Nonlinear Optics and Squeezed Light with 4WM Paul D. Lett National Institute of Standards and Technology and Joint Quantum Institute NIST and U. Maryland
outline (day 2) squeezed light non-classicallity; entanglement squeezing with a delay squeezing in multiple spatial modes 4-wave mixing in atoms an SU(1,1) interferometer
4WM in hot Rb vapor 85 Rb in a double-λ scheme ~120 C cell temp. ~1 GHz detuned ~400 mw pump ~100 µw probe - narrowband - no cavity
what do we expect from 4WM in a double-lambda system? intensity correlations (parametric process) leads to intensity difference squeezing entanglement EPR violations?
phase sum and intensity difference relations φ 0 φ 0 phase-sum squeezing φ 0 + φ 0 =φ p + φ c φ c φ p intensitydifference squeezing is really photonnumber-difference squeezing (same number of photons in each beam)
4-wave mixing frequency converter: couple 3 optical fields to make a 4th: E 4 = χ (3) E 1 E 2 E 3 requires that energy and momentum are conserved between input and output energy: ω 1 + ω 2 = ω 3 + ω 4 momentum: k 1 + k 2 = k 3 + k 4 If we are careful about how we do it, inserting 3 fields gets us a 4th with some magic properties (quantum)
light amplitude and phase or quadratures quantum version with noise coherent states have minimumsized, round fuzz-ball
squeezed light what does it mean for light to be squeezed? classical fields have symmetric noise amplitude phase different kinds of squeezing: amplitude/phase or quadrature
Normal squeezing (in time) classical or coherent light random arrivals in time and space i(t) shot noise random in time; no correlations N photons detected, on average standard deviation is N squeezing; ordered in time; reduced fluctuations i(t) intensity squeezing N photons detected, on average standard deviation is < N can be used to reduce noise in measurements
detecting squeezing classical light random arrivals in time and space i(t) we can subtract technical noise but shot noise will still remain - noise power ~ N in T
detecting squeezing classical or coherent light random arrivals in time and space vacuum input adds noise! i(t) vacuum noise is indistinguishable from shot noise so we won t see it on a classical beam - noise power ~ N in T
detecting squeezing classical or coherent light: random arrivals in time and space squeezed light gives some order to the arrivals! vacuum in; adds noise! i(t) random deletions; adds noise any loss adds noise back into measurements!
Twin beam or relativeintensity squeezing i 1 (t) i 2 (t) i 1 (t)= i 2 (t) to better than shot noise allows I can t clone a photon, but I can create 2 (nearly) identical photons or beams of them
spatial modes object lens aperture Putting a spatial filter in - an aperture in the Fourier plane - selects out the lowest order spatial mode. This is a Gaussian beam; a TEM 00 mode. A complex image can be described in terms of higherorder Hermite polynomials; orthogonal modes of the field.
spatial squeezing: quantum imaging classical or coherent light random arrivals in time and space shot noise on each pixel no correlations between pixels multi-mode squeezing: independent spatial modes are squeezed each detected current can be independently squeezed! can be used to reduce noise in spatial information
twin-beams with spatial squeezing classical or coherent light random arrivals in time and space shot noise on each pixel no correlations between pixels spatial squeezing; ordered in space, correlations between pixels i 4 (t) i 3 (t) i 2 (t) i 1 (t) we have both time and spatial squeezing: correlated, squeezed currents can be used to reduce noise in spatial information
quantum correlations from 300 mw of laser light @ 795 nm 4WM 1 cm cell of rubidium vapor @ 130 C < 1 mw of light at 795 nm + 3 GHz output: 2 amplified beams of quantum-correlated and entangled light correlated: If a particle in beam 1 has spin up, then the corresponding particle in beam 2 has spin down entangled: I get to choose the spin axis randomly
how do quantum correlations help us? It allows us to avoid some of the limitations of classical light on resolution and noise in optical systems. It is useful in position sensing, interferometry, and image processing.
