Stabilisation of an optical transition energy via nuclear Zeno dynamics in quantum dot-cavity systems

TL;DR

This paper demonstrates how quantum Zeno dynamics can stabilize the optical transition energy in quantum dot-cavity systems by using photon scattering as a nuclear spin measurement, affecting photon correlations.

Contribution

It introduces a theoretical framework showing how photon scattering induces nuclear Zeno effects, stabilizing the optical resonance in quantum dot-cavity systems.

Findings

Photon scattering acts as a nuclear spin measurement.

Quantum Zeno effect inhibits nuclear spin dynamics.

Photon autocorrelation changes with laser intensity.

Abstract

We investigate the effect of nuclear spins on the phase shift and polarisation rotation of photons scattered off a quantum dot-cavity system. We show that as the phase shift depends strongly on the resonance energy of an electronic transition in the quantum dot, it can provide a sensitive probe of the quantum state of nuclear spins that broaden this transition energy. By including the electron-nuclear spin coupling at a Hamiltonian level within an extended input-output formalism, we show how a photon scattering event acts as a nuclear spin measurement, which when rapidly applied leads to an inhibition of the nuclear spin dynamics via the quantum Zeno effect, and a corresponding stabilisation of the optical resonance. We show how such an effect manifests in the intensity autocorrelation $g^{(2)}(\tau)$ of scattered photons, whose long-time bunching behaviour changes from quadratic decay for low photon scattering rates (weak laser intensities), to ever slower exponential decay for increasing laser intensities as optical measurements impede the nuclear spin evolution.

Equations

$$H_0$$

$$H_I$$

$$+{\int }_0^{\infty }d\omega \sqrt{\kappa (\omega )}\left(b(\omega )a^{†}+b^{†}(\omega )a\right)+H_O,$$

$$\hat{\Delta }=\sum _j{A}_j{I}_j^z+\frac{1}{2\omega }\sum _{m\ne n}{A}_m{A}_n{I}_m^{+}{I}_n^{-},$$

$${\dot{\sigma }}^{-}(t)$$

$$i\dot{a}(t)$$

$$2\pi i\sqrt{\kappa }a(t)$$

$$\left(\omega -{\omega }_c+i\pi \kappa \right)a(\omega )=g{\sigma }^{-}(\omega )+\sqrt{\kappa }{b}_0(\omega ),$$

$$\int d\omega \left([\omega -{\hat{\omega }}_0(t)]{\sigma }^{-}(\omega )-ga(\omega )\right){\mathrm{e}}^{-i\omega t}=0,$$

$$\int d\omega {\mathrm{e}}^{-i\omega t}{\hat{f}}_{+}(\omega ,t)a(\omega )=\int d\omega {\mathrm{e}}^{-i\omega t}\sqrt{\kappa }[\omega -{\hat{\omega }}_0(t)]b_0(\omega ),$$

$$\int d\omega \left(b_{\mathrm{out}}(\omega ){\hat{f}}_{+}(\omega ,t)-b_{\mathrm{in}}(\omega ){\hat{f}}_{-}(\omega ,t)\right){\mathrm{e}}^{-i\omega t}=0.$$

$$\begin{array}{rl}& {\int }_{t_c-t_{\Delta }}^{t_c+t_{\Delta }}dt{\mathrm{e}}^{i{\omega }^{\prime }t}\int d\omega b_{\mathrm{out}}(\omega ){\hat{f}}_{+}(\omega ,t){\mathrm{e}}^{-i\omega t}\\ =& {\int }_{t_c-t_{\Delta }}^{t_c+t_{\Delta }}dt{\mathrm{e}}^{i{\omega }^{\prime }t}\int d\omega b_{\mathrm{in}}(\omega ){\hat{f}}_{-}(\omega ,t){\mathrm{e}}^{-i\omega t}.\end{array}$$

$$\begin{array}{rl}& \int d\omega b_{\mathrm{out}}(\omega ,t_c){\hat{f}}_{+}(\omega ,t_c)t_{\Delta }\mathrm{sinc}[t_{\Delta }(\omega -{\omega }^{\prime })]\\ =& \int d\omega b_{\mathrm{in}}(\omega ,t_c){\hat{f}}_{-}(\omega ,t_c)t_{\Delta }\mathrm{sinc}[t_{\Delta }(\omega -{\omega }^{\prime })],\end{array}$$

$${\tilde{b}}_{\mathrm{out}}(\omega )=\hat{r}(\omega ,t)\otimes {\tilde{b}}_{\mathrm{in}}(\omega ),$$

$$\begin{array}{r}\hat{r}(\omega ,t)=1+\frac{2i\pi \kappa ({\omega }_0+\hat{\Delta }(t)-\omega )}{({\omega }_c-\omega -i\pi \kappa )({\omega }_0+\hat{\Delta }(t)-\omega )-g^2},\end{array}$$

$$\begin{array}{rl}& {\tilde{b}}_{\mathrm{out}}(\omega )\equiv \int d{\omega }^{\prime }{b}_{\mathrm{out}}({\omega }^{\prime },t)t_{\Delta }\mathrm{sinc}[(\omega -{\omega }^{\prime })t_{\Delta }],\end{array}$$

$$\frac{d}{d\omega }\hat{\theta }(\omega ,t)=2\pi \frac{\kappa }{g^2},$$

$$\begin{array}{rl}& G_V^{(2)}(t,t+\tau )=\\ & {\mathrm{Tr}}_{\mathrm{tot}}\left[E_V^{(-)}(t)E_V^{(-)}(t+\tau )E_V^{(+)}(t+\tau )E_V^{(+)}(t)\chi \right],\end{array}$$

$$E_V^{(+)}(t)={\int }_0^{\infty }d\omega {\tilde{b}}_{\mathrm{out}}^V(\omega ,t),$$

$${\tilde{b}}_{\mathrm{out}}^V(\omega ,t)={\hat{r}}_{\mathrm{cr}}(\omega ,t)\otimes {\tilde{b}}_{\mathrm{in}}^H(\omega )+{\hat{r}}_{\mathrm{co}}(\omega ,t)\otimes {\tilde{b}}_{\mathrm{in}}^V(\omega ),$$

