Detecting Spin Transport in Quantum Magnets with Photons

TL;DR

This paper introduces a minimally invasive microwave photon-based method to detect spin conductance and noise in quantum magnets, enabling detailed analysis of spin transport properties.

Contribution

It proposes a novel technique using microwave resonators coupled to quantum magnets to measure spin conductance and noise through photon detection.

Findings

The method can extract both dc and finite frequency spin current noise.

It allows measurement of junction spin conductance by temperature variation.

The technique is minimally invasive and compatible with existing microwave technology.

Abstract

A minimally invasive technique is proposed for detecting the differential spin conductance and spin current noise across a junction between two quantum magnets using a high-quality microwave resonator coupled to a transmission line which is impedance matched to a photon detector downstream. Photons in the microwave resonator couple inductively to the spins in the spin subsystem, and the noise in the junction spin current imprints itself into the output photons propagating along the transmission line. The technique is capable of extracting both the dc and finite frequency noise via the output photon flux and of measuring the junction spin conductance by driving the electromagnetic environment into a different temperature regime.

Equations

$$\Phi (x)=\frac{-{\mu }_0{m}_z}{4\pi }{\int }_0^s{\int }_0^w[(x^{\prime }-x{)}^2+(y+\delta {)}^2{]}^{-3/2}d{x}^{\prime }dy,$$

$$e^{\pm i\Phi q_0/\hslash }{\varphi }_0{e}^{\mp i\Phi q_0/\hslash }={\varphi }_0\mp \Phi .$$

$$H_{\nu}=J\sum_{j=0}^{\infty}\Big{[}\tfrac{1}{2}S^{+}_{\nu,j}S^{-}_{\nu,j+1}\left(\delta_{j,i_{1}}e^{i\Phi q_{0}/\hbar}+\delta_{j,i_{2}}e^{-i\Phi q_{0}/\hbar}\right)\delta_{\nu,2}\\ +h.c.+\Delta S_{\nu,j}^{z}S_{\nu,j+1}^{z}\Big{]}\ ,$$

$$H_{TL}={\int }_0^{\infty }d{x}_t\left\{\frac{{\varphi }^2(x_t)}{2l}+\frac{[{\partial }_{x_t}q(x_t){]}^2}{2c}\right\},$$

$$H_r=\frac{{\varphi }_0^2}{2L}+\frac{q_0^2}{2C},$$

$$V(t)=-\frac{1}{2}{J}_c(S_{1,0}^{+}{S}_{2,0}^{-}{e}^{i\mu t/\hslash }+h.c.)$$

$$q(x_{t},t)=\int_{0}^{\infty}\frac{d\omega}{2\pi}\sqrt{\frac{\hbar}{2\omega z}}\ \Big{[}a_{o}(\omega)e^{i\omega(x_{t}/v-t)}\\ +a_{i}(\omega)e^{-i\omega(x_{t}/v+t)}+h.c.\Big{]}\ ,$$

$${\varphi }_0(t)$$

$${\dot{\varphi }}_0$$

$$a_o(\omega )=-\frac{{\Gamma }^{\ast }(\omega )}{\Gamma (\omega )}{a}_i(\omega )+\frac{\Phi }{\hslash L}\sqrt{\frac{2z\omega }{\hslash }}\frac{{\tilde{I}}_{i_1}^{+}(\omega )-{\tilde{I}}_{i_2}^{-}(\omega )}{\Gamma (\omega )}.$$

$${\tilde{I}}_i^{\pm }(t)=i(J/2)S_i^{+}{S}_{i+1}^{-}{e}^{\pm i\Phi q_0/\hslash }+h.c.$$

$$⟨a_i^{†}(\omega )a_i({\omega }^{\prime }){⟩}_0=2\pi n_B(\omega )\delta (\omega -{\omega }^{\prime }),$$

$$\begin{gathered} ⟨a_o^{†}(\omega )a_o(\omega )⟩=n_B(\omega ,T_t)+\frac{2E_{\Phi }\kappa \hslash \omega }{{\hslash }^4|\Gamma (\omega ){|}^2} \\ \times [(1+n_B(\omega ,T_t))(S_{i_1}(\omega ,\mu ,T)+S_{i_2}(\omega ,\mu ,T)) \\ -n_B(\omega ,T_t)(S_{i_1}(-\omega ,\mu ,T)+S_{i_2}(-\omega ,\mu ,T))], \end{gathered}$$

