# Presentation: 559481

## Quantum measurement and control of solid-state qubits and nanoresonators

**Support:**

***QUEST’04, Santa Fe, August 2004***

**Alexander Korotkov**
***University of California, Riverside***

**Outline:**

**Introduction (Bayesian approach)**
**Simple quantum feedback of a solid-state qubit**
**     ****(*****Korotkov*****, cond-mat/0404696)**
**Quadratic quantum measurements**
**     ****(*****Mao, Averin, Ruskov, Korotkov*****, PRL 93, 056803, 2004)**
**QND squeezing of a nanoresonator**
**     ****(*****Ruskov, Schwab, Korotkov*****, cond-mat/0406416)**

## Examples of solid-state qubits and detectors

**Examples of solid-state qubits and detectors**

**Two SQUIDs**

**Cooper-pair box **
**and single-electron **
**transistor (SET)**

**Double-quantum-qot**
**and quantum point**
**contact (QPC)**

**qubit**

**detector**

***I*****(*****t*****)**

*H*** = ***H***QB**** + ***H***DET**** + ***H***INT**

*H***QB ****= (********/****2****)(*****c*****1****+*****c*****1****-*****c*****2****+*****c*****2****)**** ****+**** *****H*****(*****c*****1****+*****c*****2****+*****c*****2****+*****c*****1****)**

****** ****-****  asymmetry,  *****H***** – tunneling**

****** ****= (4*****H***** ****2**** ****+**** ********2****)****1/2**** ****– ****frequency of quantum coherent ****(Rabi) oscillations**

**Two levels of average detector current:****     *****I*****1**** ****for qubit state**** |1********,  *****I*****2**** ****for**** |2******

**Response:**** *********I****** *****=***** ******I*****1***** *****-**** *****I*****2****                ****Detector noise: white, spectral density**** *****S******I***

## What happens to a qubit state during measurement?

**What happens to a qubit state during measurement?**

- For simplicity (for a moment) ***H****** *****=**** ****ε**** ****=**** ****0**, infinite barrier (frozen qubit),
- evolution due to measurement only

**“****Orthodox” answer**

**“****Conventional” (decoherence) answer (Leggett, Zurek)**

**|1> or |2>, depending on the result**

**no measurement result!  ****ensemble averaged**

**Orthodox and decoherence answers contradict each other!**

|                  applicable for: | Single quantum systems | Continuous measurements |
| --- | --- | --- |
| Orthodox | yes | no |
| Conventional (ensemble) | no | yes |
| Bayesian | yes | yes |

**Bayesian formalism describes gradual collapse of single quantum systems**
**Noisy detector output *****I*****(*****t*****) should be taken into account**

## Bayesian formalism for a single qubit

******1******** ******** *****I*****1****,  ********2******** ******** *****I*****2 **
-  *******I*****=*****I*****1*********I*****2 ****, *****I*****0****=(*****I*****1****+*****I*****2****)/2,   *****S******I***** ****– ****detector noise**

- For simulations:

**A.K., 1998**

**Similar formalisms developed earlier.  ****Key words**: **Imprecise, weak, selective, or conditional measurements,**** ****POVM, Quantum trajectories, Quantum jumps, Restricted path integral, etc.**

**Names:** **E.B. Davies, K. Kraus, A.S. Holevo, C.W. Gardiner, H.J. Carmichael, C.M. Caves, **
**              ****M.B. Plenio, P.L. Knight, M.B. Mensky, D.F. Walls, N. Gisin, I.C. Percival, G.J. Milburn,**
**              ****H.M. Wiseman, R. Onofrio, S. Habib, A. Doherty, etc****. ****(very incomplete list)**

**Averaging over ********(*****t*****) ******** master equation**

- 

**Ideal detector (****η****=1) does not decohere a single qubit (pure state remains pure),**
**then random evolution of the qubit *****wavefunction***** can be monitored**

## "Quantum Bayes theorem“ (ideal detector assumed)

-  Measurement (during time ):

-  ***H *****= ******** = 0**   **(frozen qubit)**

- Initial state:

**After the measurement during time ********, the probabilities **
**can be updated  using the ****standard Bayes formula:**

**Quantum Bayes**
**     ****formulas:**

## Nonideal detectors with input-output noise correlation 

**A.K., 2002**

***K*** – correlation between output
-        and backaction noises

**Fundamental limits for ensemble decoherence**

**Γ**** = ******** ****+ (*********I*****)****2****/4*****S******I *****, **** **** ******** ****≥**** 0  ********   ****Γ** **≥** **(*********I*****)****2****/4*****S******I****** ***

**Γ**** = ******** ****+ (*********I*****)****2****/4*****S******I *****+ *****K****** *****2*****S******I*****/4 , **** **** ******** ****≥**** 0  ********   ****Γ** **≥** **(*********I*****)****2****/4*****S******I****** *****+**** *****K****** *****2*****S******I****** *****/4**** **

- Translated into energy sensitivity:    **(****Є*****I***** ****Є*****BA*****)****1/2**** ≥  ********/2   ****or****  (****Є*****I***** ****Є*****BA ********* ****Є*****I,BA*****)****1/2**** ≥  ********/2 **

## Ideality of realistic solid-state detectors(ideal detector does not cause single qubit decoherence)

**1. Quantum point contact**         Theoretically, **ideal quantum detector, ****η**** ****=**** ****1**

**A.K., 1998 (Gurvitz, 1997; Aleiner et al., 1997)**

- Experimentally, η **>** 80%

**(using Buks et al., 1998)**

**2. SET-transistor**

***I*****(*****t*****)**

**Very non-ideal in usual operation regime, ****η**** ****‹‹**** ****1**

**Shnirman-Sch****őn, 1998; A.K., 2000, Devoret-Schoelkopf, 2000**

**However, reaches ideality, ****η**** = 1** **if:**

- - in deep cotunneling regime **(Averin, 2000, van**** ****den**** ****Brink, 2000)**

- - S-SET, using supercurrent   **(Zorin, 1996)**

- - S-SET, double-JQP peak   **(Clerk et al., 2002)**

-   ???  S-SET, usual JQP **(Johansson et al.)**, onset of QP branch **(?)**

- - resonant-tunneling SET, low bias **(Averin, 2000)**

**3. SQUID**

***V*****(*****t*****)**

**Can reach ideality, ****η**** = 1**

**(Danilov-Likharev-Zorin, 1983;**
**    ****Averin, 2000)**

**4. FET ?? HEMT ??**
-    **ballistic FET/HEMT ??    **

## Bayesian formalism for N entangled qubits 

**qb**** ****1**

**detector**

**qb**** ****2**

**qb ...**

**qb**** *****N***

***I*****(*****t*****)**

********* *****(*****t*****)**

**A.K., PRA 65, 052304 (2002); PRB 67, 235408 (2003)**

- Up to **2*****N*** levels
- of current

- (Stratonovich form)

**Averaging over ********(t) ******** master equation**** **

**Stratonovich:**

**Ito:**

**(easy derivatives and**
**  ****physical meaning)**

**(easy averaging over noise)**

## Experimental predictions and proposalsbased on the Bayesian formalism

**  ****Direct experiments on Bayesian evolution (1998)**
**  ****Measured spectral density of Rabi oscillations (1999, 2000, 2002)**
**  ****Bell-type correlation experiment (2000)**
**  ****Quantum feedback control of a qubit (2001)**
**  ****Entanglement by measurement (2002)**
**  ****Measurement and entanglement by a quadratic detector (2004) **
**  ****Simple quantum feedback via quadratures (2004) **
**  ****QND squeezing of a nanoresonator (2004)**

## Measured spectrum of qubit coherent oscillations (or spin precession)

**What is the spectral density *****S******I****** *****(****ω****)**** of detector current?**

**A.K., LT’99**
**Averin-A.K., 2000**
**A.K., 2000**
**Averin, 2000**
**Goan-Milburn, 2001**
**Makhlin et al., 2001**
**Balatsky-Martin, 2001**
**Ruskov-A.K., 2002 **
**Mozyrsky et al., 2002 **
**Balatsky et al., 2002**
**Bulaevskii et al., 2002**
**Shnirman et al., 2002**
**Bulaevskii-Ortiz, 2003**
**Shnirman et al., 2003**