Single-mode squeezing
field quadratures when you have a vacuum field it is difficult to define, say, amplitude and phase... X = x cos θ p sin θ Y = x sin θ + p cos θ
Two-mode squeezing: phase-insensitive amplifier p 1 x 1 Coupled Gain correlations p 2 x 2 two vacuum modes two noisy, but entangled, vacuum modes
Squeezing quadratures
homodyne detection mix signal with bright beam of same frequency get amplification of a small signal local oscillator fluctuations cancel out phase sensitive E +ε 2 = E 2 + ε 2 + E * ε + Eε * E ε 2 = E 2 + ε 2 E * ε Eε * E >> ε
beamsplitter magic t field amplitude relations at a beamsplitter r r*t + t*r = 0 r 2 + t 2 = 1
noise squeezed light implies, in some form, reduced fluctuations this is usually compared to shot noise N particles/second => noise ~ N 1/2 state of the art; (linear and log) 3 db = factor of 2; 10% noise = -10 db world records: (using an OPO): -9.7 db (11%) for twin beams -12. db for single-mode quadrature squeezing
Measured vs.inferred squeezing If you correct for losses in transmission and detection of photons, and assume the beams are balanced, you can project to what might have been measured in a more perfect world... from -7.1 db measured: we infer -10.2 db squeezing at the source from η = 0.95 x 0.93 = 0.89 - approximately perfect squeezing generated at the source if you allow for the injected signal!
levels of CV entanglement 1 - correlations cannot be explained by a semiclassical model 2 - information from one field provides a QND measurement of the other 3 - correlations arise from entangled or non-separable state 4 - information extracted from measuring two quadratures of one field provide values for the two quadratures of the other field that would apparently violate the Heisenberg inequality; EPR beams V X+1 X+2 V X-1 X-2 < 1 EPR criteria cannot be fulfilled with more than 3 db (50%) loss; no spy can have a better copy of the state. N. Treps and C. Fabre, arxiv: quant-ph/0407214 (2004)
" Two systems, a and b, are entangled or inseparable if it is not possible to describe them independently. " Independent of basis: Alice EPR (Einstein-Podolsky-Rosen) Paradox Bob Particle a EPR Particle b " For variables that have a continuous range of possible values (e.g. amplitude and phase):
" Need to look at collective variables " SQL: Joint quadratures " Inseparability criterion: " EPR criterion: where Amplitude difference Phase sum L.M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, PRL 84, 2722 (2000). R. Simon, PRL 84, 2726 (2000). M.D. Reid, PRA 40, 913 (1989).
squeezing from 4WM in hot Rb vapor 85 Rb in a double-λ scheme ~120 C cell temp. ~1 GHz detuned ~400 mw pump ~100 µw probe - narrowband - no cavity
strong intensity-difference squeezing measured 1 MHz detection frequency RBW 30 khz VBW 300 Hz pump detuning 800 MHz Raman detuning 4 MHz ~95% detector efficiency
intensity-difference squeezing at low frequencies better than 8 db noise suppression if backgrounds subtracted!
high frequency spectra
Slow Light EIT (less loss) spectrum with associated dispersion gain features imply dispersion and slow light as well
pulse delay 70ns FWHM Gain ~ 13 δ=10 MHz 300 mw pump 10% broadening δ=22 MHz 250 mw pump 5% broadening pulse breakup when detuned closer to Raman absorption dip
delay-locking of matched pulses probe injected conjugate generated promptly by 4WM conjugate travels at 2v g relative delay locked effective cell length probe initially travels at v g, then locks to the conjugate with velocity 2v g when delay reaches lock point determined by gain
correlation delay fluctuations on a cw beam can be looked on as a superposition of pulses of different length! cross-correlation If the conjugate photons come out ahead of the probe photons, how does this affect the squeezing?
Spectra with electronic noise floor subtracted
Spectrum of squeezing S(Ω) = dτe iωτ : z(t)z(t + τ) : z(t) = I 1 (t) I 2 (t) The spectrum of squeezing is the Fourier transform of time- and normally-ordered correlation function of the intensity difference. z(t)z(t + τ) = A I1 (τ) + A I 2 (τ) 2C I1 I 2 (τ) auto- and cross-correlations S(Ω) = S I1 (Ω)+ S I 2 (Ω) 2S I1 I 2 (Ω)
Squeezing spectrum with delay S(Ω,T ) = dτe iωτ : y(t,t )y(t + τ,t ) : y(t,t ) = I 1 (t) I 2 (t + T ) intensity difference with delay T y(t,t )y(t + τ,t ) = A I1 (τ) + A I 2 (τ) C I1 I 2 (T + τ) C I1 I 2 (T τ) Fourier transform property: f (t) F(Ω) f (t τ) e iωτ F(Ω) S(Ω,T ) = S I1 (Ω)+ S I 2 (Ω) S I1 I 2 (Ω)e iωt S I1 I 2 (Ω)e iωt S(Ω,T ) = S I1 (Ω)+ S I 2 (Ω) 2S I1 I 2 (Ω)cos(ΩT )
Squeezing with more delay (linear scale)
Squeezing with a delay single beam noise default from the 4WM cell shot noise limit optimal delay to remove differential slow-light delay (8.5 ns)
4wm vs. OPOs...or... χ (3) vs χ (2) media OPO (optical parametric oscillator) cavity restricts spatial modes 4WM (four-wave mixing) spatial modes not restricted!
