Squeezing and EPR correlation in the Mirrorless Optical Parametric ...

Squeezing and EPR correlation in the Mirrorless Optical Parametric Oscillator A. Gatti1,2 , T. Corti1 , E. Brambilla1 1

2

Istituto di Fotonica e Nanotecnologie del CNR, Piazza Leonardo da Vinci 32, Milano, Italy, Dipartimento di Scienza e Alta Tecnologia dell’Universit` a dell’Insubria, Via Valleggio 11 Como, Italy. This work analyses the quantum properties of counter-propagating twin beams generated by a Mirrorless Optical Parametric Oscillator in the continuous variable regime. Despite the lack of the filtering effect of a cavity, we show that in the vicinity of its threshold it may generate high levels of narrowband squeezing and Einstein-Podolsky-Rosen (EPR) correlation, completely comparable to what can be obtained in standard optical parametric oscillators.

arXiv:1704.09010v1 [quant-ph] 28 Apr 2017

PACS numbers: 42.50.-p,42.50.Dv,42.50.Ar,42.30.-d

Backward parametric down-conversion (PDC), where one of the twin beams back-propagates with respect to the pump laser source (Fig.1), is gaining an increasing attention in the quantum optics community. In the spontaneous regime it has a natural potentiality to generate high-purity and narrowband heralded single photons [1–3], a highly desirable and non trivial goal, which in the standard co-propagating geometry can be realized only at specific tuning points. A second appealing feature is the presence of a threshold pump intensity, beyond which the system

 signal in (vacuum)

signal out

pump

pump

idler in (vacuum)

idler out 0

Figure 1: (Color online)Scheme of backward PDC, taking place in a χ(2) crystal periodically poled with a submicrometer period Λ ≈ λp /np . Quasi-phase matching [Eq.(1)] requires then that the idler field is generated in the backward direction with respect to the signal and pump. 

in makes a transition to coherent signal oscillations, i.e. it behaves as a Mirrorless Optical parametric Oscillator (MOPO)[4]. out (vacuum) is the feedback mechanism established bysignal Responsible of this critical behaviour back-propagation and stimulated down-conversion. Ref. [5] analysed the critical behavior of twin beams below threshold, enlightening the role of the pump pairs in creating the feedback necessary to the onset of apump quantum correlation of photon classical coherence above threshold. idler out idler in In this work we turn our attention to the quantum properties of the source in the(vacuum) continuous-variable regime, so far unexplored, namely its potentiality to generate0EPR-correlated beams in the vicinity of the threshold. EPR correlation [6–8], i.e. nonclassical correlations in a pair of non-commuting field quadratures, and their associated squeezing, are features of the two -mode squeezed state produced by any down-conversion process (see e.g.[9]). However, squeezed light generated in the standard single-pass configuration is in general multimode, which is often undesirable for applications [10] Moreover high levels of squeezing are hard to be generated and detected (see e.g. [11], but also [12, 13] for recent achievements in this sense). The typical solution is to recycle the parametric light in an optical resonator, which at the same time enforces the nonlinearity and produces a sharp modal filtering, i.e. to build an optical parametric oscillator (OPO). Remarkably, this work will show that counterpropagating twin-beams, despite the lack of the filtering effect of the cavity, exhibit high levels of narrowband EPR correlation, completely comparable to what can be obtained in standard subthreshold OPOs. The role of the cavity is in the MOPO played by the distributed feedback mechanism[5], which creates a threshold where, similarly to the OPO, the quantum noise in principle diverges in some observables, allowing then noise suppression in their conjugate observables. Once technical challenges involved in its realization are overcome, this source may then represent a robust and compact alternative to the OPO. The backward geometry requires a sub-micrometer poling of the χ(2) materials, which explains why after the first theoretical prediction [14], this source had to wait forty years before being realized [4]. We consider the scheme in Fig.1, in which the laser pump at frequency ωp and the signal at frequency ωs co-propagate along the +z axis

2 in the nonlinear medium, while the idler at frequency ωi = ωp − ωs back-propagates in the −z direction. Quasiphase matching (i.e. the generalized momentum conservation) is realized when their corresponding wave numbers kj = ωj nj (ωj )/c satisfy ks − ki = kp − m

2π Λ

m = 1, 3 . . .

