JJ J - arXiv

Optical source of individual pairs of color-conjugated photons Yury Sherkunov National Graphene Institute, University of Manchester, Manchester, M13 9PL, United Kingdom

David M. Whittaker Department of Physics and Astronomy, University of Sheffield, Sheffield, S10 2TN, United Kingdom

Vladimir I. Fal’ko

arXiv:1608.07213v1 [quant-ph] 25 Aug 2016

School of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, United Kingdom and National Graphene Institute, University of Manchester, Manchester, M13 9PL, United Kingdom (Dated: August 26, 2016) We demonstrate that Kerr nonlinearity in optical circuits can lead to both resonant four-wave mixing and photon blockade, which can be used for high-yield generation of high-fidelity individual photon pairs with conjugated frequencies. We propose an optical circuit, which, in the optimal pulsed-drive regime, would produce photon pairs at the rate up to 105 s−1 (0.5 pairs per pulse) with g (2) < 10−2 for one of the conjugated frequencies. We show that such a scheme can be utilised to generate color-entangled photons.

The use of individual photons1 is one of the key elements in the implementation of quantum technologies in communications security2,3 and quantum computation4 , which stimulated a great progress in designing solid state single-photon sources5–7 . Further advancement of quantum security protocols is expected to benefit from the use of pairs of correlated photons, such as generated by the bi-exciton decay8 , spontaneous parametric down-conversion in nonlinear crystals9–13 , or four-wave mixing14–19 . To guarantee that only one photon is produced in each of the two conjugated modes, low pumping intensities had to be used in all of the above methods, leading to a low output of photon pairs10–18 . Here, we propose an optical circuit where the resonant four-wave mixing and photon blockade, both due to the same Kerr nonlinearity, create conditions for a high-yield generation of individual pairs of photons with conjugated frequencies. Moreover, for the pulsed pumping of such a circuit, the Rabi oscillations between different two-photon states can be used to optimise the output of such pairs, and we propose a scheme for using the optimally driven circuit as a source of color-entangled photon pairs20 . Spontaneous four-wave mixing (FWM) is a process that converts two photons from a coherent light source into a pair of photons with up- and down-shifted frequencies. In fibers and waveguides, such a process generates two-photon states, and additional challenge in developing devices suitable for the generation of correlated photon pairs with high fidelity and high efficiency is related to the noise due to the multi-photon generation1 associated with the increasing excitation power required for a high output of the device. At the same time, Kerr-type nonlinearity in selectively tuned microcavity-resonators may result in emission of photon pairs in spectrally well-defined modes21 , with the resonant conditions set by the photon blockade21,22 . Hence, we propose an optical circuit design where nonlinear coupling of three photon modes with conjugated frequencies ω± and ω0 ≈ 21 (ω+ + ω− ) can be used for the

J

J

J' FIG. 1. Nonlinear optical circuit with high-Q nonlinear resonators (two with frequency Ω and one with frequency Ω0 = Ω + J × δ, coupled by hoppings J and J 0 = Jx).

resonant excitation of pure two-photon pairs with optimised high-yield output, followed by the Rabi-type mixing of the two-photon states |20 , 0+ , 0− i and |00 , 1+ , 1− )i (see in Fig. 2), which might be used to produce entangled states of color-conjugated photon pairs. The proposed circuit is shown in Fig. 1. It can be envisaged as three coupled non-linear cavity resonators, each characterised by a single photonic mode: two with equal frequencies, Ω, and the third with the frequency Ω0 = Ω + J × δ. The two cavities with equal frequencies are coupled with the third cavity by hopping amplitude J and by hopping amplitude J 0 = Jx with each other, as

2 where √

2u κs 6κ , α00 = √ , α±∓ = √ , s 2 2 2s κ = α0± = √ [s ∓ (δ − x)], 2 2 κ = √ [3s2 − 12 ± (δ − x)s]. 4 2s

κ= α±0 α±±

, FIG. 2. Resonant mixing and splitting of N -photon states |n0 , n+ , n− i coupled by the FWM described by the trunˆ (2) . The detuning δ is chosen in such cated Hamiltonian H a way that the energy levels corresponding to the states |n0 + 2, 0+ , 0− i and |n0 , 1+ , 1− i are in resonance, while the energy levels of the states with multiple occupation of the modes ω+ and ω− are red-shifted. Energies of relevant states with N = 0, 1, 2, 3 photons are given in Eq. (9).

described by the Hamiltonian (~ = c = 1): ˆ (0) = Ω0 a† a1 + Ω H 1

X

a†i ai

(1)

i=2,3

+

J 0 a†2 a3

+J

X

a†1 ai

X

+ H.c. ≡

i=2,3

ωk βk† βk ,

a†i a†i ai ai

(3)

κ δ = −3x + 2J



4x2 − 1 √ (2 + n+ + n− ) 1 + 2x2 i √ + 10 2x(n+ − n− ) .

