Pulse-Enhanced Two-Photon Interference with Solid State Quantum Emitters

TL;DR

This paper demonstrates that applying a periodic sequence of $C0$ pulses can significantly improve two-photon interference and photon indistinguishability between solid-state quantum emitters affected by spectral diffusion, advancing scalable quantum networks.

Contribution

The study introduces a control protocol using periodic $C0$ pulses to enhance two-photon interference in quantum emitters with spectral diffusion, a novel approach for solid-state quantum systems.

Findings

Pulse sequence improves photon indistinguishability despite spectral diffusion.

Enhanced two-photon interference observed with external control protocols.

Spectral overlap issues mitigated by periodic pulsing in quantum emitters.

Abstract

The ability to entangle distant quantum nodes is essential for the construction of quantum networks and for quantum information processing. For solid-state quantum emitters used as qubits, it can be achieved by photon interference. When the emitter is subject to spectral diffusion, this process can become highly inefficient, impeding the achievement of scalable quantum technologies. We study two-photon interference in the context of a Hong-Ou-Mandel (HOM)-type experiment for two separate quantum emitters, with different detunings with respect to a specific target frequency. We evaluate the second order coherences that characterize photon indistinguishability between the two emitters. We find that the two-photon interference operation that is inefficient in the absence of a control protocol, when the two detunings are different and spectral overlap is lessened, can be highly improved by a periodic sequence of $\pi$ pulses at a set target frequency. Photon indistinguishability in solid state and other quantum emitters subject to spectral diffusion can thus be enhanced by the proposed pulse sequence and similar external control protocols despite the fluctuations in the environment.

Equations

$$H=\frac{\Delta }{2}{\sigma }_z$$

$$G_{34}^{(2)}(t,\theta )=⟨a_3^{†}(t)a_4^{†}(t+\theta )a_4(t+\theta )a_3(t)⟩.$$

$$g_{34}^{(2)}(\theta )=\underset{T\to \infty }{\lim }{\int }_0^T{G}_{34}^{(2)}(t,\theta )dt$$

$$a_3(t)$$

$$a_4(t)$$

$$G_{34}^{(2)}(t,\theta )$$

$$\begin{array}{rl}{\dot{\rho }}_{ee}& =i\frac{{\Omega }_x(t)}{2}({\rho }_{eg}-{\rho }_{ge})-\Gamma {\rho }_{ee},\\ {\dot{\rho }}_{gg}& =-i\frac{{\Omega }_x(t)}{2}({\rho }_{eg}-{\rho }_{ge})+\Gamma {\rho }_{ee},\\ {\dot{\rho }}_{ge}& =-i\frac{{\Omega }_x(t)}{2}\left({\rho }_{ee}-{\rho }_{gg}\right)+(i\Delta -\frac{\Gamma }{2}){\rho }_{ge},\\ {\dot{\rho }}_{eg}& =i\frac{{\Omega }_x(t)}{2}\left({\rho }_{ee}-{\rho }_{gg}\right)+(-i\Delta -\frac{\Gamma }{2}){\rho }_{eg}.\end{array}$$

$$g_i(t,\theta )=⟨{\sigma }_{+i}(t){\sigma }_{-i}(t+\theta )⟩$$

$$⟨{\sigma }_{+}(t){\sigma }_{-}(t+\theta )⟩$$

$$=$$

$$g_{1,2}(t,\theta )={\mathrm{e}}^{-\Gamma t}{\mathrm{e}}^{(-\Gamma /2+i{\Delta }_{1,2})\theta }.$$

$$G_{34}^{(2)}(t,\theta )$$

$$g_{34}^{(2)}(\theta )$$

$$g_{34}^{(2)}(\theta )=\frac{1}{4\Gamma }\left(1-\cos ({\Delta }_{21}\theta )\right){\mathrm{e}}^{-\Gamma \theta }.$$

$$g(t,\theta )=f(t,\theta ){\rho }_{gg}(t).$$

$${\rho }_{gg}(t)=1-\frac{1-(-1{)}^{M+1}{\mathrm{e}}^{-(M+1)\Gamma \tau }}{1+{\mathrm{e}}^{-\Gamma \tau }}{\mathrm{e}}^{-\Gamma (\tau -{\tau }_1)}.$$

