Exploring the Quantum-Classical Transition Using Optomechanical SystemsKorea UniversityNational Taiwan UniversityDecember 17, 2013Phys. Rev. A

(1)

Paul Nation

Exploring the Quantum-Classical Transition Using Optomechanical Systems

Korea University

National Taiwan University December 17, 2013

Phys. Rev. A 88, 053828 (2013)

(2)

How does the classical world

emerge from the underlying rules of

quantum mechanics?

(3)

?

Quantum-Classical Crossover:

(4)

?

Quantum-Classical Crossover:

Can we push the boundary higher?

(5)

?

Quantum-Classical Crossover:

Can we push the boundary higher?

- In principle yes! One of the goals in nanomechanics:

60 μm

a b

J J

S R

50 µm

O’Connell, Nature (2010) Mavalvala, MIT Etaki, Nature Phys. (2008)

(6)

Schrödinger’s Cat (1935):

- Death of cat entangled with the quantum mechanical decay of radioactive atoms.

- If atom has 50% chance of decay then state of cat is:

| i

_cat

= 1

p 2 | i + 1

p 2 | i

(7)

(8)

| i

_cat

= 1

p 2 | i + 1

p 2 | i

(9)

| i

_cat

= 1

p 2 | i + 1

p 2 | i

| i

_cat

= | i

(10)

| i

_cat

= 1

p 2 | i + 1

p 2 | i

(11)

| i

_cat

= | i

(12)

| i

_cat

= | i

- When Schrödinger looks he is making a measurement.

(13)

| i

_cat

= | i

- Is the cat simultaneous dead and alive before I measure?

- When Schrödinger looks he is making a measurement.

(14)

- Absolutely not! The Environment is always making measurements.

Gas molecules Photons

- Many diﬀerent environments, all too complicated to keep track of the dynamics.

- Interaction with the environment leads to classicality, (loss of entanglement,

superpositions, coherence,...)

- Can make quantum objects behave classical.

IBM, 2013.

- Larger objects -> more environ. interactions.

(15)

- Absolutely not! The Environment is always making measurements.

Gas molecules Photons

- Many diﬀerent environments, all too complicated to keep track of the dynamics.

- Interaction with the environment leads to classicality, (loss of entanglement,

superpositions, coherence,...)

- Can make quantum objects behave classical.

IBM, 2013.

- Larger objects -> more environ. interactions.

(16)

Quantum Eﬀects in Massive Objects:

- Must minimize the coupling to the environment.

- Low temperatures.

- In vacuum.

- Want quantum dynamics that are clearly distinguishable from classical motion.

- Want massive object, but simple to model theoretically.

Mechanical Oscillator Nonlinear Interaction

+

- Can not get rid of all environment eﬀects. Gravity may be ultimate environment!

- Must find balance between quantum dynamics and environmental eﬀects.

(17)

ˆb, !_m, _m

x

ˆ

a, !_c

E, !_p, 

Optomechanics:

- Interaction between mechanical oscillator and optical cavity via radiation pressure generated by a laser.

- Retardation eﬀects give rise to nonlinear interaction.

- Changing the laser frequency with

respect to the optical cavity resonance frequency leads to cooling or heating of the resonator.

- Same dynamics in many quantum optics related fields.

Laser detuning0 !_m

!_m

Red detuned Blue detuned

Take from

oscillator~!^m Give to oscillator~!^m

Comet “tail” due to radiation pressure of light.

(18)

5 μm a

Macroscopic Mirrors Microscopic Mirrors Suspended Pillars Trampoline Resonators

Membranes Microtoroids Double-disk Resonators Near-field Resonators

Freestanding Waveguide Optical Resonators Superconducting Circuits Photonic Crystals

Photonic Nanobeam “Zipper” Cavity Cavity Nanorods Cold Atom Cavities

(19)

5 μm a

Grams

Zeptograms

(20)

5 μm a

Grams

Zeptograms

Same physics over 20 orders of

magnitude!

(21)

- Ground state cooling of mechanical oscillators.

Applications of Optomechanics:

- Quantum limits on continuous measurements.

- Sensitive force, mass, and position detection.

- Nonclassical states of light and matter.

- Entangled states of light and matter.

- Quantum information processing and storage.

In general,

Optomechanical Interaction Nonclassical mechanical states - Want to find simple analogue quantum system that leads to nonclassical

oscillator states?

