OPEN Spatially resolved single photon

SUBJECT AREAS: detection with a quantum sensor array QUBITS A. M. Zagoskin1, R. D. Wilson1, M. Everitt1, S. Savel’ev1, D. R. Gulevich1, J. Allen1, V. K. Dubrovich2 SUPERCONDUCTING DEVICES & E. Il’ichev3

Received 1Department of Physics, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom, 2Special 26 July 2013 Astrophysical Observatory, Russian Academy of Sciences, and Cryogenic Nanoelectronics Laboratory, Nizhny Novgorod State Technical University, Russia, 3Institute for Photonic Technologies, Jena, Germany and Novosibirsk State Technical University, Accepted Novosibirsk, Russia. 25 November 2013

Published We propose a method of resolving a spatially coherent signal, which contains on average just a single photon, 10 December 2013 against the background of local noise at the same frequency. The method is based on detecting the signal simultaneously in several points more than a wavelength apart through the entangling interaction of the incoming photon with the quantum metamaterial sensor array. The interaction produces the spatially Correspondence and correlated quantum state of the sensor array, characterised by a collective observable (e.g., total magnetic moment), which is read out using a quantum nondemolition measurement. We show that the effects of local requests for materials noise (e.g., fluctuations affecting the elements of the array)pffiffiffiffi are suppressed relative to the signal from the should be addressed to spatially coherent field of the incoming photon as *1 N, where N is the number of array elements. The A.M.Z. (a.zagoskin@ realisation of this approach in the microwave range would be especially useful and is within the reach of lboro.ac.uk) current experimental techniques.

he ultimate goal and the theoretical limit of weak signal detection is the ability to detect a single photon against a noisy background. In this situation the inescapable noise produced by the measuring device itself T may be the main obstacle, but the uncertainty principle restricts possible experimental techniques of increasing the signal-to-noise ratio. For example, a weak classical signal from a remote source can be distin- guished from the local noise at the same frequency through its spatial correlations (using phase sensitive detectors; coincidence counters; etc) - i.e., by sensing its wave front. This method seems impossible in the case of a single incoming photon, since it is usually assumed that it can only be detected once. Nevertheless such a conclusion would be too hasty. In this paper we show, that a combination of a quantum metamaterial1 (QMM)- based sensor array and quantum non-demolition2,3 (QND) readout of its quantum state allows, in principle, to detect a single photon in several points. Actually, there are a few possible ways of doing this, with at least one within the reach of current experimental techniques for the microwave range. We will illustrate the possible implementations of this scheme using a simplified model, an example of which is shown in Fig. 1. Here the QMM array is modelled by a set of N qubits, which are coupled to two LC circuits: the one (A) represents the input mode, and the other (B) the readout. This model is closest to the case of microwave signal detection using superconducting qubits, which is both most interesting (at least from the point of view of radioastronomy) and most feasible, as attested by the signal progress achieved in the recent years. In particular, several schemes of microwave photon detectors based on Josephson qubits and Josephson qubit metamaterials, capable of detecting a single photon, were proposed4–6 and realized7; the entanglement detection and quantum state tomography were demonstrated as well8–11. Nevertheless our approach and conclusions apply generally, mutatis mutandis (e.g., to the case of photonic crystal decorated with two level systems13, an array of SQUIDs14 or atoms in an optical lattice12). We will therefore begin by discussing how such a detector system could work in principle. We then go to demonstrate that a clear distinction between a single incident photon and the vacuum can be seen in the response of a simple two-qubit detector array using a fully quantum mechanical model. Finally we explore the role of inter-qubit coupling and increasing the size of the QMM array using a semi-classical mean field approach.

Results Mathematical model. The system of Fig. 1 is described by the Hamiltonian

H~HazVazHqbzVbzHb: ð1Þ

SCIENTIFIC REPORTS | 3 : 3464 | DOI: 10.1038/srep03464 1 www.nature.com/scientificreports

