On the indistinguishability of Raman photons
Charles Santori, David Fattal, Kai-Mei C. Fu, Paul E. Barclay, Raymond G. Beausoleil
aa r X i v : . [ qu a n t - ph ] D ec On the indistinguishability of Raman photons
Charles Santori , David Fattal , Kai-Mei C. Fu, Paul E.Barclay, and Raymond G. Beausoleil Hewlett-Packard Laboratories, 1501 Page Mill Road, Palo Alto, CA 94304E-mail: [email protected], [email protected] Abstract.
We provide a theoretical framework to study the effect of dephasing onthe quantum indistinguishability of single photons emitted from a coherently drivencavity QED Λ-system. We show that with a large excited-state detuning, the photonindistinguishability can be drastically improved provided that the fluctuation rate ofthe noise source affecting the excited state is fast compared with the photon emissionrate. In some cases a spectral filter is required to realize this improvement, but thecost in efficiency can be made small.PACS numbers: 42.50.Ex, 42.50.Ct, 03.67.Lx, 78.67.-n n the indistinguishability of Raman photons
1. Introduction
Indistinguishable photons are required for many quantum computation schemes thatutilize their peculiar interference properties [1, 2]. They reveal a true bosonicnature and for instance tend to coalesce or “bunch” when they cross at a beam-splitter [3]. Indistinguishable photons are naturally produced during the spontaneousdecay of atoms and ions [4, 5]. Solid-state atom-like systems can also emitindistinguishable photons, with examples including quantum dots [6, 7, 8], molecules [9]and semiconductor impurities [10]; diamond nitrogen-vacancy (NV) centers also appearpromising [11]. Solid-state systems are especially interesting for large-scale photonicnetworks incorporating many devices. The temporal length of the emitted photons canbe short due to the large dipole moment and correspondingly short spontaneous emissionlifetime (0 . −
12 ns), and the lifetime can be further shortened using integrated smallmode volume optical cavities. The main drawback of solid-state systems is that they cansuffer from severe dephasing of the optical transitions caused by phonons [12], randomcharge fluctuations [13] or other mechanisms [14].One way to reduce the detrimental effect of dephasing on photon indistinguishabilityin two-level systems is to push towards shorter-lived transitions. This approachreaches its limit when the photon lifetime becomes comparable to the time jitter thatcharacterizes incoherently pumped transitions. This then necessitates true resonantexcitation, which has recently been demonstrated [15] but limits experiments tocavity geometries that minimize pump laser scattering. Alternatively, one can moveto three-level systems that can be coherently excited at a frequency different fromthat of the photon emission. Three-level Λ-type systems also potentially provide aninterface between long-lived matter qubits and “flying” photonic qubits. Schemesbased on Raman transitions in three-level systems were proposed more than a decadeago for deterministic communication in quantum networks [16], and for probabilisticentanglement generation [17] and teleportation [18]. Some more recent papers haveinvestigated the theoretical aspects of single-photon generation in three-level systemsin greater detail, including the problem of generation and trapping of photons witharbitrary waveforms [19, 20], and the problem of how best to generate indistinguishablephotons from non-identical systems [21]. Experimentally, three-level schemes have nowbeen used for single-photon generation in atoms [22], and in trapped ions [23] it wasshown that photons with arbitrary waveforms could be generated. Such experimentshave not yet been performed in solid-state systems, but following a recent demonstrationof tunable, spontaneous Raman fluorescence from a single quantum dot [24] the outlookis promising.For solid-state implementations of the Raman scheme, a critical question is how thephoton indistinguishability is affected by excited-state dephasing. Ref. [25] investigatedthis question but only for the special case of zero detuning, and furthermore the model,based on optical Bloch equations, did not allow for finite correlation timescales inthe dephasing process. The results suggested that the three-level scheme is helpful n the indistinguishability of Raman photons (single photon in waveguide) (control pulse)(cavity) Cavity mode system ! g ! e ! r Figure 1.
A Λ-system in a single-mode cavity coupled to a single-mode waveguide. for solving the time-jitter problem, but little or no improvement with respect toexcited-state dephasing was indicated. One might expect that for a large detuning,single photons generated in a Raman process will be far less sensitive to excited-stateenergy fluctuations. In this paper, we analyze theoretically the dependence of photonindistinguishability on both the detuning and the correlation timescale of the dephasingprocess affecting the excited state. We find that for a large detuning, the effectsof excited-state dephasing can be reduced by orders of magnitude beyond the usualreduction that occurs due to shortened photon lifetime in a two-level system, but only ifthe memory timescale of the noise process is short compared with the photon emissionlifetime.
