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Photon Production from the Quantum Vacuum: Taking a Break from the Bits

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Photon Production from the Quantum Vacuum: Taking a Break from the Bits Photon Production from the Quantum Vacuum: Taking a break from the bits Paul Nation RIKEN / U. Michigan 6th Winter School on Quantum Information Science & Quantum Phenomena Essence of the Quantum Vacuum: [x, p]=i 1 (Δ )2 (Δ )2 ≥ |[ ]|2 A B 4 A, B 2 (Δ )2 (Δ )2 ≥ x p 4 The quantum vacuum state is ALWAYS fluctuating: Leonhardt, 2010 position (x) time Origin of some of the most important physical processes in the universe. spontaneous emission Casimir effect Λ large-scale structure cosmological constant These are all STATIC vacuum effects DYNAMICAL vacuum amplification: Dynamics driven by energy source spontaneous emission Casimir effect Particle productionΛ large-scale structure cosmological constant Dynamical Quantum Vacuum Effects: Dynamical Casimir Hawking Radiation T ∝ a Accel Parametric Amplifier Unruh Effect Why so hard to detect? Dynamical Casimir: N∝v/c - Need small mass / massless mirror. Unruh effect: kBTU ∝ a/c - Requires very large accelerations. - Detector must be accelerating. Hawking radiation: kBTH ∝ κ/c −9 - For small black hole T H ∼ 10 K . - No way to verify photons come from black hole. Goal #1: #1: Introduce vacuum photon amplification effects, highlighting relationship to quantum parametric amplifier. I. Standard quantum optics effect. II. Workhorse of quantum optics, circuit- QED,... nonclassical states, wave-particle duality, quantum erasers, quantum teleportation, heralded photons... III. Contains the basic ingredients of all vacuum amplifiers. Goal #2: #2: Demonstrate the use of superconducting circuits to realize these effects. - Low noise and dissipation. I. Photons can travel 10km before dissipation. II. Can maintain entanglement - Controllability on the single-photon level. I. Single photons on demand. II. Controlled generation of quantum cavity states. III. Nonlinearities on the single-photon level. - Single-shot photon detection (PRL 107 217401) I. Can detect and measure single photon pairs Parametric amplifiers: Castellanos-Beltran, Nat. Phys. 4, 929 (2008) Bergeal, Nature 465, 64 (2010) Roch, PRL (In Press) (2012), arXiv:1202.1315 Hawking radiation: Nation, PRL 103, 087004 (2009) Dynamical Casimir Effect: Johansson, PRL 103, 143003 (2009) Wilson, Nature 479, 376 (2011) Lähteenmäki, arXiv:1111.5608 (2011) Rev. Mod. Phys. 84, 1 (2012) Classical Parametric Amplifier Def: A parametric amplifier is a system that amplifies an input by varying a parameter (frequency) of the system. So easy, a child can understand! Classical Parametric Amplifier Don Case, Grinnell Classical Parametric Amplifier L(0) Harmonic oscillator: θ(t)=θ(0) cos(ωst)+ sin(ωst) mωsl Modulate center of mass ( ): ωs(t)=ωs(0) + sin (ωcmt) if ω cm =2 ω s : t/2 L(0) −t/2 θ(t)=θ(0)e cos(ωst)+ e sin(ωst) mωsl Occurs only if system is initially displaced. In classical physics: θ(0) = L(0) = 0 Quantum physics: [θ, L] =0 θ(0) = L(0) = 0 Can parametrically amplify any state, even the vacuum! Closely related to particle production in quantum field theory. Prelude to quantum