OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 26 — Dec. 22, 2008
  • pp: 21462–21475
« Show journal navigation

Quantum nondemolition measurement of photon number via optical Kerr effect in an ultra-high-Q microtoroid cavity

Yun-Feng Xiao, Şahin Kaya Özdemir, Venkat Gaddam, Chun-Hua Dong, Nobuyuki Imoto, and Lan Yang  »View Author Affiliations


Optics Express, Vol. 16, Issue 26, pp. 21462-21475 (2008)
http://dx.doi.org/10.1364/OE.16.021462


View Full Text Article

Acrobat PDF (319 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We theoretically investigate a quantum nondemolition (QND) measurement with optical Kerr effect in an ultra-high-Q microtoroidal system. The analytical and numerical results predict that the present QND measurement scheme possesses a high sensitivity, which allows for detecting few photons or even single photons. Ultra-high-Q toroidal microcavity may provide a novel experimental platform to study quantum physics with nonlinear optics at low light levels.

© 2008 Optical Society of America

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

1. C. M. Caves, K. S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, “On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle,” Rev. Mod. Phys. 52, 341–392 (1980). [CrossRef]

, 2

2. F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, “Soliton backaction-evading measurement using spectral filtering,” Phys. Rev. A 66, 043810 (2002). [CrossRef]

, 3

3. P. Grangier, J. -A. Levenson, and J.-P. Poizat, “Quantum nondemolition measurement in optics,” Nature 396, 537–542 (1998). [CrossRef]

, 4

4. J. F. Roch, G. Roger, P. Grangier, J. M. Courty, and S. Reynaud, “Quantum non-demolition measurements in optics: a review and some recent experimental results,” Appl. Phys. B 55, 291–297 (1992). [CrossRef]

]. 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

1. C. M. Caves, K. S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, “On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle,” Rev. Mod. Phys. 52, 341–392 (1980). [CrossRef]

, 2

2. F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, “Soliton backaction-evading measurement using spectral filtering,” Phys. Rev. A 66, 043810 (2002). [CrossRef]

]. 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

5. G. Leuchs, C. Silberhorn, König, P. K. Lam, A. Sizmann, and N. Korolkova, in Quantum Information Theory with Continous Variables edited by S. L. Braunstein and A. K. Pati (Kluwer Academic, Dordrecht, 2002).

, 6

6. W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005). [CrossRef]

].

Since the introduction in the 1970s [7

7. K. S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, “Quantum nondemolition measurement of harmonic oscillators,” Phys. Rev. Lett. 40, 667–671 (1978). [CrossRef]

, 8

8. V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum nondemolition measurements,” Science 209, 547–557 (1980). [CrossRef] [PubMed]

], QND and BAE have been widely investigated [9

9. N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287–2292 (1985). [CrossRef] [PubMed]

, 10

10. P. Alsing, G. J. Milburn, and D. F. Walls, “Quantum nondemolition measurements in optical cavities,” Phys. Rev. A 37, 2970–2978 (1988). [CrossRef] [PubMed]

, 11

11. H. A. Haus and F. X. Kärtner, “Optical quantum nondemolition measurement and the Copenhagen Interpretation,” Phys. Rev. A. 53, 3785–3791 (1986). [CrossRef]

, 12

12. V. B. Braginsky and F. Ya. Khalili, “Quantum nondemolition measurements: the route from toys to tools,” Rev. Mod. Phys. 68, 1–11 (1996). [CrossRef]

, 13

13. J. M. Courty, S. Spälter, F. Krönig, A. Sizmann, and G. Leuchs, “Noise-free quantum-nondemolition measurement using optical solitons,” Phys. Rev. A 58, 1501–1508 (1998). [CrossRef]

]. 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

14. N. Imoto and S. Saito, “Quantum nondemolition measurement of photon number in a lossy optical Kerr medium”, Phys. Rev. A 39, 675–682 (1989). [CrossRef] [PubMed]

, 15

15. M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, “Nonideal quantum nondemolition measurements,” Phys. Rev. A. 42, 2995–3005 (1990). [CrossRef] [PubMed]

, 16

16. P. Grangier, J. M. Courty, and S. Reynaud, “Characterization of nonideal quantum nondemolition measurements,” Opt. Commun. 89, 99–106 (1992). [CrossRef]

]. Many experimental demonstrations have been performed in the fields of quantum optics [2

2. F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, “Soliton backaction-evading measurement using spectral filtering,” Phys. Rev. A 66, 043810 (2002). [CrossRef]

, 17

17. M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, “Quantum nondemolition detection of optical quadrature amplitudes,” Phys. Rev. Lett. 57, 2473–2476 (1986). [CrossRef] [PubMed]

, 18

18. N. Imoto, S. Watkins, and Y. Sasaki, “A nonlinear optical-fiber interferometer for nondemolitional measurement of photon number,” Opt. Comm. 61, 159–163 (1987). [CrossRef]

, 19

19. S. R. Friberg, S. Machida, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number of an optical soliton,” Phys. Rev. Lett. 69, 3165–3168 (1992). [CrossRef] [PubMed]

, 20

20. J. P. Poizat and P. Grangier, “Experimental realization of a quantum optical tap,” Phys. Rev. Lett. 70, 271–274 (1993). [CrossRef] [PubMed]

, 21

21. S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Back-action evading measurements for quantum non-demolition detection and quantum optical tapping,” Phys. Rev. Lett. 72, 214–217 (1994). [CrossRef] [PubMed]

, 22

22. K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, “Quantum non-demolition demonstration via repeated back-action evading measurements,” Phys. Rev. Lett. 75, 3422–3425 (1995). [CrossRef] [PubMed]

, 23

23. R. Bruckmeier, H. Hansen, and S. Schiller, “Repeated quantum non-demolition measurements of continuous optical waves,” Phys. Rev. Lett. 79, 1463–1466 (1997). [CrossRef]