2. Three-Level Equations of Motion in the Schr¨odinger Picture
Consider a 3-level system as in Fig. 1, with two ground states | e i and | g i and an excitedstate | r i where the e − r and g − r transitions are optically allowed. The g − r transitioncouples to an optical cavity mode with Rabi vacuum frequency g and detuning ∆, andthe cavity mode itself is coupled to a single-mode waveguide and has total decay rate κ .The excited state | r i can decay by emitting a photon outside of the cavity with rate γ (not desired but nevertheless present). The system starts in level | e i and is exposed to aclassical time-varying light beam which drives the e − r transition with detuning ∆ (sothat the e-to-r-to-g transition is on two-photon resonance) and complex Rabi frequencyΩ( t ). We neglect coupling of the laser to the g − r transition and coupling of the cavity n the indistinguishability of Raman photons e − r transition, good approximations only if we have either good polarizationselection rules or an energy spacing between | e i and | g i that is much larger than ∆and κ . In this calculation we include spontaneous decay only from state | r i to state | g i , so that the system cannot be re-excited after emitting a photon. This allows for alarge simplification of the dynamics, since only states with zero or one photon occur.Spontaneous decay back to the initial state | e i can, in fact, be an important decoherenceprocess, and the effect on photon indistinguishability was analyzed in Ref. [25]. Herewe are interested mainly in the effect that pure dephasing of the excited state has onthe emitted photons, so we set the r → e decay rate to zero. While there are practicallimitations, the r → e rate can be made small compared with the r → g rate eitherby having unequal strengths for the two optical transitions or by using a large Purcellenhancement of the g − r transition. Below, we shall consider the r → e decoherenceprocess separately when discussing the application of this scheme to actual solid-statesystems. Λ System
First let us consider a deterministic trajectory of the system without dephasingprocesses. A Hamiltonian describing the three-level system, cavity, and continuummodes outside of the cavity under the assumptions given above and under the rotating-wave approximation is, H = X i ω i σ ii + Ω2 e iω l t σ er + Ω ∗ e − iω l t σ re + ω c c † c + g σ gr c † + g ∗ σ rg c + X k (cid:16) ω k a † k a k + h k c a † k + h ∗ k c † a k (cid:17) + X k ′ (cid:16) ω k ′ b † k ′ b k ′ + g k ′ σ gr b † k ′ + g ∗ k ′ σ rg b k ′ (cid:17) , (1)where σ ij = | i ih j | for i, j = { e, r, g } , and c , a k and b k ′ are photon annihilation operatorsfor a single cavity mode, the continuum modes coupled to this cavity mode, and thecontinuum modes coupled to the g − r transition, respectively. The constants ω i are thefrequencies of the three levels, ω c is the cavity resonance frequency, ω l is the frequencyof the excitation laser, ω k and ω k ′ are the continuum mode frequencies, and h k and g k ′ are the coupling constants to the continuum modes. We work in a rotating frame bywriting the state of the system (assuming we begin in state | g i so that only one photoncan be emitted) as, | ψ ( t ) i = e ( t ) e − iω e t | e i + r ( t ) e − i ( ω r − ∆) t | r i + g ( t ) e − i ( ω g + ω c − δ c ) t c † | g i + X k α k ( t ) e − i ( ω g + ω k ) t a † k | g i + X k ′ α ′ k ′ ( t ) e − i ( ω g + ω k ′ ) t b † k ′ | g i , (2)where e ( t ) and r ( t ) are the amplitudes of the system in states | e i or | r i with no photonin the cavity, and g ( t ) is the amplitude of the system in state | g i with one photon inthe cavity. We have assumed that direct spontaneous emission from level r and photonescape from the cavity involve orthogonal sets of radiation modes, and α k ( t ) and α k ′ ( t )are the amplitudes for photon emission into these modes. The detuning constants for n the indistinguishability of Raman photons ω r − ω e − ω l and δ c = ω c − ω r + ω g + ∆, respectively. In thecalculation below we set δ c = 0, though this parameter could be adjusted to compensatefor a.c. Stark shifts if desired. By writing ddt | ψ ( t ) i = − iH ( t ) | ψ ( t ) i and applying theWeisskopf-Wigner approximation to remove the continuum modes from the dynamics,we find the unperturbed equations of motion,˙ e ( t ) = − i Ω2 r ( t ) , (3)˙ r ( t ) = − (cid:16) i ∆ + γ (cid:17) r ( t ) − i Ω ∗ e ( t ) − ig ∗ g ( t ) , (4)˙ g ( t ) = − ig r ( t ) − (cid:16) iδ c + κ (cid:17) g ( t ) , (5)which are