(1)

where Λ is the poling period, and nj the refraction indexes. First order interactions then require Λ ' λp /np . Our quantum model for this configuration was described in Refs.[2, 5] (see also [15]). As in the former literature, we restrict to a purely temporal description, assuming either waveguiding or a small collection angle. Below the MOPO threshold the depletion of the pump laser is negligible, and it can be described as a classical field of constant amplitude along the sample. Assuming in addition that the pump is CW, it is simply described by its complex amplitude αp = |αp |eiφp . The strength of the parametric interaction is then characterized by the dimensionless gain √ g = 2πχ|αp |lc (2) where χ is proportional to the χ(2) susceptibility of the medium and lc is the crystal length. In terms of this parameter the MOPO threshold occurs [16] at g = gthr =

π 2

(3)

The signal and idler waves are instead described by quantum field operators Aˆs (Ω, z) and Aˆi (Ω, z), for two wavepackets centered around the respective reference frequencies ωs and ωi satisfying quasi-phasematching (1) (capital Ω is the offset from the ωj ). As detailed in [5], the model is formulated in terms of linear propagation equations coupling only frequency conjugate modes ωs + Ω, ωi − Ω of the twin beams, whose solution gives a transformation linking the output operators Aˆout = Aˆs (z = lc ), Aˆout = Aˆi (z = 0) to the input ones (Fig.1), assumed in the vacuum state. Notice s i that in this geometry the boundary conditions are not the standard ones, because the signal and idler fields exit from the opposite end faces of the slab. The input-ouput relations are then the Bogoliubov transformation, characteristic of processes where particles are generated in pairs: ˆin ˆin† Aˆout s (Ω) = Us (Ω)As (Ω) + Vs (Ω)Ai (−Ω) ˆin ˆin† Aˆout i (−Ω) = Ui (−Ω)Ai (−Ω) + Vi (−Ω)As (Ω).

(4a) (4b)

The coefficients Uj (Ω) and Vj (Ω) are the trigonometric functions [5]: Us (Ω) = eiks lc eiβ(Ω) φ(Ω) sin γ(Ω) Vs (Ω) = ei(ks −ki )lc geiφp φ(Ω) γ(Ω) Ui (−Ω) = eiki lc eiβ(Ω) φ∗ (Ω) sin γ(Ω) ∗ Vi (−Ω) = geiφp φ (Ω) with γ(Ω) 1 φ(Ω) = ¯ D(Ω)l cos γ(Ω) − i 2γ(Ω)c sin γ(Ω) r D2 (Ω)lc2 γ(Ω) = g 2 + , 4

(5a) (5b) (5c) (5d)

(5e)

In these expressions: D(Ω) = ks (Ω) − ki (−Ω) − kp + kG

(6)

is the phase mismatch for two frequency conjugate signal-idler components, kj (Ω) being the wavenumber of j−th wave at frequency ωj + Ω (j = s, i). The phase β(Ω) = [ks (Ω) + ki (−Ω) − (ks + ki )]

lc 2

(7)

is a global propagation phase. Notice that the coefficients Uj (Ω) and Vj (Ω) diverge when approaching the MOPO threshold g = π/2, and, as can be easily checked, they satisfy the unitarity conditions: |Uj (Ω)|2 − |Vj (Ω)|2 = 1, and Us (Ω)Vi (−Ω) = Ui (−Ω)Vs (Ω)

3 Unlike the co-propagating case, this configuration is characterized by narrow spectral bandwidths [2, 4, 5]. Therefore, it is legitimate to retain only the first order of the Taylor expansions of the wavenumbers kj (Ω), so that lc Ω D(Ω)lc ' (ks0 + ki0 )Ω := 2 2 Ωgvs l Ω c β(Ω) ' (ks0 − ki0 ) Ω = , 2 Ωgvm where kj0 =

dkj dΩ |Ω=0

(8) (9)

, and Ω−1 gvs

≡ τgvs

  1 lc lc = + . 2 vgs vgi

(10)

is a long time scale characteristic of counterpropagating interactions, on the order of the transit time of light along the slab, involving the sum of the inverse group velocities vgj = 1/kj0 [2, 5]. In the spontaneous regime, it defines the correlation time of twin photons, while its inverse Ωgvs gives the narrow width of their spectrum, which becomes even narrower in the stimulated regime and ideally shrinks to zero on approaching threshold [5]. Conversely Ω−1 gvm ≡ τgvm =

lc lc − 2vgs 2vgi

(11)

is a short time scale related to the group velocity mismatch (GVM), and produces a small temporal offset between the signal and idler wave-packets. Clearly, |Ωgvm |  Ωgvs for any tuning conditions (see Fig.5 for a comparison in the case of LiNbO3 ). Within these linear approximations q the coefficients Uj (Ω) and Vj (Ω) basically depend on the frequency only through the ratio

Ω2 Ω2gvs ,

because γ(Ω) '

g2 +

Ω2 Ω2gvs ,

while the phase β(Ω) in (9) varies on the slow

scale |Ωgvm |  Ωgvs and remains close to zero in the spectral region where Uj (Ω) and Vj (Ω) take non trivial values. Several properties of the state of the MOPO below threshold depend solely on the Bogoliubov form (4) of the transformation, so that they are common to any linear process of photon-pair generation. In particular, if one introduces the sum and difference between frequency conjugate components of the twin beams: Cˆ± (Ω) = √12 [Aˆout s (Ω)± out Ai (−Ω)], then the transformation (4) decouples into two independent squeeze transformations[17]. The ± modes are thus individually squeezed, and their squeezing ellipses turn out oriented along orthogonal directions. As well known, this implies the simultaneous presence of correlation and anticorrelation in two orthogonal quadrature operators of the twin beams [7, 8]. In order to characterize the amount of squeezing and EPR correlation generated in this specific configuration, let us consider the quadrature operators for the individual signal and idler fields in the time domain † −iφj ˆ j (t) = Aˆout X + Aˆout (t)eiφj , j (t)e j 1 Yˆj (t) = [Aˆout (t)e−iφj − Aˆjout † (t)eiφj ] j = s, i i j

(12) (13)

ˆ j (t), Yˆk (t0 )] = δj,k δ(t − t0 ) and represent incompatible observables. The two orthogonal quadratures do not commute [X † out ˆ ˆ Notice that their Fourier transforms: Xj (Ω) = Aj (Ω)e−iφj + Aˆout (−Ω)eiφj (which are not Hermitian and hence j not observables) involve the two symmetric spectral components ωj ± Ω for each field. We then introduce proper combinations of the signal and idler quadratures: ˆ s (t) − X ˆ i (t − ∆t)] ˆ − (t) = √1 [X X 2 1 Yˆ+ (t) = √ [Yˆs (t) + Yˆi (t − ∆t)] 2

(14) (15)

where the delay ∆t between the detection of the signal and idler arms can be used as an optimization parameter. Next, we characterize the noise in the sum or difference modes by the so-called squeezing spectra D E Z +∞ ˆ + (t + τ )  δ Yˆ+ (t)δY E Σ± (Ω) = dτ eiΩτ D (16) ˆ − (t + τ ) ˆ − (t)δX  δX −∞

4 ˆ− = X ˆ − − hX ˆ−i = X ˆ − , because below the threshold the field expectation values are zero. These where e.g. δ X quantities describe the degree of correlation (”-” sign) or anticorrelation (”+” sign) existing between the field quadrature operators of the twin beams at the two crystal output faces. The value ”1” represents the shot noise level, which corresponds to two uncorrelated light beams. In the degenerate case ωs = ωi , one may also think of physically recombining the two counterpropagating beams on a beam-splitter, in order to produce two independently squeezed beams. After some long but straightforward calculations, based on the input-output relations (4), one obtains  2 1 Σ± (Ω) = Us (Ω) − Vi∗ (−Ω)eiΩ∆t ei(φs +φi ) 2 2  (17) + Us (−Ω) − Vi∗ (Ω)e−iΩ∆t ei(φs +φi ) Up to this point we used only the Bogoliubov form of the relations (4), so that Eq.(17) actually holds for any PDC process. As expected for the EPR state, the degree of correlation and anticorrelation in orthogonal quadratures are identical: Σ− (Ω) = Σ+ (Ω). The two spectral terms at r.h.s of Eq.(17) are present because detection of the temporal quadratures (12) probes the noise at ωj ± Ω for each field. In the MOPO, these terms can be made identical by setting ∆t = τgvm , which exactly compensates the temporal offset of the twin beams. However, even in the absence of such optimization, the two terms are respectively minimized by choosing φs + φi = 2θ(±Ω) = arg [Us (±Ω)Vi (∓Ω)] Ω ' ks lc + φp + arg [sincγ(Ω)] ± Ωgvm ' ks lc + φp + arg [sincγ(Ω)]

(18) (19) (20)

where the second line uses the linear approximations (8) and (9), and the last line holds because Ω/Ωgvm ≈ 0 within 2 the spectral region of interest. With this choice Σ± (Ω) → [|Us (Ω)| − |Vi (−Ω)|] reaches its minimum value at any frequency, and the noise never goes above the shot noise level ”1”. The degree of EPR correlation/anticorrelation Σ∓ (Ω) is instead plotted in Fig.2 for fixed phase angles, namely φs + φi := 2θ(0) = ks lc + φp

(21) p In this case, the noise passes from below to above the shot noise at Ω = ±Ωgvs π 2 − g 2 , where sincγ(Ω) changes sign. g=/2-0.03 g=/2-0.07 g=1.0 g=0.1

a) 2.0

b) 2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

Shot-Noise

0.0

-25 -20 -15 -10 -5

0

5

 / gvs

10 15 20 25

0

1

2

3

/gvs

4

Figure 2: (Color online) Squeezing spectra Σ± (Ω) (17), and degree of EPR correlation between the MOPO twin beams, as a function of Ω/Ωgvs , for φs + φi fixed as in Eq.(21). ∆t = 0.The inset (b) is a detail of the minima.

p These values can be used to define a bandwidth of squeezing ∆Ω = Ωgvs π 2 − g 2 ≈ 2.7Ωgvs close to threshold. Some remarks are in order: i) The EPR correlation becomes asymptotically perfect as the MOPO threshold is approached, which can be realized only close to a critical point, because the noise in the quiet quadrature can be suppressed only at the expenses of a diverging level of noise in the orthogonal one. ii) The squeezing remain significant at rather large distances from threshold, Σ± (0) ' 0.09 for g = 1, which is 36% below the MOPO threshold. ii) Excellent levels of squeezing are present in the whole emission bandwidth, that we remind is smaller than Ωgvs [5]. This is in sharp contrast

5

4500

a)

g=/2-0.03 g=/2-0.05 g=/2-0.07

3750 3000

b)

g=/2-0.03 g=/2-0.07 g=1 g=0.1

1.0 0.8 0.6

2250 1500

0.4

750

0.2

0 -0.8 -0.6 -0.4 -0.2 0.0

0.0 0.2

gvs

0.4

0.6

0.8

-6

-4

-2

0

gvs

2

4

6

Figure 3: (Color online) Antisqueezing spectra Σ± (Ω) (17) as a function of Ω/Ωgvs , for phase-angles orthogonal to those in Fig.2. The curves in (b) are normalized inside (0, 1).