, 24

24. K. Bencheick, C. Simonneau, and J. A. Levenson, “Cascaded amplifying quantum optical taps: A robust noiseless optical bus,” Phys. Rev. Lett. 78, 34–37 (1997). [CrossRef]

, 25

25. B. C. Buchler, P. K. Lam, H. -A. Bachor, U. L. Andersen, and T. C. Ralph, “Squeezing more from a quantum nondemolition measurement,” Phys. Rev. A. 65, 011803(R) (2001). [CrossRef]

], atomic physics [26

26. J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum non-demolition measurements using cold trapped atoms,” Phys. Rev. Lett. 78, 634–637 (1997). [CrossRef]

, 27

27. A. Kuzmich, L. Mandel, and N. P. Bigelow, “Generation of spin squeezing via continous quantum nondemolition measurement,” Phys. Rev. Lett. 85, 1594–1597 (2000). [CrossRef] [PubMed]

, 28

28. S. Peil and G. Gabrielse, “Observing the quantum limit of an electron cyclotron: QND measurements of quantum jumps between Fock states,” Phys. Rev. Lett. 83, 1287–1290 (1999). [CrossRef]

], and cavity-QED systems [29

29. G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999). [CrossRef]

, 30

30. C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448, 889–893 (2007). [CrossRef] [PubMed]

]. Some of these experiments such as [20

20. J. P. Poizat and P. Grangier, “Experimental realization of a quantum optical tap,” Phys. Rev. Lett. 70, 271–274 (1993). [CrossRef] [PubMed]

, 21

21. S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Back-action evading measurements for quantum non-demolition detection and quantum optical tapping,” Phys. Rev. Lett. 72, 214–217 (1994). [CrossRef] [PubMed]

] 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

22. K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, “Quantum non-demolition demonstration via repeated back-action evading measurements,” Phys. Rev. Lett. 75, 3422–3425 (1995). [CrossRef] [PubMed]

] followed by other limited number of experiments [23

23. R. Bruckmeier, H. Hansen, and S. Schiller, “Repeated quantum non-demolition measurements of continuous optical waves,” Phys. Rev. Lett. 79, 1463–1466 (1997). [CrossRef]

, 24

24. K. Bencheick, C. Simonneau, and J. A. Levenson, “Cascaded amplifying quantum optical taps: A robust noiseless optical bus,” Phys. Rev. Lett. 78, 34–37 (1997). [CrossRef]

, 29

29. G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999). [CrossRef]

, 30

30. C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448, 889–893 (2007). [CrossRef] [PubMed]

].

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

3. P. Grangier, J. -A. Levenson, and J.-P. Poizat, “Quantum nondemolition measurement in optics,” Nature 396, 537–542 (1998). [CrossRef]

]. Experiments with bright optical beams were realized in optical fibers [17

17. M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, “Quantum nondemolition detection of optical quadrature amplitudes,” Phys. Rev. Lett. 57, 2473–2476 (1986). [CrossRef] [PubMed]

, 18

18. N. Imoto, S. Watkins, and Y. Sasaki, “A nonlinear optical-fiber interferometer for nondemolitional measurement of photon number,” Opt. Comm. 61, 159–163 (1987). [CrossRef]

, 19

19. S. R. Friberg, S. Machida, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number of an optical soliton,” Phys. Rev. Lett. 69, 3165–3168 (1992). [CrossRef] [PubMed]

] and in crystals [2

2. F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, “Soliton backaction-evading measurement using spectral filtering,” Phys. Rev. A 66, 043810 (2002). [CrossRef]

, 20

20. J. P. Poizat and P. Grangier, “Experimental realization of a quantum optical tap,” Phys. Rev. Lett. 70, 271–274 (1993). [CrossRef] [PubMed]

, 21

21. S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Back-action evading measurements for quantum non-demolition detection and quantum optical tapping,” Phys. Rev. Lett. 72, 214–217 (1994). [CrossRef] [PubMed]

, 22

22. K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, “Quantum non-demolition demonstration via repeated back-action evading measurements,” Phys. Rev. Lett. 75, 3422–3425 (1995). [CrossRef] [PubMed]

, 23

23. R. Bruckmeier, H. Hansen, and S. Schiller, “Repeated quantum non-demolition measurements of continuous optical waves,” Phys. Rev. Lett. 79, 1463–1466 (1997). [CrossRef]

, 24

24. K. Bencheick, C. Simonneau, and J. A. Levenson, “Cascaded amplifying quantum optical taps: A robust noiseless optical bus,” Phys. Rev. Lett. 78, 34–37 (1997). [CrossRef]

]. 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

29. G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999). [CrossRef]

, 30

30. C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448, 889–893 (2007). [CrossRef] [PubMed]

] or measurement-induced nonlinearity [31

31. G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuring a photonic qubit without destroying it,” Phys. Rev. Lett. 92, 190402 (2004). [CrossRef] [PubMed]

]. 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

32. I. Fushman and J. Vučković, “Analysis of a quantum dondemolition measurement scheme based on Kerr nonlinearity in photonic crystal waveguides,” Opt. Express 15, 5559–5571 (2007). [CrossRef] [PubMed]

].

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

18. N. Imoto, S. Watkins, and Y. Sasaki, “A nonlinear optical-fiber interferometer for nondemolitional measurement of photon number,” Opt. Comm. 61, 159–163 (1987). [CrossRef]

, 32

32. I. Fushman and J. Vučković, “Analysis of a quantum dondemolition measurement scheme based on Kerr nonlinearity in photonic crystal waveguides,” Opt. Express 15, 5559–5571 (2007). [CrossRef] [PubMed]

]. 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.

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

42. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003). [CrossRef] [PubMed]

]. 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

9. N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287–2292 (1985). [CrossRef] [PubMed]

]. 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.

