}{e}^{-i{\varphi }_j}$ where A_j and $A_j^{\text{in}}$ are positive real constants, $A_j^{\text{out}}$ is a complex constant, φ_j and ϕ_j are the phases of the corresponding modes. Then Eqs. (4–5) become

Since Eq. (10) is a cubic equation, at least one of the roots may be unstable. Therefore, we should choose the adjustable parameters such that stable operation is maintained. The bistability is onset if $\partial {\delta }_p/\partial N_p={\partial }^2{\delta }_p/\partial N_p^2=0$ holds. This condition leads to $3N_p^2+2N_p{c}_0+c_1=3N_p+c_0=0$. Consequently, it is straightforward to show that at the critical point of bistability ${\left({\delta }_p+\text{K}\right)}^2=3{\kappa }_p^2/4$ from which we find that $2|{\delta }_p+\text{K}|>\sqrt{3}{\kappa }_p$ and $|N_p|>{\kappa }_p/2\sqrt{3}\text{K}$ should be satisfied to stay in the stable region. From Eq. (12), we find that in the stable regime with the maximum intracavity intensity for a given δ_p, the probe intensity should satisfy

3. Balanced homodyne detection of the probe field and the accuracy of QND

In this scheme balanced homodyne detection is used to measure the phase ψ_p induced in the probe field. In the following discussion we will drop the frequency dependence of the fields. As shown in Fig.1, the probe $a_p^{\text{in}}$ and the LO a_LO are prepared from the light beam at mode-u using a beamsplitter BS1 whose other input mode v is left at vacuum. The action of BS1 is given as

In order to get analytical solutions and simplify our discussion, we have assumed that the cavity-system is driven in the over-coupling regime under weak-excitation limit. One may naturally wonder how much this will deviate the results from those obtained without such assumptions. In order to clarify this issue, in Fig.2(a) we show the dependence of 〈n₂₁〉 on 〈n_s〉 by using the simplified model as described in Eq. (23) and also by direct numerical calculation using Eqs. (15), (16) and (21). In the direct numerical calculation, t_p = G_j(ω = 0) is used. The results match very well and demonstrate that the simplified model is reasonable and does not significantly affect the outcome. Figure 2(b) shows that the transmission spectrum of the signal is stable and close to unity. The transmission rate depends on the relationship between intrinsic and coupling quality factors. Higher intrinsic Q₀ will predict a larger transmission rate. Figure 2 together with Eq. (23), thus, shows without doubt that QND measurement of photon-number in the signal field is possible using an ultra-high Q microtoroidal cavity in the configuration described in Fig.1.

 

Fig. 2 (a) Photon number difference 〈n₁₂〉 versus the intracavity average signal photon number 〈n_s〉. The solid and dashed lines represent the precise numerical calculation and approximately analytical solution, respectively. (b) The transmission of the signal field versus 〈n_s〉. Parameters used in the calculations: r = 1/2, κ_j,1 = 50κ_j,0, κ_s = κ_p, Q₀ = 4×10⁸, wavelength λ = 1550 nm, third-order nonlinear susceptibility χ⁽³⁾ = 2.2 × 10⁻²² m²/V², quantization volume V = 50 μm³, and 〈n_u〉 = 10¹³.

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The statistical distribution of an operator B is most commonly expressed in terms of the mean square fluctuations (variance, or measurement error) as 〈(ΔB)²〉 ≡ 〈B²〉 – 〈B〉². From Eq. (23), we find that the variance $〈{\left(\Delta n_s^{\text{obs}}\right)}^2〉$ of $n_s^{\text{obs}}$ can be written as

 

Fig. 3 Dependence of observed variance $〈{\left(\Delta n_s^{\text{obs}}\right)}^2〉$ on (a) the probe photon number which is linearly proportional to 〈u^†u〉, and (b) the intrinsic quality factor Q₀ for the signal prepared as coherent and Fock states. The dashed and dotted lines are for coherent states with mean photon numbers 〈n_s〉 = 1 and 〈n_s〉 = 10, respectively. The curves are obtained from Eq. 25 using the parameters given in Fig.2. For (b) we used 〈n_u〉 = 10¹⁴. Lower bound of observed variance for a coherent state with 〈n_s〉 is determined by its intrinsic noise 〈n_s〉 when Q₀ ∼ 10⁸ and 〈n_u〉 ∼ 10¹⁴. For a Fock state, it is determined by Q₀, 〈n_u〉 and K, hence can be made arbitrarily small by proper choice of these parameters.