$$H_{\nu }=\frac{\hslash u}{4\pi K}{\int }_{-\infty }^{\infty }dx[{\partial }_x{\phi }_{\nu }(x){]}^2,$$

$$[{\phi }_{\nu }(x),{\phi }_{\nu }(x^{\prime })]$$

$$u$$

$$K$$

$$dS(\omega ,\mu ,T)=2{\xi }_c^2{\int }_{-\infty }^{\infty }dt\cos \left(\frac{\mu t}{\hslash }\right)D(t,T)e^{-i\omega t},$$

$$D(t,T)={\left\{\frac{\frac{\pi k_{B}T\eta }{u\hslash }}{\sin \left[\frac{\pi k_{B}T}{u\hslash }(iut+\eta )\right]}\right\}}^{2/K},$$

$$N(\Omega )=n_B(\Omega ,T_t)B+\frac{E_{\Phi }}{\hslash \Omega }\frac{\Sigma (\Omega ,\mu )}{{\hslash }^2},$$

$$\begin{gathered} \Sigma (\Omega ,\mu )\equiv [1+n_B(\Omega ,T_t)]dS(\Omega ,\mu ,T) \\ -n_B(\Omega ,T_t)dS(-\Omega ,\mu ,T). \end{gathered}$$

$$N(\Omega )\approx \frac{E_{\Phi }}{\hslash \Omega }\frac{dS(\Omega ,\mu ,T)}{{\hslash }^2}.$$

$$N\approx \frac{E_{\Phi }}{\hslash \Omega }\frac{dS(\Omega =0,\mu ,T)}{{\hslash }^2}.$$

$$F=\frac{dS(\Omega =0,\mu ,T)}{\hslash I},$$

Full text

I Introduction

Shot noise in mesoscopic conductors arises as a consequence of the quantized nature of charge transport Blanter and Büttiker (2000). Its utility far exceeds that of the equilibrium (Johnson-Nyquist) counterpart, as shot noise can be used to extract the charge Heiblum (2006) and the statistics of relevant charge carriers as well as to probe quantum many-body effects Sela et al. (2006) and entanglement Beenakker et al. (2003). In insulators, charge fluctuations are gapped out but localized electron spins can still generate fluctuations in pure spin current, or spin current noise. It is then anticipated that these pure spin currents can reveal nontrivial properties of the underlying spin system in analogy with the above-mentioned charge scenario. Indeed, recent theoretical inquiries on pure spin current noise in insulating magnets have purported its utility in revealing the quantum uncertainty associated with magnon eigenstates Kamra and Belzig (2016), the non-trivial spin scattering and heating processes generated at a detector interface Matsuo et al. (2018) and the effective spin and statistics of the tunneling spin quasiparticles Aftergood and Takei (2018).

Insulating materials present novel concerns. In metallic systems, spin current noise detection is possible via its observable effects on simultaneous charge current fluctuations Mishchenko (2003); Belzig and Zareyan (2004); Meair et al. (2011); Arakawa et al. (2015); similar methods are clearly not available for insulators. The prevailing spin current detection method in the latter has been the inverse spin Hall effect wherein spin current is detected electrically by coupling a metal with strong spin-orbit interactions to the active spin system Sinova et al. (2015). However, this spin Hall detection scheme, while reliable for spin current detection, may be unreliable for the measurement of spin current noise because spin Hall conversion processes can result in noise enhancement Dragomirova et al. (2008). Other techniques exist for detecting spin fluctuations including spin noise spectroscopy Sinitsyn and Pershin (2016), SQUID-based spectroscopy Granata and Vettoliere (2016), and quantum-impurity relaxometry van der Sar et al. (2015); Casola et al. (2018); Flebus and Tserkovnyak (2018). However, spin noise spectroscopy does not measure the current-current correlator, i.e., spin current noise, and neither SQUID-based approaches nor relaxometry are amenable to extracting current-current correlations in the spin sector either. Therefore, in connecting theory to experiment, a measurement technique capable of directly detecting spin current noise is of unique interest.

