Basics of Quantum Measurement with Quantum Light

Basics of Quantum Measurement with Quantum Light

Basics of quantum measurement with quantum light Michael Hatridge University of Pittsburgh R. Vijayaraghavan TIFR Quantum light – Fock states Quantum state with definite number of photons in it! 푛 = 1,2,3 … n is an eigenstate of number operator 푛 = 푎†푎 푛 푛 = 푛 Let’s also look at what a coherent state looks like in terms of its 푎±푎† quadratures 퐼, 푄 = (equivalent to x and p for SHO) 2 Q I Quantum light – coherent states The quantum state of a driven harmonic oscillator ∞ 2 푛 − 훼 훼 훼 = 푒 2 푛! 푛=0 This state is not a eigenstate of photon number, but instead of lowering operator w/ complex eigenvalue α 푎 훼 = 훼 훼 In I/Q space, has gaussian distribution centered on coordinates 퐼, 푄 = (푅푒 훼 , 퐼푚(훼)) Q I Quantum light – squeezed states These basic quantum states have the same area in phase space Q Q I I We can consider other, more complicated states which ‘squeeze’ the light in certain quadratures while leaving area unchanged Q Q I I What to do with quantum light? • Detect/measure it to prove quantum properties - Q-function, Wigner Tomography - time/space correlations - bunching/anti-bunching • Use it as a tool/sensor - too many to count…. • Use it for quantum information - (flying) qubit - readout of another (stationary) qubit We must match the light to its detector • ‘Click’ or photon counting detectors project flying light onto a Fock state - great for Fock states, not a good basis for discriminating coherent light Output: ‘click’ or number w/ probability 2 푃푛 = 푛 휓푙ℎ푡 1 1 휓 = ( 훼 + 훼 ) 휓 = ( 푛 + 푚 ) 2 1 2 2 Q Q I vs. I Classical (microwave) amplifier In 퐺, 푁 Out Classical I/Q plot 푄 / 퐺 Q noise 표푢푡 noise grows! I 퐼표푢푡/ 퐺 signal signal unchanged - Amplifiers DEGRADE Signal-to-Noise Ratio (SNR) - Their job is to swamp noise of later elements Amplifiers have many properties First and foremost: Noise added by amplifier - Goes by many names: noise figure, noise factor, noise temperature - Here we’ll focus on how many noise quanta the amplifier adds. - In units of added quanta this amplifier adds 830 @ 7.5 GHz Quantum limit on phase-preserving amplifier Caves 1984 Vacuum state ( 0 ) Q Operators (I/Q) or (x,p) do not commute, and so there is minimum area in phase space, associated I w/ ½ photon energy. Mininum added noise is ½ photon 푄 / 퐺 Q 표푢푡 ½ photon ½ + ½ photon I 퐼표푢푡/ 퐺 signal unchanged Outline for the remainder of our tutorials Tutorial 1: Basics of quantum measurement with quantum light - Intro to parametric amplification - Classification of amplifiers by type and interaction - Limitations of parametric amplification - An introduction to superconducting circuits Tutorial 2: Quantum measurement of coherent states and qubits with phase- sensitive amplifiers - Phase-sensitive amplification with superconducting microwave circuits - Single-shot qubit readout - Weak measurements and quantum back-action - Entanglement via sequential measurement Tutorial 3: Quantum measurement of coherent states and qubits with phase- preserving amplifiers - Phase-preserving amplification with superconducting microwave circuits - Single-shot qubit readout - Weak measurements and quantum back-action - Entanglement via amplification Introduction to parametric amplifiers How to build a classical amplifier - Inductors, capacitors, resistors, transistors + DV voltage for gain How to build a quantum amplifier? - Inductors, capacitors, resistors, transistors + DV voltage for gain Quantum harmonic oscillators Nonlinear Hamiltonian + Ports parametric drive (no internal losses) Parametrically driven couplings direct exchange parametrically driven exchange (Conv) ℋ푛푡 ℋ = 휔 푎†푎 푛푡 = 휔 푏†푏 ℏ 푎 ℏ 푏 푎 푏 푎 푏 푐 ℋ푖푛푡 † † ℏ = 푔 푎푏 + 푎 푏 - if 휔 − 휔 ≫ 휅 this term dies ℋ푖푛푡 † † † 푎 푏 푎,푏 ℏ = 푔 푎푏 푐 + 푎 푏푐 due to energy conservation (RWA) - if 휔푐 ≠ 휔푎 − 휔푏 we can drive the c- mode ‘stiffly’ - interaction also turns off slowly vs. ℋ 푖푛푡 † ∗ † detuning, limiting the on/off ratio ℏ = 푔푎푏 + 푔 푎 푏 - Parametric drive fully controls strength and phase of interaction System Dynamics – Phase preserving amplification ℋ = 휔 푎†푎 + 휔 푏†푏 + 휔 푐†푐 + 푔 푎†푏†푐 + 푎푏푐† ℏ 푎 푏 푐 Pay careful attention to coupling, this form destroys one c ‘pump’ photon to create one photon each in a and b - Modes we use for quantum signals should be driven near their resonance frequency - We need the third ‘pump’ mode to be far away from the pump frequency so the pump can be ‘stiff’ and c becomes a number System Dynamics continued ℋ = 휔 푎†푎 + 휔 푏†푏 + 휔 푐†푐 + 푔 푎†푏†푐 + 푎푏푐† ℏ 푎 푏 푐 - We’ll see this explicitly later, but for now, consider the photon exchange process - If 휔푎 + 휔푏 = 휔푐 the process works, else suppressed by energy non-conservation - This feature will remain even after c-mode an (off-resonant) pump Master Eqn - In this example we will have our pump on, as well as a ‘weak’ signal in mode a - Mode b will be left ‘idle’ 푑푎 ℋ 휅 = [ , 푎] − 푎 푎 + 휅 푎푛 푑푡 ℏ 2 푎 푑푎 휅 = −휔 푎 − 푎 푎 − 푔푏†푐 + 휅 푎푛 푑푡 푎 2 푎 푑푏 휅 = −휔 푏 − 푏 푏 − 푔푎†푐 + 휅 푏푛 (푏푛= 0) 푑푡 푎 2 푏 Switch to Fourier Domain 푎(푡) → 푎 휔 Taking Fourier transform to make clear which frequencies are linked: - assuming we drive mode a at 휔1 = 휔푎 + Δ - signal in mode b will be at 휔2 = 휔푏 − Δ 휅 −휔 푎 휔 = −휔 푎 휔 − 푎 푎 휔 − 푔푏† 휔 푐 + 휅 푎푛 휔 1 1 푎 1 2 1 2 푎 1 휅 −휔 푏 휔 = −휔 푏 휔 − 푏 푏 휔 − 푔푎† 휔 푐 2 2 푏 2 2 2 1 where we have used the following relations: +∞ +∞ +∞ +∞ +∞ 푑푎 푑 휔1푡 휔1푡 휔1푡 휔1푡 휔 푡 푒 푑푡 = (푎푒 ) 푑푡 − 휔 푎푒 푑푡 = −휔 푎(휔 ) 푎 푡 푒 푑푡 = 푎[휔1] 푏 푡 푒 2 푑푡 = 푏[휔 ] 1 1 1 2 −∞ 푑푡 −∞ 푑푡 −∞ −∞ −∞ +∞ +∞ † 휔1푡 † − 휔1+휔2 푡 휔1푡 † 푐 = 푐 푒−휔푐푡 = 푐 푒−(휔1+휔2)푡 푔푏 푐푒 푑푡 = 푔푏 푐 푒 푒 푑푡 = 푔푏 (휔2) 푐 −∞ −∞ +∞ +∞ +∞ 푛 휔1푡 푛 −휔푡 +휔1푡 푛 푎 (푡)푒 푑푡 = 푑푡 푎 휔 푒 푒 푑휔 = 푎 휔1 −∞ −∞ −∞ Input-Output Theory Transmission lines 푎푛 푏표푢푡 푎 푏 푎표푢푡 푏푛 푏 + 푏 = κ 푏 푎푛 + 푎표푢푡 = κ푎푎 푛 표푢푡 푏 Substituting back into our equations for a and b we find: κ푎 1 + α 푔 푐 − Δ = − β + κ푎 2 κ푎 κ푏 κ β∗ 푔 푐 푏 + Δ = − (1 + α∗) 2 κ푏 κ푎 ϯ 푎표푢푡 푏 where α = and β = 표푢푡 푎푛 푎푛 Closed form solution 4 4 2 2 2 2 2 2 2 2 16푔 푐 + 8푔 푐 4Δ + κ푎κ푏 + 4Δ + κ푎 4Δ + κ푏 α = 4 4 2 2 2 2 2 2 2 16푔 푐 + 8푔 푐 4Δ − κ푎κ푏 + 4Δ + κ푎 4Δ + κ푏 20 For example : κ푎 = κ푏 = 50 푀퐻푧 10 푔 = 50 푀퐻푧 푐 2 = 0.36 Gain (dB) Gain -100 -50 0.0 50 100 Δ (MHz) 퐺 − 1푒휙푝 푎 퐺 퐺 푏 퐺 − 1푒−휙푝 Gain vs. pump power When Δ = 0, gain simplifies to: 2 2 2 2 1 + ρ 2 4푔 4 α = 2 where ρ = 푐 1 − ρ κ푎κ푏 100 Where ‘safe’ solution κ푎 = κ푏 = 50 푀퐻푧 50 푔 = 50 푀퐻푧 Gain (dB) Gain 0 0.5 1.0 푐 2 Amplifier bandwidth