**Spectral peak can be seen, but**
**peak-to-pedestal ratio ****≤ 4****η**** ****≤**** 4**

- Weak coupling,* ****α****** = ******C*****/8 ****« 1**

- (result can be obtained using various
- methods, not only Bayesian method)

- Assume classical output, eV » Ω

**Contrary:**

**Stace-Barrett, 2003**
**       ****(PRL 2004)**

## Quantum feedback control of a solid-state qubit

**Goal:**** ****maintain desired phase of coherent (Rabi) oscillations**
**           ****in spite of environmental dephasing (keep qubit “fresh”) **

**Idea:** **monitor the Rabi phase ********** by continuous measurement and apply **
**          ****feedback control of the qubit barrier height,  *********H*****FB****/*****H *****= *********F*****×************** **

**To monitor phase **********  we plug detector output *****I*****(*****t*****) into Bayesian equations**

**Ruskov & A.K., 2001**

- Quantum feedback in quantum optics is discussed since 1993 (Wiseman-Milburn),
-         recently first successful experiments in Mabuchi’s group (2002, 2004).

***H*****qb****= *****H****** **********X***

## Performance of quantum feedback(no extra environment)

- Qubit correlation function

- Fidelity (synchronization degree)

***C*****=1, ********=1, ***** F*****=0, ****0.05****, ****0.5**

**For ideal detector and wide bandwidth,**
**fidelity can be arbitrary close to 100%**
**                *****D***** = exp(*********C*****/32*****F*****)**

**Ruskov & Korotkov, PRB 66, 041401(R) (2002)**

- (for weak coupling and good fidelity)

- Detector current correlation function

## Suppression of environment-induced decoherence by quantum feedback

**Big experimental problem:  ****necessity of very fast (>>********, GHz-range) **
**   ****real-time solution of the Bayesian equations; therefore wide bandwidth**

**Some help:****  “direct” (“naïve”) feedback**

**However, still wide bandwidth (****>>**** ********) required**

## Simple quantum feedback of a solid-state qubit

**(A.K., cond-mat/0404696)**

**Idea:**** use two quadrature components of the detector current *****I*****(*****t*****)**
**         ****to monitor approximately the phase of qubit oscillations **
**         **(a very natural way for usual classical feedback!)

**Anticipated problem:**** **without feedback the spectral peak-to-pedestal ratio < 4,
-                                   therefore not much information in quadratures

**We want to maintain**
**coherent (Rabi) oscillations**
**for arbitrary long time,**
******11****-********22****=cos(*********t*****), ********12****=*****i***** ****sin(*********t*****)/2**** **

- (similar formulas for a tank circuit instead of mixing with local oscillator)

**(surprisingly, situation is much better than anticipated!)**

**Advantage:**** simplicity and relatively narrow bandwidth**

***H*****qb****= *****H****** **********X***

***C*********1**

## Accuracy of phase monitoring via quadratures(no feedback yet)

**Noise improves the monitoring accuracy!**
**(purely quantum effect, “reality follows observations”)**

**C – dimensionless coupling **

- (actual phase shift, ideal detector)

- (observed phase shift)

- Best approximation

- (2/5)(411/2-1)  2.16

*****X** *2+*Y** *2****=(*S*I/*I*)2

- 1/d = 4*S*I/(*I*)2

- Noise enters the actual and observed phase evolution in a similar way

- (non-Gaussian
-   distributions)

************* *****=*********** *****-**********m**

- weak coupling *C** * 1

## Simple quantum feedback

**fidelity for different averaging ******

**nonideal detectors**

****** ****Fidelity *****F***** up to ~95% achievable (*****D****** *****~**** ****90%)**
****** ****Natural, practically classical feedback setup**
** ****Averaging ********~1/********>>1/******** (narrow bandwidth!)**
****** ****Detector efficiency (ideality) ********0.1 still OK**
****** ****Robust to asymmetry ******** and frequency shift ******
****** ****Very simple verification – just positive **
**      ****in-phase quadrature *********X********* **

**Simple experiment?!**

- (weak coupling *C*)

## Quantum feedback in optics

**Recent experiment:**** ****Science 304, 270 (2004)**

**First detailed theory:**** **
**      **H.M. Wiseman and G. J. Milburn,
-      Phys. Rev. Lett. 70, 548 (**1993**)

**No experimental attempts of quantum feedback in solid-state yet**
**(even theory is still considered controversial)**
**Experiments soon?**

## Summary on simple quantum feedbackof a solid-state qubit 

******  ****Very straightforward, practically classical feedback idea**
**    ****(monitoring the phase of oscillations via quadratures)**
**    ****works well for the qubit coherent oscillations**
******  ****Price for simplicity is a less-then-ideal operation**
**    ****(fidelity is limited by ~95%)**
******  ****Feedback operation is much better than expected **
******  ****Relatively simple experiment (simple setup, narrow **
**    ****bandwidth, inefficient detectors OK, simple verification) **

## Quadratic Quantum Measurements

**Mao, Averin, Ruskov, Korotkov; Phys. Rev. Lett. 93, 056803 (2004)**

*I*()=*I*()

**quadratic**

**Quadratic detector**

**Quadratic detection is useful for quantum error correction (Averin-Fazio, 2002)**

- Setup similar to Ruskov-Korotkov, PRB 67, 241305(R) (2003),
- but a nonlinear (instead of a linear) detector is considered

## Bayesian formalism for a nonlinear detector

**Two-qubit evolution (Ito form):**

**Recipe:**** **Coupled detector-qubits evolution and frequent collapses
-              of the number *n* of electrons passed through the detector

- Assumed: 1) weak tunneling in the detector, 2) large detector voltage
-                  (fast detector dynamics, and 3) weak response.
-            The model describes an ideal detector (no extra noises).

- (The formula happens to be the same as for linear detector)

## Two-qubit detection

**(oscillatory subspace)**

**Spectral peak at ********,**** peak/noise = (32/3)******

**Linear detector**

**Nonlinear detector**

**Quadratic detector**

**Extra spectral peaks at 2******** and 0**

**Peak only at 2********,**** peak/noise = 4******

**(******** is the Rabi frequency)**

**Mao, Averin, Ruskov, A.K., 2004**

**(Ruskov-A.K., 2002)**

**1=********, 2=********, 3=********, 4=******

**(analytical formula for weak coupling case)**

## Two-qubit quadratic detection: scenarios and switching

**Mao, Averin, Ruskov, Korotkov, 2004**

**Three scenarios:**     1) collapse into    = 1***B*** , current *I*, flat spectrum
-                                  2) collapse into    = 2***B*** , current *I*, flat spectrum
-                                      3) collapse into remaining subspace 34***B***** **, current
-                                     (*I* + *I*)/2, spectral peak at  2Ω,  peak/pedestal = 4η.

**1*****B***

**34*****B***

**2*****B***

**Switching between states due to imperfections**

- 1)  Slightly different Rabi frequencies, ****** ****=**** ********1**** ****-**** ********2**

- 2)  Slightly nonquadratic detector, ***I*****1**** ****≠**** *****I*****4**

- 3) Slightly asymmetric qubits, **ε**** ****≠**** ****0**** **

**(distinguishable by**
**  ****average current)**

## Effect of qubit-qubit interaction

****** ****- interaction between two qubits**

- First spectral peak splits (first order in ),
- second peak shifts (second order in )

**Summary on quadratic quantum measurements**

******  ****Bayesian formalism is the same as for linear detectors**
******  ****Detector nonlinearity leads to the second peak in the spectrum (at 2********),**
**        ****in purely quadratic case there is no peak at ******
**       ****(very similar to classical nonlinear and quadratic detectors)**** **
******  ****Qubits become entangled (with some probability) due to measurement,**
**      ****detection of entanglement is easier than for a linear detector (current**
**      ****instead of spectrum), imperfections lead to switching to/from entanglement**