ˆ The latter expression was obtained neglecting terms δ H (2) ˆ in H . This indicates that the resonance conditions for the states involving different occupation numbers n± can be separated, whereas the resonance conditions for the processes |n0 + 2, 0+ , 0− i ⇐⇒ |n0 , 1+ , 1− i generating a single pair of photons at conjugated frequencies ω± ,

(4)

i

δ = −3x +

2   X † † 2 ˆ = κ β+ β− β0 + H.c. + αkk0 βk† βk†0 βk0 βk + δ H, k,k0

(5)

(2)

Then, we take into account Kerr-type nonlinearity, which, for simplicity, will have the same strength, u  J, on each cavity, X

E(n0 + 2, n+ , n− ) = E(n0 , n+ + 1, n− + 1).

In this case, generation of pairs of ω± -photons is promoted by the resonance conditions for converting them from the pairs of pumped ω0 photons. Condition (5) can be obtained by tuning parameter δ in H (0) to the value

Resonators 2 and 3 are driven by a√coherent pump (at frequency ω) with the amplitudes ± 2F , which provides coupling to the mode β0 , ˆ (1) = −F eiωt β0 + H.c. H

† n+ † n− (β0† )n0 +2 (β+ ) (β− ) p |0i, (n0 + 2)!n+ !n− !

are degenerate,

where ai (a†i ) are the annihilation (creation) operators of photons in each resonator. Energies ω0,± correspond to extended eigenmodes of H (0) , 1 β0 = √ (a3 − a2 ) , c± = δ − x ± s, 2 1 β± = q [c± a1 + 2(a3 + a2 )] . 8 + c2±

|n0 + 2, n+ , n− i =

† n+ +1 † n− +1 (β † )n0 (β+ ) (β− ) |n0 , n+ + 1, n− + 1i= p0 |0i n0 !(n+ + 1)!(n− + 1)!

k=0,±

J ω0 = Ω − Jx, ω± = Ω + (δ + x ± s), 2 p 2 s = (δ − x) + 8,

ˆ (2) = u H

ˆ (2) represents interaction between The second line in H the extended modes, β0,± . Here, the first term describes resonant four-wave mixing of two ω0 photons with the pair of photons at the conjugated frequencies, ω± . The second term produces occupancy-dependent shifts in the photon frequencies. The rest of the terms generated by the canonical transformation from the single-cavity to the ˆ extended modes are combined into a perturbation δ H; under conditions which will be identified below, these terms are non-resonant for the production process of the photon pairs with conjugated frequencies, hence, they give only a small contribution (but they will be included in numerical calculations later). The ’resonant’ situation occurs when the two states

4x2 − 1 u , 4x2 + 2 J

(6)

is the same for all values of n0 . This condition sets the

3 values of the parameters in Eqs. (1) and (4) to u ; (7) ω0 = Ω − Jx; κ ≈ √ 2 1 + 2x2   p √ ω± = Ω − x ∓ 2 1 + 2x2 J √  4x2 − 1 p + 2 1 + 2x2 ∓ 2x κ; 4x + 2 √ p 3 + 12x2 ∓ 2x 2 + 4x2 2 √ α00 = 1 + 2x κ; α±± = κ; 4 1 + 2x2 p √ 3κ α±0 = α0± = [ 1 + 2x2 ± 2x]κ; α±∓ = √ . 2 1 + 2x2 To get an idea about spectral properties of this optical circuit under the resonance conditions (6)-(7), we neglect ˆ in H (2) and diagonalise the term δ H ˆ =H ˆ (0) + H ˆ (2) H

(8)

in the basis of the states {|00 , 0+ , 0− i, |10 , 0+ , 0− i, |20 , 0+ , 0− i, |30 , 0+ , 0− i, |00 , 1+ , 1− i, |10 , 1+ , 1− i}. Formally, truncation decouples this subspace of states from the rest of the low-energy spectrum with the same total number of photons. Then, the spectrum of N −photon states is: E(0) = 0, E(1) = ω0 , |0i = |00 , 0+ , 0− i, |1i = |10 , 0+ , 0− i; p √ E(2α) ≈ 2ω0 + 2 + 4x2 ± 1 2κ,

(9)

1 |2αi = √ (|20 , 0+ , 0− i ± |00 , 1+ , 1− )i; 2 p √ E(3α) ≈ 3ω0 + 6 + 12x2 ± 1 6κ, 1 |3αi = √ (|30 , 0+ , 0− i ± |10 , 1+ , 1− i). 2 Here, upper/lower signs correspond to the states A/B among |N αiN =0,1,2,3;α=A,B marked in Fig. 2.