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Returning back to Eq. (25), we see that for a signal prepared as a number state, the variance of the measured photon number becomes

Although in the above discussions, we have focused our attention on the detection of the intracavity signal photon number $n_s=a_s^{†}{a}_s$ and the statistical properties of the actually-observed intracavity signal photon number $n_s^{\text{obj}}$, one can extend the discussions to the detection of the photon number $n_s^{\text{in}}$ of an input signal pulse $a_s^{\text{in}}$ as these two are directly associated with each other through Eq. (18). In the weak-excitation limit of Kn_s ≪ κ_s, we can approximately obtain $n_s=4a_s^{\text{in}†}{a}_s^{\text{in}}/{\kappa }_s=4n_s^{\text{in}}/{\kappa }_s$, hence the relations between the statistical properties of n_s and $n_s^{\text{in}}$ can be expressed as $〈n_s^{\text{in}}〉={\kappa }_s〈n_s〉/4$ and $〈{\left(\Delta n_s^{\text{in}}\right)}^2〉={\kappa }_s^2〈{\left(\Delta n_s\right)}^2〉/16$.

4. Considerations for a practical implementation

For balanced homodyne detection, it is not necessary to use photon number resolving detectors. Instead, conventional p-i-n photodiodes with quantum efficiencies η ≥ 0.90 can be used. The generated photocurrents in the photodiodes are proportional to the number of incident photons. The subtraction is done electrically by reverse-biasing the photodiodes connected in series, and the current drawn from the middle node corresponds to photocurrent difference. Subsequently, this difference current should be amplified with an ultra-low noise amplification circuit for further processing. Thus, in an actual balanced homodyne detection, the electrical signal acquired at the output of the detection circuit contains contributions from the detected photons, the electrical noise and the photodiode dark current d. In addition to these imperfections, we should also consider the measurement induced variance, unavoidable insertion losses, and noise like thermal-optic effects in evaluating the performance of a QND measurement.

Detectors with quantum efficiencies η can be modelled by placing fictitious beamsplitters with transmission coefficients η in front of ideal detectors. Assuming that the signal state is a Fock state with photon number n_s and r = 1/2, it is easy to show that the detected photon number difference will have a signal part with mean value η〈n_u〉ζ_pn_s, and a noise contribution, due to vacuum fluctuations entering the second input port of each of the fictitious beamsplitters, with a mean value of zero and variance of η (1 − η) 〈n_u〉. In order to resolve photon number reliably, the sensitivity per photon must be much larger than the standard deviation of the noise, $\sqrt{\eta 〈n_u〉}{\zeta }_p\gg \sqrt{1-\eta }$. Error probability of making the decision that the signal has n photons if the detected photon number difference lies between η〈n_u〉ζ_p(n − 1/2) and η〈n_u〉ζ_p(n + 1/2) becomes $P_{\text{error}}=1-\text{Erf}\hspace{0.17em}\left(\frac{1}{4}{\zeta }_p\sqrt{2〈n_u〉\eta /(1-\eta )}\right)$ where a Gaussian distribution is used for noise because a coherent light with a large amplitude is used [11]. For ${\zeta }_p\sqrt{〈n_u〉}=10$, we find that P_error 10⁻⁷ and P_error 10⁻¹⁴, respectively for η equals to 0.5 and 0.7. Increasing ${\zeta }_p\sqrt{〈n_u〉}$ to 15 keeping η = 0.5 leads to P_error 10⁻¹⁴. Thus, we can safely say that the scheme allows an error-free measurement of signal photon number.

