aa r X i v : . [ qu a n t - ph ] O c t Quantum optical waveform conversion
D. Kielpinski , , J. F. Corney , and H. M. Wiseman , ARC Centre of Excellence for Coherent X-Ray Science ARC Centre of Excellence for Quantum Computer Technology Centre for Quantum Dynamics, Griffith University, Nathan QLD 4111, Australia ARC Centre of Excellence for Quantum-Atom Optics, School of Mathematics and Physics,University of Queensland, St. Lucia QLD 4072, Australia
Currently proposed architectures for long-distance quantum communication rely on networks ofquantum processors connected by optical communications channels [1, 2]. The key resource forsuch networks is the entanglement of matter-based quantum systems with quantum optical fieldsfor information transmission. The optical interaction bandwidth of these material systems is atiny fraction of that available for optical communication, and the temporal shape of the quantumoptical output pulse is often poorly suited for long-distance transmission. Here we demonstrate thatnonlinear mixing of a quantum light pulse with a spectrally tailored classical field can compressthe quantum pulse by more than a factor of 100 and flexibly reshape its temporal waveform, whilepreserving all quantum properties, including entanglement. Waveform conversion can be used withheralded arrays of quantum light emitters to enable quantum communication at the full data rate ofoptical telecommunications.
The development of long-distance quantum communication is critical for future quantum cryptography and dis-tributed quantum computing applications. Current fiber-optical quantum communication systems rely on directtransmission of quantum light pulses, but the attenuation of the fiber imposes a distance limit of tens of kilometersfor this kind of quantum communication [3] by virtue of the no-cloning theorem [4]. Quantum repeater architectures[1, 2] promise to circumvent this limit by preparing entangled states over an optical communications channel andstoring these entangled states as a resource for subsequent quantum communication. Components of quantumrepeaters have now been demonstrated with a wide variety of physical systems, including single atoms [5–7], atomicvapors [8–10], rare-earth ions in solids [11, 12], quantum dots [13], and NV-centres [14]. The common thread amongthese demonstrations is the manipulation of quantum light pulses by matter-based quantum emitters. The temporalwaveform of such emitters is typically a single-sided exponential with decay constant on the order of 1 nanosecond,which cannot be readily mode-matched to the smooth, broadband pulses desirable for telecommunications, posinga substantial disadvantage for quantum-emitter approaches to quantum networking. Recent attempts to overcomethese issues include increasing the emitter bandwidth to the GHz range by the use of nonresonant interactions[15], shaping the temporal waveform by placing the emitter in a nonlinear resonator [16], temporal modulation ofsingle-photon wavepackets [17, 18], and nonlinear frequency conversion experiments with single photons [19–22].Here we present an efficient and straightforward method of quantum optical pulse shaping and compression that im-mensely simplifies the interfacing of quantum emitters with telecommunications networks. Our method preserves thefull quantum statistics of the input field, including entanglement and any other multimode correlations, while enablingcompression by more than a factor of 100, along with flexible reshaping of the temporal waveform. In particular, ourmethod enables time/wavelength transduction of spontaneously emitted photons from quantum emitters into short,smooth pulses at telecommunications wavelengths. As shown in Fig. 1, the input field undergoes three-wave mixing(3WM) with a frequency-chirped classical laser pulse. For an appropriate choice of classical laser intensity and chirp,the 3WM product radiation has the same spectrum as the desired target mode, but receives the quantum statisticsof the input. The 3WM output is then dechirped with a second pulse shaper to match the temporal wavefunction ofthe target mode. The waveform converter extends the classical time-lens technique, which has achieved remarkableresults in compressing and stretching classical pulses [23, 24], to the quantum domain and to arbitrary pulse reshaping.We describe the 3WM process using slowly varying bosonic field operators Ψ j ( z, t ) with j = 1 the input mode and j = 2 the mode generated by 3WM. Here z measures the distance along the propagation axis in a frame comoving atthe group