$$G_V^{(2)}(t,t+\tau )=|\beta {|}^4$$

$$\mathrm{Tr}\left[{\hat{r}}_{\mathrm{cr}}^{†}(\omega ,t){\hat{r}}_{\mathrm{cr}}^{†}(\omega ,t+\tau ){\hat{r}}_{\mathrm{cr}}(\omega ,t+\tau ){\hat{r}}_{\mathrm{cr}}(\omega ,t){\rho }_N\right],$$

$$G_V^{(2)}(t,t+\tau )=\sum _{n=0}^{\infty }\sum _{\{c_i\}}$$

$${\int }_t^{t+\tau }d{t}_n{\int }_t^{t_n}d{t}_{n\text{-}1}...{\int }_t^{t_2}d{t}_1{\mathcal{G}}_{V,\{c_i\},V}^{(n)}(t,t_1,...,t_n,t+\tau ),$$

$$\begin{array}{rl}& {\mathcal{G}}_{V,\{c_i\},V}^{(n)}(t,t_1,...,t_n,t+\tau )=\\ & p_t(t,t+\tau ;n)p_t(t_1,...,t_n)p_c(V,\{c_i\},V),\end{array}$$

$$g^{(2)}(\tau )=\frac{G_V^{(2)}(t,t+\tau )}{G_V^{(1)}(t)G_V^{(1)}(t+\tau )}=\sum _{n=0}^{\infty }p(n,\tau )$$

$$\frac{n!}{{\tau }^n}{\int }_t^{t+\tau }d{t}_n{\int }_t^{t_n}d{t}_{n\text{-}1}...{\int }_t^{t_2}d{t}_1\sum _{\{c_i\}}\frac{p_c(V,\{c_i\},V)}{p_V(t)p_V(t+\tau )},$$

$$\displaystyle\tilde{b}_{\mathrm{out}}^{H}(\omega)^{\dagger}\ket{0}\otimes\ket{\delta}=\Big{[}r_{\mathrm{cr}}^{\delta}\tilde{b}_{\mathrm{in}}^{V}(\omega)^{\dagger}+r_{\mathrm{co}}^{\delta}\tilde{b}_{\mathrm{in}}^{H}(\omega)^{\dagger}\Big{]}\ket{0}\otimes\ket{\delta}.$$

$$M_c=\sum _{\delta }{r}_c^{\delta }|\delta ⟩⟨\delta |.$$

$$p(V,\!\{c_{i}\}\!,V)\!=\!\mathrm{Tr}\Big{[}\Phi_{V}\mathcal{U}_{n+1}\Big{(}\!\prod_{i=1}^{n}\Phi_{c_{i}}\mathcal{U}_{i}\Big{)}\Phi_{V}\rho_{N}(t)\Big{]},\!$$

$$\begin{array}{rl}{\Phi }_c\rho & =M_c\rho M_c^{†},\\ {\mathcal{U}}_i\rho & ={\mathrm{e}}^{-i{H}_N{\Delta }_i}\rho {\mathrm{e}}^{i{H}_N{\Delta }_i},\end{array}$$

$$g^{(2)}(\tau )\approx \frac{p_c(V,\{\},V)}{p_V(t)p_V(t+\tau )}=\frac{\mathrm{Tr}\left(O_V{\mathcal{U}}_{0\to \tau }\varrho \right)}{\mathrm{Tr}{\left(O_V{\rho }_N\right)}^2},$$

$$\mathrm{Tr}\left(O_V{\mathcal{U}}_{0\to \tau }\varrho \right)\approx \mathrm{Tr}\left(O_V^2{\rho }_N\right)-\frac{1}{2{\tau }_z^2}{\tau }^2,$$

$${\tau }_z=1/\sqrt{\mathrm{Tr}\left(O_V\left[H_N,[H_N,\varrho ]\right]\right)}.$$

Full text

I Introduction

The generation of useful entanglement between photons is the central challenge in optical quantum computing schemes. Self-assembled quantum dots (QDs) have the potential to meet this challenge, either by emitting strings of entangled photons Lindner and Rudolph (2009); Economou et al. (2010); Schwartz et al. (2016), or by mediating an effective interaction between photons via a giant phase shift Hu et al. (2008a, b); Pineiro-Orioli et al. (2013); Lodahl et al. (2015); Lodahl (2018). Current experimental efforts to utilise such schemes, however, are often hindered by noise arising due to the coupling of an electron spin to the nuclear spins in the host material McCutcheon et al. (2014); Kuhlmann et al. (2015); Stockill et al. (2016); Wuest et al. (2016); Ethier-Majcher et al. (2017). Nevertheless, the dephasing caused by these nuclear spins is qualitatively different from that caused by coupling to photon or phonon baths, as the nuclear spins evolve slowly and unitarily on the timescale set by the electron spin dynamics, which gives rise to a variety of non-Markovian effects Greilich et al. (2007); Barnes and Economou (2011); Madsen et al. (2011); Urbaszek et al. (2013); Economou and Barnes (2014); Munsch et al. (2014); Prechtel et al. (2016); Ethier-Majcher et al. (2017); Nutz et al. (2019). While this unique nature of the nuclear spin environment might make it possible to experimentally suppress nuclear spin noise and possibly even control them in a useful way, it also presents a formidable theoretical challenge to find reliable and insightful models of nuclear spin behaviour.

We consider the effect of nuclear spins in giant phase shift experiments such as those described in Ref. Androvitsaneas et al. (2016) (see Fig. 1a), in which narrowband laser photons of linear polarisation described by $\ket{H}\propto\ket{L}+\ket{R}$ scatter off a cavity containing a charged QD in a large ($\gtrsim 100$ mT) magnetic field in the Faraday configuration. Since an electronic transition couples only to one of the two circular polarisations $\ket{L}$ and $\ket{R}$, the photon polarisation state upon scattering is given by $\mathrm{e}^{i\theta_{L}}\ket{L}+\mathrm{e}^{i\theta_{R}}\ket{R}$, with the phase shift difference $\theta=\theta_{L}-\theta_{R}$ taking values of up to $180^{\circ}$ Hu et al. (2008a, b); Auffèves-Garnier et al. (2007). Hence, a linearly polarised photon $\ket{H}$ can be reflected with the orthogonal linear polarisation $\ket{V}\propto\ket{L}-\ket{R}$, as shown in Fig. 1b. The phase shift is highly sensitive to the resonance energy of the electronic transition, which in turn depends on the nuclear spin environment via the Overhauser shift Abragam (1961). As the nuclear spin system evolves, the phase shift $\theta$ drifts over time, such that high values are observed only during short intervals ($\theta>120^{\circ}$ in $100\mu\mathrm{s}$ timebins Androvitsaneas et al. (2019)) but the time-averaged phase shift is low ($\braket{\theta}\approx 6^{\circ}$ in Androvitsaneas et al. (2019)). Photon detection events in the cross polarised (orthogonal to input laser) channel are therefore bunched on a $\mu s$ timescale, such that an intensity autocorrelation function has $g^{(2)}(\tau)<1$ for $\tau<\mathrm{ns}$ due to the single photon nature of the scattered field, but $g^{(2)}(\tau)>1$ for $\tau\sim\mu\mathrm{s}$ as the nuclear spin coupling effectively leads to blinking.