Besides the non-unit efficiency of the photodiodes, balanced homodyne detection suffers from three main sources of noises: the LO shot noise, electronic noise and the mode-mismatch. In an ideal balanced homodyne detection scheme if the mode a″_p is blocked leaving only the LO, the fluctuations in the difference signal will have zero mean with the variance of (1 – r)〈n_u〉 which is the LO shot noise. Mode-mismatch effects can be minimized by proper spatial and spectral filtering as discussed in [47]. Since the detected mode in a balanced homodyne scheme is the mode picked up by the strong LO field, the mismatched part is considered as a kind of loss which decreases the overall efficiency of detection. In this regard, we can model mode-mismatch effects using the beamsplitter model we have used above to account for the non-unit detection efficiency in the system. The electrical noise originates from dark currents of the photodiodes, thermal noise and all the other noises in the amplification and data acquisition process. Then the photocurrent difference I₂₁ will include contribution from this electrical noise which fluctuates with zero mean and the variance 〈(Δn_e)²〉. Thus n₂₁ in Eq. 20 should be modified n′₂₁ ≡ n₂₁ + n_e to include the contribution from electrical noise n_e. Noting that the electrical noise contribution n_e and the contribution n₂₁ due to the field in mode a″_p are statistically independent of each other, the total variance is given by the sum of their variances. Thus, the variance $〈{\left(\Delta n_s^{\text{obs}}\right)}^2〉$ of the observed photon-number difference given in Eq. 24 should include an additional term given as $〈{\left(\Delta n_e\right)}^2〉/4{\zeta }_p^{2}r(1-r)〈n_u〉$ to account for the contribution from electrical noise. Since n_e is solely from the electronics, increasing or decreasing the LO intensity does not affect it. However, as can be seen from the above expression, we can minimize the effect of electrical noise on the variance of $〈{\left(\Delta n_s^{\text{obs}}\right)}^2〉$ by choosing a large enough LO intensity. In general, the effect of dark currents on the detected photon numbers may simply be subtracted away if photon count distributions for dark counts and the signal can be resolved. However, it is important to note that dark count subtraction can be done in repeated measurements; one cannot subtract dark counts in single-shot measurements. In short, we can say that using a strong LO will help to minimize the effects of noises other than the shot noise on the measured quantity. Moreover, it is obvious that in an efficient QND the noise introduced due to the measurement and interferometric configuration should be much smaller than the mean value of the measured photon number difference. This can be satisfied by careful adjustment of the interferometric configuration to maximize mode matching and by using ultra-high-Q cavities.

Insertion losses take place (i) before, (ii) in and (iii) after the Kerr medium as they have been studied in detail in [14]. In our scheme, (i) takes place in the coupling of the signal field into the taper-fiber which is used to couple the light to and from the WGM of the microtoroid cavity, the propagation loss in the fiber, and the coupling of the light to the WGM of the microtoroid, (ii) takes place in the form of material losses as the WGM circulates along the equatorial region of the silica toroidal cavity, and (iii) takes place during the coupling from the WGM to the fiber and propagation loss in the fiber. The loss in (i) will have a significant effect on the outcome as the QND measurement is done after this loss. In the presence of this loss, the outcome of the measurement will not reflect the actual photon number of the original signal field. Therefore, this loss should be kept as low as possible. Indeed, Imoto and Saito has shown that if the loss between the source of the signal and the Kerr medium in a QND scheme exceeds 50%, a beamsplitter, which is a linear device, will perform better than a Kerr medium [14]. In practice, coupling of light from a source to fibers can be achieved with much better efficiencies than 50%. In a typical tapering process satisfying the adiabatic tapering process, fiber-tapers with losses less than 0.2% can be achieved, thus losses due to tapering will not be a significant limitation. The loss in (iii) is not a serious limitation if the QND measurement is done only once as the loss affects the system after the measurement is performed. However, in repeated measurements, this loss should be minimized or prevented to achieve quantum repeatability. The loss in (ii) should be treated carefully because it is directly related with the scheme used to conduct the QND measurement, and it sets the ultimate limits for the feasibility of the scheme. For the toroidal microcavities we propose to use for QND measurement, the experimental quality factor Q₀ = ω_jc/2κ_j,0 reaches 4 ×10⁸ [43], and the theoretical predication even exceeds 10¹⁰ [34]. The insertion loss due to cavity is mostly due to the material losses originating from intrinsic material absorption, surface and bulk scattering and surface contaminants. The last two loss mechanisms can be prevented or minimized by adopting a well-controlled and improved fabrication methods. As we have pointed out earlier, to minimize the effect of losses due to material absorption, the system should be driven in the over-coupled regime where maximal extraction of power from the microtoroidal cavity becomes possible. In typical experiments, near complete recovery of the light field is possible. This deviation from complete recovery is due to intrinsic material loss and other losses during the propagation of the light in the taper-fiber and coupling to and from the WGM.