(22)

B

R₁

C R₂

D

Micromaser (single-atom laser):

S

Gleyzes, Nature (2007)

- Interaction between a stream of excited two-level atoms and an optical cavity.

- Only a single atom in the cavity at a given time.

- When cavity has a large quality factor, many interactions Real quantum laser.

- Crucial parameter is the “maser pump parameter”: ^{✓ =} ^p^Nexgt_int/2

- Varying pump parameter gives oscillations in cavity photon number that can be interpreted as phase transitions: “Thumbprint of the micromaser.”

- Steady states of cavity are sub-Poissonian, i.e. nonclassical oscillator states.

- Amount of time atom spends in cavity called interaction time .

atom-cavity coupling

# of atoms passing in cavity lifetime.

(23)

Sub-Poissonian States:

Oscillator Fano Factor: F = h( ˆN_b)²i/h ˆN_bi

F= 1

Fock (quantum) state _|3i

Number state

Probability

Coherent (classical) state _{|↵ =} ^p³_i

Number state

Probability

Poisson distribution

- Poisson: Variance equal to average. - Variance vanishes F= 0 - Sub-Poissonian states are quantum oscillator states with F<1.

- Strongly sub-Poissonian states characterized by negative Wigner functions.

(24)

- A quantum phase space (pseudo)probability density distribution.

Wigner Functions:

Coherent state _{|↵ =} ^p³_i Fock (quantum) state _|3i - Can possess (nonclassical) regions where distribution is negative.

- Not a true probability distribution due to .

- Negativity of Wigner function can be used as measure of nonclassicality.

Positive Wigner func. Positive Wigner func. Negative Wigner func.

(25)

H =ˆ aˆ⁺a + ˆbˆ ⁺ˆb + g₀ ⇣

ˆb + ˆb⁺⌘ ˆ

a⁺â + E â + â⁺

ˆb, !_m, _m

x

ˆ

a, !_c

E, !_p, 

Optomechanical Setup:

- Consider a single-mode, driven optomechanical system

- Coupling constant measures oscillator displacement due to a single cavity photon in units of the zero-point amplitude:

g₀

- Laser-cavity detuning given by .= (!_p !_c) /!_m

- All constants measured in units of the resonator frequency.

Key Idea: Consider high-Q resonator, , and low-Q cavity, with damping rate , driven by weak laser.

Single-photon interaction!

m = Q_m¹

h ˆN_ai ⇡ h( ˆN_a)²i ⌧ 1

x_zp = p

~/2m!^m

Cavity HO Mech. HO Radiation pressure coupling Pumping of cavity

(26)

dˆa

d⌧ = i ˆa ig₀ ⇣

ˆb + ˆb⁺⌘ ˆ

a 

2 ˆa iE dˆb

d⌧ = iˆb ig₀ˆa⁺ˆa ^m

2 ˆb p

ˆb_in

Semiclassical Dynamics:

- Input-output theory gives Langevin equations of motion for Hamiltonian operators ( ).

- Classical nonlinear eﬀects can be studied in the steady state regime.

- Steady state cavity energy given by:_N^¯_a

E² = ² + ²/4 ¯N_a 2 K ¯N_a² + K²N¯_a³ K = 2g₀²

⇣1 + ₄²^m ⌘ (Kerr constant)

“Spring-softening”

;

!_c !

- The renormalized cavity frequency can be defined by the detuning value at which is maximized.

N¯_a

⌧ = !_mt

(27)

- The semiclassical limit-cycle dynamics of both the cavity and oscillator found by assuming oscillator undergoes sinusoidal motion (Marquardt et al. PRL 2006):

x(⌧ ) = ¯x + A cos(⌧ )

Static displacement Oscillation amplitude

- Plug into Langevin equation for cavity amplitude and use Fourier series solution:

¯

a(⌧ ) = e^{i'(⌧ )}

X1 n= 1

↵_ne^in⌧

¯

a(⌧ )

↵_n = iE J_n(g₀A)

i (n + g₀x) + /2¯

- Time-averaged response peaked at discrete values:_h|¯a|²_{i =} ^X

n

|↵ⁿ|²

= n + g₀x¯ n labels oscillator sidebands, i.e. n!_m

- Lineshape is Lorentzian, but peak is shifted depending on .