(a)

(b)

12

1.2

0.8

8

0.4

4

0

0

0

1

2 1

8 4 10

100 10-1

0.5 -6 -4 -2 0 2 4 6

0.2

0.8 0.4 0 1.1 1

2 1 0.5 -15 -10 -5

0.9 0

5

10

FIG. 3. Steady state under continuous pumping, as a function of pump frequency ω. (a) γ = κ, F = 2κ; (b) γ = 8κ, F = 10κ.

In a pumped system, absorption of photons would be resonantly favoured when N incident photons have the same energy as N photons in the cavity, N ω = E(N α), and such multi-photon resonances can also be found in

weakly dissipating systems. The evolution of a quantum optical circuit, where photons experience decay at the rate γ, can be described using the master equation, X ∂ ρˆ ˆ ρˆ] + γ = −i[H, (2βi ρˆβi† − βi† βi ρˆ − ρˆβi† βi ), (10) ∂t i=0,1,2 for the density matrix, ρ, which we write in a Fock basis X ρˆ = ρ(m0 , m+ , m− ; n0 , n+ , n− ) |m0 , m+ , m− ihn0 , n+ , n− |. By solving this master equation numerically using the basis that includes states with up to Nmax ≤ 10 photons in each mode, we calculate the occupation numbers Ni = Tr[βi† βi ρˆ], zero time-delay pair correlation func(2) tions for each mode ω0,± , gi = Tr[(βi† )2 βi2 ρ]/Ni2 and the probabilities P (20 ) and P (1+ , 1− ) to find a photon pair in the mode ω0 or the conjugated modes, respectively. We checked the consistency of our calculations by converging the results upon increasing Nmax and by comparing the results of modelling where we include and ˆ neglect the interaction terms δ H. First, we solved Eq. (10) for a circuit with J 0 = J (x = 1) and γ = κ or γ = 8κ (which is typical for GaAs polaritonic microcavities26 ), with weak anti-bunching in (2) the low-flux of ω± photons demonstrated by g+ shown in Fig. 3. In contrast, for the resonant conditions set for J 0 = J (x = 1) and with γ = 0.1κ in a system continuously pumped with amplitude F , we find a much higher efficiency of production of pure two-photon states. The numerically found steady-state solutions of Eq. (10) displays resonances corresponding to the transitions in the spectrum in Fig. 2. This is illustrated in Fig. 4 by the pump-frequency ω dependence of probabilities P (1+ , 1− ) and P (20 ) to find one ω± pair or two ω0 photons in the circuit, occupation numbers for the excited ω0 pho(2) tons, and the two-photon correlation function g+ . For F  κ, P (20 ) and P (1+ , 1− ) have pronounced resonances at 2ω = E(2A) and 2ω = E(2B), corresponding to the two-photon transitions |0i → |2Ai and |0i → |2Bi. Resonant excitation of individual ω± pairs is also re(2) flected by the dips in g+ . For larger F , we identify additional resonances in the vicinity of 3ω = E(3A) and 3ω = E(3B), corresponding to the three-photon transitions |0i → |3Ai and |0i → |3Bi. Note that, at a larger F , the pump induces shifts in the resonance conditions to create multi-photon states, Eq. (9), and the maxima, e.g. in N0 shown in Fig. 4, are additionally shifted by power-dependent broadening of |0i → |1i resonance. As the pumping amplitude grows, the two-photon resonances |0i → |2Ai and |0i → |2Bi become more pronounced, accompanied by increasing P (1+ , 1− ). However, with the reference to Fig. 5, where two top panels show the steady state parameters for the circuit pumped at the resonance frequencies of |0i → |2Ai and |0i → |2Bi transitions, saturation of P (1+ , 1− ) at F ∼ κ, accompanied by pollution of photon pairs at conjugated

4

0.1 0

0.1

0.1 0

0.5

1

1.5

2 0

0.5

1.5

1

0.1 0 0.6 0.2

100

0

10-2

0

10-2

4

8

12

0

4

8

12

FIG. 5. Top: Steady-state parameters computed in the system with γ = 0.1κ under continuous pumping as a function of pumping amplitude F , for resonance frequencies of |0i → |2Ai and |0i → |2Bi transitions marked on Fig. 4. Middle: Time evolution of pumped circuit, following the switching-on of the pump F = 0.8κ. Bottom: Rabi-type oscillations of two-photon states in a circuit pumped by a Gaussian pulse (dashed line) with parameters indicated on Fig. 6.

