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)
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 d - n s y w ” edge indler W ” via: partitioned into t Sections causall by horizon at Observer confine R Observer reache Minkowski space x edge
RRW future horizon future past horizon indler W eft R ct “L In LRW, moves backward in Minkowski time In LRW, LRW Can define Rindler coordinates are hyperbolas in Mi coordinates: 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. How do mode frequencies transform?
Plane wave: φ(x, t)=exp[−iΩ(t − x/c)]
Rewrite in Rindler coordinates: Ω φ(τ) = exp i e−ατ α
mode frequency exponentially red-shifted Red-shifting gives rise to thermal spectrum. - General property of horizons Where does energy come from?
Have assumed constant acceleration.
Acceleration driven by fixed amplitude, classical unlimited source of energy.
Exact same assumption as parametric amplifier. Note: Acceleration Temperature Decoherence Testbed for relativistic quantum information. Hawking Radiation Black Hole: Region of space-time separated from the rest of the universe by a horizon inside which gravity is so strong that not even light can escape. Schwarzschild black hole; depends only on mass.
Mass
Horizon
Horizon acts like unidirectional surface can go inside; can not come out causally disconnected from outside Horizon
fast
water
slow
water Horizons in the kitchen
Horizon Horizon located at Schwarzschild radius:
2 2 rs =2GM/c ABH =4πrs
Use Einstein E = Mc 2 for energy conservation relation
2 2 κc dE = c dM = dABH 8πG c4 surface gravity: κ = 4GM
The force/mass needed to keep a small test mass stationary at the horizon as viewed at infinity. Horizon unidirectional; mass can only increase:
dABH ≥ 0
Looks like 1st-law of thermodynamics:dS ≥ 0
2 2 κc dE = c dM = dABH 8πG
κ dE = THdSBH = λdSBH 8πkBc
Implies that a black hole has a temperature Hawking (1974) showed that it does: (Hawking temp.)
γ TH = γ = κ/c characteristic frequency 2πkB Hawking radiation as particle production:
oobserver aat infinity
It is the Unruh effect! Viewed by a different observer. Hawking radiation viewed at infinity γ = κ/c includes redshift factor: = | κ Var=rs V (r)= 1 − rs/r GM a(r)= 2 1 − rs Energy r r
What does observer stationary at horizon see?
(a/c) α r → rs T → = Recover Unruh 2πkB 2πkB effect
Einstein’s Equivalence Principle in Action! Where does energy come from?
Energy in Hawking radiation must be balanced by loss of mass: 2 Mc − EH + EH
Black hole energy radiation energy
Black hole mass assumed fixed in calculation - Energy source classical & fixed amplitude
Valid only for large black holes: δM/M 1 Dynamical Casimir Effect (DCE) Dynamical Casimir (Moore) effect: - Boundary conditions lead to discrete mode structure. - “Shake” one of the mirrors:
2πc0 ωn = Modulates frequency λn changes wavelength 1D Mirror - Fulling & Davies:
- Continuum of modes - Path modifies exponent of modes negative frequency term
iωu iω(u−τ ) path u dependent e → e Can choose any mirror trajectory you like.
Experimentalist: Sinusoidal
Theorist: x(t)=−t − A exp (−κt)+B
Exponential red-shift of frequency & thermal state
γ T = 2 πkB
Dynamical Casimir Effect in Circuit-QED dc-SQUID
Φext Flux through loop changes JJ characteristic energy scale
Φext EJ (Φext)=2EJ cos π Φ0
JJ Also can be thought of as nonlinear inductor:
Φ0 Φext L (Φext)= sec π 2πIc Φ0 Recall LC-oscillator: 1 1 ω = √ ω (Φext)= LC L (Φext) C Dynamical Casimir (1D type): Wilson et al., Nature 479, 376 (2011) Dynamical Casimir SQUID gives perfect mirror (g) boundary condition: 0 Φ(x = Leff ,t)=0 Sinusoidal modulation:
(h) 0 EJ = EJ + δEJ cos(ωdt)
Effective length change: 0 0 eff = J ωd/2 δL Leff δE /EJ (i) Spectrum symmetric about: ωd/2 energy density
frequency Ω 100 a 90 80 70 20
60 Photon fluxdensity
50 15 40 –100 –200 –300 –400 –500 –600 –700 –800 10 100 b 90 5 Pump power (pW) 80 0 70 60 50 40 100 200 300 400 500 600 700 800 Digitizer detuning (MHz)
25 e b 20 Two-mode 20 Data One-mode, – Theory 15 One-mode, + 15
10 10
5
Squeezing (%) 5 Photon flux density Photon 0 20 40 60 80 100 0 Pump power (pW) 30 40 50 60 70 Pump power (pW) Dynamical Casimir (cavity type):
Microwave photons travel in medium (not vacuum) 2πcsq ωn = csq =1/ l (Φext) c λn per unit length fixed Dynamical Casimir: Lähteenmäki et al., arXiv:1111.5608 (2011) Experiment Theory detuning
Classical Cavity: 250 SQUIDs tunable by bias line Similar to Castellanos-Beltran 2007 & 2008 For single cavity mode: δLeff ωd † 2 2 HDCE = a − a DPA Hamiltonian 4deff 2
Summary Can learn a lot by playing outside! Surprising generality to quantum vacuum amplifcation. - Described by Bogoliubov transforms - Modulate system frequencies - Same quantum states - Same classical fixed amplitude * energy source approximation.
S I S I
P Thermal character of states vanishes!