velocity, and t measures the duration of the interaction between the three fields. (This coordinate conventionis to be contrasted with the classical nonlinear optics convention in which z measures the interaction length in the3WM medium and t is the time of arrival at the detector.) The escort laser pulse contains & photons and canbe approximated as a classical field that remains unaffected by 3WM. For an escort pulse much longer than the other escort laser esc ν pulseshaperquantum emitter sp ν out ν pulseshaper3WMcrystalinput output target FIG. 1: Schematic of the quantum optical waveform converter. A nonclassical light input (originating, e.g., from a quantumemitter in a high-finesse optical resonator) is combined with a highly chirped classical pulse. The combined fields undergothree-wave mixing (3WM) in a nonlinear crystal, transferring the spectral modulation of the classical pulse onto the 3WMoutput. The output is separated from the original fields and passed through a pulse shaper to remove the residual phase,producing a pulse in the target mode that inherits the quantum state of the input mode. The colours under the pulse enveloperepresent the frequency variation during the pulse length, with the variation of colours greatly exaggerated for clarity. pulses, the Hamiltonian becomes (see Methods) H = i ~ Ω Z dz e iφ ( z ) Ψ † Ψ + h.c. (1)where Ω is determined by the nonlinear coupling constant and the intensity of the escort pulse, and where φ ( z ) is thephase of the escort field. The quantum field operators evolve asΨ ( z, t ) = cos Ω t Ψ ( z,
0) + sin Ω t e iφ ( z ) Ψ ( z,
0) (2)Ψ ( z, t ) = − sin Ω t e − iφ ( z ) Ψ ( z,
0) + cos Ω t Ψ ( z,
0) (3)If the fields leave the 3WM medium after an interaction time T = π/ (2Ω), the solution for mode 2 at times t > T isjust Ψ ( z, T ) = − e − iφ ( z ) Ψ ( z, − φ ( z ) + π from mode 3. To temporally match the 3WM output to the target pulse shape,the output pulse shaper then removes the undesired relative phases of the spectral components, performing a unitarytransformation on the output field operator. The spontaneous emission from a quantum emitter, with a single-sidedexponential waveform, can be converted into a much shorter Gaussian pulse by choosing (see Methods) φ ( z ) = √ σ Z z dζ erf − ( e − ζ/ ( cτ ) ) (4)where τ is the spontaneous emission lifetime and the target amplitude is proportional to e − z / (2( cσ ) ) . The phasemodulation of Eq. (4) is visualised in Fig. 2.The ideal quantum waveform conversion described above will not be achieved with unit fidelity in real 3WM mediabecause of dispersion. We now show that the fidelity F can nevertheless exceed 99.9% for readily achievable experi-mental parameters. For pure input states, F = |h ψ ideal | ψ disp i| , where the result of ideal evolution is written | ψ ideal i and the result with dispersion included is | ψ disp i . If the input system is entangled with another quantum system,the fidelity of the final entangled state is simply the average fidelity of the eigenstates of the input density operator,weighted by their corresponding eigenvalues. A perturbative analysis of the dispersive evolution (see Methods) showsthat the error is dominated by mismatch between the group velocities v , , of the input, output, and escort fields,which we parametrise by v = ( v − v ) / v e = v − ( v + v ) /
2. For a pure input wavefunction A ( z ) withcharacteristic length scale L , we define dimensionless velocities u = v/v , u e = v e /v , where v = Ω L/ (2 π ). The errorcan be minimised by adding the compensation phase∆ opt ( z ) = 18 ( u e − u ) φ ′ ( z ) L (5) (a) (b) =στ φ a m p li t ude τ cz / τ cz / FIG. 2: a) Escort phase modulation function φ ( z ) for conversion from a single-sided exponential waveform to a Gaussianwaveform with compression ratio τ /σ = 100. b) Visualisation of the spectral chirp dφ ( z ) /dz imposed on the initial waveformby 3WM. Height indicates waveform amplitude, colour indicates local frequency after escort phase imprinting. The hue of thecolour is proportional to dφ ( z ) /dz as calculated from Eq. (4). to the initial escort phase φ ( z ). For an average photon occupation h n i , we then obtain1 − F opt = h n i u L π Z dz (cid:12)(cid:12) A ′ ( z ) − i (1 + u e /u ) φ ′ ( z ) A ( z ) (cid:12)(cid:12) (6)The ratio u e /u is set by the crystal dispersion alone, but u varies with the escort laser intensity I esc as u ∝ I − / .Thus the fidelity can be made arbitrarily close to 1 by increasing the escort laser power. Eq. (6) can then be rewrittenas 1 − F opt = ( u/u err ) for some u err .