To model the naturalpffiffiffiffiffiffiffiffi effects of decoherencepffiffiffiffiffiffiffiffiffi on the qubits we have ðÞi ðÞi ðÞi the Lindblads Lz~ 2Czs{ and Lxy~ 2Cxyszs{ acting on both qubits. These operators describe relaxation in the z-direction and A dephasing in the x-y plane of the Bloch sphere respectively. To account for the weakpffiffiffiffiffiffiffiffi continuous measurement of the output fieldDE ^ ^ we also take Lb~ 2Cbb. From the real and imaginary parts of b ~ ...... wepffiffiffiffiffiffiffiffiffiffiffiffiffi canÀÁ extract the expectation valuespffiffiffiffiffiffiffiffiffiffiffi forÀÁ the position xb z { ~ {{ 1=2vb b b and momentum pb i vb=2 b b operators. ~ B We assume the qubits act in the sz basis with vq 1 and there- ...... fore D 5 0. Typical flux qubits work at a frequency of the order of 10 GHz, whereas the relaxation and dephasing rates are usually of the order of 10 MHz17, therefore we take in the Lindblad operators 23 Figure 1 | Schematic of a section of the photon detector system. Photons Cz/vq 5 Cxy/vq 5 10 . For the input field, we take va/vq 5 0.5 (A) are incident on to the QMM array, which is comprised of N qubits. The assuming an incident photon off-resonance with the qubit. We take QMM array is also coupled to the readout tank circuit (B) in order to vb/vq 5 0.5 assuming the readout oscillator is also off-resonance perform quantum non-demolition measurement. with the qubit but resonant with the input field. For high quality tank circuits and transmission lines the lifetime is typically relatively long 18–20 compared to the operating frequency , and we therefore take Cb/ 23 a b Here vb 5 10 . Our choice of coupling parameters g /vq 5 g /vq 5 0.01 ÀÁÀÁ is also in line with the experiment19,20. H ~v a{az1=2 zftðÞa{za ð2Þ a a In the beginning, the output field is in the vacuum state and the .pffiffiffi describes the input circuit, excited by the incoming field; qubits are in the superposition stateðÞji 0 zji1 2. We then simu-  XN  late QSD trajectories with the input field initially in the vacuum state ~{1 xz z and coherent states with an average both of 1 and 5 photons. The Hqb Djsj jsj ð3Þ 2 j~1 resulting spectra of the readout mode for a typical quantum traject- ory are shown in Fig. 2. The most important conclusion is that there is the Hamiltonian of the qubits; is a clear difference to the readout depending on whether a photon is ÀÁÀÁ incident upon the detector or not. { { Hb~vb b bz1=2 zhtðÞb zb ð4Þ When the input field is initially in a coherent state the peak in the power spectral density is shifted to lower frequencies and decreases is the Hamiltonian of the output circuit with the probing field, used in magnitude compared to the clear sharp peak seen when the input in case of IMT readout; and field is in the vacuum state. The response is smeared out across the X ÀÁ X ÀÁ low frequency region leading to a higher average power when there ~ a {z x ~ b {z x Va gj a a sj , Vb gj b b sj ð5Þ photons there are photons incident upon the detector. This is par- j j ticularly clear in the case of the position operator where there is a clear distinction of approximately one order of magnitude. describe the coupling between the QMM array and the input and One could object that a coherent input state actually contains an output circuits. The effects of ambient noise can be taken into infinite number of photons, and thus cannot be considered a valid account, e.g., by adding an appropriate term H to (1) or by noise test for the proposed approach. In order to check this, we have including Lindblad operators in the master equation for the performed the same simulations for the input field in a Fock state density matrix of the system (see, e.g.15). (Fig. 3). The results show that the distinction between the responses to different input fields remains significant and still allows to reliably Numerical results for two sensor qubits. To investigate the level of distinguish between them. sensitivity of the proposed detector system, we consider the example of a QMM array comprised of two qubits coupled to the input mode Scaling of the quantum metamaterial sensor array. In order to and readout oscillator, as shown in Fig. 1. We assume that the input further investigate the role of increased number of qubits and of field has a given number of photons incident on it and is initially interqubit couplings, we consider the Hamiltonian 2 X hiX found in a coherent state, jaæ, with an average of jaj photons and 1 H~{ D sxz ðÞt sz zg szsz , ð7Þ therefore take f(t) 5 0. We also take h(t) 5 0 and assume that the 2 j j j j j jz1 intrinsic noise in the detector system is sufficient to drive the readout j j field and allow detection of the incident photons. with qubits driven by common harmonic off-resonance signal In order to fully account for the effects of decoherence and mea- (modeling the input Va of Eq.(1)) and local noise coupled through sz: surement, we make use of the quantum state diffusion formalism16 to pffiffiffiffiffiffi describe the evolution of the state vector jyæ; jðÞt ~ sinðÞvt z 2DjjðÞt : ð8Þ X DE DE ~ 1 1 Here Æj(t)æ 5 0 and Æjj(t)jl(t9)æ 5 djld(t 2 t9) and we take =D 0:01, { { { 26 jidy ~{iHjiy dtz L^ L^j{ L^ L^j{ L^ L^j jiy dt v/D 5 1.13 and D/D 5 10 . j 2 j 2 j j We represent the effects of decoherence on the evolution of the X ÂÃ ð6Þ z ^ { ^ sensor qubits by solving the Lindblad master equation Lj Lj jiy djj, X nohihi dr 1 j ~{iH½,r z L ,rL{ z L r,L{ , ð9Þ dt 2 j j j j where jdyæ and dt are the state vector and time increments respect- j ively, L^ are the Lindblad operators and dj are the stochastic Wiener j with relaxation, Lz, and dephasing, Lxy, Lindblad operators acting on ~ ~ ~ 23 increments which satisfy djidjj dji 0 and djidjj dijdt. each qubit, where Cz/D 5 Cxy/D 5 10 .

