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

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

Thomas Nutz nutzat@gmail.com Controlled Quantum Dynamics Theory Group, Imperial College London, London SW7 2AZ, United Kingdom Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol, BS8 1FD, UK    Petros Androvitsaneas    Andrew Young    Ruth Oulton    Dara P. S. McCutcheon Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Department of Electrical and Electronic Engineering, University of Bristol, BS8 1FD, UK
July 20, 2019
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 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 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.

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); Wüst et al. (2016); Éthier-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); Éthier-Majcher et al. (2017); Nutz et al. (2018). 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. (2016a)(see Fig. 1a), in which narrowband laser photons of linear polarisation described by scatter off a cavity containing a charged QD in a large ( mT) magnetic field in the Faraday configuration. Since an electronic transition couples only to one of the two circular polarisations and , the photon polarisation state upon scattering is given by , with the phase shift difference taking values of up to Hu et al. (2008a, b); Auffèves-Garnier et al. (2007). Hence, a linearly polarised photon can be reflected with the orthogonal linear polarisation , 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 and Abragam (1961). As the nuclear spin system evolves, the phase shift drifts over time, such that high values are observed only during short intervals ( in timebins Androvitsaneas et al. (2016b)) but the time-averaged phase shift is low ( in Androvitsaneas et al. (2016b)). Photon detection events in the cross polarised (orthogonal to input laser) channel are therefore bunched on a timescale, such that an intensity autocorrelation function has for due to the single photon nature of the scattered field, but for 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 to a two-time correlation function of the nuclear spin system. We show that 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 . 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.

Figure 1: (a) Experimental setup used to measure photon phase shifts due to a quantum dot inside a micropillar cavity. The states and denote horizontally and vertically polarised light, while (P)BS labels a (polarising) 50/50 beamsplitter. A variable delay between detectors and can be used to measure the intensity autocorrelation function of light scattered into cross polarised (vertical) channel, as shown in (c). (b) Right and left circularly polarised photons couple to the spin ground states and respectively. In a large magnetic field, the is far detuned, and reflects off an effectively empty cavity, while experiences a phase shift that depends on the nuclear spin Overhauser field . (c) Cross-polarised intensity correlation function for different average photon count rates obtained from a Monte-Carlo simulation of including 8 nuclear spins. While the quadratic behaviour characteristic of unitary evolution is observed for low count rates, the evolution of the nuclear spins is impeded by more frequent photon scattering, which constitutes a quantum Zeno effect. Note that the antibunching at sub-nanosecond timescales is neglected and that the weak-driving assumption is satisfied. Parameters and were randomly drawn from a Gaussian distribution with = 0.5 eV, eV, and neV.

Ii Input–output formalism with electron–nuclear spin coupling

Our aim is to calculate the cross-polarised intensity autocorrelation 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 propagating towards and away from an optical cavity with frequency and associated cavity mode operator . The cavity mode, in turn, couples to a two-level system (TLS) with ground and excited states and , respectively, which itself is coupled to a bath of nuclear spins. The total Hamiltonian describing all degrees of freedom is written , with (setting )

(1)
(2)

where , , , is the transition energy of the TLS, and is the TLS–cavity coupling strength. The nuclear Zeeman term is , with Pauli operator acting on nuclear spin and nuclear Zeeman splitting due to an external magnetic field along . The electron–nuclear coupling term is , with Overhauser shift operator

(3)

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

We approximate the cavity–port mode coupling strength as a constant over the relevant optical frequencies, and in doing so we find the Heisenberg equations of motion

(4a)
(4b)
(4c)

where we have defined the incoming and outgoing field operators as with , and with  Walls and Milburn (2008), and extended frequency integrals such that . Taking the Fourier transform of Eq. (4b) we find

(5)

where and similarly for .