In this work we develop a quantum optical treatment that relates the intensity correlation function in the cross polarised channel $g^{(2)}(\tau)$ to a two-time correlation function of the nuclear spin system. We show that $g^{(2)}(\tau)$ decreases quadratically for low laser intensities, as depicted by the blue curve in Fig. 1c. Observation of this quadratic short-time behaviour would demonstrate the coherent nature of nuclear spin noise in QDs, which could help distinguish it from other possible sources of resonance fluctuations in these systems. However, the dependence of the photon phase shift on the nuclear spin state is only one aspect of a two-way interaction, as a photon scattering event has the effect of a quantum measurement on the nuclear spin state. Incorporating this into our formalism, we find that frequent photon scattering events, corresponding to higher driving intensities, impede the nuclear spin evolution and associated drifting of the resonance energy, which leads to a broadened intensity autocorrelation function that decays linearly with $\tau$. This can be understood as a quantum Zeno effect Misra and Sudarshan (1977); Itano et al. (1990); Block and Berman (1991); Facchi et al. (2000); Pascazio (2014); Zhang and Fan (2015); Christensen et al. (2018), which is here readily observable in an optical intensity correlation function. Experimental observation of this characteristic change in the intensity autocorrelation function would demonstrate this novel quantum Zeno effect, and open up a measurement-based route to control nuclear spins in QDs.

II Input–output formalism with electron–nuclear spin coupling

Our aim is to calculate the cross-polarised intensity autocorrelation $g^{(2)}(\tau)$ for photons scattered off the QD-cavity system which incorporates the nuclear spin environment, and which we achieve using an extended input-output formalism Gardiner and Collett (1985); Walls and Milburn (2008); Auffèves-Garnier et al. (2007). We consider a continuum of optical modes described by annihilation operators $\smash{{b}(\omega)}$ propagating towards and away from an optical cavity with frequency $\omega_{c}$ and associated cavity mode operator ${a}$. The cavity mode, in turn, couples to a two-level system (TLS) with ground and excited states $\ket{\uparrow}$ and $\ket{\Uparrow}$, respectively, which itself is coupled to a bath of nuclear spins. The total Hamiltonian describing all degrees of freedom is written $H=H_{0}+H_{I}$, with (setting $\hbar=1$)

[TABLE]

where $\sigma^{z}=\left|\Uparrow\right\rangle\hskip-2.84526pt\left\langle\Uparrow\right|-\left|\uparrow\right\rangle\hskip-2.84526pt\left\langle\uparrow\right|$, $\sigma^{-}=\left|\uparrow\right\rangle\hskip-2.84526pt\left\langle\Uparrow\right|$, $\sigma^{+}=\left|\Uparrow\right\rangle\hskip-2.84526pt\left\langle\uparrow\right|$, $\omega_{0}$ is the transition energy of the TLS, and $g$ is the TLS–cavity coupling strength. The nuclear Zeeman term is $H_{Z}=\sum_{j}\omega_{j}I_{j}^{z}$, with Pauli $z$ operator $I^{z}_{j}$ acting on nuclear spin $j$ and nuclear Zeeman splitting $\omega_{j}$ due to an external magnetic field along $\hat{z}$. The electron–nuclear coupling term is $H_{O}=\frac{1}{2}\sigma^{z}\hat{\Delta}$, with Overhauser shift operator

[TABLE]

which results from a Schrieffer-Wolff transformation on the contact hyperfine Hamiltonian given in Eq. 41  Klauser et al. (2008). Note that while the contact hyperfine interaction involves two electron spin states, $\ket{\uparrow}$ and $\ket{\downarrow}$, we focus here on one of these ground states only, arbitrarily labelled $\ket{\uparrow}$. Neglecting the other spin state $\ket{\downarrow}$ is justified in a large ($\gtrsim 100$ mT) magnetic field, where energy conservation prevents flip-flops between these electron spin states which are separated by the electron Zeeman energy $\omega$.

We approximate the cavity–port mode coupling strength as a constant $\sqrt{\kappa(\omega)}\approx\sqrt{\kappa(\omega_{c})}\equiv\sqrt{\kappa}$ over the relevant optical frequencies, and in doing so we find the Heisenberg equations of motion

[TABLE]

where we have defined the incoming and outgoing field operators as ${b}_{\mathrm{in}}(t)=\int d\omega\ b_{0}(\omega)\exp(-i\omega t)$ with $b_{0}(\omega)\equiv{b}(\omega,t_{0})$, and ${b}_{\mathrm{out}}(t)=\int d\omega\ b_{1}(\omega)\exp(-i\omega t)$ with $b_{1}(\omega)\equiv{b}(\omega,t_{1})$ Walls and Milburn (2008), and extended frequency integrals such that $\int_{0}^{\infty}d\omega\rightarrow\int_{-\infty}^{\infty}d\omega\equiv\int d\omega$. Taking the Fourier transform of Eq. (4b) we find

[TABLE]

where $\sigma^{-}(t)=\int d\omega\ \sigma^{-}(\omega)\mathrm{e}^{-i\omega t}$ and similarly for ${a}(t)$.