with the single-pass co-propagating geometry, where high squeezing is difficult to observe[12], and the orientation of the squeezing ellipses varies rapidly inside the PDC bandwidth[17]. In contrast, for the MOPO the orientation of the ellipses, defined by θ(±Ω), remains practically constant inside the bandwidth Ωgvs [see Eqs.(18)-(20)]. This can be viewed as a consequence of the long (τgvs ) and short (τgvm ) time scales involved in the counterpropagating geometry. Fig.3 shows the antisqueezing of the sum or difference modes, which occurs for quadrature phases orthogonal to those in Fig.2. In this case the noise diverges on approaching threshold, which is clearly reminiscent of the critical divergence of the MOPO spectra analysed in [5]. The bandwidth of the antisqueezing spectra shrinks getting close to threshold (Fig.3b), which again reflects the shrinking of the spectra and the critical slowing down of temporal fluctuations close to the MOPO threshold[5]. , and to a The curves in Figs.2 and 3 are in sense universal for the MOPO, when plotted as a function of ΩΩ gvs very good approximation hold for any material and tuning conditions. This can be more clearly seen by deriving explicit expressions of the noise spectra. By inserting the coefficients(5) in the general result(17), using the linear approximation (8) and neglecting the contribution of the slow phase β(Ω), when φs + φi is fixed as in Eq.(21), the squeezing spectra can be written as q q 2+Ω ˜ 2 − g sin g 2 + Ω ˜2 g q Σ±S (Ω) = q (22) ˜ 2 + g sin g 2 + Ω ˜2 g2 + Ω ˜ = Ω/Ωgvs . The antisqueezing spectra, for phases orthogonal to those in Eq.(21), are just the inverse where Ω A Σ± (Ω) = 1/ΣS± (Ω). This expressions take a particularly simple form in the neighborhood of threshold and for small ˜  g. By frequencies. Let us define a distance from threshold  = gthr − g and let us consider the limit   1 and |Ω| ˜ = 0, and keeping terms at most quadratic in the expanding the various functions in Eq.(22) around  = 0 and Ω/g small quantities, we obtain ! ˜2 1 2 Ω S Σ± (Ω) −→  + 2 . (23) 1 4 gthr ˜ |Ω|g

2

˜ ≈ 2gthr This function is a parabola which reaches its minimum at Σ± (0) = 4 → 0 as  → 0, and of width ∆Ω constant close to threshold. In the same limit, the antisqueezing spectra become ΣA ± (Ω) −→

1 ˜ |Ω|g

4 2

+

˜2 Ω 2 gthr

(24)

˜ = gthr → 0, as threshold is which represents a Lorentzian peak of diverging height 42 → ∞ and of vanishing width ∆Ω approached. These approximated formula nicely reproduce the minima and the maxima in Figs.2 and 3, respectively, when not too far from threshold, and are actually valid for rather large distances from threshold, as shown by Fig.4 . We notice that such behaviors of the squeezing spectra are completely comparable to what can be obtained in standard cavity OPOs below threshold (see e.g Ref.[18], formula (7.59),page 131). Here, in the degenerate case, the

6 1.0

a)

Shot-Noise =0.03 Exact =0.03 Approx =0.4 Exact =0.4 Approx

0.8 0.6

b)

1.0

Exact 2  /4

0.8 0.6

0.4

0.4

0.2

0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0 0.0

0.3

0.6

0.9

1.2

1.5

Figure 4: (Color online)Comparison between the exact results in Eq.(17), and the approximated ones in Eq.(23). Squeezing spectra (a) as function of frequency, and (b) at zero frequency as a function of the distance from threshold .

spectrum of squeezing has the form ΣOP O (Ω) =

2 ¯2 (Athr p − Ap ) + Ω 2 ¯2 (Athr p + Ap ) + Ω

→

¯ |Ω|2

1 2 ¯ 2
 +Ω 4

(25)

where Ap is a cavity gain parameter, proportional to the pump amplitude, the χ(2) susceptibility and the photon ¯ is the frequency normalized to the cavity linewidth; the OPO threshold is at Ap = Athr = 1, lifetime in the cavity; Ω p thr and  = Ap −Ap defines also in the OPO case the dimensionless distance from threshold. Remarkably, in both MOPO and OPO cases, the behavior ∼ 2 /4 with the distance from threshold indicates that excellent level of squeezing can be obtained even at rather large distances below the threshold. i(m) 9

10 Hz 13.500

1.71

1.33

1.18

i(m) 1.09

1.04

12

10 Hz

a) gvs (GHz)

1.71

10

13.450

0

1.18

1.09

1.04

3.0

3.5

b) ’gvm (THz)

20

13.475

1.33

-10

13.425

-20

13.400 -30

13.375 1.1

1.5

2.0

2.5

s(m)

3.0

3.5

1.1

1.5

2.0

2.5

s(m)

Figure 5: (Color online) Comparison between the two spectral scales of the MOPO for PPLN pumped at 800 nm, lc = 1cm. a) Ωgvs = l2c (ks0 + ki0 )−1 ' 10Ghz is the narrow MOPO bandwidth; b) Ωgvm = l2c (ks0 − ki0 )−1 in the range 5Thz or more is the broader GVM bandwidth. For each λs , λi the poling period is chosen to realize phase-matching according to Eq.(1).