similar to those used in Refs. [19, 20]. These equations include decay rates κ and γ for the cavity and level r that can be related to the h k and g k ′ coupling coefficients,respectively, combined with the corresponding densities of states for the continuummodes. Following the cavity input-output formalism, we can also define a temporalenvelope α ( t ) of the photon emitted into the waveguide as the Fourier transform of α k ( t → ∞ ) with respect to ω k , including only the subset of modes that belong to thewaveguide. This temporal envelope is simply α ( t ) = √ κ wg g ( t ), where κ wg < κ is thecavity decay rate into the waveguide modes only. The coherence properties of α ( t ) arethus the same as those of g ( t ). The efficiency to emit a photon into the waveguide is η = R dt | α ( t ) | .For the unperturbed dynamics we shall consider only the adiabatic limits ˙ g ≪ κg and ˙ r ≪ ∆ r for which the unperturbed dynamics follow simple analytical solutions, e ( t ) = exp (cid:20) i ′ Z t −∞ dt ′ | Ω( t ′ ) | (cid:21) , (6) r ( t ) = − Ω ∗ ( t )2∆ ′ e ( t ) , (7) g ( t ) = − ig κ r ( t ) . (8)Here, we have defined ∆ ′ = ∆ − iγ p / γ p = γ + 4 | g | /κ asthe cavity-enhanced spontaneous emission rate from level r . This approximate solutionholds if {| Ω | , γ p } ≪ | ∆ | , (cid:12)(cid:12)(cid:12) Ω g ⋆ ∆ (cid:12)(cid:12)(cid:12) ≪ κ , | Ω | γ p ≪ γ , and (cid:12)(cid:12)(cid:12) | Ω | + δ c (cid:12)(cid:12)(cid:12) ≪ κ . For the deterministic Eqs. 3-5, α ( t ) is always the same, and consecutive photonsemitted by the system are perfectly indistinguishable. Now, we wish to describe thereal system which is affected by random fluctuations in its environment. Varioustreatments have been developed to include decoherence effects in the context of quantuminformation, the most commonly used being the optical Bloch equations [26] andquantum jump approaches [27]. While a quantum jump approach may be preferable interms of simplicity and physical insight, both approaches are typically used to describeinteractions with a reservoir having infinitely short correlation timescales. In solid-state n the indistinguishability of Raman photons − − s. To describe pure dephasing processes with finite memory timescales,we follow an alternative approach that uses randomly fluctuating energy shifts torepresent these processes [28, 29, 30, 31]. The time-dependent energy shifts of the systemlevels are denoted by δ e ( t ), δ r ( t ) and δ g ( t ), or ~δ for short, with correlation properties tobe defined below. If the unperturbed equations of motion are written as ddt ~x = M ( t ) ~x ,where ~x is a vector containing e , r , and g , and M ( t ) is a 3 × ddt ˜ ~x = M ( t )˜ ~x − i diag( ~δ ( t )) ˜ ~x , (9)where ˜ x i are the perturbed amplitudes. Changing variables, let us define,˜ x i ( t ) = x i ( t ) exp( − iφ i ( t )) , (10)where the φ ’s are arbitrary complex functions of time. We substitute this into Eq. 9(using Eqs. 3-5 for M ( t )) and linearize to first order in ~φ assuming that the fluctuationsare small. The linearized equations can be written as ddt ~φ − ~δ = M ( t ) ~φ , where, M ( t ) = i Ω2 re − i Ω2 re − i Ω ∗ er i Ω ∗ er + ig ∗ gr − ig ∗ gr − ig rg ig rg . (11)For the noise-induced dephasing, here we shall consider only dephasing of theexcited state, setting δ g ( t ) = δ e ( t ) = 0. Let us start with the simple case where κ is large compared with g , ∆, and the fluctuation rate of the noise source. In this limitthe cavity serves only to enhance the effective spontaneous emission rate from level r , and otherwise the cavity does not enter into the problem. Using | ˙ φ g | ≪ κ | φ g | andsubstituting Eqs. 6-8 into Eq. 11 with ~φ ( −∞ ) = 0 we find, φ e ( t ) = 14∆ ′ Z t −∞ dt ′ | Ω( t ′ ) | ¯ δ ( t ′ ) , (12) φ g ( t ) = φ r ( t ) = φ e ( t ) + ¯ δ ( t ) i ∆ ′ , (13)¯ δ ( t ) = i ∆ ′ Z t −∞ dt ′ e − i ∆ ′ ( t − t ′ ) δ r ( t ′ ) . (14)The phase φ g ( t ) that is imprinted directly onto the emitted photon has two parts. Thefirst term φ e ( t ) results from the accumulated phase error transferred to state | e i due tothe fluctuating detuning combined with the a.c. Stark shift. This term is dominant ifthe noise source fluctuates slowly, in which case ¯ δ ( t ) ≈ δ r ( t ). The second term representsa phase added directly to state | r i through its own energy fluctuation.The indistinguishability of two photons emitted by identical devices as measuredin a Hong-Ou-Mandel-type experiment [3] is, F = D(cid:12)(cid:12)R dt α ( t ) α ∗ ( t ) (cid:12)(cid:12) E(cid:0)R dt h| α ( t ) | i (cid:1) , (15) n the indistinguishability of Raman photons hi denotes an expectation value over all possible functions δ r ( t ), and α ( t ) and α ( t ) are the temporal envelopes of the two photons. It is assumed that each photonis generated through an independent but identical random process. From here on, letus consider the special case of a classical field that is turned on at t = 0 and thenheld constant, Ω( t ) = Ω θ ( t ). In this case the unperturbed, approximate solution fromEqs. 