The Heisenberg equations of motion describing the dynamics of the internal cavity mode annihilation (creation) operators aj ( aj) can be written as [45

45. D. F. Walls and G. J. Milburn, Quantum Optics(Springer-Verlag, BerlinHeidelberg, 1994)

, 46

46. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985). [CrossRef] [PubMed]

]
dajdt=i[aj,H]κj2ajκj,1ajin,
(1)
where j = s, p describes the signal and probe fields, and the cavity mode aj satisfies the boson commutation relations [aj, aj] = 1 and [aj, aj′] = 0. Here aj and ajin have different units with ajinajin and ajaj describing the impinging photon number per second and intracavity photon number, respectively. Total dissipation of the cavity modes is described as κj = κj,0 + κj,1 where κj,0 and κj,1, respectively, correspond to the cavity decay due to the intrinsic losses (material absorption, scattering, and radiation losses) and the fiber taper-cavity coupling loss. In addition to Eq. (1), we have the following relation between the incoming ajin and the outgoing ajout modes
ajout=ajin+κj,1aj.
(2)
The Hamiltonian denoted as H in Eq. (1) reads as
H=Ho+Hint,Ho=h¯ωscasas+h¯ωpcapap,Hint=K[(asas)2+(apap)2+4εasasapap],
(3)
where Ho is the free Hamiltonian of the signal and the probe cavity modes with corresponding frequencies of ωsc and ωpc, and Hint accounts for their interaction in the Kerr medium. The nonlinear coupling constant K is real and given by [h̄ωscωpc/ (2n4ε0V)] χ(3), where n denotes the linear refractive index of the silica medium, V is the effective mode volume, and ε0 is the permittivity of vacuum. The asymmetry of the nonlinear properties of the medium is described by ε which takes the value of one for materials with fully symmetric susceptibility such as silica, thus we set ε = 1. In the interaction Hamiltonian Hint, the first two terms describe the SPM of the signal and probe fields, respectively, while the last term describes XPM. In QND of signal photon number, SPM of the signal is allowed, however the SPM of the probe should be avoided as this will introduce an additional phase shift to the probe which in turn will introduce an error in the measurement of the signal photon number.

Substituting Eq. (3) into Eq. (1), we find the equations of motion as
dasdt=[i(δs+K+2Kns+4Knp)+κs2]asκs,1asin,
(4)
dapdt=[i(δp+K+2Knp+4Kns)+κp2]apκp,1apin,
(5)
after employing a transformation to a reference frame rotating at the carrier angular frequency ωj with aj = ajejt, ajin=ajineiωjt, and ajout=ajouteiωjt. In Eqs. (45), the terms 2Kns and 2Knp, respectively, correspond to the SPM of the signal and probe, nj=ajaj denotes the intracavity photon number operators, and δj = ωjcωj is the detuning between the cavity mode and the input. It is worthwhile to notice that for the accurate measurement of QND the SPM of the probe should be avoided, while the modulation of the signal due to the probe and SPM effect should be minimized or avoided so as to minimize the signal distortion.

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 aj = Ajej, ajin=Ajineϕj, and ajout=Ajouteiϕj where Aj and Ajin are positive real constants, Ajout is a complex constant, φj and ϕj are the phases of the corresponding modes. Then Eqs. (45) become
[i(δs+K+2KNs+4KNp)+κs2]As+κs,1Asinei(ϕsφs)=0,
(6)
[i(δp+K+2KNp+4KNs)+κp2]Ap+κp,1Apinei(ϕpφp)=0,
(7)
Ajout=Ajin+κj,1Ajei(φjϕj),
(8)
where we define Nj=Aj2. Following Ref. [10

10. P. Alsing, G. J. Milburn, and D. F. Walls, “Quantum nondemolition measurements in optical cavities,” Phys. Rev. A 37, 2970–2978 (1988). [CrossRef] [PubMed]

], we find the steady-state intracavity intensity in the absence of signal mode ( asin=0) as
[i(δp+K+2KNp)+κp2]Ap=κp,1Apinei(ϕpφp).
(9)
Multiplying both sides of Eq. (9) with their complex conjugates, we obtain
Np3+δp+KKc0Np2+κp2+4(δp+K)216K2c1Npκp,14K2Npin=0
(10)
where Npin=Apin2. The maximum of Np can be found by setting ∂Np/∂δp = 0, which results in
δp+K(1+2Np)=0.
(11)
Substituting Eq. (11) in Eqs. (910), we find ϕpφp = (2m + 1)π where m is an integer, and
Npin=Np4κp2κp,1,
(12)
which determines probe intensity to maximize intracavity intensity for a fixed δp.