As a final, we remark that the spectra in Figs.2 and 3 were calculated in the specific case of periodically poled Lithium Niobate ( PPLN), pumped at 800nm, with Λ = 368 nm, suitable to phase-match the type 0 process at λs = λi . The wave-numbers were evaluated using the complete Sellmeier relations in [19]. However, we did not notice appreciable differences (unless at very large frequencies Ω ≥ 15Ωgvs ) with the linearly approximated results (22), nor with curves obtained for different materials or tuning conditions, which confirms that our results are completely general for any MOPO configuration. In conclusions, the MOPO below threshold is a source of EPR entangled beams over a wide range of light frequencies, including telecom wavelengths. Our analysis has shown that this cavityless configuration of PDC can reach the same

7 narrowband, high level, and robust correlation characteristic of the cavity OPO, which represents the golden standard to for EPR beams. As such, it can be used as an alternative to the OPO, meeting the increasing demand for monolithic devices in the field of integrated quantum optics.

[1] A. Christ, A. Eckstein, P. J. Mosley, and C. Silberhorn, Opt. Expr. 17, 3441 (2009), URL http://www.opticsexpress. org/abstract.cfm?URI=oe-17-5-3441. [2] A. Gatti, T. Corti, and E. Brambilla, Phys. Rev. A 92, 053809 (2015), URL http://link.aps.org/doi/10.1103/ PhysRevA.92.053809. [3] E. Brambilla, T. Corti, and A. Gatti, ArXiv e-prints (2016), 1607.06012, URL https://arxiv.org/abs/1607.06012. [4] C. Canalias and V. Pasiskevicius, Nat. Photon. 1, 459 (2008), URL http://dx.doi.org/10.1038/nphoton.2007.137. [5] T. Corti, E. Brambilla, and A. Gatti, Phys. Rev. A 93, 023837 (2016), URL http://link.aps.org/doi/10.1103/ PhysRevA.93.023837. [6] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935), URL http://link.aps.org/doi/10.1103/PhysRev. 47.777. [7] M. D. Reid, Phys. Rev. A 40, 913 (1989), URL http://link.aps.org/doi/10.1103/PhysRevA.40.913. [8] Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Phys. Rev. Lett. 68, 3663 (1992), URL http://link.aps.org/ doi/10.1103/PhysRevLett.68.3663. [9] C. C. Gerry and P. L. Knight, Introductory Quantum Optics (2005). [10] S. Lemieux, M. Manceau, P. R. Sharapova, O. V. Tikhonova, R. W. Boyd, G. Leuchs, and M. V. Chekhova, Phys. Rev. Lett. 117, 183601 (2016), URL http://link.aps.org/doi/10.1103/PhysRevLett.117.183601. [11] A. La Porta and R. E. Slusher, Phys. Rev. A 44, 2013 (1991), URL https://link.aps.org/doi/10.1103/PhysRevA.44. 2013. [12] Y. Eto, T. Tajima, Y. Zhang, and T. Hirano, Opt. Express 16, 10650 (2008), URL http://www.opticsexpress.org/ abstract.cfm?URI=oe-16-14-10650. [13] F. Kaiser, B. Fedrici, A. Zavatta, V. D’Auria, and S. Tanzilli, Optica 3, 362 (2016), URL http://www.osapublishing. org/optica/abstract.cfm?URI=optica-3-4-362. [14] S. E. Harris, Appl. Phys. Lett. 9, 114 (1966), URL http://scitation.aip.org/content/aip/journal/apl/9/3/10.1063/ 1.1754668. [15] T. Suhara and M. Ohno, IEEE Journal of Quantum Electronics 46, 1739 (2010). [16] Y. Ding and J. Khurgin, Quantum Electronics, IEEE Journal of 32, 1574 (1996), ISSN 0018-9197. [17] A. Gatti, T. Corti, and E. Brambilla (2017), in preparation. [18] D. Walls and G. Milburn, Quantum Optics (1994). [19] D. Nikogosian, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005), ISBN 9780387220222, URL https: //books.google.it/books?id=ZW9Ynx_Z7kkC.