6-8 is a photon envelope with an abrupt rise at t = 0 followed by an exponentialdecay, α ( t ) ∝ e ( − ξ/ − iζ ) t − iφ g ( t ) where ζ ≈ − | Ω | and ξ ≈ | Ω | γ p . If the noise perturbationis small, to second order in φ g the indistinguishability can be approximated as, F ≈ ξ Z ∞ ds Z ∞ dt e − ξ ( s + t ) Re (cid:10) φ g ( s ) φ ∗ g ( t ) (cid:11) − ξ Z ∞ dt e − ξt (cid:10) φ g ( t ) φ ∗ g ( t ) (cid:11) , (16)where φ g ( t ) may be complex.
3. Finite-Memory Dephasing Processes
Next, let us consider a specific noise process with zero mean and two-time correlationfunction, h δ r ( t ) δ r ( t + τ ) i = ρ ( τ ) = σ e − β | τ | , (17)where σ is the noise amplitude and β is the fluctuation rate (inverse correlation time).This correlation function was also used in Refs. [30, 31] and can describe, for example,fluctuation of nearby traps that jump between two charge states, a common source ofspectral diffusion of the optical transitions in semiconductor quantum dots and nitrogen-vacancy centers in diamond. Suppose we have an ensemble of fluctuating traps that shiftthe energy (through the dc Stark effect, for example) of level r according to, E r ( t ) = X n c n q n ( t ) , (18)where c n are constants that depend on geometry, and q n = { , } are random variablescorresponding to the charge of each trap. The q n fluctuate independently according to,˙ p i = X j r ij p j , (19)where p i is the probability that q = i , and r ij are the transition rates. From this we candefine a zero-mean random process δ r ( t ) = E r ( t ) − h E r i with correlation function givenby Eq. 17 where, σ = X n c n r r / ( r + r ) , (20) β = r + r . (21)The number of traps involved in the dynamics will affect higher-order correlationfunctions but will not change the form of the second-order correlations needed forthe small-signal analysis presented here. In experiments, the fluctuation timescale 1 /β n the indistinguishability of Raman photons ∼ kT / ~ , but the form of thecorrelation function differs from that of Eq. 17, which corresponds to a Lorentzianpower spectrum. To simplify our discussion, here we shall focus on the simple processdescribed by Eq. 17. If a more accurate description is required, Eqs. 26-28 presentedbelow can be used to estimate the photon indistinguishability for arbitrary correlationfunctions. Pure dephasing in the δ -correlated limit, as used in the standard masterequation or quantum jump approaches, can be obtained from Eq. 17 by taking the limit β → ∞ with σ /β ≡ /T held constant.The photon indistinguishability under the process described by Eq. 17 can becalculated using Eqs. 12-14 and 16 under the assumption that σ is small so that φ g is also small. In the limit | ∆ | ≫ {| Ω | , γ p , ξ } , an approximate solution is,1 − F ≈ σ γ p (1 + βξ ) + 2 σ ∆ + β γ p + 2 βγ p . (22)The first and second terms in this expression correspond directly to φ e ( t ) and ¯ δ/i ∆ ′ ,respectively, in Eq. 13. The cross-correlation term was not included because, for large∆, it is always small compared with at least one of the terms in Eq. 22. For β small compared with the photon emission rate, the first term, corresponding to theaccumulated phase error on state | e i , is dominant. For large β , the first term becomessmall since the noise is mostly averaged out in the integral of Eq. 12 (in the frequencydomain this acts as a low-pass filter). This leaves the second term, which passes alarger noise bandwidth, to make the dominant contribution. To test the accuracy ofthis approximate result we performed Monte-Carlo simulations in which the originalEqs. 