1. It can be seen from Eq. (6) that the perturbation theory breaks down fora pulse with an arbitrarily sharp leading edge; a perturbative analysis in momentum space shows that 1 − F opt ∝ u in this limit. In practice, the time required to excite a quantum emitter is never exactly zero, so the leading edge ofthe pulse is smoothed over the excitation timescale. We also perform a full numerical simulation for a single-modesingle-photon input state (see Methods). This confirms that the effect of group-velocity dispersion is insignificant inall cases of interest.Figure 3 shows the analytic and numerical error estimates for two cases of particular experimental interest. Case 1:the conversion of 370 nm photons from a Yb + ion [5] to the 1550 nm telecommunications band using periodicallypoled lithium niobate, for which u e /u ≈ − /
3. The simulated error closely follows the perturbative result for smallvalues of u up to a compression ratio τ /σ = 200. For an escort laser pulse of energy ∼ µ J and duration of 150 ns( > τ Yb + ) in a 50 mm long crystal waveguide, one finds u = 0 .
013 and error of 1 − F = 7 × − at compressionratio of 100. Case 2: the conversion of 780 nm photons from a Rb atom to the telecommunications band, for whichone can arrange u e /u = − v < m / s for any choice of output wavelengths in the telecommunicationsband, so u < − and 1 − F ≪ − .We have shown that three-wave mixing with a modulated classical field can reshape and compress the waveform ofa quantum light pulse while faithfully maintaining the quantum information carried by the photons. A quantumlight pulse produced by a quantum emitter with a lifetime of nanoseconds can be converted to a Gaussian pulsewith a duration of tens of picoseconds that is compatible with standard telecommunications protocols. The lowerror of waveform conversion is compatible with schemes for fault-tolerant quantum communication over longdistances [1]. Quantum waveform conversion enables simultaneous time- and wavelength-division multiplexing ofthe pulses from an array of quantum emitters up to the limit of channel capacity, massively increasing quantumcommunications bandwidth. Current DWDM systems with 50 GHz channel spacing achieve their maximum capacityfor transform-limited pulses with ∼
20 ps duration, while the dispersive effects of long-haul fibre transmission requirethe pulses to have a smooth temporal waveform. As each pulse arrives from the emitter array, it can be simultaneouslyconverted to this ideal waveform and sorted into an appropriate DWDM channels. The rate of entangled pair genera-tion in a quantum network is then limited only by the telecommunications bandwidth and the size of the emitter array.
Methods
Quantum 3WM Hamiltonian
We analyze the 3WM process using the canonical quantization method [26–28]. For simplicity,the electric field polarisation vectors are assumed to lie along the 3WM crystal axes, as for a periodically poled or otherwise æ æ æ æ æ æ æ æ æ æ µ - µ - µ - µ - µ - æ æ æ µ - æ æææææææ - F u e rr (a) (b)(c) (d) =στ æ æ æ æ æ æ æ µ - µ - µ - µ - µ - æææææææ - F στ / u e rr (c) (d) u =στ FIG. 3: Error induced by quantum waveform conversion from a single-sided exponential pulse of time constant τ and rise time0 . τ to a Gaussian pulse of 1 /e time constant σ . Errors are less than 10 − for readily achievable experimental parameters(see text). a) Error at compression ratio τ /σ = 100 as a function of dimensionless group-velocity mismatch u for conversionof 370 nm photons to 1550 nm in lithium niobate. Solid line: perturbative prediction. Points: simulation results. Dashedline: Best fit of 1 − F = ( u/u err ) to simulation points. b) Error scale u err as a function of compression ratio τ /σ . Solid line:perturbative prediction. Points: simulated values computed from least-squares fits to simulation results. As expected, thesimulation results match well to the perturbative theory. c), d) are the same as a), b), but for conversion of 780 nm photons to1550 nm in type-II matched lithium niobate, near the special point v = − v e at which the escort phase term in Eq. (6) vanishes.Errors are even lower than for conversion of 370 nm photons, but at compression ratio above 10 the perturbation theory breaksdown and higher-order GVM dominates the error.noncritically phase-matched crystal. We assume conservation of momentum and energy for the carrier waves and retain onlyphase-matched processes. The χ (2) nonlinear Hamiltonian is then H = i ~ α Z dz Ψ † Ψ Ψ + h.c. (7)where the coupling constant α is determined by the material nonlinear susceptibility and the beam geometry and is readily calcu-lated in the classical limit. The escort field is taken to be a classical field of constant intensity that is phase-modulated to impartthe desired spectral modulation to mode 2. We write Ψ = ξ exp[ iφ ( z + v e t )], where ξ is the (real, positive) amplitude of theclassical escort field and φ ( z + v e t ) is the local phase. With the definition Ω ≡ αξ , Eq. (7) reduces to Eq. (1) for the case v e = 0. Phase functions for waveform shaping
