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 휓푙𝑖𝑔ℎ푡
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) ℋ𝑖푛푡 ℋ = 휔 푎†푎 𝑖푛푡 = 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. ℋ푖푛푡 † ∗ † 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 = 𝑖 [ , 푎] − 푎 푎 + 휅 푎𝑖푛 푑푡 ℏ 2 푎 푑푎 휅 = −𝑖휔 푎 − 푎 푎 − 𝑖푔푏†푐 + 휅 푎𝑖푛 푑푡 푎 2 푎 푑푏 휅 = −𝑖휔 푏 − 푏 푏 − 𝑖푔푎†푐 + 휅 푏𝑖푛 (푏𝑖푛= 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 −𝑖휔 푎 휔 = −𝑖휔 푎 휔 − 푎 푎 휔 − 𝑖푔푏† 휔 푐 + 휅 푎𝑖푛 휔 1 1 푎 1 2 1 2 푎 1 휅 −𝑖휔 푏 휔 = −𝑖휔 푏 휔 − 푏 푏 휔 − 𝑖푔푎† 휔 푐 2 2 푏 2 2 2 1
where we have used the following relations: +∞ +∞ +∞ +∞ +∞ 푑푎 푑 𝑖휔1푡 𝑖휔1푡 𝑖휔1푡 𝑖휔1푡 𝑖휔 푡 푒 푑푡 = (푎푒 ) 푑푡 − 𝑖휔 푎푒 푑푡 = −𝑖휔 푎(휔 ) 푎 푡 푒 푑푡 = 푎[휔1] 푏 푡 푒 2 푑푡 = 푏[휔 ] 1 1 1 2 −∞ 푑푡 −∞ 푑푡 −∞ −∞ −∞ +∞ +∞ † 𝑖휔1푡 † − 휔1+휔2 푡 𝑖휔1푡 † 푐 = 푐 푒−𝑖휔푐푡 = 푐 푒−𝑖(휔1+휔2)푡 𝑖푔푏 푐푒 푑푡 = 𝑖푔푏 푐 푒 푒 푑푡 = 𝑖푔푏 (휔2) 푐 −∞ −∞ +∞ +∞ +∞ 𝑖푛 𝑖휔1푡 𝑖푛 −𝑖휔푡 +𝑖휔1푡 𝑖푛 푎 (푡)푒 푑푡 = 푑푡 푎 휔 푒 푒 푑휔 = 푎 휔1 −∞ −∞ −∞ Input-Output Theory
Transmission lines
푎𝑖푛 푏표푢푡
푎 푏
푎표푢푡 푏𝑖푛
푏 + 푏 = κ 푏 푎𝑖푛 + 푎표푢푡 = κ푎푎 𝑖푛 표푢푡 푏
Substituting back into our equations for a and b we find:
κ푎 1 + α 푔 푐 − 𝑖 Δ = −𝑖 β + κ푎 2 κ푎 κ푏 κ β∗ 푔 푐 푏 + 𝑖 Δ = −𝑖 (1 + α∗) 2 κ푏 κ푎 ϯ 푎표푢푡 푏 where α = and β = 표푢푡 푎𝑖푛 푎𝑖푛 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 − 1푒𝑖휙푝
푎 퐺 퐺 푏
퐺 − 1푒−𝑖휙푝 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 mode and gain vs. bandwidth Amplifier Gain (dB) Gain (dB) 10 10 20 20 - 50
Δ (MHz) 0.0
50
κ 푎 κ κ κ 2 = 푎 푎 푎 휋 κ
= = = 퐵 푏 κ κ κ ≃ = 푏 푏 푏 50 = = = 퐺
20 50 70 푀퐻푧 κ
푀퐻푧 푀퐻푧 푀퐻푧 푎 κ 푏
Amplifier Limitations
퐺 − 1푒𝑖휙푝
1. It operates in reflection 푎 퐺 퐺 푏
퐺 − 1푒−𝑖휙푝
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 • n , P P p -1dB -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 − 1푒𝑖휙푝
푎 퐺 퐺 푏
퐺 − 1푒−𝑖휙푝
- 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
fluctuations (1/2 + 1/2); second 1/2 physically 2000 sourced in idler mode
- We’ll show how to test this with qubits in 휎 /
tutorial 3. 푚 1000 푄 푔 푒 counts
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 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
† 𝑖휙푝 † −𝑖휙푝 퐻퐺 = ℏ푔 푎 푏푒 + 푎푏 푒 I
퐶푒+𝑖휙푝 푎 푏 푎 1 − 퐶 1 − 퐶 푏
퐶푒−𝑖휙푝 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. Nature Physics (2010) Ranzani and Aumentado NJP (2015), 훷푐 PRL 푄푐 Metelmann and Clerk (2015) Sliwa et al. PRX (2015) …. Amplifier outputs are squeezed light!