3-5 are numerically integrated using a Gaussian-distributed noise source δ r ( t ) withthe required temporal correlations obtained from a pseudo-random number generatorcombined with a first-order finite-difference equation. The number of trials was set sothat the random errors in 1 − F were usually less than 10% of the mean value. Figure 2shows calculated results in the κ ≫ ∆ regime (see figure caption for parameter values),and the results are in good agreement with Eq. 22 away from ∆ = 0.It is useful also to compare the photon indistinguishability predicted for the Ramanscheme with that of an equivalent two-level system. For a two-level system that israpidly initialized into its excited state, if we neglect time jitter associated with theexcitation processes, in the limits g ≪ κ and σ → − F ) two − level ≈ σ γ p ( γ p + β ) . (23) n the indistinguishability of Raman photons (a)(b) ∆ −3 −2 −1 −5 −4 −3 −2 β / γ − F ∆: −2 −2 β / γ β / γ −5 −4 −3 −2 Figure 2. (a) Deviation from perfect photon indistinguishability (1 − F ) for thethree-level system in the large- κ regime with g = 50, κ = 1000, γ = 1, σ = 1 andΩ = 10 plotted on a logarithmic intensity scale as a function of detuning ∆ and thenormalized noise fluctuation rate ( β/γ ). The plot on the left was obtained by theMonte-Carlo method. The white region at lower-right was skipped due to excessivelylong computation times. The plot on the right was obtained from Eq.22. (b) Log-log plots of 1 − F vs. β/γ for various values of ∆ as indicated. The points (varioussymbols) are from the Monte-Carlo simulation, and the blue, solid curves are fromEq.22. The result for an instantaneously excited two-level system (Eq. 23) is alsoshown (red, dashed). This is plotted alongside the results from the three-level system in Fig. 2(b) (red-dashedcurve). The results show that the Raman scheme can provide a substantial improvementover the two-level case only when the fluctuation rate β of the noise source falls withinthe window ξ < β < ∆. When this condition is satisfied, an averaging effect occurs overthe length of the photon which improves the indistinguishability. n the indistinguishability of Raman photons
4. Indistinguishability Enhancement Using Spectral Filtering
The first and second terms in Eq. 22 can also be understood in terms of a spectralfunction describing the transfer of noise from δ r ( t ) to the emitted photon. Let us define, f ( ω + iη/
2) = 1 √ π Z ∞ dt e ( iω − η/ t φ g ( t ) , (24)which can be written in terms of δ r ( t ) as, f ( ω + iη/
2) = − (cid:0) [ M + ( iω − η/ I ] − (cid:1) √ π Z ∞ dt e ( iω − η/ t δ r ( t ) . (25)The energy spectral density h| f ( ω + iξ/ | i of noise transferred to the emitted photonis then proportional to the product of a transfer function, H ( ω, η ) = (cid:12)(cid:12)(cid:0) [ M + ( iω − η/ I ] − (cid:1) (cid:12)(cid:12) , (26)and a weighted power spectral density of δ r ( t ), S r ( ω, η ) = Z ∞−∞ dτ ρ ( τ ) e iωτ − η | τ | = σ ( η + 2 β ) ω + ( η/ β ) , (27)evaluated for η = ξ . The photon indistinguishability in the small- φ g limit given byEq. 16 can be expressed in terms of these quantities as, F = 1 + ξH (0 , ξ ) S r (0 , ξ ) − Z dω π H ( ω, ξ ) S r ( ω, ξ ) . (28)For the large- κ approximation made above, H ( ω, ξ ) receives significantcontributions from two poles at ω ≈ {− iξ/ , ∆ − iγ p / } . The first pole correspondsto broadening of the Raman line itself, while the second corresponds to spontaneousemission at the natural frequency of the r → g transition. For ∆ ≫ γ , we claim thatthis second contribution can be filtered out with little effect on the photon collectionefficiency. We then expect only the first term in Eq. 22 to survive, giving,(1 − F ) filter ≈ σ γ p (1 + βξ ) . (29)Thus, with spectral filtering, the only remaining requirement to improve theindistinguishability over the two-level case is β ≫ ξ .To demonstrate that an external spectral filter can indeed improve theindistinguishability with little reduction in efficiency, Monte Carlo simulations of Eqs. 3-5 were performed with the same parameters as in Fig. 2 but with a spectral filter appliedto the emitted photon. To keep the results experimentally relevant, only a simpleLorentzian filter was used, representing a Fabry-Perot cavity, centered on the Raman-scattering frequency with width fixed at 20 ξ . The results are shown in Fig. 3(a), andindeed the degradation of photon indistinguishability matches Eq. 29. Furthermore,there is little cost in terms of photon generation efficiency per cycle. Even for the simpleLorentzian filter used in the simulation, the filtering loss was only 5%, and the theoreticalphoton collection efficiency decreased from 90% to 85%. In theory, if the ground statesare completely free from dephasing processes then the photon indistinguishability can n the indistinguishability of Raman photons ∆:∆: (a)(b) −3 −2 −1 −5 −4 −3 −2 β / γ − F −3 −2 −1 −5 −4 −3 −2 −1 β / γ − F Figure 3.