vs. gain and mode bandwidth 20 κ푎 = κ푏 = 70 푀퐻푧 κ푎 = κ푏 = 50 푀퐻푧 10 κ푎 = κ푏 = 20 푀퐻푧 Gain (dB) Gain 20 κ푎 = κ푏 = 50 푀퐻푧 퐺 10 2휋 퐵 ≃ κ푎κ푏 Gain (dB) Gain -50 0.0 50 Δ (MHz) Amplifier Limitations 퐺 − 1푒휙푝 1. It operates in reflection 푎 퐺 퐺 푏 퐺 − 1푒−휙푝 κ κ 2. It has narrow bandwidth 2휋 퐵 ≃ 푎 푏 퐺 Limitation 3: Gain Saturation G vs Psig unsaturated saturated 20 Gain (dB) 19 Pump depletion (PD) theory: • Psig , np , G • np , P-1dB P-1dB 18 -146 -138 -130 Signal Input Power (dBm) Problem: We don’t understand well all causes Pump depletion: Abdo Phys. Rev. B (2013); C. Eichler EPJ Quantum Technology (2014) But it is quantum limited! 퐺 − 1푒휙푝 푎 퐺 퐺 푏 퐺 − 1푒−휙푝 - Consider how much noise comes out of mode a with no inputs signals - The amplifier symmetrically handles both modes; the noise out of a is double its input due to the added noise from the ‘idler’ mode irrespective of what they are - For quantum signals you will get 1 photon total strong msmt 2000 fluctuations (1/2 + 1/2); second 1/2 physically sourced in idler mode - We’ll show how to test this with qubits in 휎 / 푚 1000 tutorial 3. 푄 counts 푔 푒 0 퐼푚/휎 Phase-sensitive amplifier ℋ푖푛푡 † † † ℏ = 푔 푎 푎 푐 + 푎푏푐 - Note that this is not the coupling Vijay will present tomorrow (his has 4 a operators; 1 each for signal and idler and 2 for pump) - To see the parallel, make the subsitution (a,b) - > (au , al ). An signal to au will be both amplified and coupled to al - Essentially one half of the amplifier acts as the ‘signal’ while the other half acts as the ‘idler’ Δ input input 휔푎 휔 휔푏 푎 퐺 − 1 퐺 퐺 − 1 퐺 Δ Δ output output 휔푎 휔푏 휔푎 Phase sensitive amplifier data - At zero offset frequencies these signal and idler tones interfere - One quadrature is amplified, the other deamplified - Which quadrature will be amplified is set by pump phase Signal input w/ Signal input here constant phase Freq @ 7.75027053 GHz ) ) -68 -70 dBm dBm Amplification -80 -70 -90 Deamplification -72 Δ푓 = 1 퐻푧 Integrated Power ( Integrated Power ( -100 7.74 7.75 7.76 0.0 0.5 1.0 Freq (GHz) Pump Phase / 2π Classifying amplifiers* 3-wave coupling 4-wave coupling one mode ‘Singly degenerate’ ‘Doubly degenerate’ phase-sensitive phase-sensitive two-mode ‘Non-degenerate’ ?? phase-preserving * At optical frequencies phase-preserving => heterodyne detection phase-sensitive => homodyne detection More useful: specify parametric interaction 3-wave coupling 4-wave coupling one mode 푔푎†푎†푐 푔푎†푎†푎푎 (a) two-mode 푔푎†푏†푐 푔푎†푏†푐푐 ∗∗ (a,b) ** I’m not aware of this one being built yet Other Couplings: Photon conversion 휔푝 = 휔푎 − 휔푏 ≠ 휔푐 † 휙푝 † −휙푝 퐻퐺 = ℏ푔 푎 푏푒 + 푎푏 푒 I 퐶푒+휙푝 푎 푏 푎 1 − 퐶 1 − 퐶 푏 퐶푒−휙푝 휔푝 ≃ 휔푎 − 휔푏 푃푃 푐 2 푃 conCorr=con+20.4 퐶 = 퐶 푃 2 09 1 + 푃 푃퐶 0 0 -15 -20 9.10 9.20 9.10 9.20 Frequency (GHz) Frequency (GHz) Sliwa PRX (2015) What about still odder couplings? Ex.: 푔푎†푎†푐푐푐 ? - This will be a phase-sensitive amplifier when driven at 2휔푎 = 3휔푐 - Such couplings may limit current device performance More exciting: multiple, simultaneous couplings 푄푏 - Can make non-reciprocal devices: circulators, directional amplifiers 훷푏 훷푎 푄푎 3-mode - Can also address other shortcomings coupler - Your theory proposal here! Bergeal et al.

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