The standard procedure in input–output theory is to use the Fourier transform of Eq. (4a) to replace $\sigma^{-}(\omega)$ in Eq. (5), which is then used in the Fourier transform of Eq. (4c), $b_{\mathrm{in}}(\omega)-b_{\mathrm{out}}(\omega)=2\pi i\sqrt{\kappa}a(\omega)$ to find a relationship between frequency components of the incoming and outgoing fields $b_{\mathrm{in}}(\omega)$ and $b_{\mathrm{out}}(\omega)$. We use a similar procedure here, but note that the occurrence of the time-dependent Overhauser shift operator $\hat{\Delta}(t)$ in Eq. (4a) means there is no simple relationship between the Fourier components $\sigma(\omega)$ and $a(\omega)$. Instead we arrive at the integral equation

[TABLE]

where we have defined $\hat{\omega}_{0}(t)\equiv\omega_{0}+\hat{\Delta}(t)$ and made the approximation $\sigma_{z}\approx-1$, valid for weak driving. Combining Eqs. (5) and (6) then gives

[TABLE]

where $\hat{f}_{\pm}=\left(\omega-\omega_{c}\pm i\pi\kappa\right)\left(\omega-\hat{\omega}_{0}(t)\right)-g^{2}$. Using this in the Fourier transformation of Eq. (4c) leads to

[TABLE]

When the Overhauser term is neglected, $\hat{\Delta}(t)=0$, Eq. (8) simplifies to $\smash{{b}_{\mathrm{out}}(\omega)=r(\omega)b_{\mathrm{in}}(\omega)}$ with the scalar $r(\omega)=f_{-}(\omega)/f_{+}(\omega)$, which is the well-known cavity-QED reflection coefficient Auffèves-Garnier et al. (2007). An analogous result relating incoming and outgoing fields in the presence of nuclear spin coupling can be obtained by assuming a slowly varying Overhauser shift. To see this, we consider attempting to isolate the integrand in Eq. (8) by performing the finite-domain definite integral

[TABLE]

Choosing the integration range such that $t_{\Delta}\ll t_{\mathrm{fluc}}$, where $t_{\mathrm{fluc}}$ is the characteristic timescale of the Overhauser field fluctuations, we can approximate $\hat{\Delta}(t)\approx\hat{\Delta}(t_{c})$ in the integrands and arrive at

[TABLE]

where $b_{\mathrm{out}}(\omega,t_{c})\equiv b_{\mathrm{out}}(\omega)\mathrm{e}^{-i\omega t_{c}}$ and similarly for $b_{\mathrm{in}}(\omega,t_{c})$. If the Overhauser shift fluctuates slowly then we can choose the spectral width of the sinc function in Eq. (10), $t_{\Delta}^{-1}$, to be much narrower than the width of the function $\smash{\hat{f}_{\pm}(\omega,t_{c})}$. We then find

[TABLE]

with $\hat{r}(\omega,t)=f_{+}(\omega,t)^{-1}f_{-}(\omega,t)$ and given by

[TABLE]

while the incoming and outgoing field operators are now defined as

[TABLE]

and similarly for $\tilde{b}_{\mathrm{in}}(\omega)$. Noticing the convolution form of this expression, we see that this operator can be thought of as a broadened version of its exact frequency counterpart $b_{\mathrm{out}}(\omega)$ owing to the finite integration time $t_{\Delta}$.

The relationship between incoming and outgoing fields given in Eq. (11) is our first result, and generalises input–output theory to systems with slowly varying resonance energies. It is valid if the integration time $t_{\Delta}$ in Eq. (9) satisfies $t_{\mathrm{fluc}}\gg t_{\Delta}\gg 1/w_{f}$, where $t_{\mathrm{fluc}}$ is the fluctuation time of the Overhauser shift and $w_{f}$ is the spectral width of the phase shift feature, which is obtained by considering the frequency dependence of $\hat{f}_{\pm}(\omega,t)=\hat{f}(\omega,t)\exp(\pm i\hat{\theta}(\omega,t))$. We find that the phase factor varies most rapidly at $\omega=\omega_{c}=\hat{\omega}_{0}$, at which point

[TABLE]

and $\frac{d}{d\omega}\hat{f}(\omega,t)=0$. As such, the fractional variation $\frac{d}{d\omega}\hat{f}(\omega,t)/\hat{f}(\omega,t)$ does not exceed a bound on the order of $\kappa/g^{2}$ when considering laser–QD detunings no greater than $\omega-\hat{\omega}_{0}(t)\approx g^{2}/\kappa$, and laser–cavity detunings limited to $\omega-\omega_{c}\approx\kappa$. The functions $\smash{\hat{f}_{\pm}(\omega,t)}$ therefore vary on a frequency scale given by the linewidth $g^{2}/\kappa$ of the TLS transition. This linewidth is typically on the order of few GHz for QD experiments, while the Overhauser shift fluctuation time can be estimated to be hundreds of milliseconds based on Androvitsaneas et al. (2019), such that $t_{\mathrm{fluc}}\gg t_{\Delta}\gg 1/w_{f}$ can be satisfied and Eq. (11) is applicable to QD experiments. We interpret $t_{\Delta}$ as a parameter that adjusts the tradeoff between frequency and time resolution of our theory. Eq. (11) relates Fourier components $\tilde{b}_{\mathrm{in/out}}(\omega)$ that must be understood as averages of the exact Fourier components of the incoming and outgoing fields over a bandwidth interval $1/t_{\Delta}\gg 1/t_{\mathrm{fluc}}$. For an experiment with a QD linewidth of 1 GHz and a fluctuation time of 1 $\mu$s our theory describes effects with a resolution of up to $\sim 1$ MHz in frequency and $\sim 1$ ns in time.

III Optically measured nuclear two-time correlation function

Having established how frequency components in the incoming and outgoing fields are affected by the nuclear spin bath, we now use this result to show how a measured optical intensity autocorrelation depends on a correlation function of the nuclear spins. We consider the optical intensity autocorrelation function of the cross-polarised reflected light. Assuming a horizontally polarised input field, the correlation in the vertical polarised orientation is proportional to the second-order correlation function

[TABLE]

where $E_{V}^{(\pm)}(t)$ are the positive and negative frequency components of the vertically polarised electric field at time $t$, and the trace is performed over the total port mode–cavity–electron spin–nuclear spin system, with total initial state $\chi$, and where the Heisenberg electric field operators evolve unitarily in this complete Hilbert space.