In this work, we show that a microwave resonator circuit can be used to directly measure nonequilibrium pure spin current noise generated in a quantum magnet. Here, we illustrate this possibility in the context of a one-dimensional quantum antiferromagnet chain. As shown in Fig. 1, we consider a situation in which one spin chain (chain 1) is driven out of equilibrium by spin injection at its open end with the downstream end weakly exchange-coupled to a second spin chain (chain 2) set near a microwave cavity. The microwave cavity couples inductively to chain 2, and measurements are transmitted electrically into a transmission line where the photon number flux encodes spin current fluctuations across the coupled spin chain subsystem. We show that by coupling spins to light it is possible to measure the junction spin current noise without resort to the inverse spin Hall effect. The proposed setup also opens doors to the studies of photon counting statistics and the possible generation of non-classical radiation produced by spin current fluctuations at a quantum magnet junction.

II Heuristic picture

We first qualitatively illustrate the mode of operation. Let us consider spin current tunneling between two coupled quantum antiferromagnet chains with uniaxial spin symmetry along the $z$ axis (see Fig. 1). Spin injection, facilitated by, e.g., the spin Hall effect at the upstream end of chain 1, establishes a nonequilibrium spin bias between the two chains, and the resulting spin current flowing across the chains can be absorbed and measured at the right metal reservoir via the inverse spin Hall effect.

A rectangular wire loop with inductance $L$ is placed a distance $\delta$ away from chain 2, and is oriented so that it lies in the $xy$ plane and its bottom edge stretches from $x=x_{1}$ to $x=x_{2}$. In this geometry, the magnetic flux through the wire loop sharply increases by one unit when a single $z$-polarized spin-1 quasiparticle enters the “influence region” defined by $x_{1}<x<x_{2}$ on chain 2. The magnitude of that unit depends on the distance $\delta$ and the width $w$ of the loop. Considering the spin-1 quasiparticle as a magnetic dipole ${\bm{m}}=-\gamma\hbar\hat{\bm{z}}$ located at position $x$ on chain 2 ($\gamma$ being the gyromagnetic ratio), the flux through the loop reads

[TABLE]

where $s=x_{2}-x_{1}$ is the influence region. Fig. 2 shows a sharp increase in the flux as the quasiparticle tunnels into the influence region for various $\delta$. We find that the flux is essentially independent of the quasiparticle position in the influence region and that the dipolar fields from spins located outside of the influence region are effectively irrelevant to the total flux through the loop.

Imagine now that the $x=x_{1}$ edge of the wire loop is located at site $i_{1}$ on chain 2 and the $x=x_{2}$ edge at site $i_{2}$. Then the term in the Hamiltonian modeling the tunneling of spin-1 quasiparticles from site $i_{1}$ to $i_{1}+1$ on chain 2 (allowed by the intrachain exchange interaction) involves a term of the form $S^{-}_{2,i_{1}}S^{+}_{2,i_{1}+1}$ ($S^{+}_{2,i_{1}}S^{-}_{2,i_{1}+1}$), where $S^{\pm}_{\nu,i}$ denotes the usual spin raising (lowering) operator on chain $\nu$ at site $i$. However, the fact that every tunneling process is accompanied by a flux change $\Phi$ in the loop requires that the tunneling operator is modified to $S^{+}_{2,i_{1}}S^{-}_{2,i_{1}+1}e^{i\Phi q_{0}/\hbar}+h.c.$, where $q_{0}$ is an operator obeying $[q_{0},\phi_{0}]=i\hbar$ and translates the influence flux $\phi_{0}$ through the loop by $\Phi$, i.e.,

[TABLE]

In this configuration, $q_{0}$ is the charge on the capacitor $C$ (see Fig. 1), and $\phi_{0}$ and $q_{0}$ form a conjugate variable pair. The spin tunneling operator endowed by the flux translation operator indicates that the spin tunneling process involves interactions with the electromagnetic environment formed by the loop-capacitor subsystem. In a similar fashion, the spin tunneling term at the $x=x_{2}$ edge is also modified to S^{+}_{2,i_{2}}S^{-}_{2,i_{2}+1}e^{-i\Phi q_{0}/\hbar}+h.c.\.

Energetic considerations show that the wire loop in principle affects spin transport in chain 2. That is, for every unit of flux $\Phi$ tunneling into the influence region, the energy of the inductive system increases by $E_{\Phi}=\Phi^{2}/2L$, where $L$ is the inductance of the resonator. As a result, two regimes emerge: what we call the noninvasive and invasive regimes. In the noninvasive regime, $E_{\Phi}$ is much smaller than the non-equilibrium bias $\mu$, the temperature $T$, and the photon frequency $\Omega$, i.e., $E_{\Phi}\ll\mu,k_{B}T,\hbar\Omega$ (note that we will define the photon frequency later). In this regime, we may focus exclusively on the effect of nonequilibrium spin transport on the electromagnetic environment and neglect the back-action of the environment on the spin subsystem. A fluctuating spin current at the junction leads to a fluctuating magnetic flux through the wire loop and thus to a fluctuating electromotive force inside the resonator by Faraday’s law of induction. As mentioned previously, this fluctuating electrical signal is ultimately detected in the transmission line via an output photon number flux containing a direct imprint of the spin current noise.