The standard procedure in input–output theory is to use the Fourier transform of Eq. (4a) to replace in Eq. (5), which is then used in the Fourier transform of Eq. (4c), to find a relationship between frequency components of the incoming and outgoing fields and . We use a similar procedure here, but note that the occurrence of the time-dependent Overhauser shift operator in Eq. (4a) means there is no simple relationship between the Fourier components and . Instead we arrive at the integral equation

(6)

where we have defined and made the approximation , valid for weak driving. Combining Eqs. (5) and (6) then gives

(7)

where . Using this in the Fourier transformation of Eq. (4c) leads to

(8)

When the Overhauser term is neglected, , Eq. (8) simplifies to with the scalar , 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

(9)

Choosing the integration range such that , where is the characteristic timescale of the Overhauser field fluctuations, we can approximate in the integrands and arrive at

(10)

where and similarly for . If the Overhauser shift fluctuates slowly then we can choose the spectral width of the sinc function in Eq. (10), , to be much narrower than the width of the function . We then find

(11)

with and given by

(12)

while the incoming and outgoing field operators are now defined as

(13)

and similarly for . 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 owing to the finite integration time .

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 in Eq. (9) satisfies , where is the fluctuation time of the Overhauser shift and is the spectral width of the phase shift feature, which is obtained by considering the frequency dependence of . We find that the phase factor varies most rapidly at , at which point

(14)

and . As such, the fractional variation does not exceed a bound on the order of when considering laser–QD detunings no greater than , and laser–cavity detunings limited to . The functions therefore vary on a frequency scale given by the linewidth 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. (2016b), such that can be satisfied and Eq. (11) is applicable to QD experiments. We interpret as a parameter that adjusts the tradeoff between frequency and time resolution of our theory. Eq. (11) relates Fourier components that must be understood as averages of the exact Fourier components of the incoming and outgoing fields over a bandwidth interval . For an experiment with a QD linewidth of 1 GHz and a fluctuation time of 1 s our theory describes effects with a resolution of up to MHz in frequency and 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

(15)

where are the positive and negative frequency components of the vertically polarised electric field at time , and the trace is performed over the total port mode–cavity–electron spin–nuclear spin system, with total initial state , and where the Heisenberg electric field operators evolve unitarily in this complete Hilbert space.

To proceed we express these field operators as

(16)

where we have neglected numerical factors and retardation effects. Following Eq. (11), a cavity containing a QD with electron spin projection reflects a right circularly polarised photon according to , while a left circularly polarised photon acquires a phase shift corresponding to an empty cavity. Hence we can write

(17)

where the operators give the reflectivities into the co- and cross-polarised channels, which depend on the nuclear spin state through the dependence of on the Overhauser operator . We assume an initial state , where satisfying is a horizontally polarised coherent state of amplitude , and are states of cavity mode and nuclear spin system, respectively, and the electron is assumed to remain in state during the measurement. Substituting this state into Eq. (15) gives

(18)

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 and at . 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 , 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

(19)

where is the exclusive probability density that exactly photon scattering events take place in the interval , with the first and last photons scattered into vertical polarisation at times and , and additional photons scattered at intermediate times into polarisations labelled with being either the co- () or cross-polarised () channel. We now decompose the exclusive probability into probabilities describing the scattering times, and the scattering polarisations. We write

(20)

where is the non-exclusive probability of photon scattering events at and with intermediate scattering events at unspecified times, the probability density of these intermediate events occurring at , and the probability of these photons scattered into polarisations . For a coherent input state such as we consider, the probability of exactly scattering events occurring in the interval is given by a Poisson distribution , which depends on the coherent state amplitude (related to laser power) and the duration , while the scattering times are random and uncorrelated. This allows us to write and , where is the photon scattering rate. The normalised (non-exclusive) cross-polarised intensity correlation function can therefore be written

(21)

where with the probability that a photon scattered at time 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 and , averaged over all possible numbers, timings, and polarisation channels of intermediate events.