To proceed we express these field operators as

[TABLE]

where we have neglected numerical factors and retardation effects. Following Eq. (11), a cavity containing a QD with electron spin projection $\ket{\uparrow}$ reflects a right circularly polarised photon according to $\tilde{b}^{R}_{\mathrm{out}}(\omega,t)=\hat{r}(\omega,t)\otimes\tilde{b}_{\mathrm{in}}^{R}$, while a left circularly polarised photon acquires a phase shift $r_{0}(\omega)=\hat{r}(\omega,t)|_{g=0}$ corresponding to an empty cavity. Hence we can write

[TABLE]

where the operators $\hat{r}_{\mathrm{co/cr}}(\omega,t)=\frac{1}{2}\left(\hat{r}(\omega,t)\pm r_{0}(\omega)\right)$ give the reflectivities into the co- and cross-polarised channels, which depend on the nuclear spin state through the dependence of $\hat{r}(\omega,t)$ on the Overhauser operator $\smash{\hat{\Delta}(t)}$. We assume an initial state $\chi=\ket{H(\omega)}\bra{H(\omega)}\otimes\rho_{a}\otimes\ket{\uparrow}\bra{\uparrow}\otimes\rho_{N}$, where $\ket{H(\omega)}$ satisfying $\tilde{b}_{\mathrm{in}}^{H}(\omega^{\prime})\ket{H(\omega)}=\beta\delta(\omega-\omega^{\prime})\ket{H(\omega)}$ is a horizontally polarised coherent state of amplitude $\beta$, $\rho_{a}$ and $\rho_{N}$ are states of cavity mode and nuclear spin system, respectively, and the electron is assumed to remain in state $\ket{\uparrow}$ during the measurement. Substituting this state into Eq. (15) gives

[TABLE]

where now and in all that follows the trace is taken only over the nuclear degrees of freedom, showing that we have related an optically measured quantity to a nuclear two-time correlation function. This correlation function gives the joint probability to measure two photons scattered into the cross-polarisation channel at times $t$ and at $t+\tau$. It is a non-exclusive probability, as it does not suppose anything regarding any intermediate scattering events, into the cross-polarisation channel or otherwise Plenio and Knight (1998). Its non-exclusive nature is evidenced by the globally unitary evolution of the full Heisenberg picture operator $\smash{E_{V}^{(-)}(t)}$, which depends on the systems involved, including the photonic degrees of freedom. A non-exclusive correlation function is the correct form to make a connection with experiments, as typically one does not have access to a full scattering history, and in practice we take a statistical average over any intermediate scattering events.

However, we are interested here in how individual scattering events affect the nuclear spin environment, which in turn affects later scattering events. We therefore seek a relationship between the measured non-exclusive correlation function in Eq. (18), and an exclusive correlation function, which gives a conditional probability corresponding to a fixed number of scattering events at fixed times Plenio and Knight (1998). Such a relationship can be expressed as

[TABLE]

where $\mathcal{G}^{(n)}_{V,\{c_{i}\},V}(t,t_{1},...,t_{n},t+\tau)$ is the exclusive probability density that exactly $n+2$ photon scattering events take place in the interval $[t,t+\tau]$, with the first and last photons scattered into vertical polarisation at times $t$ and $t+\tau$, and $n$ additional photons scattered at intermediate times $t_{1},\dots,t_{n}$ into polarisations labelled $\{c_{i}\}=c_{1},\dots,c_{n}$ with $c_{i}$ being either the co- ($H$) or cross-polarised ($V$) channel. We now decompose the exclusive probability $\mathcal{G}$ into probabilities describing the scattering times, and the scattering polarisations. We write

[TABLE]

where $p_{t}(t,t+\tau;n)$ is the non-exclusive probability of photon scattering events at $t$ and $t+\tau$ with $n$ intermediate scattering events at unspecified times, $p_{t}(t_{1},...,t_{n})$ the probability density of these intermediate events occurring at $t_{1},...,t_{n}$, and $p_{c}(V,\{c_{i}\},V)$ the probability of these photons scattered into polarisations $V,c_{1},\dots,c_{n},V$. For a coherent input state $\ket{H(\omega)}$ such as we consider, the probability of exactly $n$ scattering events occurring in the interval $[t,t+\tau]$ is given by a Poisson distribution $p(n,\tau)$, which depends on the coherent state amplitude $\beta$ (related to laser power) and the duration $\tau$, while the scattering times are random and uncorrelated. This allows us to write ${p}_{t}(t,t+\tau;n)=G^{(1)}(t)G^{(1)}(t+\tau)p(n,\tau)$ and $p_{t}(t_{1},...,t_{n})=n!/\tau^{n}$, where $G^{(1)}(t)$ is the photon scattering rate. The normalised (non-exclusive) cross-polarised intensity correlation function can therefore be written

[TABLE]

where $G^{(1)}_{V}(t)=p_{V}(t)G^{(1)}(t)$ with $p_{V}(t)$ the probability that a photon scattered at time $t$ is detected in the vertically polarised channel. Written in this way, we see that the normalised cross-polarised two-time correlation function is the joint probability for two photons to scatter into the cross-polarisation at times $t$ and $t+\tau$, averaged over all possible numbers, timings, and polarisation channels of intermediate events.

The joint probability $p_{c}(V,\{c_{i}\},V)$ appearing in Eq. (21), is an exclusive quantity describing the likelihood that exactly $n+2$ photons scatter at times $t,t_{1},...,t_{n},t+\tau$ with polarisations $V,c_{1},\dots,c_{n},V$, and can be shown to depend on the nuclear spin system alone. To see this, we must understand how the detection of a photon affects the state of the nuclear system, and combine this effect with the appropriate nuclear spin evolution in between scattering events. In the former case, let us consider the implication of the scattering process described in Eq. (17). We consider an initially horizontally polarised photon and a nuclear spin state $\ket{\delta}$, giving an initial state $\tilde{b}_{\mathrm{in}}^{H}(\omega)^{\dagger}\ket{0}\otimes\ket{\delta}$, with $\ket{0}$ the vacuum. If $|\delta\rangle$ is an eigenstate of the Overhauser shift operator $\hat{\Delta}$, we can write $\hat{r}_{\mathrm{c}}(\omega)\ket{\delta}=r_{c}^{\delta}\ket{\delta}$ with the subscript indicating the co- or crossed-polarised channel. The state after scattering is then given by

[TABLE]

From this, we see that destructive (absorptive) detection of a co- or cross-polarised photon from a general nuclear state $\ket{\psi}$ then results in an (unnormalized) post-measurement state $\ket{\psi^{\prime}}=M_{c}\ket{\psi}$ with operators

[TABLE]

These operators can be interpreted as measurement operators describing the effect of a photon scattering event on the nuclear spin system. The weights of the associated POVM elements are shown in Fig. 2.