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 δp/Np=2δp/Np2=0 holds. This condition leads to 3Np2+2Npc0+c1=3Np+c0=0. Consequently, it is straightforward to show that at the critical point of bistability (δp+K)2=3κp2/4 from which we find that 2|δp+K|>3κp and |Np|>κp/23K 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 δp, the probe intensity should satisfy
|Npin|>κp383Kκp,1.
(13)
Operating the fiber-taper coupled microtoroid system under condition discussed above will eliminate the SPM of the probe while maintaining a stable operation. Moreover, from Eq. (6) we see that if the detuning δs is chosen to satisfy δs + K(1 + 4Np) = 0, phase modulation induced by the probe in the signal is also eliminated. Plugging these into Eqs. (45), we obtain
dajdt=(i2μjKns+κj2)ajκj,1ajin,
(14)
where μj takes the values of 1 and 2 for j = s and j = p, respectively. Then we take the Fourier transform x(t)=x(ω)exp(iωt)dω of Eq. (14) to get
aj(ω)=2κj,1κji2(ω2μjKns)ajin(ω),
(15)
where we use the fact that ns is a constant of motion. Using the standard input-output formalism for the cavity given in Eq. (2), we obtain the transfer functions Gj(ω)=ajout(ω)/ajin(ω) linearized to the first order in K
Gj(ω)=(12κj,1κji2ω)+i8μjκj,1Kns(κji2ω)2.
(16)
Before proceeding further, we note:
  1. ajaj actually describes the average intracavity photon number within the pulse duration T, not the average per second. This fact originates from the assumption without loss of the underlying physics, that the input signal photons construct a steady intracavity field in the adiabatic limit of κsT ≫ 1. For instance, ajaj reaches 0.1 when ns = 1 and κsT = 40.
  2. Operating the system at the well-known critical coupling point (κj,0 = κj,1) does not favor the QND measurement because QND requires that the signal light intensity to remain unchanged; however at the critical-coupling the output vanishes to zero. In order to effectively reduce the photon losses of the signal and probe, we should put the system in the deep over-coupling regime where we have κj,0κj,1 and consequently κjκj,1.
  3. In the absence of nonlinearity, K = 0, the transfer function of the system at the deep over-coupling regime becomes Gj(ω) = −(κj + i2ω)/(κji2ω) which shows that the cavity introduces a phase shift of π + 2arctan(2ω/κj) in the absence of nonlinearity. Thus, this phase shift should be compensated to measure the phase change induced only by nonlinear interaction of the signal and probe fields.
  4. In the rotating reference frame, the homodyne signal is at DC so we can evaluate the transfer function at ω = 0. Thus the measurement could be done with a local oscillator (LO) having the same angular frequency as that of the output field. Consequently, the transfer function in Eq. (16) becomes Gj(0) = −1 + i8μj(K/κj)ns which can be further arranged to give
    Gjei8μj(K/κj)ns,
    (17)
    due to the fact of 8μj(K/κj)ns ≪ 1. This expression clearly shows that the photon loss can be neglected in both signal and probe fields, and both fields accumulate phase shifts which linearly depend on the number of photons in the signal field. After compensating the π-phase shift induced by the ultra-high-quality cavity in the absence of the nonlinear effect, we obtain the fields as
    aj=ajout=ajinei8μj(K/κj)ns
    (18)
It becomes clear from the discussion above that the cavity system with the π-compensation acts as a phase-shifter with the phase-shift proportional to the number of photons in the signal, thus we can write the relation between the input and output modes as aj=ajinexp(iψj) where ψj = 8μj(K/κj)ns.

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 ψp 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 apin and the LO aLO 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
apin=iru+1rv,aLO=1ru+irv,
(19)
where r is the reflection coefficient of BS1 and u and v are the annihilation operators of the corresponding input modes of BS1. Subsequently the LO is mixed with the field ap at a 50:50 beamsplitter BS2 after the latter is π/2-phase shifted to adjust the interferometer configuration. The action of BS2 transforms the input fields as d1=(iap+aLO)/2 and d2=(ap+iaLO)/2 where ap=iap=iapout=iapinexp(iψp). Then the light fields at the output ports of BS2 are directed to the photodiodes. The generated photocurrents are subtracted to give the difference I21g(d2d2d1d1) where g is the conversion efficiency of photon-number difference into photo-current difference, and d1d1 and d2d2 are the photon-number operators for the detectors D1 and D2. From the detected photocurrent difference, we finally obtain the photon-number operator nsobs which is actually observed in homodyne detection as
nsobjd2d2d1d12ζpr(1r)nu=n212ζpr(1r)nu,
(20)
where 〈nu〉 = 〈uu〉 is the mean photon number in mode-u, and ζp = 16K/κp ≪ 1 in the weak excitation limit (Knjκj). The difference operator is calculated as
n21=ir(1r)(uuvv)(tp*tp)r(uvvu)(tp*tp)(tpuv+tp*vu)
(21)
with tp = −exp(−p). Consequently, observed signal photon number operator is expressed as
nsobs=uuvvnuns+ir(uvvu)r(1r)nunstpuv+tp*vu2ζpr(1r)nu
(22)
where we have used sin ψpψp = ζpns. Since in an actual system, the field in mode-v is generally left in vacuum state |0〉, the expectation value of nsobs is found as
nsobs=ns=n212ζpr(1r)nu
(23)
from which we see that the expectation value of the intracavity signal photon number 〈ns〉 is proportional to the difference of photon counts in the balanced detectors.

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 〈ns〉 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, tp = Gj(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 〈ns〉. The solid and dashed lines represent the precise numerical calculation and approximately analytical solution, respectively. (b) The transmission of the signal field versus 〈ns〉. Parameters used in the calculations: r = 1/2, κj,1 = 50κj,0, κs = κp, 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 〈nu〉 = 1013.

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〉 – 〈B2. From Eq. (23), we find that the variance (Δnsobs)2 of nsobs can be written as
(Δnsobs)2=(Δnu)2nunu2ns2+(1+1(1r)nu)(Δns)2+14ζp2r(1r)nu
(24)
where 〈(Δnu)2〉 and 〈(Δns)2〉 are the variances of nu and ns, respectively. In practice the field in mode-u is a strong coherent light, thus we can write 〈(Δnu)2〉 = 〈nu〉 ≫ 1 and (1 − r) 〈nu〉 ≫ 1. Consequently, Eq. (24) can be re-expressed as
(Δnsobs)2=(Δns)2+14ζp2r(1r)nu.
(25)
As shown in Eq. (25), there are two sources of noise contributing to the observed variance in the case of ideal detection. The first is the intrinsic noise of the signal 〈(Δns)2〉, and the second is the noise induced by the QND measurement scheme. The latter can be made arbitrarily small by using an ultra-high-Q cavity made of high-susceptibility material and by properly choosing 〈nu〉 and r. For a fixed r, increasing 〈nu〉 will decrease (Δnsobs)2 and increase 〈n21〉. It is easy to show that (Δnsobs)2 is minimized for r = 1/2 while 〈n21〉 attains its maximum value 〈n21〉 = ζpnu〉 〈ns〉. For a signal prepared in a number state, 〈(Δns)2〉 = 0, we find that 〈nu〉 should satisfy nu=ζp2(Δnsobs)21. As shown in Fig.3, measurement induced noise rapidly decreases with increasing 〈nu〉, and it is remarkably suppressed with increasing intrinsic Q0. It is clear that at high values of 〈nu〉, the dominant source of the observed variance is the intrinsic variance of the state to be measured. If the signal state to be measured is a coherent state, the lower bound of the observed variance is the mean photon number of the coherent state. In the case of a Fock state, the observed variance is solely due to the proposed QND scheme, and in principle it can be suppressed to negligible amounts with the use of an ultra-high-Q cavity Q0 ∼ 108 and a probe with larger average photon number 〈nu〉 ∼ 1014. The suggested probe photon number of 〈nu〉 ∼ 1014 corresponds to an average power of 13 μW in the 1550 nm band and it is much lower than the typical input power of a few mW used in experiments with microtoroidal cavities [33

33. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004). [CrossRef] [PubMed]

]. Thus, we believe that a probe with 〈nu〉 ∼ 1014 will enable significant noise suppression without causing any damage to the device.

Fig. 3 Dependence of observed variance (Δnsobs)2 on (a) the probe photon number which is linearly proportional to 〈uu〉, 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 〈ns〉 = 1 and 〈ns〉 = 10, respectively. The curves are obtained from Eq. 25 using the parameters given in Fig.2. For (b) we used 〈nu〉 = 1014. Lower bound of observed variance for a coherent state with 〈ns〉 is determined by its intrinsic noise 〈ns〉 when Q0 ∼ 108 and 〈nu〉 ∼ 1014. For a Fock state, it is determined by Q0, 〈nu〉 and K, hence can be made arbitrarily small by proper choice of these parameters.

Returning back to Eq. (25), we see that for a signal prepared as a number state, the variance of the measured photon number becomes
(Δnsobs)2=14ζp2r(1r)nu14ζp2npin
(26)
where for the second part we have used rnu=npin and r ≪ 1. On the other hand, when the signal is prepared in the coherent state with mean photon number 〈ns〉, we have
(Δnsobs)2=ns+14ζp2r(1r)nuns+14ζp2npin.
(27)
These expressions clearly show the relation between the tolerable measurement error and the 〈nu〉. For a signal with large 〈ns〉, we can tolerate a larger measurement error thus the condition on 〈nu〉 can be relaxed; however, for a smaller 〈ns〉, the tolerable measurement error will be much smaller thus 〈nu〉 should be large to satisfy this stricter error condition; as a result, the requirement for 〈nu〉 becomes more severe. Moreover, it can be shown that the (Δnsobs)2(Δns)2 coincides with (nsobsns)2 when nsobsns has no correlation with Δns.

Although in the above discussions, we have focused our attention on the detection of the intracavity signal photon number ns=asas and the statistical properties of the actually-observed intracavity signal photon number nsobj, one can extend the discussions to the detection of the photon number nsin of an input signal pulse asin as these two are directly associated with each other through Eq. (18). In the weak-excitation limit of Knsκs, we can approximately obtain ns=4asinasin/κs=4nsin/κs, hence the relations between the statistical properties of ns and nsin can be expressed as nsin=κsns/4 and (Δnsin)2=κs2(Δns)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 ns and r = 1/2, it is easy to show that the detected photon number difference will have a signal part with mean value ηnuζpns, 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) 〈nu〉. In order to resolve photon number reliably, the sensitivity per photon must be much larger than the standard deviation of the noise, ηnuζp1η. Error probability of making the decision that the signal has n photons if the detected photon number difference lies between ηnuζp(n − 1/2) and ηnuζp(n + 1/2) becomes Perror=1Erf(14ζp2nuη/(1η)) where a Gaussian distribution is used for noise because a coherent light with a large amplitude is used [11

11. H. A. Haus and F. X. Kärtner, “Optical quantum nondemolition measurement and the Copenhagen Interpretation,” Phys. Rev. A. 53, 3785–3791 (1986). [CrossRef]

]. For ζpnu=10, we find that Perror 10−7 and Perror 10−14, respectively for η equals to 0.5 and 0.7. Increasing ζpnu to 15 keeping η = 0.5 leads to Perror 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 ap is blocked leaving only the LO, the fluctuations in the difference signal will have zero mean with the variance of (1 – r)〈nu〉 which is the LO shot noise. Mode-mismatch effects can be minimized by proper spatial and spectral filtering as discussed in [47

47. S. K. Ozdemir, A. Miranowicz, M. Koashi, and N. Imoto, “Pulse-mode quantum projection synthesis: Effects of mode-mismatch on optical state truncation and preparation,” Phys. Rev. A 66, 053809 (2002). [CrossRef]

]. 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 〈(Δne)2〉. Thus n21 in Eq. 20 should be modified n21n21 + ne to include the contribution from electrical noise ne. Noting that the electrical noise contribution ne and the contribution n21 due to the field in mode ap are statistically independent of each other, the total variance is given by the sum of their variances. Thus, the variance (Δnsobs)2 of the observed photon-number difference given in Eq. 24 should include an additional term given as (Δne)2/4ζp2r(1r)nu to account for the contribution from electrical noise. Since ne 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 (Δnsobs)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.

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

44. T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12, 4742–4750 (2004). [CrossRef] [PubMed]

].

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).

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.

Y.-F Xiao and S. K. Ozdemir contributed equally to this work. The authors thank Prof. M. Koashi, Dr. T. Yamamoto and K. Azuma for fruitful discussions. This work is partly supported by a MEXT Grant-in-Aid for Scientific Research on Innovative Areas 20104003.

References and links

1.

C. M. Caves, K. S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, “On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle,” Rev. Mod. Phys. 52, 341–392 (1980). [CrossRef]

2.

F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, “Soliton backaction-evading measurement using spectral filtering,” Phys. Rev. A 66, 043810 (2002). [CrossRef]

3.