In the invasive regime, the environmental effect is not negligible and the spin transport in chain 2 should deviate from their unperturbed values. For $E_{\Phi}$ much greater than the nonequilibrium spin bias $\mu$ and temperature $T$, i.e., $\mu,k_{B}T\ll E_{\Phi}$, tunneling events become increasingly unfavorable energetically due to the resistive electromotive force emerging from Lenz’s law and leads to a suppression in the spin transmission through the chain. We refer to this phenomenon as inductive blockade, which may be thought of as the magnetic analog of the well-known Coulomb blockade studied extensively in quantum conductors Devoret et al. (1990); Ingold and Nazarov (1992). If the junction resistance is strong enough to suppress elastic scattering between the nodes, tunneling quasiparticles must have sufficient energy to excite environmental modes and proceed inelastically. However, the regime may be challenging to realize in practice due to the weakness of the spin-light interaction. In the remainder of the work, we focus on the noninvasive regime and present the technical calculations to establish the above heuristic results.

III Microscopic theory

We consider two identical semi-infinite $xxz$ quantum antiferromagnet chains coupled together at their finite ends, one additionally coupled inductively to a microwave resonator and the resonator itself placed in series with a transmission line on which measurements are performed. The spin chains are modeled by the usual $xxz$ Hamiltonian

[TABLE]

where ${\bm{S}}_{\nu,j}$ is the spin-1/2 operator on chain $\nu$ at site $j$, $J$ is the intra-chain exchange scale and $\delta_{i,j}$ is the kronecker delta. The exponential factors encode the coupling of spin chain 2 to the electromagnetic environment, as discussed previously. We assume throughout that the spin chains are in the so-called gapless phase, i.e., $|\Delta|<1$, where it can be suitably described using the Luttinger liquid formalism Giamarchi (2004).

We model the transmission line as an infinite array of parallel LC resonators with lineic capacitance $c$, lineic inductance $l$, and characteristic impedance $z=\sqrt{l/c}$. Its Hamiltonian reads

[TABLE]

where $x_{t}$ labels the position along the transmission line, $q(x_{t})$ and $\phi(x_{t})$ denote the local charge and flux, respectively, and $[q(x_{t}),\phi(x^{\prime}_{t})]=i\hbar\delta(x_{t}-x^{\prime}_{t})$. Located at the end of the transmission line, i.e., at $x_{t}=0$ (see Fig. 1), is the lumped series LC resonator with capacitance $C$ and inductance $L$, and governed by the Hamiltonian

[TABLE]

where $q_{0}\equiv q(x_{t}=0)$ and $\phi_{0}\equiv\phi(x_{t}=0)$. The total Hamiltonian for the full system then reads $H=\sum_{\nu}H_{\nu}+H_{r}+H_{TL}+V(t)$, where

[TABLE]

describes the tunneling of spin-1 quasiparticles across the spin chains allowed by the weak interchain exchange interaction $J_{c}$. The oscillatory factor $e^{i\mu t/\hbar}$ captures the fact that the spin chemical potential in chain 1 has been raised to $\mu>0$ via spin injection at its upstream end.

The standard input-output approach Clerk et al. (2010) proceeds by first expanding the local charge $q(x_{t},t)$ in Fourier series

[TABLE]

where $v=(lc)^{-1/2}$ and $a_{i,o}$ are the incoming and outgoing photon fields on the transmission line. Noting that the Heisenberg equations of motion evaluated at the lumped resonator $x_{t}=0$ give

[TABLE]

one can solve for the output photon field in terms of the known input photons,

[TABLE]

Here,

[TABLE]

is the operator for the bulk spin current in chain 2 flowing to the right at site $i$ (the exponential factor encoding the effect of the electromagnetic environment), $\Gamma(\omega)=\omega^{2}-\Omega^{2}+i\kappa\omega$, $\kappa=z/L$ is the rate at which photons decay into the transmission line and $\Omega=(LC)^{-1/2}$ is the resonance frequency.