For simplicity, we assume that the input mode and the desired target mode are both transform-limited. To match the powerspectrum of the 3WM product to the desired spectrum, φ ( z ) should satisfy | ˜ α ( k ) | = (cid:12)(cid:12)(cid:12)R dz α ( z ) e iφ ( z ) e − ikz (cid:12)(cid:12)(cid:12) , where ˜ α ( k ) isthe Fourier transform of α ( z ). In general, one can satisfy this constraint by numerical least-squares minimization. However,when the input and target bandwidths differ substantially, the method of stationary phase applies to the integral and φ ( z ) hasa closed-form solution in this limit when the target is Gaussian. Writing ˜ α ( k ) ∝ e − k / (2 σ ) , we find φ ( z ) ≈ √ σ Z z −∞ dζ erf − (cid:20) a + b Z ζ dζ α ( ζ ) (cid:21) (8)where erf is the error function, f − ( x ) = y denotes the solution of f ( y ) = x , and the constants a, b are set by the boundaryconditions of the transformation. After 3WM, the phase of ˜ α ( k ) is nontrivial and the 3WM product pulse is therefore nottransform-limited. The output pulse shaper applies a spectral compensation phase γ ( k ), implementing the unitary transforma-tion Ψ out ( z ) = 12 π Z dz Ψ ( z, t = T ) Z dk e iγ ( k ) e ik ( ζ − z ) (9) Choosing γ ( k ) = − φ (cid:18) α − (cid:18) √ b e − k /σ (cid:19)(cid:19) (10)removes the unwanted phase, so that the output pulse is transform-limited with the desired spectrum. When α ( z ) ∝ e − z / (2 µ ) is also Gaussian, equations (8) and (10) reduce to φ ( z ) = σz / (2 µ ) and γ ( k ) = − µk / (2 σ ), while for a single-sided exponentialthe solution is that given in Eq. (4). Dispersive evolution and error in state transfer
Errors in the state transfer arise from group-velocity mismatch between the three modes in the 3WM medium. In the comovingframe with velocity ¯ v = ( v + v ) /
2, the quantum Hamiltonian for group-velocity mismatch (GVM) and group-velocity dispersion(GVD) can be written as [29] H disp = X j =1 , Z dz " i ~ v j ∂ Ψ † j ∂z Ψ j + β j ∂ Ψ † j ∂z ∂ Ψ j ∂z + h.c. (11)while the 3WM Hamiltonian (1) is also modified because the phase φ ( z ) of the escort field phase evolves under dispersion (seeSupplementary Discussion). Moving to the interaction picture with respect to the original 3WM Hamiltonian (1), one derivesthe additional unitary evolution due to dispersive effects, U disp . A second-order Dyson series solution for U disp shows thatGVM mixes the vacuum noise of the initially unoccupied mode 2 into the state transfer, while GVD has a negligible effect.The removal of phase by the output pulse shaper just implements a unitary transformation on the 3WM output field, whichhas no effect on the fidelity. The fidelity is then evaluated as F = (cid:12)(cid:12) h U disp i (cid:12)(cid:12) , where the expectation value is taken with respectto the initial states of modes 1 and 2 and any other systems entangled with mode 1. Compensating the phase according to Eq.(5) is found to minimise the error independently of the input state. Taking an initial pure state in mode 1 and the vacuumstate in mode 2, we obtain Eq. (6). Numerical simulations of error in state transfer
The Heisenberg equations of motion for the field operators Ψ , Ψ are linear, so the operator of the target field after time t will be a linear combination of the initial field operators. If the input pulse has a single spatial mode A ( z ), such that | ψ (0) i = f [ a † ] | i = f hR dxA ( z )Ψ † ( z ) i | i = P n c n | n i , the quantum state at time t is | ψ (0) i = f (cid:20)Z dx n A ( z, t )Ψ † ( x ) + A ( z, t )Ψ † ( x ) o(cid:21) | i , (12)with A ( z,
0) = A ( z ) and A ( z,
0) = 0. The A n obey the same linear equations as Ψ n , but are c -number amplitudes ratherthan operators. Simulating these equations allows us to calculate the fidelity as F = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n | c n | (cid:20)Z dzA ∗ ( z ) e − iφ ( z ) A ( z, T ) (cid:21) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (13)For fidelities close to unity and a correctly compensated phase, this reproduces the linear dependence on h n i found in theperturbative calculation (6). For definiteness, we only show results for a single-photon input state. Acknowledgments
This work was supported by the Australian Research Council under DP0773354 (Kielpinski), CE0348250 (Wiseman),FF0458313 (Wiseman), and CE0348178 (Corney). We thank Geoff Pryde for helpful conversations.