$$f(t,\theta )={\mathrm{e}}^{\left(i\Delta -\frac{\Gamma }{2}\right)\theta };$$

$$f(t,\theta )=0;$$

$$f(t,\theta )={\mathrm{e}}^{-\Gamma \theta /2}{\mathrm{e}}^{i\Delta \left(\theta -m\tau \right)}.$$

$$G_{34}^{(2)}(t,\theta )$$

$$=$$

Full text

I Introduction

Recent progress in the field of Quantum Information Processsing (QIP) has used a variety of platforms as qubits. Several QIP experimental milestones have been reached using as quantum bits quantum-dotsImamoglu_Awschalom_QDOT_PRL_99 ; SantoriVuckovicYamamoto_QDOT_Nat2002 ; Kuhlmann_Warburton_QDOT_NatPhys2013 ; Gao_Imamoglu_NatComm2013 , nitrogen-vacancy (NV) centersHumphreys_Hanson_NV_entanglement ; ChildressRepeater ; Bernien_Hanson_Nature2013 ; Hanson_loopholeFree_Nature2015 or silicon-vacancy(SiV) centersSipahigil_HOM_SiV_2014 ; Rogers_SiV in diamond, trapped ionsMaunz_Monroe_HOMIons or neutral atomsRosenfeld_etal_BellTest_atoms ; Beugnon_Grangier_HOMAtoms and superconducting circuitsChou_Schoelkopf_2018 ; Kurpiers_Wallraff_2018 ; Axline_Schoelkopf_2018 ; CIbarcq_Devoret_2018 ; Zhong_Cleland_2019 . In particular, experiments have achieved teleportation of quantum states and loophole free Bell inequality tests using quantum bits separated by macroscopic distancesHanson_loopholeFree_Nature2015 ; Humphreys_Hanson_NV_entanglement ; Gao_Imamoglu_NatComm2013 ; Pfaff_Hanson_Science2014 . A central component of these experiments as well as the implementation of two-qubit gates is the two-photon interference that is essential for the generation of entanglement between distant quantum bits. This is typically achieved by interfering two photons emitted by the qubits to be entangled on a beam splitter in a Hong-Ou-Mandel (HOM)-type experimentHOM . The success of this operation is in turn tied to the indistinguishability of the photons. For solid state quantum emitters and other systems in dynamic environments, the emission/absorption spectrum can drift in an uncontrolled way away from a target frequency as a result of fluctuations in the surrounding bath (e.g. changes in local strain and motion of neighboring charges). These variations in spatial, temporal and spectral profiles can compromise photon indistinguishability and significantly hamper entanglement generation between distant quantum bits and other photon-mediated processes essential to scalable quantum networks and QIPAwschalomHansonZhou_opticsQIP ; AtatureEnglund_Wrachtrup_NatRevMat_2018 ; HansonAwschalom_QIP_ss_08 ; NV_Review_PhysRep2013 ; CarterGammonQDcavity ; JeffKimble_qtmInternet ; NielsenChuangBook .

This problem continues to receive a great deal of attention both from the point of view of material design and from the point of view of external controlAmbroseMoerner_spectralDiffusion_Nature1991 ; Schroeder_Englund_SPE_NatComm2017 ; QDOTs_HOM_Atature ; Aharonovich_Englund_Toth_SS_SPE ; Fu_Beausoleil_PRL2009 ; Acosta_Beausoleil_PRL2012 ; Dreau_Jacques ; Pfaff_Hanson_Science2014 ; Basset_Awschalom_PRL2011 ; FaraonEtalNatPhot ; Hansom_Atature_APL2014 ; Crooker_Bayer_PRL2010 ; Calajo_Passante_PRA2017 ; JSLee_Khitrin_JPhysB2008 ; IDS_2017 . In one approach proposed in recent work, it was shown that appropriate pulse sequences can be applied to quantum emitters in a dynamic environment to produce an emission or absorption spectrum that has little dependence on the fluctuations in the environmentFotsoEtal_PRL2016 ; FotsoDobrovitski_Absorption ; Fotso_JPhysB2019 . It was, for instance, shown that a periodic sequence of $\pi$-pulses can maintain the bulk of the emission spectrum at a central peak located at the pulse carrier frequency and satellite peaks at frequencies shifted from this central peak by integer multiples of $\pm\pi/\tau$; where $\tau$ is the period of the pulse sequence. This lineshape is unchanged for various detunings as long as the pulse sequences are appropriately adjustedFotsoEtal_PRL2016 ; Fotso_JPhysB2019 . In the context of quantum information processing, this naturally raises the question of how different quantum emitters, each with their own dynamic environment, would fare in a HOM-type two-photon interference when they are driven by such a pulse sequence.