The joint probability appearing in Eq. (21), is an exclusive quantity describing the likelihood that exactly photons scatter at times with polarisations , 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 , giving an initial state , with the vacuum. If is an eigenstate of the Overhauser shift operator , we can write with the subscript indicating the co- or crossed-polarised channel. The state after scattering is then given by

(22)

From this, we see that destructive (absorptive) detection of a co- or cross-polarised photon from a general nuclear state then results in an (unnormalized) post-measurement state with operators

(23)

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.

Figure 2: Probability for a photon scattering event into the co-polarised (blue) or cross-polarised channel , shown for a nuclear system in an Overhauser shift eigenstate , and as a function of the shift . For zero Overhauser shift cross-polarised photons are preferred, with this bias reversing as the Overhauser shift becomes greater than the linewidth of the electronic transition in the cavity . Parameters [eV]: , , .

In between scattering events, since the probability is conditioned on photon scattering events happening only at times , the nuclear spin evolution is the unitary evolution generated by the Hamiltonian , where the Zeeman Hamiltonian and Overhauser shift operator are defined in Eqs. (2) and (3). This allows us to write

(24)

where the superoperators and act as

(25)

and we define , , and . 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 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 vanishes, i.e. and , while the factor involving the product in Eq. (24) is the identity. Eq. (21) then gives

(26)

where we assume the nuclear system is in a steady state, i.e. , the POVM element is , and we have defined the unnormalised state . The steady state assumption allows us to take without loss of generality. Expanding the unitary propagator to second order we find

(27)

where the linear term in vanishes under the assumption that the steady state has no coherence in the Overhauser shift eigenbasis, i.e. , and we have defined the nuclear Zeno time

(28)

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 , 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 intermediate events gives the polarisation-averaged exclusive probability, which we write as

(29)

with superoperator . Here the superoperator describes a photon scattering event as a non-selective measurement, and acts as

(30)

Such a non-selective measurement takes a nuclear state and rescales all coherences in the Overhauser shift eigenstate basis by a factor

(31)

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

We calculate the probability to second order in the time intervals . To do so, we first expand the first time evolution and measurement step to arrive at

(32)

where the subscripts indicate the state after the first intermediate photon scattering event, and we have defined the first and second order contributions as and . Expanding the subsequent steps and discarding terms of cubic order yields the recursion relations

(33a)
(33b)

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

(34)

where we make use of the identities , for any operator , and . The recursion relations have solutions

(35a)
(35b)

which lead to

(36)

where we define the slope function

(37)

and where

(38)
Figure 3: Left: Representative time evolution of the nuclear spin system, here taken to be two spins spanned by the phase shift eigenstates and , with the dynamics generated following Eqs. (35a) and (35b). The plot shows the path of a Bloch vector representation of the nuclear spin state, with Overhauser amplitude and coherence operators respectively , , with . State trajectories are shown for no photon scattering events (blue), photon scattering every ns (orange), and ns (green). The dashed red line shows the surface of the Bloch sphere, a cross-section of which is shown in the inset (red section near the pole corresponds to the outer plot). Right: Exclusive joint probability following Eq. (36) of scattering into cross-polarisation at times and corresponding to the state evolution shown to the left, where . Parameters: , , , , .

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 and , given scattering events of unknown polarisation scattering at intermediate times . Following Eq. (21), averaging over the number and timing of these intermediate events gives the experimentally measurable cross-polarised intensity autocorrelation function 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 given in Eq. (35a), the linear order time evolution only affects the coherences of the unnormalised state . Each measurement reduces a coherence by a factor of , such that a particular coherence follows a sawtooth pattern shown in Fig. 3. The gradient of at time depends on a commutator of the form , where is all the coherence that has accumulated up to . This coherence has one contribution from the evolution since the last measurement at , 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 gives the number of measurements that the coherence accumulated during interval has suffered after the 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 , the slope function becomes 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 , and therefore leads to a reduced slope of . 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 , coherences are completely destroyed leading to , and we find

(39)

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 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 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 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 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 . 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).

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