Effect of two spectral filtering methods on photon indistinguishability:(a) An external spectral filter is centered on the Raman scattering wavelength toreject spontaneous emission at the natural transition frequency. Parameters: g = 50, κ = 1000, γ = 1, σ = 1 and Ω = 10. A simple Lorentzian filter was used with FWHMlinewidth set to 20 ξ (see text). The points (various symbols) are from Monte Carlosimulations, and the solid blue curves use only the first term of Eq. 22. The resultfor an instantaneously excited two-level system (Eq. 23) is also shown (red, dashed).(b) The cavity itself serves as the external filter. Parameters: g = 5, κ = 10, γ = 1, σ = 1 and Ω = 10. The points are from Monte Carlo simulations, and the solid bluecurves use Eq. 31. For the equivalent two-level system (filled red circles) a Monte Carlosimulation was used since Eq. 23 is not accurate for g ∼ κ . be improved to an arbitrary degree by decreasing the photon emission rate ξ , once ξ < β . The main penalty is the decrease of ξ itself, since this is the photon emissionrate. Decreasing ξ will eventually reduce the communication or computation speed inany quantum network application.The cavity can also serve as a spectral filter. Until now, our calculations have beenperformed in a regime κ ≫ ∆ where the cavity served only to increase the effectivespontaneous emission rate. If instead we have κ ≪ ∆, the cavity can reject spontaneousemission noise at the natural transition frequency without the need for an external n the indistinguishability of Raman photons κ the spectral transfer function H ( ω, ξ ) has three polesthat contribute. Combining Eq. 8-11 with Eq. 26 in the limits ξ ≪ { γ, γ p } ≪ ∆ withconstant Ω, we find the approximate expression, H ( ω, η ) ≈ (cid:12)(cid:12)(cid:12)(cid:12) ξγ p ( iω − η/
2) + iκ (2∆ + iκ )( iω − i ∆ − γ eff / − iκ (2∆ + iκ )( iω − κ/ (cid:12)(cid:12)(cid:12)(cid:12) , (30)where γ eff = ( κ γ p + 4∆ γ ) / ( κ + 4∆ ) is the modified spontaneous emission ratefrom level r including the cavity detuning. Neglecting the cross-terms in Eq. 30, theindistinguishability calculated using Eq. 28 becomes,1 − F ≈ σ γ p (1 + βξ ) + 2 σ ∆ + β γ eff + 2 βγ eff κ κ + 4∆ + 8 σ κ ( κ + 4∆ )( κ + 2 β ) . (31)Comparing this with Eq. 22, we see that the second term which corresponds tospontaneous emission noise at ω = ∆ is now suppressed by a factor κ / ( κ + 4∆ )due to the cavity filtering. The new, third term corresponds physically to resonantenhancement of noise by the cavity, and even for large ∆ it can make a significantcontribution. To test the validity of Eq. 31, Monte Carlo simulations of Eqs. 3-5 wereperformed for cases with κ < ∆, and the results are shown in Fig. 3(b) (see figure captionfor parameters). The results show that the cavity can indeed serve as an effective filter,and for β ≫ ξ , photon indistinguishability far exceeding that of an equivalent two-levelsystem is again obtained.