many spatial modes possible in 4WM seeded, bright modes we could, theoretically, use many spatial modes in parallel cone of vacuum-squeezed modes (allowed by phase matching)
demonstrating entanglement scan LO phase alignment and bright beam entanglement + or - probe pump conjugate pzt mirror phase stable local oscillators at +/- 3GHz from the pump pzt mirror
demonstrating entanglement vacuum squeezing unsqueezed vacuum + and - probe conjugate pumps pzt mirror 50/50 BS signal pump LO pump pzt mirror scan LO phase
twin beam vacuum quadrature entanglement measurements at 0.5 MHz
entangled images measurements at 0.5 MHz V. Boyer, A.M. Marino, R.C. Pooser, and P.D. Lett, Science 321, 544 (2008).
local oscillators for measurements of 1 db quadrature squeezed vacuum squeezed and entangled cats bright beams showing intensitydifference squeezing
multi-spatial-mode entanglement
squeezing or entanglement delay reference pulse pump pump ~10 ns delays ~30 ns extra delay 2nd cell gain adjusted to be ~1 with temperature and pump intensity
" Delay can be tuned by changing the gain: " Temperature " Pump power " Fractional delay:
E 12 = var(x 1 X 2 )var(y 1 Y 2 ) I = var(x - ) + var(y + ) " Generation of conjugate in second cell (due to gain) degrades the entanglement. A. Marino, et al., Nature, 457, 859 (2009).
" Fractional delay:
Heisenberg-limited interferometry classical vs. quantum scaling with entangled input states for same interferometer classical input 1/N 1/2 quantum input 1/N (Heisenberg limit) Holland-Burnett interferometer Hall/Pfister (JILA) - equal-number inputs for two arms - Basyean statistics analysis looking at output variance - can reach 1/N sensitivity - requires ~(1-1/N) detection sensitivity M. Holland and K. Burnett, PRL 71, 1355 (1993) SU(1,1) interferometer similar...? Yurke, McCall, Klauder, PRA 33, 4033 (1986) - atom-optical alternative
interferometers Yurke, McCall, Klauder, SU(2) and SU(1,1) interferometers, PRA 33, 4033 (1986) Passive interferometers (ex: Fabry-Perot and Mach- Zehnder) can be described by rotation group SU(2) beamsplitters and mirrors Some active interferometers can be described by SU(1,1) pump π active media such as pdc crystals or 4WM cells
ω - ω 0 phase-sensitive amplifier the phase of the injected beam, with respect to those of the pumps, will determine whether the beam will be amplified or de-amplified One can noiselessly amplify given field quadratures (maintain SNR) - very useful for signal processing! phase-insensitive ω + given the phase of 3 input beams the 4th phase is free to adjust for gain φ + = 2φ 0 - φ - gain: 0 = 2φ 0 - φ - - φ + ω 0 phase-sensitive no free ω + parameters ω -
SU(1,1) interferometer operation phase insensitive 4WM at first cell spontaneous 4WM at ω 1 phase-sensitive 4WM (deamplification) at second cell pump spontaneous 4WM at ω 2 π φ + at φ = 0 signal = any light out a π phase shift on the pump (2 x π/2 on the beam), and no phase in arms shift gives perfect deamplification and no output Yurke, et al., PRA 33, 4033 (1986)
comments on the interferometer - relies on perfect mode matching - simple intensity detection scheme works only near zero phase offset - a more sophisticated measurement strategy should give 1/N performance at all working points (similar to coherent/squeezed vacuum Mach-Zehnder interferometer; L. Pezze and A. Smerzi, PRL 100, 073601 (2008)) - note that the two arms contain light at different frequencies (nothing actually interferes in this interferometer, it is an active deamplification process)
complications - previous analysis does not include loss in system; losses add noise - mis-match of wavefronts at 2nd cell an issue due to Kerr lensing effects - slow light (dispersion effects) are different for the probe and conjugate beams - an entire cone of modes create parallel interferometers; seeding/homodyne can select one - typically we have very low gain in our system (G ~ 10)
Implementing an SU(1,1) interferometer looks not very hopeful in practice... mode matching, losses, dispersion in Rb vapor, all seem to hurt a lot. however... It does not look so bad for a sodium spinor BEC version of the SU(1,1) interferometer!