******1****-**** ****=**** ****[********2****+(********/2)]****1/2 ****- ********/2**
******1+**** ****=**** ****[********2****+(********/2)]****1/2 ****+**** ********/2**
******2**** ****=**** ****2[********2****+(********/2)]****1/2**** ****=**** ********1****-****+**** ********1+**** **

## QND squeezing of a nanoresonator

******0**** ******** ****1 GHz , *****T***** ******** ****50 mK, **
**quantum behavior *****T*****<************0**** ****or *****T*********obs****/*****Q****** *****<**** ********/2**

**Ruskov, Schwab, Korotkov, cond-mat/0406416**

- Quite similar to Hopkins, Jacobs, Habib, Schwab, PRB 2003
-    (continuous monitoring and quantum feedback to cool down)

- New feature: Braginsky’s stroboscopic QND measurement using
-     modulation of detector voltage **** **squeezing becomes possible **

- Other most important papers:
-      Doherty, Jacobs, PRA 1999 (formalism for Gaussian states)
-      Mozyrsky, Martin, PRL 2002 (ensemble-averaged evolution)

- Potential application: ultrasensitive force measurements

## Stroboscopic QND measurements

**Quantum nondemolition (QND) measurements** (Braginsky-Khalili book)
-  (a way to suppress measurement backaction and overcome standard quantum limit)
**Idea:** to avoid measuring the magnitude conjugated to the magnitude of interest

**Standard quantum limit**

**Example:** measurement of *x*(*t*2)-*x*(*t*1)

- First measurement: *p*(*t*1) > /2*x*(*t*1)*, *then even for accurate second measurement
-    inaccuracy of position difference is  *x*(*t*1)* **+** *(*t*2-*t*1)* */2*m** **x*(*t*1)* **> *(*t*2-*t*1)* */21/2*m*

**Stroboscopic QND measurements **(Braginsky *et al*., 1978; Thorne *et al*., 1978)

**Idea:**** **second measurement exactly one oscillation
-          period later is insensitive to *p*

**oscillator**

- (or *t *= *nT*/2, *T*=2/0)

**Difference in our case:**

******  ****continuous measurement**
******  ****weak coupling with detector**
******  ****quantum feedback to suppress “heating”**

## Bayesian formalism for continuousmeasurement of a nanoresonator

**Detector noise**

**Current**

**Recipe:**** frequent collapses of the number of QPC electrons**

**Nanoresonator evolution (Stratonovich form), same Eqn as for qubits:**

**Ito form (same as in many papers on conditional measurement of oscillators):**

- After that we practically follow Doherty-Jacobs (1999) and Hopkins *et al*. (2003)

## Evolution of Gaussian states

******

- Assume Gaussian states (following Doherty-Jacobs
-      and Hopkins-Jacobs-Habib-Schwab),
-      then (*x*,*x*’) is described by only 5 magnitudes:
*******x*********, *********p******* - average position and momentum (packet center),
***D******x******, D******p******, D******xp**** *– variances (packet width)
- Assume large *Q*-factor (then no temperature)

**Voltage modulation  *****f*****(*****t*****)*****V*****0****:**

**Then coupling (measurement strength) is also modulated in time:**

**Packet center evolves randomly and needs feedback (force *****F*****) to cool down**

**Packet width evolves deterministically and is QND squeezed by periodic *****f*****(*****t*****)**

## Squeezing by sine-modulation, V(t)=V0 sin(t)

**Squeezing up to 1.73 at**** ******** ****=**** ****2********0**

***D*****x****=(*********x*****)****2****,  *****D*********x********=**** *********x*********2******** ****-**** *********x*********2**

*******x*****0****= (********/2*****m*********0****)****1/2**** – **ground state width

- Squeezing obviously oscillates in time,
- maximum squeezing at maximum voltage,
- momentum squeezing shifted in phase by /2.