P. Grangier, J. -A. Levenson, and J.-P. Poizat, “Quantum nondemolition measurement in optics,” Nature 396, 537–542 (1998). [CrossRef]

4.

J. F. Roch, G. Roger, P. Grangier, J. M. Courty, and S. Reynaud, “Quantum non-demolition measurements in optics: a review and some recent experimental results,” Appl. Phys. B 55, 291–297 (1992). [CrossRef]

5.

G. Leuchs, C. Silberhorn, König, P. K. Lam, A. Sizmann, and N. Korolkova, in Quantum Information Theory with Continous Variables edited by S. L. Braunstein and A. K. Pati (Kluwer Academic, Dordrecht, 2002).

6.

W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005). [CrossRef]

7.

K. S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, “Quantum nondemolition measurement of harmonic oscillators,” Phys. Rev. Lett. 40, 667–671 (1978). [CrossRef]

8.

V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum nondemolition measurements,” Science 209, 547–557 (1980). [CrossRef] [PubMed]

9.

N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287–2292 (1985). [CrossRef] [PubMed]

10.

P. Alsing, G. J. Milburn, and D. F. Walls, “Quantum nondemolition measurements in optical cavities,” Phys. Rev. A 37, 2970–2978 (1988). [CrossRef] [PubMed]

11.

H. A. Haus and F. X. Kärtner, “Optical quantum nondemolition measurement and the Copenhagen Interpretation,” Phys. Rev. A. 53, 3785–3791 (1986). [CrossRef]

12.

V. B. Braginsky and F. Ya. Khalili, “Quantum nondemolition measurements: the route from toys to tools,” Rev. Mod. Phys. 68, 1–11 (1996). [CrossRef]

13.

J. M. Courty, S. Spälter, F. Krönig, A. Sizmann, and G. Leuchs, “Noise-free quantum-nondemolition measurement using optical solitons,” Phys. Rev. A 58, 1501–1508 (1998). [CrossRef]

14.

N. Imoto and S. Saito, “Quantum nondemolition measurement of photon number in a lossy optical Kerr medium”, Phys. Rev. A 39, 675–682 (1989). [CrossRef] [PubMed]

15.

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, “Nonideal quantum nondemolition measurements,” Phys. Rev. A. 42, 2995–3005 (1990). [CrossRef] [PubMed]

16.

P. Grangier, J. M. Courty, and S. Reynaud, “Characterization of nonideal quantum nondemolition measurements,” Opt. Commun. 89, 99–106 (1992). [CrossRef]

17.

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, “Quantum nondemolition detection of optical quadrature amplitudes,” Phys. Rev. Lett. 57, 2473–2476 (1986). [CrossRef] [PubMed]

18.

N. Imoto, S. Watkins, and Y. Sasaki, “A nonlinear optical-fiber interferometer for nondemolitional measurement of photon number,” Opt. Comm. 61, 159–163 (1987). [CrossRef]

19.

S. R. Friberg, S. Machida, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number of an optical soliton,” Phys. Rev. Lett. 69, 3165–3168 (1992). [CrossRef] [PubMed]

20.

J. P. Poizat and P. Grangier, “Experimental realization of a quantum optical tap,” Phys. Rev. Lett. 70, 271–274 (1993). [CrossRef] [PubMed]

21.

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Back-action evading measurements for quantum non-demolition detection and quantum optical tapping,” Phys. Rev. Lett. 72, 214–217 (1994). [CrossRef] [PubMed]

22.

K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, “Quantum non-demolition demonstration via repeated back-action evading measurements,” Phys. Rev. Lett. 75, 3422–3425 (1995). [CrossRef] [PubMed]

23.

R. Bruckmeier, H. Hansen, and S. Schiller, “Repeated quantum non-demolition measurements of continuous optical waves,” Phys. Rev. Lett. 79, 1463–1466 (1997). [CrossRef]

24.

K. Bencheick, C. Simonneau, and J. A. Levenson, “Cascaded amplifying quantum optical taps: A robust noiseless optical bus,” Phys. Rev. Lett. 78, 34–37 (1997). [CrossRef]

25.

B. C. Buchler, P. K. Lam, H. -A. Bachor, U. L. Andersen, and T. C. Ralph, “Squeezing more from a quantum nondemolition measurement,” Phys. Rev. A. 65, 011803(R) (2001). [CrossRef]

26.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, “Quantum non-demolition measurements using cold trapped atoms,” Phys. Rev. Lett. 78, 634–637 (1997). [CrossRef]

27.

A. Kuzmich, L. Mandel, and N. P. Bigelow, “Generation of spin squeezing via continous quantum nondemolition measurement,” Phys. Rev. Lett. 85, 1594–1597 (2000). [CrossRef] [PubMed]

28.

S. Peil and G. Gabrielse, “Observing the quantum limit of an electron cyclotron: QND measurements of quantum jumps between Fock states,” Phys. Rev. Lett. 83, 1287–1290 (1999). [CrossRef]

29.

G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, “Seeing a single photon without destroying it,” Nature 400, 239–242 (1999). [CrossRef]

30.

C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448, 889–893 (2007). [CrossRef] [PubMed]

31.

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuring a photonic qubit without destroying it,” Phys. Rev. Lett. 92, 190402 (2004). [CrossRef] [PubMed]

32.

I. Fushman and J. Vučković, “Analysis of a quantum dondemolition measurement scheme based on Kerr nonlinearity in photonic crystal waveguides,” Opt. Express 15, 5559–5571 (2007). [CrossRef] [PubMed]

33.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004). [CrossRef] [PubMed]

34.

S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71, 013817 (2005). [CrossRef]

35.

M. Yamane and Y. Asahara, Glass for Photonics (Cambridge University Press, Cambridge, 2000). [CrossRef]

36.

S. X. Qian and R. K. Chang, “Multiorder stokes emission from micrometer-size droplets,” Phys. Rev. Lett. 56, 926–929 (1986). [CrossRef] [PubMed]

37.