amplification Harmonic oscillator: H = p2/(2m)+mω2x2/2 [x, p]=m [x, x˙]=i Work in Heisenberg picture: x¨ + ω2x =0 Decompose into creation and destruction operators: complex conjugate ¯ † x(t)=f(t)a + f(t)a “mode function” (not operator) ¨ 2 Mode functions satisfy: f(t)+ω f(t)=0 Plug x ( t ) into commutator: demand = 1 for all time m m ¯˙ ¯ ˙ † [x, x˙]= f(t)f(t) − f(t)f(t) a, a =1 i i Define inner-product: (Klein-Gordon inner-product) ≡im ¯( )˙( ) − ( ) ¯˙( ) f,g f t g t g t f t orthogonal Gives: ¯ f,f =1 f,f =0 † ¯ a = −f,x a = f,x ground a = f,x mode state function Ground state:a|0 =0 Want ground state as eigenstate of H: ˙ 2 2 2 |0 = mx + mω x |0 H 2 2 2 m ˙ 2 2 2 m ˙ 2 2 = √ f(t) + ω f(t) |2 + f(t) + ω |f(t)| |0. 2 2 Solutions:( x zp = / 2 mω ) ¯ f(t)=xzp exp(−iωt) f(t)=xzp exp(+iωt) “positive frequency” “negative frequency” solution solution thus: −iωt +iωt † x(t)=xzp e a + e a let us now modulate frequency (like the swing): x¨ + ω(t)2x =0 ( →∞)= in: ω(t →−∞)=ωin out: ω t ωout ain aout |0in |0out fin(t)|t→−∞ ∼ exp (−iωint) fout(t)|t→+∞ ∼ exp (−iωoutt) In general, ¯ † ¯ † x(t)=fin(t)ain + fin(t)ain = fout(t)aout + fout(t)aout Interested in “out” state given “in” is vacuum - I know fin need 2 linearly independent solutions: ¯ ¯ f,f =0 fout = αfin + βfin using a out = f out ,x : ¯ † 2 2 aout = αain − βain ; |α| −|β| =1 Bogoliubov transformation All quantum amplifiers can be cast as Bogoliubov transformations. if “in” state | 0 in , then particle # at “out” state: † 2 Nout = 0|aoutaout|0in = |β| Particle # at “out” determined by negative frequency † ( a in ) coefficient 2 |β| ≈ 0 ωout adiabatic ωin time 2 2 1 ωin ωout ωout |β| = 1 − 4 ωout ωin δ-function ωin time Take home message: All amplifiers described by Bogoliubov transform: ¯ † 2 2 aout = αain − βain |α| −|β| =1 Vacuum defined via positive-frequency ( ae − iωt ) components by choice of mode function. Particles at “output” given by coefficient of ¯ † negative-frequency “input state”: βain Nonzero β generated by modulating mode frequency non-adiabatically. Vacuum Photon Production Methods Quantum Parametric Amplifier Simplest nonlinear photon interaction signal ( ω s ) pump ( ω p ) idler ( ω i) Energy conservation: ωp = ωs + ωi Optical par. amp. needs nonlinear medium (BBO,KTP) (2) Works for any system with effective χ nonlinearity. Will assume pump mode is classical drive,unaffected by loss of photons. i.e. unlimited, fixed amplitude supply of energy † † Hamiltonian (rotating frame): H = iη(bs bi − bsbi) Pump amplitude in coupling constant: η (frequency) Two modes of operation: Degenerate (DPA): ωs = ωi Non-degenerate (NDPA): ωs = ωi DPA Solve Heisenberg Eq. of Motion for bs = bi = b b(t)=b(0) cosh (2ηt)+b(0)† sinh (2ηt) , Bogoliubov transformation: α = cosh (2ηt) β = sinh (2ηt) If mode initially in ground state: 2 2 N = b†(t)b(t) = |β| = sinh (2ηt) # of particles from vacuum DPA If bs = bi = b ωp =2ωb (like the swing) † † 1 = + , X2 = −i(b − b ) let X b b squeezing parameter 2ηt −2ηt X1(t)=e X1(0) X2(t)=e X2(0) Squeezed State Ground State Squeezed state: 2ηt =0.5 2 2 X X momentum momentum position X1 position X1 NDPA More general case: ωs = ωi Heisenberg eqs. of motion: † bs(t)=bs(0) cosh (ηt)+bi (0) sinh (ηt) † bi(t)=bi(0) cosh (ηt)+bs(0) sinh (ηt) 2 Ns = Ni = sinh (ηt) In Schrödinger picture: two-mode squeezed state 1 ∞ |Ψ( ) = (tanh )n | ⊗| t cosh ηt n s n i ηt n=0 squeezing parameter Example of EPR state; mode-mode correlations stronger than classically allowed. NDPA What if only single-mode is observable? ? Can do a partial trace, or we can think about it ... Implicitly know energy Energy conservation of other mode Missing information Entropy Find max. entropy constrained by energy NDPA Gives entropy for single-mode thermal state: −1 ωs S = − ln 1 − e−ωs/kBT (t) − 1 − eωs/kBT (t) kBT (t) Temperature given via: 2 ωs tanh ηt = exp − kBT (t) Temperature is determined by squeezing parameter. Bipartite system so both modes in thermal state. Unruh Effect Simplest particle production method: just accelerate! Accelerating system called “Observer” Need two coordinate systems (constant a ): a =0: Minkowski Coordinates (ct, x) a>0 : Rindler Coordinates (cτ, ξ) Rindler coordinates defined by relations: aτ aτ ct = ξ sinh x = ξ cosh c c ξ = c2/a α = a/c acceleration frequency Rindler coordinates are hyperbolas in Min coordinates: ct Minkowski space- partitioned into tw LRW future horizonRRW Sections causally by horizon at x Observer confined Rindler Wedge” past horizon Observer reaches Can define “Left Rindler Wedge” via: In LRW, moves backward in Minkowski time Goal: Determine what Minkowski vacuum looks like in observers (Rindler) frame. Follow harmonic oscillator example: I. Find mode functions & vacuum states in both space-times. II. Calculate Bogoliubov transformation linking operators in both space-times. II. Evaluate number operator and determine quantum state. Minkowski Space-time: (inertial observer) = M M +¯M M,† Consider scalar-field: φ uωj aωj uωj aωj j Solution to wave equation: ωj = c|kj|−∞≤j ≤ ∞ 1 2 2 1 ∂ − ∂ =0 M = ikj x−iωj t 2 2 2 φ uωj e c ∂t ∂x 4πωj Minkowski time Vacuum state: |0M = |0 M M |0M =0 ∀ ωj aωj j j Rindler Space-time: (accel. observer) Defines vacuum in Rindler coordinates (cτ, ξ) RRW: proper time R R R |0 = |0ω ∝ exp (− ) j uωj iωjτ j LRW has its own independent vacuum state: |0L = |0 L L ωj ∝ exp ( ) uωj iωjτ j Positive and negative frequency switch roles in LRW! Need both LRW & RRW to get complete Minkowski Guess the Bogoliubov transformation: Remember vacuum defined by positive freq. terms M = R + L,† | |2 −| |2 =1 aωj C1aωj C2aωj C1 C2 positive frequency in LRW Minkowski vacuum should have particles when viewed by Rindler observer! Single mode transformations: (1),M = R cosh ( )+ †,L sinh ( ) bωj aωj r aωj r (2),M = L cosh ( )+ †,R sinh ( ) bωj aωj r aωj r Operators transform exactly like NDPA! tanh r = exp (−πωj/α) squeezing parameter Same two-mode squeezed state: 1 ∞ |0 M = (tanh )n | L ⊗| R ωj cosh r nωj nωj r n=0 particles in LRW particles in RRW Thermal state determined by squeezing parameter: 2 ω tanh (r)=e−2πω/α = exp − kBTU Defines Unruh temperature: α TU = depends only on accel. freq: α = a/c 2πkB Same temperature for all frequencies; real thermal state (not true for NDPA) Why thermal? Photons generated in pairs: one in LR Observer confined to RRW ct LRW future horizonRRW ? past horizon Thermal from lack of information due to presence of horizon.
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