0.4 0.0

0.1

0.0 -5

10-1

0.4

102

0 100 10-1 10-2 10-3

0.2

0.0 104

2

-4

-3

-2

-1

0

1

(2)

FIG. 4. Dependence of steady state values of P (1+ , 1− ), g+ , N0 and P (20 ) on the pump frequency ω for γ = 0.1κ and various amplitudes of the pump. Relevant resonance conditions correspond to multi-photon transitions sketched in Fig. 2.

frequencies with individual ω± photons (manifested by (2) increasing g+ ), suggest that a mere increase of pumping does not improve the output of correlated photon pairs. The insight into how one can increase the output of individual photon pairs comes from the pronounced beatings in the temporal evolution of P (1+ , 1− ) and (2) g+ , which follow switching-on of the excitation source F = θ(t) × const (θ is the Heaviside function), Fig. 5. These beatings are the results of two-photon Rabi oscillations23 , generated by resonance mixing of |n0 + 2, 0+ , 0− i ⇐⇒ |n0 , 1+ , 1− i states. Hence, we suggest to implement pulsed excitations, harvesting photon pairs within optimally chosen delay-time windows. Note that, √ for κ  J, the period, π/ 2κ, of Rabi oscillations |2, 0+ , 0− i ⇐⇒ |0, 1+ , 1− i, is long enough for harvesting correlated photon pairs at the time interval around the optimal delay tmax at the maximum of P (1+ , 1− ), without undermining their spectral identity. Hence, we identify time intervals of the maximal probability to find a high-fidelity conjugated photon pair (in those intervals, N+ ≈ P (1+ , 1− )). An example of time-dependent (2) P (1+ , 1− ) and g+ , produced by a Gaussian pulse of du-

ration τ , is shown in the bottom panels in Fig. 5, and in Fig. 6 we show the dependence of the size of the (2) maximum output P (1+ , 1− ), at tmax , and g+ ∼ 10−3 . The optimal choice of the duration and amplitude of the pulse offers a high yield, P (1+ , 1− ) ∼ 0.5 of an almost pure two-photon state. To achieve the desirable regime of κ/γ  1, one needs to use materials with a large non-linearity and cavities with a high quality factor, Q. Depending on the operational frequency range, these may be Q ∼ 109 superconducting microwave resonators27,28 , coupled with superconducting qbits for microwave frequencies, or trapped ions in the electromagnetically-inducestransparency regime29,30 resonantly coupled to Q ∼ 107−8 toroidal24 or microrod25 microcavities for visible or infrared frequencies. In the latter systems24,30,31 , non-linearity can reach κ ∼ 1.25 × 107 s−1 , with γ ∼ 2 × 105 s−1 , and the optimised pulses and repetition rate 0.1γ would produce pairs of colour-conjugated photon (2) pairs with gs ∼ 10−2 at the rate of up to 105 s−1 , much higher than what has been achieved using the parametric (2) down-conversion13 (pairs with gs ∼ 10−1 at the rate of 2 −1 rate 10 s ). Finally, the proposed non-linear optical circuit can be used as a color-entangled photon source20 by connecting one waveguide L to resonator 1 and another waveguide R equally coupled to both resonators 2 and 3 used for the excitation pulse. The escape of the two-photon state |1+ , 1− i into the waveguides L/R, with couplings ∼ γ, would deliver signals to the recipients at the L and R

5

2

0.60

20 5 1

0.54 0.48 1.5

3

0.6

ends,

4.5

1 

 1 x −√ √ |L+ , R− i 2 2 1 + 2x2   1 x − |R+ , L− i (11) +√ √ 2 2 1 + 2x2 1 + √ [|R+ , R− i − |L+ , L− i] , 2 1 + 2x2

|1+ , 1− i →

0 0 2

2

4

10-1

6

2

0

1 0 0

2 4

1

1

0 0

2

4

10-5

(2)

FIG. 6. P (1+ , 1− ) (top) and g+ (bottom) at t = tmax for the system pumped by a Gaussian pulse with ampli2 2 tude F = F0 e−(t−2τ ) /τ θ(t). White (black) dot marks a favourable choice of τ and F0 used in Fig. 5. Insets: (top) cor(2) related photon pair output and g+ along the line of maximal P (1+ , 1− ) at t = tmax and (bottom) parametric dependence of tmax κ.