SCIENTIFIC REPORTS | 3 : 3464 | DOI: 10.1038/srep03464 2 www.nature.com/scientificreports

104 104 ω ω > > 2 2 b 0 b 0 x 10 x 10

-4 -4 Power < 10 Power < 10

10-8 10-8 0 1 2 3 4 0 1 2 3 4 Frequency ω / ε Frequency ω / ε

104 104 ω ω > > 2 2 b 0 b 0 p 10 p 10

-4 -4 Power < 10 Power < 10

10-8 10-8 0 1 2 3 4 0 1 2 3 4 Frequency ω / ε Frequency ω / ε

2 2 Figure 2 | Power spectra for the readout field’s position xb v and Figure 3 | Power spectra for the readout field’s position xb v and 2 2 momentum pb v quadratures for a typical QSD realisation in the momentum pb v quadratures for a typical QSD realisation in the presence of local noise. The input field is in a coherent state with the presence of local noise. The input field is in a Fock state with the number of average number of photons equal 0 (black dashed line, bottom), 1 (red line, photons equal 0 (black dashed line, bottom), 1 (red line, middle) or 5 (blue middle) or 5 (blue line, top). Note the significant difference between the line, top). The difference between the data for zero and one photons in the data for zero and one photons in the incoming signal. incoming signal remains qualitatively the same as in the case of a coherent input state. In the case of N uncoupled qubits (where g 5 0), we can describe pffiffiffi the system by N independent master equations for a single-qubit jiYðÞt ~cos 2gat ji1 6ji; 6ji; density matrix, and average the observable quantities. This data is 1 2 shown in Fig. 4a,b. The spectral density of the z-component of the pffiffiffi 6 z 6 ð10Þ a ji;1 ji:2 ffiffiffiji:1 ji;2 total ‘‘spin’’ demonstrates a small, but distinct peak due to the zi sin 2g t ji0 6 p : 2 external drive, in addition to the large noise-driven signal. The ~ At the momentsffiffiffi when the first term vanishes, tn increase of the number of qubits, predictably, increases the signalffiffiffiffi p p ðÞp=2zpn 2ga, the qubits are in the maximally entangled Bell to noise ratio. The increase is in qualitative agreement with the N behaviour, expected from the analytic estimate given below in the state, and the QND measurement of their summary ‘‘spin’’ in z- Discussion section, though numerically somewhat smaller (approxi- direction realizes the observation of a single photon’s presence (a j æ mately doubling rather than tripling as N increases from 1 to 9; see Fock state 1 of the circuit A) at two spatially separated points Fig. 4a, inset). (locations of the qubits 1 and 2). The qubit-qubit coupling also increases the signal to noise ratio. A literal realisation of such a scheme for observing a single We now solve the master equation for two coupled qubits and the photon’s wavefront in multiple points is theoretically possible, but results show that while the overall signal amplitude is suppressed by hardly advisable: The resonant transfer of the incoming photon into qubit-qubit coupling, the relative amplitude of the signal signifi- the qubit array and back is vulnerable to absorption in one of the cantly increases (Fig. (4c)). qubits. A better opportunity is presented by the dispersive regime, when the mismatch between q theffiffiffiffiffiffiffiffiffiffiffiffiffiffi qubits’ and incoming photon’s res-