5. Other Decoherence Mechanisms
To estimate the total indistinguishability degradation in the Raman scheme we mustinclude two other decoherence mechanisms in addition to pure dephasing of the excited-state. The first is ground-state dephasing, against which the Raman scheme provides noprotection. The photon indistinguishability degradation due to ground-state dephasingis similar to that in a two-level system,1 − F ′ ≈ σ ′ ξ ( ξ + β ′ ) , (32)where σ ′ and β ′ are the ground-state fluctuation amplitude and rate, respectively, underthe noise model of Eq. 17. In contrast with Eq. 31, the contribution from ground-state dephasing increases as the photon emission rate ξ is decreased. In this modelthe effective decay rate 1 /T ∗ of the ground-state coherence is σ ′ /β ′ for large times( t ≫ /β ), but for short times the coherence decay follows a Gaussian function withcharacteristic timescale ∼ /σ ′ .The second additional decoherence mechanism is spontaneous decay from | r i backto | e i . As discussed in Ref. [25], this can be viewed as a time-jitter process, with the last r → e photon marking a random delay before the start of the emitted r → g photon.The indistinguishability degradation resulting from this process is,1 − F ′′ ≈ ξ e ξ e + ξ , (33) n the indistinguishability of Raman photons ξ e ≈ | Ω | γ e is the decay rate back to | e i ( γ e is the natural r → e spontaneous decayrate), and ξ is the rate for the forward process as defined above. In principle, for a three-level system ξ e can be made negligibly small by choosing system parameters such that the e − r transition is very weak, and then compensating with a stronger excitation field. Inactual systems, the achievable improvement will be limited by additional excited stateswhich, although they may be further detuned, will make a substantial contribution tothe system dynamics when the excitation field is too strong. The other way to improvethe ξ e /ξ ratio is through selective cavity enhancement of the g − r transition. Thisrequires either a cavity linewidth that is narrow compared with the e − g energy spacingor else good polarization selection rules.
6. Solid-State Sources of Indistinguishable Photons
Let us now discuss how the above results may apply to actual solid-state atom-likesystems that have two or more ground-state spin sublevels connected by strong opticaltransitions to a common excited state. Various types of optical control have beendemonstrated in semiconductor quantum dots [33, 34, 35, 36, 37, 38, 39, 24], shallowdonors [40], and diamond NV centers [41, 42, 43]. The benefit provided by a largedetuning will first of all depend on the fluctuation rate of the noise source affecting theexcited state. For spectral-diffusion processes with long correlation timescales we expectlittle improvement compared with a two-level scheme for useful photon emission rates,but a large improvement may be possible for noise processes that have sub-nanosecondcorrelation timescales.First, let us consider a charged quantum dot subject to a strong magnetic fieldthat is not aligned to the growth axis, so that two long-lived electron spin ground statesare coupled to an excited (trion) state by optical transitions. Suppose this quantumdot is placed in a state-of-the-art photonic-crystal microcavity [44] with cavity-QEDparameters g = 2 π × κ = 2 π ×
33 GHz. Because spontaneous emissioninto leaky modes in 2D photonic crystals is suppressed by a factor of 2 − γ = (0 . − . Suppose now that the excited state is subject to a phonondephasing process which we shall approximate using the above dephasing model with1 /T ,ex = σ /β = 2 π × .
16 GHz, with fluctuation rate β = 2 π ×
80 GHz (correspondingto kT at T = 4 K). If the system is excited on resonance with a short pulse, the best-case indistinguishability degradation due to excited-state dephasing is 1 − F ≈ . ∼
35 ps (the decay is non-exponentialsince the system is near the onset of strong coupling). For non-resonant excitation,possibly the only option with photonic crystal cavities due to pump laser scatter, theindistinguishability will be much worse due to the time jitter associated with the finiterelaxation time to the upper level of the optical transition. Suppose now that we insteaduse a Raman scheme with Ω = 2 π ×
16 GHz and ∆ = 2 π ×
40 GHz, and send the collectedlight through a Fabry-Perot cavity with a transmission bandwidth of 2 π × n the indistinguishability of Raman photons − F ≈ .