A. J. Campillo, J. D. Eversole, and H. B. Lin, “Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets,” Phys. Rev. Lett. 67, 437–440 (1991). [CrossRef] [PubMed]

38.

F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,” Eur. Phys. J. D 1, 235–238 (1998).

39.

S. Uetake, M. Katsuragawa, M. Suzuki, and K. Hakuta, “Stimulated raman scattering in a liquid-hydrogen droplet,” Phys. Rev. A 61, 011803 (2000). [CrossRef]

40.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621 (2002). [CrossRef] [PubMed]

41.

T. Carmon and K. J. Vahala, “Visible continuous emission from a silica microphonic device by third-harmonic generation,” Nature Physics 3, 430–435 (2007). [CrossRef]

42.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003). [CrossRef] [PubMed]

43.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip,” Appl. Phys. Lett. 85, 6113–6115 (2004). [CrossRef]

44.

T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12, 4742–4750 (2004). [CrossRef] [PubMed]

45.

D. F. Walls and G. J. Milburn, Quantum Optics(Springer-Verlag, BerlinHeidelberg, 1994)

46.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761–3774 (1985). [CrossRef] [PubMed]

47.

S. K. Ozdemir, A. Miranowicz, M. Koashi, and N. Imoto, “Pulse-mode quantum projection synthesis: Effects of mode-mismatch on optical state truncation and preparation,” Phys. Rev. A 66, 053809 (2002). [CrossRef]

48.

Mani Hossein-Zadeh and Kerry J. Vahala “Free ultra-high-Q microtoroid: a tool for designing photonic devices,” Opt. Express 15, 166–175 (2007). [CrossRef] [PubMed]

OCIS Codes
(140.4780) Lasers and laser optics : Optical resonators
(190.1450) Nonlinear optics : Bistability
(190.3270) Nonlinear optics : Kerr effect
(270.0270) Quantum optics : Quantum optics
(270.5570) Quantum optics : Quantum detectors

ToC Category:
Quantum Optics

History
Original Manuscript: July 29, 2008
Revised Manuscript: December 8, 2008
Manuscript Accepted: December 11, 2008
Published: December 15, 2008

Citation
Yun-Feng Xiao, Sahin K. Özdemir, Venkat Gaddam, Chun-Hua Dong, Nobuyuki Imoto, and Lan Yang, "Quantum nondemolition measurement of photon number via optical Kerr effect in an ultra-high-Q microtoroid cavity," Opt. Express 16, 21462-21475 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-26-21462