The goal of this paper is to answer this question. Namely, we consider the problem of photon indistinguishability for two distant quantum emitters in dynamic environments. We evaluate the intensity correlation at the detectors in a HOM-type two-photon interference when the emitters have different detunings $\Delta_{1}$ and $\Delta_{2}$ with respect to the pulse carrier frequency of an applied periodic sequence of $\pi$-pulses with period $\tau$. We find that in the situation without control protocol, when $\Delta_{1}$ and $\Delta_{2}$ are significantly different, the intensity correlation exhibits beating with vanishing values periodically as a function of the delay time $\theta$ between the two the detectors. On the contrary, in the presence of the pulse sequence, the intensity correlation vanishes at time delay $\theta=0$ but keeps a finite value for finite $\theta$. This corresponds to enhanced photon indistinguishability between two qubits that originally have different environments and spectral profiles.

II Two-Photon Interference, Spectral Diffusion: Model

The two-photon interference, pictured in Fig.1, involves two separate and independent quantum emitters. Photons from Emitters $E_{1}$ and $E_{2}$, with respective detunings $\Delta_{1}$ and $\Delta_{2}$ measured in the frame rotating at the set target frequency $\omega_{0}$, at spacetime locations $1$ and $2$, are sent to input ports of a 50:50 beam splitter and then measured at detectors $D_{1}$ and $D_{2}$, located at spacetime locations $3$ and $4$ beyond the output ports of the beams splitter. Indistinguishable photons will coalesce and emerge at the same port (Fig.1 top). This indistinguishability is synonymous with identical spatial, temporal and spectral profiles. Thus, for high efficiency, the entanglement operation requires overlapping spectra whereas it is inefficient for low spectral overlap (Fig.1 bottom).

We model each individual solid-state emitters as a two-level systems coupled to a radiation bath. The two-level system emits a photon in the course of a spontaneous transition from its excited state $|e\rangle$, located at the energy $\hbar\omega_{1}$ above the ground state $|g\rangle$ (below we will set $\hbar=1$). The optical control pulses, each of very short duration $t_{p}$, are applied at the pulse carrier frequency, $\omega_{0}$, at appropriate times. It is thus convenient to work in the rotating-wave approximation (RWA), using the basis rotating at frequency $\omega_{0}$. The system corresponding to the driven emitter in the radiation bath can then be described by a Hamiltonian of the form:

[TABLE]

where $\Delta=\omega_{1}-\omega_{0}$ is the detuning of the emitter from the target frequency, caused by the random fluctuation in the local strain or charge environment; this detuning is assumed to be static on the spontaneous emission timescale. The operators $\sigma_{z}=|e\rangle\langle e|-|g\rangle\langle g|$, $\sigma_{+}=|e\rangle\langle g|$, and $\sigma_{-}=|g\rangle\langle e|=(\sigma_{+})^{\dagger}$ are respectively, the $z$-axis Pauli matrix, the raising and the lowering operator for the two-level system. $a_{k}$ is the annihilation operator of the $k$-th photon mode, $g_{k}$ is its coupling strength to the emitter, and $\omega_{k}$ is its detuning from $\omega_{0}$. We consider pulses such that $\Omega_{x}(t)=\Omega_{x}$ during the pulses and zero otherwise. We will further assume $\Omega_{x}$ to be much larger than all other relevant energy scales so that the pulses are essentially instantaneous (i.e. $\Omega_{x}\gg\Delta,\Gamma,g_{k}$ and $t_{p}\to 0$). At the initial time, $t=0$, we will assume that both emitters have their excited states occupied and ground states unoccupied and that all radiation modes are empty. Furthermore, we assume that both emitters are driven by identical pulse sequences.

In the setup described schematically by Fig. 1, with the addition of an identical pulse sequence applied to both emitters, we want to evaluate the second order coherence equivalent to the intensity correlation at the detectors that is defined by:

[TABLE]

From this, we will extract the integrated intensity correlation corresponding to the experimentally measured cross-correlation in the Hanburry Brown and Twiss setupHanburryBrownTwiss_1956 ; KirazAtature_PRA_2004 :

[TABLE]

For a 50:50 beam splitter, the operators $a_{3}$ and $a_{4}$ at the detectors can be expressed in terms of the operators $a_{1}$ and $a_{2}$ at the emitters as:

[TABLE]

and similarly for their conjugate expressions. k-integrated operators are used because they correspond to the electric field operators. Plugging this into the equations for the coherence, we get after dropping negligible two-photon terms:

[TABLE]

If we define $g_{i}(t,\theta)=\langle a^{\dagger}_{i}(t)a_{i}(t+\theta)\rangle$ with $i=1,2$, we can rewrite $G_{34}(t,\theta)$ as:

[TABLE]

Calculating the second-order coherence then reduces to evaluating the first order coherences $g_{i}(t,0)$ and $g_{i}(t,\theta)$. For this, we will use the master equations characterizing the time-evolution of the density matrix operator for individual emitters: $\rho(t)=\rho_{ee}(t)|e\rangle\langle e|+\rho_{eg}(t)|e\rangle\langle g|+\rho_{ge}(t)|g\rangle\langle e|+\rho_{gg}(t)|g\rangle\langle g|$ with $\rho_{ge}^{*}=\rho_{eg}$ and $\rho_{ee}+\rho_{gg}=1$.

III Pulse-Driven Emitters and Solution

The master equations or optical Bloch equations governing the time-evolution of the above-defined density matrix operator is given in the rotating wave approximation by Cohen_Tannoudji_Book1992 :

[TABLE]

where $\Gamma=2\pi\int g_{k}^{2}\;\delta(\omega_{k}-\Delta)\;dk$ is the spontaneous emission rate in the absence of the control field. It is used to set the units of time and energy. Namely, we set $\Gamma=2$ and we measure energy in units of $\Gamma$ and time in units of $1/\Gamma$. Each $\pi_{x}$ pulse inverts the populations of the excited and ground state and swaps the values of $\rho_{eg}$ and $\rho_{ge}$.

In the far field region, the field operators, $a$ and $a^{\dagger}$, are related to the emitter operators, $\sigma_{+}$ and $\sigma_{-}$, by simple proportionality constants independent of time. We can accordingly redefine the coherences so thatLoudon_book1983 :

[TABLE]

The two-time correlation function $\langle\sigma_{+}(t)\sigma_{-}(t+\theta)\rangle$ can be expressed as a single-time expectation value according to the quantum regression theorem or following RF_Mollow_PhysRev1969 ; Scully_Zubairy_book1997 ; Loudon_book1983 ; Scully_Zubairy_book1997 ; Fotso_JPhysB2019 :

[TABLE]

where $\rho^{\prime}(t)=\rho(t)\sigma_{+}$, and where $\sigma_{+}$ and $\sigma_{-}$ are the time-independent operators in the Schrödinger picture. $U(t,t^{\prime})$ is the time-evolution operator from time $t$ to $t^{\prime}$ for the system described by Eqs.(8).

With the assumption that each emitter is initially prepared in its excited state and all bosonic modes are initially empty, the expression in Eq. 12 can be evaluated by integrating numerically or analytically the master equation between consecutive pulses.

III.1 No control protocol

In the absence of a control protocol, a straightforward integration of the equations yields:

[TABLE]

Plugging this into Eq.( 7), we get:

[TABLE]

with $\Delta_{21}=\Delta_{2}-\Delta_{1}$. From this, we obtain:

[TABLE]

For $T\to\infty$, this simplifies to:

[TABLE]

This is a function that takes an identical minimum at zero time delay and repeatedly in a periodic way with a period defined by $\theta=2n\pi/\Delta_{21}$, with $n$ integer..

III.2 Periodic pulse sequence

For a periodic pulse sequence, the evaluation of $g_{i}(t,\theta)$ is achieved by iteratively integrating Eqs.( 8) between consecutive pulses, using the fact that the effect of each pulse is to swap the populations of the excited and ground states as well as the coherences. This integration follows steps similar to those highlighted in Refs.FotsoDobrovitski_Absorption ; Fotso_JPhysB2019 and yields:

[TABLE]

With the times $t$ and $t+\theta$ such that $t=M\tau+(\tau-\tau_{1})$ and $t+\theta=(M+m)\tau+\tau_{2}$ as illustrated schematically in Fig.4, $\rho_{gg}(t)$ is given by:

[TABLE]