Shift due to Kerr nonlinearity (as we will see)

g₀

(28)

- Displacement and amplitude are found by self-consistently solving time averaged force balance:

and power balance equations:

¯

x = 2g₀ X

n

|↵ⁿ|²

mA = 4g₀Im X

n

↵^⇤_n+1↵_n

¯ A x

g₀x¯ / K

- In general, there are multiple solutions to these equations; multiple oscillator limit-cycles exist for a given set of parameters.

(29)

Quantum Dynamics:

- Here we are interested in the single-photon strong-coupling regime: ^g₀²^/!m & 1 - Discreteness of cavity photons becomes important.

- Radiation pressure of single-photon displaces resonator by more than its zero-point linewidth.

- Will use Master equation for full quantum dynamics to find steady state of system

L^cav = 

2 2âˆ⇢â⁺ aˆ⁺aˆˆ⇢ ⇢ˆâ⁺â L^mech = ^m

2 (¯n_th + 1) ⇣

2ˆbˆ⇢ˆb⁺ ˆb⁺ˆbˆ⇢ ˆ⇢ˆb⁺ˆb⌘

+ ^m

2 n¯_th ⇣

2ˆb⁺⇢ˆbˆ ˆbˆb⁺⇢ˆ ⇢ˆbˆbˆ ⁺⌘ - Oscillator bath characterized by avg. excitation number:

¯

n_th = [exp(~!^m/k_BT ) 1] ¹

(30)

⌘ =

P

n |wⁿ^{( )}| P

m wm⁽⁺⁾

=

P

n |wⁿ^{( )}|dxdp 1 + P

n |wⁿ^{( )}|dxdp

“Nonclassical ratio”

- To quantify the amount of “quantumness” in our oscillator states, we will take the ratio of the sum of negative Wigner densities over the positive density

elements.

- For the states considered here, this ratio is

nearly linear, a good benchmark for comparison.

- Note: You can not just count the number of negative and positive elements.

(31)

- Simulation parameters: E = 0.1,  = 0.3, Q_m = 10⁴, ¯n_th = 0

Results:

a

b

c

d

(32)

Results:

a

b

c

d

Mechanical sidebands

(33)

Results:

a

b

c

d

Nonlinear

frequency-pulling

(34)

Results:

a

b

c

d

Nonlinear

frequency-pulling

Strongest on resonance Increasing coupling leads to decrease in quantum features.

(35)

Results:

a

b

c

d

Nonlinear

frequency-pulling

Strongest on nonclassical states occur where Fano factor is larger than one!

(36)

Results:

a

b

c

d

Nonlinear

frequency-pulling

What is going on?

(37)

Results:

a

b

c

d

Nonlinear

frequency-pulling

What is going on?

(38)

- Winger functions consist of rings, one for each stable

limit-cycle.

- Circular symmetry from no phase ref. density matrix is diagonal.

- Each limit-cycle is sub-Poissonian.

- Stronger the coupling , and/or more phonons implies more limit-cycles exist.

a b c

d e f g h

a b c

d e f g h

- Strongest quantum features.

- Multiple limit-cycles means large variance

(39)

- To understand the onset, and decay, of the nonclassical oscillator properties, we fix the detuning and sweep the coupling strength_{= 0} 0  g⁰/  3

- The interplay between limit-cycles is measured by using the number state

corresponding to the maximum probability amplitude in the density matrix as an order parameter.

a b

Cavity quantum electrodynamics 1339

Figure 5. The pump curve of the one-atom maser as obtained from equation (4.3). In both the plots, the temperature is set at 0.5 K, which corresponds to n_th = 0.1 thermal photons. Top: the normalized photon number as a function of the interaction time (bottom axis), for a pump rate Nex = 40. Bottom: the normalized photon number as a function of the pump rate N^ex (bottom axis), for an interaction time t_int = 30 µs. The top axis of both plots shows the equivalent value of the pump parameter, !, for these conditions (see text). The thresholds of maser action (R_th) and quantum field statistics (Q_th) are indicated. Note that the micromaser exhibits thresholdless behaviour for the parameters of the bottom plot.

- Normalized coupling strength corresponds to the micromaser pump parameter, . Also proportional to resonator Q-factor.

g₀/

⌧_int = 1/

Optomechanical analogue Micromaser

(40)

- Why do the quantum features of the states disappear at higher couplings?

- Each limit-cycle is sub-

Poissonian in the regime where the nonclassical ratio is

nonzero.

- The merger of limit-cycles,

beginning at reduces the quantum features in the

mechanical states.