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. C. M. Caves, K, S. Thorne, R. W. P. Drever, M. Zimmermann, and V. D. Sandberg, "On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle," Rev. Mod. Phys. 52, 341-392 (1980). [CrossRef]
  2. F. König, B. Buchler, T. Rechtenwald, G. Leuchs, and S. Sizmann, "Soliton backaction-evading measurement using spectral filtering," Phys. Rev. A 66, 043810 (2002). [CrossRef]
  3. P. Grangier, J. -A. Levenson, and J.-P. Poizat, "Quantum nondemolition measurement in optics," Nature 396, 537-542 (1998). [CrossRef]
  4. J. F. Roch, G. Roger, P. Grangier, J. M. Courty, and S. Reynaud, "Quantum non-demolition measurements in optics: a review and some recent experimental results," Appl. Phys. B 55, 291-297 (1992). [CrossRef]
  5. G. Leuchs, C. Silberhorn,K¨onig, P. K.  Lam, A. Sizmann, and N. Korolkova, in Quantum Information Theory with Continous Variables edited by S. L. Braunstein and A. K. Pati (Kluwer Academic, Dordrecht, 2002).
  6. W. J. Munro, K. Nemoto, and T. P. Spiller, "Weak nonlinearities: a new route to optical quantum computation," New J. Phys. 7, 137 (2005). [CrossRef]
  7. Q1. K, S. Thorne, R. W. P. Drever, C. M. Caves, M. Zimmermann, and V. D. Sandberg, "Quantum nondemolition measurement of harmonic oscillators," Phys. Rev. Lett. 40, 667-671 (1978). [CrossRef]
  8. V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, "Quantum nondemolition measurements," Science 209, 547-557 (1980). [CrossRef] [PubMed]
  9. N. Imoto, H. A. Haus, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number via the optical Kerr effect," Phys. Rev. A 32, 2287-2292 (1985). [CrossRef] [PubMed]
  10. P. Alsing, G. J. Milburn, D. F. Walls, "Quantum nondemolition measurements in optical cavities," Phys. Rev. A 37, 2970-2978 (1988). [CrossRef] [PubMed]
  11. H. A. Haus and F. X. Kärtner, "Optical quantum nondemolition measurement and the Copenhagen Interpretation," Phys. Rev. A. 53, 3785-3791 (1986). [CrossRef]
  12. V. B. Braginsky and F. Ya. Khalili, "Quantum nondemolition measurements: the route from toys to tools," Rev. Mod. Phys. 68, 1-11 (1996). [CrossRef]
  13. J. M. Courty, S. Sp¨alter, F. Krönig, A. Sizmann, and G. Leuchs, "Noise-free quantum-nondemolition measurement using optical solitons," Phys. Rev. A 58, 1501-1508 (1998). [CrossRef]
  14. N. Imoto and S. Saito, "Quantum nondemolition measurement of photon number in a lossy optical Kerr medium", Phys. Rev. A 39, 675-682 (1989). [CrossRef] [PubMed]
  15. M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, "Nonideal quantum nondemolition measurements," Phys. Rev. A. 42, 2995-3005 (1990). [CrossRef] [PubMed]
  16. P. Grangier, J.M. Courty, and S. Reynaud, "Characterization of nonideal quantum nondemolition measurements," Opt. Commun. 89, 99-106 (1992). [CrossRef]
  17. M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, "Quantum nondemolition detection of optical quadrature amplitudes," Phys. Rev. Lett. 57, 2473-2476 (1986). [CrossRef] [PubMed]
  18. N. Imoto, S. Watkins, and Y. Sasaki, "A nonlinear optical-fiber interferometer for nondemolitional measurement of photon number," Opt. Comm. 61, 159-163 (1987). [CrossRef]
  19. S. R. Friberg, S. Machida, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number of an optical soliton," Phys. Rev. Lett. 69, 3165-3168 (1992). [CrossRef] [PubMed]
  20. J. P. Poizat, P. Grangier, "Experimental realization of a quantum optical tap," Phys. Rev. Lett. 70, 271-274 (1993). [CrossRef] [PubMed]
  21. S. F. Pereira, Z. Y. Ou, and H. J. Kimble, "Back-action evading measurements for quantum non-demolition detection and quantum optical tapping," Phys. Rev. Lett. 72, 214-217 (1994). [CrossRef] [PubMed]
  22. K. Bencheikh, J. A. Levenson, P. Grangier, and O. Lopez, "Quantum non-demolition demonstration via repeated back-action evading measurements," Phys. Rev. Lett. 75, 3422-3425 (1995). [CrossRef] [PubMed]
  23. R. Bruckmeier, H. Hansen, and S. Schiller, "Repeated quantum non-demolition measurements of continuous optical waves," Phys. Rev. Lett. 79, 1463-1466 (1997). [CrossRef]
  24. K. Bencheick, C. Simonneau, and J. A. Levenson, "Cascaded amplifying quantum optical taps: A robust noiseless optical bus," Phys. Rev. Lett. 78, 34-37 (1997). [CrossRef]
  25. B. C. Buchler, P. K. Lam, H. -A. Bachor, U. L. Andersen, and T. C. Ralph, "Squeezing more from a quantum nondemolition measurement," Phys. Rev. A. 65, 011803(R) (2001). [CrossRef]
  26. J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum non-demolition measurements using cold trapped atoms," Phys. Rev. Lett. 78, 634-637 (1997). [CrossRef]
  27. A. Kuzmich, L. Mandel, and N. P. Bigelow, "Generation of spin squeezing via continous quantum nondemolition measurement," Phys. Rev. Lett. 85, 1594-1597 (2000). [CrossRef] [PubMed]
  28. S. Peil, and G. Gabrielse, "Observing the quantum limit of an electron cyclotron: QND measurements of quantum jumps between Fock states," Phys. Rev. Lett. 83, 1287-1290 (1999). [CrossRef]
  29. G. Nogues, A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M. Raimond, and S. Haroche, "Seeing a single photon without destroying it," Nature 400, 239-242 (1999). [CrossRef]
  30. C. Guerlin, J. Bernu, S. Delglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J. M. Raimond, and S. Haroche, "Progressive field-state collapse and quantum non-demolition photon counting," Nature 448, 889-893 (2007). [CrossRef] [PubMed]
  31. G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, "Measuring a photonic qubit without destroying it," Phys. Rev. Lett. 92, 190402 (2004). [CrossRef] [PubMed]
  32. Q2. I. Fushman and J. Vučković, "Analysis of a quantum dondemolition measurement scheme based on Kerr nonlinearity in photonic crystal waveguides," Opt. Express 15, 5559-5571 (2007). [CrossRef] [PubMed]
  33. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity," Phys. Rev. Lett. 93, 083904 (2004). [CrossRef] [PubMed]
  34. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, "Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics," Phys. Rev. A 71, 013817 (2005). [CrossRef]
  35. M. Yamane and Y. Asahara, Glass for Photonics (Cambridge University Press, Cambridge, 2000). [CrossRef]
  36. S. X. Qian and R. K. Chang, "Multiorder stokes emission from micrometer-size droplets," Phys. Rev. Lett. 56, 926-929 (1986). [CrossRef] [PubMed]
  37. A. J. Campillo, J. D. Eversole, and H. B. Lin, "Cavity quantum electrodynamic enhancement of stimulated emission in microdroplets," Phys. Rev. Lett. 67, 437-440 (1991). [CrossRef] [PubMed]
  38. F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, "Evidence for intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium," Eur. Phys. J. D 1, 235-238 (1998).
  39. S. Uetake, M. Katsuragawa, M. Suzuki, and K. Hakuta, "Stimulated raman scattering in a liquid-hydrogen droplet," Phys. Rev. A 61, 011803 (2000). [CrossRef]
  40. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, "Ultralow-threshold Raman laser using a spherical dielectric microcavity," Nature 415, 621 (2002). [CrossRef] [PubMed]
  41. Q3. T. Carmon and K. J. Vahala, "Visible continuous emission from a silica microphonic device by third-harmonic generation," Nature Physics 3, 430-435 (2007). [CrossRef]
  42. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003). [CrossRef] [PubMed]
  43. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip," Appl. Phys. Lett. 85, 6113-6115 (2004). [CrossRef]
  44. T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behavior and thermal self-stability of microcavities," Opt. Express 12, 4742-4750 (2004). [CrossRef] [PubMed]
  45. D. F. Walls and G. J. Milburn, Quantum Optics(Springer-Verlag, Berlin Heidelberg, 1994)
  46. C.W. Gardiner and M. J. Collett, "Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation," Phys. Rev. A 31, 3761-3774 (1985). [CrossRef] [PubMed]
  47. S. K. Ozdemir, A. Miranowicz, M. Koashi, and N. Imoto, "Pulse-mode quantum projection synthesis: Effects of mode-mismatch on optical state truncation and preparation," Phys. Rev. A 66, 053809 (2002). [CrossRef]
  48. Mani Hossein-Zadeh, and Kerry J. Vahala "Free ultra-high-Q microtoroid: a tool for designing photonic devices," Opt. Express 15, 166-175 (2007). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited