Degenerate four-wave mixing as a low-power source of squeezed light
DDegenerate four-wave mixing as a low-powersource of squeezed light B ONNIE
L. S
CHMITTBERGER Quantum Technologies Group, The MITRE Corporation, 7515 Colshire Drive, McLean, VA, 22043 USA Approved for Public Release; Distribution Unlimited. Public Release Case Number 20-0216. ©2020 TheMITRE Corporation. ALL RIGHTS RESERVED. * [email protected] Abstract
Squeezed light is a quantum resource that can improve the sensitivity of optical measurements.However, existing sources of squeezed light generally require high powers and are not amenableto portability. Here we theoretically investigate an alternative technique for generating squeezingusing degenerate four-wave-mixing in atomic vapors. We show that by minimizing excess noise,this technique has the potential to generate measurable squeezing with low powers attainable bya small diode laser. We suggest experimental techniques to reduce excess noise and employ thisalternative nonlinear optical process to build a compact, low-power source of squeezed light.
1. Introduction
Squeezed states are a useful photonic resource impacting quantum communication, sensing, andmetrology. This type of light has unique noise properties in which the noise in one quadrature(e.g., amplitude or phase) can be reduced or “squeezed” at the expense of added noise in the other.With the appropriate detection scheme, one can take advantage of the squeezed quadrature toimprove the precision of a measurement. For example, squeezed light can be used to improve thephase sensitivity of an interferometer without increasing the optical power [1] or reduce the noiseof an imaging system [2]. Continuous-variable quantum communication and computation canalso be enabled using squeezed light, with recent notable advances including a 1.6 km free-spacetransmission of a squeezed state [3] and the generation of a one-million-mode cluster state [4].Squeezed light is generated using nonlinear optical processes, in which one or more input“pump” optical fields interact with a nonlinear material, such as atomic vapors or certainnonlinear crystals, and the mutual light-matter interaction generates optical fields with newfrequencies and/or wavevectors. Common methods for generating squeezed states today includenon-degenerate four-wave-mixing in atoms and parametric downconversion in crystals. Theseprocesses require high input optical powers (up to ∼ a r X i v : . [ qu a n t - ph ] S e p he purpose of the present work is to analyze the potential of using DFWM to generatea low-power source of squeezed light using present practical technologies. To do this, weexpand upon previous theoretical models [10, 11] by incorporating atomic decoherence effectsand optical loss. With this expanded model we demonstrate that early DFWM experimentswere unsuccessful in measuring squeezing due to high rates of collisional decoherence and/ordetection losses rather than spontaneous emission alone. Recent advances in atomic vaporcell technologies and the development of lower-cost, higher quantum efficiency photodiodesenable reduced decoherence rates and detection losses. In addition, recent experiments involvingnon-degenerate four-wave-mixing have shown that one can generate measurable squeezing evenin the presence of strong absorption [12] and that excess noise can be reduced by using techniquessuch as external optical pumping of atoms that have undergone spin-changing collisions with thewalls of the vapor cell [13]. These technological and experimental advances indicate that, byimplementing techniques to reduce excess noise, DFWM is a promising technique for observinglow-power squeezing in free space.We will outline our theoretical model for the light-atom interaction in Sec. 2. We then analyzetwo types of degenerate four-wave-mixing: phase-conjugate four-wave-mixing (Sec. 3) andforward four-wave-mixing (Sec. 4) . For both beam geometries, we derive the projected squeezinglevels as a function of loss and atomic decoherence rates and propose techniques to minimizethese sources of excess noise, thus making DFWM a viable platform for generating a low-SWaPsource of squeezed light. We present our conclusions in Sec. 5.