The overall resonator distribution, which is super-Poissonian, determines the nonclassical properties.

- In general, more phonons in resonator gives overlapping limit-cycles.

Smaller quantum signatures at mechanical sidebands.

(41)

Summary:

- Nonclassical states of a mechanical resonator can be generated in an analogue of the micromaser, if the cavity is suﬀiciently damped so as to have at most one

photon at any given time.

- This system has sub-Poissonian limit-cycles, nonclassical mechanical Wigner functions, and phonon oscillations that are also features of a micromaser.

- This is the first micromaser analogue that does not have any atom-like subsystem, only harmonic oscillators!

- Helps to understand the generation of quantum states in macroscopic mechanical systems.

- Allows for exploring the quantum-classical transition across multiple mass scales.

First single-atom laser with no atom!

But can we build it?

(42)

A cCPT optomechanical scheme for accessing the ultra strong coupling regime 3

System N ^!_^m ^g_⁰ _!^g⁰

m

g₀²

!_m

Superconducting LC oscillator [4] 1e11 60 3e 3 4e 5 1e 7 Si optomechanical crystal [5] 6e9 7 2e 3 2.5e 4 5e 7

Cold atomic gas [8] 4e4 0.06 22 340 7,500

cCPT-mechanical resonator 5e9 10 12 1.2 14

Table 1. The sideband ratio !_m/, the granularity parameter g₀/, the backaction parameter g₀/!_m and the combined quantum nonlinearity parameter g₀²/!_m, for certain demonstrated opto- and electromechanical systems, where N is the estimated number of atoms making up the mechanical resonator. Also shown for comparison are the estimated parameters of the cCPT-mechanical resonator scheme discussed in the present work.

is greater than one, then we are in the single-photon strong-coupling regime [14, 15]. In Table 1 we show a range of values for these parameters that have been realized in recent optomechanics experiments.

In the present work, we describe an optomechanical scheme involving a Cooper pair transistor (CPT) that is embedded in a superconducting microwave cavity, where a mechanically compliant, biased gate electrode couples mechanical motion to the cavity via the CPT. The basic scheme for the cavity-CPT-mechanical resonator (cCPT-MR) system is given in Figs. 1 and 2. In particular, we will show that the cCPT-MR device is capable of attaining the ultra-strong coupling regime, with relevant achievable parameters given in Table 1. Note that reference [16] discusses a very similar scheme.

This paper is organised as follows. In section 2 we describe the cCPT-MR device and give a phenomenological derivation of the e↵ective optomechanical coupling strength g₀ of the device. Next in section 3 we give a more systematic derivation of the optomechanical Hamiltonian (1), starting with a circuit model of the cCPT-MR device.

Finally, in section 4, we conclude with a discussion of our results and future work. The appendix contains the derivation of the circuit model.

2. The cCPT-MR Device

Referring to Figs. 1 and 2, the cCPT comprises two discrete components. One, the Cooper pair transistor (CPT), consists of a small superconducting island in the Coulomb blockade regime that is coupled via two Josephson junctions to macroscopic superconducting leads. The CPT has been extensively studied [17–21], and its properties are now well understood. The second component of the cCPT is a shorted quarter- wave, superconducting high-Q microwave cavity, which is flux biased to allow control over the total dc cCPT phase. The microwave cavity is based on the circuit QED architecture [22, 23] that has led to significant advances in the coherence and control of quantum superconducting circuits. The cCPT is created by embedding the CPT at the open end of the center conductor (a voltage antinode), so that it connects the central

A cCPT optomechanical scheme for accessing the ultra strong coupling regime 4

(c)

L

C

m k

C© x L_CPT

l/4

CPT gate

nanoresonator

(b) (a)

Figure 1. (a) Schematic illustration of a shorted /4, microwave resonator coupled to a feedline. (b) Detail of the CPT location and the method of coupling to a mechanical resonator. (c) Simplified circuit diagram of the device.

conductor of the cavity to the ground plane.

For our purposes, the CPT is well described by considering two charge states, |0i and |1i, corresponding to zero and one excess Cooper pairs on the island. These charge states are separated by an electrostatic energy di↵erence 2" = 4E

_c

(1 n

_g

) dependent on gate charge n

_g

, and are coupled to each other via the Josephson energy E

_J

. Introducing cavity photon annihilation and creation operators a and a

^†

, the Hamiltonian of the cCPT can be expressed as (see appendix):

H

^cCPT

= ~!