Squeezed light from a phase-sensitive amplifier
퐼𝑖푛 2 † † † Δ퐼𝑖푛 푎 푎 푐 + 푎푎푐 2 Δ푄𝑖푛
푎𝑖푛 푄𝑖푛
pump JPA 푐 퐺, 휙 퐼표푢푡
2 푎 2 퐺 Δ퐼𝑖푛 표푢푡 2 1/(2 퐺) Δ푄𝑖푛
푄표푢푡
Castellanos-Beltran et al., Nature Physics (2008) Two-mode Squeezing
Example: two-mode squeezed vacuum
퐼푎
† † † 푄푏 퐼푎 푎 푏 푐 + 푎푏푐 푄푎
푎𝑖푛 퐼−
푄+ pump JPC 퐼± = 퐼푎 ± 퐼푏 푐 퐺1, 휙1 퐼+ 푄± = 푄푎 ± 푄푏 푄− 퐼푏 푏𝑖푛
푄푏 퐼푏 푄 vacuum 푏 two-mode squeezed vacuum Yurke et al., PRA (1986) Bergeal et al., Nature Physics (2010) Amplification is a unitary transformation
퐼푎
† † † 푄푏 퐼푎 푎 푏 푐 + 푎푏푐 푎 푄푎 표푢푡
푎𝑖푛
퐺1 = 퐺2 pump JPC pump JPC 푐 퐺1, 휙1 푐 퐺2, 휙2 휙1 = 휙2 휙1 = 휙2 + 휋
퐼푏 푏𝑖푛
푄푏 푏표푢푡 퐼푏 푄 vacuum 푏
Yurke et al., PRA (1986) Flurin et al., PRL (2012) Experimental Evidence
Reflection gain vs 휙1 − 휙2, G2= 10 dB Two-mode squeezed vacuum fluctuation
analyzer alone analyzer alone noise floor
~1 dB suppression of ‘analyzer’ noise output So when do we measure?
HEMT
ωg ωe
JPC definitely by here a lot here Some here
We measure when we have lost ‘enough’ information to degrade our quantum coherence! The Story So Far
• Parametric amplifiers are quantum-limited and well suited for amplifying coherent pulese of light
• The act of amplification is quantum, ‘measurement’ occurs when sufficient info lost
• The amplifiers’ outputs are interesting states of quantum light
Quantum Measurement in Superconducting Circuits Qubits and entanglement
A Qubit Entanglement (e.g. Bell-states) 푒 푔푔 + 푒푒
α 푔 + 훽 푒 or 푔 + 푒 푔푒 + 푒푔
푔 + 𝑖 푒 • Neither qubit contains information on its own (all measurements as 푔 random as a flipped coin) • All information resides in • Two level quantum system relationship between qubits • Can be prepared in well known • Building block for quantum initial state communication, quantum • Can be placed in coherent computing superpositions states ( α 푔 + • One bit of entanglement referred 훽 푒 ) to as an ‘e-bit’ • Characterized by energy relaxation 푇1 and dephasing 푇2 time-scales What is ‘Quantum Information’? First, imagine that you have
Well behaved quantum systems (e.g. photon, electron, more generally “qubits”)
which you can
Individually control
Controllably couple (to entangle them)
Efficiently and individually measure Now put it all together
And use it to:
1. Simulate other quantum systems (Feynman 1982)
2. Perform algorithms which are hard for classical computers (Shor, Grover,…)
3. Do things we haven’t thought of/discovered yet!
Art by W. Pfaff Enter superconducting circuits
The Nobel Prize in Physics 2012 was SEM image of qubit Josephson Parametric Converter awarded jointly to Serge Haroche and David J. Wineland "for ground-breaking experimental methods that enable amplifiers measuring and manipulation of individual quantum systems"
Qubit and cavity SEM image of Josephson Ring Modulator The Josephson tunnel junction
휙 1nm 퐼 = 퐼 sin CJ 0 휑0 휙 h 0 퐼 2e
SUPERCONDUCTING LJ TUNNEL JUNCTION
nonlinear inductor shunted by capacitor
Al/AlOx/Al 200 nm tunnel junction Superconducting transmon qubit
Josephson junction with shunting capacitor anharmonic oscillator
Potential energy
푓
푒
푔
f lowest two levels form qubit
fge ~ 5.025 GHz, fef ~ 4.805 GHz
Koch et al., Phys. Rev. A (2007) Isolating the transmon from the environment
25 mm
Cavity Qubit fc,g = 7.4817 GHz fQ=5.0252 GHz 1/ = 30 ns T1 = 30 s T2R = 8 s The 8-junction Josephson Parametric Converter (a)
Signal mode Idler mode ADD SOME REF?