In between scattering events, since the probability $p(V,\{c_{i}\},V)$ is conditioned on photon scattering events happening only at times $t_{1},...,t_{n}$, the nuclear spin evolution is the unitary evolution generated by the Hamiltonian $H_{N}=H_{Z}+\hat{\Delta}/2$, where the Zeeman Hamiltonian $H_{Z}$ and Overhauser shift operator $\smash{\hat{\Delta}}$ are defined in Eqs. (2) and (3). This allows us to write

[TABLE]

where the superoperators $\Phi_{c_{i}}$ and $\mathcal{U}_{i}$ act as

[TABLE]

and we define $\Delta_{i}\equiv t_{i}-t_{i\text{-}1}$, $t_{0}\equiv t$, and $t_{n+1}\equiv t+\tau$. The probability in Eq. (24) is exclusive, and corresponds to one possible scattering history. The average over all such histories gives the measured two-time correlation function following Eq. (21). We note that it is the statistical mixture of these histories, and not their coherent superposition, that determines the observed behaviour, as for the nuclear spin system the photon scattering events are irreversible measurement processes. This formulation is analogous to the quantum jump approach Plenio and Knight (1998); Carmichael et al. (1989).

IV Zeno evolution of the nuclear two-time correlation function

We are now in a position to explore the behaviour of the normalised correlation function $g^{(2)}(\tau)$ given in Eq. (21). We begin by examining the regime of low laser power. In this regime we can assume that the probability of intermediate scattering events in a time interval $\tau$ vanishes, i.e. $p(0,\tau)\approx 1$ and $p(n\geq 1,\tau)\approx 0$, while the factor involving the product in Eq. (24) is the identity. Eq. (21) then gives

[TABLE]

where we assume the nuclear system is in a steady state, i.e. $\rho(t)=\rho(t+\tau)=\rho_{N}$, the POVM element is $O_{V}=M_{V}^{\dagger}M_{V}$, and we have defined the unnormalised state $\varrho=\Phi_{V}\rho_{N}$. The steady state assumption allows us to take $t=0$ without loss of generality. Expanding the unitary propagator $\mathcal{U}_{0\rightarrow\tau}$ to second order we find

[TABLE]

where the linear term in $\tau$ vanishes under the assumption that the steady state has no coherence in the Overhauser shift eigenbasis, i.e. $[O_{V},\rho_{N}]=0$, and we have defined the nuclear Zeno time

[TABLE]

The quadratic short-time behaviour seen in Eq. (27) is characteristic of any unitary evolution, and its experimental observation would be a signature of the non-Markovian nature of the nuclear spin bath, and help to distinguish it from other sources of resonance fluctuations. Furthermore, identification of the timescale of the nuclear spin evolution, given by the Zeno time $\tau_{z}$, would provide valuable information on the dynamical behaviour of the nuclear spin system itself.

For laser powers beyond the low intensity regime we need to take intermediate scattering events into account. Averaging over the polarisation orientation of the $n$ intermediate events gives the polarisation-averaged exclusive probability, which we write as

[TABLE]

with superoperator $V_{\tau}\equiv\mathcal{U}_{n+1}\prod_{i=1}^{n}\Phi\ \mathcal{U}_{i}$. Here the superoperator $\Phi$ describes a photon scattering event as a non-selective measurement, and acts as

[TABLE]

Such a non-selective measurement takes a nuclear state $\rho$ and rescales all coherences in the Overhauser shift eigenstate basis $\braket{\delta}{\rho}{\delta^{\prime}}$ by a factor

[TABLE]

This factor can be interpreted as an indistinguishability measure relating the states $\ket{\delta}$ and $\ket{\delta^{\prime}}$. If both states scatter a photon into the same polarisation, then they cannot be distinguished by photon scattering and $r_{\delta\delta^{\prime}}=1$. On the other hand, if the photons scattered off $\ket{\delta}$ are orthogonal to photons scattered off $\ket{\delta^{\prime}}$, then photon scattering has the effect of a projective measurement with discarded outcome. In the first case the coherence between $\ket{\delta}$ and $\ket{\delta^{\prime}}$ remains untouched, while in the latter the coherence is completely destroyed.

We calculate the probability $\mathcal{P}_{n}$ to second order in the time intervals $\Delta_{i}$. To do so, we first expand the first time evolution and measurement step to arrive at

[TABLE]

where the subscripts indicate the state after the first intermediate photon scattering event, and we have defined the first and second order contributions as $\varrho_{1}^{(1)}=-i\Delta_{1}\Phi\left([H_{N},{\varrho}]\right)$ and $\varrho_{1}^{(2)}=-(\Delta_{1}^{2}/2)\Phi\left(\left[H_{N},[H_{N},{\varrho}]\right]\right)$. Expanding the subsequent steps $\Phi\ \mathcal{U}_{i}$ and discarding terms of cubic order yields the recursion relations

[TABLE]

In terms of these density operator contributions Eq. (29) becomes

[TABLE]

where we make use of the identities $\mathrm{Tr}\left[\Phi A\right]=\mathrm{Tr}\left[A\right[$, $\mathrm{Tr}[\Phi(O_{V}A)]=\mathrm{Tr}[O_{V}\Phi(A)]$ for any operator $A$, and $\mathrm{Tr}[O_{V}\rho_{k}^{(1)}]=0$. The recursion relations have solutions

[TABLE]

which lead to

[TABLE]

where we define the slope function

[TABLE]

and where

[TABLE]

Eq. (36) constitutes the major result of this work. It gives the joint probability of two photons being detected in the vertical (crossed) polarisation channel at times [math] and $\tau$, given $n$ scattering events of unknown polarisation scattering at intermediate times $\{t_{i}\}$. Following Eq. (21), averaging over the number and timing of these intermediate events gives the experimentally measurable cross-polarised intensity autocorrelation function $g_{V}^{(2)}(\tau)$ shown in Fig. 1c.

