Quantum Nonequilibrium Statistical Mechanics Meets the Measurement Problem

David M. RogersUniversity of South Florida

APS March meeting2018

Quantum Theory of Radiation (Einstein, 1917)

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�e(t) = �e + x(t)

�g(t) = �g � x(t)

h�HA(t)i = �~!Ax(t)

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Contradicts Jaynes-Cummings:

0.5

1.0

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c) d)

Time

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a) b)

c) d)

Time

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0 100 200 300

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c) d)

Time

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0 100 200 300 400

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Time

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exactweak coupling

Exact x(t):

t

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LaTeX Editor: J. McKenzie Alexander London School of Economics

We start from the classical picture of a field interacting with a particle, and note Einstein's solution says there is a rate of emission and absorption that provides equilibrium between the particle, field, and even momentum. Then, when a Hamiltonian is added (e.g. the Rabi model), the solution tells us that there is never any energy exchange. The quantum just keeps going back and forth and what's more, the two systems are entangled so it can't even be said that there is any work being done.

When does emission actually happen?

Haroche, Nobel Lecture, 2012.

In order to resolve this contradiction, we'll have to find a way to separately describe the system and environment in a way that makes it clear the environment is infinite while the system is not.

General, classical “environment”

• Usual ?: Given a diffusive system, find its properties.

• Alternative ?: derive the transition distribution:

0.1 kJ/mol

0.0 kJ/mol

-0.1 kJ/mol

P (�x) =?

time

position, x

Rogers, Rempe, and Beck, J. Stat. Phys. 145:385 (2011).

To understand the solution, let's go back to the classical nonequilibrium theory, which I consider my work to have settled. The usual question folks ask is, given a dynamical model, to understand all its properties. My question is different. Given a current system state and a list of possible transitions, I ask ``what is the probability of each transition.?'' This leads naturally to a maximum entropy answer over the space of transitions.

My Answer: Maximum Transition Entropy

• Constrain Δx2 and ΔE

P (�x) / exp

���1�x

2 � �2�E

P = 0.0

P = 0.7

P = 0.1time

position, xP = 0.1

P = 0.2

E(x)

Rogers, Rempe, and Beck, J. Stat. Phys. 145:385 (2011).

Of course, we need constraints on those transitions. The important constraints are on exchanges of conserved quantities. That is, on the expected energy change in the system. There is also a requirement for a constraint on the rate of exchange -- which is equivalent to a Gaussian noise term.

Maximum Transition Entropy

• MaxEnt result:

P (�x) / exp

���1�x

2 � �2�E

Inverse Diffusion Constant

Inverse TemperatureKey Conclusions:

1. The environment is reduced to a random transition.

2. Temperature is a property of the environment only.

3. Value of T is determined by stationarity,

Rogers, Rempe, and Beck, J. Stat. Phys. 145:385 (2011).

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The solution of that problem leads exactly to Brownian and Langevin motion. The Lagrange multiplier for the energy always turns out to be 1/2kT in every system I have encountered. So, from the classical theory, we can conclude that the environment is a generic noise source whose exchanges of material with the system should be subject to maxent constraints. That means, in particular, that the Lagrange multipliers are instantaneous properties of the environment. Instantaneous temperature, pressure, etc. are just defined as those Lagrange multipliers that lead to zero average exchange -- picture a thermometer interacting with a turkey. Zero energy exchange means you can now read its local temperature.

• Monty Hall Problem:

• 2 steps

• 4 paths

MaxTrans vs MaxCal

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3 1/6

1/6

1/3

1/3

1/4

1/4

1/4

1/4

Rogers and Rempe, J. Phys., Conf. Ser. 402:12014 (2012).

MaxCalMaxTrans

As an aside, the original maximum caliber theory of Jaynes is modified in an essential way when considering transitions instead of paths.The Monty Hall problem illustrates this. Assigning equal a priori probability to every path creates obvious problems with branching processes.

Theory Space Classification

Gaussian Processes

Langevin Models

MaxEnt Transition Models

Fluctuation Theorems (Detailed Balance)

Chaotic Dynamical Systems

Rogers and Rempe, J. Phys., Conf. Ser. 402:12014 (2012).

I want to characterize max-trans in terms of theory space here by stating that it includes all Langevin models, while including some ideas to work on systems where FTs and chaotic hypothesis does not apply.

Spohn & Lebowitz re-tool (Adv. Chem. Phys., ’78)

a. Coupleb. Evolve

c. Projectd. Repeat

a) b)

c) d)Spohn proved long-t dynamics settles to

Einstein picture!

Rogers, PRE 95:012149, 2017.

Returning to the quantum problem, we tear a page from Spohn and Lebowitz to note that weak coupling can be derived from a strong coupling process. That strong coupling process is exactly like the Anderson thermostat in classical MD. You pick a particle and randomly assign a new velocity from a Boltzmann distribution. Eventually, the system must reach equilibrium. What's interesting is that the energy and momentum due to this exchange is exactly what Einstein described. Now let's say we pick an environmental degree of freedom `reservoir' instead. Measure and replace it with a new, thermal copy. This is the process Spohn and Lebowitz outlined.

Re-Do: a) Couple, b) Evolve

g

e

A

B

TrhHAB⇢A ⌦ ⇢B

i= 0

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Coupling: 0 work, 0 heat Isolated evolution: 0 work, 0 heat, 0 entropy

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Rogers, PRE 95:012149, 2017.

Step-by-step, if we couple to a new environment and time-evolve, there is no energy or entropy change (dynamics is unitary).

Re-Do: c) Project (measure) the environment

AB

Projective Measurement on A:

Rogers, PRE 95:012149, 2017.

• Microscopic, Landauer measurement work.

• Classically can be zero (e.g. KE of A)

• Averages >= 0 here (due to equ. source, A)

• Still 0 heat…

A B

A’ B’

But, when we measure the environmental degree of freedom, we need to do a projection. Classically we can pick a velocity, but QM-lly, everything is entangled. This leads to a new kind of quantum work -- the measurement back-action.

Re-Do: d) Repeat!

AB

Rogers, PRE 95:012149, 2017.

• Heat and work actually come from reset: A(out) -> A(in)

• _ Value(Aout

) = F 0A � FA = W

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Heat = Tenv(S0A � SA)

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A B

A’ B’

A

The last step is easy, since the environment is in an energy eigenstate. So, we capture its energy and heat using a classical thermodynamic process.

Complete Cycle

Rogers, PRE 95:012149, 2017.

• Consequence: • No FT, time-local

• No ‘use’ of

• or

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IQrej/Tenv � 0

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(except work of measurement)

We can summarize the entire cycle to show that the second law holds -- even under strong coupling. We can also derive stochastic equations of motion under random measurement intervals and get the Lindblad equation.

Second Motivation

Utsumi, Phys. Rev. B 2009• Measure electron flux • Make a histogram • Compare to theory

Rogers, PRE 95:012149, 2017.

Removing “The Environment”: Quantum MaxTrans

Rogers, J. Phys. Conf. Series. 880(1), 2017.

time

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e�it~ H

<latexit sha1_base64="qkbrXGXCO6AmiHoNbwFkmVxAu1I=">AAACBnicbVBNS8NAEN34WetX1KMgi0XwYklEUG9FLz1WMLbQxLLZbpqlmw92J0IJuXnxr3jxoOLV3+DNf+O2zUFbHww83pthZp6fCq7Asr6NhcWl5ZXVylp1fWNza9vc2b1TSSYpc2giEtnxiWKCx8wBDoJ1UslI5AvW9ofXY7/9wKTiSXwLo5R5ERnEPOCUgJZ65gG7z09cCCShOcdQ5G7oE1lgNySAm0XPrFl1awI8T+yS1FCJVs/8cvsJzSIWAxVEqa5tpeDlRAKnghVVN1MsJXRIBqyraUwiprx88keBj7TSx0EidcWAJ+rviZxESo0iX3dGBEI1643F/7xuBsGFl/M4zYDFdLooyASGBI9DwX0uGQUx0oRQyfWtmIZEZwI6uqoOwZ59eZ44p/XLun1zVmtclWlU0D46RMfIRueogZqohRxE0SN6Rq/ozXgyXox342PaumCUM3voD4zPHx0BmQY=</latexit><latexit sha1_base64="qkbrXGXCO6AmiHoNbwFkmVxAu1I=">AAACBnicbVBNS8NAEN34WetX1KMgi0XwYklEUG9FLz1WMLbQxLLZbpqlmw92J0IJuXnxr3jxoOLV3+DNf+O2zUFbHww83pthZp6fCq7Asr6NhcWl5ZXVylp1fWNza9vc2b1TSSYpc2giEtnxiWKCx8wBDoJ1UslI5AvW9ofXY7/9wKTiSXwLo5R5ERnEPOCUgJZ65gG7z09cCCShOcdQ5G7oE1lgNySAm0XPrFl1awI8T+yS1FCJVs/8cvsJzSIWAxVEqa5tpeDlRAKnghVVN1MsJXRIBqyraUwiprx88keBj7TSx0EidcWAJ+rviZxESo0iX3dGBEI1643F/7xuBsGFl/M4zYDFdLooyASGBI9DwX0uGQUx0oRQyfWtmIZEZwI6uqoOwZ59eZ44p/XLun1zVmtclWlU0D46RMfIRueogZqohRxE0SN6Rq/ozXgyXox342PaumCUM3voD4zPHx0BmQY=</latexit><latexit sha1_base64="qkbrXGXCO6AmiHoNbwFkmVxAu1I=">AAACBnicbVBNS8NAEN34WetX1KMgi0XwYklEUG9FLz1WMLbQxLLZbpqlmw92J0IJuXnxr3jxoOLV3+DNf+O2zUFbHww83pthZp6fCq7Asr6NhcWl5ZXVylp1fWNza9vc2b1TSSYpc2giEtnxiWKCx8wBDoJ1UslI5AvW9ofXY7/9wKTiSXwLo5R5ERnEPOCUgJZ65gG7z09cCCShOcdQ5G7oE1lgNySAm0XPrFl1awI8T+yS1FCJVs/8cvsJzSIWAxVEqa5tpeDlRAKnghVVN1MsJXRIBqyraUwiprx88keBj7TSx0EidcWAJ+rviZxESo0iX3dGBEI1643F/7xuBsGFl/M4zYDFdLooyASGBI9DwX0uGQUx0oRQyfWtmIZEZwI6uqoOwZ59eZ44p/XLun1zVmtclWlU0D46RMfIRueogZqohRxE0SN6Rq/ozXgyXox342PaumCUM3voD4zPHx0BmQY=</latexit><latexit sha1_base64="qkbrXGXCO6AmiHoNbwFkmVxAu1I=">AAACBnicbVBNS8NAEN34WetX1KMgi0XwYklEUG9FLz1WMLbQxLLZbpqlmw92J0IJuXnxr3jxoOLV3+DNf+O2zUFbHww83pthZp6fCq7Asr6NhcWl5ZXVylp1fWNza9vc2b1TSSYpc2giEtnxiWKCx8wBDoJ1UslI5AvW9ofXY7/9wKTiSXwLo5R5ERnEPOCUgJZ65gG7z09cCCShOcdQ5G7oE1lgNySAm0XPrFl1awI8T+yS1FCJVs/8cvsJzSIWAxVEqa5tpeDlRAKnghVVN1MsJXRIBqyraUwiprx88keBj7TSx0EidcWAJ+rviZxESo0iX3dGBEI1643F/7xuBsGFl/M4zYDFdLooyASGBI9DwX0uGQUx0oRQyfWtmIZEZwI6uqoOwZ59eZ44p/XLun1zVmtclWlU0D46RMfIRueogZqohRxE0SN6Rq/ozXgyXox342PaumCUM3voD4zPHx0BmQY=</latexit>

Rogers, arXiv:1503.01232 (2016)

However, it is still incomplete. We really need to get away from explicit representations of the environment altogether to make a truly universal theory. To accomplish this, note that if the environment contributes a random `kick', then the Hamiltonian looks like H + r*X, where r is Gaussian. Jaynes studied that in 1958. He noted that it leads to an infinite temperature steady-state. What's missing is the dissipation, and we have a max-ent idea for that. Since we want to work in transition space, we add the following `bias' which mimicks a weak energy measurement. We can show that it's classical (diagonal) analogue is exactly the max-trans answer from earlier.

Incipient Langevin Equation anyway:

Zdy e�

i

~�1(x,y) 'Z

dy e�i

~�1(x,y)e�l

2(�x

A)2

2~2

150 100 50 0 50 100

40

20

0

20

40

100 50 0 50 100 150

0.10

0.03

0.03

0.10b) c)

pre-avg e-iɸ1e-iɸ1

y

dp / dt dp / dt

c)

Figure 2: Wigner propagator (a), direct integrand (b), and pre-averaged integrand (c) for the quarticpotential energy landscape V (x) = x4/4�x2. The white dotted line in (a) shows the force, �V 0(x). Thepre-averaged integrand in (c) was computed as the convolution of Eq. 8 using the expansion of Eq. 9 upto z2.

which holds for any width, l. If l is small enough, the exponential weighting allows expanding aboutz = 0 to good approximation.