dV ~ D2z 2{v ? a Discussion onant frequencies, j j J a gj , allows to use the For a simple illustration of how a single photon can be simulta- Schrieffer-Wolff transformation. If D 5 0, the interaction term Va neously detected at several points in space, consider the case when in (1) is reduced to15,22 0  1 there is one photon in the input circuit, two identical and identically 2 X a coupled to it noiseless qubits are initially in their ground state, and B gj C ~ ~@ zA { the readout circuit is switched off. In this case the system undergoes Va sj a a: ð11Þ dVj vacuum Rabi oscillations, and its wave function is21 j

SCIENTIFIC REPORTS | 3 : 3464 | DOI: 10.1038/srep03464 3 www.nature.com/scientificreports

for the moment the rest of the system, due to the weakness of the effective coupling g2/dV in (11), we can write for the wave function of the input circuit

d ÀÁ i jiy ðÞt

Here D(a) with ð t 0 0 aðÞt ~{i dt feðÞt , ð16Þ 0 is the displacement operator

{ DðÞa ~eaa {a a: ð17Þ

Acting on a vacuum state, it produces a coherent state, D(a)j0æ 5 jaæ. Therefore, assuming that the input circuit was initially in the vacuum state, the average

{ < { < { a a t hjyaðÞt a ajiyaðÞt hjaðÞt a ajiaðÞt ð t 2 ð18Þ 2 0 0 ~jjaðÞt ~ dt feðÞt : 0 Figure 4 | (Top) Spectral density of total detector ‘‘spin’’ Sz in a qubit in the presence of noise and drive. The signal due to drive is a small thin peak Therefore the action of the incoming field on the qubits in the dis- on the left of the resonant noise response. Inset: Signal to noise ratio as the persive regime can be approximated by replacing the terms Ha and function of number of qubits. (Centre) Same in case of 8 uncoupled qubits Va in the Hamiltonian (1) with (where g 5 0). Inset: A close up of the signal-induced feature. The noise is 0  1 suppressed in case of 8 uncoupled qubits (blue) compared to the case of a 2 ! X a X single qubit (red). (Bottom) The spectral density of S in case of two B gj C z ~@ zA 2 z 2 coupled qubits, where g/D 5 2. Note that the significant shift of the htðÞ sj jjaðÞt : cjsj jjaðÞt : ð19Þ dVj resonant frequency of the system (position of the noise-induced feature). j j Inset: Signal response amplitude (left) and signal to noise ratio (right) as functions of the coupling strength. In the Heisenberg representation the ‘‘spin’’ of the jth qubit,

~~sxsxzsysyzszsz, ð20Þ Now the effect of the input field on the detector qubits is the addi- sj j j j j j j tional phase gain proportional to the number of incoming photons, satisfies the Bloch equations, which in case of zero bias and only z- which can be read out using a QND technique. noise ( ~0; j (t) 5 0), and neglecting for the moment the inter- Note that in addition to (11) we will also obtain the effective j j action with the readout circuit, take the form coupling between qubits through the vacuum mode of the oscillator (see, e.g.23). In the case of identical qubits and coupling parameters it hi d x ~ 2z y is s ðÞt 2 cjjjaðÞt gjðÞt s ðÞt ; dt j j 2 X hi ðÞga ~ x x d y 2 x z Heff ~ s s : ð12Þ s ðÞt ~{2 c jjaðÞt zg ðÞt s ðÞt {D s ðÞt ; ð21Þ 2dV j k j j j j eff j jk dt d z y If the number of qubits in the matrix is large enough, this term can be s ðÞt ~Deff s ðÞt , dt j j approximated by a ‘‘mean field’’ producingDEX an effective ‘‘tunneling’’ +~ x y for each individual qubit, D ðÞt sx~ sx sx. or, introducing sj sj +isj , eff j k k j In the following, it will be convenient to account for the ambient nohi d + ~ 2z + z z noise sources through the term s ðÞt + 2i cijjaðÞt gjðÞt s ðÞt iDeff s ðÞt ;  dt j j j X ð22Þ x z hi H ~ j ðÞt s zg ðÞt s ð13Þ d Deff z { noise j j j j szðÞt ~ s ðÞt {s ðÞt : j dt j 2i j j in the Hamiltonian. In agreement with our assumptions, these fluc- Let us initialize the qubit in an eigenstate of sx (i.e., in an eigenstate of j j 9 d j tuations in different qubits are uncorrelated: Æ j(t) k(t )æ / jk; ~ z ~ unperturbed qubit Hamiltonian, since j 0). Then sj ðÞ0 0, Æjj(t)gk(t9)æ 5 0. + ~ x Let us excite the input circuit with a resonant field, f(t) 5 fe(t) sj ðÞ0 sj ðÞ0 (i.e., 1 or 21), and, assuming the slowness of Deff(t), exp[2ivat] 1 c.c., with slow real envelope function fe(t). Neglecting the equations (22) can be solved perturbatively:

SCIENTIFIC REPORTS | 3 : 3464 | DOI: 10.1038/srep03464 4 www.nature.com/scientificreports

ð t hi Methods +ðÞ0 ~ 0 2z 0 0 x Numerical methods. In the fully quantum mechanical treatment of a detector sj ðÞt exp +2i cjjjaðÞt gjðÞt dt sj ðÞ0 ; 0 consisting of two sensor qubits we represent both the input and output fields in a ()truncated basis comprised of the first 30 number states. The deterministic part of the ð ð 0 hi t t quantum state diffusion evolution (the first two terms of (6)) is computed using a zðÞ1 ~{ x 00 2z 00 00 0 sj ðÞt Deff sj ðÞ0 sin 2 cjjjaðÞt gjðÞt dt dt <ð23Þ fifth-order Runge-Kutta formula with adaptive step size, while the stochastic part of 0 0 each time step (the third term in (6)) is integrated using the Euler method. After 150 2 ð ð periods of vb the power spectra of the output field’s position x and momentum t t0 hi b v 2 p2 quadratures are taken using a fast Fourier transform. {2D sxðÞ0 c jjaðÞt00 zg ðÞt00 dt0dt00: b v eff j i j In the semi-classical treatment of a quantum metamaterial sensor array we solve 0 0 the Lindblad master equation numerically using the Euler method. For the case of two coupled qubits, this is done using the generalized Bloch parametrization of the two- Assuming identical qubits identically coupled to the input circuit, we qubit density matrix: X 1 finally obtain for the collective variable (z-component of the total r^~ P s16s2 ð25Þ 4 ab a b qubit ‘‘spin’’ of the QMM array) a,b~0,x,y,z as in27. XN z z x S ðÞt : s ðÞt <{2cDeff s ðÞ0 N j 1. Rakhmanov, A., Zagoskin, A., Savel’ev, S. & Nori, Franco. Quantum metamaterials: j~1 Electromagnetic waves in a Josephson qubit line. Phys. Rev. B 77,144507(2008). "#ð ð ð ð ð24Þ 0 0 XN 2. Braginsky, V. B. & Khalili, F. Ya. Quantum Measurement. (Cambridge University t t t t 1 jjaðÞt00 2dt0dt00z g ðÞt00 dt0dt00 : Press, Cambridge, 1995). j 3. Lupascu, A. et al. Quantum non-demolition measurement of a superconducting 0 0 0 0 N ~ j 1 two-level system. Nature Physics 3, 119–125 (2007). 4. Romero, G., Garcia-Ripoll, J. J. & Solano, E. Microwave Photon Detector in The second term in the brackets is the result of local fluctuationspffiffiffiffi Circuit QED. Phys. Rev. Lett. 102, 137602 (2009). affecting separate qubits and is therefore, in the standard way, * N 5. Peropadre, B. et al. Approaching perfect microwave photodetection in circuit times suppressed compared to the first term (due to the regular QED. Phys. Rev. A 84, 063834 (2011). 6. Poudel, A., McDermott, R. & Vavilov, M. G. Quantum efficiency of a microwave evolution produced by the spatially coherent input photon field). photon detector based on a current-biased Josephson junction. Phys. Rev. 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Path Entanglement of Continuous-Variable Quantum 2 be O(N ) and O(N) respectively. Microwaves. Phys. Rev. Lett. 109, 250502 (2012). Though the possibility to observe a single photon’s wave-front 11. DiCandia, R. et al. Dual-Path Method for Propagating Quantum Microwaves. requires the detection of a weak, remote signal against the back- arXiv:1308.3117 [quant-ph] (2013). ground of local fluctuations, the standard signal-to-noise ratio 12. Bloch, I. Ultracold quantum gases in optical lattices. Nature Physics 1, 23 (2005). pffiffiffiffi 13. Greentree, A. D., Fairchild, B. A., Hossain, F. M. & Prawer, S. Diamond integrated N-enhancement due to the N-element coherent uncoupled quantum photonics. Materials Today 11, 22 (2008). QMM array is unlikely to be of much practical use. Noticing that 14. Stiffell, P. B., Everitt, M. J., Clark, T. D., Harland, C. J. & Ralph, J. F. Quantum downconversion and multipartite entanglement via a mesoscopic superconducting the effect of the input field