10 m
Pump coupling JRM G=20 dB 20 NR= 9 dB
BW= 9 MHz 10 D.R. ~ 10 photons @ 4.5 MHz Gain Tunability ~100’s of MHz 0
7.5 7.6 Freq (GHz) Bergeal et al Nature (2010) See also Roch et al PRL (2012) What a quantum machine looks like
Classical control lines
Dilution unit (cryogenic cooling)
50 or so readout channels
50 or so quantum bits
IBM Measurement via flying qubits
This qubit is well protected, with long coherence times
dispersive 푒 푔 interaction
phase meter
This flying “ancillary” qubit suffers the slings and arrows of outrageous fortune on its way to the detector Art by W. Pfaff If you are quiet and/or I talk too fast… on qubit state? Dispersive Measurement 푇푚 푇푚 퐼푚 = 퐼 푡 푑푡 0 푄 = 푄 푡 푑푡 푇푚 readout HEMT Ref 푚 푇푚 0 퐼 = 퐼 푡 푑푡 pulse at 푚 푄푚 = 푄 푡 푑푡 0 푓푑 0 12Q HEMT Ref cavity ring-down time Qubit + JPC vacuum Sig Idl 50Ω r resonator compact + qubit Pump readout mixer resonator pulses pulse JPC
Readout dispersive shift 휒 TR Sig Idl amplitude 푒 + qubit Pump 퐼2 + 푄2 푔 MT / isolator circulator 푚 푚 푓 pulsesSR
푓푑 width 휅 Readout 휋 transmon 푎 푔 ⊗ 훼 , 0 + 푏 푒 ⊗ 훼 , 0 2 𝑔 푒 phase 휒 푒 휗 = 2 tan−1 number of inddent −1 푄푚 휅 tan signal modes: 퐼푚 푔 휋 푓 − 2 Dispersive measurement: classical version
푔 microwave transmission cavity line phase meter
dispersive coherent cavity/pulse interaction pulse Dispersive measurement: classical version
microwave 푒 푔 cavity transmission line phase meter
dispersive coherent cavity/pulse interaction pulse Now a wrinkle: finite phase uncertainty
microwave 푒 푔 cavity transmission line phase meter
dispersive coherent cavity/pulse interaction pulse Amplifier Theory
Rsource, Ts G, N ~ Detector
Makes signal 푁𝑖푛푡 푁𝑖푛푡 + 푁푎푑푑 Classical: bigger S 퐺푆 + (Signal) Adds noise
푁푣푎푐 + 푁푣푎푐 Quantum 푁푣푎푐 Limited: S 퐺푆 (Signal) Measurement with bad meter (still classical)
noise added 푔 microwave by amp. 푒 cavity AND signal lost in transmission
phase meter
dispersive coherent cavity/pulse • Each msmt tells us only a little interaction • State after msmt not pure! pulse • This example optimistic, best commercial amp adds 20-30x noise • We fix this with quantum-limited amplification Quantum-limited amplification: projective msmt
microwave cavity only quantum 푒 푔 fluctuations
phase meter w/ P. P. pre-amp
coherent coherent superposition pulse
• state of qubit pure after each msmt • For unknown initial state 푐𝑔 푔 + 푐푒 푒 , repeat 2 2 many times to estimate 푐𝑔 , 푐푒 Quantum jumps
5
푒
휎
휎
5
0 .
푚
퐼 7
푔 -5
0 50 100 150 Time (s) • 240 ns boxcar filter • 푇1 푛 = 10 ≅ 50s • Fully linear (can see 푓 , ℎ … in IQ plane) Quantum-limited amplification: ‘partial’ msmt
microwave cavity only quantum 푒 푔 fluctuations
phase meter w/ P. P. pre-amp
coherent superposition
WEAK coherent • state of qubit pure after each pulse msmt • counter-intuitive, but is achievable in the laboratory Extra Slides Can we build such systems? “we never experiment with just one electron or atom or (small) molecule. In thought experiments we sometimes assume we do; this invariably entails ridiculous consequences […]
it is fair to state that we are not experimenting with single particles, any more that we can raise Ichtyosauria in the zoo.”*
The Nobel Prize in Physics 2012 was awarded jointly to Serge Haroche and David J. Wineland "for ground-breaking experimental methods that enable measuring and manipulation of individual *Schroedinger, Brit. J. Phil. Sci 3, 233 (1952) quantum systems"