The first term in Eq. (10) then corresponds to the reflection of incoming photons while the second term describes the emission or absorption of additional photons caused by the tunneling of spin-1 quasiparticles into and out of the influence region. The incoming photons are assumed to be equilibrated at resonator temperature $T_{t}$, which may be distinct from temperature $T$ of the spin chains, and obey

[TABLE]

where $n_{B}(\omega)=(e^{\hbar\omega/k_{B}T_{t}}-1)^{-1}$ is the Bose-Einstein distribution describing the thermal photons in the transmission line.

As discussed previously, noninvasive detection of the spin current noise is conducted in the limit where spin-photon coupling strength quantified by $\Phi$ remains sufficiently small so as to leave the spin chain subsystem approximately unperturbed while the coupling of the resonator to chain 2 is simultaneously kept strong enough for detection. Solving for the output photon flux using Eq. (10) is difficult, in principle, because the photon field itself enters the spin current operator $\tilde{I}^{\pm}_{i}(t)$ through $q_{0}(t)$ and this calls for self-consistency. However, since the correction to the output photon flux arising from the spin subsystem is anticipated to be suppressed by an overall multiplicative factor proportional to $\Phi^{2}$, c.f. second term in Eq. (10), the noninvasive assumption allows us to set the spin-photon coupling to zero, i.e., $\Phi=0$, when computing spin transport quantities, thus effectively lifting the self-consistency requirement.

With this in mind, let us now examine the output photon flux spectrum $\langle a^{\dagger}_{o}(\omega)a_{o}(\omega^{\prime})\rangle$ by inserting Eq. (10) into the expectation value while invoking the noninvasive approximation mentioned above. The output photon flux spectrum then reads

[TABLE]

where $S_{i}(\omega,\mu,T)=\int dte^{-i\omega t}\left<I_{i}(t)I_{i}(0)\right>$ denotes the spin current noise at site $i$ on chain 2; we remind the reader that the transmission line (detector) temperature $T_{t}$ is distinguished from the spin temperature $T$. Here, the local spin current operator $I_{i}(t)\equiv\tilde{I}^{\pm}_{i}(t,\Phi=0)$ is now defined in the absence of the electromagnetic environment and the expectation values are taken by setting $\Phi=0$ in $H_{2}$ [see Eq. (3)]. In obtaining Eq. (13), we also ignored cross-correlations between spin current fluctuations at $x_{1}$ and $x_{2}$. This should be a good approximation because nonlocal spin correlations are expected to decay exponentially with distance at finite temperature 111Note that for $T=4~{}{\rm K}$ Eq. (19) allows for an estimate of the length scale of non-local correlations, and we find that correlations are suppressed on the order of $\left|x_{2}-x_{1}\right|>10~{}{\rm nm}$..

In the gapless phase $|\Delta|<1$, each semi-infinite spin chain can be described as a chiral Luttinger liquid Aftergood and Takei (2018); Giamarchi (2004)

[TABLE]

where the chiral boson field $\varphi_{\nu}(x)$ obeys

[TABLE]

the boson speed and the Luttinger parameter, respectively Johnson et al. (1973), are given by

[TABLE]

and $a$ is the lattice constant. In this case, spin current inside chain 2 is essentially carried by (ballistic) noninteracting bosonic modes so the spin current fluctuations produced at the junction remain unmodified as they propagate downstream to sites $i_{1}$ and $i_{2}$ Aftergood and Takei (2018); Crépieux et al. (2003). Therefore, the local spin current noise at any point in the bulk of chain 2, i.e., $S_{i}(\omega,\mu)$, is given by the background Johnson-Nyquist noise $S_{0}(T)$ plus the noise generated at the tunnel junction between the two spin chains $dS(\omega,\mu,T)$. The total spin current noise at any site $i$ is then given by $S_{i}(\omega,\mu,T)=S_{0}(T)+dS(\omega,\mu,T)$, where Aftergood and Takei (2018)

[TABLE]

with

[TABLE]

$\xi_{c}=J_{c}a/2\pi\eta$, $\eta\sim k_{F}^{-1}$ is the short distance cutoff of the theory, and $k_{F}$ is the Fermi wavevector Sirker (2012).