It is important to note that the material absorption is also the main source of the thermal effects, which includes the thermo-optic effect and the thermal expansion of the cavity. Such effects cannot be neglected in our scheme as the circulating light intensity in the cavity is greatly enhanced due to the ultra-high Q and micro-size of the cavity. The absorption of the circulating light field heats up the cavity leading to a red-shift in the resonant cavity modes for increasing signal power. Surprisingly, the thermal-induced shift has the same direction (red or blue) as the Kerr effect suggesting that, in principle, thermal effect can be exploited to amplify the signal for improved detection sensitivity. A more detailed description of the thermal effects requires the study of the dynamical thermal behavior and the thermal self-stability similar to the one carried out in Ref. [44].

Another issue that may come up in practice is the cross-talk when measuring signal of a few photons with a strong probe field. Linear cross-talk may take place due to the leakage of the probe photons into the signal channel, while the non-linear cross-talk may be due to four-wave mixing in silica. In principle, their effects can be minimized by choosing the probe and signal such that they could be separated from each other at least in one degree of freedom such as wavelength and polarization. Cross-talk due to four-wave mixing and similar nonlinear processes may be prevented by making it difficult for the fields to satisfy phase-matching condition. For instance, we can choose two largely de-tuned cavity modes for the signal and probe or we can adjust the polarization of the signal and probe orthogonal to each other. This remedy will also work to minimize the effect of linear cross-talk on the signal. The signal and probe at different wavelengths and polarizations can be separated from each other by using wavelength and polarization selective elements with high extinction coefficients (e.g., dichromic mirrors, switches and polarizing beamsplitters) after the fields leave the cavity (see M2 in Fig.1).

Although here we have analyzed the proposed scheme considering only one use, the scheme allows for repeated measurements if cascaded in series with the output signal from an earlier stage is directed to the next stage as an input signal. The analysis and experimental construction can be extended to include repeated uses of the scheme similar to those in [11, 22, 23, 24]. We envision that a single optical fiber with multiple tapered-sections runs along an array of toroidal cavities which are placed along the tapered sections in such a way that over-coupling operation is satisfied. After each stage a fiber grating (or a conventional add-drop filter or one constructed using toroidal cavity [48]) separates the probe from the signal output, and the signal is directed to the next stage while the probe is directed to homodyne detection. In such an all-fiber construction, the success of the scheme will be limited by the loss during the coupling of the signal into the fiber at the initial stage and also by the extinction coefficients of the gratings used between two stages. The former is a problem for fiber based schemes and coupling losses must be minimized using efficient coupling mechanisms, while the latter is a problem for all QND schemes using optical Kerr effect.

5. Conclusion

In conclusion, we have theoretically studied QND measurement of photon number with an ultra-high-Q toroidal microcavity. Based on the current experimental parameters, the present scheme exhibits a high feasibility to detect a few or even single intracavity photons without destroying them. This study suggests that on-chip toroidal microcavities with their wafer-scale integration and control can be incorporated as a sensitive QND detector into quantum circuits integrated on a single chip. Therefore the nonlinear interaction in a microtoroid cavity at low input light levels shows a great potential as a new experimental platform for quantum optics and quantum information processing.