Author contributions
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Dispersive evolution
Error in the state transfer arises from dispersion in the 3WM evolution. To calculate the error, we treat all dispersion-related terms as perturbations to the nondispersive 3WM Hamiltonian H ≡ i ~ Ω Z dz Ψ † ( z, t )Ψ ( z, t ) e iφ ( z ) + h.c. (14)which induces ideal state transfer by the unitary evolution U ( T ) over the time T = π/ (2Ω). Mode 2 is assumed tobe initially unoccupied throughout the calculation, as for the applications discussed in the main paper.We consider dispersion up to second order, i.e., group velocity mismatch (GVM) between modes 1, 2, and 3, andgroup velocity dispersion (GVD) within each of these modes. In nondegenerate mixing, as considered here, GVM hasmuch larger effects than GVD. The GVD and GVM parameters are expressed using the variables v = ( v − v ) / v e = v − ( v + v ) / v i = dω i /dk i ) and β i = d ω i /dk i for GVD. We define ǫ ≪ v, v e ∼ O ( ǫ ), β i ∼ O ( ǫ ). The dispersion of the quantum modes 1 and 2 is describedby the Hamiltonian H disp ≡ i ~ v Z dz " ∂ Ψ † ( z, t ) ∂z Ψ ( z, t ) − ∂ Ψ † ( z, t ) ∂z Ψ ( z, t ) + h.c. (15)+ ~ Z dz " β ∂ Ψ † ( z, t ) ∂z Ψ ( z, t ) + β ∂ Ψ † ( z, t ) ∂z Ψ ( z, t ) + h.c. (16)The dispersion of the escort laser (mode 3) affects the quantum state transfer indirectly through the 3WM Hamiltonian.At O ( ǫ ), the escort mode is governed by the evolution equation d Ψ dt = − v e d Ψ dz + iβ d Ψ dz (17)As will be seen, the error can be minimised by precompensation of the phase function. For an escort pulse of constantintensity, Ψ ∝ e − iχ ( z,t ) , we write the compensated phase function at t = 0 as χ ( z,
0) = φ ( z ) + ∆( z ). The solution ofEq. (17) is χ ( z, t ) = φ ( z − v e t ) + ∆( z − v e t ) − β t (cid:2) ( φ ′ ( z )) − iφ ′′ ( z ) (cid:3) (18)where the prime applied to functions (as in φ ′ ) indicates differentiation with respect to z . Including the effect ofdispersion, the 3WM Hamiltonian becomes H ( d )3WM ≡ H + (cid:20) i ~ Ω Z dz Ψ † ( z, t )Ψ ( z, t ) (cid:16) e iχ ( z,t ) − e iφ ( z ) (cid:17) + h.c. (cid:21) (19)We go to an interaction frame with respect to H to obtain the perturbation Hamiltonian V I ( t ) ≡ U † ( H disp + H ( d )3WM − H ) U ≈ V ( t ) + V ( t ) (20)where V ( t ) = O ( ǫ ) and V ( t ) = O ( ǫ ). The Dyson series gives the unitary operator U I ( t ) describing the perturbedevolution in the interaction frame: U I ( t ) = 1 − i ~ Z t dt V I ( t ) − ~ Z t dt ′ Z t ′ dt ′′ V I ( t ′ ) V I ( t ′′ ) + . . . (21)Expanding the Dyson series to O ( ǫ ) gives U eff ( t ) ≈ U ( t ) + U ( t ) + U ( t ) (22) U ( t ) ≡ − i ~ Z t dt V ( t ) U ( t ) ≡ − i ~ Z t dt V ( t ) U ( t ) ≡ − ~ Z t dt ′ Z t ′ dt ′′ V ( t ′ ) V ( t ′′ ) (23)It can be shown that V and V are both Hermitian, so