**Analytics (weak coupling):**

-  - detector efficiency, *C*0 – coupling

**Quantum feedback:**

- (same as in Hopkins *et al*.; without modulation
-    it cools the state down to the ground state)

- Feedback is sufficiently efficient, ***D*********x************ *****D*****x**

**Squeezing *****S****** ***

***D*****x****, *****D*********x******

*f*(*t*)

**Ruskov-Schwab-Korotkov**

## Squeezing by stroboscopic (pulse) modulation

**pulse modulation**

**Efficient squeezing at ******** ****=**** ****2********0****/*****n***

**Squeezing by stroboscopic (pulse) modulation**

***D*****x****, *****D*********x******

**Squeezing *****S****** ***

*f*(*t*)

**(natural QND condition)**

**Momentum squeezing as well**

- using
- feedback

***D*****x****=(*********x*****)****2**

***S*********1**

**Ruskov-Schwab-Korotkov**

## Squeezing by stroboscopic modulation

**Squeezing *****S***

*f*(*t*)

**Analytics (weak coupling, short pulses)**

-   Maximum squeezing               Linewidth

***C*****0** – dimensionless coupling with detector
*******t*** – pulse duration,  ***T*****0**** **= 2/0
**** – quantum efficiency of detector
-      (long formula for the line shape)

**Squeezing *****S***** ****=**** ****(*********x*****0****/*********x*****)****2**

**Finite *****Q*****-factor** limits the time we can afford to
-   wait before squeezing develops, ******wait****/*****T*****0****~*****Q*****/******

- Squeezing saturates as ~exp(-*n*/*n*0) after
-                                      measurements

- Therefore, squeezing cannot exceed

## Observability of nanoresonator squeezing

**Procedure:** 1) prepare squeezed state by stroboscopic measurement,
-                     2) switch off quantum feedback
-                     3) measure in the stroboscopic way

- For instantaneous measurements (*t*0) the variance of *X**N* is

*S* – squeezing,
- *x*0 – ground state width

- Then distinguishable from ground state (*S*=1)
-   in one run for *S** * 1 (error probability ~*S** *-1/2)

- Not as easy for continuous measurements because of extra “heating”.
*D**X,N* has a minimum at some *N* and then increases.
- However, numerically it seems                                                (only twice worse)

**Example:**

- for *C*0=0.1, =1, *t*/T0=0.02, 1/*S*=0.036

**Squeezed state is distinguishable in one run (with small **
**error probability), therefore suitable for ultrasensitive **
**force measurement beyond standard quantum limit**

**Ruskov-Schwab-Korotkov**

## Summary on QND squeezing of a nanoresonator

******  ****Periodic modulation of the detector voltage modulates**
**    ****measurement strength and periodically squeezes the **
**    ****width of the nanoresonator state (“breathing mode”)**
******  ****Packet center oscillates and is randomly “heated” by**
**    ****measurement; quantum feedback can cool it down**
**    ****(keep it near zero in both position and momentum)**
******  ****Sine-modulation leads to a small squeezing (<1.73),**
**    ****stroboscopic (pulse) modulation can lead to a strong**
**    ****squeezing (>>1) even for a weak coupling with detector**
******  ****Still to be done: correct account of *****Q*****-factor and temperature**
******  ****Potential application: force measurement beyond standard**
**    ****quantum limit**

## Conclusions

**●****  ****Bayesian formalism for solid-state quantum measurements**
**     ****is being used to produce various experimental predictions**
**     ****(though still not well-accepted in solid-state community)**
**●****  ****Simple, practically classical feedback using quadratures of**
**    ****the detector current should work well for qubit oscillations;**
**    ****relatively simple experiment **
**●****  ****Measurements by nonlinear (quadratic) detectors are described**
**    ****by the Bayesian formalism (same formulas as for linear detector),**
**    ****nonlinearity leads to the spectral peak at double frequency**
**    ****and makes easier qubit entanglement by measurement**
**●****  ****Measurement of a nanoresonator with strength modulated in time**
**    ****(modulating detector voltage) can produce a squeezed state;**
**    ****squeezed state is measurable and potentially useful**
**●****  ****No solid-state experiments yet; hopefully, reasonably soon**