002 while the lifetime for the Raman transition is0 . /T ⋆ is limited to <
10 ns due to hyperfinecoupling between the electronic and nuclear spins. However, a recent experiment withpositively charged quantum dots [39] suggests that T ⋆ values of hundreds of nanosecondsor more may be achievable. If we take T ⋆ to be 500 ns, for β ′ ≫ ξ one could hope for1 − F ′ = 2 / ( ξT ⋆ ) = 0 . β ′ ≪ ξ the situation is better since the dephasingwill follow a Gaussian time dependence, and in the first 0 . ξ − dependence in Eq. 32 for β ′ → | e i . If we have a perfectly balanced Λ system,the degradation is given approximately by the inverse Purcell factor (cid:16) g κγ (cid:17) − giving1 − F ′′ ∼ .
01. It is tempting to try to reduce γ e by moving to an imbalanced Λ systemusing a magnetic field that is nearly aligned to the growth axis, or by using a very largemagnetic field. However there are in fact two excited (trion) states, and if γ e decreasesfor one of them, it will increase for the other. Furthermore, at practical magnetic fieldstrengths the energy separation between the two trion states cannot be made much largerthan the value of ∆ used in this example. Thus, 1 − F total ∼ .
01 is probably close to thebest possible performance. Nevertheless, we note several improvements in the Ramanscheme compared with the two-level scheme. First, the overall indistinguishability issubstantially improved, especially if we consider realistic excitation schemes in the two-level case. Additionally, the Raman scheme can tolerate much more high-frequencyexcited-state dephasing than was included here. Finally, the Raman scheme includesa long-lived matter qubit coupled to a single-photon emitter as needed for quantumnetworking applications. In this example we assumed that the ground-state splittingcould be made large compared with the cavity linewidth κ . If this cannot be achievedin a photonic-crystal cavity using practical magnetic field strengths, then it may beadvantageous instead to use microdisk cavities, which tend to have higher quality factors,can also reach a strong coupling regime, and can be coupled efficiently to a taperedoptical fiber [47].Next, let us briefly discuss the case of diamond NV centers. This systemseems attractive for quantum computation since in isotopically purified diamond thespin coherence lifetime (as measured in a spin-echo experiment) can reach severalmilliseconds, and T ⋆ (without spin echo) can be several microseconds or longer [48].A Λ-type system can be obtained either by using a magnetic field to mix twoof the ground states [41] or by using strain or an electric field to mix some ofthe excited states [42, 43, 49]. Furthermore, at low temperatures, lifetime-limitedspectral linewidths of 13 MHz have been observed in high-purity samples using laserspectroscopy [11]. However, while much recent progress has been made towards n the indistinguishability of Raman photons Q , small-mode-volume cavities in diamond [50, 51], it seemsunrealistic to expect extremely large factors for total spontaneous emission lifetimemodification as have been realized in quantum dots. Apart from technological issues,the main difficulty in this system is that only ∼
4% of the spontaneous emission isinto the zero-phonon line, and only this emission can be efficiently coupled to a high- Q cavity. This amounts to a factor of 25 penalty in converting the Purcell factorinto total spontaneous emission lifetime modification. Thus, the best approach toobtain a photon indistinguishability above 0 . r → e process. Since the excited-state levelsplittings near an anticrossing are only ∼
200 MHz [49], there is not much room for largedetunings. Furthermore, the most problematic excited-state dephasing process in thissystem seems to be a slow spectral diffusion process related to charge trap fluctuationsas was mentioned above. For these reasons, and because the spontaneous emissionlifetime (12 ns) is already rather long, the best performance can probably be obtainedby operating the system on resonance.
7. Conclusion
In summary, we have shown that if photon indistinguishability is limited by excited-state dephasing, it can in some cases be drastically improved using a three-level Ramanscheme with a large detuning. The dependence of the photon indistinguishabilityon the noise fluctuation rate was examined, and it was shown that a large detuningprovides an improvement when the noise fluctuation rate is fast compared with thephoton emission rate. To realize this benefit it is usually necessary to use an internal orexternal spectral filter to remove a weak spontaneous emission component at the naturaltransition frequency. We have also considered the practical application of this scheme tosolid-state systems such as semiconductor quantum dots and diamond nitrogen-vacancycenters. Due to additional mechanisms which degrade the photon indistinguishability,a large detuning appears most helpful in the quantum-dot case where large Purcellenhancements can be achieved. With this approach, we are optimistic that solid-state single-photon sources can be improved to a level where they will be truly usefulfor quantum network applications such as creating distributed entanglement betweenmatter qubits.This material is based upon work supported by the Defense Advanced ResearchProjects Agency under Award No. HR0011-09-1-0006 and The Regents of the Universityof California. [1] E. Knill, R. Laflamme, and G. J. Millburn. A scheme for efficient quantum computation withlinear optics.
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