The linear and quadratic terms in Eq. (36) can be understood in terms of a generalized quantum Zeno effect. As can be seen by the vanishing trace of $\varrho_{k}^{(1)}$ given in Eq. (35a), the linear order time evolution only affects the coherences of the unnormalised state ${\varrho}$. Each measurement $\Phi$ reduces a coherence $\braket{\delta^{\prime}}{\varrho_{k}^{(1)}}{\delta}$ by a factor of $r_{\delta\delta^{\prime}}$, such that a particular coherence follows a sawtooth pattern shown in Fig. 3. The gradient of $\mathcal{P}_{n}$ at time $\tau$ depends on a commutator of the form $[H_{N},\varrho_{\mathrm{coh}}]$, where $\varrho_{\mathrm{coh}}\sim\Phi^{n-j+1}[H_{N},\varrho]$ is all the coherence that has accumulated up to $\tau$. This coherence has one contribution from the evolution since the last measurement at $t_{n}$, which leads to the quadratic term in Eq. (36), and another contribution due to all the coherence that has partially ‘survived’ the previous measurements, and is given by the linear term. The exponent of $\Phi$ gives the number of measurements that the coherence accumulated during interval $\Delta_{j}$ has suffered after the $n$th measurement.

In the limiting case of the polarisation of scattered photons being independent of the nuclear spin state, the nuclear spin coherence is not affected by scattering. It is readily seen that for $\Phi([H_{N},\rho_{N}^{V}])=[H_{N},\rho_{N}^{V}]$, the slope function becomes $S_{n}=t_{n}/\tau_{Z}^{2}$ and the quadratic time evolution of Eq. (27) is recovered. In general, however, an intermediate scattering event and associated measurement reduces the coherence, which decreases $S_{n}$, and therefore leads to a reduced slope of $\mathcal{P}_{n}$. This process can be interpreted as the system partially loosing its ‘memory’ of the previous time evolution stored as coherence.

In the opposite limit, in which projective measurements are made at evenly spaced time intervals $\Delta=\tau/n$, coherences are completely destroyed leading to $S_{n}=0$, and we find

[TABLE]

This linear short-time evolution is characteristic of the Markovian regime, in which the system ‘forgets’ all previous time evolution with every scattering event. The slope of this linear time evolution decreases with the number $n$ of measurements, such that frequent photon scattering can stabilise the nuclear system in a state that maintains resonance. This constitutes a novel nuclear quantum Zeno effect. The results of a Monte-Carlo simulation of $g^{(2)}(\tau)$ which averages over the intermediate scattering histories are shown in Fig. 1c, and demonstrate the characteristic flattening of the correlation function with increasing laser power, which is the experimental signature of this nuclear quantum Zeno effect.

V Nuclear quantum Zeno dynamics in the presence of Markovian noise

The nuclear quantum Zeno dynamics described above arise from unitary evolution of the nuclear spin system and the resulting non-Markovian behaviour of the fluctuating excitonic resonance energy. However, in typical QD experiments other sources of noise may be present which lead to dephasing of the excition, and which are Markovian and memoryless on the timescale of the nuclear spin evolution. In particular fluctuating charges in the vicinity of the QD can dephase the excitonic state Kuhlmann et al. (2015), and also phonons can perturb the excitonic transition Nazir and McCutcheon (2016). To investigate the effects of Markovian dephasing noise, we add a random, time varying shift $s(t)$ to the resonance energy $\omega_{0}$ that takes on a particular value with probability $p(s)$. This shift leads to dephasing of the excitonic state, as the phase of that state evolves in proportion to the exciton energy. Given the random shift of this energy, the phase undergoes a random walk and the average state dephases at a rate $\gamma=t_{c}\sigma^{2}/2$, where $\sigma^{2}$ is the variance of the shift probability distribution $p(s)$ and $t_{c}$ is the characteristic timescale of the fluctuations. In our case it is the typical magnitude $\sigma$ of the shift rather than the dephasing rate $\gamma$ which affects the nuclear spin evolution, as explained below.

Given a nuclear spin state $\ket{\delta}$ and a random energy shift $s$, a horizontally polarised photon $\ket{H}$ is scattered into a polarisation state $r_{\mathrm{cr}}^{\delta+s}\ket{V}+r_{\mathrm{co}}^{\delta+s}\ket{H}$ with probability $p(s)$, where $r_{\mathrm{co/cr}}^{\delta+s}$ are the reflection coefficients into co/cross-polarisation, respectively (cf. Eq. 22). The probability $p_{\mathrm{cr}}$ of photon scattering into cross-polarisation is then associated with a modified POVM element $\tilde{O}_{V}=\sum_{\delta}\sum_{s}p(s)|r_{\mathrm{cr}}^{\delta+s}|^{2}\ket{\delta}\bra{\delta}$ and the nuclear spin state upon scattering is obtained by the quantum operation $\tilde{\Phi}_{V}$ with operators $\{M_{V}^{s}=\sqrt{p(s)}\sum_{\delta}r_{cr}^{\delta+s}\ket{\delta}\bra{\delta}\}$, i.e.

[TABLE]

(cf. Eq. 23 and 26). The random resonance shift $s$ effectively broadens the resonant feature in the polarization rotation. Replacing the POVM element $O_{V}$ and quantum operation $\Phi$ in Eq. 29 by their stochastic versions $\tilde{O}_{V}$ and $\tilde{\Phi}$ then yields a modified cross-polarized intensity autocorrelation $\smash{g_{V}^{(2)}(\tau)}$.