2. Theoretical model of the light-atom interaction
In this section we define our theoretical model of the light-atom interaction by deriving andsolving the density matrix equations of motion. This model differs from previous models [10, 11]in that we include optical loss and atomic decoherence processes.To model the effects of decoherence, we go beyond the two-level system which is typicallyused to describe DFWM and consider an empirical four-level model in which atoms can decay toan additional state from which they cannot directly participate in the four-wave-mixing process.We emphasize that this model, detailed below, is not intended to provide a complete physicaldescription of atomic decoherence. Rather, it is used here as a convenient tool through which wecan impose an approximate “decoherence rate” on the atoms, which reduces the efficiency of thefour-wave-mixing process, while still allowing decohered atoms to be optically pumped.Our model accounts for different types of relaxation processes in the atoms, some of whichinduce decoherence. Uniform relaxation (decay) is described by the standard spontaneousemission rates in the density matrix equations of motion. Atom-atom and atom-wall collisionscan result in an overall phase shift, decay into an unwanted spin state, and/or a rotation of theatomic polarization axis [14, 15]. We note that atoms which have undergone a spin-changingcollision or some other form of decoherence must be optically pumped before they can participatein a four-wave-mixing process; this in turn reduces the effective atomic density and results inadditional spontaneously emitted light.To incorporate the effects of collisional decoherence, we must modify the closed two-levelatom description used in previous works to also include additional energy levels into which theatom can decay. We employ the energy level scheme shown in Fig. 1(a), where | (cid:105) and | (cid:105) areanalogous to the ground and excited states used in a traditional two-level atom description, andlevel | (cid:105) represents an energy level into which the atoms can decay (such as a separate Zeemanstate), but which does not directly participate in the four-wave-mixing process. The atoms mustbe optically pumped through state | (cid:105) and into state | (cid:105) before they can participate in the DFWMprocess.The density matrix equations for a four-level non-degenerate four-wave-mixing process arederived in Ref. [16]. We use the same methods here and apply the relevant detuning and decay ۧȁ1 ۧȁ2ۧȁ3 ۧȁ4 E p1 E p2 ∆ 𝝌 (𝟑) F(z,x) B(z,x)f(z) b(z) 𝜙𝜙 𝝌 (𝟑) F(z) 𝛼 f - (z,x)f + (z,x)f - (z,x) (a) Energy diagram (b) Phase conjugation(c) Forward four-wave-mixing Fig. 1. (a) The four-level energy scheme considered here. The detunings ∆ and ∆ define the frequency difference between the pump field and the resonance frequency ofthe atomic transition. Fields E p and E p may in fact be the same pump field dependingon the beam geometry, but the subscript is used as an aid to track the field componentthat is pumping decohered atoms into the desired ground state versus that which is astandard optical pumping cycle. (b) The phase-conjugate/backward four-wave-mixingbeam geometry. (c) The forward four-wave-mixing beam geometry. rate parameters for our degenerate four-wave-mixing system. We define Ω j as the Rabi frequencyfor field j , ∆ n as the optical field detuning from level n , and Γ nm as the decay rate from level n tolevel m . The coupled density matrix equations for the system shown in Fig. 1(a) with densitymatrix elements σ ij in the rotating frame are as follows: ∂σ ∂ t = i (cid:16) Ω ∗ p σ − Ω p σ (cid:17) + Γ σ + Γ σ , (1) ∂σ ∂ t = i (cid:16) Ω ∗ p σ − Ω p σ (cid:17) + Γ σ + Γ σ , (2) ∂σ ∂ t = i (cid:16) Ω p σ − Ω ∗ p σ (cid:17) − Γ σ , (3) ∂σ ∂ t = i (cid:16) Ω p σ − Ω ∗ p σ (cid:17) − Γ σ , (4) ∂σ ∂ t = i (cid:16) Ω p σ − Ω ∗ p σ (cid:17) + ( i ∆ − i ∆ − Γ ) σ , (5) ∂σ ∂ t = i (cid:0) Ω p σ − Ω p σ (cid:1) + ( i ∆ − Γ ) σ , (6) ∂σ ∂ t = i (cid:0) Ω p σ − Ω p σ (cid:1) + ( i ∆ − Γ ) σ , (7) ∂σ ∂ t = i (cid:0) Ω p σ − Ω p σ (cid:1) + ( i ∆ − Γ ) σ , (8) ∂σ ∂ t = i (cid:0) Ω p σ − Ω p σ (cid:1) + ( i ∆ − Γ ) σ , (9)nd ∂σ ∂ t = i (cid:16) Ω ∗ p σ − Ω p σ (cid:17) + ( i ∆ − i ∆ − Γ ) σ . (10)To conserve population, we have ∂ ( σ + σ + σ + σ )/ ∂ t =