the expectation values of U ( T ) and U ( T ) are purely imagi-nary. U ( T ) contributes to the error only as O ( v , v e ) ∼ O ( ǫ ), showing that we must include U ( T ) in a consistentperturbative expansion of the error to lowest order in GVM. Moreover, U ( T ) contributes only as O ( β i ) ∼ O ( ǫ ).We consider only error terms up to O ( ǫ ), so we discard U ( T ). Hence GVD has no effect on the error in this analysis.The solution of the nondispersive Hamiltonian lets us express Ψ ( z, t ) and Ψ ( z, t ) in terms of Φ( z ) ≡ Ψ ( z, t = 0)and Υ( z ) ≡ Ψ ( z, t = 0), giving V ( t ) = − ~ Z dz (cid:20)(cid:0) Ω sin 2Ω t ∆ + (cid:0) v sin Ω t − v e Ω t sin 2Ω t (cid:1) φ ′ (cid:1) Φ † Φ − iv cos 2Ω t d Φ † dz Φ − e iφ cos 2Ω t (∆ − v e tφ ′ ) Φ † Υ + ie iφ v sin 2Ω t (cid:18) d Φ † dz Υ − Φ † d Υ dz (cid:19)(cid:21) + h.c. (24)Since V ( t ) is normally ordered and mode 2 is initially unoccupied, terms that involve only mode 2 do not affect thefidelity and are omitted from the expressions. We wish to evaluate the error at the end of the state transfer, whichoccurs at time t = T ≡ π/ (2Ω), giving U ( T ) = − i Z dz h ( z )Φ † ( z )Φ( z ) (25) h ( z ) ≡ z ) + π ( v − v e ) φ ′ ( z ) (26)so that ∆ ∼ O ( ǫ ) or higher if ∆ is to minimise the error in state transfer. The remaining term in U eff ( T ) is computedto be U ( T ) = 164Ω Z dz dz (cid:8) − h ( z ) h ( z )Φ † ( z )Φ( z )Φ † ( z )Φ( z ) − v e cos [ φ ( z ) − φ ( z )] φ ′ ( z ) φ ′ ( z )Φ † ( z )Υ( z )Υ † ( z )Φ( z ) − ive i ( φ ( z ) − φ ( z )) (cid:20) g + ( z )Φ † ( z )Υ( z ) (cid:18) d Υ † ( z ) dz Φ( z ) − Υ † ( z ) d Φ( z ) dz (cid:19) − g − ( z ) (cid:18) d Φ † ( z ) dz Υ( z ) − Φ † ( z ) d Υ( z ) dz (cid:19) Υ † ( z )Φ( z ) (cid:21) − v e i ( φ ( z ) − φ ( z )) (cid:20) d Φ † ( z ) dz Υ( z )Υ † ( z ) d Φ( z ) dz + Φ † ( z ) d Υ( z ) dz d Υ † ( z ) dz Φ( z ) − Φ † ( z ) d Υ( z ) dz Υ † ( z ) d Φ( z ) dz − d Φ † ( z ) dz Υ( z ) d Υ † ( z ) dz Φ( z ) (cid:21)(cid:27) (27)Here g ± ( z ) ≡ π Ω∆( z ) − ( π ± v e φ ′ ( z ) and we have eliminated terms with purely imaginary expectation values, asthese terms do not contribute to the error at O ( ǫ ). Fidelity calculation
We quantify the error in the state transfer by computing the fidelity F between the actual output state of the waveformconverter and the ideal dispersion-free output state. The final pulse shaping just implements a unitary transformationon the 3WM output field, which has no effect on the fidelity. For many applications one wishes to convert entangledstates involving both mode 1 and some other modes, so the full state before waveform conversion takes the form | Ψ i i = X j α j | ψ ( j ) i i ⊗ | χ ( j ) i (28)where the | ψ ( j ) i i are orthonormal states of mode 1 and the | χ ( j ) i are orthonormal states over the other systems.Writing