is nothing but a simple one-qubit quantum interference device ring. Phys. Rev. B 72,014508(2005). quantum gate applied to each element of the array and that intro- 15. Zagoskin, A. M. Quantum Engineering: Theory and Design of Quantum Coherent ducing a simple qubit-qubit coupling scheme can improve matters, Structures. (Cambridge University Press, Cambridge, 2011). we can ponder a more sophisticated approach. By performing on a 16. Percival, I. Quantum State Diffusion. (Cambridge University Press, Cambridge, 1998). group of qubits a set of quantum manipulations, which would realize 17. Astafiev, O. et al. Resonance Fluorescence of a Single Artificial Atom. Science 327, a quantum error correction routine, one can hope to improve the 840–843 (2010). sensitivity of the system. 18. Il’ichev, E. et al. 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Blais, A., Huang, R.-S., Wallraff, A., Girvin, S. M. & Schoelkopf, R. J. Cavity quantum metrology ). The seeming violation is due to the fact that quantum electrodynamics for superconducting electrical circuits: An architecture in26 the systems are made to interact with a common ‘‘quantum bus’’ for quantum computation. Phys. Rev. A 69, 062320 (2004). (and are no longer independent), and it is the state of the bus that is 23. Makhlin, Yu., Scho¨n, G. & Shnirman, A. Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys. 73, 357–400 (2001). consequently measured. 24. Gardiner, C. W. & Zoller, P. Quantum Noise. Springer, Berlin (2004). In conclusion, we have shown the possibility in principle to detect 25. Giovannetti, V., Lloyd, S. & Maccone, L. Quantum Metrology. Phys. Rev. Lett. 96, the wavefront of a spatially coherent signal containing on average a 010401 (2006). single photon, by using the quantum coherent set of spatially sepa- 26. Braun, D. & Martin, J. Heisenberg-limited sensitivity with decoherence-enhanced measurements. Nature Communications 2, 223 (2011). rated qubits (a quantum metamaterial sensor array). The key feature 27. Savel’ev, S. et al. Two-qubit parametric amplifier: Large amplification of weak of this approach is the combination of the nonlocal photon inter- signals. Phys. Rev. A 85, 013811 (2012). action with the collective observable of the QMM array and its QND measurement. The realisation of our approach would allow to greatly improve the sensitivity of radiation detectors by suppressing the Acknowledgments effects of local noise as well as lowering the detection bar to the AMZ, RDW, ME and SS were supported by the John Templeton Foundation. EI minimum allowed by the uncertainty principle. acknowledges the financial support of the EU through the project iQIT.

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Author contributions Additional information A.M.Z., V.K.D. and D.R.G. formulated the problem. A.M.Z. and S.S. proposed the detection Competing financial interests: The authors declare no competing financial interests. strategies and models. A.M.Z., E.I. and M.E. suggested the possible implementations and How to cite this article: Zagoskin, A.M. et al. Spatially resolved single photon detection with evaluated their feasibility. A.M.Z., D.R.G. and J.A. performed the analytical calculations. a quantum sensor array. Sci. Rep. 3, 3464; DOI:10.1038/srep03464 (2013). S.S., M.E. and R.D.W. performed the numerical calculations. All authors contributed to the discussion of the physical content of the investigated effects and their possible applications, This work is licensed under a Creative Commons Attribution- and to the writing of the paper. NonCommercial-ShareAlike 3.0 Unported license. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0

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