0. For succinctness, we define˜ E j = E j e i (cid:174) k j ·(cid:174) r , σ ii , j j = σ ii − σ j j , ξ = i ∆ − Γ , and ξ = i ∆ − Γ . We also define d as thetransition dipole moment between states | (cid:105) and | (cid:105) , (cid:126) as Planck’s constant divded by 2 π , (cid:15) asthe permittivity of free space, and c as the speed of light.The steady-state solution for the component of the density matrix describing the | (cid:105) → | (cid:105) transition is σ = − i (cid:126) ξ d ˜ E p σ , . (11)The corresponding susceptibility is given by χ = − in a (cid:126) (cid:15) ξ | d | σ , . (12)The intensity of the input pump optical field is defined as I p = (cid:15) c | E p | . By solving thecoupled density matrix equations to determine the steady state value of σ , , we find that theoff-resonant saturation intensity is I s ∆ = (cid:15) c (cid:126) (cid:2) Γ Γ ( Γ + Γ ) | ξ | + Γ Γ ( Γ + Γ ) | ξ | (cid:3) | d | Γ Γ ( Γ + Γ ) . (13)When I p (cid:28) I s ∆ , the susceptibility in Eq. 12 can be written as the sum of the linear and nonlinearcomponents, χ ≈ χ lin + χ NL , where σ , ≈ Γ Γ ( Γ + Γ ) | ξ | Γ Γ ( Γ + Γ ) | ξ | + Γ Γ ( Γ + Γ ) | ξ | (cid:18) − I p I s ∆ (cid:19) , (14) χ lin = − in a (cid:126) (cid:15) ξ | d | Γ Γ ( Γ + Γ ) | ξ | Γ Γ ( Γ + Γ ) | ξ | + Γ Γ ( Γ + Γ ) | ξ | , (15)and the nonlinear susceptibility is χ NL = − χ lin I p I s ∆ . (16)We use the parameter Γ to describe the rate at which atoms decay into an unwanted groundstate, e.g., due to collisions or other decoherence mechanisms. The experimental values of Γ and Γ are related to the atomic velocity (i.e., the rates at which atoms collide with the walls andpass through the pump beam after collisions, respectively). The average velocity of the atomsis given by v avg ≈ (cid:112) k B T / m , where k B is Boltzmann’s constant, T is the atomic temperature,and m is the mass of the atom [15]. For a vapor cell temperature of 110 ◦ C, v avg ≈
308 m/sfor rubidium atoms. For a 1 cm vapor cell size and in the absence of other collisions, atomswould collide with the walls at a rate of approximately Γ coll ≈ π ×
62 kHz ≈ . Γ , where Γ = π × . Γ reported in Ref. [16]. This high rate may be due to other inelastic collisions or decoherencemechanisms such as the interaction of atoms with background magnetic fields. Atoms that losecoherence immediately after traversing the pump field will have an even higher decoherence rate(approximately 0 . Γ for typical beam widths). We therefore expect that a realistic value of Γ will lie in the range of approximately 0 . Γ ≤ Γ ≤ . Γ .e note that Γ → | (cid:105) ↔ | (cid:105) path. With this assumption,Eq. 12 simplifies to χ (cid:12)(cid:12)(cid:12) Γ = = − n a | d | ( ∆ / Γ − i ) (cid:126) (cid:15) Γ (cid:16) + ∆ / Γ (cid:17) + | E | /| E s ∆ | , (17)with | E s ∆ | = (cid:126) Γ (cid:0) + ∆ / Γ (cid:1) / (cid:0) | d | (cid:1) , which is the susceptibility of a two-level atom [17].To summarize our model, previous works on degenerate four-wave-mixing often assume atwo-level atom structure where atom number is conserved. Here, we add a separate groundstate in order to empirically model loss of atomic population from coherence with the relevantfour-wave-mixing transition (| (cid:105) → | (cid:105)) . This technique allows us to tune both the rates at whichatoms decohere and that for which they are optically pumped back into state | (cid:105) in order toinvestigate the regimes under which squeezing may occur.
3. Phase conjugate degenerate four-wave-mixing
We first consider the field geometry shown in Fig. 1(b), which is often referred to as backwardfour-wave-mixing or phase conjugation. We restrict this to the case where the pump-probeangle φ is large enough such that the only type of four-wave-mixing process that occurs is inthe backward (phase-conjugate) geometry. We consider small angles that give rise to forwardfour-wave-mixing in Sec. 4. The first step is to solve for the propagation of the pump fieldsthrough the atomic vapor in the absence of the weak (generated) fields. The wave equation is ∇ (cid:174) E − c ∂ (cid:174) E ∂ t = (cid:15) c ∂ (cid:174) P ∂ t . (18)Below threshold for DFWM, the electric field amplitudes of the forward and backward pumpbeams have exponential solutions F ( z ) = ˜ Fe i δ z and B ( z ) = ˜ Be − i δ z where δ = k χ lin (cid:2) − I p / I s ∆ (cid:3) and ˜ F and ˜ B are independent of position.Above threshold for DFWM, we can define the total electric field −→ ˜ E ( t ) = −→ E e − i ω t + c.c., where −→ E = F ( z , x ) e ik ( cos θ z + sin θ x ) ˆ x + B ( z , x ) e − ik ( cos θ z + sin θ x ) ˆ x + f ( z ) e ikz ˆ x + b ( z ) e − ikz ˆ x . (19)Here, all fields are frequency degenerate, k is the wavenumber in vacuum, and we assume f and b are weak fields.The solution to the wave equation in the case of a single seed beam ( f ( ) > b ( L ) =
0) iswell-known and was originally derived in Ref. [18]. In the quantized treatment, the solutions forthe forward and backward field operators are a f ( z ) = cos [| κ |( z − L )] cos (| κ | L ) a f ( ) + i | κ | κ ∗ sin (| κ | z ) cos (| κ | L ) a † b ( L ) (20)and a † b ( z ) = cos (| κ | z ) cos (| κ | L ) a † b ( L ) + i κ ∗ | κ | sin [| κ |( z − L )] cos (| κ | L ) a f ( ) , (21)where κ = − k χ NL . (22)The commutation relations are [ a i , a j ] = [ a † i , a † j ] =
0, and [ a i , a † j ] = δ ij . The field quadraturesare defined as X i = Re [ a i ] = a i + a † i / Y i = Im [ a i ] = a i − a † i /( i ) . We also define themplitude sum/difference X ± = X f ± X b and the phase sum/difference Y ± = Y f ± Y b . Wenote that for some operator O , (cid:10) ∆ O (cid:11) = (cid:10) ( O − (cid:104) O (cid:105)) (cid:11) = (cid:10) O (cid:11) − (cid:104) O (cid:105) . We take the inputfields a f ( ) and a b ( L ) to be coherent states, such that (cid:68) ∆ X f ( ) (cid:69) = (cid:68) ∆ Y f ( ) (cid:69) = / (cid:10) ∆ X b ( L ) (cid:11) = (cid:10) ∆ Y b ( L ) (cid:11) = /
4. We note that the joint measurement of the fields can be squeezed,but the indivdual quadratures are thermal. For example, the noise of the amplitude quadrature for a f at z = L is (cid:68) ∆ X f ( L ) (cid:69) = (cid:2) sec (| κ | L ) + tan (| κ | L ) (cid:3) . (23)Therefore, when | κ | L = (cid:68) ∆ X f ( L ) (cid:69) = , andthe output is just coherent. For 0 < | κ | L < π /
2, the variance is bigger than that of a coherentstate. As | κ | L → π /
2, the variance approaches ∞ . To predict the squeezing associated with a joint quadrature measurement, we calculate thevariance associated with the output fields a f ( L ) and a b ( ) . The quadrature operator for theforward field is j f = e − i θ f a f ( L ) + e i θ f a † f ( L ) and the quadrature operator for the backward fieldis j b = e − i θ b a b ( ) + e i θ b a † b ( ) , where θ f and θ b are the phases of the homodyne detectors. Thejoint quadrature operator is j = j f + j b . In the ideal case of no loss, the variance is (cid:10) ∆ j (cid:11) = (cid:12)(cid:12)(cid:12)(cid:12) sec (| κ | L ) + ie − i ( θ f − θ b ) κ | κ | tan (| κ | L ) (cid:12)(cid:12)(cid:12)(cid:12) . (24)We note that in this ideal case, the noise is independent of the input field strength a f ( ) ; i.e., anynoise on the seed beam is present in each generated beam and then canceled by the nature of thejoint measurement. The measurable quadrature squeezing in units of decibels (dB) is given by M Q =
10 log (cid:20) ∆ j ∆ j SN (cid:21) , (25)where j SN corresponds to the noise that would be measured using coherent fields. For homodynedetection, where the local oscillator power dominates the measurement, we can take shot noise ofthe joint homodyne measurement to be the noise in the case where | κ | L →
0, i.e., where there isno four-wave-mixing [19]. The results of Eqs. 24 and 25 are shown in Fig. 2 for example valuesof | κ | L .Here, higher values of | κ | L correspond to stronger light-atom interactions and hence moresqueezing. In practice, one can increase | κ | L by increasing the atomic density, applied opticalintensity, or the length of the vapor cell, for example. In the limit where | κ | L → π / θ f − θ b = π / M Q → −∞ . In other words, in this ideal case of no loss, there is an operatingpoint for which we expect “infinite squeezing.” Of course, in practice, there always exists someloss, which quickly reduces the squeezing to finite values.To account for loss, we use the procedure outlined in the Appendix. To model the effects ofDoppler broadening, we must modify the detuning according to ∆ − (cid:174) k · (cid:174) v , where (cid:174) k is the opticalwavevector and (cid:174) v is the atomic velocity [20, 21]. For a thermal distribution, the atomic motionis described by the Maxwell-Boltzmann distribution S ( v z ) ∝ e − v z / u , where u = (cid:112) k B T / m isthe average velocity of the atoms and v z is the projection of the atomic velocty along ˆ z , i.e., theoptical axis. To determine the Doppler-broadened nonlinear coefficient, we integrate the densitymatrix elements over all velocities. The projected squeezing including loss is shown in Fig. 3(a)for typical experimental parameters, which follows the expected trend that squeezing is optimizedfor no loss, and all squeezing is lost for 100% loss. π π π π π - θ f - θ b S quee z i ng ( d B ) | κ | L = π /
3, Quadrature | κ | L = π /
6, Quadrature | κ | L = π /
3, Intensity - difference | κ | L = π /
6, Intensity - difference Fig. 2. The projected quadrature and intensity-difference squeezing generated in thephase-conjugate four-wave-mixing geometry in the ideal case of no loss and with | κ | L = π / π /
6) corresponding to red (blue) curves. The solid and dashed curvescorrespond to quadrature squeezing and the horizontal dot-dashed lines correspond tointensity-difference squeezing, which is independent of the phase of the measurement.Shot noise corresponds to 0 dB. Negative (positive) values correspond to squeezing(anti-squeezing).
The projected squeezing as a function of input pump intensity in the case of 30% loss(transmittion parameter η = .
7) is shown in Fig. 4(a). Here, we use typical experimentalparameters for atomic density, field detuning, and pump intensity, as defined in the caption.The various curves correspond to different decoherence rates, where the best squeezing isobtained for the slowest rates of decoherence. The oscillatory solutions in Eqs. 20 and 21 giverise to oscillatory squeezing results, which are not immediately intuitive. (We note that thecurves representing faster decoherence rates also oscillate, but their first minima occur at higherpump intensities.) In this phase-conjugate beam geometry, the counterpropagating pump beamsgenerate a sinusoidal interference pattern across the atoms, which in turn creates a sinusoidalintensity-depedent index of refraction. One can therefore interpret this process as a type of“nonlinear” Bragg scattering, where the efficiency of scattering depends on the index of refractionand the angle of the optical field relative to the spatially varying index of refraction. In theforward four-wave-mixing beam geometry described in Sec. 4, there is only a single pump beam,and the projected squeezing correspondingly has a non-oscillatory solution.
It is also worthwhile to analyze the projected squeezing of an alternative detection scheme, knownas “intensity-difference detection,” which is generally simpler to implement than homodynedetection in practice. In this case, one sends each beam generated by the four-wave-mixingprocess to one of two photodiodes on a balanced detector. This type of direct measurementthat does not require a local oscillator is considered to be particularly useful for simplifying thedetection apparatus for certain applications in quantum imaging and metrology.To model intensity-difference squeezing, we consider the number operators of the output fields, n f = a † f ( L ) a f ( L ) and n b = a † b ( ) a b ( ) . The intensity difference is N − = n f − n b . The noise ofthe intensity-difference measurement is therefore (cid:104) N − (cid:105) = γ f − γ b , where γ f ( γ b ) is the seedphoton number for the forward (backward) mode. In the vacuum-seeded case, γ f = γ b =
0. Forthe seeded case considered below, I assume the forward mode is seeded with a weak beam ofphoton number γ f = γ (cid:29) γ b = M ID = −
10 log (cid:20) ∆ N − ∆ ˆ n SN (cid:21) , (26) a) (b) Fig. 3. Squeezing as a function of η for (a) the phase-conjugate beam geometry and (b)the forward beam geometry. For both figures, we assume the following parameters: Γ = . Γ , Γ = π × Γ = π ×
30 kHz (approximately the rate at whichan atom will traverse the vapor cell), d = . × − C · m, ∆ ≈ ∆ = ∆ = Γ ,L = I p = , n a = atoms/m . where ∆ ˆ n SN is the shot noise, i.e., the variance of the measurement of two coherent beamshaving the same intensities as the forward and backward beams generated in the four-wave-mixingprocess.To calculate shot noise, we need to first calculate the number of photons in each beam at theoutput of the four-wave-mixing process, i.e., (cid:68) a † f ( L ) a f ( L ) (cid:69) and (cid:68) a † b ( ) a b ( ) (cid:69) . For the coherentbeams used for our shot noise measurement, the variance is simply equal to the average value,and thus (cid:10) ∆ ˆ n (cid:11) = (cid:10) ˆ n f (cid:11) + (cid:104) ˆ n b (cid:105) . In the single-seeded case, (cid:10) ∆ ˆ n (cid:11) = γ sec (| κ | L ) + ( γ + ) tan (| κ | L ) . (27)Comparing the noise (cid:10) ∆ N − (cid:11) = γ to the shot noise, one can see that the four-wave-mixing processhas not altered the average fluctuations on the output beams. However, it has increased the photonnumber while keeping the noise unchanged, hence giving rise to a squeezing of M SQ,PC ,η = = −
10 log (cid:20) γγ sec (| κ | L ) + ( γ + ) tan (| κ | L ) (cid:21) . (28)In the limit where | κ | L → π /
2, the trend approaches infinite squeezing in this ideal case of noloss. This limiting behavior agrees with the results of Ref. [11]. The expected intensity-differencesqueezing for example values of | κ | L is shown in Fig. 2. It is interesting to note that the expectedintensity-difference squeezing is less than that expected for the optimal quadrature measurement.This result can be attributed to the fact that the intensity-difference measurement essentiallyneglects all phase information, and hence performs a measurement related to a projection of theamplitude quadrature rather than the full noise ellipse. This result is consistent with squeezinggenerated by non-degenerate four-wave-mixing [19].The intensity-difference squeezing in the presence of loss in the case where η f = η b = η and γ (cid:29) M ID,PC = (cid:20) − η (cid:18) + ( | κ | L ) − (cid:19)(cid:21) . (29)The projected squeezing as a function of the transmission parameter η is shown in Fig. 3(a)for example experimental parameters defined in the caption. The projected squeezing level a) (b)(c) (d) Fig. 4. Quadrature and intensity-difference squeezing for the (a,b) phase-conjugate and(c,d) forward four-wave-mixing beam geometries as a function of pump intensity forthe example case of η = .