The result of a Monte Carlo simulation including these Markovian processes is shown in Fig. 4, and we see that they have a twofold effect on the intensity autocorrelation. Firstly, the bunching seen at $\tau=0$ is reduced, which is a consequence of the Markovianity of the noise on the timescale of the nuclear spin evolution. A cross-polarised photon detection projects the nuclear spin system into a state with significantly increased probability weight on resonant configurations, such that a photon scattering event immediately afterwards has a high chance of scattering into the cross-polarised channel. In the presence of stochastic noise, however, a cross-polarised photon detection yields less information regarding the nuclear spin state, and therefore leads to a smaller increase in the likelihood of a second cross-polarised scattering event, which results in weaker bunching and thus a decreased $\smash{g^{(2)}_{V}(0)}$. Secondly, the effect of intermediate photon scattering events impeding the nuclear spin evolution, and thereby the decay of $g^{(2)}_{V}(\tau)$ via the quantum Zeno effect, is reduced. This reduced effect of a photon scattering event on the nuclear spin evolution can be attributed to the decreased precision of the measurement performed on the nuclear spin system by the photon due to the averaging over stochastic shifts $s$. As can be seen in Fig. 4, however, neither of these effects alter the qualitative behaviour of the correlation function. The decay of the intensity autocorrelation still changes from relatively fast and initially quadratic for low laser intensities, to slow and exponential for higher intensities. Observation of this quantum Zeno effect should therefore be possible if the broadening of the excitionic transition due to dephasing is significantly smaller than the broadening due to nuclear spins, such that a photon scattering event still yields sufficient information of the nuclear spin state.

The time evolution of $g^{(2)}_{V}(\tau)$ in the above analysis is due to nuclear spin evolution alone, and the electron spin is assumed to remain in an eigenstate of the electron Zeeman Hamiltonian. In an isolated electron–nuclear spin system with a magnetic field $B_{\mathrm{ext}}\gtrsim 100$ mT Cywiński et al. (2009), this assumption is well justified, as electron spin relaxation by electron–nuclear spin flip flops is energetically forbidden. In practice, however, there are additional mechanisms that may lead to electron spin relaxation, which in turn will lead to an exponentially decaying intensity correlation function that cannot be stabilized by the nuclear quantum Zeno effect. For example, co-tunneling of electrons in and out of the QD can lead to such electron spin relaxation, although we note that this mechanism can be strongly suppressed by tuning of the QD energy with an external electric field Dreiser et al. (2008).

Another electron spin relaxation mechanism that might obscure the nuclear quantum Zeno dynamics is given by second-order electron-nuclear flip flops, which arise when an environment such as the phonon bath supplies or absorbs the flip flop energy Latta et al. (2009). Such environment-assisted flip flops arise from the contact hyperfine Hamiltonian given by

[TABLE]

when also in the presence of electron spin dephasing at a rate $\eta$, and lead to electron spin relaxation at a rate $\sim\eta\frac{\alpha^{2}}{\omega_{e}^{2}}$, where $\omega_{e}$ is the electronic Zeeman splitting as before and $\alpha$ gives the interaction energy with the unpolarised nuclear spin bath Nutz et al. (2019). This relaxation mechanism is therefore suppressed by a strong external magnetic field and tuning of the temperature and electrostatic environment to minimise the dephasing rate, however a theoretical estimate of $\eta$ and $\alpha$ is beyond the scope of this work. Experimental measurement of the electron spin relaxation rate has confirmed the $\omega_{e}^{-2}$ suppression of relaxation by a magnetic field and achieved spin lifetimes of hundreds of $\mu$s at sub-Tesla magnetic fields Dreiser et al. (2008), which is significantly slower than the resonance fluctuations that are important in this work Androvitsaneas et al. (2019). Putting these observations together, we conclude that the nuclear quantum Zeno effect should therefore be observable for magnetic fields of $\sim 1$ T, temperatures of $T=4~{}\mathrm{K}$, and using tuning of the charge state to maximise the electron spin lifetime.

VI Discussion and Conclusion

In order to experimentally demonstrate the nuclear quantum Zeno effect predicted here, it would suffice to observe the characteristic change of the cross polarised intensity autocorrelation function from quadratic to linear short-time behaviour, as the intensity of the input laser light increases. The non-Markovian quadratic regime is the most challenging to observe, since the intensity must be low enough that unobserved intermediate scattering events have vanishing probability, which in turn implies a long integration time of the experiment. Increasing the input laser intensity will introduce intermediate photon scattering events that take place during a delay time $\tau$ of interest. If these photons can be detected, the polarisation outcomes of these detection events need to be averaged over to calculate the degree of second-order coherence $g^{(2)}(\tau)$. If these events are lost and not detected, this averaging is automatically performed. Loss does therefore not invalidate the measurement as long as there is an estimate of the photon scattering rate for a given intensity. Following Eq. (36) one expects a broadening as well as a change from quadratic to linear behaviour of the intensity autocorrelation function with intensity, which is a second experimental feature of the nuclear quantum Zeno effect.

Our result paves the way for experimental demonstration of a novel nuclear spin effect in quantum dots, with implications for both fundamental theoretical investigations and photonic quantum computing. Importantly, our formulation of the quantum Zeno effect in terms of a two-time correlation function has the advantage that it is possible to observe the effect without initialising the system in a particular state. The intensity autocorrelation considers pairs of cross-polarised photon detection events, the first of which effectively initializes the nuclear system in a state with increased likelihood of being close to resonance. The second photon count then probes how far the system has evolved away from this initial state, and intermediate photon counts disturb this evolution. This generalized description of a quantum Zeno effect in terms of imperfect measurements and two-time correlation functions likely applies to other experimentally accessible quantum systems. Another interesting theoretical aspect of the nuclear quantum Zeno process is the explicit connection between a measurement and the physical process of photon scattering. This connection shows that it is the coherence-destroying effect of measurements that impedes coherent evolution and gives rise to the quantum Zeno effect. The formulation of coherence reduction of photon scattering as a measurement is merely a convenient formalism, making it clear that a coherence-removing process that gives rise to a quantum Zeno effect does not need to be a measurement.

Beyond these implications the nuclear quantum Zeno effect may be relevant to the experimental realization of a quantum dot-based source of entangled photons. A weak laser could be used to stabilise the nuclear system in a state for which the electronic transition is close to resonance and where high phase shifts can be achieved. If a method was found to simultaneously keep the electron spin in a superposition then the nuclear Zeno effect could be used to realize photonic states with useful entanglement properties as proposed in Hu et al. (2008b), even in the presence of a nuclear spin environment.

Acknowledgements.

The authors thank John Rarity, Terry Rudolph, Sophia Economou, Ed Barnes, Will McCutcheon, and Gary Sinclair for interesting and useful discussions. T.N. acknowledges financial support from the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/2007-2013) under REA Grant agreement 317232. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 703193. This work was also funded by the Frontiers in Quantum Technologies programme and by the Engineering and Physical Sciences Research Council (EP/M024156/1, EP/N003381/1 and EP/L024020/1).