The function $f(t,\theta)$ is such that:

• •

For $t$ and $t+\theta$ in the same pulse interval, we have:

[TABLE]

• •

For $t$ and $t+\theta$ separated by an odd number of pulse intervals, we have:

[TABLE]

• •

For $t$ and $t+\theta$ separated by an even number of pulse intervals, we have:

[TABLE]

These expressions, combined with those of $g(t,0)=\rho_{gg}(t)$ and $g(t+\theta,0)=\rho_{gg}(t+\theta)$ can be brought into the expression of $G^{(2)}_{34}(t,\theta)$ and the integral for $g^{(2)}_{34}(\theta)$ completed numerically.

IV Results

Fig.2 and Fig.3 are the main results of this paper. They present, for two emitters with detunings $\Delta_{1}=3.0$ and $\Delta_{2}=-4.0$, the intensity correlation at the detectors in the absence of a control protocol and in the presence of a periodic pulse sequence of period $\tau=0.2$ respectively. The inserts show in both situations the emission spectra of the individual emitters. In the absence of a control protocol, the emission spectra have Lorentzian lineshapes centered around the emitters detunings (in the frame rotating at $\omega_{0}$). Thus, in this situation, the spectral overlap is limited for $|\Delta_{21}|=|\Delta_{1}-\Delta_{2}|>\Gamma$. In this case, the intensity correlation vanishes periodically with a period defined by the value $\Delta_{21}$ of the difference between the frequencies of the two emittersLegero_Kuhn_PRL2004 , namely, it vanishes at $\theta=2n\pi/\Delta_{21}$, $n$ integer.

In the second situation, i.e when the two emitters are driven by the same periodic pulse sequence of period $\tau=0.2$, as shown in the insert, spectral overlap is enhanced with both emission spectra having overlapping central peaks despite $|\Delta_{21}|>\Gamma$. The intensity correlation vanishes at $\theta=0$ and remains finite for $\theta\neq 0$ indicating enhanced photon indistinguishability.

In Fig.5 and Fig.6, we show that the intensity correlations for $\tau=0.2$ and for $\tau=0.3$ respectively, for $\Delta_{1}=3.0$ and $\Delta_{2}=-4.0$ (blue line), $\Delta_{1}=3.0$ and $\Delta_{2}=-2.0$ (red line), $\Delta_{1}=3.0$ and $\Delta_{2}=2.0$ (green line). It clearly exhibits little dependence on the individual emitters detunings. Furthermore, these figures show the dependence of the intensity correlations on the pulse period $\tau$. The function vanishes at $\theta=0$ and remains finite elsewhere. Note that the small oscillations away from the minimum correspond to beating at the pulse sequence period. Note that the intensity correlation is nearly identical for $\Delta_{21}$ spanning a range of width $\sim 10\Gamma$. These figures demonstrate an enhanced photon indistinguishability as long as $\Delta_{1,2}\;\times\tau\leq 1$.

The results presented here are obtained using the transient expression of $\rho_{gg}$ (Eq. 18) and $f(t,\theta)$ (Eqs. 19, 20, 21) for a total time equal to $4.8$ but one could also use the long time stationary regime where $\rho_{gg}\approx 1/2$ under a periodic pulse sequence. In this case, we will get:

[TABLE]

The results are overall similar with the only difference that the beating at long times is strongly suppressed in the steady state regime.

V Conclusion

Photon indistinguishability is essential for a variety of photon-mediated operations in quantum information processing. For quantum emitters in the solid state or in other dynamic environments, this can be compromised by spectral diffusion due to fluctuations in the surrounding bath of the emitter resulting in low efficiency for the aforementioned operations. We have examined the two-photon interference in the context of a HOM-type experiment when the two involved quantum emitters are subject to spectral diffusion that reduces the spectral overlap. We have evaluated the intensity correlation at the detectors when both emitters are driven by a periodic sequence of $\pi$ pulses of period $\tau$. In the absence of the control protocol, the intensity correlation exhibits beating with vanishing values periodically as a function of the delay time $\theta$ between the two the detectors. Under the control sequence, the intensity correlation vanishes at zero time delay but remains finite otherwise. These results indicate that the control protocols can indeed enhance photon indistinguishability and thus improve the efficiency of fundamental operations that are central to the development of scalable quantum information processing and quantum networks for qubits susceptible to spectral diffusion.