7. In order from the shortest to longest dashing, the curvesshow the cases Γ = . Γ , Γ = . Γ , Γ = . Γ , and Γ = . Γ , with the latterbeing the solid curve. All other parameters are the same as that for Fig. 3. follows closely to the projected quadrature squeezing for these particular parameters where thelight-atom interaction is strong. It is worthwhile to note that | κ | L < π / Γ > . Γ ), onerequires fairly high pump intensities to observe measurable squeezing. However, for sufficientlylow decoherence rates, we project that more than 3 dB of squeezing is attainable at low pumpintensities (less than 10 W/cm ).Phase-conjugate DFWM is therefore a promising technique for generating squeezing usinglow optical powers. We note that, in practice, this process requires careful alignment of thecounterpropagating pump fields to ensure optimal gain in the four-wave-mixing process, wheregain is defined as the ratio of the power in the output probe beam to the input seed beam. Inaddition, the present work employs a single-mode plane wave description of the optical fields,whereas in practice, multimode emission can occur [22]. In the high-gain regime, multimodeemission can give rise to cross-correlations among the generated fields which potentially suppresssqueezing in a given spatial mode. We therefore anticipate that it will be beneficial to workwith fairly low gains in practice to suppress multimode four-wave-mixing. The phase-conjugatefour-wave-mixing process also requires a sufficiently small angle between the pump and probebeams to maximize their overlap and hence the effective length L , but it should be kept sufficientlylarge to suppress additional four-wave-mixing processes. . Forward four-wave mixing In the case of very small pump-probe angles, another type of four-wave-mixing process can occur.Unlike the phase conjugation process, the forward four-wave-mixing process shown in Fig. 1(c)is only nearly phase-matched, but it can also give rise to high gain under certain conditions.The electric field above threshold for forward four-wave-mixing is −→ E = −→ F ( z ) e ikz + −→ f − ( z , x ) e ik ( cos θ z − sin θ x ) + −→ f + ( z , x ) e ik ( cos θ z + sin θ x ) , (30)where k is the wavenumber in vacuum, and f + and f − are weak fields. In this case, with f ± ( z , x ) = f (cid:48)± e i δ k z , δ k = ( δ + k − k cos α ) , and operating with the pump-probe angle α ≈ (cid:112) χ lin I p / I s ∆ (typically 3-5 mrad) for optimizing the phase-matching condition, we find the solutions to thewave equation for the probe and conjugate fields in the quantized treatment are a − ( z ) = i ν | ν | sinh (| ν | z ) a † + ( ) + cosh (| ν | z ) a − ( ) (31)and a † + ( z ) = cosh (| ν | z ) a † + ( ) − i ν ∗ | ν | sinh (| ν | z ) a − ( ) , (32)where ν = − k θ χ lin I p I s ∆ . (33)These solutions agree with the results of Ref. [23].Using the same calculation methods presented above for phase-conjugate four-wave-mixing,the quadrature noise in the ideal case of no loss for forward four-wave-mixing goes as (cid:10) ∆ j (cid:11) = (cid:12)(cid:12)(cid:12)(cid:12) − i cosh (| ν | L ) + e − i ( θ + + θ − ) ν | ν | sinh (| ν | L ) (cid:12)(cid:12)(cid:12)(cid:12) , (34)where θ ± represent the homodyne detector phases for the f ± fields. The projected quadraturesqueezing is shown in Fig. 3(b) as a function of loss and in Fig. 4(c) as a function of pump intensity.The squeezing shows the expected trend, where it is optimized for low loss and higher pumpintensities. At low intensities, we find that the forward four-wave-mixing case is not projectedto produce as much squeezing as the phase-conjugate beam geometry for these parameters, butit can still potentially produce more than 3 dB of squeezing with less than 10 W/cm of pumpintensity for a decoherence rate of Γ = . Γ .For the intensity-difference measurement, the noise in the ideal, no loss, single-seeded case isgiven by (cid:10) ∆ N − (cid:11) = γ , which is the same as that for the phase-conjugate four-wave-mixing beamgeometry. The shot noise is (cid:10) ∆ ˆ n (cid:11) = − + ( + γ ) cosh ( | ν | L ) . (35)Hence, the intensity-difference squeezing for this forward geometry in the limit where γ (cid:29) M ID,FFWM = (cid:20) ( | ν | L ) (cid:21) . (36)In the presence of loss where η = η + = η − , M ID,FFWM = [ − η + η sech ( | ν | L )] . (37)The intensity-difference squeezing as a function of η is shown in Fig. 3(b) for example experimentalparameters defined in the caption. We also show the projected squeezing as a function of pumpntensity in Fig. 4(d). We find that for both quadrature and intensity-difference measurements inthe forward four-wave-mixing case, the nonlinearity saturates above some pump intensity (justabove 10 W/cm for Γ = . Γ ). For pump intensities above this saturated value, the squeezingdoes not substantially increase, and hence it is not necessary to work well above the saturationintensity.