⁰

a

^†

a + "

_z

E

_{J x}

cos ⇥

0

(a + a

^†

) + ⇡

_ext

/

₀

⇤

, (2)

where

_x

and

_z

are the Pauli matrices, !

₀

is the cavity frequency,

_ext

is an external flux bias, and

₀

is the flux quantum. The first two terms in equation (2) describe the cavity photons and the CPT charge. The third term describes the coupling between the CPT charge states and the cavity photons. In a standard CPT, this term would read E

_{J x}

cos '/2 where ', the total superconducting phase di↵erence between the source and drain, can be treated as a classical variable [20, 21]. In the cCPT, however, quantum fluctuations of the cavity photon field must be accounted for via the identification

ˆ

'/2 =

₀

(a + a

^†

), which is proportional to the electric field in the cavity at the location of the CPT. The dimensionless parameter

₀

= p

Z

₀

/R

_K

⇡ 0.04, where R

_K

= h/e

²

= 25.8 k⌦ is the resistance quantum, describes the strength of the quantum phase fluctuations of the cavity field, which can be important for large cavity photon numbers [24, 25]. Experimental study [25, 26] indicates that equation (2) accurately models the cCPT.

The above-described cCPT functions as a sensor by capacitively coupling the CPT island to a system of interest, in our case a mechanical resonator (MR) consisting of a doubly clamped beam (made for example of SiN and coated with Al [27, 28]) as in figure 1(b). An important property of the CPT is that it acts as a charge-tunable quantum inductor L

_CPT

when biased on its supercurrent branch; L

_CPT

is the kinetic inductance associated with the CPT’s gate charge dependent supercurrent [18] [see figure 3(a) and (b)]. When the CPT is embedded in a microwave cavity, L

_CPT

appears in parallel with the cavity’s e↵ective inductance L at resonance, as in figure 1(c), and can therefore cause a dispersive shift of the cavity resonant frequency. When

A cCPT optomechanical scheme for accessing the ultra strong coupling regime 10

Figure A1. Simplified model of the cCPT-MR system, where the cavity center conductor has length L, and the Josephson junctions are assumed to have equal capacitances C_J and critical currents I_c. The cavity inductance and capacitance per unit length are denoted L^c, C^c, respectively. The cavity is capacitively coupled to a probe/transmission line.

while the junction condition at x = L is

2 +(t) c(L, t) = 2⇡n + 2⇡ / 0, (A.5)

where n is an integer and is the flux threading the superconducting loop formed out of the center conductor, ground plane, and the CPT. In the following, we will approximate the flux as ⇡ ^ext, i.e., assume that the induced flux in the loop due to the circulating super current can be neglected. We also “freeze” out the MR motion, so that C_m is fixed and non-dynamical; the mechanical component is straightforwardly introduced once we have obtained the cCPT Hamiltonian (2).

We now use equation (A.5) to eliminate + from the dynamical equations;

equations (A.2) and (A.1) become respectively 2CJ

0

2⇡

d²

dt² + 2Ic cos [ c(L, t)/2 + ⇡ ext/ 0] sin + C_g dVg

dt + C_mdVMR

dt = 0 (A.6)

and

0c(L, t) + CJ

2C^c

00c(L, t) = 2⇡L^cIc 0

sin [ _c(L, t)/2 + ⇡ _ext/ ₀] cos + ⇡L^c

0

⇣CgV˙g + CmV˙MR

⌘ , (A.7)

where have set n = 0 since it does not a↵ect the observable dynamics and we have used the cavity wave equation (A.3) to replace ¨c with ⁰⁰_c. Equation (A.7) is interpreted as a (rather nontrivial) boundary condition on the cavity field _c(x, t) at the x = L end that couples the cavity to the CPT.

We now formally solve the cCPT equations (A.3) and (A.6), subject to the boundary conditions (A.4) and (A.7), using the approximate eigenfunction expansion method, with equation (A.7) replaced by the following simpler boundary condition at x = L:

0c(L, t) + CJ

2C^c

00c(L, t) ⇡ ⁰c(x, t)|_x=L+C_J_/(2_C_c₎ = 0, (A.8)

Cavity-Cooper Pair Transistor:

- Most diﬀicult part is single-photon strong coupling:

- Motion of mechanical resonator modulates charging energy of electrons on the Cooper- pair transistor island.

- Causes measurable frequency shift of cavity.

(43)