1 2

3 4

5

6

7 8

9 10 11

12

Shields, A. Nature Photonics 1, 215–223 (2007). Bennett, C. H. and Brassard, G. International Conference on Computers, Systems & Signal Processing, Bangalore, India, 10-12 December 1984, pp. 175-179. Kimble, H.J. Nature 453, 1023–1030 (2008). Knill, E. Laflamme, R. and Milburn, G. J. Nature 409, 46 – 52 (2001). Kurtsiefer, C., Mayer, S., Zarda, P. and Weinfurter, H. Phys. Rev. Lett. 85, 290–293 (2000). Santori, C., Fattal, D., Vuckovic, J., Solomon, G. S. and Yamamoto, Y. Nature 419, 594–597 (2002). Wei, Y.-J.et al. Nano Letters 14, 6515–6519 (2014). Stevenson, R. M., Young, R. J., Atkinson, P., Cooper, K., Ritchie, D. A. and Shields, A. J. Nature 439, 179–182 (2006). Fasel, S. et al. New Journal of Physics 6, 163 (2004). Soujaeff, A. et al. Opt. Express 15, 726 –734 (2007). Chen, J., Pearlman, A.J., Ling, A., Fan, J. and Migdall, A. Opt. Express 17, 6727–6740 (2009). Hunault, M., Takesue, H., Tadanaga, O., Nishida, Y. and

who would detect arrival of photons, distinguishing their color. Projecting the wave function Eq.(11) onto the subspace with one photon at L and one at R, recipients L and R would be able to use the color-entangled photon pairs similarly to what was suggested for the polarisationentangled photon pairs1,8,11 . Then, optimal output of the color-entangled states would be achieved in a circuit with x → 0 (J 0  J, corresponding to a simple √ linear chain of three cavity-resonators with ω± ≈ ω0 ± 2J, well separated from both ω0 mode and the pumping field), for which the two-photon states in the first two lines of Eq. (11) would be nothing but a Bell pair. Acknowledgements We thank P. Kok, S. Flach, M. Skolnick, and L. Glazman for useful discussions. This work was supported by EPSRC Programme Grant EP/J007544.

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15 16 17 18 19

20 21

22

23

24

25

Asobe, M. Opt. Lett. 35, 1239–1241 (2010). Fortsch, M., et al. Nature Communications 4, 1818, (2013). Li, X., Voss, P.L., Sharping, J.E. and Kumar, P. Phys. Rev. Lett. 94, 053601 (2005). Sharping, J. E. et al. Opt. Express 14, 12388–12393 (2006). Takesue, H. et al. Appl. Phys. Lett. 91, 201108 (2007). Collins, M. et al. Nature Communications 4, 3582 (2012). Davanco, M. et al. Appl. Phys. Lett. 100, 261104 (2012). Sherkunov, Y., Whittaker, D. M. and Fal’ko, V. Phys. Rev. A 93, 043842 (2016). Ramelow, S., et al. Phys. Rev. Lett. 103, 253601 (2009). Sherkunov, Y., Whittaker, D.M., Schomerus, H. and Falko V. Phys. Rev. A 90, 033845 (2014). Verger, A., Ciuti, C. and Carusotto, I. Phys. Rev. B 73, 193306 (2006). Sherkunov, Y., Whittaker, D. M. and Fal’ko, V.I. Phys. Rev. A 93, 023843 (2016). Armani, D.K., Kippenberg, T.J., Spillane, S.M. and Vahala, K.J. Nature 421, 925 (2003). DelHaye, P., Diddams, S.A. and Papp, S.B. Appl. Phys.

6

26 27 28

Lett. 102, 221119 (2013). Dufferwiel, S.et al. Appl. Phys. Lett. 104, 192107 (2014). Barends, R. et al. Appl. Phys. Lett. 97, 023508 (2010). Geerlings, K. et al. Appl. Phys. Lett. 100, 192601 (2012).

29

30

31

Imamoglu, A., Schmidt, H., Woods, G. and Deutsch, M. Phys. Rev. Lett. 79, 1467–1470 (1997). Hartmann, M.J. and Plenio, M.B. Phys. Rev. Lett. 99, 103601 (2007). Aoki, T. et al. Nature 443, 671 (2006).