5. Conclusions and Outlook
We have shown that degenerate four-wave-mixing in the phase-conjugate and forward beamgeometries can produce more than 3 dB of squeezing for low ( (cid:46)
10 W/cm ) pump intensities forsufficiently low loss and long coherence times. To understand why squeezing using DFWM hasnot yet been observed experimentally, it is necessary to examine typical values of loss (1 − η )and decoherence rates ( Γ ). Loss arises from the vapor cell windows, optics, the quantumefficiency of the detectors, and, in the case of homodyne detection, the visibility of the fringesgenerating by interfering the generated field with a local oscillator. Typically, AR-coated opticswill only give rise to a few percent loss, largely due to polarization imperfections. The glass wallat the exit of an AR-coated cell should have <
1% loss, but in practice can have more and bedetuning-dependent due to atoms coating the cell walls. It is critical to heat the vapor cells insuch a way that the coldest part of the cell does not coincide with the optical paths. One of thelargest sources of loss can arise from the photodiode in the detection apparatus, and it is criticalto work with the highest possible quantum efficiency detector to minimize loss.We hypothesize that previous experimental works were unable to observe squeezing usingdegenerate four-wave-mixing due to a combination of optical loss and high rates of decoherence,especially from wall collisions. Additional sources of noise in experiments may also arise fromasymmetries in the optical fields and imperfect alignment along with nonlinear lensing effectsof the pump beam(s) that can generate multimode coupling. By incorporating techniques forreducing decoherence and minimizing optical loss, we predict that squeezing is attainable usingdegenerate four-wave-mixing even with low ( ≈
10 mW) optical powers. Techniques for reducingdecoherence include the use of OTS-coated cells or a separate pump beam for optically pumpingatoms that have collided with the walls of the vapor cell [13]. These techniques allow atomsto immediately participate in four-wave-mixing with the primary pump beam, which reducesspontaneous emission from optical pumping in the spatial modes of interest. With the morerecent advances in vapor cell quality and lower-cost high-quantum-efficiency detectors, we expectthat degenerate four-wave-mixing can enable the development of an efficient, low-power sourceof squeezed light.
Acknowledgements
We thank Dr. Brielle Anderson for many helpful discussions in recent years on squeezing andnoise. We gratefully acknowledge funding from the MITRE Innovation Program.©2020 The MITRE Corporation. ALL RIGHTS RESERVED.
References
1. LIGO Collaboration, “Enhanced sensitivity of the ligo gravitational wave detector by using squeezed states of light,”Nat. Photonics , 613 EP – (2013).2. A. Kumar, H. Nunley, and A. M. Marino, “Observation of spatial quantum correlations in the macroscopic regime,”Phys. Rev. A , 053849 (2017).3. C. Peuntinger, B. Heim, C. R. Müller, C. Gabriel, C. Marquardt, and G. Leuchs, “Distribution of squeezed statesthrough an atmospheric channel,” Phys. Rev. Lett. , 060502 (2014).4. J.-i. Yoshikawa, S. Yokoyama, T. Kaji, C. Sornphiphatphong, Y. Shiozawa, K. Makino, and A. Furusawa, “Invitedarticle: Generation of one-million-mode continuous-variable cluster state by unlimited time-domain multiplexing,”APL Photonics , 060801 (2016).5. N. V. Corzo, Q. Glorieux, A. M. Marino, J. B. Clark, R. T. Glasser, and P. D. Lett, “Rotation of the noise ellipse forsqueezed vacuum light generated via four-wave mixing,” Phys. Rev. A , 043836 (2013).. A. Schönbeck, F. Thies, and R. Schnabel, “13 db squeezed vacuum states at 1550 nm from 12 mw external pumppower at 775 nm,” Opt. Lett. , 110–113 (2018).7. A. M. C. Dawes, L. Illing, J. A. Greenberg, and D. J. Gauthier, “All-optical switching with transverse optical patterns,”Phys. Rev. A , 013833 (2008).8. M. W. Maeda, P. Kumar, and J. H. Shapiro, “Observation of optical phase-sensitive noise on a light beam transmittedthrough sodium vapor,” Phys. Rev. A , 3803–3806 (1985).9. M. D. Reid and D. F. Walls, “Quantum theory of nondegenerate four-wave mixing,” Phys. Rev. A , 4929–4955(1986).10. R. S. Bondurant, P. Kumar, J. H. Shapiro, and M. Maeda, “Degenerate four-wave mixing as a possible source ofsqueezed-state light,” Phys. Rev. A , 343–353 (1984).11. M. D. Reid and D. F. Walls, “Generation of squeezed states via degenerate four-wave mixing,” Phys. Rev. A ,1622–1635 (1985).12. J. D. Swaim and R. T. Glasser, “Squeezed-twin-beam generation in strongly absorbing media,” Phys. Rev. A ,033818 (2017).13. L. Zhu, X. Guo, C. Shu, H. Jeong, and S. Du, “Bright narrowband biphoton generation from a hot rubidium atomicvapor cell,” Appl. Phys. Lett. , 161101 (2017).14. M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D. Budker, E. B. Alexandrov, M. V. Balabas, andV. V. Yashchuk, “Relaxation of atomic polarization in paraffin-coated cesium vapor cells,” Phys. Rev. A , 023401(2005).15. D. Budker, L. Hollberg, D. F. Kimball, J. Kitching, S. Pustelny, and V. V. Yashchuk, “Microwave transitions andnonlinear magneto-optical rotation in anti-relaxation-coated cells,” Phys. Rev. A , 012903 (2005).16. M. T. Turnbull, P. G. Petrov, C. S. Embrey, A. M. Marino, and V. Boyer, “Role of the phase-matching conditionin nondegenerate four-wave mixing in hot vapors for the generation of squeezed states of light,” Phys. Rev. A ,033845 (2013).17. R. Boyd and D. Prato, Nonlinear Optics , Nonlinear Optics Series (Elsevier Science, 2008).18. A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,”Opt. Lett. , 16–18 (1977).19. B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements withbright- and vacuum-seeded su(1,1) interferometers,” Phys. Rev. A , 063843 (2017).20. W. J. Brown, J. R. Gardner, D. J. Gauthier, and R. Vilaseca, “Amplification of laser beams propagating through acollectionof strongly driven, doppler-broadened two-level atoms,” Phys. Rev. A , R1601–R1604 (1997).21. M. Pinard, R. W. Boyd, and G. Grynberg, “Third-order nonlinear optical response resulting from optical pumping:Effects of atomic motion,” Phys. Rev. A , 1326–1336 (1994).22. A. M. C. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science ,672–674 (2005).23. P. Kumar and J. H. Shapiro, “Squeezed-state generation via forward degenerate four-wave mixing,” Phys. Rev. A ,1568–1571 (1984).24. T. Li, B. E. Anderson, T. Horrom, B. L. Schmittberger, K. M. Jones, and P. D. Lett, “Improved measurement oftwo-mode quantum correlations using a phase-sensitive amplifier,” Opt. Express , 21301–21311 (2017). ppendix: Optical loss We model loss using the same method presented in Ref. [24], in which “beam splitters" oftransmission η f , b are inserted into the path of the probe and conjugate beams, as shown in Fig. 5.Here we describe the case of the phase-conjugate four-wave-mixing beam geometry, but it isstraightforward to apply this procedure to the forward four-wave-mixing case. We define theinitial optical state before the four-wave-mixing process as v = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) a f ( ) a † f ( ) a b ( L ) a † b ( L ) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) . (38)The output states are defined as (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) a f ( L ) a † f ( L ) a b ( ) a † b ( ) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = M · v , (39)where M = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) sec (| κ | L ) i | κ | κ ∗ tan (| κ | L ) (| κ | L ) − i | κ | κ tan (| κ | L ) i κ | κ | tan (| κ | L ) sec (| κ | L ) − i κ ∗ | κ | tan (| κ | L ) (| κ | L ) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) . (40)The transmissions of the beam splitters model linear optical loss, such as reflections off surfacesand imperfect detector quantum efficiencies. These beam splitters add vacuum noise, as shownin Fig. 5 by the vacuum states v and v . We define L = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) √ η f √ η f √ η b
00 0 0 √ η b (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , (41)which describes the transmission through the beam splitters, and the vector V = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) i (cid:112) − η f v − i (cid:112) − η f v † i (cid:112) − η f v − i (cid:112) − η f v † (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , (42) ig. 5. A schematic of the theoretical model for incorporating loss that models beamsplitters of transmission η f , b , which also introduce vacuum noise. The vacuum fieldsare represented by the field operators v , . which describes the coupling of vacuum noise into the optical fields. Then the final state is v fin = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) a f a † f a b a † b (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = L · (M · v ) + V . (43)For the phase-conjugate four-wave-mixing geometry, the squeezing as a function of thetransmission parameter η is shown in Fig. 3 in the case ηη
00 0 0 √ η b (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , (41)which describes the transmission through the beam splitters, and the vector V = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) i (cid:112) − η f v − i (cid:112) − η f v † i (cid:112) − η f v − i (cid:112) − η f v † (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , (42) ig. 5. A schematic of the theoretical model for incorporating loss that models beamsplitters of transmission η f , b , which also introduce vacuum noise. The vacuum fieldsare represented by the field operators v , . which describes the coupling of vacuum noise into the optical fields. Then the final state is v fin = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) a f a † f a b a † b (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = L · (M · v ) + V . (43)For the phase-conjugate four-wave-mixing geometry, the squeezing as a function of thetransmission parameter η is shown in Fig. 3 in the case ηη f = ηη
00 0 0 √ η b (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , (41)which describes the transmission through the beam splitters, and the vector V = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) i (cid:112) − η f v − i (cid:112) − η f v † i (cid:112) − η f v − i (cid:112) − η f v † (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) , (42) ig. 5. A schematic of the theoretical model for incorporating loss that models beamsplitters of transmission η f , b , which also introduce vacuum noise. The vacuum fieldsare represented by the field operators v , . which describes the coupling of vacuum noise into the optical fields. Then the final state is v fin = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) a f a † f a b a † b (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = L · (M · v ) + V . (43)For the phase-conjugate four-wave-mixing geometry, the squeezing as a function of thetransmission parameter η is shown in Fig. 3 in the case ηη f = ηη b = ηη