U full ( T ) as the unitary evolution under the full dispersive Hamiltonian, we have F = X j | α j | (cid:12)(cid:12)(cid:12)D ψ ( j ) i (cid:12)(cid:12)(cid:12) U † full U (cid:12)(cid:12)(cid:12) ψ ( j ) i E(cid:12)(cid:12)(cid:12) (29)Thus, since we can always find the transfer fidelity for entangled input states by taking an appropriate weighted sumover pure-state fidelities, we need only calculate the transfer fidelity of a pure input state | ψ i i . In many cases, such astate is characterised by a mode creation operator a † = Z dz A ( z )Φ † ( z ) (30)where Φ( z ) = Ψ ( z, t = 0) and the mode wavefunction A ( z ) is normalised as R dz | A ( z ) | = 1. An initial k -photonnumber state in the mode is given by ( a † ) k | i and a general single-mode initial pure state | ψ i i = P ∞ k =0 c k | k i canbe written as | ψ i i = f ( a † ) | i , where | k i denotes a number state and f ( x ) is defined through the series expansion f ( x ) = P ∞ k =0 c k x k / √ k !.The pure-state fidelity is given by the squared overlap between the state obtained from phase-compensated dispersiveevolution and that obtained from uncompensated nondispersive evolution. In an interaction frame with respect tothe nondispersive 3WM Hamiltonian, we have F = (cid:12)(cid:12)(cid:12)D ψ i (cid:12)(cid:12)(cid:12) U † eff ( T ) (cid:12)(cid:12)(cid:12) ψ i E(cid:12)(cid:12)(cid:12) (31)and using the matrix elements derived in the Appendix, we find h ψ i | U ( T ) | ψ i i = i h n i Z dz h ( z ) | A ( z ) | (32) h ψ i | U ( T ) | ψ i i = 164Ω Z dz dz (cid:8) − h n ( n − i h ( z ) h ( z ) | A ( z ) | | A ( z ) | + 2 h n i h ( z ) h ( z ) A ∗ ( z ) A ( z ) δ ( z − z ) − h n i v e cos [ φ ( z ) − φ ( z )] φ ′ ( z ) φ ′ ( z ) A ∗ ( z ) A ( z ) δ ( z − z ) − i h n i ve i ( φ ( z ) − φ ( z )) [ g + ( z ) ( A ∗ ( z ) A ( z ) ∂ z δ ( z − z ) − A ∗ ( z ) A ′ ( z ) δ ( z − z )) − g − ( z ) ( A ′∗ ( z ) A ( z ) δ ( z − z ) − A ∗ ( z ) A ( z ) ∂ z δ ( z − z ))] − h n i v e i ( φ ( z ) − φ ( z )) [ A ′∗ ( z ) A ′ ( z ) δ ( z − z ) + A ∗ ( z ) A ( z ) ∂ z ∂ z δ ( z − z ) − A ∗ ( z ) A ′ ( z ) ∂ z δ ( z − z ) − A ′∗ ( z ) A ( z ) ∂ z δ ( z − z )] } (33)= 164Ω ( − h n ( n − i (cid:18)Z dz h | A | (cid:19) + 2 h n i Z dz h | A | − h n i v e Z dz | φ ′ A | +8 h n i vv e Z dz φ ′ [ A ∗ ( iA ′ + φ ′ A ) + h.c.] − h n i v Z dz | A ′ − iφ ′ A | (cid:27) (34)= 132Ω ( −h n ( n − i (cid:18)Z dz h | A | (cid:19) + h n i Z dz h | A | − h n i Z dz (cid:12)(cid:12) vA ′ − i ( v + v e ) φ ′ A (cid:12)(cid:12) ) (35)where n is the number operator of the mode, h n i denotes the expectation value of n , and we have retained only realterms. The fidelity is optimised when h ( z ) = 0 so that∆ opt ( z ) = π ( v e − v ) φ ′ ( z )4Ω (36) F opt = 1 − h n i Z dz (cid:12)(cid:12) vA ′ ( z ) − i ( v + v e ) φ ′ ( z ) A ( z ) (cid:12)(cid:12) (37)which is just Eq. (6).0 Appendix: Matrix elements for pure states