atom analogy Instead of doing 4WM with light, we can make an analogy with matter waves... While light requires matter to provide the nonlinear mixing interaction, matter can collide with itself and therefore can does not require an additional medium to mediate the interaction.
spinor BECs A spinor BEC is one where the atoms are in a superposition of internal states. Think of an F = 1 BEC in an optical trap, confined to a single spatial mode; there are three condensates with m F = -1,0,+1 that are coupled. You can also describe this as a single condensate with a vector (spinor) order parameter: ρ 1 e iφ 1 ρ 0 e iφ 0 ρ +1 e iφ +1
4WM in spinors instead of just mixing one spin state and several k-vectors, mix spin states spinor condensate; F = 1, m F = -1, 0, +1 a pair of m = 0 atoms can exchange for a m = +1 / m= -1 pair m = +1 m = 0 m = -1 m = 0 linear Zeeman cancels; quadratic Zeeman and collisional interactions left
spinor 4WM a number of groups have recently made interesting measurements on matter-wave 4WM in Rb spinor BECs Klempt/Ertmer, Stamper-Kurn, Bongs/Sengstock C. Klempt, et al., Phys. Rev. Lett. 103, 195302 (2009).! C. Klempt, et al., Phys. Rev. Lett. 104, 195303 (2010).! S. Leslie, et al., Phys. Rev. A 79, 043631 (2009).! L. Sadler, et al., Nature 443, 312 (2006).! J. Kronjager, et al., Phys. Rev. Lett. 97, 110404 (2006).! recent work:! C. Gross, et al., Nature, 480, 219 (2011) (Oberthaler)! B. Lücke, et al., Science 334, 773 (2011) (Ertmer/Klempt)! inspiring us to look at the 4WM analogy in sodium
spinor BEC interferometry initialize with all atoms in the m = 0 state in optical trap dress the m = 0 state (microwave field) to be highest E use 4WM interaction to produce correlated input state of +1 / -1 atoms for interferometry tight trapping ensures mode-matching (sacrifice enclosed path geometry) 87 Rb F=2 approximates F=1 4WM for low transfer; F=1 in 87 Rb or Na requires rf field dressing
Energy quadratic Zeeman shifts F = 1 0 > + 0 > + > + - > - > 0 > 1> m F + > + - > 0 > + 0 > Magnetic field Magnetic field
rf field dressing F = 2 0 > + 0 > + > + - > F = 1 0 > + 0 > + > + - > -1 > 0 > +1> m F rf field allows spontaneous creation of +/-1 pairs
atom interferometer option spinor condensate in an optical trap m = +1 atoms m = 0 BEC pump m= -1 atoms collisional interaction: +1 φ 0 π 0-1 m = 0 m = 0-1 +1 differential light shifts allow phase-shifting in trap m = +1 m = -1 pair creation collisional interaction in spinor BEC formally equivalent to 4WM for small transfer
Na spinor SU(1,1) advantages mode matching is forced by confinement in laser dipole trap; force into single mode(?) dispersion issues are removed in atom case losses are minimal in Na: (20 sec trap lifetime; 20 ms experiment time) shifts with rf dressing allow for easy phase shifting of pump m = 0 phase with respect to probe and conjugate (m F = +/-1 condensates) potential for high efficiency detection of small m = +/-1 condensates
observation techniques Make population measurements by Stern- Gerlach technique: - drop condensate in a magnetic field gradient - let expand and take image - destructive measurement of populations - possible state-selective ionization techniques for small atom numbers Β + 0 - only allows you to see populations
amplification of atom seed Single-sided seed to avoid phase sensitivity Phase-insensitive coherent growth of a seed condensate (small m = +1 condensate generated by rf transition from m = 0)
Credits Photons: Alberto Marino Kevin Jones (Williams) Ryan Glasser Neil Corzo (CINVESTAV) Quentin Glorieux Jeremy Clark Ulrich Vogl Zhifan Zhou (ECNU) Atoms: Jonathan Wrubel Paul Griffin (Strathclyde) Hyewon Pechkis Ryan Barnett (UMD) Eite Tiesinga Extras: PDL William D Phillips