IV Noise detection

Equation (13) shows that a single mode electromagnetic environment can imprint transport quantities generated at a junction coupling two quantum magnets on the number of total output photons propagating along an attached transmission line. Reference Inomata et al., 2016 provides an example of a single microwave photon detector based on a superconducting flux qubit capable of detecting individual photons propagating through a transmission line with a refresh rate of $\sim 400$ ns, a narrow bandwidth, a well-characterized efficiency of $\sim 0.66\pm 0.06$, and a low dark count rate of $\sim 0.01$. If such a detector with bandwidth $B$ is attached at the end of the transmission line in Fig. 1 and is designed to count propagating microwave photons with a central frequency $\Omega$, Eq. (13) gives the expected rate of photons arriving at the detector as

[TABLE]

where

[TABLE]

Here, we have assumed a high-quality resonator obeying $\kappa\lesssim B\ll\Omega$. The first term in $N(\Omega)$ is the number of background thermal photons while the second term represents photon emission and absorption during interchain tunneling. In the limit of low transmission line/detector temperatures $k_{B}T_{t}\ll\hbar\Omega$, we have $n_{B}(\Omega,T_{t})\ll 1$ and thus obtain

[TABLE]

This is a central result of this work, i.e., the output photon flux gives a direct measurement of the ac spin current noise generated at the quantum magnet junction.

If the spin temperature is held higher than the resonance frequency, i.e., $\hbar\Omega\ll k_{B}T$, the output photon flux directly measures the dc component of the spin current noise, i.e.,

[TABLE]

Since the spin current $I$ across the junction can be measured directly using inverse spin Hall effect at the right metallic reservoir (see Fig. 1), the proposed system allows one to extract a quantity directly proportional to the dc spin Fano factor, defined as the noise normalized by the current

[TABLE]

recently studied in Refs. Kamra and Belzig, 2016; Matsuo et al., 2018; Aftergood and Takei, 2018.

Figure 3 shows a plot of photon flux $N(\Omega)$, Eq. (20), as a function of the detector temperature $T_{t}$. It exhibits the same qualitative behavior for essentially all spin biases $\mu$, thus allowing one to quantify the noise using this proposed extraction method for various values of $\mu$. The plot in the figure is generated for a spin tunnel junction with coupling strength $\xi_{c}\sim 0.01J$ and spin bias $\mu=0.03J$, intrachain exchange scale of $J/k_{B}\sim 10^{3}~{}\mathrm{K}$ Hirobe et al. (2017), spin temperature of $T=4$ K , and an inductor loop of inductance $L\sim 1$ nH with dimensions $x_{2}-x_{1}=w=10~{}\mathrm{\mu m}$ placed $\delta\sim 1$ nm from spin chain 2. Under these conditions, we estimate $E_{\Phi}\sim 10^{-11}~{}\mathrm{K}\sim 2~{}\mathrm{Hz}$. For these parameters and at resonance frequency $\Omega/2\pi=1$ GHz 222Here, we assumed a cut off length in the Luttinger theory of $\eta/a\approx 10$, we expect the chain-to-cavity interaction to produce approximately 26 photons per second in the low temperature regime (shaded in blue), where the photon flux converges to the quantity in Eq. (22). We believe the detection of this number of output photons is within the reported capabilities of a single microwave photon detector impedance matched to a transmission line Inomata et al. (2016).

V Conclusion

We have shown that by placing a high-quality microwave resonator circuit in close proximity to a pair of exchange-coupled spin chains it is possible to directly measure spin transport quantities at a junction between the two spin systems. When the electromagnetic environment interacts weakly with the spin system, the setup allows measurement of both finite frequency and dc spin current fluctuations by examining the total number of output photons produced by interactions with the environment. This work opens doors to the possibilities of exploring the photodetection statistics of radiation produced at a junction between two quantum magnets and the generation of antibunched photons by exploiting the similarity between the current spin subsystem and a tunnel junction between two quantum conductors Beenakker and Schomerus (2001); *beenakkerPRL04. Our theory can also be extended to describe spin transport between tunnel-coupled gapped spin systems (e.g., quantum Ising chains) that may mimic the dynamical Coulomb blockade physics of superconducting Josephson junctions coupled to electromagnetic environments Ingold and Nazarov (1992). A final intriguing possibility would be to recast the detection methodology proposed here in terms of a generalized full counting statistics formalism as expressed in, e.g., Ref. Nazarov and Kindermann, 2003.

Acknowledgments. The authors thank J. Yuan, and M. Dartiailh for valuable discussions. This research was supported by Research Foundation CUNY Project #90922-07 10.