To evaluate the matrix elements involved in the fidelity calculation, we first observe that[Φ( z ) , a † ] = Z dζ A ( ζ ) [Φ( z ) , Φ † ( ζ )] = Z dζ A ( ζ ) δ ( ζ − z ) = A ( z ) (38)[ a, a † ] = Z dζ A ∗ ( ζ ) [Φ( ζ ) , a † ] = Z dζ | A ( ζ ) | = 1 (39)[Φ( z ) , ( a † ) k ] = kA ( z )( a † ) k − (40) (cid:10) (cid:12)(cid:12) a j ( a † ) k (cid:12)(cid:12) (cid:11) = k ! δ jk (41)Writing h n i for the expectation value of the photon number in | ψ i i , we find M ≡ (cid:10) ψ i (cid:12)(cid:12) Φ † ( z )Φ( z ) (cid:12)(cid:12) ψ i (cid:11) = ∞ X k =0 ∞ X j =0 c ∗ k c j √ k ! j ! (cid:10) (cid:12)(cid:12) a j Φ † ( z )Φ( z )( a † ) k (cid:12)(cid:12) (cid:11) (42)= A ∗ ( z ) A ( z ) ∞ X k =0 | c k | k ! k (cid:10) (cid:12)(cid:12) a k − ( a † ) k − (cid:12)(cid:12) (cid:11) (43)= A ∗ ( z ) A ( z ) ∞ X k =0 k | c k | (44)= h n i A ∗ ( z ) A ( z ) (45) M ≡ (cid:10) ψ i (cid:12)(cid:12) Φ † ( z )Φ( z )Φ † ( z )Φ( z ) (cid:12)(cid:12) ψ i (cid:11) = ∞ X k =0 ∞ X j =0 c ∗ k c j √ k ! j ! (cid:10) (cid:12)(cid:12) a j Φ † ( z )Φ( z )Φ † ( z )Φ( z )( a † ) k (cid:12)(cid:12) (cid:11) (46)= A ∗ ( z ) A ( z ) ∞ X k =0 ∞ X j =0 c ∗ k c j √ k ! j ! jk (cid:10) (cid:12)(cid:12) a j − Φ( z )Φ † ( z )( a † ) k − (cid:12)(cid:12) (cid:11) (47)= A ∗ ( z ) A ( z ) ∞ X k =0 ∞ X j =0 c ∗ k c j √ k ! j ! jk (cid:10) (cid:12)(cid:12) a j − (cid:0) Φ † ( z )Φ( z ) + δ ( z − z ) (cid:1) ( a † ) k − (cid:12)(cid:12) (cid:11) (48)= A ∗ ( z ) A ( z ) ∞ X k =0 | c k | k ! k (cid:0) ( k − ( k − A ∗ ( z ) A ( z ) + δ ( z − z )( k − (cid:1) (49)= h n ( n − i A ∗ ( z ) A ( z ) A ∗ ( z ) A ( z ) + h n i A ∗ ( z ) A ( z ) δ ( z − z ) (50)Then (cid:28) ψ i (cid:12)(cid:12)(cid:12)(cid:12) Φ † ( z ) d Φ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ψ i (cid:29) = ∂ z M = h n i A ∗ ( z ) A ′ ( z ) (51) (cid:28) ψ i (cid:12)(cid:12)(cid:12)(cid:12) d Φ † ( z ) dz ! d Φ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ψ i (cid:29) = ∂ z ∂ z M = h n i ( A ∗ ) ′ ( z ) A ′ ( z ) (52) (cid:28) ψ i (cid:12)(cid:12)(cid:12)(cid:12) Φ † ( z )Φ( z )Φ † ( z ) d Φ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) ψ i (cid:29) = ∂ z M = h n ( n − i A ∗ ( z ) A ( z ) A ∗ ( z ) A ′ ( z ) − h n i A ∗ ( z ) A ′ ( z ) δ ( z − z )(53)and similarly for the other matrix elements involving mode 1. For mode 2, which is initially unoccupied, we calculate M ≡ (cid:10) (cid:12)(cid:12) Υ( z )Υ † ( z ) (cid:12)(cid:12) (cid:11) = δ ( z − z ) (54) (cid:28) (cid:12)(cid:12)(cid:12)(cid:12) d Υ( z ) dz Υ † ( z ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) = ∂ z M = ∂ z δ ( z − z ) (55) (cid:28) (cid:12)(cid:12)(cid:12)(cid:12) d Υ( z ) dz d Υ † ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) = ∂ z ∂ z δ ( z − z2