�1

(x, y � z) = �1

(x, y)� z · �xA� 1

2z · @yz · �xA+O(z3), (9)

�xA ⌘ 1

2

✓�A(x+ y/2)

�x+

�A(x� y/2)

�x

◆

Retaining the term linear in z exhibits exponential fall-o↵ away from Newton’s law,Z

dy e�i

~�1(x,y) 'Z

dy e�i

~�1(x,y)e�l

2(�x

A)2

2~2 . (10)

Note that the factor (�xA)2 generates the Langevin equation in the classical limit, y ! 0.

P (x, p|x0, p0) = Z�1

kin

(x0, p0)�(�x� ✏p/m) exp

(� (�xA)2

2�2✏+

�

2mp · �xA

)(11)

Eq. 11 clearly has a maximum entropy structure, with constraints on on h(�xA)2|x0, p0i and hdE|x0, p0i =h��x/✏·�xAi – a connection that was first introduced in Ref. [16]. That derivation identified a normaliza-tion factor Z

kin

(x0, p0) which is termed the kinetic partition function. Averages obtained by di↵erentiating(logZ

kin

) are identical to the results that would be obtained from the usual Stratonovich form of thestochastic calculus for the Langevin equation.

The structure of the Langevin equation is thus embedded in quantum mechanics itself. Quantummechanics is still reversible because of the phase factor. Classically, however, the loss of the phaseinformation should be modeled with random jumps in momentum on the scale of ~/l. Since z3 andhigher terms depend on 3rd order and higher derivatives of the potential. l = l(x, y) can be safely set tothe largest length over which the action functional remains quadratic.[25, 27]

Figure 2a plots the Wigner propagator for the symmetric quartic potential, V (x) = x4/4 � x2 at✏ = 10�1 in atomic units (~ = electron mass = 1). Although the full propagator is 4-dimensional,Eq. 7 shows that it only depends on the combination x and �p/✏. The integrals over y were carried outnumerically using 200-point Gauss-Hermite quadrature after pre-averaging (integrating over z in Eq. 8)using the expansion of Eq. 9 up to z2 (rather than the first-order Eq. 10). It was verified that the resultwas not a↵ected by the particular choice of width, l (for which we used 1 Bohr). The dramatic e↵ect ofpre-averaging is shown in Fig. 2b and c, which plot the integrand before and after.

The phase structure visible in Airy-function like oscillations of �p is physically smoothed by theuncertainty principle. Although it is not shown here, propagating a coherent state is identical to smearingthe propagator with a Gaussian function. Further smearing over �p/✏ is achieved by averaging overrandom forces (introduced in Sec. 4).

4

(Sabo, Doll, Freeman. J. Chem. Phys. 116:3509, 2002.

Rogers, J. Phys. Conf. Series. 880(1), 2017.Rogers, arXiv:1503.01232 (2016)

Bonus: A Kinetic Partition Function

Z[⇢] = Tr [G[⇢]]

h�Hi = �@ logZ[⇢]

@�/2 0

2

4

6

8

10

0 5 10 15 20

−20

−16

−12

−8

−4

0

Po

wer

/ 1

0−

3

Po

wer

β

n = 0

1

2

dH > 0 dH < 0

Hot Cold

P (x0|x) = e

��x

2/2�2���E/2

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⇢0 =X

r

prZ�1/2W (Ur)⇢W (Ur)

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classical:

quantum:

Rogers, arXiv:1503.01232 (2016)

Interestingly though, this propagator has a matrix-valued normalization constant. It has the mathematical properties of a partition function, but its log-derivative generate cumulants of the work distribution.

Conclusions

Thanks To: Students: Guy Dayhoff, Selamawi Ambaye Collaborators: Sameer Varma, Joshua Vermaas,

Brian Space

• A consistent maxent approach uses a classical environment to define nonequilibrium quantum heat and work.

• The measurement action destroys notions of system temperature and purity. — It’s incorrect to apply Boltzmann / Lindblad at low-T!

• Confirms ‘back-action’ effects noted by S. Deffner, J. Paz, and W. Zurek (but from an environment-perspective).

• Closely parallels frameworks of K. Funo (projection+feedback), J. Horowitz (unravelling), M. Esposito (repeated projection).

parallelscience.com